Operator Theory And Analysis Of Infinite Networks (Contemporary Mathematics And Its Applications: Monographs, Expositions And Lecture Notes) 9811265518, 9789811265518

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Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes Print ISSN: 2591-7668 Online ISSN: 2591-7676 Series Editor M Zuhair Nashed (University of Central Florida) Editorial Board Guillaume Bal (University of Chicago) Gang Bao (Zhejiang University) Liliana Borcea (University of Michigan) Raymond Chan (The Chinese University of Hong Kong) Adrian Constantin (University of Vienna) Willi Freeden (University of Kaiserslautern) Charles W Groetsch (The Citadel) Mourad Ismail (University of Central Florida)

Palle Jorgensen (University of Iowa) Marius Mitrea (University of Missouri Columbia) Otmar Scherzer (University of Vienna) Frederik J Simons (Princeton University) Edriss S Titi (Texas A&M University) Luminita Vese (University of California, Los Angeles) Hong-Kun Xu (Hangzhou Dianzi University) Masahiro Yamamoto (University of Tokyo)

This series aims to inspire new curriculum and integrate current research into texts. Its aims and main scope are to publish: – Cutting-edge Research Monographs – Mathematical Plums – Innovative Textbooks for capstone (special topics) undergraduate and graduate level courses – Surveys on recent emergence of new topics in pure and applied mathematics – Advanced undergraduate and graduate level textbooks that may initiate new directions and new courses within mathematics and applied mathematics curriculum – Books emerging from important conferences and special occasions – Lecture Notes on advanced topics Monographs and textbooks on topics of interdisciplinary or cross-disciplinary interest are particularly suitable for the series. Published Vol. 7

Operator Theory and Analysis of Infinite Networks: Theory and Applications by Palle E T Jorgensen & Erin P J Pearse

Vol. 6

Generalized Radon Transforms and Imaging by Scattered Particles: Broken Rays, Cones, and Stars in Tomography by Gaik Ambartsoumian

Vol. 5

Tensor Algebra and Analysis for Engineers: With Applications to Differential Geometry of Curves and Surfaces by Paolo Vannucci More information on this series can also be found at https://www.worldscientific.com/series/cmameln

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Library of Congress Cataloging-in-Publication Data Names: Jørgensen, Palle E. T., 1947– author. | Pearse, Erin P. J., 1975– author. Title: Operator theory and analysis of infinite networks : theory and applications / Palle E.T. Jorgensen (University of Iowa, USA), Erin P.J. Pearse (California Polytechnic State University, USA). Description: New Jersey : World Scientific, [2023] | Series: Contemporary mathematics and its applications: monographs, expositions and lecture notes, 2591-7668 ; vol. 7 | Includes bibliographical references and index. Identifiers: LCCN 2022051320 | ISBN 9789811265518 (hardcover) | ISBN 9789811265525 (ebook for institutions) | ISBN 9789811265532 (ebook for individuals) Subjects: LCSH: System analysis. | Operator theory. | Hilbert space. Classification: LCC QA402 .J67 2023 | DDC 511/.5--dc23/eng20230117 LC record available at https://lccn.loc.gov/2022051320 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/13120#t=suppl Desk Editors: Sanjay Varadharajan/Kwong Lai Fun Typeset by Stallion Press Email: [email protected] Printed in Singapore

Erin is grateful to his wife and family for their patience and support while working on this volume. He also expresses his gratitude to his mentors Palle Jorgensen and Michel Lapidus for their insight, support, and generosity of ideas. Dedicated to the memory of Professors Ka-Sing Lau, Derek W. Robinson, and Robert S. Strichartz.

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Preface

We often hear that mathematics consists mainly of ‘proving theorems’. Is a writer’s job mainly that of ‘writing sentences?’ — G.-C. Rota It can be shown that a mathematical web of some kind can be woven about any universe containing several objects. The fact that our universe lends itself to mathematical treatment is not a fact of any great philosophical significance. — B. Russell

Note to the Reader In this book, we wish to present operators in Hilbert space (with an emphasis on the theory of unbounded operators) from the vantage point of a relatively new trend, the analysis of infinite networks. This in turn involves such hands-on applications as infinite systems of resistors and random walk on infinite graphs. Other such “infinite” systems include mathematical models of the Internet. This new tapestry of applications offers a special appeal and has the further advantage of bringing into play additional tools from both probability and metric geometry. While we have included some fundamentals of operator theory in the appendices, readers will first be treated to the fundamentals of infinite networks and their operator theory. Throughout the exposition, we will make continual use of the axioms of Hilbert space and such standard tools as the Schwarz inequality, Riesz’s lemma, projections, and the lattice of subspaces, all of which are available in any introductory functional vii

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analysis book. Readers not already familiar with this material may wish to consult the appendices for the axioms of Hilbert space, embedding, and isomorphism theorems (Appendix A); bounded and unbounded linear operators, the geometry of projections, infinite banded matrices, Hermitian and selfadjoint operators with dense domain, adjoint operators and their graphs, and deficiency indices (Appendix B). Some of the material is motivated by deeper aspects of Hilbert space theory: The Gel’fand triple construction in Chapter 7, deficiency indices of unbounded operators in Chapter 5, parallels between Kolmogorov consistency and the GNS construction in Chapter 16, and the relation of KMS states to long-range order in Chapter 16. Familiarity with these topics are not a prerequisite for this book! Conversely, we hope that the present setting allows for a smooth introduction to these areas (which may otherwise be dauntingly technical) and have correspondingly provided extensive introductory material at the relevant locations in the text. By using the intrinsic inner product (associated with the effective resistance), we are able to obtain results which are more physically realistic than many found elsewhere in the literature. This inner product is quite different from the standard 2 inner product for functions defined on the vertices of a graph and holds many surprises. Many of our results apply much more generally than those already present in the literature. The next section elaborates on these rather vague remarks and highlights the advantages and differences inherent in our approach in a variety of circumstances. This work is uniquely interdisciplinary, and as a consequence, we have made an effort to address the union (as opposed to the intersection) of several disparate audiences: graph theory, resistance networks, spectral geometry, fractal geometry, physics, probability, unbounded operators in Hilbert space, C*-algebras, and others. It is inevitable that parts of the background material there will be unknown to some readers, so we have included the appendices to mediate this. After presenting our results at various talks, we felt that the inclusion of this material would be appreciated by most. The subject of operator theory enjoys periodic bursts of renewed interest and progress, often because of impulses and inspiration from neighboring fields. We feel that these recent trends and interconnections in discrete mathematics are ready for a self-contained presentation; a presentation we hope will help both students and researchers gain access to operator theory as well as some of its more exciting applications.

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The literature on Hilbert space and linear operators frequently breaks into a dichotomy: axiomatic vs. applied. In this book, we aim at linking the two sides: After introducing a set of axioms and using them to prove some theorems, we provide examples with explicit computations. For any application, there may be a host of messy choices for inner product, and often, only one of them is right (despite the presence of some axiomatic isomorphisms). The most famous example of a nontrivial isomorphism stems from the birth of quantum theory. The matrix model of Werner Heisenberg was in fierce competition with the PDE model of Erwin Schr¨odinger until John von Neumann ended the dispute in 1932 by proving that the two Hilbert space models are in fact unitarily equivalent. However, despite the presence of such an axiomatic equivalence, one must still do computations in whichever one of the two models offers solutions to problems in the laboratory. Brief Overview of Contents Therefore, either the reality on which our space is based must form a discrete manifold or else the reason for the metric relationships must be sought for, externally, in the binding forces acting on it. — G. F. B. Riemann

Among the subjects in mathematics, functional analysis and operator theory are special in several respects: They are relatively young (measured in the historical scale of mathematics), and they often have a more interdisciplinary flavor. While the axiomatic side of the subjects has matured, there continues to be an inexhaustible supply of exciting applications. We focus here on a circle of interdisciplinary areas: weighted graphs and their analysis. The infinite cases are those that involve operators in Hilbert space and entail potential theory, metric geometry, probability, and harmonic analysis. Of the infinite weighted graphs, some can be modeled successfully as systems of resistors, but the resulting mathematics has much wider implications. In the following, we sketch some main concepts from resistance networks. The following rather terse sequence section is an abstract for the reader who wishes to get an idea of the contents after just reading a page or so. A more detailed description is given in Introduction. A resistance network is a weighted graph (G, c). The conductance function cxy weights the edges, which are then interpreted as resistors

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of possibly varying strengths. The effective resistance metric R(x, y) is the natural notion of distance between two vertices x, y in the resistance network. While it has some counterintuitive properties (for example, it is not a geodesic metric), it is well suited to problems involving energy, probability, and the graph Laplacian. The space of functions of finite energy (modulo constants) is a Hilbert space with inner product E, which we call the energy space HE . The evaluation functionals on HE give rise to a reproducing kernel {vx } for the space. Once a reference vertex o is fixed (as an “origin”), these functions vx satisfy Δvx = δx − δo , where Δ is the network Laplacian. This kernel yields a detailed description of the structure of HE = F in ⊕ Harm, where F in is the closure of the space of finitely supported functions and Harm . is the closed subspace of harmonic functions. These “dipoles” {vx .. x ∈ G} form a total system of vectors in HE and also a Green’s function for the Laplacian Δ. The energy E splits accordingly into a “finite part,” expressed as a sum taken over the vertices, and an “infinite part,” expressed as a limit of sums. Intuitively, the latter part corresponds to an integral over some sort of boundary bdG, which is developed explicitly in Chapter 5. The kernel {vx } also allows us to recover easily many known (and sometimes difficult) results about HE . As HE does not come naturally equipped with a natural orthonormal basis, we provide candidates for frames (and dual frames) when working with an infinite resistance network. In particular, the presence of nonconstant harmonic functions of finite energy leads to different plausible definitions of the effective resistance metric on infinite networks. We characterize the free resistance RF (x, y) and the wired resistance RW (x, y) in terms of Neumann or Dirichlet boundary conditions on a certain operator. (In the literature, these correspond to the limit current and minimal current, respectively) We develop a library of equivalent formulations for each version. Also, we introduce the “trace resistance” RS (x, y), computed in terms of the trace of the Dirichlet form E to finite subnetworks. This provides a finite approximation, which is more accurate from a probabilistic perspective and gives a probabilistic explanation of the discrepancy between RF and RW . For R = RF or R = RW , the effective resistance is shown to be negative semidefinite so that it induces an inner product on a Hilbert space into which it naturally embeds. We show that for (G, RF ), the resulting Hilbert space is HE , and for (G, RW ) it is F in. Under the free embedding, each vertex x is mapped to the element vx of the energy kernel; under the wired

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embedding, it is mapped to the projection fx of vx to F in. This establishes HE as the natural Hilbert space in which to study the effective resistance. We obtain an analytic boundary representation for elements of Harm in a sense analogous to that of Poisson or Martin boundary theory. We construct a Gel’fand triple S ⊆ HE ⊆ S  and obtain a probability measure P and an isometric embedding of HE into L2 (S  , P). This gives a concrete representation of the boundary in terms of the measures (1 + vxn )dP ∈ S  /F in, where {xn } is a sequence tending to infinity. The spectral representation for the graph Laplacian Δ on HE is drastically different from the corresponding representation on 2 . Since the ambient Hilbert space HE is defined by the energy form, many interesting phenomena arise, which are not present in 2 ; we highlight many examples and explain why this occurs. In particular, we show how the deficiency indices of Δ as an operator on HE indicate the presence of nontrivial boundary of a resistance network and why the 2 operator theory of Δ does not see this. Along the way, we prove that Δ is always essentially self-adjoint on the 2 space of functions on an resistance network and examine the conditions for the network Laplacian and its associated transfer operator to be bounded, compact, essential self-adjoint, etc. We consider two approaches to measures on spaces of infinite paths in a resistance network. One arises from considering the transition probabilities of a random walk as determined directly by the network, i.e., p(x, y) =  cxy / y∼x cxy . The other applies only to transient networks and arises from considering the transition probabilities induced by a unit flow to infinity. The latter leads to the notion of forward-harmonic functions, for which we also provide a characterization in terms of a boundary representation. Using our results, we establish precise bounds on correlations in the Heisenberg model for quantum spin observables, and we improve the earlier results of Powers. Our focus is on the quantum spin model on the rank-3 lattice, i.e., the resistance network with Z3 as vertices and with edges between nearest neighbors. This is known as the problem of long-range order in the physics literature and refers to KMS states on the C ∗ -algebra of the model.

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About the Authors

Palle Jorgensen is a professor at the University of Iowa. He has previously held academic or teaching positions at the University of Pennsylvania, USA; Stanford University, USA; and Aarhus University, Denmark. He has authored more than 300 highly cited research papers and more than 10 books. He has received numerous honors and awards and has delivered several lectures, including one in 2018 where Jorgensen was the NSF/CBMS speaker, giving 10 lectures on Harmonic Analysis: Smooth and Non-smooth, which was published in volume 128 of the AMS/CBMS book series. His research is interdisciplinary and lies at the crossroads of pure and applied mathematics. Jorgensen is a frequently invited speaker, giving colloquial and conference presentations at universities and centers in both the US and abroad, most recently at the University of Illinois, USA; the University of Oslo, Norway; the University of Colorado, USA; Rutgers University, USA; the University of Central Florida, USA; Oberwolfach, Germany; Harvard University, USA; Cornell University, USA; Bar-Ilan University, Israel; Ben Gurion University, Israel; and Stockholm University, Sweden. Jorgensen has mentored more than 10 postdocs and has directed more than 30 PhD theses.

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Erin Pearse is a professor at California Polytechnic State University, San Luis Obispo (Cal Poly). He has previously held positions at Cornell University, the University of Iowa, and the University of Oklahoma, all in the US. He received his degree at UC Riverside, USA, under Michel Lapidus. He has co-authored 24 articles and edited three proceedings volumes; this is his first book. He is also a managing editor of the Journal of Fractal Geometry and holds a patent for “Iterated geometric harmonics for data imputation and reconstruction of missing data,” a technology used in machine learning. Since 2016, Erin has focused his attention on the climate crisis. He is the director of the Initiative for Climate Leadership and Resilience (ICLR) at Cal Poly, an organizer of the Climate Solutions Now conference series, and a member of the leadership team at the SLO Climate Coalition.

Acknowledgments

While working on the project, the coauthors have benefited from interaction with colleagues and students, and we thank them for generously suggesting improvements as our book progressed. The authors are grateful for stimulating comments, helpful advice, and valuable references from Daniel Alpay, John Benedetto, Donald Cartwright, Il-Woo Cho, Raul Curto, Dorin Dutkay, Alexander Grigor’yan, Dirk Hundertmark, Richard Kadison, Keri Kornelson, Michel Lapidus, Russell Lyons, Diego Moreira, Peter M¨orters, Paul Muhly, Massimo Picardello, Bob Powers, Marc Rieffel, Karen Shuman, Sergei Silvestrov, Jon Simon, Myung-Sin Song, Bob Strichartz, Andras Telcs, Sasha Teplyaev, Elmar Teufl, Ivan Veselic, Lihe Wang, Wolfgang Woess, and Qi Zhang. The authors are particularly grateful to Russell Lyons for several key references and examples, and to Jun Kigami for several illuminating conversations and for suggesting the approach in (3.51). Initially, the first named author (PJ) learned of discrete potential theory from Robert T. Powers at the University of Pennsylvania in the 1970s, but interest in the subject has grown exponentially since.

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Contents

Preface

vii

About the Authors

xiii

Acknowledgments

xv

List of Figures

xxi

List of Symbols and Notation

xxiii

Introduction

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1

2

Resistance Networks 1.1 The Electrical Resistance Network Model . . . . 1.2 The Energy Form E . . . . . . . . . . . . . . . . 1.3 Currents and Potentials on Resistance Networks 1.4 Potential Functions and Their Relationship to Current Flows . . . . . . . . . . . . . . . . . . 1.5 The Compatibility Problem . . . . . . . . . . . . 1.6 Remarks and References . . . . . . . . . . . . . The Energy Hilbert Space 2.1 Evaluation Functional Lx and Reproducing Kernel vx . . . . . . . . . . . . . . . . . . . . 2.2 Finitely Supported Functions and Harmonic Functions . . . . . . . . . . . . . . . . . . . . 2.3 Discrete Gauss–Green Formula . . . . . . . . 2.4 More About Monopoles and the Space M . . xvii

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10 14 19 23

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27

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2.5 2.6 3

4

5

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Applications and Extensions . . . . . . . . . . . . . . . . Remarks and References . . . . . . . . . . . . . . . . . .

The 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Resistance Metric Resistance Metric on Finite Networks . . . . . . . . . Resistance Metric on Infinite Networks . . . . . . . . Free Resistance . . . . . . . . . . . . . . . . . . . . . . Wired Resistance . . . . . . . . . . . . . . . . . . . . Harmonic Resistance . . . . . . . . . . . . . . . . . . Trace Resistance . . . . . . . . . . . . . . . . . . . . . Projections in Hilbert Space and the Conditioning of the Random Walk . . . . . . . . . . . . . . . . . . . . 3.8 Comparison of Resistance Metric to Other Metrics . . 3.9 Generalized Resistance Metrics . . . . . . . . . . . . . 3.10 Remarks and References . . . . . . . . . . . . . . . .

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53 54 57 58 63 66 68

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78 81 83 85

Schoenberg–von Neumann Construction of the Energy Space HE 4.1 Schoenberg and von Neumann’s Embedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 HE as an Invariant of G . . . . . . . . . . . . . . . . . . . 4.3 Remarks and References . . . . . . . . . . . . . . . . . . The 5.1 5.2 5.3 5.4

Laplacian on HE Properties of Δ on HE . . . . . . . . . Harmonic Functions and the Domain of The Defect Space of ΔV . . . . . . . . . Remarks and References . . . . . . . .

The 6.1 6.2 6.3 6.4 6.5 6.6 6.7

2 Theory of Δ and the Transfer Operator Essential Self-Adjointness of the Laplacian on 2 (1) The Spectral Representation of Δ . . . . . . . . . . Frames and Duality . . . . . . . . . . . . . . . . . . Spectral Reciprocity . . . . . . . . . . . . . . . . . . The Transfer Operator on 2 (1) . . . . . . . . . . . The Laplacian and Transfer Operator on 2 (c) . . . Remarks and References . . . . . . . . . . . . . . .

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45 50

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107 109 116 117 125 135 141 150

Contents

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The 7.1 7.2 7.3 7.4 7.5

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Boundary and Boundary Representation Motivation and Outline . . . . . . . . . . . . . . . Gel’fand Triples and Duality . . . . . . . . . . . . The Resistance Boundary of a Transient Network The Structure of SG . . . . . . . . . . . . . . . . . Remarks and References . . . . . . . . . . . . . .

Multiplication Operators on the Energy Space 8.1 Bounded Multiplication Operators . . . . . . . 8.2 Algebras of Multiplication Operators . . . . . 8.3 Bounded Functions of Finite Energy . . . . . . 8.4 Remarks and References . . . . . . . . . . . .

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151 151 155 165 169 171

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173 175 176 187 191

Symmetric Pairs 9.1 Symmetric Pairs and Closability . . . . . . . . . . . . . 9.2 Applications of Symmetric Pairs to Laplace Operators on Infinite Networks . . . . . . . . . . . . . . . . . . . . 9.3 Gaussian Fields and the Malliavin Derivative . . . . . . 9.4 Tomita–Takesaki Theory . . . . . . . . . . . . . . . . . 9.5 Remarks and References . . . . . . . . . . . . . . . . . Dissipation Space HD The Structure of HD . . . . . . . . . . . . . . The Divergence Operator . . . . . . . . . . . . Analogy with Calculus and Complex Variables Solving Potential-Theoretic Problems with Operators . . . . . . . . . . . . . . . . . . . . . 10.5 Remarks and References . . . . . . . . . . . .

10 The 10.1 10.2 10.3 10.4

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203 206 213 218

219 . . . . . . 220 . . . . . . 225 . . . . . . 227 . . . . . . 229 . . . . . . 233

11 Probabilistic Interpretations 235 11.1 Path Space of a General Random Walk . . . . . . . . . . 236 11.2 Forward-Harmonic Functions . . . . . . . . . . . . . . . . 242 11.3 Remarks and References . . . . . . . . . . . . . . . . . . 247 12 Spectral Comparisons 249 12.1 Comparing Graphs with Different Conductance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 12.2 Comparing Different Conductance Functions . . . . . . . 250

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12.3 12.4 12.5

Moments of Δ(c) and Monotonicity of Spectral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 Remarks and References . . . . . . . . . . . . . . . . . . 261

13 Examples and Applications 13.1 Finite Graphs . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Infinite Graphs . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Remarks and References . . . . . . . . . . . . . . . . . . 14 Lattice Networks 14.1 Lattice Networks with Constant Conductances 14.2 Non-Simple Integer Lattice Networks . . . . . 14.3 An Example of When Δ is Not Essentially Self-Adjoint on HE . . . . . . . . . . . . . . . 14.4 Remarks and References . . . . . . . . . . . .

263 263 268 275

277 . . . . . . 278 . . . . . . 291 . . . . . . 297 . . . . . . 300

15 Infinite Trees 303 15.1 Remarks and References . . . . . . . . . . . . . . . . . . 310 16 Application to Magnetism and Long-Range 16.1 Kolmogorov Construction of L2 (Ω , P) . . . 16.2 The GNS Construction . . . . . . . . . . . 16.3 Magnetism and Long-Range Order in Resistance Networks . . . . . . . . . . . . . 16.4 KMS States . . . . . . . . . . . . . . . . . 16.5 Remarks and References . . . . . . . . . .

Order 311 . . . . . . . . 312 . . . . . . . . 315 . . . . . . . . 319 . . . . . . . . 322 . . . . . . . . 324

17 Future Directions

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Appendix A

Some Functional Analyses

331

Appendix B

Some Operator Theories

339

Appendix C

Navigation Aids for Operators and Spaces

355

Appendix D

A Guide to the Bibliography

357

Bibliography

361

Index

389

List of Figures

1 Thematic relations among the central ideas in the text with chapter numbers. . . . . . . . . . . . . . . . . . . . . . . . . .

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1.1 A Dirac mass on an edge of Z2 . . . . . . . . . . . . . . . . . . 14 1.2 A failed attempt at constructing a potential to match Figure 1.1. 15 3.1 Effective resistance as network reduction to a trivial network. This basic example uses parallel reduction followed by series reduction; see Remark 3.4. . . . . . . . . . . . . . . . . . . . . 3.2 Comparison of free and wired exhaustions for the example of a binary tree; see Definitions 3.8 and 3.16. Here, the vertices of Gk are all those which lie within k edges (“steps”) of the origin. If the edges of G all have a conductance of 1, then so do all the W edges of each GF k and Gk , except for the edges incident upon ∞k = ∞Gk , which have a conductance of 2. . . . . . . . . . .

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9.1 A sequence {ϕn } ⊆ C(X) for which ϕn |K = 1 and  lim X ϕn dλ = 0. See Example 9.5. . . . . . . . . . . . . . . . . 196 10.1 The action of d and d on the orthogonal components of HE and HD . See Theorem 10.8 and Definition 10.9. . . . . . . . . 221 12.1 Construction of a “horizontally connected binary tree” of Example 12.15. . . . . . . . . . . . . . . . . . . . . . . . . . . 255 13.1 The solution v as represented on R. . . . . . . . . . . . . . . . 267 13.2 One-sided ladder network, constant conductances. Edge labels here indicate current flow. . . . . . . . . . . . . . . . . . . . . 270 xxi

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13.3 One-sided ladder network, exponentially decaying conductances. Edge labels here indicate conductances. . . . . . . . . . 270 13.4 One-sided ladder network, geometric conductances. Edge labels here indicate conductances, and α > 1 > β > 0. . . . . . . . . 271 14.1 The function vx , a solution to Δv = δx − δ0 in (Z, 1). . . . . . 14.2 “Nerd Sniping” from xkcd by Randall Munroe (all rights reserved by the author), whom we gratefully acknowledge for allowing us to reproduce his comic. See Theorem 14.7 for the solution, and see Section 16.3 for implications with regard to magnetism in R2 and R3 . . . . . . . . . . . . . . . . . . . . . . 14.3 In Z3 , it may happen that R(x, y) > R(x, ∞), where R(x, ∞) = limz→∞ R(x, z). This phenomenon is represented here schematically as a “black hole.” . . . . . . . . . . . . . . . . . . . . . . 14.4 The function vx for x = 1, 2, 3 in Z, as obtained from the Fej´er kernel. See Example 14.16. . . . . . . . . . . . . . . . . . . . . 14.5 The energy kernel element vn on the integer network (Z, 1). . 14.6 The functions v1 , v2 , and v3 on (Z, cn ). Also, the monopole wo and the projection f1 = PFin v1 . See Lemma 14.31. . . . . . . . 14.7 The projection of the Dirac mass −δo onto Fin2 ; see Example 14.36 and also Lemmas 14.31 and 14.31. . . . . . . . . . . 14.8 A Mathematica plot of the defect vector u on (Z+ , 2n ); see Example 14.37 and Lemma 14.39. The left plot shows u(x) for x = 0, 1, . . . , 10, and the plot on the right shows data points for u(x), x = 10, 11, 12, . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 The homogeneous tree of degree 3 (left) and the binary tree from symbolic dynamics (right). The root of the tree is labeled xo . If the gray branch is pruned from the homogeneous tree, the two become isomorphic. . . . . . . . . . . . . . . . . . . . 15.2 The reproducing kernel on the tree with c = 1. For a vertex x which is adjacent to the origin o, this figure illustrates the elements vx , fx = PFin vx , and hx = PHarm vx , as discussed in Example 15.2. This figure should be compared with Figure 14.6 (with c = 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Approximants to the reproducing kernel on the tree with c = 1; see Example 15.2. . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 An unbounded harmonic function of finite energy. See Example 15.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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285 286 289 295 297

299

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304 305 309

List of Symbols and Notation

TA ; the operator induced by the matrix A, 108 V ; infinite matrix with entries vx , vy E , 119 Xn ; location of the random walk at time n, 236 Def λ (A); the defect space Eigλ (A ), 103, 134 E; expectation, 161, 206 Fin2 ; range of ΔV , 36, 99 F (α, ω); set of current flows from α to ω, 9 Ff ; collection of flows specific by f , 232 ΔM ; the closure of the Laplacian when taken to have the dense domain M, 33 Δc ; probabilistic Laplacian, 144 M; the span of the monopoles, 33 Γ(a); the space of paths beginning at a, 70 P(α, ω); set of paths in G from α to ω, 10, 139 SG ; “Schwartz space” of test functions (of rapid decay), 153

(G, R); the network G as a metric space under R, 90 AT ; the matrix of the operator T in the basis {δx }, 108 E; projection-valued measure, 116 J; the operator mapping 2 to HE by Jδx = [δx ], 142 Lx ; evaluation functional, 27 Mf ; multiplication operator (by f : X → C), 174 PFo ; projection to span PF δo , 128 PX ; projection to subspace X, 31 QV (ξ); quadratic form induced by V , 135 R; resistance metric, 54 RF (x, y); free resistance metric, 59 W R (x, y); wired resistance metric, 63 tr R (x, y); trace resistance metric, 70 RH (x, y); effective resistance as computed in subnetwork H, 58, 63, 70 xxiii

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SG ; space of distributions, 170 V(F ); the vector space span{vx }x∈F , for finite F ⊆ G0 , 117 V0 ; subset of V with balanced coefficients, 133 bd H; boundary of a subgraph, 36 βbd ; boundary form, 105 ; H  is the complement of H in G, 71 deg(x); degree of the vertex x, 98 D(I); dissipation of the current I, 9 distTV (μ, ν); total variation metric, 84 distγ (x, y); geodesic distance, shortest-path metric, 81 ∂v ∂ ; normal derivative of v on bd H, 36 2  (c); the 2 space of functions f : (G, c) → F with counting measure, 108 2 (1); the 2 space of functions f : (G, 1) → F with counting measure, 107 δx ; Dirac mass at the vertex x, 29 ζ; cocycle, 238 I; identity operator, 103 int H; interior of a subgraph, 36 ker T ; kernel of the operator T , 98 ·, ·W ; (Wiener) inner product on L2 (SG , P), 159 ·, ·V ; inner product induced by V , 177 ·, ·c ; standard (unweighted) inner product on 2 (c), 108

·, ·E ; energy inner product E(·, ·), 25 ·, ·1 ; standard (unweighted) inner product on 2 (1), 107 P[a → b]; probability of the random walk started at a reaching b, 71 P(c) ; a measure on the space of all paths in G, 236 PQ ; (quotient) probability measure induced by canonical projection, 165 ran T ; range of the operator T , 98 B; the set of balanced functions, 133  bd G ; boundary sum, 37 {Gk }; exhaustion of a network, 26 fx ; projection of vx to Fin, 31 hx ; projection of vx to Harm, 31 k(x, dy); Poisson kernel, 151 o; origin (fixed reference vertex), 25 p(x, y); transition probability x → y, 70 p(x, y); transition probability of the random walk on the network: cxy /c(x), 236 vx ; dipole, energy kernel, 28 wx ; monopole at x, 36 E (b) ; energy form for conductance function b, 251 Fin; E-closure of span{δx }, 29 D; grounded energy space, 43, 173 G; graph, 2 HD ; dissipation Hilbert space, 219

List of Symbols and Notation

HE ; energy Hilbert space, 25 HE (b) ; energy Hilbert space for conductance b, 250 Harm; the harmonic functions of finite energy, 29 Δ; graph Laplacian, 4 ΔF ; the closure of the Laplacian with domain F , 50 ΔH ; the extension of ΔV to Harm, 101 ; a self-adjoint extension of Δ ΔM , 158 ΔV ; the closure of the Laplacian when taken to have the dense domain V, 98 Δ1 ; the closure Δ on span{δx } under ·, ·1 , 109 (b) Δ ; Laplacian for conductance function b, 251 L; set of all cycles, 9 Γ; set of paths, 19, 243 Γ; the space of all infinite paths, 236 PCyc ; projection to Cyc, 222 Pd ; projection to d HE , 222 PFin ; projection to Fin, minimizing projection, 222 PHarm ; projection to Harm, 222 PKir ; projection to Kir, 222 PN bd ; projection to N bd, minimizing projection, 222

xxv

P; probabilistic transition operator, 70 T; transfer operator, 4, 136 V; the vector space span{vx }x∈G0 , 98 div; divergence operator, 225, 242 V; Hilbert space under ·, ·V , 177 c; edge conductance, 2 γ; path, 3, 60, 167, 236, 243 I; current; a skew-symmetric function on edges, 8 ϑ; cycle, 9 d; drop operator, 220 e; oriented edge, 223 ϕe ; normalized Dirac mass on an edge, 224 E; (Dirichlet) energy form, 5  forward Laplacian, 244 Δ; Φ; the operator sending δx → vx , 128 ; modular operator, 216 1; the constant function 1, the vacuum vector, 2, 25, 107, 163 F ; finitely supported functions , 50 I; inclusion embedding, 251 (b) vx ; dipole in HE (b) , 251 M; von Neumann algebra, 213 X; X = G0 \ {o}, 173

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Introduction

... an apt comment on how science, and indeed the whole of civilization, is a series of incremental advances, each building on what went before. — Stephen Hawking

The subject of resistance networks has its origins in electrical engineering applications, and over decades, it has served to motivate a number of advances in discrete mathematics, such as the study of boundaries, percolation, stochastic analysis, and random walks on graphs; each of these is a well-established school of research, with its own striking scientific advances. It is premature to attempt to summarize the vast variety of new theorems — they are still appearing at a rapid rate in research journals! A common theme in the study of boundaries on infinite discrete systems X (weighted graphs, trees, Markov chains, or discrete groups) is the focus on a suitable subspace of functions on X, usually functions which are harmonic in some sense (i.e., fixed points of a given transfer operator). We are interested in the harmonic functions of finite energy, as this class of harmonic functions comes equipped with a natural inner product and corresponding Hilbert space structure. This will guide our choice of topics and emphasis, from an otherwise vast selection of possibilities. This volume is dedicated to the construction of unified functionalanalytic framework for the study of these potential-theoretic function spaces on graphs and an investigation of the resulting structures. The primary object of study is a resistance network: a graph with weighted edges. Our foundation is the effective resistance metric as the intrinsic notion xxvii

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of distance, and we approach the analysis of the resistance network by studying the space of functions on the vertices which have finite (Dirichlet) energy. There is a large existing literature on this subject, but ours is unique in several respects, most of which are due to the following: • We use the effective resistance metric to find canonical Hilbert spaces of functions associated with the resistance network. • We adhere to the intuition arising from the metaphor of electrical resistance networks, including Kirchhoff’s law and Ohm’s law. • We relate the results of our Hilbert space construction to the isotropic Heisenberg ferromagnet and obtain some results on long-range order in quantum statistical mechanics. • It is known (see Ref. [LP16] and the references therein) that the resistance metric is unique for finite graphs and not unique for certain infinite networks. We are able to clarify and explain the difference in terms of certain Hilbert space structures and also in terms of Dirichlet versus Neumann boundary conditions for a certain operator. Additionally, we introduce trace resistance and harmonic resistance and relate these to the aforementioned. A large portion of this volume is dedicated to developing an operatortheoretic understanding of a certain boundary which appears in diverse guises. The boundary appears first in Chapter 2 in a crucial but mysterious way, as an agent responsible for the misbehavior of a certain formula relating the Laplace operator to the energy form. It reappears in Chapter 3 as an agent responsible for the failure of various formulations of the effective resistance R(x, y) to agree for certain infinite networks. In Chapter 7, we pursue the boundary directly, using tools from operator theory and stochastic integration. The pedagogical aim behind this approach is to demonstrate operator theory via a series of applications. Many examples are given throughout the book. These may serve as independent student projects, although they are not exercises in the traditional sense. Prerequisites We have endeavored to make this book as accessible and self-contained as possible. Nonetheless, readers coming across various ideas for the first time may wish to consult the following books: Ref. [DS84] (resistance networks), Refs. [AF09, LPW08] (probability), and Ref. [DS88] (unbounded operators).

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xxix

Our aim is to offer an accessible introduction to a present and active interdisciplinary subject. It is of current interest, and at the same time, it has exciting roots, covering applied as well as pure areas. A personal perspective: As our book gradually took shape, we have been influenced by many recent pioneering papers and research monographs. While we include a more complete discussion of the literature in connection with the many cited references, we wish to especially call attention to the following: Refs. [Chu07, CR06, CW92, Kig03, KL12, LP16, PW90, PW87, SC04, Soa94, Tho90, Woe00, Woe09, Zem91]. Detailed Description of Contents Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. — D. Hilbert

Chapter 1. Resistance Networks. We introduce the resistance network as a connected simple graph G = {G0 , G1 } equipped with a positive weight function c on the edges. The edges G1 ⊆ G0 × G0 are ordered pairs of vertices, so c is required to be symmetric. Hence, each edge (x, y) ∈ G1 is interpreted as a conductor with conductance cxy (or a resistor with resistance c−1 xy . Heuristically, smaller conductances (or larger resistances) correspond to larger distances; see the discussion in Chapter 3. We make  frequent use of the weight that c defines on the vertices via c(x) = y∼x cxy , where y ∼ x indicates that (x, y) ∈ G1 . The graphs we are most interested in are infinite graphs, but we do not make any general assumptions of regularity, group structure, etc. We require that c(x) is finite at each x ∈ G0 , but we do not generally require that the degree of a vertex be finite, nor that c(x) be bounded. In the “cohomological” tradition of von Neumann, Birkhhoff, Koopman, and others [vN32c, Koo36b, Koo57], we study the resistance network by analyzing spaces of functions defined on it. These are constructed rigorously as Hilbert spaces in Section 4.1; in the meantime, we collect some results about functions u, v : G0 → R defined on the vertices. The network Laplacian (or discrete Laplace operator ) operates on such a function by taking v(x) to a weighted average of its values at neighboring points in the graph, i.e.,  v(x) − v(y)  cxy (v(x) − v(y)) = , (1) (Δv)(x) := c−1 xy y∼x y∼x

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where x ∼ y indicates that (x, y) ∈ G1 . (The rightmost expression in formula (1) is written so as to resemble the familiar difference quotients from calculus.) This is the usual second-difference operator of numerical analysis when adapted to a network. There is a large literature on discrete harmonic analysis (basically, the study of the graph/network Laplacian), which include various probabilistic, combinatoric, and spectral approaches. It would be difficult to give a reasonably complete account, but the reader may find an enjoyable approach to the probabilistic perspective in Refs. [Spi76, Tel06a], the combinatoric in Ref. [ABR07], the analytic in Ref. [Fab06], and the spectral in Refs. [Chu96, GIL08b]. More sources are peppered about the relevant sections in the following. Our formulation (1) differs from the stochastic formulation often found in the literature, but the two may easily be reconciled; see (1.6). Together with its associated quadratic form, the bilinear (Dirichlet) energy form 1   cxy (u(x) − u(y))(v(x) − v(y)) (2) E(u, v) := 2 0 y∼x x∈G

acts on functions u, v : G0 → R and plays a central role in the (harmonic) analysis on (G, c). (There is also the dissipation functional D, a twin of E, which acts on functions defined on the edges G1 and is introduced in the following section.) The first space of functions we study on the resistance network is the domain of the energy, that is, .

dom E := {u : G0 → R .. E(u) < ∞}.

(3)

For finite graphs, we prove the simple and folkloric key identity, which relates the energy and the Laplacian: E(u, v) = u, Δv1 = Δu, v1 , u, v ∈ dom E, (4)  2 inner where u, Δv1 = x∈G0 u(x)Δv(x) indicates the standard  product. The formula (4) is extended to infinite networks in Theorem 2.40 (see (9) for a preliminary discussion), where a third term appears. Indeed, understanding the mysterious third term is the motivation for most of this investigation. In Section 1.3, we collect several well-known and folkloric results and reprove some variants of these results in the current context. Currents are introduced as skew-symmetric functions on the edges; the intuition is that I(x, y) = −I(y, x) > 0 indicates electrical current flowing from x to y. In marked contrast to common tradition in geometric analysis [ABR07, PS07],

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we do not fix an orientation. For us, an orientation is a choice of one of {(x, y), (y, x)} for each edge and hence just a notation to be redefined as convenient. In particular, any nonvanishing current defines an orientation; one makes the choice so that I is a positive function. At this point, we give the definition of the dissipation, an inner product defined for functions on the edges, and its associated quadratic form: 1  2 c−1 (5) D(I) = xy I(x, y) . 2 1 (x,y)∈G

Most of our results in this section are groundwork for the sections to follow; several results are folkloric or obtained elsewhere in the literature. We include items which relate directly to results in later sections; the reader seeking a more well-rounded background is directed to Refs. [LPW08, LP16, Soa94, CdV98, Bol98] and the excellent elementary introduction Ref. [DS84]. After establishing the Hilbert space framework of Chapter 2, we exploit the close relationship between the two functionals E and D and use operators to translate a problem from the domain of one functional to the domain of the other. We also introduce Kirchhoff’s law and Ohm’s law, and in Section 1.5, we discuss the related compatibility problem: Every function on the vertices induces a function on the edges via Ohm’s Law, but not every function on the edges comes from a function on the vertices. This is related to the fact that most currents are not “efficient” in a sense which can be made clear variationally (cf. Theorem 1.40) and which is important in the definition of effective resistance metric in Theorem 3.2. We recover the well-known fact that the dissipation of an induced current is equal to the energy of the function inducing it in Lemma 1.30; this is formalized as an isometric operator in Theorem 10.12. We show that the equation Δv = δα − δω

(6)

always has a solution; we call such a function a dipole. In (6) and everywhere else, we use the notation δx to indicate a Dirac mass at x ∈ G0 , that is,  1, y = x, (7) δx = δx (y) := 0, else. Proving the existence of dipoles allows us to fill the gaps in Refs. [Pow76a, Pow76b] (see Section 7) and extend the definition of effective resistance . metric in Theorem 3.2 to infinite dimensions. The set of dipoles {vx .. x ∈ G} is a total system of vectors in HE and also a Green’s function for the Laplacian Δ.

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As is discussed at length in Remark 1.25, the study of dipoles, monopoles, and harmonic functions is a recurring theme of this book: Δv = δα − δω ,

Δw = −δω ,

Δh = 0.

As mentioned above, for any network G and any vertices x, y ∈ G0 , there is a dipole in dom E. However, dom E does not always contain monopoles or nonconstant harmonic functions; the existence of monopoles is equivalent to transience of the network [Lyo83]; we give a new criterion for transience in Lemma 2.53. In Theorem 14.5, we show that the integer lattice networks (Zd , 1) support monopoles iff d ≥ 3, but in Theorem 14.17, we show all harmonic functions on (Zd , 1) are linear and hence do not have finite energy. (Both of these results are well known; the first is a famous theorem by Polya — we include them for the novelty of the method of proof.) In contrast, the binary tree in Example 15.4 supports monopoles and nontrivial harmonic functions, both of finite energy (any network supporting nontrivial harmonic functions also supports monopoles, cf. Ref. [Soa94, Thm. 1.33]). It is apparent that monopoles and nontrivial harmonic functions are sensitive to the asymptotic geometry of (G, c). Chapter 2. The energy Hilbert space HE : We use the natural Hilbert space structure on the space of finite-energy functions (with inner product given by E) to reinterpret previous results as claims about certain operators and thereby clarify and generalize results from the beginning of Chapter 1 up to Section 1.3. This is the energy space HE . We construct a reproducing kernel for HE from first principles (i.e., via Riesz’s Lemma) in Section 2.1. If o ∈ G0 is any fixed reference point, define vx to be the vector in HE which corresponds (via Hilbert space duality) to the evaluation functional Lx : Lx u := u(x) − u(o). Then, the functions {vx } form a reproducing kernel, and vx is a solution to the discrete Dirichlet problem Δvx = δx − δo . Although these functions are linearly independent, they are usually neither an orthonormal basis (orthonormal basis) nor a frame. However, the span of {vx } is dense in dom E and appears naturally when the energy Hilbert space is constructed from the resistance metric by von Neumann’s method; cf. Section 4.1. Note that the Dirac masses {δx }G0 , which are the usual candidates for an orthonormal basis, are not orthogonal with respect to the energy inner product (2); cf. (1.11). In fact, Theorem 2.49 shows that {δx }G0 may not

Introduction

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even be dense in the energy Hilbert space! Thus, {vx } is the only canonical choice for a representing set for functions of finite energy. In Section 2.2, we use the Hilbert space structure of HE to better understand the role of the nontrivial harmonic functions. In particular, Lemma 2.20 shows that we may decompose HE into the functions of finite support (Fin) and the harmonic functions of finite energy (Harm): HE = Fin ⊕ Harm.

(8)

In Section 2.3, we prove a discrete version of the Gauss–Green formula (Theorem 2.40), which appears to be absent from the literature:   u(x)Δv(x) + u(x) ∂∂v (x), ∀u ∈ HE , v ∈ M (9) E(u, v) = x∈G0

x∈bd G

where denotes the normal derivative of v and M is a space containing span{vx }; see Section 2.3 for precise definitions. For the moment, both the boundary and the normal derivatives are understood as limits (and hence vanish trivially for finite graphs); we will be able to define these objects more concretely via techniques of Gel’fand in Chapter 7. It turns out that the boundary term (that is, the rightmost sum in (9)) vanishes unless the network supports nontrivial harmonic functions (that is, nonconstant harmonic functions of finite energy). More precisely, in  Theorem 2.49, we prove that there exist u, v ∈ HE for which bd G u ∂∂v = 0 if and only if the network is transient. That is, the random walk on the network with transition probabilities p(x, y) = cxy /c(x) is transient. We also give several other equivalent conditions for transience in Section 2.4. It is easy to prove (see Corollary 2.73) that nontrivial harmonic functions cannot lie in 2 (G0 ). This is why we do not require u, v ∈ 2 (G0 ) in general and why we stringently avoid including such a requirement in the definition of the domain of the Laplacian. Such a restriction would remove the nontrivial harmonic functions from the scope of our analysis, and we will see that they are at the core of some of the most interesting phenomena appearing on an infinite resistance network. ∂v ∂ (x)

Chapter 3. Effective resistance metric: The effective resistance metric R is foundational to our study, instead of the shortest-path metric more commonly used as graph distance. The shortest-path metric on a weighted graph is usually defined to be the sum of the resistances in any shortest path between two points. The effective resistance metric is also defined via c but in a more complicated way. The crucial difference is that the effective resistance metric reflects both the topology of the graph and the

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weighting c; two points are closer together when there is more connectivity (more paths and/or paths with greater conductance) between them. The effective resistance metric is a much more accurate way to measure distance when travel from point x to point y can be accomplished simultaneously through many paths, e.g., flow of electrical current, fluid diffusion through porous media, or data transfer over the internet. In Section 3.1, we give a multifarious definition of the effective resistance metric R, which may be physically characterized as the voltage drop between two vertices when electrical leads with a fixed current are applied to the two vertices. Most of these formulations appear elsewhere in the literature, but some appear to be specific to the physics literature, some to probability, and some to analysis. We collect them and prove their equivalence in Theorem 3.2, including a couple new formulations that will be useful in later sections. It is somewhat surprising that when these formulas are extended to an infinite network in the most natural way, they are no longer equivalent. (Note that each of the six formulas has both a free and wired version, but some appear much less natural in one version than in the other.) Some of the formulas lead to the “free resistance” RF , and others lead to the “wired resistance” RW ; here, we follow the terminology of Ref. [LP16]. In Section 3.2, we precisely characterize the types of extensions that lead to each and explain this phenomenon in terms of projections in Hilbert space, Dirichlet versus Neumann boundary conditions, and via probabilistic interpretation. Additionally, we discuss the “trace resistance” given in terms of the trace of the Dirichlet form E, and we study the “harmonic resistance,” which is the difference between RF and RW and is not typically a metric. Chapter 4. Construction of the energy space HE : The energy space HE was constructed as a reproducing kernel Hilbert space in §2.1, but there is another approach. In Section 4.1, we use a theorem of von Neumann to construct a Hilbert space from the metric space (G, R) and produce an isometric embedding of (G, R) into HE ; cf. Theorem 4.2. For infinite networks, (G, RF ) embeds into HE and (G, RW ) embeds into Fin. In Section 4.2, we discuss how this enables one to interpret HE as a canonical invariant of the original resistance network (G, c). Chapter 5. The Laplacian on HE : We study the operator theory of the Laplacian in some detail in Section 5.1, examining the various domains and self-adjoint extensions. We identify one domain for the Laplacian, which allows for the choice of a particular self-adjoint extension for the

Introduction

xxxv

constructions in Chapter 7. Also, we give technical conditions which must be considered when the graph contains vertices of infinite degree and/or the conductance functions c(x) is unbounded on G0 . This results in an extension of the Royden decomposition to HE = Fin1 ⊕ Fin2 ⊕ Harm, where Fin2 is the E-closure of span{δx − δo } and Fin1 is the orthogonal complement of Fin2 within Fin. Example 14.36 shows a case where Fin2 is not dense in Fin. In Section 5.3, we study the defect space of ΔV , that is, the space spanned by solutions to Δu = −u. In Section 5.3.1, we relate the boundary term of (9) to the boundary form βbd (u, v) := 21 (ΔV u, vE − u, ΔV vE ) ,

u, v ∈ dom(ΔV )

(10)

of classical functional analysis; cf. Ref. [DS88, Section XII.4.4]. In Theorem 5.22, we show that if Δ fails to be essentially self-adjoint,1 then Harm = {0}. In general, the converse does not hold: Corollary 6.89 shows that Δ has no defect when deg(x) < ∞ and c(x) is bounded. (Thus, any homogeneous tree of degree 3 or higher with constant conductances provides a counterexample to the converse.) Chapter 6. The 2 theory of Δ and T: We consider some results for Δ and T as operators on 2 (1), where the inner product is given by u, Δv1 :=  u(x)Δv(x) and on 2 (c), where the inner product is given by u, Δvc :=  c(x)u(x)Δv(x). We prove that the Laplacian is essentially self-adjoint on 2 (1) under very mild hypotheses in Section 6.1. The subsequent spectral representation allows us to give a precise characterization of the domain of the energy functional E in this context. In Section 6.3, we study the relation between the reproducing kernel {vx } and the spectral properties of Δ and its selfadjoint extensions. In particular, we examine the necessary conditions for {vx } to be a frame for HE and the relation between vx and δx . This allows for the study of a type of spectral reciprocity in Section 6.4. In Section 6.5, we examine boundedness and compactness of Δ and T in terms of the decay properties of c. The space 2 (c) considered in Section 6.6 is essentially a technical tool; it allows for a proof that the terms of the Discrete Gauss– Green formula are absolutely convergent and hence independent of any exhaustion. However, it is also interesting in its own right, and we show an interesting connection with the probabilistic Laplacian c−1 Δ. Results from 1 See

Definition B.8.

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this section imply that Δ is also essentially self-adjoint on HE , subject to the same mild hypotheses as the 2 (1) case. The energy Hilbert space HE contains much different information about a given infinite graph system (G, c) than does the more familiar 2 sequence space, even when appropriate weights are assigned. In the language of Markov processes, HE is better adapted to the study of (G, c) than 2 . One reason for this is that HE is intimately connected with the resistance metric R. Chapter 7. The boundary bd G and boundary representation: We study the boundary bd G in terms of the Laplacian by reinterpreting the boundary term of (9) as an integral over a space which contains HE . This gives a representation of bd G as a measure space whose structure is well studied. In Theorem 7.1 of Section 7.1, we observe that an important consequence of (9) is the following boundary representation for the harmonic functions:  x u ∂h + u(o) (11) u(x) = ∂ bd G

for u ∈ Harm, where hx = PHarm vx is the projection of vx to Harm; see (8). This formula is in the spirit of Choquet theory and the Poisson integral formula and is closely related to Martin boundary theory. Unfortunately, the sum in (11) is only understood in a limiting sense and so provides limited insight into the nature of bd G. This motivates the development of a more concrete expression. We use a self-adjoint of Δ to construct a Gel’fand triple SG ⊆ HE ⊆ S  and a extension Δ G ∞ ) is a suitable dense Gaussian probability measure P. Here, SG := dom(Δ (Schwartz) space of “test functions” on the resistance network, and SG is the corresponding dual space of “distributions” (or “generalized functions”). This enables us to identify bd G as a subset of SG , and in Corollary 7.32, we rewrite (11) more concretely as  u(ξ)hx (ξ) dP(ξ) + u(o), (12) u(x) =  SG

again for u ∈ Harm and with hx = PHarm vx . Thus, we study the metric/measure structure of G by examining an associated Hilbert space of random variables. This is motivated in part by Kolmogorov’s pioneering work on stochastic processes (see Section 16.1) as well as on a powerful refinement of Minlos. The latter is in the context of the Gel’fand triples mentioned just above; see Ref. [Nel64] and Section 7.2. Further applications

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to harmonic analysis and to physics are given in sections starting from Chapter 11 up to Section 16.3. Chapter 8. Multiplication operators on the energy space: We study multiplication operators (Mf u)(x) = f (x)u(x), where f : G0 \ {o} → C. Due to the necessary reliance on pointwise values of f in defining a multiplication operators, this chapter uses the grounded energy product u, vo = u(o)v(o) + u, vE discussed in Section 2.4.1. In Section 8.1, we show that no nontrivial multiplication operators are self-adjoint (in striking contract to multiplication operators in L2 spaces) and that the adjoint of Mf is given on the energy kernel by M  vx = f (x)vx . We also characterize the boundedness of multiplication operators in terms of positive semidefiniteness of a related function in Theorem 8.7 and in terms of a uniform bound on a family of related operators in Theorem 8.21. In Section 8.2, we study algebras of multiplication operators and compute the operator norm of Mδx in terms of the conductance and effective resistance at x; see Theorem 8.12. The duality between {δx } and {vx } reappears in Theorem 8.16. In Section 8.3, we study the Banach algebra AE of bounded functions of finite energy and find that AFin = Fin ∩ AE is a closed ideal in AE and that there is a close relation between the Gel’fand space of AHarm = AE /AFin and bd G. When Harm = 0, the Gel’fand space of AE turns out to be the one-point compactification of G. Chapter 9. Symmetric pairs: We introduce a generalization of the notion of a symmetric operator: a pair of operators, each of which is contained in the adjoint of the other. We flesh out the relationship between symmetric pairs and a single symmetric operator in Section 9.1.1. This device facilitates studying the relationship between the energy space and the 2 spaces associated with a network, as is discussed in Section 9.2. The apparatus of symmetric pairs helps clarify the relationship between the Laplacian and inclusion operator (see Theorems 9.25 and 9.26) and provides for a way to prove the closability of the energy form and construct the Krein extension of the energy Laplacian (Remark 9.30). Another wellknown self-adjoint extension, the Friedrichs extension, is constructed in Section B.3, and an alternative approach to the Krein extension is given in Section B.4. We provide applications of symmetric pairs to network problems and also provide some applications to problems in stochastic processes in Section 9.3 (Malliavin derivative and Skorokhod integral) and the theory

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of von Neumann algebras in Section 9.4 (Tomita–Tamesaki classification of Type III factors). Chapter 10. HE and HD : The dissipation space HD is the Hilbert space of functions on the edges when equipped with the dissipation inner product. We solve problems in discrete potential theory with the use of the drop operator d (and its adjoint d ), where dv(x, y) := cxy (v(x) − v(y)).

(13)

The drop operator d is, of course, just an implementation of Ohm’s law and can be interpreted as a weighted boundary operator in the sense of homology theory. The drop operator appears elsewhere in the literature, sometimes without the weighting cxy ; see Refs. [Chu96, Tel06a, Woe00]. However, we use the adjoint of this operator with respect to the energy inner product, instead of the 2 inner product used by others. This approach appears to be new, and it turns out to be more compatible with physical interpretation. For example, the displayed equation preceding Ref. [Woe00, (2.2)] shows that the 2 adjoint of the drop operator is incompatible with Kirchhoff’s node law. Since the resistance metric may be defined in terms of currents obeying Kirchhoff’s laws, we elect to make this break with the existing literature. Additionally, this strategy will allow us to solve the compatibility problem described in Section 1.5 in terms of a useful minimizing projection operator Pd , discussed in detail in Section 10.4. Furthermore, we believe our formulation is more closely related to the (co)homology of the resistance network as a result. We decompose HD into the direct sum of the range of d and the currents which are sums of the characteristic functions of cycles HD = ran d ⊕ cl span χϑ ,

(14)

where ϑ is a cycle, i.e., a path in the graph which ends where it begins. In (14) and elsewhere, we indicate the closed linear span of a set by cl span χϑ := cl span{χϑ }. From (8) (and the fact that d is an isometry), it is clear that the first summand of (14) can be further decomposed into weighted edge neighborhoods dδx and the image of harmonic functions under d in Theorem 10.8. After a first draft of this book was complete, we discovered that the same approach is taken in Ref. [LP16]. One of us (PJ) recalls conversations with Raul Bott concerning an analogous Hilbert space operator-theoretic approach to electrical networks, apparently attempted in the 1950s in the engineering literature. We could not find details in any journals; the closest we could come is the fascinating paper Ref. [BD49]

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by Bott et al. An additional early source of influence is Norbert Wiener’s paper, Ref. [WR46]. In Section 10.4, we describe how d solves the compatibility problem and may be used to solve a large class of problems in discrete potential theory. Also, we discuss the analogy with complex analysis. Chapter 11. Probabilistic interpretations: In Refs. [LP16, DS84, Tel06a, Woe00] and elsewhere, the random walk on a resistance network is defined by the transition probabilities p(x, y) := cxy /c(x). In this context, the probabilistic transition operator is P = c−1 T, and one uses the stochastically renormalized Laplacian Δc := c−1 Δ, where c is understood as a multiplication operator; see Definition 1.3. This approach also arises in the discussion of trace resistance in Section 3.6 and allows one to construct currents on the graph as the average motion of a random walk. As an alternative to the approach described above, we discuss a probabilistic interpretation slightly different from those typically found in the literature: We begin with a voltage potential as an initial condition and consider the induced current I. The components of such a flow are called current paths and provide a way to interpret potential-theoretic problems in a probabilistic setting. We study the random walks where  the transition probability is given by I(x, y)/ z∼x I(x, z). We consider the harmonic functions in this context, which we call forward-harmonic functions and the associated forward-Laplacian of Definition 11.16. We give a complete characterization of forward-harmonic functions as cocycles, following Ref. [Jor06]. Chapter 12. Spectral comparisons: In this chapter, we consider the effect of changing the conductance function; for two different conductance functions b and c, we compare: 1. the energy forms E (b) and E (c) and the respective energy Hilbert spaces HE (b) and HE (c) that they define; 2. the systems of dipole vectors that form by reproducing kernels for the two Hilbert spaces; see Definition 2.11; 3. the respective Laplace operators Δ(b) and Δ(c) and their spectra; 4. the spaces of finite-energy harmonic functions on HE (b) and HE (c) ; and 5. the effective resistance metrics on HE (b) and HE (c) . Chapter 13. Examples: We collect an array of examples that illustrate the various phenomena encountered in the theory and work out many concrete

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examples. Some elementary finite examples are given in Section 13.1 to give the reader an idea of the basics of resistance network theory. In Section 13.2, we move on to infinite graphs. Chapter 14. Integer lattices: The lattice resistance network (Zd , c) have vertices at the points of Rd , which have integer coordinates, and the edges between every pair of vertices (x, y) with |x − y| = 1. The case for c = 1 is amenable to Fourier analysis, and in Section 14.1, we obtain explicit formulas for many expressions: • Lemma 14.4 gives a formula for the potential configuration functions {vx }. • Theorem 14.7 gives a formula for the resistance distance R(x, y). • Theorem 14.9 gives a formula for the resistance distance to infinity in the sense R(x, ∞) = limy→∞ R(x, y). • Theorem 14.5 gives a formula for the solution w of Δw = −δo on Zd ; it is readily seen that this w has finite energy (i.e., a monopole) iff d ≥ 3. In Ref. [P´ol21], P´olya proved that the random walk on this graph is transient if and only if d ≥ 3; see Ref. [DS84] for a nice exposition. We offer a new characterization of this dichotomy (there exist monopoles on Zd if and only if d ≥ 3), which we recover in this section via a new (and completely constructive) proof. In Remark 14.18, we describe how in the infinite integer lattices, functions in HE may be approximated by functions of finite support. Chapter 15. Trees: When the resistance network is a tree (i.e., there is a unique path between any two vertices), the resistance distance coincides with the geodesic metric, as there is always exactly one path between any two vertices; cf. Lemma 3.52 and the preceding discussion. When the tree has exponential growth, as in the case of homogeneous trees of degree ≥ 3, one can always construct nontrivial harmonic functions and monopoles of finite energy. In fact, there is a very rich family of each, and this property makes this class of examples a fertile testing ground for many of our theorems and definitions. In particular, these examples highlight the relevance and distinctions between the boundary (as we construct it), the Cauchy completion, and the graph ends of Refs. [PW90, Woe00]. In particular, they enable one to see how adjusting decay conditions on c affects these things.

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Chapter 16. Magnetism: The integer lattice networks (Zd , 1) investigated in Chapter 14 comprise the framework of infinite models in thermodynamics and quantum statistical mechanics. In Section 16.3, we employ these formulas in the refinement of an application to the theory of the (isotropic Heisenberg) model of ferromagnetism, as studied by R. T. Powers. In addition to providing an encapsulated version of the Heisenberg model, we give a commutative analogue of the model, extend certain results of Powers from Refs. [Pow75, Pow76a, Pow76b, Pow78, Pow79], and discuss the application of the resistance metric to the theory of ferromagnetism and “long-range order.” This problem was raised initially by Powers and may be viewed as a noncommutative version of Hilbert spaces of random variables. Ferromagnetism in quantum statistical mechanics involves algebras of noncommutative observables and may be described with the use of states on C  -algebras. As outlined in the cited references, the motivation for these models draw on thermodynamics; hence, the notions of equilibrium states (formalized as KMS states, see Section 16.4). These KMS states are states in the C  -algebraic sense (that is, positive linear functionals with norm 1), and they are indexed by absolute temperature. Physicists interpret such objects as representing equilibria of infinite systems. In the current case, we consider spin observables arranged in a lattice of a certain rank, d = 1, 2, 3, . . . , and with nearest-neighbor interaction. Rigorous mathematical formulation of phase transitions appears to be a hopeless task with current mathematical technology. As an alternative avenue of inquiry, much work has been conducted on the issue of long-range order, i.e., the correlations between observables at distant lattice points. These correlations are measured relative to states on the C  -algebra, in this case, in the KMS states for a fixed value of temperature. While we shall refer to the literature, e.g. Refs. [BR79, Rue69], for formal definitions of key terms from the C  -algebraic formalism of quantum spin models, physics, and KMS states, we include a minimal amount of background and terminology from the physics literature. Chapter 17. Future directions: We conclude with a brief discussion of several projects which have arisen from work on this book as well as some promising new directions that we have not yet had time to pursue. Appendices: We give some background material from functional analysis and operator theory in Appendices A and B. In Appendix C, we include some diagrams to help clarify the properties of the many operators and spaces we discuss and the relations between them.

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What This Book is About ‘Obvious’ is the most dangerous word in mathematics. — E. T. Bell

For organization of the main parts in the book, readers are referred to Figure 1. The effective resistance metric provides the foundation for our investigations because it is the natural and intrinsic metric for a resistance network, as the work of Kigami has shown; see Ref. [Kig01] and the extensive list of references by the same author therein. Moreover, the close relationship between diffusion geometry (i.e., geometry of the resistance metric) [MM08, SMC08, CKL+ 08, CM06] and random walks on graphs leads us to expect or hope that there will be many applications of our results to several other subjects, in addition to fractals: models in

Fig. 1.

Thematic relations among the central ideas in the text with chapter numbers.

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quantum statistical mechanics, analysis of energy forms, interplay between self-similar measures and associated energy forms, certain discrete models arising in the study of quasicrystals (e.g., Refs. [BM00, BM01]), and multiwavelets (e.g. Refs. [BJMP05, DJ06, DJ07, Jor06]), among others. A general theme of these areas is that the underlying space is not sufficiently regular to support a group structure yet is “locally” regular enough to allow analysis via probabilistic techniques2 Consequently, the analysis of functions on such spaces is closely tied to Dirichlet energy forms and the graph Laplacian operator associated with the graph. This appears prominently in the context of this book as follows: 1. The embedding of the metric space ((G, c), R) into the Hilbert space HE of functions of finite energy, in such a way that the original metric may be recovered from the norm, i.e., R(x, y) = E(vx − vy ) = vx − vy 2E , where vx ∈ HE is the image of x under the embedding. 2. The relation of the energy form to the graph Laplacian via the equation   u(x)Δv(x) + u(x) ∂∂v (x), (15) E(u, v) = x∈G0

x∈bd G

introduced just above in the discussion in Chapter 1. Each summation on the right-hand side of (15) is more subtle than it appears. These details for the first sum are given in Theorem 2.40, and the details for the second sum are the focus of almost all of Chapter 7. 3. The presence of nonconstant harmonic functions of finite energy. These are precisely the objects which support the boundary term in (15) and imply RW (x, y) < RF (x, y). They are also responsible for the boundary described in Section 7. 4. The solvability of the Dirichlet problem Δw = −δy , where δy is a Dirac mass at the vertex y ∈ G0 . The existence of finite-energy solutions w is equivalent to the transience of the random walk on the network. Such functions are called monopoles, and (via Ohm’s law) they induce a unit flow to infinity, as discussed in Refs. [DS84, LP16, LPW08]. 2 Roughly, the general idea is to exploit the correspondence from potential theory between a differential operator and its Dirichlet form; for example, consider the following equivalent problems:  . Δu = 0, on Ω , ⇐⇒ u = argmin E(v), {v : Ω → R .. v = f on ∂Ω }. u = f, on ∂Ω ,

Since the Dirichlet form E is determined by the stochastic process, knowledge of the stochastic process allows one to indirectly find (weak) solutions to differential equations.

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Remark 1 (Relation to numerical analysis). In addition to uses in graph theory and electrical networks, the discrete Laplacian Δ has other uses in numerical analysis: Many problems in partial differential equation (PDE) theory lend themselves to discretization in terms of subdivisions or grids of refinements in continuous domains. A key tool in applying numerical analysis to solving PDEs is discretization and use of repeated differences, especially for using the discrete Δ in approximating differential operators and PDOs. See, for example, Ref. [AH05]. One picks a grid size δ and then proceeds in steps: 1. Start with a partial differential operator, then study an associated discretized operator with the use of repeated differences on the δ-lattice in Rd . 2. Solve the discretized problem for δ fixed. 3. As δ tends to zero, numerical analysts evaluate the resulting approximation limits, and they bound the error terms. When discretization is applied to the Laplace operator in d continuous variables, the result is our Δ for the network (Zd , c); see Chapter 14 for details and examples. However, when the same procedure is applied to a continuous Laplace operator on a Riemannian manifold, the discretized Δ will be the network Laplacian on a suitable infinite network (G, c), which in general may have a much wilder geometry than Zd . This yields numerical algorithms for the solution to PDEs, and in the case of second-order PDEs, the discretized operator is the discrete Laplacian studied in this investigation. Motivation and applications A drunk man will eventually return home, but a drunk bird will lose its way in space. — G. Polya

Applications to infinite networks of resistors serve as a motivation, but our theorems have a wider scope, have other applications, and are, we believe, of independent mathematical interest. Our interest originates primarily from three sources3: 3 These three sources served as our introduction, but of course, once diving in, we discovered a host of other marvelous references, including those by Lyons & Peres,

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1. a series of papers written by Powers in the 1970s in which he introduced infinite systems of resistors into the resolution of an important question from quantum statistical mechanics in Refs. [Pow75, Pow76a, Pow76b, Pow78, Pow79]; 2. the pioneering work of Kigami on the analysis of PCF self-similar fractals, viewing these objects as rescaled limits of networks, see Ref. [Kig01]; 3. Doyle and Snell’s lovely book Random Walks and Electrical Networks, which gives an excellent elementary introduction to the connections between resistance networks and random walks, including a resistancetheoretic proof of Polya’s famous theorem on the transience of random walks in Zd . Indeed, our larger goal is the cross-pollination of these areas, and we hope that the results of this book may be applicable to analysis on fractal spaces. A first step in this direction is given in Theorem 17.3. To this end, a little more discussion of each of the above two subjects is in order. Powers was interested in magnetism and the appearance of “long-range order,” which is the common parlance for the correlation between spins of distant particles; see Chapter 16 for a larger discussion. Consequently, he was most interested in graphs such as the integer lattice Zd (with edges between vertices of distance 1 and all resistances equal to 1) or other regular graphs that might model the atoms in a solid. Powers established a formulation of resistance metric that we adopt and extend in Chapter 3, where we also show it to be equivalent to Kigami’s formulation(s). Also, the proofs of Powers’ original results on effective resistance metric contain a couple of gaps that we fill. In particular, Powers does not seem to have had been aware of the possibility of nontrivial harmonic functions until Ref. [Pow78], where he mentions them for the first time. It is clear that he realized several immediate implications of the existence of such functions, but there are more subtle (and just as important!) phenomena that are difficult to see without the clarity provided by Hilbert space geometry. Powers studied an infinite graph G by working with an exhaustion, that  is, a nested sequence of finite graphs G1 ⊆ G2 ⊆ . . . ⊆ Gk ⊆ G = k Gk . For example, Gk might be all the vertices of Zd lying inside the ball Kaimanovich, Woess, and too many others to mention. Our apologies to those not named here.

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of Euclidean radius k and the edges between them. Powers used this approach to obtain certain inequalities for the resistance metric, expressing the consequences of deleting small subsets of edges from the network. Although he makes no reference to it, this approach is very analogous to Rayleigh’s “short-cut” methods, as it is called in Ref. [DS84]. Powers’ use of an expanding sequence of graphs may be thought of as a “limit in the large,” in contrast to the techniques introduces by Kigami, which may be considered “limits in the small.” Self-similarity and scale renormalization are the hallmarks of the theory of fractal analysis as pursued by Kigami, Strichartz, and others (see Refs. [HKK02, Kig01, Kig03, Hut81, Str06, BHS05, Bea91, JP94, Jor04], for example), but these ideas do not enter into Powers’ study of resistors. One aim of the current work is the development of a Hilbert space framework suitable for the study of limits of networks defined by a recursive algorithm which introduces new vertices at each step and rescales the edges via a suitable contractive rescaling. As is known from, for example Refs. [JP94, Jor06, Str98a, Str06, Tep98], there is a spectral duality between “fractals in the large” and “fractals in the small.” The significance of Hilbert spaces ‘How large the World is!!’ said the ducklings, when they found how much more room they now had compared to when they were confined inside the egg-shell. ‘Do you imagine this is the whole world?’ asked the mother, ‘Wait till you have seen the garden; it stretches far beyond that to the parson’s field, but I have never ventured to such a distance’. — H. C. Andersen (from The Ugly Duckling)

A main theme of this book is the use of Hilbert space technology in understanding metrics, potential theory, and optimization on infinite graphs, especially through finite-dimensional approximation. We emphasize those aspects that are intrinsic to infinite resistance networks, and our focus is on analytic aspects of graphs, as opposed to the combinatorial and algebraic sides of the subject, etc. Those of our results stated directly in the framework of graphs may be viewed as discrete analysis, yet the continuum enters via spectral theory for operators and the computation of probability of sets of infinite paths. In fact, we display a rich variety of possible spectral types, considering the spectrum as a set (with multiplicities), as well as the associated spectral measures and representations/resolutions.

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Related issues for Hilbert space completions form a recurrent theme throughout our book. Given a resistance network, we primarily study three spaces of functions naturally associated with it: HE , HD , and, to a lesser extent, 2 (G0 ). Our harmonic analysis of functions on G is studied via operators between the respective Hilbert spaces, as discussed in Chapter 10, and the Hilbert space completions of these three classes are used in an essential way. In particular, we obtain the boundary of the graph (a necessary ingredient of (15) and the key to several mysteries) by analyzing the finite energy functions on G, which cannot be approximated by functions of finite support. However, this metric space is naturally embedded inside the Hilbert space HE , which is already complete by definition/construction. Consequently, the Hilbert space framework allows us to identify certain vectors as corresponding to the boundary of (G, c) and thus obtain a concrete understanding of the boundary. However, the explicit representations of vectors in a Hilbert space completion (i.e., the completion of a pre-Hilbert space) may be less than transparent; see Ref. [Yoo07]. In fact, this difficulty is quite typical when Hilbert space completions are used in mathematical physics problems. For ´ example, in Refs. [JO00, Jor00], one begins with a certain space of smooth functions defined on a subset of Rd with certain support restrictions. In relativistic physics, one must deal with reflections, and there will be a separate positive definite quadratic form on each side of the “mirror.” As a result, one ends up with two startlingly different Hilbert space completions: a familiar L2 -space of functions on one side and a space of distributions on ´ the other. In Refs. [JO00, Jor00], one obtains holomorphic functions on one side of the mirror, and the space of distributions on the other side is spanned by the derivatives of the Dirac mass, each taken at the same specific point x0 . It is the opinion of the authors that most interesting results of this book arise primarily from three things:

1. differences between finite approximations and infinite networks, and how & when these differences vanish in the limit; 2. the phenomena that result when one works with a quadratic form whose kernel contains the constant functions; 3. the boundary (which is not a subset of the vertices) that naturally arises when a network supports nonconstant harmonic functions of finite energy, and how it explains other topics mentioned above.

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In classical potential theory, working with modulo constant functions amounts to working with the class of functions satisfying f  2 < ∞ but abandoning the 2 requirement f 2 < ∞. This has some interesting consequences, and the nontrivial harmonic functions play an especially important role; see Remark 2.74. What would one hope to gain by removing the 2 condition? 1. From the natural embedding of the metric space (G, R) into the Hilbert space HE of functions of finite energy given by x → vx , the functions vx are not generally in 2 . See Figure 14.5 of Example 14.16 for an illustration. 2. The resistance metric does not behave nicely with respect to 2 conditions. Several formulations of the resistance distance R(x, y) involve optimizing over collections of functions which are not necessarily contained in 2 , even for many simple examples. 3. Corollary 2.73 states that nontrivial harmonic functions cannot lie in 2 (G0 ). Consequently, imposing an 2 hypothesis removes the most interesting phenomena from the scope of study; see Remark 2.74. The infinite trees studied in Examples 15.2–15.6 provide examples of these situations. While there are earlier books in the general area, e.g., Refs. [Kig10, Lig99, LP16, Per99, Woe00], we stress that our current focus is different in a number of ways, especially in its emphasis on the use of the theory of operators in Hilbert space, both bounded and unbounded, and their corresponding spectral analysis. As will become clear, it is in fact the theory of unbounded operators which is of more critical use in our current analysis of transient random walks. In physical terms, the intuitive reason for this is that signals interact with “boundaries” only if they travel to “infinity” in finite time. It is also our hope that the interaction between the two areas, dynamical systems on the one hand and operator theory on the other, will be of independent interest to readers; for example, we present a number of implications for operator theory coming from our current analysis on infinite graphs. One general sense in which operator theory can benefit from our current graph analysis is by giving specific geometric meaning to otherwise abstract notions, such as the different self-adjoint extensions of unbounded symmetric/Hermitian operators which have dense domain but are not (yet) self-adjoint. For a Laplace operator Δ associated with

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a given network, we discuss the geometric explanations of von Neumann’s deficiency indices and self-adjoint extensions of Δ, provided by an analysis of possible boundary constructions associated with the path space of the network and its associated Markov transition operator; this is discussed in Chapters 7, 9, and 11; see also Ref. [JP19]. Measures and measure constructions A reader glancing at our book will note a number of incarnations of measures on infinite sample spaces: It may be a suitable space of paths (Sections 11.1–11.2 and Section 16.1) or an analogue of the Schwartz space of tempered distributions (Section 7.2). The latter case relies on a construction of “Gel’fand triples” from mathematical physics. The reader may wonder why they face yet another measure construction, but each construction is dictated by the problems we solve. Taking limits of finite subsystems is a universal weapon used with great success in a variety of applications; we use it here in the study of resistance distances on infinite graphs (Section 3.2); boundaries and boundary representations for harmonic functions (Sections 6.3, 7.2, and 11.1–11.2); and equilibrium states and phase-transition problems in physics (Sections 16.1–16.2). (1) HE as an L2 space: The central Hilbert space in this study, the energy space HE , appears with a canonical reproducing kernel but without any canonical basis, and there is no obvious way to see HE as an L2 (X, μ) for some X and μ. Therefore, a major motivation for our measure constructions is just to be able to work with HE as an L2 space. In Section 16.1, we use a construction from probability to write HE = L2 (Ω , μ) in a way that makes the energy kernel {vx }x∈G0 into a system of (commuting) random variables. Here, Ω is an infinite Cartesian product of a chosen compact space S; one copy of S for each point x ∈ G0 . In Section 16.2, we use a noncommutative version of this probability technology: Rather than Cartesian products, we use infinite tensor products of C ∗ -algebras A, one for each x ∈ G0 . The motivation here is an application to a problem in quantum statistical mechanics. The “states” on the C ∗ -algebra of all observables are the quantum mechanical analogues of probability measures in classical problems. Heuristically, the reader may wish to think of them as noncommutative measures; see, for example, Ref. [BR97].

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(2) Boundary integral representation of harmonic functions: As it sometimes happens, the path to bd G is somewhat circuitous: We begin with the discovery of an integral over the boundary, which leads us to understand functions on the boundary, which in turn points the way to a proper definition of the boundary itself. A closely related motivation for a measure is the formulation of an integral representation of harmonic functions u ∈ HE :  u(ξ)hx (ξ) dP(ξ) + u(o), (16) u(x) =  SG

where hx = PHarm vx . Thus, the focus of Section 7.2 is a formalization of  x the imprecise “Riemann sums” u(x) = bd G u ∂h ∂ +u(o) of Section 2.3 as an integral of a bona fide measure. To carry this out, we construct a Gel’fand triple SG ⊆ HE ⊆ SG , where SG is a dense subspace of HE and SG ’ is its dual but with respect to a strictly finer topology. We are then able to produce a Gaussian probability measure P on SG and isometrically embed HE into L2 (SG , P). In fact, L2 (SG , P) is the second quantization of HE . However, the focus here is not on realizing HE as an L2 space (or subspace) but in obtaining the boundary integral representation of harmonic functions as in (16). Our aim is then to build formulas that allow us to compute values of harmonic functions u ∈ HE from an integral representation which yields u(x) as an integral over bd G ⊆ SG . Note that this integration in (16) is with respect to a measure depending on x just as in the Poisson and Martin representations. (3) Concrete representation of the boundary: We would like to realize bd G as a measure space defined on a set of well-understood elements; this is the focus of the constructions in Chapter 7. The goal is a measure on the space of all infinite paths in G which yields the boundary bd G in such a way that G ∪ bd G is a compactification of G, which is compatible with the energy form E and the Laplace operator Δ and hence also the natural resistance metric on (G, c). This type of construction has been carried out with great success for the case of bounded harmonic functions (e.g., Poisson representation and the Fatou–Primalov theorem) and for nonnegative harmonic functions (e.g., Martin boundary theory), but our scope of inquiry is the harmonic functions of finite energy. Finally, we would like to use this Gaussian measure on SG to clarify bd G as a subspace of SG . Such a relationship

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is a natural expectation, as the analogous thing occurs in the work of Poisson, Choquet, and Martin. What This Book is Not About Many of the topics discussed in this book may appear to have been previously discussed elsewhere in the literature, but there are certain important subtleties which actually make our results quite different. This section is intended to clarify some of these. While there already is a large literature on electrical networks and on graphs (see, for example, Refs. [CW92, CW07, DK88, Dod06, DS84, Pow76b, CdV04, CR06, Chu07, FK07] and the preprint Ref. [Str10], which we received after the first version of this book was completed), we believe that our current operator- or spectral-theoretic approach suggests new questions and new theorems and allows many problems to be approached in greater generality. The literature on the analysis of graphs breaks down into a variety of overlapping subareas, including: combinatorial aspects, systems of resistors on infinite networks, random-walk models, operator algebraic models [DJ08, Rae05], probability on graphs (e.g., infinite particle models in physics [Pow79]), Brownian motion on self-similar fractals [Hut81], Laplace operators on graphs, finite-element approximations in numerical analysis [BS08]; and more recently, use in Internet search algorithms [FK07]. Even just the study of Laplace operators on graphs subdivides further due to recently discovered connections between graphs and fractals generated by an iterated functions system (IFS); see, for example, Refs. [Kig03, Str06]. Other major related areas include discrete Schr¨odinger operators in physics, information theory, potential theory, uses of the graphs in scaling analysis of fractals (constructed from infinite graphs), probability and heat equations on infinite graphs, graph C  -algebras, groupoids, Perron– Frobenius transfer operators (especially as used in models for the Internet); multiscale analysis, renormalization, and operator theory of boundaries of infinite graphs (more current and joint research between the coauthors.) The motivating applications from Refs. [Pow75, Pow76a, Pow76b, Pow78, Pow79] include the operator algebra of electrical networks of resistors (lattice models, C  -algebras, and their representations), and more specifically, KMS-states from statistical mechanics. While working and presenting our

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results, we learned of even more such related research directions from experts working in these fields, and we are thankful to them all for taking the time to explain some aspects of them to us. The main point here is that the related literature is vast, but our approach appears to be novel and our results, while reminiscent of classical theory, are also new. We now elucidate certain specific differences. Spectral theory Our approach differs from the extensive literature on spectral graph theory (see Ref. [Chu96] for an excellent introduction and an extensive list of further references) due to the fact that we eschew the 2 basis for our investigations. We primarily study Δ as an operator on HE and with respect to the energy inner product. The corresponding spectral theory is radically different from the spectral theory of Δ in 2 . Most other work in spectral graph theory takes place in 2 , even implicitly when working with finite graphs: The adjoint of the drop operator (see Definition 10.2) is taken with respect to the 2 inner product and consequently violates Kirchhoff’s laws. In fact, the discussion preceding Ref. [Woe00, (2.2)] shows how this version of the adjoint is incompatible with Kirhhoff’s law, as mentioned in the summary of Chapter 10. Additionally, Ref. [Chu96] and others work with the spectrally renormalized Laplacian Δs := c−1/2 Δc−1/2 . However, Δs is a bounded Hermitian operator (with spectrum contained in [0, 2]) and so is unsuitable for our investigations of bd G based on defect indices, etc. As we have only encountered relatively few instances where the complete details are worked out for spectral representations in the framework of discrete analysis, we have attempted to provide several explicit examples. These are likely folkloric, as the geometric possibilities of graphs are vast and so is the associated range of spectral configurations. A list of recent and past papers of relevance includes Refs. [Str10, Car72, Car73a, Car73b, CR06, Chu07, CdV99, CdV04, Jor83] and Wigner’s original paper on the semicircle law [Wig55]. The current investigation also led to a spectral analysis of the binary tree from the perspective of dipoles in Ref. [DJ08]; this study discovered that the spectrum of Δ on the binary tree is also given by Wigner’s semicircle law. There is also literature on infinite/transfinite networks and generalized Kirchhoff laws using nonstandard analysis, etc.; see Refs. [Zem91, Zem97].

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However, this context allows for edges with a resistance of 0, which we do not allow (for physical as well as theoretical reasons). One can neglect the resistance of wires in most engineering applications but not when considering infinite networks (the epsilons add up!). The resulting theory therefore diverges rapidly from the observations of this book; according to our definitions, all networks support currents satisfying Kirchhoff’s law, and in particular, all induced currents satisfy Kirchhoff’s law. Operator algebras There are recent papers in the literature which also examine graphs with tools from operator algebras and infinite determinants. The papers Refs. [GIL08b, GIL09, GIL08a] by Guido et al. are motivated by questions on fractals and study the detection of periods in infinite graphs using the Ihara zeta function, a variant of the Riemann zeta function. There are also related papers with applications to the operator algebra of groupoids [Cho08, FMY05] and Refs. [BM00, BM01], which apply infinite graphs to the study of quasi-periodicity in solid-state physics. However, the focus in these papers is quite different from ours, as are the questions asked and the methods employed. While periods and quasi-periods in graphs play a role in our current results, they enter our picture in quite different ways, for example via spectra and metrics that we compute from energy forms and associated Laplace operators. There does not seem to be a direct comparison between our results and those of Guido et al. Boundaries of graphs There is also no shortage of papers studying boundaries of infinite graphs: Refs. [PW90, Saw97, Woe00] discuss the Martin boundary, Refs. [PW90, Woe00] also describe the more geometrically constructed “graph ends,” and Refs. [Car72, Car73a, Car73b] use unitary representations. There are also related results in Refs. [CdV99, CdV04] and Refs. [Kai98, Kai92a, Kai92b, KW02]. While there are connections to our study, the scope is different. Martin boundary theory is really motivated by constructing a boundary for a Markov process, and the geometry/topology of the boundary is rather abstract and a bit nebulous. Additionally, one needs a Green’s function, and it must satisfy certain hypotheses before the construction can proceed. Furthermore, the focus of Martin boundary theory is the nonnegative

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harmonic functions. Our boundary construction is more general in that it applies to any electrical network, as in Definition 1.7, and it remains correct for all harmonic functions of finite energy, including constant functions and harmonic functions which change sign. However, it is also more restrictive in the sense that a resistance network may support functions which are bounded below but do not have finite energy. We should also point out that our boundary construction is related to, but different from, the “graph ends” introduced by Freudenthal and others. The ends of a graph are the natural discrete analogue of the ends of a minimal surface (usually assumed to be embedded in R3 ), a notion which is closely related to the conformal type of the surface. Starting with the central book [Woe00] by Wolfgang Woess, the following references will provide the reader with an introduction to the study of harmonic functions on infinite networks and the ends of graphs and groups: Refs. [Woe86, Woe87, Woe89] and Ref. [Woe95] on Martin boundaries, Ref. [PW90] on ends, Ref. [Woe96] on Dirichlet problems, and Ref. [Woe97] on random walk. A comparison of the examples in Chapters 14 and 15 illustrates that varying the resistances produces dramatic changes in the topology of the boundary. Roughly, our boundary essentially consists of those infinite paths which can be distinguished by harmonic functions of finite energy; see Section 7.3 for details. It follows that transient networks with no nontrivial harmonic functions have exactly one boundary point (corresponding to the unique monopole). In particular, the integer lattices (Zd , 1) have precisely one boundary point for d ≥ 3 and have zero boundary points for d = 1, 2. The Martin boundary of (Z2 , 1) consists of two points; similarly, (Z2 , 1) has two graph ends; cf. Ref. [PW90].

General Remarks Remark 2 (Real- and complex-valued functions). Throughout the introductory discussion of resistance networks beginning from Chapter 1 up to Section 1.3, we discuss collections of real-valued functions on the vertices or edges of the graph G. Such objects are most natural for the heuristics of the physical model and additionally allow for induced orientation/order and make certain probabilistic arguments possible. However, in the later portions of this book, we need to incorporate complex-valued functions into the discussion in order to make full use of spectral theory and

Introduction

lv

other methods. Thus, C-valued functions are considered in Chapter 2 in preparation for these later situations. Remark 3 (Symbols glossary). For the aid of the reader, we have included a list of symbols and abbreviations used in this document; it can be found on p. 384. Wherever possible, we have attempted to ensure that each symbol has only one meaning. In cases of overlap, the context should make things clear. In Appendix C, we also include some diagrams which we hope clarify the properties of the many operators and spaces we discuss and the relations between them.

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Chapter 1

Resistance Networks While electrical networks are only a different language for reversible Markov chains, the electrical point of view is useful because of the insight gained from the familiar physical laws of electrical networks. — Y. Peres The excitement that a gambler feels when making a bet is equal to the amount he might win times the probability of winning it. — B. Pascal

Resistance networks are the basic object of study throughout this volume. The basic idea is that a graph with weighted edges makes a good discrete model for diffusions, when the weights are interpreted as “sizes” or “capacities” for transfer in some sense. Such a model is useful for understanding the flow of heat in perforated media, diffusion of water in porous matter, and the transfer of data through the Internet. However, due to its intuitive appeal and historical precedent, we have chosen to stick predominantly with the metaphor of electricity flowing through a network of conductors. In this situation, the weights correspond to conductances (recall that conductance is the reciprocal of resistance), functions on the vertices may be interpreted as voltages, and corresponding functions on the edges of the graph may be interpreted as currents. This context also provides a natural interpretation for the energy E which will be central to our study: If v is a function on the vertices of the graph (i.e., a voltage), then E(v) is a number representing the potential energy of this configuration and, equivalently, the power dissipated by the electrical current induced by v.

1

2

Operator Theory and Analysis of Infinite Networks

1.1 The Electrical Resistance Network Model This section contains the basic definitions used throughout this book; we introduce the mathematical model of an electrical resistance network (RN) as a graph G whose edges are understood as conductors and whose vertices are the nodes at which these resistors are connected. The conductance data is specified by a function c so that c(x, y) is the conductance of the edge (resistor) between the vertices x and y. With the network data (G, c) fixed, we begin the study of functions defined on the vertices. We define many basic terms and concepts used throughout the book, including the Dirichlet energy form E and the Laplace operator Δ. Additionally, we prove a key identity relating E to Δ for finite graphs: Lemma 1.13. In Theorem 2.40, this will be extended to infinite graphs, in which case it is a discrete analogue of the familiar Gauss–Green identity from vector calculus. The appearance of a somewhat mysterious boundary term in the Theorem 2.40 prompts several questions, which are discussed in Remark 2.6. Answering these questions comprises a large part of the book; cf. Chapter 7. In fact, Theorem 2.40 provides much of the motivation for an energy-centric approach we pursue throughout our study; the reader may wish to look ahead to Remark 2.74 for a preview. Definition 1.1. A graph G = (G0 , G1 ) is a set of vertices G0 and a set of edges G1 ⊆ G0 × G0 . Two vertices are neighbors (or are adjacent ) iff there is an edge (x, y) ∈ G1 connecting them, and this is denoted by x ∼ y. This relation is symmetric, as (y, x) ∈ G1 whenever (x, y) ∈ G1 . The set of neighbors of x ∈ G0 is G(x) = {y ∈ G0 .. y ∼ x}. .

(1.1)

Definition 1.2. The conductance cxy is a symmetric function c : G0 × G0 → [0, ∞)

(1.2)

in the sense that cxy = cyx . It is our convention that x ∼ y if and only if cxy = 0, that is, there is an edge (x, y) ∈ G1 if and only if 0 < c(x, y) < ∞. The notation c = 1 will be used as a more convenient notation for c = χG1 . In particular, c = 1 does not mean that G is a complete graph. Conductance is the reciprocal of resistance, and this is the origin of the name “resistance network.” It is important to note that c−1 xy gives the resistance between adjacent vertices; this feature distinguishes c−1 xy from the

Resistance Networks

3

effective resistance R(x, y) discussed later, for which x and y need not be adjacent. Definition 1.3. The conductances define a measure or weighting on G0 by  cxy . (1.3) c(x) := y∼x

Whenever G is connected, it follows that c(x) > 0 for all x ∈ G0 . The notation c will also be used, on occasion, to indicate the multiplication operator (cv)(x) := c(x)v(x). Definition 1.4. A (finite) path γ from α ∈ G0 to ω ∈ G0 is a sequence of adjacent vertices (α = x0 , x1 , x2 , . . . , xn = ω), i.e., xi ∼ xi−1 for i = 1, . . . , n. The path is simple if any vertex appears at most once (so that a path is simply connected). An infinite path in G is a sequence {xi }∞ i=1 such that xi ∼ xi−1 for i = 1, 2, . . . . Definition 1.5. A graph G is connected iff for any pair of vertices α, ω ∈ G0 , there exists a finite path γ from α to ω. Remark 1.6. Note that for resistors connected in series, the resistances are simply added up, so this condition implies there is a path of finite resistance between any two points. We emphasize that all graphs and subgraphs considered in this volume are connected. At this point, the reader may wish to peruse some of the examples in Chapter 13. Definition 1.7. An electrical resistance network is a connected graph (G, c) whose conductance function satisfies c(x) < ∞ for every x ∈ G0 . We interpret the edges as being defined by the conductance: x ∼ y iff cxy > 0. Note that c need not be bounded in Definition 1.7. Also, we typically assume an RN to be simple in the sense that there are no self-loops, and there is at most one edge from x to y. This is mostly for convenience: Basic electrical theory says that two conductors c1xy and c2xy connected in parallel can be replaced by a single conductor with conductance cxy = c1xy + c2xy . Also, electric current will never flow along a conductor connecting a node to itself. Nonetheless, such self-loops may be useful for technical

4

Operator Theory and Analysis of Infinite Networks

considerations: One can remove the periodicity of a random walk by allowing self-loops. This can allow one to obtain a “lazy walk” which is ergodic and hence amenable to application of tools such as the Perron– Frobenius theorem. See, for example, Refs. [LPW08, LP16, AF09]. We are interested in certain operators that act on functions defined on electrical resistance networks. Definition 1.8. The Laplacian on G is a linear difference operator that acts on a function v : G0 → R by  (Δv)(x) := cxy (v(x) − v(y)). (1.4) y∼x

A function v : G0 → R is called harmonic iff Δv ≡ 0. Definition 1.9. The transfer operator on G is the linear operator T, which acts on a function v : G0 → R by  (T v)(x) := cxy v(y). (1.5) y∼x

Hence, the Laplacian may be written as Δ = c − T, where (cv)(x) := c(x)v(x). We won’t worry about the domain of Δ or T until Chapter 5. For now, consider both of these operators as defined on any function v : G0 → R. Readers familiar with the literature will note that the definitions of the Laplacian and transfer operator given here are normalized differently from those found elsewhere in the literature. For example, Ref. [DS84] and other probabilistic references use 1  cxy (v(x) − v(y)), (1.6) Δc := c−1 Δ = 1 − P, so (Δc v)(x) := c(x) y∼x where P := c−1 T is the probabilistic transition operator corresponding to the transition probabilities p(x, y) = cxy /c(x). For another example, Ref. [Chu96] and other spectral-theoretic references use Δs := c−1/2 Δc−1/2 = 1 − c−1/2 T c−1/2 , so (Δs v)(x) := v(x) −

 cxy v(y)  . c(y) y∼x

(1.7)

However, these renormalized versions are much more awkward to work with in the current context, especially when dealing with the inner product

Resistance Networks

5

and kernels of the Hilbert spaces we shall study. Not only are (1.4) and (1.5) better suited to the electrical resistance network framework (as will be evinced by the operator theory developed in Chapter 2 and succeeding sections), but both Δc and Δs are bounded operators and hence do not allow for the delicate spectral analysis carried out in Chapters 5–7. 1.2 The Energy Form E In this section, we study the relationship between the energy E and Laplacian Δ on finite networks, as expressed in Lemma 1.13. This formula will be used prolifically, as it also holds for infinite networks in many circumstances. In fact, a noticeable portion of Chapter 2 is devoted to determining when this is so. Definition 1.10. The graph energy of an electrical resistance network is the quadratic form defined for functions u : G0 → R by 1  E(u) := cxy (u(x) − u(y))2 . (1.8) 2 0 x,y∈G

There is also the associated bilinear energy form 1  cxy (u(x) − u(y))(v(x) − v(y)) E(u, v) := 2 0

(1.9)

x,y∈G

so that E(u, u) = E(u) (see Proposition 1.12 as follows). For both (1.8) and (1.9), note that cxy = 0 for vertices which are not neighbors and hence only pairs for which x ∼ y contribute to the sum; the normalizing factor of 12 corresponds to the idea that each edge should only be counted once. The domain of the energy is dom E = {u : G0 → R .. E(u) < ∞}. .

(1.10)

The close relationship between the energy and the conductances is highlighted by the simple identities  −cxy , x = y, (1.11) E(δx , δy ) = c(x), x = y, where δx is a (unit) Dirac mass at x ∈ G0 . Proving these identities is easy and is left as an exercise. A significant upshot of (1.11) is that the Dirac masses are not orthogonal with respect to energy.

6

Operator Theory and Analysis of Infinite Networks

Remark 1.11. It is immediate from (1.8) that E(u) = 0 if and only if u is a constant function. The energy form is positive semidefinite, but if we use modulo constant functions, it becomes positive definite and hence an inner product. We formalize this in Definition 2.1 and again in Section 4.1. In classical potential theory (or Sobolev theory), this would amount to working with a class of functions satisfying f  2 < ∞ but abandoning the requirement that f 2 < ∞. As a result of this, the nontrivial harmonic functions play an especially important role in this book. In particular, it is precisely the presence of nontrivial harmonic functions that prevents the functions of finite support from being dense in the space of functions of finite energy; see Section 2.2. ¯ Traditionally (e.g., Refs. [Kat95, FOT94]), the study of quadratic forms 2 would combine E(u, v) and u, v . In our context, this is counterproductive and would eclipse some of our most interesting results. Some of our most intriguing questions about elements v ∈ HE involve boundary considerations, and in these cases, v is not in 2 (G0 ) (Corollary 2.73). One example of this arises in the discrete Gauss–Green formula (Theorem 2.40); another arises in the study of forward-harmonic functions in Section 11.2. The reader can readily check the results of the following proposition, which may be found in Ref. [Str06, Section 1.3] and Ref. [Kig01, Ch. 2], for example. Proposition 1.12. The following properties hold : (1) (2) (3) (4)

E(u, u) = E(u). (Polarization) E(u, v) = 14 [E(u + v) − E(u − v)]. (Markov property) E([u]) ≤ E(u), where [u] is any contraction of u. E is closable (see Section 9.2).

For example, let [u] := min{1, max{0, u}}. The following result relates the Laplacian to the graph energy on finite networks and can be interpreted as a relation between dom E and 2 (G0 ). Lemma 1.13. Let G be a finite electrical resistance network. For u, v ∈ dom E, E(u, v) =

 x∈G0

u(x)Δv(x) =

 x∈G0

v(x)Δu(x).

(1.12)

Resistance Networks

7

Proof. Direct computation yields E(u, v) =

  1  cxy u(x)v(x) − u(x)v(y) − u(y)v(x) + u(y)v(y) 2 0 x,y∈G

=

1  1  c(x)u(x)v(x) + c(y)u(y)v(y) 2 2 0 0 x∈G

y∈G

1  1  − u(x) T v(x) − u(y) T v(y) 2 2 0 0 =



x∈Gn

c(x)u(x)v(x) −

x∈G0

=





y∈G

u(x) T v(x)

x,y∈G0

u(x) (c(x)v(x) − T v(x))

x∈G0

=



u(x)Δv(x).

(1.13)

x∈G0

Of course, the computation is identical for

 x∈G0

v(x)Δu(x).



We include the following well-known result for completeness. Corollary 1.14. On a finite electrical resistance network, all harmonic functions of finite energy are constant.  Proof. If h is harmonic, then E(h) = x∈G0 h(x)Δh(x) = 0. See Remark 1.11.  Connectedness is implicit in the calculations in both Lemma 1.13 and Corollary 1.14; recall that all electrical resistance networks considered in this work are connected. We extend Lemma 1.13 to infinite graphs in Theorem 2.40, where the formula is more complicated:  u(x)Δv(x) + {“boundary term”}. E(u, v) = x∈G0

It is shown in Theorem 2.49 that the presence of the boundary term corresponds to the transience of the random walk on the underlying network. In fact, one can interpret Corollary 1.14 as the reason why the boundary term alluded to above vanishes on finite networks. We study the interplay between E and Δ further in Sections 6.5 and 6.6.

8

Operator Theory and Analysis of Infinite Networks

1.3 Currents and Potentials on Resistance Networks The potential theory for an electrical resistance network is studied via an experiment in which 1 A of current is passed through the network, inserted into one vertex, and extracted at some other vertex. The voltage drop measured between the two nodes is the effective resistance between them, see Chapter 3. When the voltages are fixed at certain vertices, it induces a current in the network in accordance with the laws of Kirchhoff and Ohm. This induced current is introduced formally in Definition 1.31. Induced currents are important for studying flows of minimal dissipation and will also be useful in the study of forward-harmonic functions in Section 11.2. If a voltage drop of 1 V is imposed between two vertices, the effective resistance between these two vertices is the reciprocal of the dissipation of the induced current. In Theorem 1.41, we show that there always exists an harmonic function satisfying the boundary conditions implied by the above described experiment in order to fill a gap in Ref. [Pow76b]. In Theorems 1.40 and 3.2, it is shown that these harmonic functions correspond to currents that minimize energy dissipation. Definition 1.15. A current is a skew-symmetric function I : G0 × G0 → R. Definition 1.16. An orientation is a subset of the edges which includes exactly one of each pair {(x, y), (y, x)}. For a given current I, one may pick an orientation by requiring that I(x, y) > 0 on every edge for which I is nonzero and arbitrarily choosing (x, y) or (y, x) outside the support of I. We refer to this as an orientation induced by the current ; this will be used in Section 11.2 to study the forward-harmonic functions. The energy is a functional defined on functions v : G0 → R, which gives the voltages between different vertices in the network. The associated notion defined on the edges of the network is the dissipation of a current. Definition 1.17. The dissipation of a current may be thought of as the energy lost as the current flows through an electrical resistance network. More precisely, for I, I1 , I2 : G1 → R, D(I) :=

1 2

 (x,y)∈G1

2 c−1 xy I(x, y) .

(1.14)

Resistance Networks

The associated bilinear form is the dissipation form: 1  c−1 D(I1 , I2 ) := xy I1 (x, y)I2 (x, y). 2 1

9

(1.15)

(x,y)∈G

Again, the normalizing factor of 12 corresponds to the idea that each edge only contributes once to the sum. The domain of the dissipation is .

dom D := {I .. D(I) < ∞}.

(1.16)

Remark 1.18. It is easy to see that dom D is a Hilbert space under the inner product (1.15). Indeed, dom D = 2 (O, c). This is studied at length in Chapter 10. Definition 1.19. A cycle ϑ is a set of n edges corresponding to a sequence of vertices (x0 , x1 , x2 , . . . , xn = x0 ) ⊆ G0 for which (xk , xk+1 ) ⊆ G1 for each k. Denote the set of cycles in G by L. Definition 1.20. For physical realism, we often require that a current flow satisfy Kirchhoff ’s node law, i.e., the total current flowing into a vertex must equal the total current flowing out of a vertex:  I(x, y) = 0, ∀x ∈ G0 . (1.17) y∼x

This is indeed the version of Kirchhoff’s law you would find in a physics textbook, with our convention I(x, y) > 0 indicating that the current flows from x to y. However, if we are performing the experiment described above, then there are boundary conditions at α, ω to take into account, and Kirchhoff’s node law takes the nonhomogeneous form ⎧ ⎪ x = α, ⎪ ⎨1,  (1.18) I(x, y) = δα − δω = −1, x = ω, ⎪ ⎪ y∼x ⎩0, else, where δx is the usual Dirac mass at x ∈ G0 . Definition 1.21. A current flow from α to ω is a current I ∈ dom D that satisfies (1.18). The set of all current flows is denoted by F (α, ω). We usually use α to denote the beginning of a flow and ω to denote its end. Shortly, we will see that the currents corresponding to potentials are precisely the current flows.

10

Operator Theory and Analysis of Infinite Networks

Remark 1.22. Although trivial, it is important to note that the characteristic function of a current path χγ : G1 → {0, 1} trivially satisfies (1.18). Also, the characteristic function of a cycle satisfies (1.17) in much the same way. As a consequence, if I ∈ F (α, ω), then I + tχϑ ∈ F (α, ω) for any t ∈ R by a brief computation. In other words, perturbation on a cycle preserves the Kirchhoff condition. However, the dissipation will vary because D(χϑ ) > 0. 1.4 Potential Functions and Their Relationship to Current Flows From the proceeding section, it is clear that a special role is played by functions v : G0 → R, which satisfy the equation Δv = δα − δω . Such a function is the solution to a discrete Dirichlet problem where the “boundary” has been chosen to be α and ω (not to be confused with the boundary term discussed in Remark 2.6). Definition 1.23. A dipole is a function v ∈ dom E that satisfies Δv = δα − δω

(1.19)

for some vertices α, ω ∈ G0 . The collection of all such functions is denoted by P(α, ω). Note that when G is finite, P(α, ω) contains only a single element. This follows from Corollary 1.14 because the difference of any two solutions to (1.19) is harmonic. Remark 1.24. The definition of a monopole that we give here is a heuristic definition; we give the precise definition in Definition 2.30. A monopole at ω is a function w : G0 → R that satisfies Δw = kδω ,

w ∈ dom E, k ∈ C.

(1.20)

Hereinafter, we are primarily concerned with monopoles wo , where o = ω is some fixed vertex which acts as a point of reference or “origin.” Also, we typically take k = −1, as the induced current of such a monopole is a unit flow to infinity in the language of Ref. [DS84]. Remark 1.25. The study of dipoles, monopoles, and harmonic functions is a recurring theme of this book: Δv = δα − δω ,

Δw = −δω ,

Δh = 0.

Resistance Networks

11

In Theorem 1.41, we show that P(α, ω) is nonempty for any α and ω on any network (G, c); the existence of monopoles and nontrivial harmonic functions is a much more subtle issue. In Corollary 2.19, we offer a more refined proof of the existence of dipoles, using Hilbert space techniques. Perhaps a more interesting question is when P(α, ω) contains more than element; the linearity of Δ shows immediately that any two dipoles in P(α, ω) differ by a harmonic function of finite energy.1 We have shown that when a connected graph is finite, the only harmonic functions are constant (Corollary 1.14); therefore, P(α, ω) consists only of a single function up to the addition of a constant. The situation for monopoles is similar, as the difference of two monopoles at ω is also a harmonic function. Not all electrical resistance networks support monopoles; the current induced by a monopole is a finite flow to infinity and hence indicates that the random walk on the network is transient, as shown by Ref. [Lyo83]. See also Refs. [DS84, LP16] for terminology and proofs. It is well known that for a reversible Markov chain, if a random walk starting at one vertex is transient, then it is transient when starting at any vertex. We give a very brief proof of this in Lemma 1.42 and a new criterion for transience in Lemma 2.53. On some networks, a monopole can be understood as the limit of a sequence of dipoles vxn , where Δvxn = δxn − δo and xn → ∞. In such a situation, a monopole can be considered as a dipole where one of the Dirac masses “sits at ∞.” However, this is not possible on all networks, as is illustrated by the binary tree in Example 15.4. Again, the linearity of Δ shows immediately that any two monopoles at ω differ by a harmonic function. When these monopoles correspond to a “distribution of dipoles at  infinity” (i.e., a limit of sums ax vx , where the vx s are dipoles with x → ∞ in the limit), the addition of a harmonic function transforms the distribution at infinity. It will take some work to make these ideas precise; for now, the reader can consider this remark simply as a preview of the coming attractions. The presence of monopoles is also extremely closely related to the existence of “long-range order,” and the theoretical foundation of magnetism in R3 ; see Section 16.3. 1 The emphasis is added here because the vast majority of work done in this area concerns positive harmonic functions, especially as pertains to connections with probability and potential theory. However, just as positivity is an appropriate condition for probability spaces, orthogonality is more germane to Hilbert spaces, and so, our primary restriction is the finiteness of energy.

12

Operator Theory and Analysis of Infinite Networks

Furthermore, it is possible for an electrical resistance network to support monopoles but not nontrivial harmonic functions. In Chapter 14, we show that the integer lattice networks (Zd , 1) support monopoles (Theorem 14.5). However, all harmonic functions are linear and hence do not have finite energy; cf. Theorem 14.17. Both of these results are well known in the literature in different contexts and/or with different terminology. Lemma 1.26. The dipoles P(α, ω) and the current flows F (α, ω) are convex sets. Furthermore, if v ∈ P(α, ω), then v + h ∈ P(α, ω) for any harmonic function h; similarly, if I ∈ F (α, ω), then I + J ∈ F (α, ω) for any function J satisfying (1.17).  ci = 1, then the linearity of Δ gives Proof. If vi ∈ P(α, ω), ci ≥ 0 and 

  Δ ci vi = ci (δα − δω ) = δα − δω . ci Δvi = The computation for the other parts is similar.



Theorem 1.27. E attains its minimum for some unique v ∈ P(α, ω), and D attains its minimum for some unique I ∈ F (α, ω). Proof. Each of these is a quadratic form on a convex set by Lemma 1.26, so the result is an immediate application of Ref. [Rud87, Thm. 4.10] or Ref. [Nel69]. For example, to underscore the uniqueness, suppose that . E(v1 ) = E(v2 ). Then, with ε := inf{E(v) .. v ∈ P(α, ω)}, the parallelogram law gives E(v1 − v2 ) = 2E(v1 ) + 2E(v2 ) − 4ε2 = 0 since E(vi ) = ε because vi were chosen to be minimal.



Definition 1.28. Ohm’s law (V = RI) appears in the current context as v(x) − v(y) =

1 cxy

I(x, y).

(1.21)

Remark 1.29. It will shortly become evident (if it isn’t already) that the current flows satisfying Kirchhoff’s law correspond to harmonic functions via Ohm’s law, and that the current flows satisfying the nonhomogeneous Kirchhoff’s law (1.18) correspond to dipoles, that is, the solutions to the Dirichlet problem (1.19) with Neumann boundary condition. To make this precise, we need the notion of induced current given in Definition 1.31 and justified by Lemma 1.30.

Resistance Networks

13

Lemma 1.30. Every function v : G0 → R induces a unique current via I(x, y) := cxy (v(x) − v(y)), and the dissipation of this current is the energy of v: D(I) = E(v).

(1.22)

Moreover, if v ∈ P(α, ω), then I ∈ F(α, ω). Proof. It is clear that Ohm’s law defines a current. The equality (1.22) requires a very brief calculation and follows straight from the definitions; see (1.9) and (1.14). A proof of (1.22) is also given in Ref. [DS84]. If v ∈ P(α, ω), then Δv = δα − δω , and   cxy (v(x) − v(y)) = I(x, y) (1.23) (δα − δω )(x) = (Δv)(x) = y∼x

y∼x

verifies the nonhomogeneous form of Kirchhoff’s law.



Definition 1.31. Given v ∈ P(α, ω), the induced current is defined via Ohm’s law as in the statement of Lemma 1.30. That is, I(x, y) := cxy (v(x) − v(y)).

(1.24)

Remark 1.32. Note that (1.22) holds when the current I is induced by v. It makes no sense to attempt to apply the same equality to a general current: There may be NO associated potential because of the compatibility problem described in the following. Nonetheless, Theorem 10.27 provides a way to give the identity analogous to (1.22) for general currents by using the adjoint of the operator implicit in (1.24). Remark 1.33. If Δv = δα − δω has a solution v0 , then any other solution is of the form v = v0 + h, where h is harmonic by linearity of Δ. So, to minimize energy, one must consider such perturbations: d [E(v0 + th)]t=0 = 0 dt Conversely, if E(v, h) = 0, then

⇐⇒

E(v0 , h) = 0.

E(v + th) = E(v) + 2tE(v, h) + t2 E(h) ≥ E(v), which shows that the energy is minimized at t = 0. In particular, energy is minimized for v, which contains no harmonic component. In Lemma 2.20, this important principle is restated in the language of Hilbert spaces: Energy is minimized for the v which is orthogonal to the space of harmonic functions with respect to E.

14

Operator Theory and Analysis of Infinite Networks

Analogous remarks hold for I which minimizes D(I). However, note that Kirchhoff’s law is blind to conductances, so I ∈ F (α, ω) does not imply that D(I) is minimal. In the next section, we show that induced currents are minimal with respect to D when they are induced by a minimal potential v. 1.5 The Compatibility Problem The converse to Lemma 1.30 is not always true, but a partial converse is given by Theorem 1.40. Given an electrical resistance network (G, c), one can always attempt to construct a Ohm’s function by fixing the value v(x0 ) at some point x0 ∈ G0 and then applying Ohm’s law to determine the value of v for other vertices x ∼ x0 . However, this attempt can fail if the network contains a cycle (see Example 13.2 for an example) because the existence of a cycle is equivalent to the existence of two distinct paths from one point to another. This phenomenon is worked out in detail for a simple case in Example 1.34. In general, it may happen that there are two different paths from x0 to y0 , and the net voltage drop v(x0 ) − v(y0 ) computed along these two paths is not equal. Such a phenomenon makes it impossible to define v. Note that Kirchhoff’s law does not forbid this because (1.17) and (1.18) are expressed without reference to the conductances c. We refer to this as the compatibility problem: A general current function may not correspond to a potential, even though every potential induces a welldefined current flow (see Lemma 1.30). In this section, we provide the following answer: For any current, there exists a unique associated current which does correspond to a potential. Example 1.34 (The Dirac mass on an edge). Consider a Dirac mass on an edge of the network (Z2 , 1), as depicted in Figure 1.1. We use such a current here to illustrate the compatibility problem. To find a potential corresponding to this current, consider the following dilemma: I(x, y) = 1

x y

Fig. 1.1.

A Dirac mass on an edge of Z2 .

Resistance Networks

x y

Fig. 1.2.

15

w z

A failed attempt at constructing a potential to match Figure 1.1.

and I ≡ 0 elsewhere correspond to a potential (up to a constant) which has v(x) = 1 and v(y) = 0, as in Figure 1.2. Since I(x, w) = 0, we have v(w) = v(x), and since I(y, z) = 0, we have v(z) = v(y). But then, v(z) = < 1 = 0 = v(z), contradicting the fact that I(w, z) = 0!  Definition 1.35. A current I satisfies the cycle condition iff D(I, χϑ ) = 0 for every cycle ϑ ∈ L. (We call χϑ a cycle.) Remark 1.36. From the preceding discussion, it is clear that for a current satisfying the cycle condition, the voltage drop between vertices x and y may be measured by summing up the currents along any single path from x to y, and the result will be independent of the chosen path. In the Hilbert space interpretation presented in Chapter 10, the cycle condition is restated as “I is orthogonal to cycles.” The next two results must be folklore (perhaps, dating back to the 19th century?), but we include them for their relevance in Chapter 10, especially the Hilbert space decomposition in Theorem 10.8 (see also Figure 10.1). While writing the second draft of this book, the authors discovered a similar treatment in Ref. [LP16, Chapter 9]. Lemma 1.37. I is an induced current if and only if I satisfies the cycle condition. Proof. (⇒) If I is induced by v, then for any ϑ ∈ L, the sum  (x,y)∈ϑ

1 cxy

I(x, y) =



(v(x) − v(y)) = 0

(1.25)

(x,y)∈ϑ

since every term, v(xi ), appears twice, once positive and once negative, whence D(I, χϑ ) = 0. (⇐) Conversely, to prove that there is such a v, we must show that v(x0 ) − v(y0 ) is independent of the path from x0 to y0 used to compute it. In a direct analogy to basic vector calculus, this is equivalent to the fact

Operator Theory and Analysis of Infinite Networks

16

that the net voltage drop around any closed cycle is 0: 

(v(x) − v(y)) =

(x,y)∈ϑ

 (x,y)∈ϑ

1 cxy

I(x, y) = D(I, χϑ ) = 0.

Now, define v by fixing v(x0 ) for some point x0 ∈ G0 and then coherently 1 use v(x) − v(y) = cxy I(x, y) to compute v at any other point.  The presence of cycles is not always obvious! As an exercise, we invite the reader to determine the cycles involved in Example 1.34. The following lemma is named in honor of Zemanian [Zem97]. Lemma 1.38 (Resurrection of Kirchhoff ’s law). Let I be a current induced by v. Then, v ∈ P(α, ω) if and only if I satisfies the nonhomogeneous Kirchhoff ’s law. Proof. (⇒) Computing directly,   I(x, y) = cxy (v(x) − v(y)) = Δv(x) = δα − δω . y∼x

(1.26)

y∼x

(⇐) Conversely, to show Δv = δα − δω ,  Δv(x) = cxy (v(x) − v(y))

def of Δ

y∼x

=



I(x, y)

(1.21)

y∼x

= δα − δω ,

I ∈ F (α, ω).



Corollary 1.39. Let I be a current induced by v. Then, v is harmonic if and only if I satisfies the homogeneous Kirchhoff ’s law. Proof. Mutatis mutandis, this is the same as the proof of Lemma 1.38.  Theorem 1.40. I minimizes D on F (α, ω) if and only if I is induced by the potential v that minimizes E on P(α, ω). Proof. (⇒) Since I minimizes D, we have d [D(I + tJ)]t=0 = 0 dt

(1.27)

Resistance Networks

17

for any current J satisfying the homogeneous Kirchhoff’s law. From Remark 1.22, this applies in particular to J = χϑ , where ϑ is any cycle in L. d [D(I + tχϑ )]t=0 = 0. To see this, replace I Note that D(I, χϑ ) = 0 iff dt χ by I + t ϑ in (1.14), differentiate D(I + tχϑ ) term by term with respect to t, and evaluate at t = 0 to obtain that (1.27) is equivalent to   1 1 I(x, y)χϑ (x, y) = I(x, y) = 0, ∀ϑ ∈ L. c cxy 1 xy (x,y)∈G

(x,y)∈ϑ

By Lemma 1.37, this shows that I is induced by some v; and by Lemma 1.38, we know that v ∈ P(α, ω). From Lemma 1.30, it is clear that v must also be the energy-minimizing element of P(α, ω). (⇐) Since I is induced by v ∈ P(α, ω), the only thing we need to check is that I is minimal with respect to any harmonic current (i.e., a current induced by a harmonic function); this follows from Lemma 1.37 and the first part of the proof. If h is any harmonic function on G0 , denote the induced current by H as before. Then, Lemma 1.30 gives d d [D(I + tH)]t=0 = [E(v + th)]t=0 = 0 dt dt by the minimality of v.



Theorem 1.41. P(α, ω) is never empty. Proof. It is clear that F (α, ω) = ∅ because one always has the characteristic function of a current path from α to ω (since we are assuming the underlying graph is connected); see Definition 11.12 and Remark 1.22. From Theorem 1.27, one sees that there is always a flow which minimizes dissipation. By Theorem 1.40, this minimal flow is induced by an element of P(α, ω).  The following result is well known in probability (see, for example, Ref. [Str05]), but we include it here for completeness and the novel method of proof. Corollary 1.42. If the random walk on (G, c) is transient when starting from y ∈ G0 , then it is transient when starting from any x ∈ G0 . Proof. According to Ref. [Lyo83], the hypothesis means there is a monopole wy ∈ dom E with Δw = δy . But then, by Theorem 1.41 and  the linearity of Δ, v + wx is a monopole at x for any v ∈ P(x, y).

18

Operator Theory and Analysis of Infinite Networks

Remark 1.43. There are examples for which the elements of P(α, ω) do not lie in 2 (G0 ); see Figure 14.5 of Example 14.2. Proposition 1.44. If G is finite and v ∈ P(α, ω), then v(ω) ≤ v(x) ≤ v(α) for all x ∈ G0 . Proof. This is immediate from the maximum principle for harmonic functions on the finite set G0 with boundary {α, ω}. See, for example, Ref. [LP16, Section 2.1] or Ref. [LPW08].  Remark 1.45. In Chapter 3, we will see that Proposition 1.44 extends to a more general result: If v is a unique element of P(α, ω) of minimal energy, then the same conclusion follows. One way to see this is to define u(x) = Px [τα < τω ] (i.e., the probability that the random walk started at x reaches α before ω). By Theorem 3.18, v is defined by v(x) = u(x)RW (α, ω), where RW (α, ω) is the (wired) effective resistance between α and ω. 1.5.1 Current paths It is intuitively obvious that for a connected graph, current should be able to flow between any two points. Indeed, it is a basic result in graph theory that for any connected graph, one can find a path of minimal length between any two points, and from Remark 1.6, we know that the resistance along such a path is finite. In Theorem 1.48, we show a stronger result: that one can always find a path along which the potential function decreases monotonically. In other words, there is always at least one “downstream path” between the two vertices. This fact is easiest to demonstrate by an appeal to a basic fact about probability (Lemma 1.47). Our definition of an electrical resistance network is mathematical (Definition 1.7) but is motivated by engineering; modification of the conductors (c) will alter the associated probabilities and thus change which current flows are induced in the sense of Definition 1.31. We are interested in quantifying this dependence. Obviously, on an infinite graph, the computation of current paths involves all of G, and it is not feasible to attempt to compute these paths directly. Consequently, we feel our proof of Theorem 1.48 may be of independent interest. Definition 1.46. Let v : G0 → R be given, and suppose we fix α and ω for which v(α) > v(ω). Then, a current path γ (or simply, a path) is an edge path from α to ω with the extra stipulation that v(xk ) < v(xk−1 ) for each k = 1, 2, . . . , n. Denote the set of all current paths by Γ = Γα,ω (dependence

Resistance Networks

19

on the initial and terminal vertices is suppressed when context precludes confusion). Also, define Γα,ω (x, y) to be the subset of current paths from α to ω which pass through the edge (x, y) ∈ G1 . The following lemma is immediate from elementary probability theory, as it represents the probability of a union of disjoint events, but it will be helpful. Lemma 1.47. Suppose (G, c) is an electrical resistance network and v:G0 → R satisfies Δv = δα − δω . Then, if I is the current associated with v by I(x, y) = cxy (v(x) − v(y), then I satisfies  P(γ). (1.28) I(x, y) = γ∈Γα,ω (x,y)

The method of proof in the next proposition is a bit unusual in that it uses a probability to demonstrate existence. This result fills a hole in the proof of distΔ (x, y) = distD (x, y) in Ref. [Pow76b] (recall (3.1) and (3.3)). Theorem 1.48. If v ∈ P(α, ω), then Γα,ω = ∅. Moreover, v(α) > v(ω). Proof. Theorem 1.41 ensures that we can find v ∈ P(α, ω); let I be the current flow associated with v. Then, Δv(α) = 1 implies that there is some y ∼ α for which I(α, y) > 0. By Lemma 1.47,  P(γ) > 0, I(α, y) = γ∈Γα,ω (α,y)

which implies that there must exist a positive term in the sum and hence a γ ∈ Γα,ω . Since we may now choose a path γ ∈ Γα,ω , the second claim follows.  1.6 Remarks and References Of the cited references in this chapter, some are more specialized. However, for prerequisite material (if needed), the reader may find the book by Doyle and Snell [DS84] especially relevant. It has several editions and is available for free on arXiv (math/0001057). While it is a gold mine of ideas and illuminating examples, it remains accessible to undergraduates. We have been much inspired by Doyle and Snell’s book on electrical networks [DS84] and by Kigami’s work on effective resistance and discrete potential theory (especially as it pertains to renormalization and

20

Operator Theory and Analysis of Infinite Networks

scaling limits) [Kig01, Kig03, Kig08]. We are similarly indebted to Woess’ book [Woe00], covering probability and analysis on infinite networks, Markov chains, and especially the theory of boundaries, as developed in Refs. [Woe86, Woe87, Woe89, Woe95, Woe97, Woe96, Tho90] and elsewhere. The reader will find Refs. [AF09, LP16, LPW08] to be excellent references for the random walks on graphs and Markov chains in general (with an emphasis on reversible chains). The main themes in this and later chapters are also tangentially related to the fascinating work by Chung on the spectral theory of transfer operators on infinite graphs [Chu07, CR06]. Much of this chapter is just based on high-school physics, but a couple of key references for us were the papers by Powers [Pow75, Pow76a, Pow76b, Pow78, Pow79] and an early (and often overlooked) paper by Bott and Duffin [BD49]. We especially recommend the discussion of networks and resistance distance in Powers’ paper [Pow76b]. While the main focus there is a problem for quantum spin systems on lattices, Powers develops the elementary properties of effective resistance from scratch in this paper and adapts them to the Heisenberg model. Since our first chapter serves in part as an overview of the material in the book and some results in the literature but not in our book, there are quite a number of papers and books that are appropriate to cite, so here is a partial list: Refs. [Woe00, CW04, Woe03, KW02, Kig03, Kig01, Ter78, PW76, Sve56, Cra52, Sch91]. A partial list of further pioneering work: • Albeverio and Collaborators: Dirichlet spaces, potential theory, and mathematical physics [AHeKS77, AR89, AKR93, AKR12, ABK06, AOS07, AC07]. • Aldous and Fill: reversible Markov chains and random walks on graphs [AF09]. • Benjamini, Lyons, Pemantle, Peres, and Schramm (separately or in various combinations): effective resistance, probability on trees, percolation, analysis, and probability on infinite graphs [LP16, LPW08, Per99, BLPS01, BLPS99, BLS99, LPS03, ALP99, LPS06, LP03, LPP96, NP08b, Lyo03]. • Cartwright: random walks, Dirichlet functions, and spectrum [CSW93, CW92, CW04, CW07]. • Chung: spectral theory of transfer operators on infinite graphs [Chu07, CR06]. • Doob: martingales and probabilistic boundaries [Doo53, Doo55, Doo58, Doo59].

Resistance Networks

21

• Doyle and Snell: electrical networks [DS84], and Doyle: Ref. [Doy88]. • Hida: use of Hilbert space methods in stochastic integration [Hid80]. • Kigami: effective resistance and discrete potential theory [Kig01, Kig03, Kig08]. • Kolmogorov: foundations of probability theory [Kol56]. • Liggett: infinite spin models [Lig93, Lig95, Lig99]. • von Neumann: the theory of unbounded operators, quantum mechanics, and metric geometry [vN32a, vN32b, vN32c]. • Powers: use of resistance distance in the estimation of long-range order in quantum statistical models [Pow75, Pow76a, Pow76b, Pow78, Pow79]. • Saloff-Coste: harmonic analysis and probability and random walks in relation to groups [SCW06, SCW09]. • Schroeder: harmonic analysis and signal processing on fractals [Sch91]. • Soardi: harmonic analysis and potential theory on infinite graphs [Soa94], a substantial influence. • Schrader and collaborators: quantum graphs, metric graphs, and Brownian motion on graphs and on fractals [Sch01, Sch09, KS06, KPS12b, KPS07, KPS12a]. • Spitzer: random walk [Spi76]. • Stroock: Markov processes [Str05]. • Telcs, random walks, graphs, and fractals [Tel06a, Tel06b, Tel03, Tel01]. • Yin and Zhang: use of stochastic integration in renormalization theory [YZ05]. While we present a number of theorems related in one way or the other to earlier results, the material is developed here from simple axioms and from a unifying point of view: We make use of fundamental principles in the theory of operators in Hilbert space. Using this, we develop a variety of results on networks, scaling relations, renormalization, twospin models, long-range order, and discrete potential theory. Our aim and emphasis is to develop the material from first principles: Riesz duality, reproducing kernels, metric embedding into Hilbert space, and stochastic integral models. As a bonus, we are able to point out how basic principles from operator theory lead to the unification of a variety of existing results and, in some cases, their extensions. See Ref. [JP19] for a more abstract approach. Remark 1.49. Part of the motivation for Theorem 1.40 is to fix an error in Ref. [Pow76b]. The author was not apparently aware of the possibility of nontrivial harmonic functions and hence did not see the need for taking the

22

Operator Theory and Analysis of Infinite Networks

element of P(α, ω) with minimal energy. This becomes especially important in Theorem 3.2. Theorem 1.40 is generalized in Theorem 10.27, where we exploit certain operators to obtain, for any given current I, an associated minimal current. This minimal current is induced by a potential, even if the original is not, and provides a resolution to the compatibility problem described at the beginning of Section 1.5. In Section 10.4.1, we revisit this scenario and show how the minimal current may be obtained by the simple application of a certain operator once it has been properly interpreted in terms of Hilbert space theory. See Theorem 10.27 and its corollaries in particular. Remark 1.50. Theorem 1.41 fills a gap in Ref. [Pow76b]. A key point is that a finite dissipation of the flow ensures the finite energy of the inducing voltage function by Lemma 1.30. A different proof of Theorem 1.41 is obtained in Corollary 2.19 by the application of Hilbert space techniques. Theorem 1.41 also follows from the results of Ref. [Soa94, Section III.4] since the difference of two Dirac masses corresponds to a “balanced” flow, i.e., the same amount of current flows in as flows out. Remark 1.51. In Corollary 1.42, we refer to Ref. [Lyo83] for the equivalence of transience with the existence of a finite flow to infinity. This is the most common reference for this result, but we should point out that a slightly different version of it (restated in terms of the Green function) appears earlier in the often overlooked paper, Ref. [Yam79]. (The results presented in this volume were discovered independently).

Chapter 2

The Energy Hilbert Space I would like to make a confession which may seem immoral: I do not believe in Hilbert space anymore. — J. von Neumann I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce. — F. Dyson

A number of tools used in the theory of weighted graphs, especially for the infinite case, were first envisioned in the context of resistance networks. We introduce them here, but it is helpful to keep in mind that they apply to a variety of other problems outside the original context of resistance networks. In particular, these concepts and tools lead to the introduction of metrics. These in turn have applications to neighboring fields; see, for example, Chapters 3, 14, and 16. For the analysis of resistance networks, the (Dirichlet) energy form E is a natural tool, and so, it is helpful study the Hilbert space HE of functions on the network where the inner product is given by E. While one’s first instinct may be to select 2 (X) as the preferred Hilbert space, we show in Chapter 4 that the energy space HE is in some ways a natural choice. The relationship between the two Hilbert spaces 2 (X) and HE is subtle and is explored further in Chapters 5, 6, and elsewhere. For example, the Laplacian operator Δ has quite different properties depending on the choice of Hilbert space.

23

24

Operator Theory and Analysis of Infinite Networks

In this section, we study the Hilbert space HE of (finite-energy) voltage functions, that is, equivalence classes of functions u : G0 → C, where u  v iff u − v is constant. On this space, the energy form is an inner product, and there is a natural reproducing kernel {vx }x∈G0 indexed by the vertices; see Theorem 2.9. Since we work with respect to the equivalence relation defined just above, most formulas are given with respect to differences of function values; in particular, the reproducing kernel is given in terms of differences with respect to some chosen “origin.” Therefore, for any given resistance network, we fix a reference vertex o ∈ G0 to act as an origin. It will readily be seen that all results are independent of this choice, and this affords the convenience of working with a single-parameter reproducing kernel. When working with representatives, we typically abuse notation and use u to denote the equivalence class of u. One natural choice is to take u so that u(o) = 0; a different but no less useful choice is to pick k so that v = 0 outside a finite set, as discussed further in Definition 2.14. In Theorem 2.40, we establish a discrete version of the Gauss–Green formula which extends Lemma 1.13 to the case of infinite graphs. The appearance of a somewhat mysterious boundary term prompts several questions, which are discussed in Remark 2.6. Answering these questions comprises a large part of the book; cf. Chapter 7. We are able to prove in Lemma 2.76 that this boundary term vanishes for finitely supported functions on G and in Corollary 2.73 that nontrivial harmonic functions cannot be in 2 (G0 ). Later, we will see that the boundary term vanishes precisely when the random walk on the network is recurrent. This is discussed further in Remark 2.74 and provides the motivation for an energycentric approach we pursue throughout our study. The energy Hilbert space HE will facilitate our study of the resistance metric R in Chapter 3. In particular, it provides an explanation for an issue stemming from the “nonuniqueness of currents” in certain infinite networks; see Refs. [LP16, Tho90]. This disparity leads to differences between two apparently natural extensions of the effective resistance to infinite networks, which are greatly clarified by the geometry of Hilbert space. Also, HE presents an analytic formulation of the type problem for random walks on an electrical resistance network: The transience of the random walk is equivalent to the existence of monopoles, that is, finite-energy solutions to a certain Dirichlet problem. In fact, this approach will readily allow us to obtain explicit formulas for effective resistance on integer lattice networks in Chapter 14, with applications to a physics problem given in Ref. [Pow76b] in Chapter 16. Most results in this section appeared in Ref. [JP13a].

The Energy Hilbert Space

25

Definition 2.1. The energy form E is symmetric and positive definite, and its kernel is the set of constant functions on G. Let 1 denote the constant function with value 1 and recall that ker E = C1. Then, the energy space HE := dom E/C1 is a Hilbert space with the inner product and corresponding norm, respectively, given by u, vE := E(u, v) and uE := E(u, u)1/2 .

(2.1)

It can be checked directly that the above completion consists of (equivalence classes of) functions on G0 via an isometric embedding into a larger Hilbert space as in Refs. [LP16, MYY94] or by a standard Fatou’s lemma argument as in Ref. [Soa94]; we provide a completeness argument in Theorem 10.13. Fix a reference vertex o ∈ G0 to act as an “origin.” It will readily be seen that all results are independent of this choice. Remark 2.2. In Chapter 4, we provide an alternative construction of HE via the techniques of von Neumann and Schoenberg. This provides for a more explicit description of the structure of HE and its relation to the metric geometry of (G, R) and shows that HE is the natural Hilbert space in which to embed (G, R). However, this must be postponed until after the introduction of the effective resistance metric. Remark 2.3 (Four warnings about HE ). (1) HE has no canonical orthonormal basis; the usual candidates {δx } are not orthogonal, and typically, their span is not even dense, as we discuss further in the following. (2) Multiplication operators are not Hermitian; see Lemma 8.3 and Remark 8.32. (3) There is no natural interpretation of HE as an 2 -space of functions on the vertices G0 or edges G1 of (G, c). HE does contain the embedded image of 2 (G0 , μ) for a certain measure μ, but these spaces are not typically dense in HE . Also, HE embeds isometrically into a subspace of 2 (G1 , c), but it is generally nontrivial to determine whether a given element of 2 (G1 , c) lies in this subspace. HE may also be understood as a 2 space of random variables (see Section 16.1) or realized as a subspace of L2 (S  , P), where S  is a certain space of distributions (see Chapter 7). (4) Pointwise identities should not be confused with Hilbert space identities; see Remark 2.17 and Lemma 2.36. To elaborate on the last point, note that elements of HE are technically equivalence classes of functions which differ only by a constant; this is

26

Operator Theory and Analysis of Infinite Networks

what is meant by the notation dom E/R1. In other words, if v1 = v2 + k for k ∈ C, then v1 = v2 in HE . When working with representatives, we typically abuse notation and use u to denote the equivalence class of u. Often, we choose u so that u(o) = 0 (occasionally without warning). A different but no less useful choice is to pick k so that v = 0 outside a finite set when v is a function of finite support (see Definition 2.14). Definition 2.4. An exhaustion of G is an increasing sequence of finite and  connected subgraphs {Gk } so that Gk ⊆ Gk+1 and G = Gk . Definition 2.5. The notation  x∈G0

:= lim

k→∞



(2.2)

x∈Gk

is used whenever the limit is independent of the choice of exhaustion {Gk } of G. We typically justify this independence by proving the sum to be absolutely convergent. Remark 2.6. One of the main results in this section is a discrete version of the Gauss–Green theorem presented in Theorem 2.40:   u(x)Δv(x) + u(x) ∂∂v (x), u, v ∈ HE . (2.3) u, vE = x∈G0

x∈bd G

This differs from the literature, where it is common to find E(u, v) = u, Δv2 given as a definition (of E or of Δ, depending on the context), e.g., Refs. [Kig01, Kig03, Str10].  We refer to bd G u ∂∂v as the “boundary term” by analogy with classical PDE theory. This terminology should not be confused with the notion of boundary that arises in the discussion of the discrete Dirichlet problem. In particular, the boundary discussed in Refs. [Kig03, Kig08] refers to a subset of G0 . By contrast, when discussing an infinite network G, our boundary bd G is never contained in G. Green’s identity follows immediately from (2.3) in the form     (u(x)Δv(x) − v(x)Δu(x)) = v(x) ∂∂u (x) − u(x) ∂∂v (x) . (2.4) x∈G0

x∈bd G

Note that our definition of the Laplace operator is the negative of that often found in the PDE literature, where one will find Green’s identity written as   (uΔv − vΔu) = (u ∂∂ v − v ∂∂ u). Ω

∂Ω

The Energy Hilbert Space

27

As the boundary term may be difficult to contend with, it is extremely useful to know when it vanishes. We have several results concerning this: (i) Lemma 2.76 shows that the boundary term vanishes when either argument of u, vE has finite support; (ii) Lemma 2.49 gives the necessary and sufficient conditions on the electrical resistance network for the boundary term to vanish for any u, v ∈ HE ; (iii) Lemma 2.71 shows that the boundary term vanishes when both arguments of u, vE and their Laplacians lie in 2 . In fact, Lemma 2.49 expresses the fact that it is precisely the presence of monopoles that prevents the boundary term from vanishing. An example with a nonvanishing boundary term is given in Example 15.5. 2.1 Evaluation Functional Lx and Reproducing Kernel vx In this section, we show how the energy space can be constructed as a reproducing kernel Hilbert space; cf. Section A.4. An approach based on the Schoenberg–von Neumann theorem is described in Chapter 4 and is based on properties of the effective resistance metric (investigated in Chapter 3). Definition 2.7. For any vertex x ∈ G0 , define the linear evaluation functional Lx on HE by Lx u := u(x) − u(o).

(2.5)

Lx is a relative evaluation functional in that it yields values of the function relative to the value at the origin. Lemma 2.8. For any x ∈ G0 , one has |Lx u| ≤ kE(u)1/2 , where k depends only on x. Proof. Since G is connected, choose a path {xi }ni=0 with x0 = o, xn = x,  1/2 n −1 and cxi ,xi−1 > 0 for i = 1, . . . , n. For k = c , the Schwarz i=1 xi ,xi−1 inequality yields

n 

2

 c

x ,x



i i−1 |Lx u|2 = |u(x) − u(o)|2 =

(u(xi ) − u(xi−1 )) ≤ k 2 E(u).



cxi ,xi−1 i=1



28

Operator Theory and Analysis of Infinite Networks

Theorem 2.9. HE is a (relative) reproducing kernel Hilbert space. Proof. By Lemma 2.8 and Riesz representation theorem, we are justified in defining vx to be the unique element of HE for which vx , uE = u(x) − u(o) for every u ∈ HE . Thus, {vx }x∈G0 \{o} forms a relative reproducing kernel for HE .

(2.6) 

Corollary 2.10. span{vx } is dense in HE . Proof. It is an immediate consequence of Theorem 2.9 that HE = span{vx }x∈G0 , where the closure is taken with respect to E.

(2.7) 

Definition 2.11. The family of functions {vx }x∈G0 \{o} is called a energy kernel because of Theorem 2.9. Note that vo is omitted because it corresponds to a constant function since vo , uE = 0 for every u ∈ HE . There is a rich literature dealing with reproducing kernels and their manifold application to both continuous analysis problems (see Section A.4 and also, for example, Refs. [Aro50, AD06, AL08, AAL08, BV03, Zha09, Yoo07]) and infinite discrete stochastic models. One of the differences between these studies and our current work is the approach we take in Definition 2.11, i.e., the use of “relative” reproducing kernels. Remark 2.12. Definition 2.11 is justified by Theorem 2.9. In this book, the functions vx will play a role analogous to fundamental solutions in PDE theory; see Section 10.3. The functions vx are R-valued. This can be seen by first constructing the energy kernel for the Hilbert space of R-valued functions on G and then using the decomposition of a C-valued function u = u1 + u2 into its real and imaginary parts. Alternatively, see Lemma 2.27. Reproducing kernels will help with many calculations and explain several of the relationships that appear in the study of resistance networks. They also extend the analogy with complex function theory discussed in Section 10.3. The reader may find Refs. [Aro50, Yoo07, Jor83] to provide a helpful background on reproducing kernels. Remark 2.13 (Probabilistic interpretation of vx ). The energy kernel {vx } is intimately related to effective resistance distance R(x, y). In fact,

The Energy Hilbert Space

29

R(x, o) = vx (x) − vx (o) = E(vx ), and similarly, R(x, y) = E(vx − vy ). This is discussed in detail in Chapter 3, but we give a brief summary here to help the reader get a feeling for vx . For a random walk (RW) starting at the vertex y, let τx be the hitting time of x (i.e., the time at which the RW first reaches x) and define the function ux (y) = P[τx < τo | RW starts at y]. Here, the RW is governed by the transition probabilities p(x, y) = cxy /c(x); cf. Remark 2.35. One can show that vx = R(x, o)ux is the representative of vx with vx (o) = 0. Since the range of ux is [0, 1], one has 0 ≤ vx (y)−vx (o) ≤ vx (x)−vx (o) = R(x, o). Many other properties of vx are similarly clear from this interpretation. For example, it is easy to compute vx completely on any tree. 2.2 Finitely Supported Functions and Harmonic Functions Definition 2.14. For v ∈ HE , one says that v has finite support iff there / F . Equivalently, is a finite set F ⊆ G0 for which v(x) = k ∈ C for all x ∈ the set of functions of finite support in HE is .

span{δx } = {u ∈ dom E .. u(x) = k for some k, for all but finitely many x ∈ G0 },

(2.8)

where δx is the Dirac mass at x, i.e., the element of HE containing the characteristic function of the singleton {x}. It is immediate from (1.11) that δx ∈ HE . Define Fin to be the closure of span{δx } with respect to E. Definition 2.15. The set of harmonic functions of finite energy is denoted by .

Harm := {v ∈ HE .. Δv(x) = 0 for all x ∈ G0 }.

(2.9)

Note that this is independent of the choice of representative for v in virtue of (1.4). Lemma 2.16. For any x ∈ G0 , one has δx , uE = Δu(x). Proof. Compute δx , uE = E(δx , u) directly from formula (1.9).



Remark 2.17. When the Laplacian is discussed as an operator on HE , Lemma 2.16 has the interpretation that the Dirac masses {δx }x∈G0 form

30

Operator Theory and Analysis of Infinite Networks

a reproducing kernel for Δ. Note that one can take the definition of the Laplacian to be the operator A defined via the equation δx , uE = Au(x). This point of view is helpful, especially when distinguishing between identities in Hilbert space and pointwise equations. For example, if h ∈ Harm, then Δh and the constant function 1 are identified in HE because u, ΔhE = u, 1E = 0 for any u ∈ HE . However, one should not consider a (pointwise) solution to Δu(x) = 1 as a harmonic function. Lemma 2.18. For any x ∈ G0 , Δvx = δx − δo . Proof. Using Lemma 2.16, Δvx (y) = δy , vx E = δy (x) − δy (o) = (δx −  δo )(y). By applying Lemma 2.18 to vα − vω , we see the following. Corollary 2.19. The space of dipoles P(α, ω) is nonempty. Lemma 2.16 is extremely important. Since Fin is the closure of span{δx }, it implies that the finitely supported functions and the harmonic functions are orthogonal. This result is called the “Royden Decomposition” in Ref. [Soa94, Ch. VI] and also appears elsewhere, e.g., Ref. [LP16, Section 9.3]. Theorem 2.20. HE = Fin ⊕ Harm. Proof. For all v ∈ HE , Lemma 2.16 gives δx , vE = Δv(x). Since Fin = span{δx }, this equality shows v ⊥ Fin whenever v is harmonic. Conversely, if δx , vE = 0 for every x, then v must be harmonic. Recall that constants  functions are 0 in HE . Corollary 2.21. span{δx } is dense in HE iff Harm = 0. Remark 2.22. Corollary 2.21 is immediate from Theorem 2.20, but we wish to emphasize the point, as it is not the usual case elsewhere in the literature. Part of the importance of the energy kernel {vx } arises from the fact that the Dirac masses are generally inadequate as a representing set for HE . This leads to unusual consequences, e.g., for u ∈ HE , one may have  u = u(x)δx , in HE . x∈G0

The Energy Hilbert Space

31

 More precisely, u − x∈Gk u(x)δx E may not tend to 0 as k → ∞ for any exhaustion {Gk }. Indeed, one may have u ∈ HE with u −  x∈Gk u(x)δx E → ∞ as k → ∞; see Examples 15.3 and 15.9. Definition 2.23. Let fx = PFin vx denote the image of vx under a projection to Fin. Similarly, let hx = PHarm vx denote the image of vx under a projection to Harm. For future reference, we state the following immediate consequence of orthogonality. Lemma 2.24. With fx = PFin vx , {fx }x∈G0 is a reproducing kernel for Fin, but fx ⊥ Harm. Similarly, with hx = PHarm vx , {hx }x∈G0 is a reproducing kernel for Harm, but hx ⊥ Fin. Remark 2.25. The role of vx in HE with respect to ·, ·E is directly analogous to the role of the Dirac mass δx in 2 with respect to the usual 2 inner product. This analogy will be developed further when we show that vx is the image of x ∈ G0 under a certain isometric embedding into HE in Chapter 4. It is obvious that δx ∈ HE , and the following result shows that δy is always in span{vx } when deg(y) < ∞. However, it is not true that vy is always in span{δx } or even in its closure. This is discussed further in Chapter 4.  Lemma 2.26. For any x ∈ G0 , δx = c(x)vx − y∼x cxy vy .  Proof. Lemma 2.16 implies δx , uE = c(x)vx − y∼x cxy vy , uE for every u ∈ HE , so we apply this to u = vz , z ∈ G0 . Since δx , vx ∈ HE , it must also   be that y∼x cxy vy ∈ HE . 2.2.1 Real- and complex-valued functions on G0 While we need complex-valued functions for some later results concerning spectral theory, it will usually suffice to consider R-valued functions elsewhere. Lemma 2.27. The reproducing kernels vx , fx , and hx are all R-valued functions. Proof. Computing directly, 1  (vz (x) − vz (y))(u(x) − u(y)) = vz , uE . vz , uE = 2 0 x,y∈G

32

Operator Theory and Analysis of Infinite Networks

Then, applying the reproducing kernel property, vz , uE = u(x) − u(o) = u(x) − u(o) = vz , uE . Thus, vz , uE = vz , uE for every u ∈ Harm, and vz must be R-valued.  The same computation applies to fz and hz . Definition 2.28. A sequence of functions {un } ⊆ HE converges pointwise in HE iff ∃k ∈ C such that un (x) − u(x) → k for each x ∈ G0 . Lemma 2.29. If {un } converges to u in E, then {un } converges to u pointwise in HE . Proof. Define wn := un − u so that wn E → 0. Then, n→∞

|wn (x) − wn (o)| = |vx , wn E | ≤ vx E · wn E −−−−−→ 0 so that lim wn exists pointwise and is a constant function.



2.3 Discrete Gauss–Green Formula A key difference between our development of the relationship between the Laplace operator Δ and the Dirichlet energy form E (embodied in the discrete Gauss–Green formula of Theorem 2.40) is that Δ is Hermitian but not necessarily self-adjoint in the current context. This is in sharp contrast to the literature on resistance forms [Kig03], the general theory ¯ of Dirichlet forms and probability [FOT94, BH91], and Dirichlet spaces in potential theory [Bre67, CC72]. In fact, the “gap” between Δ and its selfadjoint extensions comprises an important part of the boundary theory for (G, c) and accounts for the features of the boundary terms in the discrete Gauss–Green identity of Theorem 2.40. Before completing the extension of Lemma 1.13 to infinite networks, we need some definitions. Definition 2.30. A monopole at x ∈ G0 is an element wx ∈ HE which satisfies Δwx (y) = δxy , where k ∈ C and δxy is Kronecker’s delta. When nonempty, the set of monopoles at the origin is closed and convex, so E attains a unique minimum here; let wo always denote the unique energyminimizing monopole at the origin.

The Energy Hilbert Space

33

When HE contains monopoles, let Mx denote the vector space spanned by the monopoles at x. This implies that Mx may contain harmonic functions; see Lemma 2.44. We indicate the distinguished monopoles as wxv := vx + wo

and wfx := fx + wo ,

(2.10)

where fx = PFin vx . (Corollary 2.45 confirms that wxv = wfx for all x iff if Harm = 0.) Remark 2.31. Note that wo ∈ Fin, whenever it is present in HE and, similarly, that wfx is an energy-minimizing element of Mx . To see this, suppose wx is any monopole at x. Since wx ∈ HE , write wx = f + h by Theorem 2.20 and get E(wx ) = E(f ) + E(h). Projecting away the harmonic component will not affect the monopole property, so wfx = PFin wx is a unique monopole of minimal energy. Also, wo corresponds to the projection of 1 to D0 ; see Section 2.4.1. Definition 2.32. The dense subspace of HE spanned by monopoles (or dipoles) is M := span{vx }x∈G0 + span{wxv , wfx }x∈G0 .

(2.11)

Let ΔM be the closure of the Laplacian when taken to have the dense domain M. Note that M = span{vx } when there are no monopoles (i.e., when all solutions to Δw = δx have infinite energy) and that M = span{wxv , wfx } when there are monopoles; see Lemma 2.44. The space M is introduced as a dense domain for Δ and for its use as a hypothesis in our main result, that is, as the largest domain of validity for the discrete Gauss–Green identity of Theorem 2.40. Note that while a general monopole need not be in dom ΔM (see Example 15.4 or 14.36), we show in Lemma 2.36 that it is always the case that it lies in dom ΔM . Definition 2.33. A Hermitian operator S on a Hilbert space H is called semibounded iff v, Sv ≥ 0

for every v ∈ D

so that its spectrum lies in some half line [κ, ∞).

(2.12)

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Operator Theory and Analysis of Infinite Networks

Lemma 2.34. ΔM is Hermitian; a fortiori, ΔM is semibounded.   by wy , Proof. Suppose we have two finite sums u = ax wx and v = writing wx for wxv or wfx . We may assume that o appears neither in the sum u nor for v; see Definition 2.11. Then, Lemma 2.16 gives u, ΔvE = =

 



ax by wx , Δwy E = ax by Δwx (y) =



ax by wx , δy E

ax by δxy .

 Of course, Δu, vE = ax by δxy in exactly the same way. The argument for linear combinations from {vx } is similar, so ΔM is Hermitian. Then, u, ΔuE =

 x,y

ax ay δxy =



|ax |2 ≥ 0

x

shows that ΔM is semibounded. The argument for {vx } is similar.



Remark 2.35 (Monopoles and transience). The presence of monopoles in HE is equivalent to the transience of the underlying network, that is, the transience of a simple random walk on the network with transition probabilities p(x, y) = cxy /c(x). To see this, note that if wx is a monopole, then the current induced by wx is a unit flow to infinity with finite energy. It was proved in Ref. [Lyo83] that the network is transient if and only if there exists a unit current flow to infinity; see also Ref. [LP16, Thm. 2.10]. It is also clear that the existence of a monopole at one vertex is equivalent to the existence of a monopole at every vertex: consider vx + wo . The corresponding statement about transience is well known. Since Δ agrees with ΔM pointwise, we may suppress the reference to the domain for ease of notation. When given a pointwise identity Δu = v, there is an associated identity in HE , but the next lemma shows that one must use the adjoint. Lemma 2.36. For u, v ∈ HE , Δu = v pointwise if and only if v = ΔM u in HE . Proof. We show that u ∈ dom ΔM for simplicity, so let ϕ ∈ span{vx } be  given by ϕ = ni=1 ai vxi ; the proof for ϕ ∈ span{wxv , wfx } is similar. Then,

The Energy Hilbert Space

35

Lemmas 2.16 and 2.18 give Δϕ, uE =

n 

ai δxi − δo , uE =

i=1

n 

ai (Δu(xi ) − Δu(o)).

i=1

Since Δu(x) = v(x) by hypothesis, this may be continued as Δϕ, uE =

n 

ai (v(xi ) − v(o)) =

i=1

n 

ai vxi , vE = ϕ, vE .

i=1

Then, the Schwarz inequality gives the estimate |Δϕ, uE | = |ϕ, vE | ≤  ϕE vE , which means u ∈ dom ΔM . The converse is trivial. Remark 2.37 (Monopoles give a reproducing kernel for ran ΔM ). Lemma 2.36 means that wx , ΔuE = δx , uE

for all u ∈ dom ΔM

(2.13)

and for every wx ∈ Mx . Combined with Lemma 2.16, this immediately gives wx , ΔuE = Δu(x).

(2.14)

If {wx }x∈G0 is a collection of monopoles which includes one element from each Mx , then this collection is a reproducing kernel for ran ΔM . Note that the expression Δu(x) is defined in terms of differences, so the righthand side is well defined even without reference to another vertex, i.e., independent of the choice of representative. As a special case, let wxo be the representative of wfx which satisfies o wx (o) = 0. Then, the Green function is g(x, y) = wyo (x), and {wxo }x∈G0 \{o} gives a reproducing kernel for ran ΔM ⊆ Fin. Therefore, M contains an extension of the Green kernel to all of HE . In Definition 2.32, we give a domain M for Δ which ensures that ran ΔM contains all finitely supported functions and is thus dense in Fin. However, even when Δ is defined so as to be a closed operator, one may not have clo Fin ⊆ ran Δ; in general, the containment ran(S clo ) ⊆ (ran S) may be strict. The operator closure S clo is done with respect to the graph norm, and the closure of the range is done with respect to E. We note that in Ref. [MYY94, (G.1)], it is claimed that the Green function is a reproducing kernel for all of Fin. In our context, at least, the Green function is a reproducing kernel only for ran Δ, where Δ has been chosen with a suitable dense domain. In general, the containment ran Δ ⊆ Fin may be strict. In fact, it is true that ran ΔM ⊆ Fin, and even this containment may be strict.

36

Operator Theory and Analysis of Infinite Networks clo

Note that wfx is the only element of Mx that lies in (ran ΔM ) , and it may not lie in ran ΔM . A different choice of domain for Δ can exacerbate the discrepancy between ran Δ and Fin: If one were to define ΔV to be the closure of Δ when taken to have dense domain V := span{vx }, then ran ΔV is dense in Fin2 , the E-closure of span{δx − δo }; see Definition 5.5. However, it can happen that Fin2 is a proper orthogonal subspace of Fin (the E-closure of span{δx }). An example of f ∈ Fin1 := Fin  Fin2 is computed in Example 14.36. The domain of Δ can thus induce a refinement of the Royden decomposition: HE = Fin1 ⊕ Fin2 ⊕ Harm. See Theorem 2.20 and the comment preceding it. Fin2 is studied in Section 5.1.1. Note that a monopole need not be in dom ΔV ; see Example 14.40. However, it is always the case that wx ∈ dom ΔV , which is the content of the next lemma. Definition 2.38. If H is a subgraph of G, then the boundary of H is .

bd H := {x ∈ H .. ∃y ∈ H  , y ∼ x}.

(2.15)

The interior of a subgraph H consists of the vertices in H whose neighbors also lie in H: .

int H := {x ∈ H .. y ∼ x =⇒ y ∈ H} = H \ bd H.

(2.16)

For vertices in the boundary of a subgraph, the normal derivative of v is  ∂v cxy (v(x) − v(y)) for x ∈ bd H. (2.17) ∂  (x) := y∈H

Thus, the normal derivative of v is computed like Δv(x), except that the sum extends only over the neighbors of x which lie in H. Definition 2.38 will be used primarily for subgraphs that form an exhaustion of G in the sense of Definition 2.4: an increasing sequence of  finite and connected subgraphs {Gk } so that Gk ⊆ Gk+1 and G = Gk .   Also, recall that bd G := limk→∞ bd Gk from Definition 2.39.

The Energy Hilbert Space

37

Definition 2.39. A boundary sum (or boundary term) is computed in terms of an exhaustion {Gk } by   := lim (2.18) bd G

k→∞

bd Gk

whenever the limit is independent of the choice of exhaustion, as in Definition 2.5. The boundary bd G is examined more closely as an object in its own right in Chapter 7. The key point of the following result is that for u, v in a specified set, the two sums are both finite. The decomposition is true for all u, v ∈ HE by taking limits of (2.20) but is clearly meaningless if it takes the form ∞− ∞. Theorem 2.40 (Discrete Gauss–Green formula). If u ∈ HE and v ∈ M, then   u(x)Δv(x) + u(x) ∂∂v (x). (2.19) u, vE = x∈G0

x∈bd G

Proof. It suffices to work with R-valued functions and then complexify afterward. By the same computation as in Lemma 1.13, we have 1  cxy (u(x) − u(y))(v(x) − v(y)) 2 x,y∈Gk



=

u(x)Δv(x) +

x∈int Gk



u(x) ∂∂v (x).

(2.20)

x∈bd Gk

Taking limits of both sides as k → ∞ gives (2.19). It remains to be seen that one of the sums on the right-hand side is finite (and hence that both are). For this part, we work just with u and polarize afterward. Note that if v = wz is a monopole, then   u(x)Δv(x) = u(x)δz (x) = u(z). x∈G0

x∈G0

This is obviously independent of exhaustion and immediately extends to v ∈ M.  Remark 2.41. It is clear that (2.19) remains true much more generally than under the specified conditions. Clearly, the formula holds whenever  x∈G0 |u(x)Δv(x)| < ∞. Unfortunately, given any hypotheses more specific than this, the limitless variety of infinite networks almost always allows one to construct a counterexample, i.e., one cannot give a condition for which the formula is true for all u ∈ HE for all networks. To see this,

38

Operator Theory and Analysis of Infinite Networks

suppose that v = Then,

∞ i=1

ai wxi with each wxi a monopole at the vertex xi .



u(x)Δv(x) =

x∈G0

∞ 

ai u(xi ),

i=1

and one would need to provide a condition on sequences {ai } that would ∞ ensure i=1 ai u(xi ) is absolutely convergent for all u ∈ HE . Such a hypothesis is not likely to be useful (if it is even possible to construct) and would depend heavily on the network under investigation. Nonetheless, the formula remains true in many specific contexts. For example, it is clearly valid whenever v is a dipole, including all those in the energy kernel. We will also see that it holds for the projections of vx to Fin and to Harm. Consequently, for v which are limits of elements in M, we often use this result in combination with ad hoc arguments. After reading a preliminary version of the paper on which this chapter is based, a reader pointed out to us that a formula similar to (2.19) appears in Ref. [DK88, Prop 1.3]; however, the authors of the latter apparently did not pursue the extension of this formula to infinite networks. We were also directed toward Ref. [KY89, Thm. 4.1], in which the authors give some conditions under which Lemma 1.13 extends to infinite networks. The main differences here are that the scope of Kayano and Yamasaki’s theorem is limited to a subset of what we call Fin and that Kayano and Yamasaki are interested in when the boundary term vanishes; we are more interested in when it is finite and nonvanishing; see Theorem 2.49, for example. Since Kayano and Yamasaki do not discuss the structure of the space of functions they consider, it is not clear how large the scope of their result is; their result  requires the hypothesis x∈G0 |u(x)Δv(x)| < ∞, but it is not so clear what functions satisfy this. By contrast, we develop a dense subspace of functions on which to apply the formula. Furthermore, we show in the next chapter that these functions are relatively easy to compute. Recall that span{hx } is a dense subspace of Harm; the following boundary representation of harmonic functions in this space is the focus of Chapter 7. Corollary 2.42 (Boundary representation of harmonic functions). For all u ∈ span{hx },  x u(x) = u ∂h (2.21) ∂  + u(o). bd G

The Energy Hilbert Space

39

Proof. Recall from Lemma 2.24 that {hx } is a reproducing kernel for  x Harm. Therefore, u(x) − u(o) = hx , uE = u, hx E = bd G u ∂h ∂  because   G0 uΔhx = 0. Note that hx = hx by Lemma 2.27.   ∂u Lemma 2.43. For all u ∈ dom ΔV , G0 Δu = − bd G ∂  . Thus, the discrete Gauss–Green formula (2.19) is independent of representatives. Proof. On each (finite) Gk in any given exhaustion,    ∂u Δu(x) + (x) = cxy (u(x) − u(y)) = 0 ∂ x∈int Gk

x∈bd Gk

x,y∈Gk

since each edge appears twice in the sum, once with each sign (orientation). For the second claim, we apply the formula of the first to see that the result remains true when u is replaced by u + k:   (u + k)Δv + (u + k) ∂∂v bd G G0

       ∂v ∂v  = uΔv + u ∂ + k +  ∂ . Δv   bd G bd G G0 G0 2.4 More About Monopoles and the Space M This section studies the role of the monopoles with regard to the boundary term of Theorem 2.40 and provides several characterizations of transience of the network in terms of the operator-theoretic properties of ΔM . Note that if h ∈ Harm satisfies the hypotheses of Theorem 2.40, then   E(h) = bd G h ∂∂h . In Theorem 2.49, we show that E(u) = G0 uΔu for all u ∈ HE iff the network is recurrent. With respect to HE = Fin ⊕ Harm, this shows that the energy of finitely supported functions comes from the sum over G0 , and the energy of harmonic functions comes from the boundary sum. However, for a monopole wx , the representative specified  by wx (x) = 0 satisfies E(w) = bd G w ∂w ∂  , but the representative specified  by wx (x) = E(wx ) satisfies E(w) = G0 wΔw. Roughly, a monopole is therefore “half of a harmonic function” or halfway to being a harmonic function. A further justification for this comment is given by Corollary 2.45: The proof shows that a harmonic function can be constructed from two monopoles at the same vertex. A different perspective on the same theme is given in Remark 2.58. The general theme of this section is the ability of monopoles to “bridge” the finite and the harmonic.

40

Operator Theory and Analysis of Infinite Networks

Lemma 2.44. When the network is transient, M contains the spaces span{vx }, span{fx }, and span{hx }, where fx = PFin vx and hx = PHarm vx . Proof. The first two are obvious, since vx = wxv − wo and fx = wfx − wo by Definition 2.30. For the harmonics, note that these same identities give wxv − wo = vx = fx + hx = wfx − wo + hx , which implies that hx = wxv − wfx . (Of course, wxv = wfx when Harm = 0.)  Corollary 2.45. Harm = 0 iff there is more than one monopole at x. Proof. As usual, if this is true for any x, it is true for all. Suppose HE contains a monopole wx = wxv . Then, h := wxv − wx is a nonzero harmonic  function in HE . Theorem 2.46 ([Soa94, Thm. 1.33]). Let u be a nonnegative function on a recurrent network. Then, u is superharmonic if and only if u is constant. Corollary 2.47. If Harm = 0, then there is a monopole in HE . Proof. If h ∈ Harm and h = 0, then h = h1 − h2 , with hi ∈ Harm and hi ≥ 0, by Ref. [Soa94, Thm. 3.72]. (Here, hi ≥ 0 means that hi is bounded below, and so, we can choose a representative which is nonnegative.) Since hi cannot both be 0, Theorem 2.46 implies that the network is transient. Then, by Ref. [Lyo83, Thm. 1], the network supports a monopole.  Definition 2.48. The phrase “the boundary sum is nonvanishing” indicates that (2.19) holds with nonzero boundary sum when applied to u, vE for every representative of u except the one specified by u(x) = u, wE for w ∈ Mx . Recall from Remark 2.35 that the network is transient iff there are monopoles in HE . Theorem 2.49. The network is transient if and only if the boundary sum is nonvanishing. Proof. (⇒) If the network is transient, then as explained in Remark 2.35, there is a w ∈ HE with Δw = δz . Now, let wz := PFin w so that for any u ∈ dom ΔV , (2.19),  z u, wz E = u(z) + u ∂w ∂ . bd G

The Energy Hilbert Space

41

 z It is immediate that bd G u ∂w ∂  = 0 if and only if the computation is done with the representative of u specified by u(z) = u, wz E . (⇐) Suppose that there does not exist w ∈ HE with Δw = δz for any z ∈ G0 . Then, M = span{vx }, as discussed in Definition 2.30. Therefore, it suffices to show that  uΔvx , u, vx E = x∈G0

but this is clear because both sides are equal to u(x) − u(o) by (2.6) and Lemma 2.18.  Remark 2.50. It follows from Theorem 2.49 that a monopole wx cannot lie in ran ΔV . However, one can have wx ∈ ran ΔV , as in Example 14.40. Lemma 2.51. The network is transient if and only if there is a sequence {εk } with εk → 0 and supk (εk + Δ)−1 δx E ≤ B < ∞. Proof. For both directions of the proof, we let fk := (εk + Δ)−1 δx .

be any self-adjoint extension1 of ΔV , and let E(dλ) be the (⇒) Let Δ corresponding projection-valued measure. Then,  ∞ 1 −1

) E(dλ)u, (2.22) Rε u = (ε + Δ u= ε + λ 0

)−1 for the resolvent. Note that where we use the notation Rε := (ε + Δ

Rε ⊆ (Δ

Rε ) = Δ

 R = Δ

Rε . On the other hand, Δ

⊆ Δ ; therefore, Δ ε V  

⊆ Rε Δ . Combining these gives Δ

Rε ⊆ Rε Δ . Now, we apply this Rε Δ V V and (2.22) to u = Δ w to get

−1 −1  −1

)

(εk + Δ

)

) δx = (εk + Δ ΔV w = Δ w fk = (εk + Δ  ∞ λ E(dλ)w. = ε k +λ 0

Note that Rε is bounded, and so, w ∈ dom Rε automatically. This integral implies 2  ∞  ∞ λ 2 E(dλ)w2E ≤ E(dλ)w2E = w2E . fk E ≤ εk + λ 0 0 Thus, we have supk (εk + Δ)−1 δx E = sup fk E ≤ B = wE < ∞. 1 For concreteness, one may take the Friedrichs extension (see Section B.3 and (B.8)), but this is not necessary. See also Definition 7.7 and Section 5.1 in this regard.

42

Operator Theory and Analysis of Infinite Networks

(⇐) We show the existence of a monopole at x. Since εk fk + Δfk = δx , the bound sup fk E ≤ B implies that Δfk − δx E = εk fk  ≤ εk B → 0. Let w be a weak- limit of {fk }. Then, for ϕ ∈ dom ΔV , Δϕ, wE = lim Δϕ, fk E = lim ϕ, Δfk E k→∞

k→∞

= lim ϕ, δx − εk fk E = ϕ, δx E k→∞



so that w is a monopole at x. clo

Lemma 2.52. On any network, (ran ΔM ) ker ΔM .

⊆ Fin and hence Harm ⊆

Proof. If v ∈ M, then clearly ΔM v ∈ Fin. To close the operator, we consider sequences {un } ⊆ M, which are Cauchy in E and for which {Δun } is also Cauchy in E, and then include u := lim un in dom ΔM by defining ΔM u := lim ΔM un . Since fn := ΔM un has finite support for each n, the E-limit of {fn } must lie in Fin. Since Fin is closed, the first claim follows. The second claim follows upon taking orthogonal complements.  Theorem 2.53. The network is transient if and only if (ran ΔM )c = Fin. Proof. (⇒) If the network is transient, we have a monopole at every vertex; see Remark 2.35. Then, any u ∈ span{δx } is in ran ΔM because the monopole wx is in dom ΔM , and so Fin ⊆ ran ΔM . The other inclusion is Lemma 2.52. (⇐) If δx ∈ ran ΔM for some x ∈ G0 , then ΔM w = δx for w ∈ dom ΔM ⊆ dom E, and so, w is a monopole. Then, the induced current dw is a unit flow to infinity, and the network is transient, again by Ref. [Lyo83].  2.4.1 Comparison with the grounded energy space There are some subtleties in the relationship between HE and D, as discussed in Refs. [LP16, KY89, KY84, MYY94, Yam79], so we take a moment to give the details. We have attempted to match the notation of

The Energy Hilbert Space

43

these sources. We will return to the definitions and notations used here when considering multiplication operators in Chapter 8. Definition 2.54. The inner product u, vo := u(o)v(o) + u, vE makes dom E into a Hilbert space D, which we call the grounded energy space. Let D0 be the closure of span{δx } in D, and let HD be the space of harmonic functions in D. Throughout this section (only), we use the notation uo := u(o) for u ∈ D. Definition 2.55. With regard to D, we define the vector subspace . M− o := {u ∈ D . Δu = −uo δo }. .

(2.23)

Note that M− o contains the harmonic subspace .

HDo := {u ∈ D .. Δu = 0 and uo = 0}.

(2.24)

The previous definition is motivated by the following lemma. − − Lemma 2.56. D⊥ 0 = Mo and hence D = D0 ⊕ Mo .

Proof. With uo := u(o), we have u ∈ D⊥ 0 iff u ⊥ span{δx }, which means that 0 = u, δx o = uo δx (o) + u, δx E = uo δxo + Δu(x),

∀x ∈ G0 ,

(2.25) 

which means Δu = −uo δo .

⊥ Let PD0 denote the projection of D to D0 and PD the projection to D⊥ 0. 0

Remark 2.57. The constant function 1 decomposes into a linear combi⊥ 1 = 1 − v, and observe nation of two monopoles: Let v = PD0 1 and u = PD 0 that Δu = −uo δo by Lemma 2.56 and that Δv = Δ(1 − u) = −Δu = uo δo , so uo = 1 − vo gives Δv = (1 − vo )δo . In general, the constant function k1 ⊥ k1, where decomposes into v = PD0 k1 and u = PD 0 Δv = (k − uo )δo

and

Δu = −uo δo .

With respect to the decomposition D = D0 ⊕ M− o given by Lemma 2.56, (1) (2) there are two monopoles, wo ∈ D0 and wo ∈ M− o (which may be equal),

44

Operator Theory and Analysis of Infinite Networks (1)

(2)

such that 1 = uo wo − uo wo . When one passes from D to HE by modding out constants, these components of 1 add up to form (possibly constant) harmonic functions. An example of this is given in Example 14.34. Consequently, Lemma 2.56 yields a short proof of Ref. [LP16, Exc. 9.6c]: Prove that the network is recurrent iff 1 ∈ D0 . To see this, observe that if u is the projection of 1 to D⊥ 0 , then u = 0 iff there is a monopole. This result first appeared (in a more general form) in Ref. [Yam77, Thm. 3.2]. Remark 2.58. Despite the fact that Theorem 2.20 gives HE = Fin ⊕ Harm, note that D = D0 ⊕ HD. This is a bit surprising since HE = D/C1, etc., and this mistake has been made in the literature, e.g., Ref. [Yam79, Thm. 4.1]. The discrepancy results from the way 1 behaves with respect to PD0 ; this is easiest to see by considering .

D0 + k := {f + k1 .. f ∈ D0 , k ∈ C},

k = 0.

If the network is transient and f ∈ D0 + k, k = 0, then f = g + k1 for some g ∈ D0 , and ⊥ f = (g + kPD0 1) + kPD 1 0

shows that f ∈ / D0 . Nonetheless, it is easy to check that D0 + k is equal to the o-closure of span δx + k and hence that (D0 + C1)/C1 = Fin. This appears in Ref. [LP16, Exc. 9.6b]. Similarly, note that for a general h ∈ HD, ⊥ h + k1 so that h ∈ / D⊥ one has h = PD 0. 0 We conclude with a curious lemma that can greatly simplify the computation of monopoles of the form PD0 1; it is used in Example 14.34. In the next lemma, uo = u(o), as above. ⊥ Lemma 2.59. Let u ∈ D⊥ 0 . Then, u = PD0 1 if and only if uo = E(u) + 2 uo ∈ [0, 1).

Proof. From u2o +1−u2o = 12o = 1, one obtains E(u)−uo +|uo |2 = 0. 2 From u, 1 − uo = 0, one obtains  E(u) − uo + |uo | = 0. Combining the 1 equations gives uo = uo = 2 (1 ± 1 − 4E(u)) so that uo ∈ [0, 1]. However, uo = 1, or else, E(u) = 0 would imply 1 ∈ D⊥ 0 in contradiction to (2.25). The converse is clear.  Remark 2.60. The significance of the parameter uo is not clear. However, it appears to be related to the overall “strength” of the conductance of the network; we will see in Example 14.34 that uo ≈ 1 corresponds to a rapid

The Energy Hilbert Space

45

growth of c near ∞. Also, it follows from Remark 2.57 and Lemma 2.59 that uo = 0 corresponds to the recurrence. There is probably a good interpretation of uo in terms of probability and/or the speed of the random walk, but we have not yet determined it. The existence of conductances ⊥ 1) = 14 is similarly intriguing and even attaining maximal energy E(PD 0 more mysterious. Example 14.34 shows that the maximum is attained on (Z, cn ) for c = 2.

2.5 Applications and Extensions In Section 2.5.1, we use the techniques developed above to obtain new and succinct proofs of four known results, and in Section 2.5.2, we give some useful special cases of our main result, Theorem 2.40. Definition 2.61. For an infinite graph G, we say u(x) vanishes at ∞ iff for any exhaustion {Gk }, one can always find k and a constant C such that / Gk . One can always choose the representative u(x) − C∞ < ε for all x ∈ of u ∈ HE so that C = 0, but this may not be compatible with the choice u(o) = 0. Definition 2.62. Say γ = (x0 , x1 , x2 , . . . ) is a path to ∞ iff xi ∼ xi−1 for each i, and for any exhaustion {Gk } of G, ∀k, ∃N

/ Gk . such that n ≥ N =⇒ xn ∈

(2.26)

2.5.1 More about Fin and Harm The next two results are almost converse to each other, although the exact converse of Lemma 2.63 is false; consider the dipoles on the “geometric integers,” as depicted in Figure 14.6. Lemma 2.63 is related to Ref. [Soa94, Thm. 3.86], in which the result is stated as holding almost everywhere with respect to the notion of extremal length. Lemma 2.63. If u ∈ HE and u vanishes at ∞, then u ∈ Fin. Proof. Let u = f + h ∈ HE vanish at ∞. This implies that for any exhaustion {Gk } and any ε > 0, there is a k and C for which h(x)−C∞ < ε outside Gk . A harmonic function can only obtain its maximum on the boundary, unless it is constant, so in particular, ε bounds h(x) − C∞ on  all of Gk . Letting ε → 0, h tends to a constant function and u = f .

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Operator Theory and Analysis of Infinite Networks

Lemma 2.64. If h ∈ Harm is nonconstant, then from any x0 ∈ G0 , there is a path to infinity γ = (x0 , x1 , . . . ), with h(xj ) < h(xj+1 ) for all j = 0, 1, 2, . . . . Proof. Abusing notation, let h be any representative of h. Since h(x) =  cxy y∼x c(x) h(y) ≤ supy∼x h(y) and h is nonconstant, we can always find y ∼ x for which h(y1 ) > h(x0 ). This follows from the maximal principle for harmonic functions; cf. Ref. [LP16, Section 2.1], Ref. [LPW08, Ex. 1.12], or Ref. [Soa94, Thm. 1.35]. Thus, one can inductively construct a sequence which defines the desired path γ. Note that γ is infinite, so the condition h(xj ) < h(xj+1 ) eventually forces it to leave any finite subset of G0 , so Definition 2.62 is satisfied.  It is instructive to prove the contrapositive of Lemma 2.63 directly as follows. Lemma 2.65. If h ∈ Harm\{0}, then h has at least two different limiting values at ∞. Proof. Choose x ∈ G0 for which hx = PHarm vx ∈ HE is nonconstant. Then, Lemma 2.64 gives a path to infinity γ1 along which hx is strictly increasing. Since the reasoning of Lemma 2.64 works just as well with the inequalities reversed, we also get γ2 to ∞ along which hx is strictly decreasing. This gives two different limiting values of hx and hence hx cannot vanish at ∞.  Corollary 2.66. If h ∈ Harm is nonconstant, then h ∈ / p (G0 ) for any 1 ≤ p < ∞. Proof. Lemma 2.65 shows that no matter what representative is chosen   for h, the sum hpp = x∈G0 |h(x)|p has the lower bound x∈F εp = εp |F |  for some infinite set F ⊆ G0 . 2.5.2 Special applications of the discrete Gauss–Green formula In this section, we use Lemma 1.13 to infinite networks to establish that Δ is Hermitian when its domain is correctly chosen (Corollary 2.69) and that Lemma 1.13 remains correct on infinite networks for vectors in span{vx } (Theorem 2.76).

The Energy Hilbert Space

47

Lemma 2.67. If v ∈ span{vx }, then u, vE =

 x∈G0

u(x)Δv(x).

Proof. It suffices to consider v = vx , whence   u(y)Δvx (y) = u(y)(δx − δo )(y) = u(x) − u(o) = u, vx E G0

G0

by Lemma 2.18 and the reproducing property of Theorem 2.9. Theorem 2.68. For u, v ∈ span{vx },  u, ΔvE = Δu(x)Δv(x). Furthermore,



(2.27)

x∈G0

 x∈G0

Δu(x) = 0 for u ∈ span{vx }.

 Proof. Let u ∈ span{vx } be given by the finite sum u = x ξx vx . Since vo is a constant, we may assume that the sum does not include o. Then,   ξx Δvx (y) = ξx (δx − δo )(y) = ξy . (2.28) Δu(y) = x

x

Now, we have u, ΔuE =



ξx ξy vx , Δvy E =

x,y



ξx ξy vx , δy − δo E .

x,y

Since it is easy to compute vx , δy − δo E = δxy + 1 (Kronecker’s delta), we have



 2  



2 ξx ξy (δxy + 1) = |ξx | +

ξx

(2.29) u, ΔuE =



x,y

x

=

 x

x



2





|Δu(x)| +

Δu(x)

x

2

(2.30)

by (2.28). Since u ∈ span{vx }, Δu ∈ span{δx − δo } (see (2.28)) so that  u, ΔuE < ∞ and (2.29) is convergent. Therefore, x Δu(x) is absolutely convergent, hence independent of exhaustion. Since  Δvy (x) = 1 − 1 = 0 x∈G0

 by Lemma 2.18, it follows that x Δu(x) = 0, and the second sum in (2.29) vanishes. Then, (2.27) follows from polarization. 

Operator Theory and Analysis of Infinite Networks

48

Corollary 2.69. The Laplacian ΔV is Hermitian and even semibounded on dom ΔV (see Definition 2.33) with  |Δu(x)|2 ≤ u, ΔuE < ∞. (2.31) 0≤ x∈G0

Proof. For u, v ∈ span{vx }, two applications of Lemma 2.68 yield   Δu, vE = Δu(x)Δv(x) = Δu(x)Δv(x) = Δv, uE . x∈G0

x∈G0

This property is clearly preserved under closure of the operator. Now, let u ∈ dom ΔV and choose {un } ⊆ V with limn→∞ un − uE = limn→∞ Δun − ΔuE = 0. Then, Fatou’s lemma [Mal95, Thm. I.7.7] yields   |Δu(x)|2 = lim |Δun (x)|2 ≤ lim un , Δun E = u, ΔuE , (2.32) x∈G0

n→∞

x∈G0

which gives the central inequality in (2.31) and hence semiboundedness.   Remark 2.70. The notation u ∈ 1 means x∈G0 |u(x)| < ∞, and the  notation u ∈ 2 means x∈G0 |u(x)|2 < ∞. When discussing an element u of HE , we say that u lies in 2 if it has a representative which does, i.e., if u + k ∈ 2 for some k ∈ C. This constant is clearly necessarily unique on an infinite network if it exists. The next result is a partial converse to Theorem 2.40.  Lemma 2.71. If u, v, Δu, Δv ∈ 2 , then u, vE = x∈G0 u(x)Δv(x), and u, v ∈ dom E. Proof. If u, Δv ∈ 2 , then uΔv ∈ 1 , and the following sum is absolutely convergent:  1  1  u(x)Δv(x) = u(x)Δv(x) + u(y)Δv(y) 2 2 0 0 0 x∈G

x∈G

y∈G

1   = cxy u(x)(v(x) − v(y)) 2 0 y∼x x∈G

−

1   cxy u(y)(v(x) − v(y)) 2 0 x∼y y∈G

=

1   cxy (u(x) − u(y))(v(x) − v(y)), 2 0 y∼x x∈G

The Energy Hilbert Space

49

which is (1.9). The absolute convergence justifies the rearrangement in the last equality; the rest is merely algebra. Substituting u for v in the identity  just established, uΔu ∈ 1 shows u ∈ dom E and similarly for v. Remark 2.72. All that is required for the computation in the proof of Lemma 2.71 is that uΔv ∈ 1 , which is certainly implied by u, Δv ∈ 2 . However, this would not be sufficient to show that u or v lies in dom E. We will see in Theorem 2.63 that if h ∈ Harm is nonconstant, then h + k is bounded away from 0 on an infinite set of vertices for any choice of constant k. So, the next result should not be surprising. Corollary 2.73. If h ∈ HE is a nontrivial harmonic function, then h cannot lie in 2 .   Proof. If h ∈ 2 , then E(h) = x∈G0 h(x)Δh(x) = x∈G0 h(x)·0 = 0 by <  Lemma 2.71. But since h is nonconstant, E(h) > 0.  Remark 2.74 (Restricting to 2 misses the most interesting bit). When studying the graph Laplacian, some authors define dom Δ = {v ∈ . 2 .. Δv ∈ 2 }. Our philosophy is that dom E is the most natural context for the study of functions on G0 , and this is described in detail in Section 4.1. Some of the most interesting phenomena in dom E are due to the presence of nontrivial harmonic functions, as we show in this section and the examples of Chapters 14 and 15. Consequently, Corollary 2.73 shows why one loses some of the most interesting aspects of the theory by only studying those v which lie in 2 . Example 15.2 illustrates the situation of Corollary 2.73 on a tree network. In general, if a at least two connected components of G \ {o} / 2 for vertices x in these components. are infinite, then vx ∈ 2.5.3 Discrete Gauss–Green formula for networks with vertices of infinite degree If there are vertices of infinite degree in the network, then it does not necessary follow that span{δx } ⊆ span{vx } or that span{δx } ⊆ M. However, we do have the following version of Theorem 2.40. When all vertices have finite degree, Theorem 2.76 follows from Theorem 2.40 by Lemma 2.26. Definition 2.75. Let F := span{δx }x∈G0 denote the vector space of the functions of a finite support, and let ΔF be the closure of the Laplacian when taken to have the domain F .

50

Operator Theory and Analysis of Infinite Networks

Note that F is a dense domain only when Harm = 0 by Corollary 2.21. Again, since Δ agrees with ΔF pointwise, we may suppress the reference to the domain for ease of notation. The next result extends Lemma 1.13 to infinite networks.  Theorem 2.76. If u or v lies in dom ΔF , then u, vE = x∈G0 u(x) Δv(x). Proof. First, suppose u ∈ dom ΔF and choose a sequence {un } ⊆ span{δx } with un −uE → 0. From Lemma 2.16, one has δx , vE = Δv(x), and hence,  un , vE = un (x)Δv(x) x∈G0

holds for each n. Define M := sup{un E }, and note that M < ∞ since this sequence is convergent (to uE ). Moreover, |un , vE | ≤ M · vE by the Schwarz inequality. Since un converges pointwise to u in HE by Lemma 2.29, this bound will allow us to apply Fatou’s lemma (as stated in Ref. [Mal95, Lem. 7.7], for example), as follows: u, vE = lim un , vE n→∞

= lim

n→∞

=





un (x)Δv(x)

hypothesis un ∈ span{δx }

x∈G0

u(x)Δv(x).

x∈G0

Note that the sum over G0 is absolutely convergent, as required by Definition 2.4. Now, suppose that v ∈ dom ΔF and observe that this implies v ∈ Fin also. By Theorem 2.20, one can decompose u = f + h, where f = PFin u and h = PHarm u, and then, u, vE = f, vE + h, vE = f, vE since h is orthogonal to v. Now, apply the previous argument to f, vE .



2.6 Remarks and References For background material and applications of reproducing kernel Hilbert spaces, we suggest the standard references, Refs. [PS72, Aro50] as well as

The Energy Hilbert Space

51

Refs. [AD06, AL08, Kai65, MYY94, Yoo07, Zha09, BV03, Arv97, Arv76c, ADV09]. Of the cited references for this chapter, some are more specialized. However, for prerequisite material (if needed), the reader may find key facts used above on operators in Hilbert space in the books by Dunford-Schwartz [DS88] and Kato [Kat95]. Soardi’s book [Soa94] on potential theory is also helpful. The space of finite-energy functions (often called Dirichlet or Dirichletsummable functions) on a space is studied widely throughout the literature. In the context of graphs and networks, we recommend Ref. [Soa94] (especially Ch. III), Ref. [LP16] (especially Ch. 9), and Refs. [Kig03, Yam79, Yam77, MYY94, CW92, Woe96, KY89, KY84, KY82], although we first learned about it from Ref. [Kig01] and Ref. [Str06]. Throughout most of this literature, the authors study the grounded energy space, and it is the purpose of Section 2.4.1 to clarify the relations between E(u, v)

and E(u, v) + u(o)v(o)

and hence also between HE = Fin ⊕ Harm

and D = D0 ⊕ M− o .

Remark 2.77. Theorem 2.20, which shows that HE = Fin ⊕ Harm, is often called the “Royden Decomposition” in the literature, in honor of Royden’s analogous result for Riemann surfaces. In many contexts that admit a Laplace operator or suitable analogue, the ensuing decomposition into “finite” and “harmonic” function spaces is typically called the Royden decomposition, even though the actual contributions of Royden are related only in spirit. Note that in Ref. [Soa94, Thm. 3.69] (and see Ref. [LP16, Section 9.3]), the author uses the grounded inner product, and hence, the decomposition D = D0 + HD is not orthogonal. Of course, the energy form E is in Dirichlet form, and the reader seeking more background on the general theory of Dirichlet forms and ¯ probability should see Refs. [FOT94, BH91] and on Dirichlet spaces in potential theory, Refs. [Bre67, CC72]. The best reference for Dirichlet forms in the current context would be Kigami’s treatment of resistance forms in Ref. [Kig03]. However, one should also see Ref. [RS95] and the lovely volume Ref. [JKM+ 98]. For further material on harmonic functions of finite energy, see Ref. [CW92].

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Operator Theory and Analysis of Infinite Networks

Remark 2.78. In Ref. [Kig03], Kigami constructs the Green kernel elex (y) = gB (x, y) using potential-theoretic methods. In this context, ments gB B is a nonempty finite set which is considered as the boundary of a Dirichlet problem. In the case when B = {o}, one has gbx = vx , where vx is an element of the energy kernel, as defined in Definition 2.11. However, the construction we give here is entirely in terms of Riesz duality and the Hilbert space structure of HE , as opposed to the discrete potential theory and was discovered independently. While Refs. [Kig01, Kig03] are often considered to pertain specifically to self-similar fractals, there are large parts of both works which are applicable to discrete potential theory more broadly. In particular, many key properties of the resistance metric and its interrelations with the Laplacian and energy form were first worked out in Ref. [Kig03]. Remark 2.79 (Comparison with Kuramochi kernel). After a first draft of this book was completed, the authors were referred to Ref. [MYY94], in which the authors construct a reproducing kernel very similar to ours, which they call the Kuramochi kernel. Indeed, the Kuramochi kernel element kx corresponds to a representative of vx that takes the value 0 at o. This makes the Kuramochi kernel a reproducing kernel for the space of functions .

D(G; o) := {u ∈ dom E .. u(o) = 0}. As advantages of the current approach, we note that our formulation puts the Green kernel in the same space as the reproducing kernel. This will be helpful in the following; for example, the kernel elements vx and fx = PHarm vx can be decomposed in terms of the Green kernel. See Definition 2.30 and Remark 2.37. The reader will find that we put the energy kernel to very different uses than the Kuramochi kernel.

Chapter 3

The Resistance Metric The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science. — D. Hilbert

We now study a natural notion of distance on (G, c): the resistance metric R. While not as intuitive as the more common shortest-path metric, it reflects the topology of the graph more accurately and may be more useful for modeling and practical applications. The effective resistance is intimately related to the random walk on (G, c), the Laplacian, and the Dirichlet energy form [Kig03, LP16, LPW08, Soa94, Kig01, Str06, DS84]. In Section 3.1, we give several formulations of this metric (Theorem 3.2), each with its own advantages. Many of these are familiar from the literature: (3.1) from Ref. [Pow76b] and Ref. [Per99, Ch. 8], (3.2) from Ref. [DS84], (3.3) from Refs. [DS84, Pow76b], and (3.4) and (3.5) from Refs. [Kig03, Kig01, Str06]. In Section 3.2, we extend these formulations to infinite networks. Due to the possible presence of nontrivial harmonic functions, some care must be taken when adjusting these formulations. It turns out that there are two canonical extensions of the resistance metric to infinite networks which are distinct precisely when Harm = 0 (cf. Ref. [LP16] and the references therein): the “free” resistance and the “wired” resistance. We are able to clarify and explain the difference in terms of the reproducing kernels for HE and Fin and also in terms of Dirichlet versus Neumann boundary conditions; see Remark 3.19. We also explain the discrepancy in terms 53

Operator Theory and Analysis of Infinite Networks

54

of projections in HE and attempt to relate this to conditioning of the random walk on the network; see Section 3.7 and Remark 3.38. Additionally, we introduce trace resistance and harmonic resistance and relate these to the free and wired resistances. (Note: Unlike the others, harmonic resistance is not a metric.) The trace resistance coincides with the free resistance and provides a mechanism to compute it explicitly. Most results in this section appeared in Ref. [JP10a]. 3.1 Resistance Metric on Finite Networks We make the standing assumption that the network is finite in Section 3.1. However, the results actually remain true on any network for which Harm = 0. Definition 3.1. If 1 A of current is inserted into an electrical resistance network at x and withdrawn at y, then the (effective) resistance R(x, y) is the voltage drop between the vertices x and y. Theorem 3.2. The resistance R(x, y) has the following equivalent formulations: .

R(x, y) = distΔ (x, y) := {v(x) − v(y) .. Δv = δx − δy } .

= distE (x, y) := {E(v) .. Δv = δx − δy }

(3.1) (3.2)

.

= distD (x, y) := min{D(I) .. I ∈ F(x, y)} .

= distR (x, y) := 1/ minv∈dom E {E(v) .. v(x) = 1, v(y) = 0}

(3.3) (3.4)

= distκ (x, y) := minv∈dom E {κ ≥ 0 .. |v(x) − v(y)|2 ≤ κE(v)} (3.5) .

= dists (x, y) :=

sup {|v(x) − v(y)|2 .. E(v) ≤ 1}. .

v∈dom E

(3.6)

Proof. (3.1) ⇐⇒ (3.2). We may choose v satisfying Δv = δx − δy by Theorem 1.41. Then,   v(z)Δv(z) = v(z)(δx (z) − δy (z)) = v(x) − v(y), (3.7) E(v) = z∈G0

z∈G0

where the first equality is justified by Theorem 1.13. (3.2) ⇐⇒ (3.3). Note that every v ∈ P(x, y) corresponds to an element I ∈ F (x, y) via Ohm’s law by Lemma 1.30, and E(v) = D(I) by the same lemma. Also, this current flow is minimal by Theorem 1.40.

The Resistance Metric

55

(3.2) ⇐⇒ (3.4). Suppose that Δv = δx − δy . Since E(v + k) = E(v) and Δ(v + k) = Δv for any constant k, we may adjust v by a constant so that v(y) = 0. Define u :=

v − v(x) v(x) − v(y)

so that u(x) = 0 and u(y) = 1. Observe that (3.1) gives E(v) = v(x) − v(y), whence E(u) = E(v)/(v(x) − v(y))2 = (v(x) − v(y))−1 ≥ min E(u). This shows that E(v) ≤ [min E(u)]−1 and hence distE ≤ distR . For the other inequality, suppose u minimizes E(u) subject to u(x) − u(y) = 0. Then, by Theorem 1.13 and the same variational argument as described in Remark 1.33, we have  ρ(z)Δu(z) = 0 E(ρ, u) = z∈G0

for every function ρ for which ρ(x) = ρ(y) = 0. It follows that Δu(z) = 0 for z = x, y and hence Δu = ξδx + ηδy . Observe that E(u) = E(−u) = E(1 − u), and so, the same result follows from minimizing E with respect to the conditions u(y) = 1 and u(x) = 0. This symmetry forces η = −ξ, and we have Δu = ξδx − ξδy . Now, for v = 1ξ u, one has Δv = δx − δy , and so, E(u) = ξ 2 E(v) = ξ 2 (v(x) − v(y)) = ξ(u(x) − u(y)) = ξ, where the second equality follows from (3.1). Then, ξ = whence distE ≥ distR . (3.4) ⇐⇒ (3.5). Starting with (3.5), it is clear that . |v(x)−v(y)|2 E(v)

distκ (x, y) = inf{κ ≥ 0 ..

1 E(v)

= E(u),

≤ κ, v ∈ dom E}

2

= sup{ |v(x)−v(y)| , v ∈ dom E, v nonconstant}. E(v) v Given a nonconstant v ∈ dom E, one can substitute u := |v(x)−v(y)| into the previous line to obtain 2 |v(x)−v(y)| ((((2 , v ∈ dom E, v nonconstant} ( distκ (x, y) = sup{ |u(x)−u(y)| E(u)( |v(x)−v(y)| ((((2 1 = sup{ E(u) , u ∈ dom E, |u(x) − u(y)| = 1}

= 1/ inf{E(u), u ∈ dom E, |u(x) − u(y)| = 1}.

56

Operator Theory and Analysis of Infinite Networks

Since we can always add a constant to u and multiply by ±1 without changing the energy, this is equivalent to letting u range over the subset of dom E for which u(x) = 1 and u(y) = 0, and we have (3.4). (3.5) ⇐⇒ (3.6). It is immediate that (3.5) is equivalent to   |v(x) − v(y)|2 .. . E(v) < ∞ . sup E(v)  For any v ∈ dom E, define w := v/ E(v) so that |w(x) − w(y)|2 = |v(x) − v(y)|2 E(v)−1/2 , with E(w) = 1. Clearly, then, |w(x) − w(y)|2 ≤ dists (x, y). The other inequality is similar.  The equivalence of (3.3) and (3.1) is shown elsewhere (e.g., see Ref. [Pow76b, Ch. II]), but the reader will find some gaps, so we have included a complete version of this proof for completeness. The terminology “effective resistance metric” is common in the literature (see, for example, Refs. [Kig01, Str06]), where it is usually given in the form (3.4). The formulation (3.5) will be helpful for obtaining certain inequalities later. It is also clear that dists of (3.6) is the norm of the operator Lxy defined by Lxy u := u(x) − u(y), see Lemma 2.8 and Theorem 3.12. Remark 3.3. Taking the minimum (rather than the infimum) in (3.3), etc., is justified by Theorem 1.27. The same argument implies that the energy kernel on G is uniquely determined. Remark 3.4 (Resistance distance via network reduction). Let H be a (connected) planar subnetwork of a finite network G and pick any x, y ∈ H. Then, H may be reduced to a trivial network consisting only of these two vertices and a single edge between them via the use of the following three basic transformations: (i) series reduction, (ii) parallel reduction, and (iii) the ∇–Y transform. Each of these transformations preserves the resistance properties of the subnetwork, that is, for x, y ∈ G \ H, R(x, y) remains unchanged when these transformations are applied to H. The effective resistance between x and y may be interpreted as the resistance of the resulting single edge. An elementary example is shown in Figure 3.1. A more sophisticated technique of network reduction is given by the Schur complement construction defined in Remark 3.38. The following result is not new (see, for example, Ref. [Kig01, Section 2.3]), but the proof given here is substantially simpler than most others found in the literature.

The Resistance Metric

57

Fig. 3.1. Effective resistance as network reduction to a trivial network. This basic example uses parallel reduction followed by series reduction; see Remark 3.4.

Lemma 3.5. R is a metric. Proof. Symmetry and positive definiteness are immediate from (3.2), and we use (3.1) to check the triangle inequality. Let v1 ∈ P(x, y) and v2 ∈ P(y, z). By superposition, v3 := v1 + v2 is in P(x, z). For s ∼ t, it is clear that v3 (s) − v3 (t) = v1 (s) − v1 (t) + v2 (s) − v2 (t). By summing along any path from x to z, one sees that this remains true for s ∼ / t, whence R(x, z) = v3 (x) − v3 (z) = v1 (x) − v1 (z) + v2 (x) − v2 (z) ≤ v1 (x) − v1 (y) + v2 (y) − v2 (z) = R(x, y) + R(y, z), where the inequality follows from Proposition 1.44.



3.2 Resistance Metric on Infinite Networks There are difficulties with extending the results of the previous section to infinite networks, but it can be done (and has attracted some interest; see Figure 14.2). The existence of nonconstant harmonic functions h ∈ dom E implies the nonuniqueness of solutions to Δu = f , and hence, (3.1)–(3.3) are no longer well defined. Two natural choices for the extension lead to the free resistance RF and the wired resistance RW . In this section, we attempt to explain the somewhat surprising phenomenon where one may have RW (x, y) < RF (x, y): 1. In Theorem 3.12, we show how RF corresponds to choosing solutions to Δu = δx − δy from the energy kernel and how it corresponds to currents which are decomposable in terms of paths. The latter leads to a probabilistic interpretation, which provides for a relation to the trace of the resistance discussed in Section 3.6.

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2. In Theorem 3.18, we show how RW corresponds to the projection to Fin. Since this corresponds to the minimization of energy, it is naturally related to capacity. See also Remark 3.7. Both of these notions are methods of specifying a unique solution to Δu = f in some way. The disparity between RF and RW is thus explained in terms of boundary conditions on Δ as an unbounded self-adjoint operator on HE in Remark 3.19. For an alternative approach, see Ref. [Kig03], where the author uses a potential-theoretic formulation (axiomatic harmonic analysis) to explain the discrepancy between RF and RW in terms of domains. (This will also be apparent from our approach, see Remark 3.61.) To compute the effective resistance in an infinite network, we need three notions of the subnetwork: free, wired, and trace. Strictly, these may not actually be subnetworks of the original graph; they are networks associated with a full subnetwork. Throughout this section, we use H to denote a finite full subnetwork of G, H 0 to denote its vertex set, and H F , H W , and H tr to denote the free, wired, and trace networks associated with H, respectively (these terms are defined in other sections as follows). Definition 3.6. If H is a subnetwork of G which contains x and y, define RH (x, y) to be the resistance distance from x to y as computed within H. In other words, compute RH (x, y) by any of the equivalent formulas of Theorem 3.2 but extremizing over only those functions whose support is contained in H. We always use the notation {Gk }∞ k=1 to denote an exhaustion of the infinite network G. Recall from Definition 2.4 that this means each Gk is  a finite connected subnetwork of G, Gk ⊆ Gk+1 , and G0 = G0k . Since x and y are contained in all but finitely many Gk , we may always assume that x, y ∈ Gk . Also, we assume in this section that the subnetworks are full — this is not necessary but simplifies the discussion and causes no loss of generality. Definition 3.7. Let H 0 ⊆ G0 . Then, the full subnetwork on H 0 has all the edges of G for which both endpoints lie in H 0 with the same conductances.

3.3 Free Resistance Definition 3.8. For any subset H 0 ⊆ G0 , the free subnetwork H F is just the full subnetwork H. That is, all edges of G with endpoints in H 0 are

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edges of H with the same conductances. Let RH F (x, y) denote the effective resistance between x and y as computed in H = H F , as in Definition 3.6. The free resistance between x and y is then defined to be RF (x, y) := lim RGFk (x, y), k→∞

(3.8)

where {Gk } is any exhaustion of G. Remark 3.9. The name “free” comes from the fact that this formulation is free of any boundary conditions or considerations of the complement of H, in contrast to the wired and trace formulations of the next two sections; see Figure 3.2. See Ref. [LP16, Ch. 9] for further justification of this nomenclature. One can see that RH F (x, y) has the drawback of ignoring the conductivity provided by all paths from x to y that pass through the complement of H. This provides some motivation for the wired and trace approaches in the following.

Fig. 3.2. Comparison of free and wired exhaustions for the example of a binary tree; see Definitions 3.8 and 3.16. Here, the vertices of Gk are all those which lie within k edges (“steps”) of the origin. If the edges of G all have a conductance of 1, then so do all W the edges of each GF k and Gk , except for the edges incident upon ∞k = ∞Gk , which have a conductance of 2.

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Definition 3.10. Fix x, y ∈ G and define the linear operator Lxy on HE by Lxy v := v(x) − v(y). Remark 3.11. Theorem 3.12 is the free extension of Theorem 3.2 to infinite networks; it shows that R(x, y) = Lxy and that R(x, o) is the best possible constant k = kx in Lemma 2.8. In the proof, we use the notation χγ for a current which is the characteristic function of a path, that is, a current which takes a value of 1 on every edge of γ ∈ Γ(x, y) and 0 on  all other edges. Then, I = ξγ χγ indicates that I decomposes as a sum of currents supported on paths in G. Theorem 3.12. For an infinite network G, the free resistance RF (x, y) has the following equivalent formulations: RF (x, y) = (vx (x) − vx (y)) − (vy (x) − vy (y))

(3.9)

= E(vx − vy )

(3.10) .

= min{D(I) .. I ∈ F (x, y) and I =



γ∈Γ(x,y) ξγ γ }

χ

.

(3.11)

= 1/ min{E(u) .. u(x) = 1, u(y) = 0, u ∈ dom E}

(3.12)

= inf{κ ≥ 0 .. |v(x) − v(y)|2 ≤ κE(v), v ∈ dom E}

(3.13)

= sup{|v(x) − v(y)|2 .. E(v) ≤ 1, v ∈ dom E}.

(3.14)

.

.

Proof. To see that (3.10) is equivalent to (3.8), fix any exhaustion of G and note that 1  cst ((vx − vy )(s) − (vx − vy )(t))2 E(vx − vy ) = lim k→∞ 2 s,t∈Gk

= lim RGFk (x, y), k→∞

where the latter equality is from Theorem 3.2. Then, for the equivalence of formulas (3.9) and (3.10), simply compute E(vx − vy ) = vx − vy , vx − vy E = vx , vx E − 2 vx , vy E + vy , vy E and use the fact that vx is R-valued; cf. Lemma 2.27. To see that (3.11) is equivalent to (3.8), fix any exhaustion of G and define   . F (x, y) H := {I ∈ F (x, y) .. I = γ⊆H ξγ χγ }.

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From (3.3), it is clearly true for each Gk that  . RGFk (x, y) = min{D(I) .. I ∈ F(x, y) and I = γ⊆Gk ξγ χγ }.    Since F (x, y) G = k F (x, y) G , formula (3.11) follows. Note that D is a k  quadratic form on the closed convex set F (x, y) G , and hence, it attains its minimum. The equivalence of (3.12) and (3.14) is given by Ref. [Kig01, Thm. 2.3.4]. As for (3.13) and (3.14), they are both clearly equal to Lxy (as described in Remark 3.11) by the definition of operator norm; see Ref. [Rud87, Section 5.3], for example. To show that these are equivalent to RF , as defined in (3.8), define a subspace of HE consisting of those voltages whose induced currents are supported in a finite subnetwork H by F . (3.15) HE H = {u ∈ dom E .. u(x) − u(y) = 0 unless x, y ∈ H}. This is a closed subspace, as it is the intersection of the kernels of a collection of continuous linear functionals Lst , and so, we can let Qk be the projection to this subspace. Then, it is clear that Qk ≤ Qk+1 and that limk→∞ u − Qk u E = 0 for all u ∈ HE , so RGFk (x, y) = Lxy HE |Gk →C = Lxy Qk ,

(3.16)

where the first equality follows from (3.5) (recall that Gk is finite); therefore, L Q RF (x, y) = lim RGFk (x, y) = lim Lxy Qk = lim xy k = Lxy . k→∞

k→∞

k→∞

 In view of the previous result, the free case corresponds to the consideration of only those voltage functions whose induced current can be decomposed as a sum of currents supported on paths in G. The wired case considered in the next section corresponds to considering all voltage functions whose induced current flow satisfies Kirchhoff’s law in the form (1.18); this is clear from comparison of (3.11) to (3.21). See also Remark 3.20. Formula (3.9) turns out to be useful for explicit computations. Explicit formulas for the effective resistance metric on Zd are obtained from (3.9) in Theorem 14.7; compare with Ref. [Soa94, Section V.2]. Remark 3.13. In Theorem 3.12, the proofs that RF is given by (3.11) or (3.13) stem from essentially the same underlying martingale argument.

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In a Hilbert space, a martingale is an increasing sequence of projections {Qk } with the martingale property Qk = Qk Qk+1 . Recall that conditional expectation is a projection. In this context, Doob’s theorem [Doo53] then states that if {fk } ⊆ H is such that fk = Qk fj for any j ≥ k, then the following are equivalent: (i) There is a f ∈ H such that fk = Qk f for all k. (ii) supk fk < ∞. For (3.11), we are actually projecting to subspaces of HD , the Hilbert space of currents introduced as the dissipation space in Chapter 10. In Ref. [LP16, Section 9.1], the free resistance RF (x, y) is defined directly via this approach (and similarly for RW (x, y)). In view of the previous result, the free case corresponds to the consideration of only those voltage functions whose induced current can be decomposed as a sum of currents supported on paths in G. The wired case considered in the next section corresponds to considering all voltages functions whose induced current flow satisfies Kirchhoff’s law (1.20); this is clear from comparison of (3.11) to (3.21). See also Remark 3.20. Formula (3.9) turns out to be useful for explicit computations; we use it to obtain explicit formulas for the effective resistance metric on Zd in Theorem 14.7. The following result is also a special case of Ref. [Kig01, Thm. 2.3.4]. Proposition 3.14. RF (x, y) is a metric. Proof. One has RGFk (x, z) ≤ RGFk (x, y) + RGFk (y, z) for any k, so take the limit.  From Theorem 3.12, it is clear that the triangle inequality also has the formulation E(vx − vz ) ≤ E(vx − vy ) + E(vy − vz ),

∀x, y, z ∈ G0 ,

which is easily shown to be equivalent to vx (z) + vy (z) ≤ vz (z) + vx (y),

∀x, y, z ∈ G0 ,

using the convention vx (o) = 0. The next result is immediate from (3.13) and also appears in Ref. [Kig03].

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63

Corollary 3.15. Every function in HE is H¨ older continuous with exponent 1 with respect to the free resistance. 2 It is known from Ref. [Nel64] that the Gaussian measure of Brownian motion is supported on the space of such functions, and this will be useful later; cf. Remark 7.33 and the beginning of Section 7.2. It is somewhat subtle to determine if R(x, ·) is in HE . 3.4 Wired Resistance Definition 3.16. Given a finite full subnetwork H of G, define the wired subnetwork H W by identifying all vertices in G0 \H 0 to a single, new vertex labeled ∞. Thus, the vertex set of H W is H 0 ∪ {∞}, and the edge set of H W includes all the edges of H, with the same conductances. However, if x ∈ H 0 has a neighbor y ∈ G0 \ H 0 , then H W also includes an edge from x to ∞ with conductance  cx∞ := cxy . (3.17) y∼x, y∈H 

Let RH W (x, y) denote the effective resistance between x and y as computed in H W , as in Definition 3.6. The wired resistance is then defined to be (x, y), RW (x, y) := lim RGW k k→∞

(3.18)

where {Gk } is any exhaustion of G. Remark 3.17. The wired subnetwork is equivalently obtained by “soldering together” (shorting) all vertices of H  , and hence, it follows from Rayleigh’s monotonicity principle that RW (x, y) ≤ RF (x, y); cf. Ref. [DS84, Section 1.4] or Ref. [LP16, Section 2.4]. The reader will see by comparison to Theorem 3.18 that the wired resistance RW is also the effective resistance associated to the resistance form (E, Fin) of Ref. [Kig03]; see Remark 3.61. However, the wired resistance is not related to the “shorted resistance form” of Ref. [Kig03, Ch. 3] (see Prop. 3.6 in particular). The justification for (3.17) is that the identification of vertices in Gk may result in parallel edges. Then, (3.17) corresponds to replacing these parallel edges by a single edge according to the usual formula for resistors in parallel.

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Theorem 3.18. The wired resistance may be computed by any of the following equivalent formulations: .

(3.19)

= min{E(v) .. Δv = δx − δy , v ∈ dom E}

(3.20)

RW (x, y) = min{v(x) − v(y) .. Δv = δx − δy , v ∈ dom E} v

.

v

.

= min{D(I) .. I ∈ F(x, y), D(I) < ∞} I

.

(3.21)

= 1/ min{E(u) .. u(x) = 1, u(y) = 0, u ∈ Fin}

(3.22)

= inf{κ ≥ 0 .. |v(x) − v(y)|2 ≤ κE(v), v ∈ Fin}

(3.23)

= sup{|v(x) − v(y)|2 .. E(v) ≤ 1, v ∈ Fin}.

(3.24)

.

.

Proof. Since (3.23) and (3.24) are both clearly equivalent to the norm of Lxy : Fin → C (where, again, Lxy u = u(x) − u(y), as in Definition 3.10), we begin by equating them to (3.18). From Definition 2.14, we see that W HE H := {u ∈ HE

.. .

spt u ⊆ H}

(3.25)

is a closed subspace of HE . Let Qk be the projection to this subspace. Then, it is clear that Qk ≤ Qk+1 and that limk→∞ PFin u − Qk u E = 0 for all ˜ on G whose u ∈ HE . Each function u on H W corresponds to a function u support is contained in H; simply define ⎧ ⎨u(x), x ∈ H, u ˜(x) = ⎩u(∞H ), x ∈ / H. It is clear that this correspondence is bijective and that RGW (x, y) = Lxy HE |W k G

k

→C

= Lxy Qk ,

where the first equality follows from (3.5) (recall that Gk is finite); therefore, RW (x, y) = lim RGW (x, y) = lim Lxy Qk = Lxy PFin , k k→∞

k→∞

which is equivalent to (3.23). To see that (3.19) is equivalent to (3.20), note that the minimal energy solution to Δu = δx − δy lies in Fin since any two solutions must differ by a harmonic function. Let u be a solution to Δu = δx − δy and define

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65

f = PFin u. Then, f ∈ Fin and Δf = δx − δy imply f 2E =

 z∈G0

f (z)Δf (z) =



f (z)(δx − δy )(z) = f (x) − f (y).

(3.26)

z∈G0

To see (3.19)≤ (3.23), let κ be the optimal constant from (3.23). If u ∈ Fin is the unique solution to Δu = δx − δy , then   |u(x) − u(y)|2 |u(x) − u(y)|2 = u(x) − u(y), κ = sup ≥ E(u) E(u) u∈Fin where the last equality follows from E(u) = u(x) − u(y) by the same computation as in (3.26). For the reverse inequality, note that with Lxy , as just above,

2   2 |u(x) − u(y)|2     = Lxy E(u)u1/2  =  vx − vy , E(u)u1/2  , E(u) E for any u ∈ Fin. Note that Lemma 3.21 allows one to replace vx by fx = PFin vx , whence 

|u(x) − u(y)|2 ≤ E(fx − fy )E E(u)u1/2 = E(fx − fy ) E(u) by Cauchy–Schwarz. The infimum of the left-hand side over nonconstant functions u ∈ Fin gives the optimal κ in (3.23) and thus shows that (3.23) ≤ (3.20). To see (3.20) is equivalent to (3.21), recall that I minimizes D over F (x, y) if and only if I = du for u, which minimizes E over {v ∈ . dom E .. Δv = δx − δy }. Apply this to I = df , where f = PFin u is the minimal energy solution to Δu = δx − δy . The equivalence of (3.22) and (3.24) is directly parallel to the finite case and may also be obtained from Ref. [Kig01, Thm. 2.3.4].  Remark 3.19 (RF vs. RW explained in terms of boundary conditions on Δ). Observe that both spaces F . HE H = {u ∈ HE .. u(x) − u(y) = 0 unless x, y ∈ H} and W HE H = {u ∈ HE

.. .

spt u ⊆ H}

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consist of functions which have no energy outside of H. The difference is that if the complement of H consists of several connected components, then u ∈ HE |F H may take a different constant value on each one; this is not F allowed for elements of HE |W H . Therefore, HE |H corresponds to Neumann boundary conditions, and HE |W H corresponds to Dirichlet boundary conditions. That is, from the proofs of Theorems 3.12 and 3.18, we see that: 1. RH F (x, y) = u(x) − u(y), where u is the solution to Δu = δx − δy with Neumann boundary conditions on H  , and 2. RH W (x, y) = u(x) − u(y), where u is the solution to Δu = δx − δy under Dirichlet boundary conditions on H  . Remark 3.20. While the wired subnetwork takes into account the conductivity due to all paths from x to y (see Remark 3.9), it is overzealous in that it may also include paths from x to y that do not correspond to any path in G (see Remark 3.13). On an infinite network, this leads to current flows in which some of the current travels from x to ∞ and then from ∞ to y. Consider the example in Ref. [Mor03]: Let G be Z with cn,n+1 = 1 for each n. Then, define J by ⎧ ⎨1, n = 1 J(n, n − 1) = ⎩0, n = 1. If a unit current flow from 0 to 1 is defined to be a current satisfying  y∼x I(x, y) = δx − δy , then J is such a flow which “passes through ∞.” Here, J is certainly not of finite energy. However, tweaking the conductances allows one to construct examples where there are finite flows through ∞; see Remark 14.32 for a graphed example of this phenomenon. Theorem 3.21. RW (x, y) is a metric. Proof. This follows from the finite case, exactly as in Theorem 3.14.



3.5 Harmonic Resistance Definition 3.22. For an infinite network (G, c), define the harmonic resistance between x and y as Rha (x, y) := RF (x, y) − RW (x, y).

(3.27)

The next result is immediate upon comparing Theorems 3.12 and 3.18.

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67

Theorem 3.23. With hx = PHarm vx , as in Remark 3.12, the harmonic resistance is equal to Rha (x, y) = (hx (x) − hx (y)) − (hy (x) − hy (y)) = E(hx − hy ) =

(3.28) (3.29)

1 . min{E(v) .. v(x)=1,v(y)=0}

−

1 . min{E(f ) .. f (x)=1,f (y)=0,f ∈Fin}

(3.30) = inf{κ ≥ 0 .. |h(x) − h(y)|2 ≤ κE(h), h ∈ Harm}

(3.31)

= sup{|h(x) − h(y)|2 .. E(h) ≤ 1, h ∈ Harm}.

(3.32)

.

.

Remark 3.24. Note that Rha is not the effective resistance associated with a resistance form, as in Remark 3.61, since (RF5) may fail. If Rha were the effective resistance associated with a resistance form, then Ref. [Kig03, Prop. 2.10] would imply that Rha (x, y) is a metric, but this can be seen to be false by considering basic examples. See Example 14.30, for example. The same remarks also apply to the boundary resistance R∂ (x, y) discussed as follows. Definition 3.25. For an infinite network (G, c), define the boundary resistance between x and y by 1 . (3.33) R∂ (x, y) := W −1 R (x, y) − RF (x, y)−1 Intuitively, some portion of the wired/minimal current from x to y passes through infinity; the quantity R∂ (x, y) gives the voltage drop “across infinity”; see Remark 3.53. From this perspective, infinity is “connected in parallel.” The boundary bd G in Chapter 7 (see also the latter part of Section 8.3) is a more rigorous definition of the set at infinity. Theorem 3.26. The boundary resistance is equal to R∂ (x, y) =

RW (x, y)RF (x, y) . Rha (x, y)

(3.34)

In particular, the resistance across the boundary is infinite if Harm = 0. Proof. From (3.27), one has RF (x, y) = 1/(RW (x, y)−1 − R∂ (x, y)−1 ), which gives 1 1 1 = − E(vx − vy ) E(fx − fy ) R∂ (x, y)

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by Theorems 3.12 and 3.18, and hence, 1 1 1 = − . R∂ (x, y) E(fx − fy ) E(vx − vy ) Now, solving for R∂ gives R∂ (x, y) =

E(fx − fy )E(vx − vy ) , E(hx − hy )

and the conclusion follows from (3.10), (3.20), and (3.29).

(3.35) 

3.6 Trace Resistance A third approach to effective resistance comes from considering subnetworks that take into account the connectivity of the complement of the subnetwork but which do not add anything extra. It is shown in Corollary 3.43 that this trace resistance coincides with free resistance. The name “trace” is due to the fact that this approach comes from considering the trace of the ¯ Dirichlet form E to a subnetwork; see Ref. [FOT94]. Several of the ideas in this section were explored previously in Refs. [Kig01, Kig03, Met97]; see also Ref. [Kig12]. A discussion of the trace resistance and trace subnetworks requires some definitions relating the transition operator (i.e., Markov chain) P to the probability measure P(c) on the space of (infinite) paths in G, which start at a fixed vertex a. Such a path is a sequence of vertices {xn }∞ n=0 , where x0 = a and xn ∼ xn+1 for all n. Recall the path space of Definition 3.30 and the associated measure of Definition 3.32. Definition 3.27. Let Xm be a random variable which denotes the (vertex) location of the random walker at time m. Then, let τx be the hitting time of x, that is, the random variable, which is the expected time at which the walker first reaches x: .

τx := min{m ≥ 0 .. Xm = x}.

(3.36)

More generally, τH is the time at which the walker first reaches the subnetwork H. For a walk starting in H, this gives τH = 0. 3.6.1 The trace subnetwork It is well known that networks {(G, c)} are in bijective correspondence with reversible Markov processes {P}; this is immediate from the detailed balance

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69

equations which follow from the symmetry of the conductance: c(x)p(x, y) = cxy = cyx = c(y)p(y, x). It follows from Δ = c(1 − P) that networks are thus in bijective correspondence with Laplacians if one defines a Laplacian as in (1.4). That is, a Laplacian is a symmetric linear operator which is nonnegative definite, has kernel consisting of the constant functions, and satisfies (Δδx )(y) ≤ 0 for x = y. In other words, every row (and column) of tr(Δ, H) sums up to 0. (This is the negative of the definition of a Laplacian as in Refs. [Kig01, CdV98].) In this section, we exploit the bijection between Laplacians and networks to define the trace subnetwork. For H 0 ⊆ G, the idea is as follows: take the trace to H 0

G ←→ Δ −−−−−−−−−−−−−−→ tr(Δ, H 0 ) ←→ H tr . Definition 3.28. The trace of G to H 0 is the network whose edge data is defined by the trace of Δ to H 0 , which is computed as the Schur complement of the Laplacian of H with respect to G. More precisely, write the Laplacian of G as a matrix in block form, with the rows and columns indexed by vertices, and order the vertices so that those of H appear first:   H A BT , (3.37) Δ= H B D 

where B T is the transpose of B. If (G) := {f : G0 → R}, the corresponding mappings are A : (H) → (H), B : (H) → (H  ),

B T : (H  ) → (H), D : (H  ) → (H  ).

(3.38)

It turns out that the Schur complement tr(Δ, H 0 ) := A − B T D−1 B

(3.39)

is the Laplacian of a subnetwork with vertex set H 0 ; cf. Ref. [Kig01, Section 2.1] and Remark 3.371 . A formula for the conductances (and hence 1 It

will be clear from (3.51) that D −1 always exists in this context, and hence, (3.39) is always well defined. Furthermore, the existence of the trace is given in Ref. [Kig03, Prop. 2.10]; it is known from Ref. [Kig01, Lem. 2.1.5] that D is invertible and negative semidefinite.

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the adjacencies) of the trace is given in Theorem 3.35. Denote this new subnetwork by H tr . If H 0 ⊆ G0 is finite, then for x, y ∈ H, the trace of the resistance on H is denoted by RH tr (x, y) and defined as in Definition 3.1. The trace resistance is then defined to be Rtr (x, y) := lim RGtr (x, y), k

(3.40)

k→∞

where {Gk } is any exhaustion of G. Remark 3.29. The name “trace” is due to the fact that this approach comes from considering the trace of the Dirichlet form E on a subnetwork; ¯ see Ref. [FOT94]. Recall that Δ = c − T = c(I − P), where T is the transfer operator and P = c−1 T is the probabilistic transition operator defined pointwise by Pu(x) =



p(x, y)u(y) for p(x, y) =

y∼x

cxy . c(x)

(3.41)

The function p(x, y) gives the transition probabilities, i.e., the probability that a random walker currently at x will move to y with the next step. Since c(x)p(x, y) = c(y)p(y, x),

(3.42)

the transition operator P determines a reversible Markov process with state space G0 ; see Refs. [LPW08, LP16]. Note that the harmonic functions (i.e., Δh = 0) are precisely the fixed points of P (i.e., Ph = h). The proof of the next theorem requires a couple more definitions which relate P to the probability measure P(c) on the space of paths in G. Recall from Definition 1.4 that a path is a sequence of vertices {xn }∞ n=0 , where x0 = a and xn ∼ xn+1 for all n. Definition 3.30. Let Γ(a) denote the space of all paths γ beginning at the vertex a ∈ G0 . Then, Γ(a, b) ⊆ Γ(a) consists of those paths that reach b and before returning to a: .

Γ(a, b) := {γ ∈ Γ(a) .. b = xn for some n, and xk = a, 1 ≤ k ≤ n}. (3.43)

The Resistance Metric

If a, b ∈ bd H, then we write  . Γ(a, b) H  := {γ ∈ Γ(a, b) .. xi ∈ H  , 0 < i < τb }

71

(3.44)

for the set of paths from a to b that do not pass through any vertex in H 0 . Remark 3.31. Note that if x, y ∈ bd H are  adjacent, then any path of the form γ = (x, y, . . . ) is trivially in Γ(a, b) H  . Definition 3.32. The space Γ(a) carries a natural probability measure P(c) defined by  P(c) (γ) := p(xi−1 , xi ), (3.45) xi ∈γ

where p(x, y) is as in (3.41). The construction of P(c) comes from applying Kolmogorov consistency to the natural cylinder set Borel topology that makes Γ(a) into a compact Hausdorff space; cf. Chapter 11 for further discussion. Definition 3.33. Let P[a → b] denote the probability that a random walk started at a will reach b before returning to a. That is, P[a → b] := P(c) (Γ(a, b)).

(3.46)

Note that this is equivalent to P[a → b] = Pa [τb < τa ] := P[τb < τa | x0 = a],

(3.47)

where τa is the hitting time of a, i.e., the expected time of the first visit to a after leaving the starting point. If a, b ∈ bd H, then we write     (3.48) P[a → b] H  := P(c) Γ(a, b) H  , that is, the probability that a random walk started at a will reach b via a path lying outside H (except for the start and end points, of course). Remark 3.34 (A more probabilistic notation). The formulation in (3.48) is conditioning P(c) (Γ(a, b)) on avoiding H; the notation is intended to evoke something like “P[a → b | γ ⊆ H  ].” However, this would not be correct because a, b ∈ H and γ may pass through H after τb . In Theorem 3.35, we use the following common notation as in Ref. [Spi76] or Ref. [Woe00], for example. All notations are for the random

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walk starting at x: Pn (x, y) = p(n) (x, y) = Px [Xn = y] prob. the walk is at y after n steps,  (n) G(x, y) = ∞ (x, y) exp. number of visits to y, n=0 p f (n) (x, y) = Px [τy = n] prob. the walk first reaches y

F (x, y) =

∞

n=0 f

on the nth step, (n)

(x, y) prob. the walk ever reaches y.

Note that if the walk is killed when it reaches y, then p(n) (x, y) = f (n) (x, y) because the first time it reaches y is the only time it reaches y. Therefore, when the walk is conditioned to end upon reaching a set S, one has G(x, y) = F (x, y) for all y ∈ S. Theorem 3.35. For H 0 ⊆ G0 , the conductances in the trace subnetwork H tr are given by   ctr (3.49) xy = cxy + c(x)P[x → y] H  . Consequently, the transition probabilities in the trace subnetwork are given by  ptr (x, y) = p(x, y) + P[x → y] H  . (3.50) Proof. Using subscripts to indicate the block decomposition corresponding to H and H  as in (3.37), the Laplacian may be written as     −cA PB T cA (1 − PA ) H cA Δ= . for c = cD H −cD PB cD (1 − PD ) 

Then, the Schur complement is tr(Δ, H) = cA − cA PA − cA PB T (I − PD )−1 c−1 D cD PB  ∞    n = cA − cA PA + PB T PD PB n=0

= cA (I − PX ).

(3.51)

Note that PD is substochastic, and hence, the RW has positive probability of hitting bd Gk , whose vertices act as absorbing states. This means that the expected number of visits to any vertex in H  is finite, and hence, the matrix PX has finite entries.

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Meanwhile, using PA (x, y) to denote the (x, y)th entry of the matrix PA + and τH as in Definition 3.33, we have     P[x → y] H  = P(c) Γ(x, y) H  ∞   (c) {γ ∈ Γ(x, y) H  =P

 .. .

+ τH

= k}

k=1

 = P(c) {γ ∈ Γ(x, y) H  +

∞ 

.. .

+ τH = 1}

 P(c) {γ ∈ Γ(x, y) H 

.. .



+ τH = k}

(3.52) 

k=2

= PA (x, y) +

∞  

PB T (x, s)PnD (s, t)PB (t, y)

(3.53)

n=0 s,t

= PX (x, y). To justify (3.53), note that by (3.38), PnD corresponds to the steps taken in H  . Therefore,  ∞    n PB T PD PB (x, y) = PB T P0D PB (x, y) + PB T P1D PB (x, y) + · · · n=0

is the probability of the random walk taking a path that steps from x ∈ H to H  , meanders through H  for any finite number of steps, and finally steps into y ∈ H. Since y ∈ / H , PB T PkD PB (x, y) = Px [Xk+2 = y] = Px [τy = k + 2] because the walk can only reach y on the last step, as in Remark 3.34. It follows from classical theory (see Ref. [Spi76], for example) that the sum in (3.53) is a probability (as opposed to an expectation, etc.) and justifies the probabilistic notation PX in (3.51). Note that PA (x, y) corresponds to the one-step path from x to y, which is trivially in Γ(x, y) H  by (3.44). Since PA (x, y) = p(x, y) = cxy /c(x), the desired conclusion (3.49) follows from combining (3.51), (3.53), and (3.48). Of course, (3.50) follows immediately by dividing through out by c(x).  The authors are grateful to Jun Kigami for helpful conversations and guidance regarding the proof of Theorem 3.35.

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Remark 3.36. It is clear from (3.49) that the edge sets of int H and int H tr are identical, but the conductance between two vertices x, y ∈ bd H tr is greater iff there is a path from x to y that does not pass through H. Indeed, if there is a path from x to y which lies entirely in H  except for the endpoints, then x and y will be adjacent in H tr , even if they were not adjacent in H. Remark 3.37 (The trace construction is valid for general subsets of vertices). While Definition 3.28 applies to a (connected) subnetwork of G, it is essential to note that Theorem 3.35 applies to arbitrary subsets H 0 of G0 . It is clear from (3.49) that the edge sets of int H and int H tr are identical, but the conductance between two vertices x, y ∈ bd H tr is greater iff there is a path from x to y that does not pass through H. Indeed, if there is a path from x to y which lies entirely in H  except for the endpoints, then x and y will be adjacent in H tr , even if they were not adjacent in H. Remark 3.38 (Resistance distance via Schur complement). A theorem by Epifanov states that every finite planar network with vertices x, y can be reduced to a single equivalent conductor via the use of three simple transformations: parallel, series, and ∇–Y; cf. Refs. [Epi66, Tru89] as well as Ref. [LP16, Section 2.3] and Ref. [CdV98, Section 7.4]. More precisely: (1)

(2)

(i) Parallel: Two conductors cxy and cxy connected in parallel can be (1) (2) replaced by a single conductor cxy = cxy + cxy . (ii) Series: If z has only the neighbors x and y, then z may be removed from the network and the edges cxz and cyz should be replaced by a −1 −1 single edge cxy = (c−1 . xz + cyz ) (iii) ∇–Y: Let t be a vertex whose only neighbors are x, y, z. Then, this “Y” may be replaced by a triangle (“∇”) which does not include t with conductances: cxy =

cxt cty , c(t)

cyz =

cyt ctz , c(t)

cxz =

cxt ctz . c(t)

This transformation may also be inverted to replace a ∇ with a Y and introduce a new vertex. It is a fun exercise to obtain the series and ∇–Y formulas by applying the Schur complement technique to remove a single vertex of degree 2 or 3 from

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a network. Indeed, these are both special cases of the following: Let t be a vertex of degree n, and let H be the (star-shaped) subnetwork consisting only of t and its neighbors. If we write the Laplacian for just this subnetwork with the tth row and column last, then ⎤ ⎡ 0 −cx1 t cx 1 t · · · ⎢ .. . . .. .. ⎥ ⎥ ⎢ . ⎢ . . . ⎥, Δ|H = ⎢ ⎥ ⎢ 0 ··· cxn t −cxnt ⎥ ⎦ ⎣ −cx1 t · · · −cxn t c(t) and the Schur complement is ⎡ cx 1 t · · · ⎢ . . tr(Δ|H , H \ {t}) = ⎢ ⎣ .. . . 0

⎤ ⎤ ⎡ cx 1 t 0 ⎥! " 1 ⎢ .. ⎥ ⎥ ⎢ . ⎥ . ⎦ − c(t) ⎣ .. ⎦ cx1 t · · · cxn t , · · · cx n t cx n t

whence the new conductance from xi to xj is given by cxi t ctxj /c(t). It is interesting to note that the operator being subtracted corresponds to the projection to the rank-one subspace spanned by the probabilities of leaving t: ⎡ ⎤ cx 1 t ⎥! " 1 ⎢ ⎢ .. ⎥ cx t · · · cx t = c(t)|v v| 1 n . ⎣ ⎦ c(t) cx n t using Dirac’s ket–bra notation for the projection to a rank-1 subspace spanned by v, where " ! v = p(t, x1 ) · · · p(t, xn ) . In fact, |v v| = PX in the notation of (3.51). In general, the trace construction (Schur complement) has the effect of probabilistically projecting away the complement of the subnetwork. In Remark 3.4, we described how the effective resistance can be interpreted as the correct resistance for a single edge which replaces a subnetwork. The following corollary of Theorem 3.35 formalizes this interpretation by exploiting the fact that the Schur complement construction is viable for arbitrary subsets of vertices; see Remark 3.37. In this case, 0 one takes ! 1 the "trace of the (typically disconnected) subset {x, y} ⊆ G ; note −1 that −1 1 is the Laplacian of the trivial two-vertex network when the edge between them has unit conductance. The following result is also an extension of Ref. [Kig01, (2.1.4)] to infinite networks.

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76

Corollary 3.39. Let H 0 = {x, y} be any two vertices of a transient network G. Then, the trace resistance can be computed via   1 1 −1 tr(Δ, H) = tr (3.54) = A − B T D−1 B. 1 R (x, y) −1 Proof. Take H = {x, y} in Theorem 3.35. As discussed in Remark 3.37, it is not necessary to have x ∼ y. Note that in this case,  (PB T n PnD PB ) (x, y) corresponds to all paths from x to y that consist of more than one step:   ∞    n  P[x → y]  = PA (x, y) + PB T PD PB (x, y) = p(x, y) + P(γ). H

n=0

|γ|≥2

(3.55)  Corollary 3.40. The trace resistance Rtr (x, y) is given by Rtr (x, y) =

1 . c(x)P[x → y]

(3.56)

Proof. Again, take H 0 = {x, y}. Then,  S  Rtr (x, y)−1 = cH xy = cxy + c(x)P[x → y] H     = c(x) p(x, y) + P[x → y] H  = c(x)P[x → y], where Corollary 3.39 gives the first equality and Theorem 3.35 gives the second.  Remark 3.41 (Effective resistance as “path integral”). Corollary 3.40 may also be obtained by the more elegant (and much shorter) approach of Ref. [LP16, Section 2.2], where it is stated as follows: The mean number of times a random walk visits a before reaching b is P[a → b]−1 = c(a)R(a, b). We give the current proof to highlight and explain the underlying role of the Schur complement with respect to network reduction; see Remarks 3.37 and 3.38. A key point of the current approach is to emphasize the expression of effective resistance R(a, b) in terms of a sum over all possible paths from a to b. By Remark 3.20, it is apparent that this “path-integral” interpretation makes Rtr much more closely related to RF than to RW , as seen by the following results.

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Corollary 3.42 (Ref. [Kig01, Prop. 2.1.11]). Let H1 ⊆ H2 be finite subnetworks of a transient network G. Then, for a, b ∈ H20 , one has F F (a, b) = RH (a, b). RH 1 2 Corollary 3.43. On any network, Rtr (a, b) = RF (a, b). Proof. By Corollary 3.42, it is clear that RGtr (a, b) = RGtr (a, b) for all k. k k+1 Meanwhile, any path from a to b will lie in Gk for sufficiently large k, so it is F (a, b)}∞ clear that by Theorem 3.40, the sequence {RG k=0 is monotonically k F tr  decreasing with limit R (a, b) = R (a, b). Remark 3.44. Writing [x → y | γ ⊆ H] to indicate a restriction to paths from x to y that lie entirely in H, as in Remark 3.34, one has (x, y) = RGtr k ≤

1 c(x) (P[x → y | γ ⊆ Gk ] + P[x → y | γ  Gk ]) 1 = RGFk (x, y). c(x)P[x → y | γ ⊆ Gk ]

Essentially, Corollary 3.42 is an expression of the first equality, and Corollary 3.43 is a consequence of the inequality and how it tends to an equality as k → ∞.

3.6.2 The shorted operator It is worth noting that the operator D defined in (3.38) is always invertible as in the discussion following (3.51). However, the Schur complement construction is valid more generally. As is pointed out in Ref. [BM88], the shorted operator generalizes the Schur complement construction to positive operators on a (typically infinite dimensional) Hilbert space H; see Refs. [And71, AT75, Kre47]. In general, let T = T  be a positive operator, so ϕ, T ϕ ≥ 0 for all ϕ ∈ H, and let S be a closed subspace of H. Partition T analogously to (3.38) so that A : S → S, B : S → S  , B T : S  → S, and D : S → S. Theorem 3.45 ([AT75]). With respect to the usual ordering of self-adjoint operators, there exists a unique operator Sh(T ) such that     L 0 . Sh(T ) = sup L ≥ 0 .. ≤T , 0 0 L

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and it is given by Sh(T ) = lim

ε→0+

  A − B T (D + ε)−1 B .

In particular, the shorted operator coincides with the Schur complement whenever the latter exists. There is another characterization of the shorted operator due to Ref. [BM88]. satisfying Theorem 3.46 ([BM88]). Suppose {ψn } ⊆ H is a! sequence ϕ " θ ]. Then, = [ some M ∈ R, and lim T

ψn , Dψn  ≤ M for n→∞ ψ 0 n   Sh(T )ϕ = limn→∞ Aϕ + B T ψn . 3.7 Projections in Hilbert Space and the Conditioning of the Random Walk In Remark 3.19, we gave an operator-theoretic account of the difference between RF and RW . The foregoing probabilistic discussions might lead one to wonder if there is a probabilistic counterpart. An alternative approach is given in Ref. [Kig03, App. B]. On a finite network, it is well known that vx = R(o, x)ux ,

(3.57)

where ux (y) is the probability that a random walker (RW) starting at y reaches x before o: ux (y) := Py [τx < τo ].

(3.58)

Here again, τx denotes the hitting time of x, as in Definition 3.27. vx Note that (3.2) gives ux = E(v . The relationship (3.57) is discussed in x) Refs. [DS84, LPW08, LP16]. Theorem 3.47 is a wired extension of (3.57) to transient networks. The corresponding free version appears in Conjecture 3.48. Theorem 3.47. On a transient network, let fx be the representative of PFin vx specified by fx (o) = 0. Then, for x = o, fx is computed probabilistically by # (3.59) fx (y) = RW (o, x) Py [τx < τo ] $ GW c(x) G G k + Py [τo = τx = ∞]Px [τo = ∞] lim c(∞k ) P∞k [τ∞k < τ{x,o} ] . k→∞

(3.60)

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Proof. Fix x, y and an exhaustion {Gk }∞ k=1 , and suppose without loss of generality that o, x, y ∈ G1 . Since vx = fx on any finite network, the (k) (k) (k) (o, x)ˇ ux , where fx is the unique solution identity (3.57) gives fx = RGW k to Δv = δx − δo on the finite (wired) subnetwork GW k , and GW

k u ˇ(k) x (y) := Py [τx < τo ],

where the superscript indicates the network in which the random walk (k) travels. As in the previous case, we just need to check the limit of u ˇx , for which we have GW

GW

k k uˇ(k) x (y) = Py [τx < τo & τx < τ∞k ] + Py [τx < τo & τx > τ∞k ]. (3.61)

The first probability in (3.61) is GW

Py k [τx < τo & τx < τ∞k ] = PG y [τx < τo & τx < τG ] k

k→∞

PG y [τx

−−−−−→

< τo & τx < ∞] = PG y [τx < τo ],

where the last equality follows because τx < τo implies τx < ∞. The latter probability in (3.61) measures the set of paths which travel from y to ∞k without hitting x or o and then to x without passing through o and hence can be rewritten as GW

GW

GW

Py k [τ∞k < τx < τo ] = Py k [τ∞k < τ{o,x} ]P∞kk [τx < τo ]

W GW G = Py k [τ∞k < τ{o,x} ] P∞kk [τ∞k < τ{x,o} ] GW

GW



× P∞kk [τx < τ{∞k ,o} ] + P∞kk [τx < τ{∞k ,o} ]

since a walk starting at ∞k may or may not return to ∞k before reaching x. First, consider only those walks which do not loop back through ∞k (i.e., multiply out the above expression and take the second term) to observe GW

GW

Py k [τ∞k < τ{o,x} ]P∞kk [τx < τ{∞k ,o} ] GW

GW

c(x) = Py k [τ∞k < τ{o,x} ]Px k [τ∞k < τo ] c(∞ (3.62) k)



 GW GW c(x) = 1 − Py k [τ{o,x} < τ∞k ] 1 − Px k [τo < τ∞k ] c(∞ k)

  c(x) = 1 − PG 1 − PG y [τ{o,x} < τGk ] x [τo < τGk ] c(∞k )     k→∞ c(x) −−−−−→ 1 − PG 1 − PG y [τ{o,x} < ∞] x [τo < ∞] lim c(∞k ) k→∞

=

PG y [τo

= τx =

∞]PG x [τo

=

c(x) ∞] lim c(∞ . k) k→∞

(3.63)

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80

Note that (3.62) comes from the reversibility of the walk and the way probability is computed for paths from ∞k to x, which avoid o and ∞ ∞k . Since the network is transient, k=1 c(∞k )−1 is summable by Nash– c(x) = 0 causes (3.63) to vanish. William’s criterion, and so, limk→∞ c(∞ k) Now, for walks which do loop back through ∞k , the same arguments as above yield GW

GW

GW

Py k [τ∞k < τ{o,x} ]P∞kk [τ∞k < τ{x,o} ]P∞kk [τx < τ{∞k ,o} ] GW c(x) P∞kk [τ∞k c(∞ ) k k→∞

k→∞

G −−−−−→ PG y [τo = τx = ∞]Px [τo = ∞] lim

< τ{x,o} ], 

and the conclusion follows.

The following conjecture expresses a free extension of (3.57) to infinite networks. We offer an erroneous “proof” in the hope that it may inspire the reader to find a correct proof. The error is discussed in Remark 3.49. In the statement of Conjecture 3.48, we use the notation |γ| < ∞

(3.64)

to denote the event that the walk is bounded, i.e., the trajectory is contained in a finite subnetwork of G. Conjecture 3.48. On an infinite resistance network, let vx be the representative of an element of the energy kernel specified by vx (o) = 0. Then, for x = o, vx is computed probabilistically by vx (y) = RF (o, x)Py [τx < τo | |γ| < ∞],

(3.65)

that is, the walk is conditioned to lie entirely in some finite subnetwork as in (3.64). Proof. [“Nonproof”] Fix x, y and suppose without loss of generality that (k) (k) o, x, y ∈ G1 . One can write (3.57) on Gk as vx = RGFk (o, x)ux . In other (k)

words, vx is an unique solution to Δv = δx − δo on the finite subnetwork F (x, y) by (3.8), it only remains to check GF k . Since R (x, y) = limk→∞ RGF k (k)

the limit of ux . Using a superscript to indicate the network in which the random walk travels, we have GF

G F k lim u(k) x (y) = lim Py [τx < τo ] = lim Py [τx < τo | γ ⊆ Gk ].

k→∞

k→∞

k→∞

(3.66)

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81

Here again, the notation [γ ⊆ H] denotes the event that the random walk never leaves the subnetwork H, i.e., τH  = ∞. The events [γ ⊆ GF k ] are nested and increasing, so the limit is the union, and (3.65) follows. Note F  that GF k is recurrent, so γ ⊆ Gk implies τx < ∞. Remark 3.49. As indicated, the argument outlined above is incomplete due to the second equality of (3.66). While the set of paths from y to x in GF k is the same as the set of paths from y to x in G which lie in Gk , the probability of a given path may differ when computed in a network or the other. This happens precisely when γ passes through a boundary point: The transition probability away from a point in bd Gk is strictly larger in GF k than it is in Gk . 3.8 Comparison of Resistance Metric to Other Metrics 3.8.1 Comparison to geodesic metric On a Riemannian manifold (Ω , g), the geodesic distance is % distγ (x, y) := inf γ

0

1

 . g(γ  (t), γ  (t))1/2 dt .. γ(0) = x, γ(1) = y, γ ∈ C 1 .

Definition 3.50. On (G, c), the geodesic distance from x to y is .

distγ (x, y) := inf{r(γ) .. γ ∈ Γ(x, y)}, where r(γ) :=



(3.67)

 −1 (for resistors in series, the total resistance is c (x,y)∈γ xy

the sum). Remark 3.51. Definition 3.50 differs from the definition of the shortest path metric found in the literature on general graph theory; without weights on the edges, one usually defines the shortest path metric simply as the minimal number of edges in a path from x to y. (This corresponds to taking c = 1.) Such shortest paths always exist. According to Definition 3.50, shortest paths may not exist (cf. Example 13.11). Of course, even when they do exist, they are not always unique. It should be observed that the effective resistance is not a geodesic metric in the usual sense of metric geometry; it does not correspond to a length structure in the sense of Ref. [BBI01, Chapter 2].

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Lemma 3.52. The effective resistance is bounded above by the geodesic distance. More precisely, RF (x, y) ≤ distγ (x, y) with equality for all x, y if and only if G is a tree. Proof. If there is a second path, then some positive amount of current will pass along it (i.e., there is a positive probability of getting to y via this route). To make this precise, let v = vx − vy and let γ = (x = x0 , x1 , . . . , xn = y) be any path from x to y: RF (x, y)2 = |v(x) − v(y)|2 ≤ r(γ)E(v) by the exact same computation as in the proof of Lemma 2.8 but with u = v. The desired inequality then follows by dividing both sides by E(v) = RF (x, y). The other claim follows by observing that trees are characterized by the property of having exactly one path γ between any x and y in G0 . By (3.11), RF (x, y) can be found by computing the dissipation of the unit current which runs entirely along γ from x to y. This means that I(xi−1 , xi ) = 1 on γ, and I = 0 elsewhere, so RF (x, y) = D(I) =

n 

1

i=1

cxi−1 xi

I(xi−1 , xi )2 =

n  i=1

1 = r(γ). cxi−1 xi



This type of inequality is explicitly calculated in Example 13.3. Remark 3.53. It is clear from the end of the proof of Lemma 3.52 that on a tree, vx − vy is locally constant on the complement of the unique path from x to y. However, this may not hold for fx − fy , where fx = PFin vx ; see Example 15.2. This is an example of how the wired resistance can “cheat” by considering currents which take a shortcut through infinity; compare (3.11) with (3.21). 3.8.2 Comparison with Connes’ metric The formulation of R(x, y) given in (3.1) may evoke Connes’ maxim that a metric can be considered as the inverse of a Dirac operator; Ref. cf. [Con94]. This does not appear to have a literal incarnation in the current context, but we do have the inequality of Lemma 3.54 in the case when c = 1. In this formulation, v ∈ HE is considered as a multiplication operator defined

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83

on u by (vu)(x) := v(x)u(x),

∀x ∈ G0 ,

(3.68)

and both v and Δ are considered as operators on 2 (G0 ∩ dom E. We use the commutator notation [v, Δ] := vΔ − Δv, and [v, Δ] is understood as the usual operator norm on 2 . Lemma 3.54. If c = 1, then for all x, y ∈ G0 , one has √ . R(x, y) ≤ sup{|v(x) − v(y)|2 .. [v, Δ] ≤ 2, v ∈ dom E}.

(3.69)

Proof. We compare (3.69) with (3.6). Writing Mv for multiplication by v, it is straightforward to compute from the definitions,  (Mv Δ − ΔMv )u(x) = (v(y) − v(x))u(y) y∼x

so that the Schwarz inequality gives 2       2 [Mv , Δ]u 2 = (v(y) − v(x))u(y)    x∈G0 y∼x       ≤ |v(y) − v(x)|2 |u(y)|2 . x∈G0

y∼x

y∼x

By extending the sum of |u(x)|2 to all x ∈ G0 (an admittedly crude estimate), this gives [v, Δ]u 22 ≤ 2 u 22E(v), and hence, [v, Δ] 2 ≤ 2E(v).  3.9 Generalized Resistance Metrics In this section, we describe a notion of effective resistance between probability measures, of which R(x, y) (or RF and RW ) is a special case. This concept is closely related to the notion of total variation of measures and hence is related to mixing times of Markov chains; cf. Ref. [LPW08, Section 4.1]. When the Markov chain is taken to be a random walk on an ERN, the state space is just the vertices of G. Definition 3.55. Let μ and ν be two probability measures on G0 . Then, the total variation distance between them is distTV (μ, ν) := 2 sup |μ(A) − ν(A)|. A⊆G0

(3.70)

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Proposition 3.56 ( [LPW08, Prop. 4.5]). Let μ and ν be two probability measures on the state space Ω of a (discrete) Markov chain. Then, the total variation distance between them is  & '      .. u(x)μ(x) − u(x)ν(x) . u ∞ ≤ 1 . (3.71) distTV (μ, ν) = sup    x∈Ω

x∈Ω

Here, u ∞ := supx∈G0 |u(x)|.

3.9.1 Effective resistance between measures If we think of μ as a linear functional acting on the space of bounded functions, then it is clear that (3.71) expresses distTV (μ, ν) as the operator norm μ − ν . That is, it expresses the pairing between μ ∈ 1 and u ∈ ∞ . We can therefore extend RF directly (see (3.13) and (3.14) and Remark 3.11). Definition 3.57. The free resistance between two probability measures is ⎧ ⎫ 2  ⎨  ⎬   .  distRF (μ, ν) := sup  u(x)μ(x) − u(x)ν(x) .. u E ≤ 1 .  ⎩ ⎭ 0 0 x∈G

x∈G

(3.72) It is clear from this definition (and Remark 3.11) that RF (x, y) = distRF (δx , δy ). This extension of RF to measures was motivated by a question by Rieffel in Ref. [Rie99]. Since then, Rieffel has developed a noncommutative theory of effective resistance in Ref. [Rie14]. 3.9.2 Total variation spaces Definition 3.58. Since dom E is a Banach space, we may define a new pairing via the bilinear form  u(x)μ(x), (3.73)

u, μTV := x∈G0

where μ is an element of TV := {μ : G0 → R .. ∃kμ s.t.| u, μTV | ≤ kμ · E(u)1/2 , ∀u ∈ dom E}. (3.74) .

The Resistance Metric

85

Then, TV = dom u, ·TV is the dual of dom E with respect to the total variation topology induced by (3.73). Also, the norm in TV is given by μ TV := inf{k .. | u, μTV | ≤ k · E(u)1/2 , ∀u ∈ dom E}. .

(3.75)

Remark 3.59. Since TV is a Banach space, which is the dual of a normed space, the unit ball .

{μ ∈ TV .. μ TV ≤ 1}

(3.76)

is compact in the weak- topology by Alaoglu’s theorem. Lemma 3.60. The Laplacian Δ maps HE into TV with Δv TV ≤ PFin v E . Proof. For u, v ∈ HE , write v = f + h with f = PFin v and h = PHarm v so that                    u(x)Δv(x) ≤  u(x)Δf (x) +  u(x)Δh(x)           0 x∈G0 x∈G0 x∈G   = | u, f E | ≤ u E f E , by Theorem 2.76 followed by the Schwarz inequality. The mapping is contractive relative to the respective norms because v E is an element of the set on the right-hand side of (3.75) and hence at least as big as the  infimum, whence Δv TV ≤ f E ≤ v E . 3.10 Remarks and References A key reference for this chapter is Ref. [Kig03]; the relationship between the free and wired resistances can be elegantly phrased in terms of resistance forms, as we describe in the following remark. Additionally, the role of the trace resistance is apparent in Kigami’s work [Kig01, Kig03, Kig95, Kig94, Kig93]. However, the potential-theoretic approach can be intimidating to the uninitiated, and we hope that our treatment of effective resistance from the first principles of Hilbert space theory will provide a gentle introduction as well as new insights. As an example of this, we feel that the probabilistic proof of Theorem 3.35 (to which we are indebted to Jun Kigami) offers insight as to why the operation of Schur complement should correspond to taking the trace.

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After Powers’ papers in the 1970s (starting with Ref. [Pow76b]), there has been an explosive interest in metric geometry, potential theory [Bre67], spectral theory [Chu96], and harmonic analysis [Car73a] on infinite graphs. As illustrated in Ref. [Kig03], a good deal of the motivation arose from a parallel research track dealing with analysis of fractals. In addition, some of the early work was motivated by problems in statistical mechanics (see, for example, Refs. [Rue69, Rue04] on thermodynamic formalism). Remark 3.61 (Comparison with resistance forms). In Ref. [Kig03, Def. 2.8], a resistance form is defined as follows: Let X be a set and let E be a symmetric quadratic form on (X), the space of all functions on X, and let F denote the domain of E. Then, (E, F ) is a resistance form iff: (RF1) F is a linear subspace of (X) containing the constant functions, and E is nonnegative on F with E(u) = 0 iff u is constant. (RF2) F / ∼ is a Hilbert with inner product E, where ∼ is the equivalence relation defined on F by u ∼ v iff u − v is constant. (RF3) For any finite  subset V ⊆ U and for any v ∈ (V ), there is u ∈ F such that u V = v. (RF4) For any p, q ∈ X, the effective resistance associated to the form (E, F ) is the (finite) number + , 2 . .. u ∈ F , E(u) > 0 . (3.77) RE,F (p, q) := sup |u(p)−u(q)| E(u) (RF5) If u ∈ F , then u defined by u(x) := min{1, max{0, u(x)}} (the unit normal contraction of u in the language of Dirichlet forms) is also in F . Upon comparison of (3.12) and (3.13) with (3.22)–(3.24), one can see that RF is the effective resistance associated with the resistance form (E, HE ) and that RW is the effective resistance associated with the resistance form (E, Fin). We are grateful to Jun Kigami for pointing this out to us. Note that the wired resistance is not related to the “shorted resistance form” of Ref. [Kig03, Chapter 3] (see Prop. 3.6 in particular). See also Remark 3.24. The reader will also find Ref. [Soa94] to be a good reference for effective resistance. While Refs. [Soa94, DS84, LP16, LPW08, Per99] do not especially emphasize the metric aspect of effective resistance, they provide an exceptional description of the relationship between effective resistance and random walks. The books Refs. [Kig01] and [Str06] are also useful for

The Resistance Metric

87

understanding connections between effective resistance and the energy form and Laplace operator on graphs and on self-similar fractals. For different formulations of the effective resistance appearing in the literature, see Ref. [Pow76b] and Ref. [Per99, Ch. 8] for (3.1), Ref. [DS84] for (3.2), Refs. [DS84, Pow76b] for (3.3), and Refs. [Kig03, Kig01, Str06] for (3.4) and (3.5). For investigations of the “limit current” (corresponding to free resistance) and “minimal current” (corresponding to wired resistance), one should consult Ref. [Soa94] and earlier sources [Fla71, Tho90, Zem76]. The role of effective resistance in combinatorics (dimer configurations, percolation on finite sets, etc.) is discussed in Refs. [BK05, Rie99, KW09]. The role of Schur complement in the trace of a resistance form appears in Ref. [Kig03] and also less specifically in Refs. [Met97, KW09, BM88, AT75, And71], where it is sometimes called the shorted operator. Also, see Ref. [KdZLR08] for the role of Schur complement in regard to Dirichlet-toNeumann maps. Remark 3.62 (Open problem). Prove Conjecture 3.48.

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Chapter 4

Schoenberg–von Neumann Construction of the Energy Space HE If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. — John von Neumann

In Section 2.1, we showed that HE is a (relative) reproducing kernel Hilbert space with reproducing kernel {vx }x∈G0 and how HE can therefore be constructed as a reproducing kernel Hilbert space (see Remark 2.10). In this section, we describe how HE can be constructed as a Hilbert space from the metric space structure of G under an effective resistance metric. Studying the geometry of state space X through vector spaces of functions on X is a fundamental idea, and its variations can be traced back to several areas of mathematics. In the setting of Hilbert space, it originates with a suggestion by Koopman [Koo27, Koo36b, Koo57] in the early days of “modern” dynamical systems, ergodic theory, and the systematic study of representations of groups. A separate impetus in 1932 were the two ergodic theorems, the L2 variant due to von Neumann [vN32c] and the pointwise variant due to Birkhoff. While Birkhoff’s version is deeper, von Neumann’s version really started a whole trend: mathematical physics, quantization [vN32c], and operator theory, especially the use of the adjoint operator and the deficiency indices which we find useful in Chapters 5–7; cf. Refs. [vN32a, vN32b]. Furthermore, there is an interplay between Hilbert space, on the one hand, and pointwise results in function theory,

89

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Operator Theory and Analysis of Infinite Networks

on the other: In fact, the L2 -mean ergodic theorem by von Neumann is really a corollary to the spectral theorem in its deeper version (spectral resolution via projection-valued measures), as developed by Stone and von Neumann in the period 1928–1932; cf. Refs. [vN32b] and [Arv76a, Ch. 2]. This legacy motivates the material in this section as well as our overall approach.

4.1 Schoenberg and von Neumann’s Embedding Theorem In Theorem 4.2, we show that a resistance network equipped with a resistance metric may be embedded in a Hilbert space in such a way that R is induced from the inner product of the Hilbert space. As a consequence, we obtain an alternative and independent construction of the Hilbert space HE of finite-energy functions. This provides further justification for HE as the natural Hilbert space for studying the metric space (G, RF ) = ((G, c), RF ) and Fin as the natural Hilbert space for studying the metric space (G, RW ); see Remark 4.4. Although we are interested in both (G, RF ) and (G, RW ), for brevity, we sometimes refer to both as (G, R) when the distinction is not important. It is a natural question to ask whether or not a metric space may be naturally represented as a Hilbert space, and Schoenberg and von Neumann proved a general result which provides an answer; cf. Theorem A.17 and Refs. [BCR84, SW49]. Theorem 4.1 (Schoenberg–von Neumann). Suppose (X, d) is a metric space. There exists a Hilbert space H and an embedding u : (X, d) → H sending x → ux and satisfying d(x, y) = ux − uy H

(4.1)

if and only if d2 is conditionally negative semidefinite (cf. Definition A.7). We apply Theorem 4.1 to the metric space (G, R) to obtain a Hilbert space H and a natural embedding (G, R) → H. The reader may find Refs. [VNS41, vN32a, BCR84, Ber96, Sch38b] to be helpful. One aspect of the following theorem that contrasts sharply with the classical theory is that the embedding is applied to the metric R1/2 instead of R for each of R = RF and R = RW ; recall from (3.10) that RF (x, y) = vx − vy 2E , etc.

Schoenberg–von Neumann Construction of the Energy Space HE

91

Theorem 4.2. (G, RF ) may be isometrically embedded in a Hilbert space. Proof. According to Theorem 4.1, we need only to check that RF is conditionally negative semidefinite (see Definition A.7). Let f : G0 → C   satisfy x∈G0 f (x) = 0. We must show that x,y∈F f (x)RF (x, y)f (y) ≤ 0 for any finite subset F ⊆ G0 . From (3.10), we have   f (x)RF (x, y)f (y) = f (x)E(vx − vy )f (y) x,y∈F

x,y∈F

=



f (x)E(vx )f (y) − 2

x,y∈F



+



f (x)E(vy )f (y)

x,y∈F



= −2

f (x)vx , vy E f (y)

x,y∈F



x∈F

f (x)vx ,

 y∈F

 f (y)vy

 2     = −2  f (x)vx  ≤ 0.   x∈F

E

(4.2)

E

For (4.2), note that the sum containing E(vx ) and the sum containing E(vy ) vanished by the assumption on f .  Corollary 4.3. (G, RW ) may be isometrically embedded in a Hilbert space. Proof. Because the energy minimizer in (3.20) is fx = PFin vx , we can repeat the proof of Theorem 4.2 with fx in place of vx to obtain the result.  Theorem 4.4. The energy space HE is unique (up to unitary equivalence), regardless of whether it is constructed as a reproducing kernel Hilbert space (using dipoles and evaluation functionals, as in Section 2.1) or as the Hilbert space embedding of a metric space (using the Schoenberg–von Neumann Theorem 4.1). Proof. Since RF (x, y) = vx − vy 2E by (3.10), Theorem A.18 shows that the embedded image of (G, RF ) is unitarily equivalent to the E-closure of span{vx }, which is HE . However, what is even more is that since both the RKHS and Schoenberg–von Neumann approaches use the correspondence x → vx (this is apparent in the latter where we use R(x, y) = vx − vy 2E ), one can see that the Hilbert spaces corresponding to these two approaches are actually the same and not just unitarily equivalent. 

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Remark 4.5. Similarly to Theorem 4.4, RW (x, y) = fx − fy 2E , where fx := PFin vx , by (3.20), whence the embedded image of (G, RW ) is unitarily equivalent to the E-closure of span{fx}, which is Fin. Remark 4.6. The energy space is shown to be complete under  · E by independent means in Theorem 10.13. One can choose any vertex o ∈ G0 to act as the “origin,” and its embedded image becomes the origin of the new Hilbert space. As a quadratic form defined on the space of all functions v : G0 → C, the energy is indefinite and hence allows one to define only a quasinorm. There are ways to deal with the fact that E does not “see constant functions.” One possibility is to adjust the energy so as to obtain a true norm as follows: E o (u, v) := E(u, v) + u(o)v(o).

(4.3)

The corresponding quadratic form is immediately seen to be a norm; this ¯ approach is followed in Ref. [FOT94], for example, and also occasionally in the work of Kigami. This is discussed in Section 2.4.1 under the name “grounded energy form.” For most of this volume,1 we have instead elected to use “modulo constants”; the kernel of E is the set of constant functions, and inspection of von Neumann’s embedding theorem shows that it is precisely these functions which are “modded out” in von Neumann’s construction. However, the constant functions resurface as multiples of the vacuum vector in the Fock space representation of Section 7.3. 4.2 HE as an Invariant of G In this section, we show that HE may be considered as an invariant of the underlying graph. Definition 4.7. Let G and H be resistance networks with respective conductances cG and cH . A morphism of resistance networks is a function ϕ : (G, cG ) → (H, cH ) between the vertices of the two underlying graphs for which G cH ϕ(x)ϕ(y) = rcxy ,

for some fixed r and all x, y ∈ G0 . 1 Essentially,

everywhere aside from Chapter 8.

0 < r < ∞,

(4.4)

Schoenberg–von Neumann Construction of the Energy Space HE

93

Two resistance networks are isomorphic if there is a bijective morphism between them. Note that this implies .

H 1 = {(ϕ(x), ϕ(y)) .. (x, y) ∈ G1 }.

(4.5)

Definition 4.8. A morphism of metric spaces is a homothetic map, that is, an isometry composed of a dilation: ϕ : (X, dX ) → (Y, dY ),

dY (ϕ(a), ϕ(b)) = rdX (a, b),

0 < r < ∞, (4.6)

for some fixed r and all a, b ∈ X. An isomorphism is, of course, an invertible morphism. We allow for a scaling factor r in each of the previous definitions because an isomorphism amounts to a relabeling, and rescaling is just a relabeling of lengths. More formally, an isomorphism in any category is an invertible mapping, and dilations are certainly invertible for 0 < r < ∞. Theorem 4.9. For each of R = RF , RW , there is a functor R : (G, c) → ((G, c), R) from the category of resistance networks to the category of metric spaces. Proof. We check that an isomorphism ϕ : (G, cG ) → (H, cH ) of resistance networks induces an isomorphism of the corresponding metric spaces, i.e., ϕ preserves E. Let x, y denote vertices in G and s, t denote vertices in H. We compare the energy of u : H 0 → C with that of u ◦ ϕ : G0 → C as follows:  u ◦ ϕ, v ◦ ϕE = cG xy (u ◦ ϕ(x) − u ◦ ϕ(y))(v ◦ ϕ(x) − v ◦ ϕ(y)) x,y

= r−1



cH ϕ(x)ϕ(y) (u(ϕ(x)) − u(ϕ(y)))(v(ϕ(x)) − v(ϕ(y)))

x,y

= r−1



cH st (u(s) − u(t))(v(s) − v(t))

s,t

= r−1 u, vE ,

(4.7)

where we can change to summing over s, t because ϕ is a bijection. Therefore, the reproducing kernel {vx } of (G, RF ) (or {PFin vx } of (G, RW )) is preserved and hence so is the metric. 

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Operator Theory and Analysis of Infinite Networks

Corollary 4.10. If ϕ : G → H is an isomorphism of resistance networks with scaling ratio r, then for v : H 0 → C and v ◦ ϕ : G0 → C, Δ(v ◦ ϕ) = r−1 Δ(v) ◦ ϕ.

(4.8)

Proof. Compute Δ(v ◦ ϕ)(x) exactly as in (4.7).



Corollary 4.11. An isomorphism ϕ : (G, cG ) → (H, cH ) of resistance networks induces an isomorphism of metric spaces (where the resistance networks are equipped with their respective effective resistance metrics). We use the notation [S] to denote the closure of the span of a set of vectors S in a Hilbert space, where the closure is taken with respect to the norm of the Hilbert space. The following restatement of Theorem A.18 is included here for convenience. Theorem 4.12. Suppose H and K are Hilbert spaces and that there exist embeddings h : X → H, with hx − hy H = d(x, y), and k : X → K, with kx − ky K = d(x, y), where {hx }x∈X has a dense span in H and {kx }x∈X has a dense span in K. Then, there exists a unique unitary isomorphism U : H → K, and it is induced by U (hx ) = kx . Remark 4.13. Theorem 4.12 indicates that there is an essentially unique functor from the category of metric spaces (with negative semidefinite metrics) to the category of Hilbert spaces. Thus, the composition is a functor from resistance networks to Hilbert spaces; therefore, HE = HE (G) is an invariant of G. Remark 4.14. To obtain a first quantization, one would need to prove that a contractive morphism between resistance networks induces a contraction between the corresponding Hilbert spaces. In other words, f : G1 → G2

=⇒

Tf : HE (G1 ) → HE (G2 ),

with Tf vE ≤ vE whenever f is contractive. The embedding of HE under the Wiener isometry W is the second quantization discussed in Remark 7.24. 4.3 Remarks and References Of the results in the literature which are of relevance to the current chapter, Refs. [Bar04, Ale75, PS72, Sch38b, Sch38a] are especially relevant. Some of the cited references in this chapter are more specialized, but for prerequisite

Schoenberg–von Neumann Construction of the Energy Space HE

95

material (if needed), the reader may find the books by Guichardet [Gui72], Hida [Hid80], and Parthasarathy & Schmidt [PS72] especially relevant. Standard applications of a negative definite function include either the construction of an abstract Hilbert space [vN55, vN32c] or the construction of measures on a path space [PS72, Min63]. We use the terminology “Schoenberg–von Neumann embedding” to denote a set of general principles, both classical and modern: Geometry/Physics

[Koo57]

/ Hilbert space

[Nel73a]

/ L2 (Ω , P),

where the right-hand side is an L2 space of random variables. Koopman [Koo57] is credited with making precise a direct link between point transformations in dynamical systems, on the one hand, and associated Hilbert space operators, on the other. By now, this correspondence is taken for granted. The idea is simply to lift a point transformation, say T , to an associated operator acting in a space of functions. The relevant function spaces may be taken to be a Hilbert space of the form L2 (μ) when μ is an invariant measure for T . (Naturally, there are more subtle Koopman correspondences, but this one suffices for the moment). At the time, Koopman’s idea also served to make a link between classical and quantum ideas [Koo36a, Koo40b, Koo40a]. Some examples of the Schoenberg–von Neumann embedding include Brownian motion [Nel64, Nel69], second quantization and quantum fields [Min63, Gro70, Hid80], stochastic integrals [Mal95], spin models [Lig93], and quantum spin models [Pow67, Pow76a, Pow76b]. See Chapter 16 for further details, especially Theorem 16.9.

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Chapter 5

The Laplacian on HE I have tried to avoid long numerical computations, thereby following Riemann’s postulate that proofs should be given through ideas and not voluminous computations. — D. Hilbert

In this chapter, we study the operator theory of the Laplacian in some detail, examining the various domains and self-adjoint extensions (see also Chapter 9 and Sections B.3 and B.4). Also, we give the technical conditions which must be considered when a graph contains vertices of infinite degree and/or the conductance function c(x) is unbounded on G0 . A technical obstacle must be overcome: While 2 (G0 ) has a canonical orthonormal basis, this is not so for HE . Instead, the analysis of HE is carried out with the use of an independent and spanning system {vx } in HE ; these vectors are non-orthogonal, but this non-orthogonality is a rich source of information. In Section 5.3.1, we relate the boundary term of (2.19) to a boundary form akin to that of classical functional analysis; see Definition 5.20. In Theorem 5.22, we show that if Δ fails to be essentially self-adjoint,1 then Harm = {0}. In general, the converse does not hold: Any homogeneous tree of degree 3 or higher with constant conductances provides a counterexample; cf. Corollary 6.89. Definition 5.1. Let V := span{vx }x∈G0 denote the vector space of finite linear combinations of dipoles. Then, let ΔV be the closure of the Laplacian

1 See

Definition B.8 for the definition of essential self-adjointness. 97

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when taken to have the dense domain V. We also continue to use the notation M := span{vx }x∈G0 + span{wxv , wfx }x∈G0 , and let ΔM be the closure of the Laplacian when taken to have the dense domain M, as in Definition 2.32. In situations where results do not depend on the choice of domain, we use Δ to denote either ΔM or ΔV . We use ran T to denote the range of the operator T and ker T to denote its kernel (null-space).

5.1 Properties of Δ on HE Definition 5.2. The network (G, c) satisfies the Powers bound iff c := supx∈G0 c(x) < ∞. Also, we denote the degree of x ∈ G0 by deg(x) := . |{y .. cxy > 0}|. The Powers bound is used more in Chapter 6 (see Definition 6.69 and the surrounding discussion); we include it here for use in a couple of technical lemmas. Lemma 5.3. If the Powers bound is satisfied, then Δ maps HE into ∞ (G0 ). Proof. By Lemma 2.16 and (1.11), |Δv(x)| = |δx , vE | ≤ δx E · vE =  c(x)1/2 vE . Lemma 5.4. If deg(x) < ∞ for every x ∈ G0 , or if c < ∞, then ran ΔV ⊆ dom ΔV , and similarly for ΔM . Proof. It suffices to show that ΔV vx = δx −δo ∈ dom ΔV for every x ∈ G0 , and this will be clear if we show that δx ∈ dom ΔV . By Lemma 2.26,  δx = c(x)vx − y∼x cxy vy . If deg(x) is always finite, then we are done. If  not, we need to see why y∼x cxy vy ∈ dom ΔV for any fixed x ∈ G0 .  exhaustion, Fix x ∈ G0 and denote ϕ := y∼x cxy vy . Let (Gk )∞ k=1 be an  as in Definition 2.4 (see also Definition 2.5), and define ϕk := y∈Gk cxy vy .  It is clear that ϕ − ϕk E → 0. Next, we show that ΔV ϕk − y∼x cxy (δy − δo )E → 0, from which it follows that {ΔV ϕk } is Cauchy and that ϕ ∈

The Laplacian on HE

dom ΔV with ΔV ϕ =



y∼x (δy

99

− δo ):

2       cxy (δy − δo ) ΔV ϕk −   y∼x

E

 2       = cxy (δy − δo ) cxy δy − δo 2E  ≤ c(x) y∈G  y∈G E

k

⎛

= c(x) ⎝ ⎛ = c(x) ⎝



k

cxy δy 2E − 2

y∈Gk





cxy δy , δo E +

y∈Gk

cxy c(y) + 2

y∈Gk

≤ c(3c + c(o))



 y∈Gk

cxy coy + c(o)

 y∈Gk



⎞ cxy δo 2E ⎠ ⎞

cxy ⎠

y∈Gk

cxy ,

(5.1)

y∈Gk

which tends to 0 as k → ∞; observe that for a fixed x ∈ G0 , one has cxy < 1 for y ∈ Gk with k sufficiently large and that (5.1) follows from the 1/2 1/2  Cauchy–Schwarz inequality applied to cxy cxy (δy − δo ). 5.1.1 Finitely supported functions and the range of Δ It follows from Remark 2.37 that one always has ran Δ ⊆ Fin and hence Harm ⊆ ker Δ for Δ = ΔV or Δ = ΔM . The rest of this section is roughly an examination of the reverse containment, i.e., what conditions give ran Δ = Fin. Determining when ran Δ = Fin essentially boils down to the following technical question: When is span{δx − δo } dense in Fin? It is curious that this never happens on a finite network (Lemma 6.17) but is often true on an infinite network.2. However, see Example 14.36 Definition 5.5. Let Fin2 be the E-closure of span{δx −δo }, and let Fin1 be the orthogonal complement of Fin2 in Fin. This extends the decomposition HE = Fin ⊕ Harm, in some cases, to HE = Fin2 ⊕ Fin1 ⊕ Harm.

2 Compare

with Lemma 6.63!

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Lemma 5.6. Let (G, c) be an infinite network. If c < ∞, then Fin = Fin2 . Proof. It suffices to approximate the single Dirac mass δo by linear (n) combinations of differences. For each n, fix n vertices {xk }nk=1 , no two  n of which are adjacent. Therefore, define ϕn := n1 k=1 (δo − δx(n) ) and k compute 2  n n 2  1  1      2 δx(n)  = 2 δo − ϕn E =  δx(n)  k  k n n E E

k=1

≤

k=1

1 c (n) sup c(xk ) ≤ → 0, n 1≤k≤n n

where the second equality comes from orthogonality; for j = k, δx(n) and k δx(n) are not adjacent, hence δx(n) , δx(n) E = 0 by (1.11). Now, it is trivial j

k

to approximate δz = (δz − δo ) + δo .

j



Remark 5.7. The analogous result of Lemma 5.6 does not hold when one replaces the hypothesis c < ∞ with the hypothesis deg(x) < ∞; see Example 14.36 for such an example (in which Fin2 is not dense in Fin). 5.2 Harmonic Functions and the Domain of Δ Curiously, even though Δh(x) = 0 pointwise for every x ∈ G0 , it may happen that h is not in the domain of Δ. However, harmonic functions are always in the domain of the adjoint ΔM by Lemma 2.52. ˜ V is any Hermitian extension of ΔV whose domain Lemma 5.8. If Δ ˜ V u ∈ Fin ˜ V h = 0 for any h ∈ Harm. Moreover, Δ contains Harm, then Δ ˜ for any u ∈ dom ΔV . ˜ V ⊆ Δ . Proof. Recall the following ordering of operators: ΔV ⊆ Δ V  ˜ ˜ Since ΔV is an extension of ΔV and Harm ⊆ ΔV , the first claim follows immediately from Lemma 2.52. The second claim now follows from the ˜ V ) hE = 0 for every h ∈ Harm since ˜ V v, hE = v, (Δ first because Δ  ˜ ˜ ΔV ⊆ (ΔV ) .  We have a partial converse of Lemma 2.52. Note that if span{δx − δo } is dense in Fin (as discussed in Lemma 5.6), then Lemma 5.9 implies ker ΔV = Harm.

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101

Lemma 5.9. ker ΔV is the orthogonal complement of span{δx − δo }. Proof. Suppose u ∈ ker ΔV so that ΔV u = 0. Then, 0 = ΔV u, vx E = u, ΔV vx E = u, δx − δo E . This shows u is orthogonal to span{δx − δo }.



The Lemma 5.6 gives an idea of when the hypotheses of Lemma 5.9 are satisfied. In fact, a weaker hypothesis will suffice: One just needs to be able to find an infinite subset of nonadjacent vertices on which c(x) is bounded. Definition 5.10. Define ΔH to be the extension of ΔV to the domain dom ΔV + Harm by ΔH (v + h) := ΔV v. By abuse of notation, let ΔH denote the closure of ΔH with respect to the graph norm; see Definition B.5. Lemma 5.11. ΔH is well defined, Hermitian, and semibounded. Proof. We must check ΔH (0) = 0, so suppose v + h = 0 for v ∈ V and h ∈ Harm. Then, Lemma 2.52 gives ΔV (v + h) = 0, whence ΔV v =  −ΔV h = 0. Theorem 5.12. ΔH is self-adjoint. Proof. Let w ∈ HE satisfy ΔH w = −w. To see that w = 0, note that w ∈ dom ΔH , so ΔV w ∈ Fin by Lemma 5.13 presented next. But then, w = −ΔH w = −ΔV w ∈ Fin, so w2E = w, wE =



wΔw = −

G0



|w|2 ≤ 0

G0

so that w = 0 in HE . This shows ΔH is essentially self-adjoint, but ΔH is closed by definition, so it is self-adjoint.  .

Lemma 5.13. dom ΔH = {w ∈ dom ΔV .. ΔV w ∈ Fin}. Proof. For purposes of this proof, it is permissible to work with HE as a real vector space and complexify afterward. (⊆) Suppose that w ∈ dom ΔH , i.e., we have the estimate |w, ΔH (v + h)E | ≤ C1 v + hE

for all v ∈ V and h ∈ Harm.

(5.2)

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Then, for all t ∈ R, 2

|w, ΔH vE | ≤ C12 v + th2E ≤ C12 v2E + 2t|v, hE |2 + t2 h2E for all v ∈ V and h ∈ Harm. This quadratic polynomial in t is nonnegative, and hence, its discriminant must be nonpositive so that

2 C14 |v, hE |2 ≤ C12 h2E C12 v2E − |w, ΔH vE | Ph v2E =

|v, hE |2 ≤ v2E − C2 |w, ΔH vE |2 , h2E

where Ph is a projection to the rank-1 subspace spanned by h and C2 = If we let {hi } be an orthonormal basis for Harm, then 2

C22 |w, ΔH vE | ≤ v2E − Ph1 v2E

1 C1 .

for all v ∈ V.

Inductively substituting v = v − h2 , v = v − (h2 + h3 ), etc., we have 2

C22 |w, ΔH (v − h2 )E | ≤ v − Ph2 v2E − Ph1 v2E  = v2E − Ph2 v2E + Ph1 v2E .. .   2 2 C2 |w, ΔH (v + i hi )E | ≤ v2E − i Phi v2E = v2E − PHarm v2E . By the definition of ΔH , all the terms on the left-hand side are equal to C22 |w, ΔH vE |2 = C22 |w, ΔV vE |2 . Since PFin vE = v2E − PHarm v2E , we have established that |w, ΔV vE | ≤ C3 PFin vE

for all v ∈ V.

Now, Riesz’s lemma gives an f ∈ Fin such that w, ΔV vE = f, PFin vE

for all v ∈ V.

However, orthogonality allows one to remove the projection (since the first argument is already in Fin), whence ΔV w, vE = f, vE for all v ∈ V, and so, ΔV w = f ∈ Fin. (⊇) Let w be in the set on the right-hand side. To see w ∈ dom ΔH , we need the estimate (5.2), but |w, ΔH (v + h)E | = |w, ΔV vE | = |ΔV w, vE | = |ΔV w, PFin vE | ,

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103

where the last equality follows from the hypothesis ΔV w ∈ Fin. This gives |w, ΔH (v + h)E | ≤ ΔV wE · PFin (v + h)E , but PFin vE = PFin (v +  h)E ≤ v + hE , so (5.2) follows. Corollary 5.14. A closed extension of ΔV is self-adjoint if and only if Harm is contained in its domain. Proof. It is helpful to keep in mind the operator ordering ΔV ⊆ ΔH = ΔH ⊆ ΔV . ˜ then the result ˜ be a self-adjoint extension of ΔV . If ΔH ⊆ Δ, (⇒) Let Δ   ˜ = Δ, ˜ and the result ˜ ⊆ ΔH , then again ΔH ⊆ Δ ⊆ (Δ) is obvious, and if Δ H

is equally obvious. ˜ then ΔH ⊆ Δ, ˜ ˜ is a closed extension of ΔV with Harm ⊆ dom Δ, (⇐) If Δ so clo  clo ˜ ˜  Δclo H ⊆ Δ ⊆ (Δ) ⊆ (ΔH ) ⊆ ΔH ,

˜ is closed and the last by where the first inclusion holds because Δ Theorem 5.12.  5.3 The Defect Space of ΔV Let ΔV continue to denote the graph closure of the operator Δ on the (dense) domain V := span{vx }. Definition 5.15. Since ΔV is Hermitian on its domain by Corollary 2.69, Definition B.10 and Theorem B.11 imply that the defect space of ΔV is .

Def −1 (ΔV ) := {v ∈ dom ΔV .. ΔV v = −v}.

(5.3)

Observe also that Def −1 (ΔV )⊥ = ran(I + ΔV ). In Section 6.4.3, we use a certain quadratic form to study the defect space Def (ΔV ). Lemma 5.16. u is a defect vector of ΔV if and only if there is a constant k such that Δu(x) = −u(x) + k at each x ∈ G0 . Proof. Recall that the meaning of such a pointwise identity is that u ∈ dom ΔV and ΔV u = −u+k in HE ; see Lemma 2.36. The reverse implication is obvious; for the obverse, it suffices to check the claim against the (dense)

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energy kernel: 0 = vx , Δ u + uE = δx − δo , uE + vx , uE = Δu(x) − Δu(o) + u(x) − u(o) by Lemma 2.16, which proves the claim with k = Δu(o) + u(o).



Remark 5.17 (Defect vectors and the Gauss–Green formula). We have introduced the defect space of ΔV here to alleviate any concerns  regarding the convergence of x∈G0 u(x)Δu(x) in (2.19); the reader will note that if u is a defect vector, then   u(x)Δu(x) = − |u(x)|2 , x∈G0

x∈G0

which must equal −∞ since there are no defect vectors in 2 . This is a reasonable concern, as there do exist networks with nontrivial defect; see Example 14.37. However, such defect vectors are proscribed by the hypotheses of Theorem 2.40 by the following lemma. Lemma 5.18. dom ΔV ∩ Def −1 (ΔV ) = 0. Proof. Suppose u ∈ dom ΔV ∩ Def −1 (ΔV ). Note that ΔV is an extension of ΔV , so such a u satisfies ΔV u = −u. However, since ΔV is semibounded on its domain by Corollary 2.69, this implies 0 ≤ u, ΔV uE = ΔV u, uE = −u, uE = −u2E , 

whence u = 0. Corollary 5.19. For u ∈ Def −1 (ΔV )⊥ ,  u ∂∂u = PHarm (I + Δ)−1 u2E .

(5.4)

bd G

Proof. By Lemma B.13, we can take u to be of the form u = v + Δv with v ∈ dom ΔV , so u = (I + Δ)v and v = (I + Δ)−1 u. By the discrete Gauss–Green formula (Theorem 2.40) and the Royden decomposition (Theorem 2.20), we have   u2E = uΔu + u ∂∂u = PFin v2E + PHarm v2E , G0

bd G

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105

whence 

u ∂∂u = PHarm v2E = PHarm (I + Δ)−1 u2E ,

bd G



and we have (5.4). 5.3.1 The boundary form βbd

In this section, we relate the defect of Δ to the boundary term of the discrete Gauss–Green formula (Theorem 2.40), thereby extending Theorem 2.49. We return to the consideration of ΔM , the closure of the Laplacian when taken to have the dense domain M := span{vx }x∈G0 + span{wxv , wfx }x∈G0 , as in Definition 2.32. The reader may find Ref. [DS88, Section XII.4.4] to be a useful reference. Definition 5.20. Define the boundary form βbd (u, v) :=

1 2

(ΔM u, vE − u, ΔM vE ) ,

u, v ∈ dom(ΔM ).

(5.5)

To see the significance of βbd for the defect spaces, note that if ΔM f = zf , where z ∈ C with Im z = 0, then βbd (f, f ) = (Im z)f 2E . Lemma 5.21. The boundary form βbd (u, v) vanishes if u or v lies in dom(ΔM ). Proof. For v ∈ dom(ΔM ), ΔM u, vE = u, ΔM vE by the definition of the adjoint and u, ΔM vE = u, ΔM vE by the fact that ΔM extends ΔM . Hence, both terms of (5.5) are equal for u, v ∈ dom(ΔM ). The proof  is identical if u ∈ dom(ΔM ). The following result extends Theorem 2.49. Theorem 5.22. If Harm = {0}.

ΔM

fails

to

be essentially

self-adjoint,

then

Proof. We prove that the boundary form βbd (u, v) vanishes identically whenever Harm = {0}. Since the boundary sum can only be nonzero when Harm = {0}, the conclusion will follow once we show that   ∂u ∂v  (5.6) βbd (u, v) = 21 ∂  (ΔM v) − (ΔM u) ∂  . bd G

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To see this, apply Theorem 2.40 to obtain   ΔM u, vE = ΔM uΔM v + ΔM u ∂∂v G0

=



bd G

ΔM uΔM v +

G0

for any u, v ∈ pointwise:

dom(ΔM ).



ΔM u ∂∂v

bd G

The second equality follows because Δ = Δ

ΔM u(x) − ΔM u(o) = vx , ΔM uE = ΔM vx , uE = δx − δo , uE = Δu(x) − Δu(o), where the last equality comes from Lemma 2.16. Also, note that u ∈ dom(ΔM ) implies ΔM u ∈ HE so that Theorem 2.40 applies and both terms are finite. Consequently, the two sums over G0 cancel and the theorem follows.  Remark 5.23. There is an alternative, more elementary way to prove Theorem 5.22. Suppose w = 0 is a nonzero defect vector with ΔM w = w. Then, we can find a representative for w such that      wΔw + w ∂w |w|2 + Re w ∂w w ∂w w, wE = ∂ = ∂  + Im ∂ . G0

bd G

G0

bd G

bd G

(5.7) Since w2E = w, wE is real (and strictly positive, by hypothesis), this implies the boundary sum is nonzero, and Theorem 2.49 gives the existence of nontrivial harmonic functions. It also follows from (5.7) that such a nonzero defect vector satisfies   |w|2 = − Im w ∂w ∂ > 0 so that Im

G0

 bd G

bd G

w ∂w ∂  < 0.

5.4 Remarks and References The literature on various incarnations of the Laplace operator is both large and diverse; for example, these operators in mathematical physics go by the name “discrete Schr¨odinger operators.” Readable introductions include Refs. [CdV99, Chu96, Dod06, Soa94, Web08]. The reader may also find Refs. [HKLW07, Woe00, Woe03, RS95, JKM+ 98] to be useful.

Chapter 6

The 2 Theory of Δ and the Transfer Operator One geometry cannot be more true than another; it can only be more convenient. — H. Poincare

This chapter is devoted to the study of the graph Laplacian Δ and the transfer operator T, when considered as acting on the space 2 of squaresummable functions on the vertices. The measure weighting the vertices will always be the counting measure, but we consider different weights on the edges. The operators Δ and T have a profoundly different spectral theory with respect to the 2 inner product than the E inner product. We investigate certain properties of the Laplacian and transfer operator on .  (6.1) 2 (1) := {u : G0 → C .. x∈G0 |u(x)|2 < ∞}, with the inner product u, v1 :=



u(x)v(x),

(6.2)

x∈G0

including self-adjointness and boundedness and relations to their HE counterparts. The constant function 1 appears in the notation to specify the weight involved in the inner product, in contrast to c. This is necessary because we will also be interested in Δ and T as operators on . (6.3) 2 (c) := {u : G0 → C .. x∈G0 c(x)|u(x)|2 < ∞}, 107

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with the inner product u, vc :=



c(x)u(x)v(x).

(6.4)

x∈G0

While the pointwise definition of Δ and T remains the same on 2 (1) and 2 (c), they are different operators with different domains and different spectra! It is important to keep in mind that in general, none of HE , 2 (1) or 2 (c) are contained in any of the others. However, we provide some conditions under which embeddings exist in Section 6.5.2. We give only some selected results, as this subject is well documented elsewhere in the literature. In Section 6.3, we consider the systems {vx } and {δx } and a kind of spectral reciprocity between them in terms of frame duality. In previous parts of this book, we approximated infinite networks by truncating the domain; this is the idea behind the definition of Fin and the use of exhaustions. This approach corresponds to a restriction on span{δx }x∈F , where F is some finite subset of G0 . In Section 6.3, we consider truncations to sets of the form span{vx }x∈F , i.e., essentially restricting to the dual variable. Note that an element of this set generally will not have finite support. In Section 6.6, we consider a map J : 2 (c) → HE as a quotient map induced by the equivalence relation discussed in Remark 2.3. It turns out that J is an embedding of 2 (c) into Fin and that its range is dense in Fin. In this section, we have occasion to consider a Laplacian as an infinite matrix; see Definition 6.7. A function u on G0 will be viewed as a column vector. If A = (ax,y )x,y∈G0 is an R-valued function on G0 ×G0 , then TA (u) = Au is defined by  ax,y u(y) (6.5) TA : u → Au, where (Au)(x) = y∈G0

(i.e., by matrix multiplication) with the understanding that the summation in the right-hand side of (6.5) is absolutely convergent. Similarly, for a linear operator T on 2 (1) or 2 (c), we write AT for the matrix of T with respect to the orthonormal basis {δx }x∈G0 . Much of the material in this chapter first appeared in Ref. [JP11b]. See Refs. [KL12, KL10, Woj11, Woj07] for closely related material, some of which appeared earlier.

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109

6.1 Essential Self-Adjointness of the Laplacian on 2 (1) Dealing with unbounded operators always requires a bit of care; the reader is invited to consult Appendix B.1 to refresh on some principles of selfadjointness of unbounded operators. Recall that for S to be self-adjoint, it must be Hermitian and satisfy dom S = dom S  , where .

dom S  := {v ∈ H .. |v, Su| ≤ Kv u , ∀u ∈ dom S}. In the unbounded case, it is not unusual for dom S  dom S  . Some good references for this section are Refs. [vN32a, Nel69, RS75, Rud91, DS88, Jør78]. Due to Corollary 2.73, we can ignore the possibility of nontrivial harmonic functions while working in this context. Combining Theorem 2.40 with Theorem 2.49, one can relate the inner products of HE and 2 (1) by u, Δv1 = u, vE

(6.6)

for all u, v ∈ span{δx }. Observe that span{δx } is dense in 2 (1) with respect to (6.2) and dense in HE in the E norm when Harm = 0. Then, (6.6) immediately implies that the Laplacian is Hermitian on 2 (1) because, again, for all u, v ∈ span{δx }, u, Δv1 = u, vE = v, uE = v, Δu1 = Δu, v1 .

(6.7)

This may seem trivial, but it turns out that Δ is not Hermitian on 2 (c); cf. Lemma 6.93. Theorem 6.70 shows that if c is uniformly bounded (6.68), then Δ is a bounded operator and hence self-adjoint. However, in Theorem 6.1, we are able to obtain a much stronger result, assuming a bound on deg(x) but not on Δ: The Laplacian on any electrical resistance network is essentially self-adjoint on 2 (1). (Recall that Δ is essentially self-adjoint iff it has a unique self-adjoint extension; cf. Definition B.8.) In Theorem 6.8, we prove a stronger result (the condition on deg(x) is removed). This is a sharp contrast to the case for HE , as seen from Theorem 5.22. In the latter parts of this section, we also derive several applications of Theorem 6.1. We begin with the operator Δ defined on span{δx }, the dense domain consisting of functions with finite support. Then, let Δ1 denote the closure of Δ with respect to (6.2), that is, its minimal self-adjoint extension to 2 (1).

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Theorem 6.1. If deg(x) < ∞ for every x ∈ G0 , Δ1 is essentially selfadjoint on 2 (1). In Corollary 6.89, we apply Theorem 6.1 to get an essential selfadjointness result for Δ on HE . The matrix for the operator Δ1 on 2 (1) is banded (cf. Section B.2): ⎧ ⎪ ⎪ ⎨c(x), y = x, (6.8) AΔ1 (x, y) = δx , Δ1 δy 1 = −cxy , y ∼ x, ⎪ ⎪ ⎩0, else. The bandedness of AΔ1 is a crucial element in the proof of Theorem 6.1; Example B.14 shows how this proof technique can fail without bandedness. See also Example B.18 for what can go awry without bandedness. However, bandedness is not sufficient to guarantee essential selfadjointness. In fact, see Example B.18 for a Hermitian operator on 2 which is not self-adjoint, despite having a uniformly banded matrix, that is, there is some n ∈ N such that each row and column has no more than n nonzero entries. The essential self-adjointness of Δ1 in this context is likely a manifestation of the fact that the banding is geometrically/topologically local; the nonzero entries correspond to the vertex neighborhood of a point in G0 . By working directly with the matrix A = AΔ1 , we remove the bandedness hypothesis from Theorem 6.1 in the improved result of Theorem 6.8. The proof of Theorem 6.1 requires a technical lemma. Lemma 6.2. The Laplacian Δ1 is semibounded on dom Δ1 . A fortiori, for any u, v ∈ 2 (1), u, Δ1 v1 =



c(x)u(x)v(x) −

x∈G0



cxy u(x)v(y).

Proof. For any u, v ∈ Fin, a straightforward computation shows   u, Δ1 u1 = c(x)|u(x)|2 − cxy u(x)u(y), x∈G0

(6.9)

x,y∈G0

(6.10)

x,y∈G0

whence the equality in (6.9) follows by taking limits and polarizing. To see that Δ1 is semibounded, apply the Schwarz inequality first with respect

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111

to y, then with respect to x, to compute  

1/2        2 cxy u(x)u(y) ≤ c(x)|u(x)| cxy |u(y)|    0 y∼x 0 y∼x x∈G

x∈G

≤



1/2 ⎡ c(x)|u(x)|2

x∈G0

=



⎣



⎤1/2 cxy |u(y)|2 ⎦

x,y∈G0

c(x)|u(x)|2

x∈G0

so that the difference on the right-hand side of (6.10) is nonnegative.



Proof (Proof of Theorem 6.1). Lemma 6.2 shows Δ1 is semibounded on 2 (1), so by Theorem B.11, it suffices to show the implication Δ1 v = −v

=⇒

v = 0,

v ∈ dom Δ1 .

(6.11)

Suppose that v ∈ 2 is a solution to Δ1 v = −v. Then, clearly, Δ1 v ∈ 2 , and then, by Lemma B.16, 0 ≤ v, AΔ1 v1 = v, Δ1 v1 = −v, v1 = − v 21 ≤ 0

=⇒

v = 0,

where AΔ1 is the matrix of Δ1 in the orthonormal basis {δx }x∈G0 . To justify the first inequality, consider that we may find a sequence {vn } ⊆ Fin with v − vn 1 → 0. Because the matrix AΔ1 is banded (see Section B.2 and (6.8)), it suffices to ensure that AΔ1 vn → AΔ1 v, and hence, (vn , AΔ1 vn ) converges to (v, AΔ1 v) in the graph norm, and so, vn , AΔ1 vn 1 converges to v, AΔ1 v1 . Then, vn , AΔ1 vn 1 = E(vn ) ≥ 0 for each n, and positivity  is maintained in the limit (even though lim E(vn ) may not be finite). See Ref. [Web08] for a similar result. It follows from Theorem 6.1 that the closure of the operator Δ1 is self-adjoint on 2 (1) and hence has a unique spectral resolution, determined by a projection valued measure on the Borel subsets of the infinite half-line R+ . This is in sharp contrast with the continuous case; in Example B.14, we illustrate this by indicating how d2 Δ1 = − dx 2 fails to be an essentially self-adjoint operator on the Hilbert space L2 (R+ ).

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112

6.1.1 Matrix Laplace operators on 2 Definition 6.3. We denote the set of finitely supported functions on G0 by c0 (G0 ) := span{δx }x∈G0 .

(6.12)

This alternative notation is intended to emphasize the distinction with Fin. Lemma 6.4. If A = (ax,y )x,y∈G0 is an infinite matrix, then matrix multiplication (6.5) defines an operator TA : c0 (G0 ) → 2 (G0 )

(6.13)

if and only if for any fixed y ∈ G0 , the function x → ax,y is in 2 (G0 ). In this case, TA is Hermitian if and only if ax,y = ay,x for all x, y ∈ G0 . Proof. This is clear because Aδy = a·,y is the column in A with index y  and δx , Aδy  = ax,y . The latter claim is standard. Lemma 6.5. Let A = (ax,y )x,y∈G0 be an infinite matrix which defines an operator TA : c0 (G0 ) → 2 (G0 ) as in Lemma 6.4. Then, the following two conditions are equivalent for two vectors v and w in 2 (G0 ):  (i) w(y) = x∈G0 ax,y v(x) is absolutely convergent for each y ∈ G0 , and w ∈ 2 (G0 ). (ii) v ∈ dom(TA ) and TA v = w. In particular, the action of the operator TA is given by formula (6.5). Proof. (i) =⇒ (ii). To show that v ∈ dom TA , note that TA u, v1 is equal to      TA u(x)v(x) = ax,y u(y)v(x) = ax,y v(x)u(y) x∈G0

x∈G0 y∈Y

=



w(y)u(x)

y∈G0 x∈G0

(6.14)

y∈G0

by Fubini–Tonelli theorem. This gives the estimate |TA u, v| ≤ w 1 u 1 by the Cauchy–Schwarz inequality, which means v ∈ dom TA . The equality TA v = w follows from (6.14). For the converse, note that w ∈ 2 (G0 ) because v ∈ dom(TA ).   = Then, the same calculation in reverse gives x∈G0 u(x)TA v(x)    x∈G0 y∈G0 ax,y u(y)v(x).

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113

Corollary 6.6. There exists w ∈ HE such that v, TA u1 = w, u1 holds for all u ∈ dom TA if and only if v ∈ dom TA and TA v = w. If we additionally assume that A is symmetric, the pointwise identity (Av)(x) = w(x) holds for all x ∈ G0 . Definition 6.7. We say that the infinite matrix A = (ax,y )x,y∈G0 defines a (matrix) Laplacian iff A satisfies (i) ax,y = ay,x for all x, y ∈ G0 ; (ii) ax,y ≤ 0 if x = y; and  0 (iii) y∈G0 ax,y = 0 for all x ∈ G . In this case, we write ΔA for the corresponding Hermitian operator ΔA : c0 (G0 ) → 2 (G0 ) defined by matrix multiplication, as in Lemma 6.4. Note that it follows immediately from (ii) and (iii) that  ax,x = − y∈G0 \{x} ax,y ≥ 0 for each x ∈ G0 , so the sum in (iii) is automatically absolutely convergent. Theorem 6.8 asserts that the three elementary conditions identified in Definition 6.7 are sufficient to ensure the Laplacian is essentially self-adjoint and hence has a well-defined and unique spectral representation. Theorem 6.8 (Essential self-adjointness of matrix Laplacians on 2 (G0 )). If the infinite matrix A = (ax,y )x,y∈G0 defines a matrix Laplacian on G0 , then the corresponding the Hermitian operator ΔA : c0 (G0 ) → 2 (G0 ) is essentially self-adjoint. The proof of Theorem 6.8 requires Lemma 6.10, variants of which appear in the literature in different contexts, e.g., Refs. [Kig03, Cor. 6.9] and ¯ [FOT94, Thm. 1.3.1]. Remark 6.9. Keller and Lenz provide an extended and improved version of this result to the situation of more general measures in Refs. [KL12, KL10], as long as the measure gives weight ∞ to infinite paths. (This is true automatically for the counting measure, which we use exclusively.) Note also that the results of Refs. [KL12, KL10] allow for positive potentials (denoted therein by c). Consequently, one cannot hope to study the deficiency spaces of Δ unless one considers (i) 2 spaces with respect to a measure which violates this axiom or (ii) some other Hilbert space entirely. In this volume, we elect to go with the latter option and focus primarily on the energy Hilbert space. Related but less general results also appear in Refs. [Web08, Woj07]; see also Ref. [Woj09].

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Lemma 6.10 (Semiboundedness of ΔA on 2 (G0 )). If the infinite matrix A = (ax,y )x,y∈G0 defines a matrix Laplacian on a countably infinite set G0 , then ΔA is semibounded and positive semidefinite with  u, ΔA u1 = 12 (−ax,y )|u(x) − u(y)|2 for all u ∈ c0 (G0 ). (6.15) x,y∈G0

Proof. Note that the right-hand side of (6.15) is a sum of nonnegative terms by Definition 6.7(ii) and that it is a finite sum by (6.12). The double summation on the right-hand side of (6.15) is 

ax,y |u(x) − u(y)|2 =

x,y∈G0

 

ax,y |u(x)|2

x∈G0 y∈G0

−2



ax,y Re(u(x)u(y))

x,y∈G0

+

  y∈G0

ax,y |u(y)|2 .

(6.16)

x∈G0

The last sum on the right-hand side vanishes by Definition 6.7(iii), and similarly, the first sum vanishes by combining parts (i) and (iii) of the same definition. Thus, the computation (6.16) continues as = −2



ax,y Re(u(x)u(y))

x,y∈G0

=−



ax,y u(x)u(y) −

x,y∈G0



ay,x u(x)u(y) = −2u, ΔA u1 ,

x,y∈G0

which gives (6.15). In view of assumption (i), we further get that u, ΔA u1 ≥ 0 for all u ∈ c0 (G0 ). Hence, the operator ΔA is semibounded and positive semidefinite.  We now turn to the proof of Theorem 6.8. Proof (Proof of Theorem 6.8). Assume that some v ∈ 2 (G0 ) satisfies 

ax,y v(y) = −v(x).

(6.17)

y∈G0

By applying Lemma 6.5, we must prove that v = 0 to complete the proof of Theorem 6.8. First, observe that (i)–(iii) imply that each of the following

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115

functions on G0 × G0 is summable, i.e., it is in 1 (G0 × G0 ): ax,y |v(x)|2 ,

ax,y |v(y)|2 ,

v(x)ax,y v(y),

and ax,y |v(x) − v(y)|2 .

Note that with (i)–(iii), Fubini’s theorem applies to the double summations of each of these functions. Pick an exhaustion {Fk }∞ k=1 as in Definition 2.4, then (6.17) gives          = 0 and lim v + a v(y) ax,y v(y) = −v(x), ∀x ∈ G0 . lim  x,y   k→∞  k→∞  y∈Fk y∈Fk 1

(6.18) The argument in the proof of Lemma 6.10 now yields the following:     (−ax,y )|v(x) − v(y)|2 = 2 v(x)ax,y v(y) x∈G0 y∈Fk

x∈G0 y∈Fk

−



|v(x)|2

x∈G0

−



y∈Fk



ax,y

y∈Fk

|v(y)|2



ax,y .

(6.19)

x∈G0

Combining (iii) with (6.18) and Fatou’s lemma, we can pass to the limit in (6.19). To compute this limit, note that for the first term on the right-hand side in (6.19), equation (6.18) gives      k→∞ v(x)ax,y v(y) = v(x) ax,y v(y) −−−−−→ − |v(x)|2 x∈G0 y∈Fk

x∈G0

= − v 21 .

y∈Fk

x∈G0

(6.20)

The second term on the right-hand side of (6.19) vanishes by Defini tion 6.7(iii) because limk→∞ y∈Fk ax,y = 0. Consequently, one obtains the identity   (−ax,y )|v(x) − v(y)|2 = −2 v 21 . (6.21) x∈G0 y∈G0

Since the left-hand side in (6.21) is nonnegative (as noted initially) and the right-hand side is nonpositive, it must be the case that v 21 = 0, whence v = 0.  For future use, we note the following corollary, which follows easily from a known characterization of positive semidefinite infinite matrices.

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0 Corollary 6.11. If {Fk }∞ k=1 is an exhaustion of G , as in Definition 2.4, and A(Fk ) := (ax,y )x,y∈Fk is the finite submatrix of A corresponding to Fk , then det A(Fk ) ≥ 0 for every k.

6.2 The Spectral Representation of Δ It is clear from Lemma 2.71 that v ∈ dom E whenever v and Δ1 lie in 2 (1). However, this condition is not necessary, and the precise characterization of dom E is more subtle. ˆ 1/2 u Theorem 6.12. For all u ∈ 2 (1) ∩ dom E, u E = Δ 1 ˆ 1 . Therefore, HE can be characterized in terms of the spectral resolution of Δ1 as  vˆ 1 < ∞}, 2 (1) ∩ dom E = {v : G0 → C .. Δ 1 .

1/2

(6.22)

where vˆ is the image of v in the spectral representation of Δ1 . Proof. For the proof, we write Δ = Δ1 . Theorem 6.1 gives a spectral resolution  (6.23) Δ = λE(dλ), E : B(R+ ) → P roj(2 ). √ Applying the functional calculus to the Borel function r(x) = x, we have   . |λ| · E(dλ)v 2 < ∞}. Δ1/2 = λ1/2 E(dλ), dom Δ1/2 = {v ∈ 2 .. (6.24) This gives v ∈ 2 (1) ∩ dom E if and only if v + k ∈ dom Δ1/2 for some k ∈ C. Since Δ(v + k) = Δv, the same is true for Δ1/2 by functional calculus.  Remark 6.13. It is important to observe that dom E is not simply the . ˆ 1/2 vˆ < ∞}. The ˆ 1/2 = {ˆ v .. vˆ ∈ L2 and Δ spectral transform of dom Δ 2 restriction vˆ ∈ L must be removed because there are many functions of finite energy which do not correspond to L2 functions. For an elementary yet important example, see Figure 14.5 of Example 14.16. Indeed, recall from Corollary 2.73 that no nontrivial harmonic function can be in 2 ; see Example 15.2. In this example, v is equal to the constant value 1 on one infinite subset of the graph and equal to the constant value 0 on another. Remark 6.14. For the example of the integer lattice Zd , Remark 14.18 shows quite explicitly why the addition of a constant to v ∈ HE has no effect on the spectral (Fourier) transform. In this example, one can see directly

The 2 Theory of Δ and the Transfer Operator

117

that addition of a constant k before taking the transform corresponds to the addition of a Dirac mass after taking the transform. As the Dirac mass is supported where the transform of the function vanishes, it has no effect. We can also give a reproducing kernel for Δ on 2 (1). Recall from (1.1) . that the vertex neighborhood of x ∈ G0 is G(x) := {y ∈ G0 .. y ∼ x} ⊆ G0 . Also, recall from Definition 1.7 that x ∈ / G(x) and from Definition 1.3 that  the conductance function is c(x) := y∼x cxy . Lemma 6.15. The functions {Δδx }x∈G0 = {c(x)δx − c(x·) χG(x) }x∈G0 give a reproducing kernel for Δ on 2 (1). Proof. Since δx , u1 = u(x), the result follows by recapitulating (6.8):  Δv(x) = c(x)v(x) − cxy v(y) = c(x)δx , v1 − c(x·) χG(x) , v1 = Δδx , v1 . y∼x

Since c(x) < ∞, it is clear that Δδx ∈ 2 (1).



6.3 Frames and Duality In previous parts of this book, we have approximated infinite networks by truncating the domain; this is the idea behind the definition of Fin in Definition 2.14 and in the use of exhaustions for various arguments (Definition 2.4). This approach corresponds to a restriction to span{δx }x∈F , where F is some finite subset of G0 . In this section, we consider truncations in the dual variable, i.e., restrictions to sets of the form V(F ) = span{vx }x∈F .1 This is directly analogous to the usual time–frequency duality in Fourier theory and the use of cutoff functions as Fourier multipliers. Note that we do not restrict the support of the functions under consideration: We restrict the index set of the representing functions {vx }x∈G0 , in the spirit of Karhunen–Lo`eve, leads to a form of spectral reciprocity between the associated Laplace operator and its “inverse” V in the sense described in Section 6.4. The support of vx ∈ V(F ) typically extends outside of F ; examples are given in Chapter 14. Indeed, elements of V(F ) do not typically have finite support; cf. Definition 2.14 and Figure 14.5 of Example 14.2. 1 Recall

from Definition 5.1 that V := span{vx }x∈G0 and ΔV is the closure of the Laplacian when taken to have the dense domain V.

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The energy kernel {vx } generally fails to be a frame for HE , as shown by Lemma 6.19 and the ensuing remarks. However, things improve when restricting to a finite subset. We shall approach the infinite case via a compatible system of finite dual frames, one for each finite subset F ⊆ G0 \ {o}; see Definition 6.18. In Theorem 6.23, we show that {δx }x∈F and {vx }x∈F form a dual frame system, and so, we refer to the subscript indexing {vx }x∈F as the “dual variable.” The text Ref. [Chr08] is suggested for readers with further interests in frame bounds. We obtain optimal frame bounds in Corollary 6.24. In Theorem 6.27, we show that the boundedness of ΔV is equivalent to both the existence of a global upper frame bound (i.e., one can let F → G) and the existence of a spectral gap. We begin with two lemmas whose parallels serve to underscore the theme of this section.  Lemma 6.16. For a finite sum ψ = x=o ξx vx , with ξx ∈ C, ξx = δx , ψE = Δψ(x).

(6.25)

Consequently, the vectors {vx } are linearly independent. Proof. We compute (6.25) directly, for y = o,    δy , ψE = ξx δy , vx E = ξx (δy (x) − δy (o)) = ξx δy (x) = ξy . x=o

x=o

x=o



If ψ = 0, then this calculation shows that ξy = 0 for each y.  Lemma 6.17. For a finite sum ψ = x=o ξx δx , with ξx ∈ C, ξx = vx , ψE = ψ(x) − ψ(o).

(6.26)

Consequently, the vectors {δx } are linearly independent. Proof. We compute (6.26) directly, for y = o,    vy , ψE = ξx vy , δx E = ξx (δx (y) − δx (o)) = ξx δx (y) = ξy . x=o

x=o

x=o

If ψ = 0, then this calculation shows that ξy = 0 for each y.



Definition 6.18. Recall that c0 (F ) := span{δx }x∈F , and let Δc0 (F ) be the Laplacian when taken to have this (non-dense) domain. Let ΔV(F ) denote the Laplacian when taken to have the domain V(F ), even though it not dense in HE .

The 2 Theory of Δ and the Transfer Operator

119

Then, c0 (F ) is a dual frame for V(F ) if there are constants 0 < A ≤ B < ∞ (called frame bounds) for which  |δx , ψE |2 ≤ B ψ 2E , ∀ψ ∈ V(F ). (6.27) A ψ 2E ≤ x∈F

Lemma 6.19. {vx } is a frame for HE if and only if 2 (G0 ) and HE are isomorphic. Proof. Since {vx } is a reproducing kernel, the frame inequalities take the form  |w(x)|2 ≤ B w 21 . (6.28) A w 21 ≤ Each inequality indicates a (not necessarily isometric) embedding.



Remark 6.20. The second inequality fails if Δ does not have a spectral gap; this is discussed further in Section 6.3. See also Lemma 10.17 and the examples in Chapters 13 and 14. A spectral gap is present for the example of the binary tree. Definition 6.21. Define a (presumably infinite) V Hermitian matrix by [V ]x,y∈G0 := vx , vy E ,

(6.29)

and denote the |F | × |F | submatrix obtained by deleting rows and columns from F C by VF . Definition 6.22. For ψ ∈ HE , define X : HE → (G0 ), where (G0 ) is the space of all functions on G0 by X ψ (x) := δx , ψE .

(6.30)

By Remark 2.17, X is morally identical to the Laplacian when defined formally on all of HE ; note that X ψ may not lie in HE . In the proof of Theorem 6.23, the notations · , ·1 and · 1 refer to the space 2 (1), that is,  f, g1 = f (x)g(x) x∈G0 2

is the unweighted  inner product, etc. This notation is used again in the discussion of these spaces in this chapter. We also use the notation λmin (A) := min spec(A) for any Hermitian matrix A.

and λmax (A) := max spec(A)

(6.31)

Operator Theory and Analysis of Infinite Networks

120

Theorem 6.23. For any finite F ⊆ G0 , one has λmin (VF ) > 0 for the minimal eigenvalue of Definition 6.21, and {δx }x∈F is a dual frame for V(F ) with frame bounds  1 1 ψ 2E ≤ ψ 2E . |δx , ψE |2 ≤ λmax (VF ) λmin (VF )

(6.32)

x∈F

Proof. First, to show that λmin > 0, we show that 0 is not in the spectrum of VF . By way of contradiction, suppose ∃ξ : F → C such that ξ ≡ 0 and  vx , vy E ξy = 0. (VF ξ)(x) = y∈F

The vector ψ =

 y∈F

vy ξy ∈ V(F ) is nonzero by Lemma 6.16, and yet,

ψ(x) − ψ(o) = vx , ψE =



vx , vy E ξy = (VF ξ)(x) = 0.

y

< Hence, ψ is constant; therefore, ψ = 0 in HE .  So, 0 is not in the spectrum of VF . Then, in the notation of (6.30),

ψ 2E

 2      = δx , ψE vx  = δx , ψE vx , vy E δy , ψE   x∈F

x,y∈F

E

=



X ψ (x)vx , vy E X ψ (y)

x,y∈F

= X ψ , VF X ψ 1 , whence λmin X ψ 21 ≤ ψ 2E ≤ λmax X ψ 21 , and the conclusion (6.32)   follows from X ψ 21 = x∈F |δx , ψE |2 . Corollary 6.24. The frame bounds in (6.32) are optimal. Proof. Let ξ ∈ spec(VF ) and ξ : F → C with VF ξ = λξ. The vector  ξ = x∈F ξx vx is in HE by the proposition and ξ = δx , ψE = X ψ (x) for each x ∈ F by Lemma 6.16 and (6.30). Moreover, ψ 2E = ξ, VF ξ1 = λ ξ 21 =



|δx , ψE |2 .

x∈F

Applying this to λmin and to λmin shows that the bounds are optimal.



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121

In the next lemma, we use Δ specifically to indicate that the Laplacian is considered pointwise and without regard to domains. Lemma 6.25. X represents ΔV on (G0 ), i.e., Δ(Xψ) = X(ΔV ψ) for all ψ ∈ dom ΔV . Proof. Fix ψ ∈ dom ΔV and x ∈ G0 . Then, Δ(Xψ)(x) = c(x)X ψ (x) −



cxy X ψ (y)

y∼x

= c(x)δx , ψE −

cxy δy , ψE

y∼x

 =



c(x)δx −





cxy δy , ψ

y∼x

E

= Δδx , ψE . Now, since δx ∈ dom ΔV , we have Δ(Xψ)(x) = ΔV δx , ψE = δx , ΔV ψE =  X(ΔV ψ)(x). Lemma 6.26. For any ψ ∈ V(F ), we have ψ, ΔV(F ) ψE =  2  ψ 2 ψ   x∈F |X (x)| + x∈F X (x) . In particular, ΔV(F ) is nonnegative. Proof. Writing Δ for ΔV(F ) , this follows from ψ, ΔψE =



X ψ (x)X ψ (y)vx , Δvy E

x,y∈F

=



X ψ (x)X ψ (y)((δy (x) − δy (o)) − (δo (x) − δo (o)))

x,y∈F

=



X ψ (x)X ψ (y)(δy (x) + 1)

x,y∈F

=



 X ψ (x)X ψ (x) +

x∈F

 x∈F

⎞ ⎛  X ψ (x) ⎝ X ψ (y)⎠,

(6.33)

y∈F

where (6.33) follows because o ∈ / F. Incidentally, Lemma 6.26 offers a proof of Theorem 2.69.



Operator Theory and Analysis of Infinite Networks

122

Theorem 6.27. The following are equivalent : (i) ΔV is a bounded operator on HE . (ii) There is a global upper frame bound B < ∞ in (6.32), i.e.,  |δx , ψE |2 ≤ B ψ 2E , ∀ψ ∈ HE .

(6.34)

x=o

(iii) There is a spectral gap inf spec(VF ) > 0, where F runs over the set F of all finite subsets of G0 \ {o}. Proof. (i) =⇒ (ii): If ΔV is bounded, then by Lemma 2.16 followed by Corollary 2.69,   |δx , ψE |2 = |Δψ(x)|2 = ψ, ΔψE ≤ B ψ 2E . x=o

x=o

 (ii) =⇒ (i): First, fix ε > 0. Note that  x∈G0 Δψ(x) = 0 by  √ Corollary 2.68, so choose F so that  x∈F Δψ(x) < ε. The hypothesis of the global upper frame bound B gives   |Δψ(x)|2 = |δx , ΔψE |2 ≤ B ψ 2E x∈F

x∈F

so that Lemma 6.26 implies |ψ, ΔψE | ≤

 x∈F

 2     |Δψ(x)| +  Δψ(x) < B ψ 2E + ε,   2

x∈F

and we get |ψ, ΔψE | ≤ B ψ 2E as ε → 0. (i) ⇐⇒ (iii): Observe that (6.32) and Lemma 6.24 imply that λmin1 (F ) ≤ B, and hence, λmin (F ) ≥ 1/B, ∀F ∈ F . If we have an exhaustion F1 ⊆ F2 ⊆  · · · Fk = G0 \{o}, then the minimax theorem indicates that λmin (Fk+1 ) ≤ λmin (Fk ), so ΔV −1 = sup{ B1 ≥ 0 .. ψ, ΔV ψE ≤ B ψ 2E , .

= lim λmin (Fk ). k→∞

∀ψ ∈ V } 

Corollary 6.28. If {δx } is a dual frame for {vx }, then the upper and lower frame bounds A and B provide bounds on the free resistance metric: 2 2 ≤ RF (x, y) ≤ . B A

(6.35)

The 2 Theory of Δ and the Transfer Operator

123

Proof. By (3.10), we are motivated to apply the frame inequalities applied to the function vx − vy ∈ HE via Theorem 6.27:  A vx − vy 2E ≤ |δz , vx − vy E |2 ≤ B vx − vy 2E . z∈G0 \{o}

The result now follows by (3.10) since

 z∈G0 \{o}

|δz , vx − vy E |2 = 2.



Remark 6.29. Note that in Corollary 6.28, it is possible that A → 0 as F increases to G0 so that the upper bound tends to ∞. Lemma 6.30. For finite F ⊆ G0 , Harm ∩ V(F ) = ∅. A fortiori, ΔV has a spectral gap. n Proof. Let h = i=1 ci vxi . If h is harmonic, then    ci , ci δx i − δo 0 = Δh = ci (δxi − δo ) = which implies ci = 0 for each i since the Dirac masses are linearly independent vectors. The second claim follows because 0 is not in the point  spectrum of ΔV on the finite-dimensional space V(F ). The symmetry of formula (6.36) in x and y provides another proof that Δc0 (F ) is Hermitian. Lemma 6.31. For all x, y ∈ G0 , Δδx , δy E = −(c(x) + c(y))cxy +



cxz czy .

(6.36)

z∼x,y

Proof. For Δc0 (F ) , use z ∼ x, y to denote that z is a neighbor of both x and y, and compute    Δδx , δy E = c(x)δx − cxz δz , δy z∼x

= c(x)δx , δy E −



E

cxz δz , δy E

z∼x

= c(x)δx , δy E − cxy δy , δy E − = −c(x)cxy − cxy c(y) +





cxz δz , δy E

z∼x z=y

cxz czy

z∼x z=y

= −(c(x) + c(y))cxy +



z∼x,y

cxz czy .



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Operator Theory and Analysis of Infinite Networks

6.3.1 Projecting with respect to the dual variable In this section, we continue to use the notation VF (from Definition 6.21) for the |F | × |F | matrix with entries [VF ]x,y = vx , vy E for x, y ∈ F . Also, note that combining Lemmas 2.16 and 2.18 yields δx , vy E = δx (y) − δo (y),

x, y ∈ F,

(6.37)

which is a Kronecker delta under the additional assumption that o ∈ / F. This observation will help when studying the following projection operator. Definition 6.32. For a finite set F ⊆ G0 \ {o}, denote the operator PF : HE → V(F ). Theorem 6.33. For w ∈ HE and a finite set F ⊆ G0 \ {o}, let w|F denote the vector with entries [w|F ]x = w(x) − w(o) for x ∈ F . Then,  PF w = x∈F ξx vx is given by   ξx = ((VF )−1 w|F )(x) = (VF )−1 x,y (w(y) − w(o)). (6.38) y∈F

Proof. We require w − PF w, vx E = 0 for all x ∈ F , whence  w(x) − w(o) = PF w, vx E = ξy vy , vx E

(6.39)

y∈F

 so that ξ is the solution to VF ξ = w|F . See Remark 6.38 for an expression for PF in terms of an orthonormal basis of V(F ). In the following result, Δw(y) is computed pointwise via (1.4) and without consideration of domains. .

Corollary 6.34. For a finite set F ⊆ G0 \ {o} and w ∈ span{vx .. x ∈ F }, we have  Δw(x)vx . (6.40) w= x∈F

Proof. From Lemma 2.16, we have     ξy vy = ξy δx , vy E = ξx , Δw(x) = δx , wE = δx , y

E

where the last equality comes from (6.37).

(6.41)

y



The 2 Theory of Δ and the Transfer Operator

125

6.4 Spectral Reciprocity The exact relationship between V (or rather, its diagonalization D) and Δ is made precise in Corollary 6.54; see also Remarks 6.45 and 6.59. In this section and in later sections, we write 2 (G0 , 1) and 2 (F, 1) to clarify the support of the functions in question as needed. Definition 6.35. Denote the spectrum of VF by ΛF = {λF j } for some enumeration j = 1, 2, . . . , |F |. Note that λmin (VF ) > 0 by Theorem 6.23 and that VF is diagonalizable with eigenfunctions ξj = ξjF ∈ 2 (F, 1). That is, the spectral theorem provides an orthonormal basis (ONB) {ξjF } with VF ξj = λj ξj

for each j, F.

(6.42)

For convenience, we suppress the index and write (6.42) as VF ξλ = λξλ . Definition 6.36. For a finite F ⊆ G0 and VF ξλ = λξλ as above, define 1  uλ := √ ξλ (x)vx . λ x∈F

(6.43)

Lemma 6.37. The operator ΨF : 2 (F, 1) → V(F ) defined by ΨF (ξλ ) = uλ is unitary, and consequently, {uλ }λ∈ΛF is an orthonormal basis of V(F ). Proof. For x, y ∈ F , compute   1 1 ξj (x) ξk (y) vx , vy E = ξj (x)(VF ξk )(x), uj , uk E = λj λk x,y∈F λj λk x∈F and since ξk is an eigenvector, this continues as  √ λk  λk uj , uk E = ξj (x) ξk (x) = ξj , ξk  = δjk , λj λj 2

x∈F

where δjk is the Kronecker delta since {ξλ } is an ONB for 2 (F, 1).



Remark 6.38. By Lemma 6.37, we have an expression for the projection PF : HE → V(F ) of Definition 6.32 in terms of an orthonormal basis. In Dirac notation, this is  |uλ uλ |. (6.44) PF = λ∈ΛF

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Operator Theory and Analysis of Infinite Networks

Lemma 6.39. With respect to the ONB {uλ }, one has  λ1/2 ξλ (x)uλ for all x ∈ F. PF vx =

(6.45)

λ∈ΛF

Proof. Let x ∈ F . Then, compute   PF vx = |uλ uλ |vx E = uλ , vx E uλ λ∈ΛF

=

 λ∈ΛF

λ∈ΛF

1  √ ξλ (y) vy , vx E uλ λ y∈F

by (6.44) followed by (6.43). Continuing, PF vx =

 λ∈ΛF

 λ 1 √ (VF ξλ )(x)uλ = √ ξλ (x)uλ λ λ λ∈ΛF

since ξλ is an eigenvector. Note that λ ∈ R+ since V is positive semidefinite by assumption. It remains to observe that PF vx = vx for x ∈ F , but this follows from Definition 8.24.  Remark 6.40. In the language of Theorems A.17 and A.18, Equation (6.45) takes the following form:  √  v= (6.46) λ ξλ ⊗ uλ , λ∈ΛF

where {ξλ } is an ONB for 2 (F, 1) and {uλ } is an ONB for V(F ). The significance of this symmetric expression of v is that it allows us to compute a norm in HE (where the sum would be over x ∈ F ) by instead computing an 2 norm (where the sum is over λ ∈ ΛF ). For an example, see Corollary 6.58. In Chapter 7, we construct a Gel’fand triple SG ⊆ HE ⊆ SG and isometrically embed HE → L2 (SG , P). Here, SG is a space of “test functions” which is dense in HE but comes equipped with a strictly finer Fr´echet topology, and SG is a space of “distributions” obtained by taking the dual with respect to this topology. Elements u ∈ HE can then be extended to ˜(ξ) = u, ξE for ξ ∈ SG . As P is a probability measure, functions on SG via u one can then interpret {vx }x∈G as a stochastic process, i.e., a system of random variables indexed by the vertices of the underlying graph. In this context, (6.46) becomes an instance of the Karhunen–Lo`eve decomposition (see, for example, Ref. [AG75]) of a stochastic process into its random and

The 2 Theory of Δ and the Transfer Operator

deterministic components:  √  v˜x (ξ) = λ ξλ (x) ⊗ u ˜λ (ξ) ,

127

x ∈ G, ξ ∈ (SG , P).

(6.47)

λ∈ΛF

In fact, it turns out that {˜ uλ }λ∈ΛF is a system of independent identically distributed Gaussian N (0, 1) random variables for any finite F ⊆ G0 . See also Section 6.4.1 for more relations to Karhunen–Lo`eve. Definition 6.41. Denote the diagonalization of VF by ⎤ ⎡ λ1 ⎥ ⎢ ⎥ ⎢ λ2 ! ⎥ ⎢ λPuλ = ⎢ DF := ⎥, .. ⎥ ⎢ . λ∈ΛF ⎦ ⎣ λ|F |

(6.48)

where Puλ is a projection to span{uλ }λ∈ΛF . Note that DF−1 is a well-defined operator on 2 (F, 1) of rank |F | < ∞. Definition 6.42. If {ξλ } is an ONB of eigenvectors of VF , denote the expectation of ξλ by  E(ξλ ) = ξλ (x) = χF , ξλ  . (6.49) 2

x∈F

Lemma 6.43. If δo is a Dirac mass at the origin, the expansion of PF δo with respect to {uλ } is given by PF δo = −

 E(ξλ ) √ uλ . λ λ∈ΛF

(6.50)

Proof. Using PF = PF∗ , (6.43) and the fact that uλ ∈ V(F ), we compute the coefficients: 1  uλ , PF δo E = PF uλ , δo E = uλ , δo E = √ ξλ (x) vx , δo E λ x∈F 1  = −√ ξλ (x), λ x∈F where the last line follows from the dipole property since x = o.



Definition 6.44. Define Φ : 2 (G0 , 1) → HE Φδx = vx

on dom Φ = span{δx }x∈G0 .

For a finite set F ⊆ G0 \ {o} and ξ ∈ c0 (F ), we have Φξ =

 x∈G0

(6.51) ξ(x)vx .

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Operator Theory and Analysis of Infinite Networks

Remark 6.45. Observe that in terms of Φ, (6.25) can be rewritten as (ΔΦ(ξ))(x) = ξ(x), whence Φ is a right-inverse for Δ. On the other hand, δx , Φ Φδy  = vx , vy E = [V ]xy , 2

∀x, y ∈ G0 .

Therefore, since ξ ∈ c0 (F ) is finitely supported, we have     (V ξ)(x) = vx , vy E ξy = vx , ξy vy = vx , ΦξE y∈F

y∈F

(6.52)

(6.53)

E

so that (V ξ)(x) = Φξ(x) − Φξ(o) by the dipole property. This shows how close V is to being a right-inverse for Δ (i.e., a Green operator). Remark 6.46. The operator Φ is typically not closable. To see this, we show why the adjoint is not typically densely defined. First, pick ξ ∈ c0 (G0 ) and u ∈ V, and compute Φ :   ξ, Φ u1 = Φξ, uE = ξx vx , uE = ξx (u(x) − u(o)). x∈G0

x∈G0

So, for u ∈ HE , note that Φ u is the representative of u that vanishes at o. For u ∈ HE , let us denote by u(0) the representative of u specified by u(o) = 0 so that Φ u = u(0) . Then, 

dom Φ = {u ∈ HE .. u(0) ∈ 2 (G0 , 1)}. .

It is easy to see that this class is not dense in HE . Definition 6.47. Let PFo : HE → HE be the projection to the span of PF δo as in (6.50). That is, PFo = |PF δo PF δo | in Dirac notation. Definition 6.48. The compression of Δ to F is the restricted action of the operator Δ to V(F ), and it is given by PF ΔPF . Theorem 6.49 (Spectral reciprocity). If F ⊆ G0 \ {o} is nonempty and finite, then PF ΔPF = ΦDF−1 Φ + PFo .

(6.54)

Proof. For λ, κ ∈ ΛF , we have uλ , PF ΔPF uκ E = uλ , Δuκ E because uκ ∈ V(F ). Then, 1  uλ , PF ΔPF uκ E = √ ξλ (x)ξκ (y) vx , Δvy E λκ x,y∈F 1  =√ ξλ (x)ξκ (y)(δxy + 1) λκ x,y∈F

(6.55)

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129

by (6.43) and (12.6). The computation of (6.55) continues as 1  1  1  ξλ (x)ξκ (x) + √ ξλ (x) √ ξκ (y) = √ κ λκ x∈F λ x∈F y∈F 1  1 1 = √ ξλ (x)ξκ (x) + √ E(ξλ ) √ E(ξκ ). κ λκ x∈F λ

(6.56)

Since uλ is in dom Φ automatically for finite F , the right-hand side of (6.54) is $

uλ , (ΦDF−1 Φ + PFo )uκ

% E

% $ = Φ uλ , DF−1 Φ uκ  (F, 1) + uλ , PFo uκ E , 2

which matches with the right-hand side of (6.56) by (6.50). This verifies (6.54) on the orthonormal basis of Lemma 6.37 and hence for all of V(F ).  Remark 6.50. We refer to Theorem 6.49 as the spectral reciprocity theorem because it relates the eigenvalues of Δ to the reciprocal eigenvalues of its inverse on any finite F ⊆ G0 . Suppose one writes the matrix for Δ as in Definition 6.7 so that the ˜ be the matrix rows and columns are indexed by points of G0 . Let Δ which results from deleting the row and column corresponding to a chosen point o. Corollary 6.54 makes precise the well-known statement that one can invert the Laplacian after deleting the row and column corresponding to a point o.2 In particular, without deleting the row and column of o, one is forced to contend with an auxiliary term 1 in (12.6) (which corresponds to the projection P o = |oo| to the one-dimensional subspace spanned by δ0 ). Lemma 6.51. For every nested sequence of finite sets {Fn }n∈N with  Fn = G0 \ {o}, the limit of PFn ΔPFn exists and with dom Δ = V, Δ = lim PFn ΔPFn n→∞

(6.57)

in the strong operator topology, that is, limn→∞ PFn ΔPFn v − Δv E for all v ∈ dom Δ. 2 Recall that if M is a Hermitian matrix acting on a finite-dimensional Hilbert space H, then the restriction of M to the orthocomplement of the zero eigenspace is invertible.

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130

Proof. Let f ∈ V so that there is some finite set F ⊆ G0 \ {o} for which  ξx vx . f= x∈F 0 Without loss of generality, let {Fn }∞ n=1 be an exhaustion of G \ {o} with F ⊆ F1 . Then, PF f = f , and n→∞

PFn ΔPFn f = PFn Δf −−−−−→ Δf since PFn increases to the identity operator n→∞

PFn ΔPFn f − Δf E = (PFn − I)Δf E −−−−−→ 0.



Remark 6.52. Note that one does not expect the analogue of Lemma 6.51 to hold for the closure of the operator Δ with dom Δ = V for the same reason that one does not expect the boundary sum in Theorem 2.40 to vanish for general u ∈ HE . Indeed, for harmonic functions u, one cannot find n with F ⊆ Fn and PF f = f , as in the proof. See also Remark 6.59. Definition 6.53. Let P o := |δo δo | = proj δo be the rank-1 projection on H defined by u, P o wE = u, δo E δo , wE . Corollary 6.54. For dom Δ = V, the limit Δ − P o = limn→∞ ΦDF−1 Φ n exists for any exhaustion {Fn } of G0 \ {o}. Proof. Since arguments exactly analogous to those in Lemma 6.51 give n→∞ PFon −−−−−→ P o , we have Φ = lim PFn ΔPFn + lim PFon = Δ − P o lim ΦDF−1 n

n→∞

n→∞

n→∞



by applying Theorem 6.49 and then Lemma 6.51. 6.4.1 Spectral measures

In this section, we continue to study VF , where VF is the |F | × |F | matrix whose rows and columns are indexed by vertices of F ⊆ G0 and whose entries are given by [VF ]x,y = vx , vy E for a finite set F ⊆ G0 . Recall from Definition 6.42 that {ξλ } is an ONB of eigenvectors of VF corresponding to the spectrum ΛF = spec(VF ) = {λj }Jj=1 and that the eigenvectors have expectations  ξj (x) = χF , ξj  . E(ξj ) = 2

x∈F

The 2 Theory of Δ and the Transfer Operator

131

We also use the result from Lemma 6.37 that 1  uλ := √ ξλ (x)vx λ x∈F is an ONB for V(F ) and that from Lemma 6.43 that the expansion of PF δo with respect to {uλ } is PF δo = −

 E(ξλ ) √ uλ . λ λ∈ΛF

(6.58)

Definition 6.55. Since PF ΔPF = DF−1 + PF δo is the J × J matrix TF whose (j, k)th entry is given by τj,k =

E(ξj )E(ξj ) δj,k + , λj λj λk

denote the spectrum of this matrix TF = [τj,k ] by S F = spec(TF ) = {σjF }Jj=1 . Remark 6.56. In Definition 6.55, it is important to note that τj,k , ξj , and λj all depend on the choice of F . However, for ease of notation, we suppress this dependence and also, henceforth, write σj = σjF . Corollary 6.57. For any finite subset F ⊆ G0 \ {o} with |F | = J, one has 1 J 2 j=1 E(ξj ) = 1. J Proof. Since χF coincides with the constant vector 1 on F , we use Pξj u = ξj , u ξj to compute directly     2 χF , ξj  2 = |E(ξj )| = Pξj χF 2 = χF 21 = |F | = J.   2

2

j

j

j

Corollary 6.58. For any finite subset F ⊆ G0 \ {o} with |F | = J, one has J J o max λ ≤ PF ≤ min λ . Proof. Using Corollary 6.57 and Definition 6.47, PFo = PF δo 2E = J 2 j=1 E(ξj ) = J. See Remark 6.40. Then, the double inequality follows by estimating the largest (but clearly finite) eigenvalue and the smallest (but clearly strictly positive) eigenvalue.  Remark 6.59. When Δ is not essentially self-adjoint, the presence of PFo (as in (6.54), for example) makes it impossible to obtain self-adjoint extensions of Δ via a filtration by finite subsets. This obstacle can only be overcome by passing to spectral measures. If dom Δ = V, then the spectral

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measure of some self-adjoint extension of Δ comes from the weak- limit of linear combinations of equally weighted Dirac masses: μF =

J 1 δσ . J j=1 j

(6.59)

Here, μF refers to the spectral representation of PF ΔPF , and we are relying on standard tools from the literature. Indeed, approximation of measures with the use of spectral sampling is a versatile and powerful tool. For approximation in the weak- topology on measures (as in the current context), see the excellent reference book Ref. [Bil13] for details. When applied to spectral measures, these approximations were first studied in the book by Stone; see Ref. [Sto51, Ch. X]. The approach in Refs. [Sto51, Sto90] is especially amenable to our current applications: A main theme is the study of unbounded operators in Hilbert space, realized concretely as banded infinite matrices. This is illustrated in the following diagram. PF ΔP O F  μF

F →G0

weak-*

/Δ

⊆

/Δ ˜ O

(6.60)

 /μ ˜

In the limit of (6.59), as F → G0 , μF may become a smooth measure. The key point is that considering the limit of PF ΔPF as F → G0 does not take one far enough. However, a consideration of the spectral measures ˜ is the spectral measure of μF of PF ΔPF shows that each weak-* limit μ ˜ some self-adjoint extension Δ of Δ, and by general theory, every self-adjoint extension of Δ arises in this way. In the preceding discussion, F → G0 refers implicitly to a limit with ∞ respect to an exhaustion {Fn }n∈N , where Fk ⊆ Fk+1 and n=1 Fk = G0 \ {o}. Note that the limit Δ = limF →G0 PF ΔPF is unique (see Lemma 6.51) and hence independent of the choice of exhaustion {Fn }n∈N . However, the non-uniqueness of weak-* limits corresponds to the fact that μ ˜ = limF →G0 μF may depend on the choice of exhaustion. Different weak-* ˜ of Δ. limits may correspond to different self-adjoint extensions Δ 6.4.2 Spectral reciprocity for balanced functions Balanced functions are functions which sum to 0. In the context of resistance networks, a balanced function is the divergence of a current flow with no

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133

transient component; these functions are mentioned briefly in Ref. [Soa94, Section III.3]. Throughout this section, we make the standing assumption that F ⊆ G0 does not contain o. Definition 6.60. A function ξ : G0 → C is balanced iff ξ has finite support  and x∈G0 ξ(x) = 0. Denote the space of such functions by B. For any finite F ⊆ G0 , let BF denote the collection of functions in B whose support is contained in F . Recall from Definition 5.1 that V := span{vx }x∈G0 \{o} and V(F ) := .

span{vx .. x ∈ F } and from Definition 6.44 that Φ : 2 (G0 , 1) → H is given by Φ(δx ) = vx on dom Φ = c0 (G0 ) = span{δx }x∈G0 . Definition 6.61. Denote the subspace of V with balanced coefficients by .

V0 := Φ(B) = {Φ(ξ) .. ξ ∈ B},

(6.61)

.

and similarly for V0 (F ) := Φ(BF ) = {Φ(ξ) .. ξ ∈ BF }. The following curious fact can be found in most introductory books on functional analysis; also, see the discussion of Fin2 in Section 5.1.1. Proposition 6.62. Let X be a topological vector space, and let X0 be a dense linear subspace. If f is a linear functional on X0 , then ker f is dense in X if and only if f is discontinuous. Lemma 6.63. B is dense in 2 (G0 , 1) if and only if G0 is infinite.  Proof. Define f : B → C by f (ξ) = x∈G0 ξx . Note that G0 is finite if and only if the constant function 1 is in 2 (G0 , 1), which (by Riesz duality) holds if and only if f is continuous on 2 (G0 , 1). The result now follows from Proposition 6.62.  The next lemma indicates how Φ “intertwines” the spectral densities of Δ and V ; it expresses spectral reciprocity for the Rayleigh quotients of V and a flavor of Δ. Compare with Theorem 6.49. Theorem 6.64. For all ξ ∈ B, one has Φ(ξ), ΔΦ(ξ)E = ξ 21 and ξ, V ξ1 = Φ(ξ) 2E , and hence, ξ 21 Φ(ξ), ΔΦ(ξ)E = . 2 Φ(ξ) E ξ, V ξ1

(6.62)

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134

Proof. The first identity is immediate for ξ ∈ B by (2.68). For the second, note that       2 ξ(x)vx , ξ(y)vy = ξx ξy vx , vy E Φ(ξ) E = x∈G0

y∈G0

E

x∈G0 y∈G0

= ξ, V ξ1 .



Definition 6.65. We say that Δ has a spectral gap α > 0 iff α ψ 2E ≤ ψ, ΔψE

for all ψ ∈ V0 .

(6.63)

Theorem 6.66 (Spectral gap). Δ has a spectral gap α > 0 if and only if (6.52) defines a bounded self-adjoint operator V : 2 (G0 , 1) → 2 (G0 , 1) with V ≤ √1α . Proof. This follows immediately when either side of (6.62) is bounded from below by α > 0.  6.4.3 The quadratic form QV In Section 5.3, we studied the defect space Def −1 (ΔV ); we now turn to Def (ΔV ). Recall from Definition B.10 that Def (ΔV ) = {v ∈ . dom ΔV .. ΔV v = v}, and recall from Theorem B.11 that ΔV fails to be essentially self-adjoint precisely when Def (ΔV ) = ∅. In Theorem 6.68, we give a characterization of Def (ΔV ) = ∅ in terms of a quadratic form QV (ξ) (see (6.65)), which may be readily computed for a given electrical resistance network. For ξ : G0 → R, we denote  ξx vx . (6.64) w := Φ(ξ) = x=o

A priori, there is no reason for (6.64) to correspond to a function or an element of HE . Definition 6.67. We also consider the quadratic form associated with the countably infinite matrix V from Definition 6.21: ' &  .. ξx vx , vy E ξy < ∞ . F ⊆ G0 \ {o}, |F | < ∞ . (6.65) QV (ξ) := sup F

x,y

We consider the corresponding Hilbert space in Section 8.2; see Definition 8.10.

The 2 Theory of Δ and the Transfer Operator

135

Theorem 6.68. Let w be as in (6.64). Then, w is a defect vector iff ξ ∈ EigΔ ( ) ∩ dom QV . That is, w ∈ HE

Δ∗V w = w

and

(6.66)

if and only if QV (ξ) < ∞

and

Δξ = ξ.

(6.67)

Proof. We switch between function notation ξ(x) (as in the right-hand side of (6.67)) and vector notation ξx in the following computation:  (Δξ)(x) = c(x)ξ(x) − cxy ξ(y) by (1.4) y∼x

= c(x)δx , wE −

cxy δy , wE

y∼x

 =



c(x)δx −



Lemma 6.16



cxy δy , w

y∼x

linearity, cxy > 0 E

= Δδx , wE

by (1.4)

= δx , Δ wE

Definition of adjoint.

From this, it is clear that Δ∗V w = w holds if and only if Δξ = ξ. For the finiteness conditions, observe that for any finite set F , linearity gives 2       ξx vx  = ξx vx , vy E ξy .    x∈F

E

x,y∈F

Taking the supremum over finite sets F gives w ∈ HE iff QV (ξ) < ∞.



6.5 The Transfer Operator on 2 (1) Definition 6.69. We say the graph (G, c) satisfies the Powers bound iff c ∞ := sup c(x) < ∞.

(6.68)

x∈G0

The terminology “Powers bound” stems from Ref. [Pow76b], wherein the author uses this bound to study the emergence of long-range order in statistical models from quantum mechanics. Our motivation is somewhat different, and most of our results do not require such a uniform bound. However, when satisfied, it implies the boundedness of the graph Laplacian

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(and hence its self-adjointness) and the compactness of the associated transfer operator; see Section 6.5. The fact that the Powers bound entails the inclusion 2 (1) ⊆ HE (see Theorem 6.79) illustrates how strong this assumption really is. While the Laplacian may be unbounded for infinite networks in general, Theorem 6.70 gives one situation in which Δ is always bounded. To see the sharpness, note that this bound is obtained in the integer lattices of Example 14.2. In particular, for d = 1, we have Δ = sup |4(sin2 2t )| = 4 = 2 c . Theorem 6.70. As an operator on 2 (1), the Laplacian satisfies Δ 1 ≤ 2 c and hence is a bounded self-adjoint operator whenever the Powers bound holds. Moreover, this bound is sharp. Proof. Since Δ = c − T, this is clear by the following lemma.



Recall from Definition 1.9 that the transfer operator T acts on an element of dom Δ by  cxy v(y). (6.69) (T v)(x) := y∼x

One should not confuse T with the (bounded) probabilistic transition operator P = c−1 T; recall that the Laplacian may be expressed as Δ = c − T, where c denotes the associated multiplication operator. Note that T = c−Δ is Hermitian on 2 (1) by (6.7). This is a bit of a surprise since transfer operators are not generally Hermitian. Unfortunately, T1 may not be self-adjoint. In fact, the transfer operator of Example B.18 is not even essentially self-adjoint; see also Refs. [vN32a, Rud91, DS88]. Lemma 6.71. T1 ≤ c . Proof. Recall that T1 = T pointwise. The triangle inequality and Schwarz inequality give   √ √   cxy cxy f (y) |f (x)| |f, T f 1 | ≤ x∈G0

≤



y∼x

|f (x)|c(x)

x∈G0



≤

 1/2

 x∈G0



1/2 cxy |f (y)|

y∼x

2

⎞1/2 1/2 ⎛  ⎝ c(x)|f (x)|2 cxy |f (y)|2 ⎠ . x,y∈G0

The 2 Theory of Δ and the Transfer Operator

137

 1/2 Since both the factors above may be bounded above by c · f 21 (using another application of Schwarz for the one on the right), we have  |f, T f c | ≤ c · f 21 . Remark 6.72. When d = 1, Example 14.2 (the simple integer lattice) shows that the bound of Corollary 6.71 is sharp. From the proof of Lemma 14.3, one finds that T = sup |2 cos t| = 2 = 1 + 1 = c(n),

∀n ∈ Z.

Definition 6.73. Let cx be defined by cx (y) = cxy , so  cx · cx := c2xy .

(6.70)

y∼x

We denote this with the shorthand c2x = cx · cx . Theorem 6.74. If c is bounded, then T1 : 2 (1) → 2 (1) is bounded and self-adjoint. If T1 is bounded, then c2x is a bounded function of x. Proof. (⇒) The boundedness of T is Lemma 6.71. Any bounded Hermitian operator is immediately self-adjoint; see Definition B.2. (⇐) For the converse, suppose that c2x is unbounded. It follows that 0 2 there is a sequence {xn }∞ n=1 ⊆ G with cxn → ∞ and a path γ passing through each xn exactly once. Consider the orthonormal sequence {δxn }:    cxy δz (y) = czy δz (y), and T1 δz 21 = c2yz . T1 δz (x) = y∼x

y∼z

y∼z

Then, letting z run through the vertices of γ, it is clear that T1 δz 21 → ∞.  Recall from Definition 2.61 that u(x) vanishes at ∞ iff for any exhaustion {Gk }, one can always find k such that u(x) ∞ < ε for all x∈ / Gk . Using a nested sequence as described in Definition 2.61, it is not difficult to prove that T1 is always the weak limit of the finite-rank operators Tn . defined by Tn := Pn T1 Pn , where Pn is projection to Gn = span{δx .. x ∈ Gn } so that  cxy v(y). (6.71) Tn v(x) = χGn (x) (T1 v)(x) = y∼x y∈Gn

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Norm convergence does not hold without further hypotheses (see Example 14.24), but we do have Theorem 6.75, which requires a lemma. Theorem 6.75. If c ∈ 2 and deg(x) is bounded on G, then the transfer operator T1 : 2 (1) → 2 (1) is compact. If T1 is compact, then c2x vanishes at ∞. Proof. (⇐) Consider any nested sequence {Gk } of finite connected subsets  of G, with G = Gk , and the restriction of the transfer operator to these subgraphs, given by TN := PN T1 PN , where PN = PGN is projection to GN . Then, for DN := T1 − TN , consider the operator norm    0 PN T1 PN⊥    DN =  (6.72) ,  P ⊥ T1 PN P ⊥ T1 P ⊥  N

N

N

where the orthonormal basis for the matrix coordinates is given by {δxk }∞ k=1 for some enumeration of the vertices. Since deg(x) is bounded, the matrices for Δ1 and hence also T1 are uniformly banded; whence DN is uniformly bounded with band size bN and Lemma B.17 applies. Since the first N entries of DN +bN v are 0, we have b 2 ∞ ∞ m    cmnk = c(xm )2 , DN +bN v 2 = m=N +1

k=1

m=N +1

2

which tends to 0 for c ∈  (1). (⇒) For the converse, suppose that c2x does not vanish at ∞. It follows 0 2 that there is a sequence {xn }∞ n=1 ⊆ G , with cxn 1 ≥ ε > 0, and a path γ passing through each of them exactly once. By passing to a subsequence if necessary, is also possible to request that the sequence satisfies G(xn ) ∩ G(xn+1 ) = ∅,

∀n,

since the sequence need not contain every point of γ. Consider the orthonormal sequence {δxn }. We show that {T1 δxn } contains no convergence subsequence:   cx n y δx n − cx m z δx m T 1 δx n − T 1 δx m = y∼xn

z∼xm

T1 δxn − T1 δxm 2 = T1 δxn 2 + T1 δxm 2   = c2xn y + c2xm z ≥ 2ε. y∼xn

z∼xm

The 2 Theory of Δ and the Transfer Operator

139

There are no cross terms in the final equality by orthogonality; xn+1 was chosen to be far enough past xn that they have no common neighbors.  Corollary 6.76. If c vanishes at ∞ and deg(x) is bounded, then T1 is compact. Proof. The proof of the forward direction of Theorem 6.75 just uses the hypotheses to show that supx,y cxy can be made arbitrarily small by restricting x, y to lie outside of a sufficiently large set.  6.5.1 Fredholm property of the transfer operator A stronger form of the Theorem 6.78 was already obtained in Corollary 2.19, but we include this brief proof for its radically contrasting flavor. Definition 6.77. A Fredholm operator L is one for which the kernel and co-kernel are finite dimensional and whose range is therefore closed. In this case, the Fredholm index is dim ker L − dim ker L . Alternatively, L is a ˆ is invertible in the Calkin algebra. The Fredholm operator if and only if L Calkin algebra is the quotient of B(H), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H by the ideal K(H) of compact operators. Theorem 6.78. If c vanishes at infinity, then P(α, ω) is nonempty. Proof. When the Powers bound is satisfied, the previous results show Δ is a bounded self-adjoint operator, and T is compact. Consequently, Δ is a Fredholm operator. By the Fredholm alternative, ker Δ = 0 if and only if ran Δ = 2 (1). Modulo the harmonic functions, ker Δ = 0, so δα − δω has a  pre-image in 2 (1). 6.5.2 Some estimates relating HE and 2 (1) In this section, we make the standing assumption that the functions under consideration lie in HE ∩ 2 (1). Strictly, the elements of HE are equivalence classes, but each has a unique representative in 2 , and it is understood that we always choose this one. Our primary tool will be the identity E(u, v) = u, Δv1 from (6.6), which is valid on the intersection HE ∩ 2 (1). For example, note that this immediately gives v, ΔvE = Δv 21

and E(v) = Δ1/2 v 21 ,

(6.73)

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Operator Theory and Analysis of Infinite Networks

where the latter follows from the spectral theorem. Theorem 1.41 showed that P(α, ω) = ∅ for any choice of α = ω. It is natural to ask other questions in the same vein: (i) Is 2 (1) ⊆ HE ? No: Consider the one-dimensional integer lattice described in Example 14.26. (ii) Is HE ⊆ 2 (1)? No: Consider the function f defined on the binary tree in Example 15.3, which takes the value 1 on half the tree and −1 on the other half (and is 0 at o). This function has energy E(f ) = 2, but it is easily seen that there is no k for which f + k ∈ 2 (1). (iii) Does Δv ∈ 2 imply v ∈ HE or v ∈ 2 (1)? Neither of these are true, by the example in the previous item. (iv) Is P(α, ω) ⊆ 2 ? No: Consider again the one-dimensional integer lattice, with α < ω. Then, if v ∈ P(α, ω), it will be constant (and equal to v(α)) for xn to the left of α, and it will be constant (and equal to v(ω)) for xn right of ω. We have some results in these directions under further restrictions: (i) When the Powers bound (6.68) is satisfied, then 2 (1) ⊆ HE . See Theorem 6.79. (ii) When the Powers bound (6.68) is satisfied, then v ∈ HE ⇐⇒ Δv ∈ HE . See Theorem 6.104. (iii) If Δv ∈ 2 (1), then v ∈ HE . Moreover, if Δ has a spectral gap, then the converse holds. See Theorem 6.105. Lemma 6.79. v E ≤ Δ1/2 · v 1 for every v ∈ HE . If the Powers bound (6.68) is satisfied, then 2 (1) ⊆ HE . Proof. Since v 2E = v, Δv1 , this is immediate from Lemma 6.70.



Lemma 6.80. If Δ is bounded on 2 (1), then it is bounded with respect to E. Proof. The hypothesis implies Δ is self-adjoint on 2 so that one can ˆ on L2 (G0 , dν) and perform the following take the spectral representation Δ computation: ˆ ∞ · Δ1/2 v 1 = Δ ˆ ∞ · v E . Δv E = ΔΔ1/2 v 1 ≤ Δ



Lemma 6.81. Let v ∈ 2 (1). If v ≥ 0 (or v ≤ 0), then v E ≤ v 1 . If v is bipartite and alternating, then v E ≥ v 1 .

The 2 Theory of Δ and the Transfer Operator

141

Proof. Both statements follow immediately from the equality  cx,y v(x)v(y). E(v) = v, Δv1 = v, v1 − v, T v1 = v 1 −



y∼x

6.6 The Laplacian and Transfer Operator on 2 (c) In Sections 6.1–6.5, we studied Δ and T as operators on the unweighted space 2 (1). In this section, we consider the renormalized versions of these operators and attempt to carry over as many results as possible to the context of 2 (c): . (6.74) 2 (c) := {u : G0 → C .. x∈G0 c(x)|u(x)|2 < ∞}, with the inner product u, Δvc :=



c(x)u(x)Δv(x).

(6.75)

x∈G0

6.6.1 The Laplacian on 2 (c) Take the operator Δ defined on span{δx }, the dense domain consisting of functions with finite support. Then, let Δc denote the closure of Δ with respect to (6.75), that is, its minimal self-adjoint extension to 2 (c). Lemma 6.82. For u ∈ 2 (c) and v ∈ HE ,  √ |u(x)Δv(x)| ≤ 2 u c · v E .

(6.76)

x∈G0

Proof. Apply the Schwarz inequality twice, first with respect to the x summation, then with respect to y:   √   cxy u(x)√cxy (v(x) − v(y)) |u(x)Δv(x)| = x∈G0

x,y∈G0

≤





y∈G0

⎛ ≤⎝



1/2  cxy |u(x)|

2

x∼y

 x,y∈G0



1/2 cxy |v(x) − v(y)|

2

x∼y

⎞1/2 ⎛ cxy |u(x)|2⎠

⎞1/2  ⎝2 cxy |v(x) − v(y)|2⎠ , 2 0 x,y∈G

and the resulting inequality retroactively justifies the implicit initial Fubination. 

Operator Theory and Analysis of Infinite Networks

142

Definition 6.83. For u ∈ c0 (G0 ), define a linear functional Λu on HE by  Λu (v) := u(x)Δv(x). (6.77) x∈G0

Observe that Λu is continuous because |Λu (v)| ≤ u c · v E by Lemma 6.82, whence Riesz’s lemma gives a w ∈ Fin for which Λu (v) = w, vE

holds for every v ∈ Fin.

(6.78)

Definition 6.84. Let J : 2 (c) → HE denote the map which sends u → w, where w is as in (6.78). In other words, J is defined by Ju, vE = Λu (v).

(6.79)

Definition 6.84 allows one to see directly that Jδx = [δx ]:  Jδx , vE = δx (y)Δv(y) = Δv(x) = δx , vE for all v ∈ HE . y∈G0

This idea is the reason for Definition 6.84 and also Theorem 6.85. It is also easy to see that δx 2c = c(x) = δx 2E , although the two norms · c and  · E are clearly different in general.3 In fact, if ϕ = x∈F ξx δx ∈ c0 (G0 ) (so F is finite), then one may easily compute   ϕ 2c = c(x)|ξ|2 , whereas Jϕ 2E = ϕ 2c − cxy ξx ξy . x∈F y∼x

x∈F

Theorem 6.85. The map J is the quotient map induced by the equivalence relation u  v iff u − v = const and gives a continuous embedding of 2 (c) into Fin with √ (6.80) Ju E ≤ 2 u c , ∀u ∈ 2 (c). Furthermore, the closure of ran J with respect to E is Fin. Proof. The formulation of J in (6.79) gives   Ju, vx E = Λu (vx ) = u(y)Δvx (y) = u(y)(δx − δo )(y) x∈G0

x∈G0

= u(x) − u(o). 3 Note

that we have suppressed the J in this equation; it is more properly written as δx 2c = c(x) = Jδx 2E .

The 2 Theory of Δ and the Transfer Operator

143

This shows that J is the quotient map as claimed. The bound (6.80) follows immediately upon combining (6.76) with (6.77). Now, let w = Ju for any u ∈ 2 (c), and apply Λu to v = fx = PFin vx ∈ Fin to get  w(x) − w(o) = fx , wE = Λu (fx ) = u(y)(δx − δo )(y) = u(x) − u(o). y∈G0

Since {fx } is thus a reproducing kernel for any element of ran J, this shows that ran J ⊆ Fin, and hence, √ √ |Ju, vE | = |Λu (v)| ≤ 2 u c · v E =⇒ Ju E ≤ 2 u c. The E-closure of ran J is equal to Fin because ran J contains span{δx }.



Lemma 6.86. The adjoint map J  : HE → 2 (c) is given by J  u = u − Pu,

(6.81)

where P is the probabilistic transition operator defined in (3.41). Proof. First, let u ∈ span{δx } and v ∈ HE , and note that Ju, vE = u, Δv1 by (6.6). Then,     1  u(x)Δv(x) = u(x)c(x) v(x) − cxy v(y) , u, Δv1 = c(x) y∼x 0 0 x∈G

x∈G

whence Ju, vE = u, (I − P)vc on the subspace span{δx }, which is dense  in Fin in the norm · E and dense in 2 (c) in the norm · c . Remark 6.87. Lemma 6.86 provides another proof of Theorem 6.85. Proof (Alternative proof of Theorem 6.85). Suppose that v ∈ ran(J)⊥ ⊆ HE so that Ju, vE = 0 for all u ∈ 2 (c). Then, Ju, vE = u, J  vc = u, v − Pvc ,

∀u ∈ 2 (c),

by (6.81) so that v −Pv = 0 in 2 (c). Then, recall that v ∈ Harm iff Δv = 0 clo = iff Pv = v. This shows that ran(J)⊥ = Harm, and hence ran(J) ⊥ ⊥⊥ = Harm = Fin.  ran(J) Corollary 6.88. If Δu = −u, then O( c(x)), as x → ∞.



1 2 c(x) |u(x)|

< ∞ and u(x) =

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Proof. Recall that a defect vector u satisfies Δu = −u ∈ HE and hence u − Pu = − 1c u. The result follows by substituting the latter into (6.82). 1 |u(x)|2 ≤ B, which gives the final This immediately implies a bound c(x) claim.  Recall that ΔV denotes the closure of the Laplacian when taken to have the dense domain V := span{vx }x∈G0 \{o} of finite linear combinations of dipoles. Corollary 6.89. If c(x) is bounded on G0 and deg(x) < ∞, then ΔV is essentially self-adjoint on HE . Proof. Suppose w ∈ dom ΔV satisfying Δ w = −w. This means there is a K (possibly depending on w) such that |w, ΔvE | ≤ K v E for all v ∈ V.  Then, for u ∈ 2 (c) and v ∈ span{vx }, set Λu (v) = x∈G0 u(x)Δv(x). As in the proof of Theorem 6.85, Λu extends to a continuous linear functional on HE , so applying it to w gives                   u(x)w(x) =  u(x)(−w(x)) =  u(x)Δw(x) = |Λu (w)|        0 0 0 x∈G

x∈G

x∈G

√ ≤ 2 u c · w E .

However, if c(x) is bounded by c , then  u c =



1/2 c(x)|u(x)|

2

≤ c 1/2 u 1.

x∈G0

Combining the two displayed equations above yields the inequality   √   u(x)w(x) ≤ 2 c 1/2 u 1 · w E .   This shows that u → x∈G0 u(x)w(x) is a continuous linear functional on 2 (c) so that Riesz’s lemma puts w ∈ 2 (1). However, now that w is a defect  vector in 2 (1), Theorem 6.1 applies, and hence, w = 0. Definition 6.90. Let Δc := c−1 Δ = I − P = JJ  denote the probabilistic Laplace operator on HE , as in (1.6). Note that we abuse notation here in the suppression of the quotient map so that I − P denotes an operator on HE and a mapping HE → 2 (c).

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145

Corollary 6.91. For any v ∈ HE , Δc is contractive on HE and (I − P)v ∈ 2 (c) with (I − P)v E ≤ v E . Proof. Since J is contractive, it follows that J  is contractive by basic operator theory; this is a consequence of the polar decomposition applied to J. Then, Δc = JJ  is certainly continuous with  c(x)|v(x) − Pv(x)|2 = J  v 2c ≤ 2 v 2E < ∞ (6.82) x∈G0

for every v ∈ HE . Furthermore, as an operator into HE , √ (I − P)v E = JJ  v E ≤ 2 J  v c ≤ 2 v E , which shows that Δc is even contractive in this context.

(6.83) 

Remark 6.92. It is intriguing to note that for v ∈ HE , one has v − Pv ∈ 2 (c), even though it is quite possible that neither v nor Pv lies in 2 (c) (for an extreme example, consider v ∈ Harm). Observe that (6.82) implies a bound c(x)|v(x) − Pv(x)|2 ≤ B 2 , whence |v(x) − Pv(x)| ≤ Bc(x)−1/2 . Consequently, if {Gk } is any exhaustion of G, then v ≈ Pv on Gk for large k (in the sense of 2 (c)). Roughly, one can say that any v ∈ HE tends to being a harmonic function at ∞, and the faster c grows, the better the approximation. Lemma 6.93. Δc is Hermitian if and only if c(x) is a constant function on the vertices. Proof. This can be seen by computing the matrix representation of Δc with respect to the orthonormal basis { √δx }, in which case the (x, y)th c(x)

entry is 

[AΔc ]x,y

 δy δx , Δ = c(x) c(y) c    δt (z) δx (z) c(y)δy (z)  − = c(z) cty c(x) c(y) c(y) 0 t∼y z∈G

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=

 z∈G0



δt (z) c(z)δx (z) c(y)δy (z) c(z)δx (z)  cty − c(x) c(y) c(x) t∼y c(y)

= c(x)δxy −

 δt (x) c(x) cty c(y) t∼y

= c(x)δxy −

 δy , c(x) cxy c(y) y∼x

which is not symmetric in x and y. We used Δδy = c(y)δy − which follows easily from Lemma 2.26.





t∼y cty δt ,



6.6.2 P and T on 2 (c) Theorem 6.94. Let J : 2 (c) → HE by Jδx = δx , and let J  be the adjoint taken with respect to these inner products. Then: (i) As an operator on 2 (c), P = I − J  J is self-adjoint with −I ≤ P ≤ I. (ii) As an operator on HE , P = I − JJ  is self-adjoint with −I ≤ P ≤ I. Proof. (i) Since J is bounded and hence closed, a theorem of von Neumann implies J  J is self-adjoint. Then, u, Puc = u, (I − J  J)uc = u, uc − Ju, JuE = u 2c − Ju 2E . Since Ju 2E ≤ 2 u 2c by (6.80), this establishes − u 2c ≤ u, Puc ≤ u 2c . (ii) By the same argument as in part (i), JJ  is self-adjoint. Then,  JJ  = J  J gives the same bound for P on HE . Remark 6.95. One can also see that P is self-adjoint by independent  arguments. For 2 (c), we have u, Pvc = x,y u(x)cxy v(y) = Pu, vc , and for HE , we have Pvx , vy E = vx − c−1 (δx − δo ), vy E = vx , vy E − c−1 (δx − δo )(y) + c−1 (δx − δo )(o) = vx , vy E −

1 c(o)

where δxy is the Kronecker delta.

−

δxy c(x) ,

(6.84)

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147

Remark 6.96. von Neumann’s ergodic theorem implies that N 1  n P u = PHarm u. N →∞ N n=1

lim

In general, it is difficult to know when one has the stronger result that Pn u → PHarm u. A Perron–Frobenius–Ruelle theorem would require an invariant measure with certain properties not satisfied in the current context. Nonetheless, one can see that limn→∞ Pn u lies in Harm; this is shown in Theorem 6.99. Lemma 6.97. For all ϕ ∈ span{vx }, one has Pϕ, ϕE = ϕ 2E −

 |Δϕ(x)|2 . c(x) 0

(6.85)

x∈G

Proof. For some finite set F ⊆ G0 and ϕ = Pϕ, ϕE =

 x,y∈F

 x∈F

ax vx ,

 2  |ax |2 1    ax ay vx , vy E − ax  −   c(o)  c(x) 0 0 x∈G

x∈G

2     |Δϕ(x)|2 1   , Δϕ(x) − = ϕ 2E −   c(o)  c(x) 0 0 x∈G

x∈G

where we have used (2.28) and (6.84) for the last step. Then, Theorem 2.68 shows that the middle sum vanishes, and we have (6.85).  Corollary 6.98. For ϕ ∈ span{vx },

 x∈G0

|Δϕ(x)|2 c(x)

≤ 2 ϕ 2E .

Proof. Apply (6.85) to the bound −I ≤ P ≤ I from Theorem 6.94.



Theorem 6.99. P is strictly contractive on Fin. Proof. Note that no harmonic functions lie in span{vx } (by Lemma 2.67, 2  for example), so x∈G0 |Δϕ(x)| > 0.  c(x) For Theorem 6.101, we need to consider iterates Pn of the probabilistic (n) transition operator, the induced conductances cxy , and the corresponding

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energy spaces HE (c(n) ). From      2 (P u)(x) = P p(x, y)u(y) = p(x, y) p(y, z)u(z) y∼x

  cxy cyz = u(z), c(x) c(y) y∼x z∼y 

we take p2 (x, y) = ating, we obtain c where

y∼x

z∼y

(2)

z∼x,y (n)

p(x, z)p(z, y) and define cxy := c(x)p2 (x, y). Iter(n)

and the spaces HE

E (n) (u) :=

.

:= {u : G0 → C .. E (n) (u) < ∞},

1  (n) cxy |u(x) − u(y)|2 . 2 0

(6.86)

x,y∈G

Here, we use E (n) as shorthand for E (b) in the case when b = c(n) (see (12.2)) (n) in order to avoid the heavy notation E (c ) . Lemma 6.100. For each n = 1, 2, . . . , one has c(x) =



(n) y∈G0 cxy .

  (2) (2) (x, y) = c(x) shows Proof. From above, y∈G0 cxy = c(x) y∈G0 p that the sum at x does not change. The case for general n follows by iterating.  In the proof of the following theorem, we write Δn := c(I − Pn ), which is not the same as Δn . Also, we abuse notation and write 2 (c) for J(2 (c)), in accordance with Theorem 6.85. Theorem 6.101. P is densely defined on HE with Pu c ≤ u c + u E ,

∀u ∈ J(2 (c)).

(6.87)

In fact, P(n) u c ≤ u c + u E (n) for every n ≥ 1. Proof. Note that it immediately follows from Lemma 6.100 that 2 (c) ⊆ ( (n) (n) and hence that there is a Jn : 2 (c) → HE for which n HE Jn u, vE (n) =



u(x)Δn v(x) =

x∈G0

= u, vc − u, Pn vc .

 x∈G0

u(x)c(x) (v(x) − Pn v(x))

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149

Observe that the rearrangement in the last step is justified by the convergence of Jn u, vE (n) and u, vc . The Schwarz inequality gives Pn v ∈ 2 (c) and |u, Pn vc | ≤ |u, vc | + |Jn u, vE (n) | ≤ u c v c + u c v E (n) . This shows that Pn v is in the dual of 2 (c) and hence in 2 (c). By Riesz’s theorem, Pu c is the best constant possible in the above inequality, and so, (6.87) follows.  Theorem 6.102. The adjoint of T1 : HE → HE is Δ−1 1 T1 Δ1 . Proof. Working directly with the inner product, −1 Δ−1 1 T Δ1 u, vE = Δ1 T Δ1 u, Δ1 v1 = T Δ1 u, v1

= Δ1 u, T v1 = u, T vE .



Recall from Definition B.5 that the graph of Δ is v Graph(Δ) := {[ Δv ] .. v, Δv ∈ 2 (1)} ⊆ 2 ⊕ 2 . .

(6.88)

v Theorem 6.103. The map ϕ : Graph(Δ) → HE by ϕ : [ Δv ] → v is contractive. If G is infinite, then ker ϕ = 0.

Proof. Observe that v ∈ dom ϕ iff v, Δv ∈ 2 (1), so Lemma 1.13 guarantees v ∈ HE . Contractivity follows by combining Lemma 2.71 with the Schwarz inequality: v 2E = v, Δv1 ≤ v 1 Δv 1 ≤

1 2

 v 21 + Δv 21 .

The map ϕ can only fail to be injective if there exist elements of 2 (1) which differ only by a constant. However, this cannot happen when G is infinite,   as x (v(x) + k) can be finite for at most one value of k. Corollary 6.104. Assume (G, c) satisfies the Powers bound (6.68). Then, Δ1 v E ≤ c · v E . If Δ1 has a spectral gap δ := inf spec Δ1 > 0, then v E ≤ δ Δ1 v E . Proof. For the first claim, Theorem 1.13 gives 3/2

1/2

Δ1 v 2E = Δ1 v, Δ21 v1 = Δ1 v 21 ≤ Δ1 2 · Δ1 v 21 = Δ1 2 · v 2E .

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For the second claim, let δ denote the spectral gap as in the hypotheses. The Powers bound ensures an upper bound K on the spectrum of Δ1 . Since s ≥ 1δ if and only if s ≤ δs2 , we have  K  K 1/2 2 2 2 s|ˆ v | dξ ≤ δs2 |ˆ v |2 dξ = δ Δ1 v 21 . v E = Δ1 v 1 =  δ

δ

Corollary 6.105. Assume (G, c) satisfies the Powers bound (6.68). Then, v ∈ 2 (1) if and only if Δv ∈ 2 (1). One may also define the action of Δ on a function v ∈ HE by (Δv)(u) := E(u, v) = u, Δv1 .

(6.89)

Proposition 6.106. The formula (6.89) extends the usual definition of Δ as defined on [vx ] ⊆ 2 (1), where Δvx = δx − δo . Proof. Suppose v ∈ HE and u = δx . Then, (Δv)(u) = δx , Δv1 = Δv(x).



6.7 Remarks and References Of the results in the literature of relevance to this chapter, Refs. [KL12, KL10, Dod06, DR08, Sto08, DEIK07, BB05, Chu07, CR06, BLS07, CM20] are especially relevant. The reader may also wish to consult Refs. [Nel73a, vN32a], and good background references include Refs. [DS88, RS75, Arv02, Chu96, LP89].

Chapter 7

The Boundary and Boundary Representation Nature is an infinite sphere of which the center is everywhere and the circumference nowhere. — B. Pascal

Boundary theory is a well-established subject; the deep connections between harmonic analysis, probability, and potential theory have led to several notions of boundary; see the Remarks and References section at the end of this chapter. Most material in this section originally appeared in Ref. [JP11a]. 7.1 Motivation and Outline Recall the classical result of Poisson that gives a kernel k : Ω × ∂Ω → R from which a bounded harmonic function can be given via  u(y)k(x, dy), x ∈ Ω , y ∈ ∂Ω . (7.1) u(x) = ∂Ω

Here, the probabilist’s notation k(x, dy) = k(x, y)dy is used to emphasize the fact that the kernel incorporates a measure on ∂Ω , which may or may not be a function on ∂Ω . The material in Chapter 7 is motivated by the discrete analogue of the Poisson boundary representation of a harmonic function which appeared in Corollary 2.42 and which we restate here for convenience.

151

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Theorem 7.1 (Boundary sum representation of harmonic functions). For all u ∈ Harm and hx = PHarm vx ,  x u(x) = u ∂h + u(o). (7.2) ∂ bd G

Up to this point, the boundary sum in (7.2) has been understood only as a limit of sums. Comparison of (7.2) and (7.1) makes one optimistic that bd G can be realized as some compact set which supports a “measure” ∂hx ∂ , thus giving a nice representation of the boundary sum of (7.2) as an integral. In Theorem 7.32, we extend Theorem 7.1 to such an integral representation. Boundary theory of harmonic functions can roughly be divided into three ways: the bounded harmonic functions (Poisson theory), the nonnegative harmonic functions (Martin theory), and the finite-energy harmonic functions studied in this book. While Poisson theory is a subset of Martin theory,1 the relationship between Martin theory and the study of HE is more subtle. For example, there exist unbounded functions of finite energy; cf. Examples 14.27 and 15.9 and the illustration on the front of this volume. Some details are given in Ref. [Soa94, Section 3.7]; see also Refs. [Woe09, KV83, Kai96, Kai92b]. Whether the focus is on the harmonic functions which are bounded, nonnegative, or finite-energy, the goals of the associated boundary theory are essentially the same: (1) Compactify the original domain D by constructing/identifying a boundary bd D. Then, D = D ∪ bd D, where the closure is with respect to some (hopefully natural) topology. (2) Define a procedure for extending harmonic functions u from D to bd D. This extension u ˜ is typically a measure (or other linear functional) on bd D; it may not be representable as a function. (3) Obtain a kernel (x, β) defined on D × bd D, against which one can integrate the extension u ˜ so as to recover the value of u at a point in D:  u(x) = (x, β)˜ u (dβ), ∀x ∈ D, bd D

whenever u is a harmonic function of the given class. The difference between our boundary theory and that of Poisson and Martin is rooted in our focus on HE rather than 2 : Both of these classical 1 Alas! This is an oversimplification; Poisson boundary is fundamentally a measuretheoretic construction, while Martin boundary is a topological construction.

The Boundary and Boundary Representation

153

theories concern harmonic functions with growth/decay restrictions. By contrast, provided they neither grow too wildly nor oscillate too wildly, elements of HE may remain positive or even be unbounded. See Example 15.9 for a function h ∈ Harm which is unbounded. Our boundary essentially consists of infinite paths which can be distinguished by monopoles, i.e., two paths are not equivalent iff there is a monopole w with different limiting values along each path. It is an immediate consequence that recurrent networks have no boundary, and transient networks with no nontrivial harmonic functions have exactly one boundary point (corresponding to the fact that the monopole at x is unique). In particular, the integer lattices (Zd , 1) each have one boundary point for d ≥ 3 and an empty boundary for d = 1, 2. In contrast, the Martin boundary of (Zd , 1) is homeomorphic to the unit sphere S d−1 (where S 0 = {−1, 1}), and each (Zd , 1) has only one graph end (except for (Z, 1), which has two); cf. Ref. [PW90, Section 3.B]. In our version of the program outlined above, we follow the steps in the order (2)–(3)–(1). A brief summary is given here; further introductory material and technical details appear at the beginning of each section: For (2), we construct a Gel’fand triple SG ⊆ HE ⊆ SG to extend the energy o integration form to a pairing · , · W on SG × SG and then use Itˆ to extend this new pairing to HE × SG . This yields a suitable class of linear functionals ξ on HE , and we can extend a function u on ˜ on SG by duality, i.e., u ˜(ξ) := u, ξ W . We need to expand HE to u the scope of our inquiry to include SG because HE will not be sufficient; no infinite-dimensional Hilbert space can support a σfinite probability measure by a theorem by Nelson [Nel64, Gro67, Gro70, Min63]. For (3), we use the Wiener transform to isometrically embed HE into L2 (SG , P). Applying this isometry to the energy kernel {vx }, we get a reproducing kernel (x, dP) := hx dP, where hx = PHarm vx and P is a version of the Wiener measure. In fact, P is a Gaussian probability measure on SG whose support is disjoint from Fin. For (1), we consider certain measures μx , defined in terms of the kernel and the Wiener measure just introduced, which are supported on SG /Fin and indexed by the vertices x ∈ G0 . Then, the elements of the boundary bd G correspond to the limits of sequences {μxn }, where xn → ∞, modulo a suitable equivalence relation. This is the content of Section 7.3.

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Items (2) and (3) are the content of Section 7.2, and the main result is Theorem 7.20 (and its corollaries). Due to the close relationship between the Laplacian and the random walk on a network, there are good intuitive reasons why one would expect stochastic integrals (by which we mean the Wiener transform) to be related to the boundary. “Going to the boundary” of (G, c) involves a suitable notion of the limit, and it is a well-known principle that suitable limits of the random walk yield Brownian motion realized in L2 -spaces of global measures (e.g., the Wiener measure). However, before this program can proceed, we need a suitable dense subspace SG ⊆ HE of “test functions” for the construction of a Gel’fand triple. The basic idea is to use the “smooth functions,” that is, u ∈ HE for which Δ(. . . Δ(u)) ∈ HE for any number of applications of Δ. Making this precise requires a certain amount of attention to technical details concerning the domain of Δ, which relies on the results of Section 5.2. Finally, we examine the connection between the defect spaces of Δ and bd G via the use of the boundary form introduced in Section 5.3.1, in parallel to those of classical functional analysis. The reader is directed to Appendix B for a brief review of some of the pertinent ideas from operator theory, especially regarding the graph of an operator (Definition B.5) and von Neumann’s theorem characterizing essential self-adjointness (Theorem B.11). Note: In several parts of this section, we use vector space ideas that are not so common when discussing Hilbert spaces, e.g., finite linear span and (not necessarily orthogonal) linear independence. Most of the material in this chapter originally appeared in Ref. [JP11a]. Remark 7.2. In Chapter 11, we return to the three-way comparison of harmonic functions which are bounded, nonnegative, or finite-energy but for a different purpose: the construction of measures on spaces of (infinite) paths in (G, c). In the case of bounded harmonic functions on (G, c), the associated probability space is derived directly as a space of infinite paths in G, and the measure is constructed via the standard Kolmogorov consistency method, that is, as a projective limit constructed via cylinder sets. While the current construction is also implicitly in terms of cylinder sets (due to Minlos’ framework), the reader will note by comparison that the two probability measures and their support are quite different. As a result, the respective kernels take different forms. However, both techniques yield a way to represent the values h(x) for h harmonic and x ∈ G0 as an integral over “the boundary.”

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155

While Doob’s martingale theory works well for harmonic functions in L∞ or L2 , the situation for HE is different. The primary reason is that HE is not immediately realizable as an L2 space. A considerable advantage of our Gel’fand–Wiener–Itˆo construction is that HE is isometrically embedded into L2 (SG , P) in a particularly nice way: It corresponds to the polynomials of degree 1. See Remark 7.24. Recall that Corollary 2.10 shows that span{vx } is dense in HE and that {vx } is a reproducing kernel for HE . Throughout Chapter 7, we implicitly use the version of Δ introduced in Definition 2.32, which we now recall for convenience. Definition 7.3. Let M := span{vx , wxv , wfx }x∈G0 denote the vector space of finite linear combinations of dipoles. Let ΔM : HE → HE be the closure of the Laplacian when taken to have the dense domain M. Caution: When studying an operator, an important subtlety is that “the” adjoint Δ depends on the choice of domain, i.e., the linear subspace dom(Δ) ⊆ H. In Chapter 6, we considered Δ as an operator mapping between different Hilbert spaces; the adjoint there is different from what we consider here. Note that since Δ agrees with ΔM pointwise, we may suppress the reference to the domain for ease of notation. Recall from Corollary 2.69 that ΔM is Hermitian and even semibounded on its domain. See Chapter 5 for more about the properties of ΔM , including its range, domain, and self-adjoint extensions. 7.2 Gel’fand Triples and Duality According to the program outlined above, we would like to obtain a (probability) measure space to serve as the boundary of G. It is shown in Refs. [Nel64, Gro67, Gro70, Min63] that no infinite-dimensional Hilbert space of functions H is sufficient to support a Gaussian measure P (i.e., it is not possible to have 0 < P(H) < ∞ for a σ-finite measure). However, it is possible to construct a Gel’fand triple (also called a rigged Hilbert space): a dense subspace S of H with S ⊆ H ⊆ S ,

(7.3)

where S is dense in H and S  is the dual of S. Additionally, S and S  must also satisfy some technical conditions: S is a Fr´echet space in its

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own right but realized as dense subspace in H, with density referring to the Hilbert norm in H. However, S  is the dual of S with respect to a Fr´echet topology defined via a specific sequence of seminorms. Finally, it is assumed that the inclusion mapping of S into H is continuous in the respective topologies. It was Gel’fand’s idea to formalize this construction ˇ abstractly using a system of nuclearity axioms [GMS58, Min58, Min59]. Our presentation here is adapted from quantum mechanics, and the goal is to realize bd G as a subset of S  . There is a concrete situation when the Gel’fand triple construction is especially natural: H = L2 (R, dx) and S is the Schwartz space of functions of rapid decay. That is, each f ∈ S is C ∞ smooth functions which decay (along with all its derivatives) faster than any polynomial. In this case, S  is the space of tempered distributions, and the seminorms defining the Fr´echet topology on S are .

pm (f ) := sup{|xk f (n) (x)| .. x ∈ R, 0 ≤ k, n ≤ m},

m = 0, 1, 2, . . . ,

where f (n) is the nth derivative of f . Then, S  is the dual of S with respect to this Fr´echet topology. One can equivalently express S as .

˜ 2 )n f ∈ L2 (R), ∀n}, S := {f ∈ L2 (R) .. (P˜ 2 + Q

(7.4)

˜ are the Heisenberg operators discussed in Example B.18. where P˜ and Q ˜ 2 is most often called the quantum mechanical The operator P˜ 2 + Q Hamiltonian, but some others (e.g., Hida, Gross) would call it a Laplacian, and this perspective tightens the analogy with the current study. In this sense, (7.4) could be rewritten as S := dom Δ∞ ; compare with (7.8). The duality between S and S  allows for the extension of the inner product on H to a pairing of S and S  : ·, · H : H × H → C

to

 ·, · ∼ H : S × S → R.

In other words, one obtains a Fourier-type duality restricted to S. Moreover, it is possible to construct a Gel’fand triple in such a way that P(S  ) = 1 for a Gaussian probability measure P. When applied to H = HE , the construction yields two main outcomes: 1. the next best thing to a Fourier transform for an arbitrary graph; 2. a concrete representation of HE as an L2 measure space HE ∼ = L2 (S  , P). As a next step, we state Bochner’s theorem, which characterizes the Fourier transform of a positive finite Borel measure on the real line. The reader may find Refs. [RS75, Ber96] helpful for further information.

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157

Theorem 7.4 (Bochner). Let G be a locally compact abelian group. Then, ˆ where M(G) is there is a bijective correspondence F : M(G) → PD(G), ˆ is the set of positive definite the collection of measures on G, and PD(G) functions on the dual group of G. Moreover, this bijection is given by the Fourier transform  eξ,x dν(x). (7.5) F : ν → ϕν by ϕν (ξ) = G

In our applications to the resistance network (Zd , 1) in Chapter 14, the underlying group structure allows us to apply the above version of Bochner’s theorem. For our representation of the energy Hilbert space HE in the case of a general resistance network, we need Minlos’ generalization of Bochner’s theorem from Refs. [Min63, Sch73]. This important result states that a cylindrical measure on the dual of a nuclear space is a Radon measure iff its Fourier transform is continuous. In this context, however, the notion of Fourier transform is infinite dimensional and is dealt with by the introduction of Gel’fand triples [Lee96]. Theorem 7.5 (Minlos). Given a Gel’fand triple S ⊆ H ⊆ S  , Bochner’s theorem may be extended to yield a bijective correspondence between the positive definite continuous2 functions on S and the Radon probability measures on S  . Moreover, in a specific case, this correspondence is uniquely determined by the identity  1 eu,ξH˜ dP(ξ) = e− 2 u,uH , (7.6) S

where ·, · H is the original inner product on H and ·, · H˜ is its extension to the pairing on S × S  . Remark 7.6. Formula (7.6) may be interpreted as defining the Fourier transform of P; the function on the right-hand side is positive definite and plays a special role in stochastic integration, and it is used in quantization [AJL11, AJ12]. 7.2.1 A space of test functions SG on G To apply Minlos’ theorem in the context of (G, c), we first need to construct a Gel’fand triple for HE ; we begin by identifying a certain 2 Here,

continuity refers to the Fr´ echet topology on S.

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subspace of the domain of ΔM . Recall from Definition 7.3 that M := span{vx , wxv , wfx }x∈G0 . be a self-adjoint extension3 of ΔM . Definition 7.7. Let Δ p u := (Δ Δ ...Δ )u be the p-fold product of Δ applied to u ∈ HE . Let Δ p Define dom(Δ ) inductively by

.

p p−1 ) := {u .. Δ )}. dom(Δ u ∈ dom(Δ

(7.7)

Remark 7.8. Observe that ΔM is Hermitian and commutes with conjugation (since c is R-valued); therefore, a theorem by von Neumann’s states is well defined. that it has a self-adjoint extension. This ensures that Δ However, our results can be complexified eventually; see Remark 7.23. Definition 7.9. The (Schwartz) space of functions of rapid decay is ∞

∞ ), SG := dom(Δ

(7.8)

∞ ) := p where dom(Δ p=1 dom(Δ ) consists of all R-valued functions u ∈ HE p u ∈ HE for any p. for which Δ The space of Schwartz distributions or tempered distributions is the dual space SG of R-valued continuous linear functionals on SG . Therefore, v ∈ SG iff there is a C and p such that

p sE |s, v W | ≤ CΔ

for all s ∈ SG .

(7.9)

Remark 7.10. A good choice of self-adjoint extension in Definition 7.7 is the operator ΔH discussed in Section 5.2. It is critical to make the unusual step of taking a self-adjoint extension of ΔM for several reasons. Most importantly, we need to apply the spectral theorem to extend the energy inner product ·, · E to a pairing on SG × SG . In fact, it will turn out that for p u, Δ −p v E , u ∈ SG , v ∈ SG , the extended pairing is given by u, v W = Δ p −p 4 where p is any integer large enough to ensure Δ u, Δ v ∈ HE . This relies crucially on the self-adjointness of the operator appearing on the right-hand side. Moreover, without self-adjointness, we would be unable to prove that SG is dense in HE ; see Lemma 7.14. Additionally, the self-adjoint extensions of ΔM are in bijective correspondence with the isotropic subspaces of dom(ΔM ), and we will see that these are useful for understanding the boundary of G in terms of defect; 3 Elsewhere in this volume, the notation Δ ˜ is used for a generic self-adjoint extension  here to avoid entangled superscripts. of Δ; we use Δ 4 Here, Δ  −p v refers to a pth primitive of v; see the proof of Theorem 7.15.

The Boundary and Boundary Representation

159

see Theorem 5.22. Recall that a subspace U ⊆ dom(ΔM ) is isotropic iff βbd (u, v) = 0, ∀u, v ∈ U, where βbd is as in Definition 5.20. Since dom(ΔM ) is isotropic (cf. Theorem 5.21), we think of U as a subspace of the quotient (boundary) space B = dom(ΔM )/ dom(ΔM ). Remark 7.11. Note that SG and SG consist of R-valued functions. This technical  u,· detail is important because we do not expect the integral ˜ W e dP from (7.6) to converge unless it is certain that u, · is S R-valued. This is the reason for the last conclusion of Lemma 7.16. ) with respect to the graph Remark 7.12. Note that SG is dense in dom(Δ norm by standard spectral theory. For each p ∈ N, there is a seminorm on SG defined by

p uE . up := Δ

(7.10)

) into It follows from standard theory that the seminorms  · p turn dom(Δ a Fr´echet space; see Refs. [Tre06, Nel72, Sch85b, Sch85a].

Definition 7.13. Let χ[a, b] denote the usual indicator function of the interval [a, b] ⊆ R, and let S be the spectral transform in the spectral , and let E be the associated projection-valued measure. representation of Δ Then, define En to be the spectral truncation operator acting on HE by  n En u := S χ[ n1 , n] Su = E(dt)u. 1/n

Lemma 7.14. SG is a dense subspace of HE (with respect to E), and so, SG ⊆ HE ⊆ SG is a Gel’fand triple. Proof. This essentially follows immediately once it is clear that En maps HE into SG . For u ∈ HE and for any p = 1, 2, . . . ,  n p 2 Δ En uE = λ2p E(dλ)u2E ≤ n2p u2E , (7.11) 1/n

so En u ∈ SG . It follows that u − En uE → 0 by standard spectral theory.  Theorem 7.15. The energy form ·, · E extends to a pairing on SG × SG defined by p −p u, Δ v E , u, v W := Δ

where p is any integer such that |v(u)| ≤ KΔp uE for all u ∈ SG .

(7.12)

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160

p s) := s, v W Proof. For v ∈ SG , we have C and p satisfying (7.9). Set ϕ(Δ to obtain a continuous linear functional on HE (after extending to the p s} by 0 if necessary). Now, Riesz’s orthogonal complement of span{Δ p s, w E for all s ∈ SG , and lemma gives a w ∈ HE for which s, v W = Δ −p v := w ∈ HE to make the meaning of the right-hand side of we define Δ (7.12) clear. 

Lemma 7.16. The pairing on SG × SG is equivalently given by u, ξ W = lim ξ(En u), n→∞

(7.13)

where the limit is taken in the topology of SG . Moreover, u ˜(ξ) = u, ξ W is R-valued on SG . . This is a standard result in spectral theory, Proof. En commutes with Δ as En and Δ are unitarily equivalent to the two commuting operations of truncation and multiplication, respectively. Therefore,

p −p p −p En s, Δ s, Δ ξ E = En Δ ξ E ξ(En u) = En u, ξ W = Δ p −p s, En Δ = Δ ξ E .

Standard spectral theory also gives En v → v in HE , so p −p p −p s, En Δ u, Δ lim ξ(En u) = lim Δ ξ E = Δ v E .

n→∞

n→∞

Since the pairing · , · W is a limit of real numbers, it is R-valued.



Corollary 7.17. En extends to a mapping E˜n : SG → HE defined via ˜n ξ E := ξ(En u). Thus, we have a pointwise extension of · , · W to u, E HE × SG given by ˜n ξ E . u, ξ W = lim u, E n→∞

(7.14)

Lemma 7.18. If deg(x) is finite for each x ∈ G0 or if c < ∞, then one has vx ∈ SG . Proof. This is immediate from Lemma 5.4.



Remark 7.19. When the hypotheses of Lemma 7.18 are satisfied, note that span{vx } is dense in SG with respect to E but not with respect to the Fr´echet topology induced by the seminorms (7.10), nor with respect to the

The Boundary and Boundary Representation

graph norm. One has the inclusions       s vx u ⊆ ⊆ , u s ΔM vx Δ Δ

161

(7.15)

where s ∈ SG and u ∈ HE , with the second inclusion dense and the first inclusion not dense. We have now obtained a Gel’fand triple SG ⊆ HE ⊆ SG , and we are ready to apply the Minlos’ theorem to a particularly lovely positive definite function on SG in order that we may obtain a particularly nice measure on SG . Recall that we constructed HE from the resistance metric in Chapter 4 by making use of negative definite functions. In the proof of the following theorem, we apply the Schoenberg–von Neumann theorem (see Theorem A.17) with t = 12 to the resistance metric in the form RF (x, y) = vx − vy 2E from (3.10). The proof of Theorem 7.20 also uses the notation Eξ (f ) := S  f (ξ) dP(ξ). G

Theorem 7.20. The Wiener transform W : HE → L2 (SG , P) defined by W : v → v˜,

v˜(ξ) = v, ξ W ,

(7.16)

is an isometry. Here, P is defined by (7.6); see Remark 7.6. The extended reproducing kernel {˜ vx }x∈G0 is a system of Gaussian random variables which gives the resistance distance by vx − v˜y )2 ). RF (x, y) = Eξ ((˜

(7.17)

Moreover, for any u, v ∈ HE , the energy inner product extends directly as 

 u, v E = Eξ u ˜v˜ = u ˜v˜ dP. (7.18)  SG

Proof. Since RF (x, y) is conditionally negative semidefinite by the proof of Theorem 4.2, we may apply the Schoenberg–von Neumann theorem (Theorem A.17) and deduce that exp(− 12 u − v2E ) is a positive definite function on HE × HE . Consequently, an application of the Minlos’ correspondence to the Gel’fand triple established in Lemma 7.14 yields a Gaussian probability measure P on SG . Moreover, (7.6) gives Eξ (eu,ξW ) = e− 2 u E , 1

2

(7.19)

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whence one computes

 1 1 2 1 + u, ξ W − u, ξ W + · · · dP(ξ) = 1 − u, u E + · · · . (7.20)  2 2 SG Now, it follows that E(˜ u2 ) = Eξ (u, ξ 2W ) = u2E for every u ∈ SG by comparing the terms of (7.20), which are quadratic in u. Therefore, W : HE → SG is an isometry, and (7.20) gives Eξ (|˜ vx − v˜y |2 ) = Eξ (vx − vy , ξ 2 ) = vx − vy 2E ,

(7.21)

whence (7.17) follows from (3.10). Note that by comparing the linear terms, (7.20) implies Eξ (1) = 1 so that P is a probability measure, and Eξ (u, ξ ) = 0 and Eξ (u, ξ 2 ) = u2W so that P is actually Gaussian N (0, u2W ). Finally, use polarization to compute  1

u + v2E − u − v2E u, v E = 4    1  2 2 = u + v˜| − Eξ |˜ u − v˜| by (7.21) Eξ |˜ 4  1 2 2 = |˜ u + v˜| (ξ) − |˜ u − v˜| (ξ) dP(ξ) 4 SG  u ˜(ξ)˜ v (ξ) dP(ξ). =  SG

This establishes (7.18) and completes the proof.



Remark 7.21. With the embedding HE → L2 (SG , P), we obtain a maximal abelian algebra of Hermitian multiplication operators L∞ (SG ) acting on L2 (SG , P); see Ref. [Rud91]. By contrast, see (ii) of Remark 2.3. Remark 7.22. It is important to note that since the Wiener transform W : SG → SG is an isometry, the conclusion of Minlos’ theorem is stronger than usual: The isometry allows the energy inner product to be extended isometrically to a pairing on HE × SG instead of just SG × SG . Remark 7.23. The reader will note that we have taken pains to keep  everything R-valued in this chapter (especially  u,ξthe elements of SG and SG ), W dP(ξ) in (7.19). However, primarily to ensure the convergence of S  e  ˜v˜ dP now that we have established the fundamental identity u, v E = S  u in (7.18) and extended the pairing ·, · W to HE × SG , we are at liberty to complexify our results via the standard decomposition into real and complex parts: u = u1 + u2 with ui R-valued elements of HE , etc.

The Boundary and Boundary Representation

163

Remark 7.24. The polynomials are dense in L2 (SG , P). More precisely, if ϕ(t1 , t2 , . . . , tk ) is an ordinary polynomial in k variables, then   (7.22) ϕ(ξ) := ϕ u1 , ξ W , u2 , ξ W , . . . uk , ξ W is a polynomial on SG and   . u1 (ξ), u 2 (ξ), . . . u k (ξ) , deg(ϕ) ≤ n, .. uj ∈ HE , ξ ∈ SG } Polyn := {ϕ  (7.23) is the collection of polynomials of degree at most n, and {Polyn }∞ n=0 is an increasing family whose union is all of SG . One can see that the monomials u, ξ W are in L2 (SG , P) as follows: Compare the like powers of u from either

 = 0 and side of (7.20) to see that Eξ u, ξ 2n+1 W 

 (2n)! Eξ u, ξ 2n = |u, ξ W |2n dP(ξ) = n u2n (7.24) W E ,  2 n! SG and then, apply the Schwarz inequality; see Ref. [AJL17] for a derivation. 2  To see why the polynomials {Polyn }∞ n=0 should be dense in L (SG , P), ∞ observe that the sequence {PPolyn }n=0 of orthogonal projections increases ˜} forms a martingale for any u ∈ HE to the identity; therefore, {PPolyn u (i.e., for any u ˜ ∈ L2 (SG , P)) and so converges to u ˜ as n → ∞ by Doob’s martingale convergence theorem [Doo53]. If we denote the “multiple Wiener integral of degree n” by .

Hn := Polyn − Polyn−1 = cl span{u, · nW .. u ∈ HE },

n ≥ 1,

and H0 := C1 for a vector 1 with 12 = 1, then we have an orthogonal decomposition of the Hilbert space L2 (SG , P) =

∞ 

Hn .

(7.25)

n=0

See Ref. [Hid80, Thm. 4.1] for a more extensive discussion. A physicist would call (7.25) the Fock space representation of L2 (SG , P) with “vacuum vector” 1; note that Hn has a natural (symmetric) tensor product structure. Familiarity with these ideas is not necessary for the remainder, but the decomposition (7.25) is helpful for understanding two key things: (i) The Wiener isometry W : HE → L2 (SG , P) identifies HE with the subspace H1 of L2 (SG , P); in particular, L2 (SG , P) is not isomorphic to HE . In fact, it is the second quantization of HE .

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(ii) The constant function 1 is an element of L2 (SG , P) but does not correspond to any element of HE . In particular, constant functions in HE are equivalent to 0, but this is not true in L2 (SG , P). It is somewhat ironic that we began this story by removing the constants (via the introduction of E) only to reintroduce them with a certain amount of effort, muchlater. Item (ii) explains why it is not nonsense to write things like P(SG ) = S  1 dP = 1 and will be helpful when discussing boundary G elements in Section 7.3. Corollary 7.25. For ex (ξ) := evx ,ξW , one has Eξ (ex ) = e− 2 R hence,  1 F ex (ξ)ey (ξ) dP(ξ) = e− 2 R (x,y) . Eξ (ex ey ) = 1

 SG

F

(o,x)

, and

(7.26)

Proof. Substitute u = vx or u = vx −vy in (7.19) and apply Theorem 3.12.  Remark 7.26. A concrete example of Corollary 7.25 appears in Remark 14.13 for the integer lattice G = (Zd , c). Remark 7.27. Remark 3.41 discusses the interpretation of the free resistance as the reciprocal of an integral over a path space; Corollary 7.25 provides a variation on this theme:  F ex (ξ)ey (ξ) dP. (7.27) R (x, y) = −2 log Eξ (ex ey ) = −2 log  SG

Observe that Theorem 7.20 was carried out for the free resistance, but all the arguments go through equally well for the wired resistance; note that RW is similarly conditionally negative semidefinite by Theorem A.17 and Corollary 4.3. Thus, there is a corresponding Wiener transform W : Fin → L2 (SG , P) defined by W : v → f˜,

f = PFin v

and f˜(ξ) = f, ξ W .

(7.28)

Again, {f˜x }x∈G0 is a system of Gaussian random variables, which gives the wired resistance distance by RW (x, y) = Eξ ((f˜x − f˜y )2 ). Remark 7.28. According to Theorem 7.20, the system {ex }x∈G0 of Corollary 7.25 forms a system of Gaussian random variables and the Wiener transform is an isometry, so Eξ [ex ] = 0 and Eξ [ex ey ] = δx,y .

(7.29)

The Boundary and Boundary Representation

165

Since the independence of Gaussian random variables is determined by the first two moments, it follows that {ex }x∈G0 forms a system of i.i.d. Gaussian random variables with a mean of 0 and a variance of 1. This is noteworthy because while independence implies orthogonality, the converse does not hold without the additional hypothesis that the distributions be Gaussian. 7.3 The Resistance Boundary of a Transient Network Recall that we began this section with a comparison of the Poisson boundary representation  u(y)k(x, dy), u bounded and harmonic on Ω ⊆ Rd , (7.30) u(x) = ∂Ω

with the E boundary representation  x u(x) = u ∂h + u(o), u ∈ Harm, ∂

and hx = PHarm vx .

(7.31)

bd G

Remark 7.29. For u ∈ Harm and ξ ∈ SG , let us abuse notation and write u for u˜. That is, u(ξ) := u˜(ξ) = u, ξ W . Unnecessary tildes obscure the presentation and the similarities to the Poisson kernel. Definition 7.30. Define PQ to be the image measure on SG /Fin induced by the standard projection π : SG → SG /Fin, i.e., PQ (B) := P(π −1 B) for B ∈ B(SG /Fin). Now, PQ is a probability measure on the quotient, and Theorem 7.20 gives a corresponding energy integral representation. Remark 7.31. We suppress tildes as in Remark 7.29 when context precludes confusion; for example, in the proof of Theorem 7.32, in Remark 7.33, and in Definition 7.36. Theorem 7.32 (Boundary integral representation for harmonic functions). For any u ∈ Harm and with hx = PHarm vx ,  u(x) = u(ξ)hx (ξ) dPQ (ξ) + u(o). (7.32)  /Fin SG

Proof. Starting with Lemma 2.24, compute  u(x) − u(o) = hx , u E = u, hx E =

 SG

uhx dPQ ,

(7.33)

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where the last equality comes from substituting v = hx in (7.18); recall  from Lemma 2.27 that hx = hx . Remark 7.33 (A Hilbert space interpretation of bd G). In view of Corollary 7.32, we are now able to “catch” the boundary between SG and SG by using ΔM and its adjoint. The boundary of G may be considered as (a possibly proper subset of) SG . Corollary 7.32 suggests that (x, dξ) := hx (ξ)dPQ is the discrete analogue in HE of the Poisson kernel k(x, dy), and a comparison of (7.2) with (7.32) provides a way of understanding a boundary integral as a limit of Riemann sums:   x u hx dPQ = lim u(x) ∂h (7.34) ∂ (x). k→∞

 SG

bd Gk

(We continue to omit the tildes as in Remark 7.29.) By a theorem by Nelson, older-continuous with PQ is fully supported on those functions which are H¨ 1 1 exponent α = 2 , which we denote by Lip( 2 ) ⊆ SG ; see Ref. [Nel64, App. A, Thm. 4]. Recall from Corollary 3.15 that HE ⊆ Lip( 12 ). See also Refs. [MP10, Arv76b, Arv76c, Min63, Nel69]. We are finally able to give a concrete representation of elements of the boundary. We continue to use the measure PQ from Definition 7.30. Recall the Fock space representation of L2 (SG , P) discussed in Remark 7.24: 

∞ SG Q ∼  ⊗n 2 ,P HE , (7.35) L = Fin n=0 where HE⊗0 := C1 for a unit “vacuum” vector 1 corresponding to the constant function and HE⊗n denotes the n-fold symmetric tensor product of HE with itself. Observe that 1 is orthogonal to Fin and Harm but is not the zero element of L2 (SG , PQ ).  Lemma 7.34. For all v ∈ HE , S  v dPQ = 0. Proof. The integral



 SG

G

v dPQ =



 SG

1v dPQ is the inner product of two

elements in L2 (SG , PQ ), which lie in different (orthogonal) subspaces; see (7.25).  Remark 7.35. Alternatively, Lemma 7.34 holds because the expectation of every odd-power monomial vanishes by (7.20); see also (7.24) and the surrounding discussion of Remark 7.24.

The Boundary and Boundary Representation

167

Note that Lemma 7.34 is a statement about balanced functions (in a sense similar to that of Definition 6.60. ˜ x for elements Recall that we abuse notation and write hx = hx , · W = h  of SG ; see Remark 7.31. Definition 7.36. Denote the measure appearing in Corollary 7.32 by dμx := (1 + hx ) dPQ .

(7.36)

The function 1 does not show up in (7.32) because it is orthogonal to Harm:   Q u(1 + hx ) dP = u dPQ + u, hx E = u, hx E for u ∈ Harm,  SG

 SG

where we used Lemma 7.34. Nonetheless, its presence is necessary:     1dμx = 1(1 + hx ) dPQ = 1 dPQ + 1hx dPQ = 1,  SG

 SG

 SG

 SG

again by Lemma 7.34. Remark 7.37. We have shown that as a linear functional, μx [1] = 1. It follows from standard functional analysis that μx ≥ 0 PQ -a.e. on SG . Thus, μx is absolutely continuous with respect to PQ with Radon–Nikodym dμx derivative dP Q = 1 + hx . Definition 7.38. Recall that a path in G is an infinite sequence of successively adjacent vertices. We say that a path γ = (x0 , x1 , x2 , . . . ) is a path to infinity, and write γ → ∞ iff γ eventually leaves any finite set F ⊆ G0 , i.e., ∃N

such that n ≥ N =⇒ xn ∈ / F.

(7.37)

If γ1 = (x0 , x1 , x2 , . . . ) and γ2 = (y0 , y1 , y2 , . . . ) are two paths to infinity, define an equivalence relation by γ1  γ2

⇐⇒

lim (h(xn ) − h(yn )) = 0,

n→∞

for every h ∈ M.

(7.38)

In particular, all paths to infinity are equivalent when Harm = 0. If β = [γ] is any such equivalence class, pick any representative γ = (x0 , x1 , x2 , . . . ) and consider the associated sequence of measures {μxn }

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Operator Theory and Analysis of Infinite Networks

from Definition 7.36. As probability measures, these lie in the unit ball, so Alaoglu’s theorem gives a weak- limit: νβ := lim μxn .

(7.39)

n→∞

For any h ∈ Harm, this measure satisfies   n→∞ ˜ dμx −− h(xn ) = − − − → h n  SG

˜ dνβ . h

(7.40)

bd G

Thus, we define bd G to be the collection of all such β and extend harmonic functions to bd G via  ˜ dνβ . ˜ (7.41) h h(β) := bd G

Definition 7.39. For u ∈ HE , denote u∞ := supx∈G0 |u(x) − u(o)| and say u is bounded iff u∞ < ∞. Lemma 7.40. If v ∈ HE is bounded, then PFin v is also bounded. Proof. Choose a representative for v with 0 ≤ v ≤ K. Then, by Corollary 7.32 and (7.36),   Q v(ξ)hx (ξ) dP (ξ) + u(o) = v(ξ) dμx (ξ) + u(o). PHarm v(x) =  SG

 SG

Since μx is a probability measure (cf. Remark 7.37), we have PHarm v ≥ 0, and hence, the finitely supported component PFin v = v − PHarm v is also bounded.  Lemma 7.41. Every v ∈ M is bounded. In particular, vx ∞ ≤ RF (o, x). Proof. According to Definition 2.32, it suffices to check that vx , wxv , and wfx are bounded for each x. Furthermore, wfx ∞ = PFin wxv ∞ ≤ wxv ∞ by Lemma 7.40, and wxv = vx + wo by definition, so it suffices to check vx and wo . By Ref. [Soa94, Lem. 3.70], wo has a representative which is bounded, taking only values between 0 and wo (o) > 0. It remains only to check vx . The following approach is taken from the “proof” of Conjecture 3.48. Fix x, y ∈ G0 and an exhaustion {Gk }, and suppose without loss of generality that o, x, y ∈ G1 . Also, let us consider the representative of vx specified by vx (o) = 0. On a finite network, it is well known (see (3.57) and the surrounding discussion) that vx = R(o, x)ux ,

(7.42)

The Boundary and Boundary Representation

169

where ux (y) is the probability that a random walker starting at y reaches x before o, that is, ux (y) := Py [τx < τo ], where τx denotes the hitting time of x. This idea is discussed in Refs. [DS84, LPW08, LP16]. (k) (k) Therefore, one can write (7.42) on Gk as vx = RGFk (o, x)ux . In other (k)

words, vx is the unique solution to Δv = δx − δo on the finite subnetwork (k) (o, x) for all y ∈ Gk . GF k . Consequently, for every k, we have vx (y) ≤ RGF k Since RF (x, y) = limk→∞ RGFk (x, y) by Definition 3.8, we have vx ∞ ≤ RF (o, x) for every x ∈ G0 .  Theorem 7.42. Let β ∈ bd G, and let γ = (x0 , x1 , x2 , . . . ) is any representative of β. Then, β ∈ bd G defines a continuous linear functional on SG via  β(v) := lim v˜ dμxn , v ∈ SG . (7.43) n→∞

 SG

In fact, the action of β is equivalently given by β(v) = lim PHarm v(xn ) − PHarm v(o), n→∞

v ∈ SG .

(7.44)

Proof. To see that (7.43) and (7.44) are equivalent, compute      v˜(1 + hxn ) dPQ = v˜ 1 dPQ + v˜hxn dPQ     SG S S G G = v, hxn E = PHarm v(xn ) − PHarm v(o) because 1 is orthogonal to HE in L2 (SG , PQ ); see (7.25). Now, to see that (7.43) or (7.44) defines a bounded linear functional, . we only need to check that supv∈SG {β(v) .. vE = 1} is bounded, but this is the content of Lemma 7.41. Note that the equivalence relation (7.38) ensures that the limit is independent of the choice of representative γ.  Remark 7.43. In light of (7.44), one can think of νβ in (7.39) as a Dirac mass at β. Thus, β ∈ bd G is a boundary point, and integrating a function f against νβ corresponds to evaluation of f at that boundary point.  7.4 The Structure of SG

The next results are structure theorems akin to those found in the classical theory of distributions; see Ref. [Str03, Section 6.3] or Ref. [AG92, Section 3.5].

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Theorem 7.44. The distribution space SG is p + u, v E .. u ∈ SG , v ∈ HE , p ∈ Z }. SG = {ξ(u) = Δ .

(7.45)

p u, v E defines Proof. It is clear from the Schwarz inequality that ξ(u) = Δ a continuous linear functional on SG for any v ∈ HE and nonnegative integer p. For the other direction, we use the same technique as in Lemma 7.15. p uE Observe that if ξ ∈ SG , then there exists K, p such that |ξ(u)| ≤ KΔ p u → ξ(u) is continuous for every u ∈ SG . This implies that the map ξ : Δ .. p + u . u ∈ HE , p ∈ Z }. This can be extended on the subspace Y = span{Δ to all of HE by precomposing with the orthogonal projection to Y . Now, p u, v E .  Riesz’s lemma gives a v ∈ HE for which ξ(u) = Δ

As noted in the proof of Theorem 7.15, one can consider the element −p v. ξ ∈ SG appearing at the end of the proof of Theorem 7.44 to be ξ = Δ ∞  −p This implies SG = p=0 Δ (HE ); see Ref. [Nel72]. Definition 7.45. Extend Δ to SG by defining Δξ(vx ) := δx , ξ W so that Δξ(vx ) =

(7.46)



Now, extend Δ so that

y∼x cxy (ξ(vx )−ξ(vy )) follows readily from Lemma 2.26.  ˜ vx )(ξ) := Δv x (ξ) ˜ defined on v˜x ∈ L2 ( SG , PQ ) by Δ(˜ to Δ Fin

˜ : v˜x → c(x)˜ Δ vx −



cxy v˜y .

(7.47)

y∼x

Since vx → v˜x is an isometry, it is no great surprise that  ˜ ˜ vx , Δ˜ vy L2 = v˜x (ξ)˜ vy (Δξ) dPQ (ξ) = vx , Δvy E .  SG

(7.48)

We now provide two results enabling one to recognize certain elements of SG . Lemma 7.46. A linear functional f : SG → C is an element of SG if and only if there exists p ∈ N and F0 , F1 , . . . , Fp ∈ HE such that f (u) =

p 

k u E , Fk , Δ

k=0

∀u ∈ HE .

(7.49)

The Boundary and Boundary Representation

171

Proof. By definition, f ∈ SG iff ∃p, C < ∞ for which |f (u)| ≤ Cup for every u ∈ SG . Therefore, the linear functional p 2 p k u, . . . Δ u) = f (u) ) → C u, Δ Φ: by Φ(u, Δ dom(Δ k=0 p is continuous, and Riesz’s lemma gives F = (Fk )pk=0 ∈ k=0HE with p u, . . . Δ u)  H = f (u) = F, (u, Δ E

p 

k u H . Fk , Δ E



k=0 : HE → HE is bounded, then S  = HE . Corollary 7.47. If Δ G is Proof. We always have the inclusion HE → SG by taking p = 0. If Δ  is also bounded, and (7.49) gives bounded, then the adjoint Δ  p    k ) Fk , u f (u) = (Δ , ∀u ∈ SG . (7.50)

k=0

 HE

Since SG is dense in HE by Lemma 7.14, we have f =

p

 k k=0 (Δ ) Fk

∈ HE . 

is Remark 7.48. One can also prove Corollary 7.47 as follows: If Δ p p bounded, then so is Δ for any p, whence dom Δ = HE and SG = ∞   p p=0 dom Δ = HE . Then, SG = HE = HE .

7.5 Remarks and References Boundary theory is a well-established subject; see, for example, Refs. [Bre67, Doo59, Doo66]. The deep connections between harmonic analysis, probability, and potential theory have led to several notions of boundary, and we will not attempt to give complete references. However, we recommend Refs. [Saw97, Woe09] for introductory material on Martin boundary and Refs. [Car73a, Woe00] for further information. The papers Refs. [Yam79, Lyo83] and [NW59] are foundational with regard to connections between energy and transience. With regard to infinite graphs and finite-energy functions, see Refs. [Soa94, SW91, CW92, Dod06, PW90, PW88, Woe86, Tho90]. An attractive and modern presentation especially well suited to the needs of our current chapter is Ref. [CSW93] by Cartwright, Soardi, and Woess. An excellent book for what we need on path-space integrals is Ref. [Hid80]. Related material on fractals can be found in Refs. [Kig03, Kig10, Kig95].

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The boundary representation given in Corollary 7.32 above is related to a large number of analogous representations in the literature; see, for example, Refs. [Soa94, Spi76, Str05], Ref. [Woe00, Theorem, 24.7], or Ref. [Saw97, Theorem 3.1 and Theorem 4.1]. There are two primary differences between these more traditional approaches and the one adopted here: 1. We focus on the harmonic functions of finite energy (as opposed to the nonnegative or bounded harmonic functions). 2. Our representation is developed via Hilbert spaces. In fact, the latter is made possible by the former. However, there are no easy ways of relating pointwise bounded functions to finite-energy functions on an infinite weighted graph. Hence, Corollary 7.32 does not immediately compare with analogous theorems in the literature. In this chapter, we obtain a stochastic process indexed by a set of . vertices in G using the fact that the energy kernel {vx .. x ∈ G0 } ⊆ HE is isometrically and naturally embedded into a Gaussian probability space and thus into a Gaussian stochastic process. It is worth noticing that the use of probability spaces and stochastic processes in this manner has uses in other areas of mathematics; for example, in the study of Gaussian power series [PV05], see especially the discussion on p. 11 of the author’s extension of the traditional Poisson kernel to a white noise stochastic integral. The reader may additionally wish to consult Refs. [Woe00, Woe89, Woe95, SCW09, SCW06, KW02, Kig10, IR08, BW05, Gui72, OP19, CM20].

Chapter 8

Multiplication Operators on the Energy Space

Recall from Section 2.4.1 the grounded energy space D, which comprises the functions of finite energy endowed with the inner product u, vo := u(o)v(o) + u, vE .

(8.1)

Throughout this chapter, given u ∈ dom E, we work with the representative of u (which we also denote by u) for which u(o) = 0.

(8.2)

It should be noted that under this convention, Fin is the E-closure of the class of functions on G which are constant (but not necessarily 0) outside of a finite set. Also, this convention allows (2.6) to be written as vx , uE = u(x) for every u ∈ HE .

(8.3)

X = G0 \ {o}

(8.4)

We also write

for brevity. We shall have occasions to use basic tools from the theory of matrix order, that is, the usual ordering of finite Hermitian matrices in which A ≥ 0 iff ξ, Aξ ≥ 0; see Appendix A. Also, we use ξ to denote the coefficients indexed by the vertices and write ξx := ξ(x). Most material in this chapter originally appeared in Ref. [JP13b].

173

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Operator Theory and Analysis of Infinite Networks

Definition 8.1. For a function f : X → C, we denote by Mf the corresponding multiplication operator: (Mf u)(x) := f (x)u(x),

∀x ∈ X.

(8.5)

When context precludes confusion, we suppress the dependence on f and just write M . The norm of M is the usual operator norm .

M  := M D→D = sup{f uD .. uD ≤ 1}.

(8.6)

Remark 8.2. The multiplication operators on D are a little unusual, especially in comparison with the more familiar Hilbert spaces of L2 functions. Lemma 8.3 shows that R-valued functions do not define Hermitian multiplication operators, except in the trivial case of f = 0, and Theorem 8.4 shows that Mf vx = f (x)vx , which is not equal to Mf vx . See Remark 8.5. Additionally, the operator norm of Mf is not related to the sup norm of f . In Example 14.23, we show that the energy kernel on (Z, 1) provides an example of a bounded function f : X → C for which the Mf , as an operator on D, is unbounded. Lemma 8.3. For f : X → C, the multiplication operator M = Mf is Hermitian if and only if f = 0. Proof. Applying (8.1) to u, v ∈ D yields  1  cxy f (x)u(x)v(x) − f (x)u(x)v(y) M u, vo = 2 x,y∈G  − f (y)u(y)v(x) + f (y)u(y)v(y) + f (o)u(o)v(o). By comparison with the corresponding expression, this is equal to u, M vE iff (f (y) − f (x))u(y)v(x) = (f (y) − f (x))u(x)v(y) holds for all x, y ∈ G. However, since we are free to vary u and v, it must be the case that f is  constant and f = f . Since Lemma 8.3 shows that the adjoint of a multiplication operator is not what one would expect, one immediately wonders what the adjoint is, and this is the subject of our next result. Theorem 8.4. Let M  be the adjoint of the multiplication operator M = Mf with respect to the energy inner product (8.1). Then, the adjoint of M is defined by its action on the energy kernel : M  vx = f (x)vx ,

∀x ∈ X.

(8.7)

Multiplication Operators on the Energy Space

175

Proof. Since the energy kernel is dense in D, it suffices to show that vy , M  vx − f (x)vx o = 0 for every y ∈ X. Using (8.1), we have vy , M  vx o = M vy , vx o = f · vy , vx E + f (o)vy (o)vx (o) by Lemma 2.27. Then, the reproducing property and (8.2) give vy , M  vx o = f (x)vy (x) − f (o)vy (o) + f (o)vy (o)vx (o) = f (x)vy (x). On the other hand, vy , f (x)vx o = f (x)vy , vx E + vy (o)f (x)vx (o), = f (x)vy (x) 

by (8.2).

Remark 8.5. Note that M  multiplies vx by the scalar f (x), not the function f . 8.1 Bounded Multiplication Operators Lemma 8.6. If L is an operator on a Hilbert space H, then the following are equivalent : (i) L : H → H is bounded with L ≤ b. (ii) b2 − L L ≥ 0. (iii) b2 − LL ≥ 0. In Lemma 8.6, L ≥ 0 means u, Lu ≥ 0 for all u in some dense subset of H, L means LH→H, and of course b2 means b2 I. The only nontrivial part of the proof of Lemma 8.6 is (ii) ⇐⇒ (iii), which uses polar decomposition; see Ref. [KR97], Ref. [Rud91], or Ref. [RS72], for example. Theorem 8.7. M = Mf is bounded on HE with M  ≤ b if and only if   sf (x, y) := b2 − f (x)f (y) vx , vy E

(8.8)

is a positive semidefinite function on X × X. Proof. We work with the dense linear subspace V := span{vx }x∈X of the energy space. By Lemma 8.6, the first hypothesis in the statement of

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Theorem 8.7 is equivalent to u, (b2 − M M  )uE ≥ 0, Since u ∈ V means u = evaluate (8.9): u, (b2 − M M  )uE =

 x∈F

 x,y∈F

=



x,y∈F

=



x,y∈F

=



∀u ∈ V.

(8.9)

ξx vx for some finite set F ⊆ X, we can

ξx ξy vx , (b2 − M M  )vy E   ξx ξy vx , b2 vy E − M  vx , M  vy E   ξx ξy b2 vx , vy E − f (x)vx , f (y)vy E   ξx ξy b2 − f (x)f (y) vx , vy E ,

x,y∈F

where Lemma 8.4 was used to obtain the third equality. In view of (A.1), it is now clear that (8.9) holds for every choice of coefficients ξ if and only if sf (x, y), as defined in (8.8), is a positive semidefinite function on X × X.  Corollary 8.8. If f1 and f2 are functions on X and Mfi  ≤ bi < ∞ for i = 1, 2, then   sf1 f2 (x, y) := b1 b2 − (f1 f2 )(x)(f1 f2 )(y) vx , vy E (8.10) is a positive semidefinite function on X ×X, where (f1 f2 )(x) := f1 (x)f2 (x). Proof. Since Mf1 Mf2 = M(f1 f2 ) , we get M(f1 f2 )  ≤ b1 b2 . Now, the result follows from Lemma 8.6.  Remark 8.9. It is rather difficult to prove Corollary 8.8 from first principles. 8.2 Algebras of Multiplication Operators We continue to use X := G \ {o} in conjunction with convention (8.2), as in (8.4). We begin by considering the multiplication operators Mx := Mδx , that is, the special case of multiplication operators corresponding to the

Multiplication Operators on the Energy Space

function

 f := δx =

1, y = x, 0, y = x.

177

(8.11)

As in Definitions 6.21 and 6.67, we consider the countably infinite matrix V , with entries given by Vxy := vx , vy E ,

(8.12)

and the associated quadratic form   .. ξx vx , vy E ξy < ∞ . F ⊆ G0 \ {o}, |F | < ∞ . (8.13) QV (ξ) := sup F

x,y

Definition 8.10. The polarization of (8.13) yields an inner product  ξ, ηV = ξx ηy Vxy (8.14) x,y∈X

and a subsequent Hilbert space .

V = {(ξx )x∈X .. ξ : X → C, ξV < ∞}, (8.15)

where ξV = ξ, ξV . The summation in (8.14) is understood as shorthand for the supremum over such finite sums, as in (8.13).  Remark 8.11. Since V is psd, one has that x,y∈F ξx ξy Vxy ≥ 0 for every finite subset F of X, and so, ξ, ξV can be defined by (8.14) as the supremum of the finite sums over F . Then, ξ, ηV is obtained by polarization. Note also that (8.12) defines a self-adjoint operator V with dom V 1/2 = V by Kato’s theorem [DS88]. We see in Lemma 8.25 that V is unitarily equivalent to HE . Recall from Definition 3.8 that R(x) denotes the (free) effective resistance between x and o, and note that R(x) = vx (x) under the convention (8.3). Recall also from Definition 3.33 that P[x → o] denotes the probability that a random walk starting at x reaches o before returning to x. Theorem 8.12. For any x ∈ X, the multiplication operator Mx = Mδx is bounded on HE with

(8.16) Mx  = c(x)R(x) = P[x → o]−1/2 .

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Proof. Define an operator on HE via (Df V Df¯)xy := f (x)Vxy f (y),

∀x, y ∈ X × X,

(8.17)

where Df is the diagonal operator whose xth diagonal entry is f (x). Consequently, sf (x, y) = (1 − f (x)f (y))vx , vy E = Vxy − (Df V Df¯)xy = (V − Df V Df¯)xy . (8.18) By Theorem 8.7, we need to show that sδxo is psd, but f = δxo changes (8.18) into sδxo (x, y) = Vxy − Vxo ,xo δ(x,y),(xo ,xo ) ,

(8.19)

where δ(x,y),(xo ,xo ) is a Kronecker delta at matrix position (xo , xo ). To check that (8.19) is psd, suppose that F ⊆ X is any finite subset containing xo so that positive semidefiniteness is equivalent to V − Po V Po ≥ 0,

(8.20)

where Po is the projection in V onto the one-dimensional subspace of {f : F → C} spanned by δxo . So, the boundedness of Mx follows from (A.5). It remains to compute the norm. First, note that for any u ∈ HE , one immediately has Mxu2E = c(x)|u(x)|2 from (1.9). Then, (8.2), (8.3), the Schwarz inequality, and (3.9) give

|u(x)| = |u(x) − u(o)| = |vx , uE | ≤ vx E uE = R(x)uE ,

and (8.16) follows upon multiplying across by c(x) and applying a combination of Corollaries 3.40 and 3.43.  8.2.1 The multiplier C  -algebra of HE Our work in this section is inspired in part by the work on quantum graphs as systems of coherent state configurations on countable graphs. See Refs. [BK13, AC07, ABK06, AOS07, KS06, KPS07], for example. Definition 8.13. Define the multiplier C  -algebra of HE to be the C  subalgebra of B(HE ) generated by the bounded multiplication operators Mf . We denote this algebra by  . C  (HE ) := {Mf , Mf .. f : X → C and Mf is bounded}, (8.21) and the relations defining this algebra are given in Corollary 8.18. In (8.21), the symbol indicates that the linear span is closed in the operator topology, i.e., the uniform norm of bounded operators.

Multiplication Operators on the Energy Space

179

Remark 8.14. There is an important distinction between the abelian algebra generated by Mf (with f such that sf is psd, as in Theorem 8.8), and the C  -algebra generated by Mf . The first is abelian and the second very non-abelian. Remark 8.15. Theorem 8.12 shows that Mx ∈ C  (HE ) and hence that Mf ∈ C  (HE ) for every finitely supported function f : X → C. Recall that |uv| is Dirac’s notation for the rank-1 operator that sends v to u, and it is a projection if and only if u = v. Theorem 8.16. For any x ∈ X, Mx and Mx are the rank-1 operators expressed in Dirac notation by Mx = |δx vx |

and

Mx = |vx δx |.

(8.22)

Proof. It suffices to verify the second identity in (8.22) on the dense set span{vx }. Since Mx vy = δx (y)vy is 0 unless x = y, we have  |vx δx |vy , y = x, Mx vy = δx (y)vy = (δy (x) − δy (o))vx = vx δx , vy E = 0, else, where we have used (8.3). Now, the first identity in (8.22) follows from the second. For an alternative proof, note that Mf vx = f (x)vx by Lemma 8.4, which implies that Mx = |vx δx |. Then, Mx = (Mx ) = |vx δx | = |δx vx |.  Note that (1.9) immediately gives  −cxy , x = y, (8.23) δx , δy E = c(x), x = y. Remark 8.17. One can prove Theorem 8.12 from Lemma 8.16:



Mx HE →HE = |δx vx | HE →HE = δx E vx E = c(x) R(x). (8.24) Corollary 8.18. C  (HE ) is the C  -subalgebra of B(HE ) with generators {Mx , Mx }x∈X and relations Mx My = δx , δy E |vx vy |,

(8.25)

Mx My = vx , vy E |δx δy |,

(8.26)

where δx , δy E is as in (8.23).

Operator Theory and Analysis of Infinite Networks

180

Proof. The computations are direct applications of (8.22) and Dirac’s notation:   Mx My = |vx δx | (|δy vy |) = |vx δx |δy vy | = δx , δy E |vx vy |, 

and similarly for Mx My .

Remark 8.19. Corollary 8.18 shows that C  (HE ) contains all the rank-1 projections corresponding to the functions {vx }. Since the span of this set is dense in HE , this implies that C  (HE ) contains all finite-rank operators and hence all the compact operators (since the compact operators are obtained by closing the space of finite-rank operators). Thus, Corollary 8.18 shows that C  (HE ) is quite large. Remark 8.20. Let us introduce the normalized functions ux :=

vx vx E

and dx :=

δx δx E

(8.27)

and the corresponding rank-1 projections onto the spans of these elements: 1 M  Mx = (P[x → o])Mx Mx c(x)R(x) x (8.28)

Ux := |ux ux | = proj span ux = and Dx := |dx dx | = proj span dx =

1 Mx Mx = (P[x → o])Mx Mx . c(x)R(x) (8.29)

Then, one has two systems of orthonormal projections satisfying the relations vx , vy E Ux Uy =

|ux uy |, R(x)R(y)

Ux Dy = ux , dy E |ux dy |,

Dx Uy = dx , uy E |dx uy |,

δx , δy E Dx Dy =

|dx dy |. c(x)c(y)

Moreover, one also has  ran Ux = HE where



x∈X

and



ran Dx = Fin,

x∈X

indicates that one takes the closed linear span.

(8.30)

Multiplication Operators on the Energy Space

181

Theorem 8.21 gives a necessary and sufficient condition for determining whether or not an operator is bounded. In the statement and proof, the ordering is as defined by (A.2). It will also be helpful to keep in mind that ⎤ ⎡ ⎤⎡V Vx1 x2 Vx1 x3 · · · x1 x1 f (x1 ) 0 0 ··· ⎢ ⎥ ⎢ 0 0 ···⎥ Vx2 x1 Vx2 x2 Vx2 x3 · · · ⎥ f (x2 ) ⎢ ⎥⎢ ⎢ ⎥ Df V Df = ⎢ 0 0 f (x3 ) · · · ⎥ Vx3 x1 Vx3 x2 Vx3 x3 · · · ⎥ ⎣ ⎦⎢ ⎣ ⎦ .. .. .. .. .. .. .. . . . . . . . ⎡ ⎤ f (x1 ) 0 0 ··· ⎢ ⎥ ⎢ 0 f (x2 ) 0 ···⎥ ⎢ ⎥ ×⎢ ⎥ ⎢ 0 0 f (x3 ) · · · ⎥ ⎣ ⎦ .. .. .. . . . ⎡ ⎤ f (x1 )Vx1 x1 f (x1 ) f (x1 )Vx1 x2 f (x2 ) f (x1 )Vx1 x3 f (x3 ) · · · ⎢ ⎥ ⎢ f (x2 )Vx2 x1 f (x1 ) f (x2 )Vx2 x2 f (x2 ) f (x2 )Vx2 x3 f (x3 ) · · · ⎥ ⎢ ⎥ =⎢ ⎥, ⎢ f (x3 )Vx3 x1 f (x1 ) f (x3 )Vx3 x2 f (x2 ) f (x3 )Vx3 x3 f (x3 ) · · · ⎥ ⎣ ⎦ .. .. .. .. . . . . (8.31) as in (8.17), and that VF and DF = (Df )F are the finite submatrices of V and Df obtained by taking only the rows and columns corresponding to those vertices x which lie in the finite subset F ⊆ X. The limit of the filter {TF }F ⊆X of the operators defined in (8.32) will be computed in Corollary 8.26. Theorem 8.21. The multiplication operator M = Mf is bounded on HE if and only if the family of operators 1/2

−1/2

TF = VF DF VF

(8.32)

is uniformly bounded as F ranges over all finite subsets of X, i.e., there exists a constant b < ∞ such that TF V→V ≤ b,

∀F ⊆ X, |F | < ∞.

Here, VF and DF are the truncated operators with entries [VF ]x,y∈F := vx , vy E

and

[DF ]x,y∈F := f (x)δxy ,

(8.33)

as in Definition 6.21 (but DF here is not to be confused with that in Definition 6.41).

Operator Theory and Analysis of Infinite Networks

182

In the case when these equivalent conditions are satisfied, Mf HE →HE = sup TF V→V ≤ b,

(8.34)

F

where the sup is taken over all finite subsets F ⊆ X. Proof. From Theorem 8.7, we know that M is bounded iff sf (x, y) in (8.8) is semidefinite, and this inequality can be written in terms of matrices as b2 V − DV D ≥ 0 with respect to the ordering (A.2); see Lemma A.6. This transforms a difficult condition (positive semidefiniteness) into an easier condition to check: b2 ξ, VF ξV − ξ, DF VF DF ξV ≥ 0,

∀ξ ∈ V.

(8.35)

Note that V is psd (essentially by definition):       ξx ξy Vx,y = ξx ξy vx , vy E = ξx vx , ξy vy x,y∈F

x,y∈F

x∈F

2



= ξx vx ≥ 0,

x∈F

y∈F

E

E

and so, we have V = (V 1/2 )2 by Lemma A.5. Then, (8.35) gives 1/2

1/2

1/2

∀ξ ∈ V.

1/2

1/2

VF DF ξ2V = TF VF ξ2V ≤ b2 VF ξ2V

(8.36)

Thus, there is a bounded operator sending VF ξ to VF DF ξ for any ξ ∈ V. Less grandiosely, this means there is an n × n matrix TF satisfying 1/2

TF VF

1/2

= VF DF

and TF V→V ≤ b.

(8.37)

From (8.37), it is clear that TF is given by (8.32), and the independence of b from F follows by the uniform boundedness principle.  Remark 8.22. Note that TF is not self-adjoint for general finite F (even in the case when f is R-valued) because   1/2 −1/2 −1/2 1/2 VF DF VF = VF DF VF . However, one can still compute the operator norm of TF as the square root of the largest eigenvalue of TF TF .

Multiplication Operators on the Energy Space

183

Remark 8.23. Even in the case when Mz = Mδz , it may be very difficult to use (8.32) to compute Mz  and preferable to use Theorem 8.12 instead. In this situation, one has only −1/2

1/2

TF = VF (δx,z δy,z )VF

, 1/2

but it is even difficult to compute the entries of VF

−1/2

and VF

.

Our next goal is to compute the limit of the filter {TF }F ⊆X in Corollary 8.26, where the ordering is the usual partial order of set containment on the finite sets F . However, this will require some further discussion of V from Definition 8.15. Definition 8.24. Given a finite subset F ⊆ X, define PF to be the . projection to the subspace spanned by {vx .. x ∈ F }. The purpose of J in the following lemma is that it serves to intertwine Mf with a more computable operator, see (8.41) in the following corollary and also (8.42). Recall that V is defined in Definition 6.21 and discussed in Remark 8.11. Lemma 8.25. A unitary equivalence between V and HE is given by the operator  ξx vx , (8.38) J : HE → V by Ju = V 1/2 ξ for u = x∈X

where the convergence of the sum in (8.38) is with respect to E. Proof. Let u, w ∈ span{vx }x∈F be given by   u= ξx vx and w = ηx vx , x∈F

(8.39)

x∈F

where F is some finite subset of X. Then, (8.14) gives      u, wE = ξx vx , ηy vy = ξx ηy vx , vy E = ξ, ηV . (8.40) x∈F

y∈X

E

x,y∈F

Now, for general u, w ∈ HE , let PF be as in Definition 8.24, and compute 1/2

1/2

u, wE = lim PF u, PF wE = lim VF ξ, VF ηV = V 1/2 ξ, V 1/2 ηV , F →X

F →X

where the middle equality comes from (8.40).



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184

Corollary 8.26. Let TF be defined as in (8.32), and let J be defined as in (8.38). In the case when the equivalent conditions of Theorem 8.21 are satisfied, one has T = lim TF = JMf J 

T  = lim TF = JMf J  ,

and

F →X

F →X

(8.41)

where Mf is the adjoint with respect to E, T  is the adjoint with respect to V, and the limit is taken in the strong operator topology. Thus, Mf ∼ = limF →X TF . Proof. To get (8.41), first pick a finite F ⊆ X and with PF as in Definition 8.24:  Mf PF u2E = PF u, Mf Mf PF uE = ξx ξy f (x)f (y)Vxy x,y∈F

= ξ, DF VF DF ξV 1/2

= VF DF ξ2V . 1/2

However, (8.32) means that TF VF continues with

1/2

= VF DF , and so, the computation

1/2

1/2

Mf PF u2E = VF DF ξ2V = TF VF ξ2V . Now, let F → X on both sides, and the proof follows by Theorem 8.21.  Consequently, one has a commutative square as follows. HE J

 V

Mf

/ HE

(8.42)

J

T

 /V

In light of Remark 8.23, it will be helpful to have a condition which is only sufficient to ensure the boundedness of Mf (not necessary) but is much easier to check. Theorem 8.27. The operator M = Mf satisfies Mf HE →HE ≤





|f (x)|

|f (x)| c(x)R(x) = P[x → o] x∈X x∈X

(8.43)

and is hence a bounded operator on HE whenever the right-hand side of (8.43) converges.

Multiplication Operators on the Energy Space

185

Proof. For F ⊆ X finite, let f |F = f χF be the restriction of f to F . Then, Theorem 8.12 and (8.22) give M f |F =



f (x)Mx =

x∈F



f (x)|δx vx |,

(8.44)

x∈F

where the summation is finite so that Mf |F is clearly bounded. Now, we show that Mf |F converges to Mf in norm, as F → X. Since Mf |F HE →HE ≤



|f (x)| |δx vx | H

E →HE

x∈F

=



x∈F

=



|f (x)| δx E vx E

|f (x)| c(x)R(x),

x∈F

we have (8.43). Moreover, when the right-hand side of(8.43) converges, then for any ε > 0, there exists an F0 such that 

|f (x)| c(x)R(x) < ε, x∈X\F0

which shows that limF →X Mf |F − Mf HE →HE = 0, and combining Corollary 3.40 with Corollary 3.43 completes the proof.  One result appearing in the proof of Theorem 8.27 will be helpful on its own. Corollary 8.28. If Mf satisfies (8.43), then Mf |F converges to Mf in norm, where Mf |F is as in (8.44). In particular, (8.43) implies Mf =

 x∈x

f (x)Mx =



f (x)|δx vx |,

(8.45)

x∈X

where the sum converges in the norm operator topology. It seems doubtful that Mf |F converges to Mf in norm, in general. However, we do have a partial result in this direction in Theorem 8.30. Lemma 8.29. Let {Fn }∞ n=1 be an exhaustion of X, and define Pn to be . the projection to span{vx .. x ∈ Fn } for each n ∈ N. If Mf is bounded, then

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for each n, there is an m = mn with Pn Mf Pn = Pn Mfmn Pn ,

(8.46)

where fk = f |Fk = f χFk is the restriction of f to Fk . Proof. Since the energy kernel has dense span in HE , we can apply the Gram–Schmidt algorithm to obtain an orthonormal basis. {ψx }x∈X .1 Thus, we can write  Pn Mf Pn = f (x)|ψy ψy | |δx vx | |ψz ψz | x y≤x z≤y

=



f (x)ψy , δx E vx , ψz E |ψy ψz |.

(8.47)

x y≤x z≤y

However, for all n, there exists an m ≥ n (which we write as mn to C emphasize the dependence on n) such that, for x ∈ Fn and y, z ∈ Fm , n one has ψy , δx E = vx , ψz E = 0. This essentially follows from the finite range of c and the nature of the Gram–Schmidt algorithm and shows that the sum in (8.47) is finite.  Theorem 8.30. Let {Fn }∞ n=1 be an exhaustion of X, and for a fixed f : X → C, and let fn = f |Fn = f χFn be the restriction of f to Fn . If Mf is bounded, then Pn Mfmn Pn converges to Mf in the strong operator topology, where Mfmn is a finite-dimensional suboperator of Mf as in Lemma 8.29. Proof. Note that Pn M Pn converges strongly to M whenever M is a bounded operator by general operator theory. Then, by Lemma 8.29, the right-hand side of (8.46) also converges to Mf in the strong operator topology.  Corollary 8.31. If Mf is bounded, then the range of Mf lies in Fin. Proof. Since Mx = |δx vx | by (8.22) and ran Mx ⊆ Cδx , this follows immediately from Theorem 8.30.  1 This

is carried out in more detail in Ref. [JP11a, Section 3.1]. Note that vo = vx0 is not included in the enumeration.

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187

8.3 Bounded Functions of Finite Energy In the preceding section, we considered the functions f for which Mf is a bounded operator. In this section, we consider the algebra of bounded functions f in HE ; cf. Definition 7.39. Neither of these spaces of functions is contained in the other, as illustrated in the examples of Chapter 13; see also Examples 14.23 and 15.9. In Chapter 7, we gave two ways of constructing a Gel’fand triple SG ⊆ HE ⊆ SG for the energy space. This allowed for the isometric embedding of HE into the Hilbert space L2 (SG , P) given in Theorem 7.20. Remark 8.32. The Wiener transform gives a representation of the Hilbert space HE as an L2 space of functions on a probability “sample space” (SG , P). This is useful in many ways: (a) While direct computation in HE is typically difficult (when solving equations, for example), passing to the transform allows us instead to convert geometric problems in HE into manipulation of functions on SG or on a subspace of it. (b) As we show in this section, problems involving bounded operators in HE can be subtle. The Wiener transform immediately offers a maximal abelian algebra of bounded operators, viz., multiplication by L∞ functions on SG .2 Definition 8.33. Denote the collection of bounded functions of finite energy by .

AE := {u ∈ HE .. u is bounded}.

(8.48)

Define multiplication on AE by the pointwise product (u1 u2 )(x) := u1 (x)u2 (x)

(8.49)

uA := u∞ + uE .

(8.50)

and a norm on AE by

2 The

multiplication operator on HE “before the transform” (discussed in Sections 8.1  , P) “after the transform.” and 8.2) should not be confused with those in L2 (SG

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Lemma 8.34. (AE ,  · A ) is a Banach algebra. Proof. It is obvious that u1 u2 is bounded; one checks that u1 u2 ∈ dom E by directly computing u1 u2 2E =

1 2



cxy |u1 u2 (x) − u1 u2 (y)|2

x,y

=

1 2



cxy |(u1 (x) − u1 (y))u2 (x) + (u2 (x) − u2 (y))u1 (x)|2

x,y

≤

1 2



cxy (|(u1 (x) − u1 (y)||u2 (x)| + |u2 (x) − u2 (y)||u1 (x)|)2

x,y

=

1 2



cxy (|(u1 (x) − u1 (y)|2 |u2 (x)|2

x,y

+

1 2



cxy |u2 (x) − u2 (y)|2 |u1 (x)|2

x,y

+

1 2



cxy |u1 (x)||u2 (x)||u1 (x) − u1 (y)||u2 (x) − u2 (y)|

x,y

≤ u22 ∞ u1 2E + 2u1 ∞ u2 ∞ |u1 , u2 E | + u21 ∞ u2 2E , which is clearly finite. This estimate also implies that (u, v) → uvA is a closed linear functional on the product space AE × AE . The closed graph theorem then implies that it is continuous, i.e., uvA ≤ CuA vA ,

for all u, v ∈ AE .

It is a standard argument that one can then find an equivalent norm for which the same inequality holds with C = 1; see Ref. [KR97], for example.  Definition 8.35. By the Gel’fand space of a Banach algebra A, we mean the spectrum spec(A) realized as either the collection of maximal ideals of A or as the collection of multiplicative linear functionals on A. See Refs. [Arv02, Arv76a]. Let ζ ∈ spec(AE ) denote a multiplicative linear functional on AE so that ker ζ is a maximal ideal of AE , and let ΦA : AE → C(spec(AE )) denote the Gel’fand transform so that ΦA (v)(ζ) := ζ(v).

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189

There is a norm equivalent to the one given in (8.50) with respect to which AE becomes a Banach algebra (see Ref. [KR97], for example), and we are concerned with the Gel’fand space of this one. Lemma 8.36. As a Banach algebra, AE is isometrically isomorphic to C(spec(AE )). Proof. We need to show that ker ΦA = 0. This is equivalent to showing that AE is semisimple, i.e., the intersection of all the maximal ideals is 0. It therefore suffices to show that an intersection of a subcollection of the maximal ideals is 0. Let Lx denote the multiplicative linear functional defined by Lx u := u(x). Since Lx u = u, vx E under convention (8.2) and  {vx } is dense in HE (and therefore the total), it follows that ker Lx = 0.  Definition 8.37. Recall from Definition 2.14 that span{δx } is the collection of functions of finite support; see also the first paragraph of Section 8.1. If we complete span{δx } in the sup norm, we obtain the collection of bounded functions on G, and if we complete in E, we obtain Fin. Therefore, the closure of span{δx } in the norm of AE is AFin := Fin ∩ AE .

(8.51)

Lemma 8.38. AFin is a closed ideal in AE . Proof. Fix x ∈ G and let δx ∈ AFin be the characteristic function of {x}, as defined in Definition 2.14. Take any finite set F ⊆ G and any  linear combination f = x∈F ξx δx . Since v · δx = v(x)δx , one has v · f =  x∈F ξx v(x)δx , which is clearly supported in F again. This shows that the collection of all finitely supported functions on G is an ideal. Now, for f ∈ AFin , take {fn } where each fn has finite support and f − fn A → 0. This is possible in view of Definition 8.37. Since v · fn ∈ span{δx } by the first part, (v · f ) − (v · fn )A = v · (f − fn )A ≤ vA f − fn A → 0, which shows that v · f ∈ AFin (by Definition 8.37 again).

(8.52) 

Definition 8.39. Since AFin is a closed ideal, it is standard that AHarm := AE /AFin

(8.53)

is an algebra and in fact a Banach algebra under the usual norm .

[u]Harm := inf{u + f A .. f ∈ BFin }.

(8.54)

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Theorem 8.40. The Gel’fand space of AHarm contains bd G, and there is an isometric embedding AHarm → C(bd G). Proof. Recall from Theorem 7.20 that for vx ∈ HE , one defines v˜x ∈ L2 (SG , P) by v˜x (ξ) = vx , ξG = lim vx,n , ξE , n→∞

(8.55)

where {vx,n }n∈N is any sequence in SG converging to vx and that with this extension, harmonic functions in HE have the boundary representation  ˜ h v˜x (ξ) ∂∂  (ξ) dP(ξ) + h(o). (8.56) h(x) =  SG

˜ = 0 on bd G, then It follows immediately from this representation that if h h = 0 everywhere on G. (Here, ˜ h = 0 on bd G means limn→∞ h(xn ) = k for some k ∈ C and any sequence {xn } with limn→∞ xn = ∞.) ˜(β) defines a multiplicaGiven any β ∈ bd G, the evaluation χβ (u) := u tive linear functional on AHarm so that bd G is contained in the Gel’fand  space of AHarm . Theorem 8.41. If Harm = 0, then the Gel’fand space of AE is G ∪ {∞}. Proof. Let χ ∈ spec(AE ) and apply it to both sides of v · δx = v(x)δx (the left-hand side is a product in AE , and the right-hand side is a scalar multiple of δx ) to obtain χ(v) · χ(δx ) = v(x)χ(δx ), and hence, χ(δx ) · (χ(v) − v(x)) = 0,

∀x ∈ G, ∀v ∈ AE .

(8.57)

This implies (i) χ(δx ) = 0 for all x or else (ii) ∃y ∈ G for which χ(δy ) = 0. Since Harm = 0, Theorem 2.20 implies that χ is determined by its action on {δx }x∈G . Thus, only the zero functional satisfies χ(δx ) = 0 for all x ∈ G, and we may safely ignore case (i). For case (ii), it follows that χ(δx ) = 0 for all x = y, so χ(v) = v(y) by (8.57). This shows that χ corresponds to evaluation at the vertex y; note that the uniqueness of y for which χ(δy ) = 0 is implicit. Observe that C(G) is not unital because the constant function 1  0 in HE . We unitalize AE in the usual way: A˜E = AE ×C with (a1 , λ1 )(a2 , λ2 ) := (a1 a2 +λ2 a1 +λ1 a2 , λ1 λ2 ). (8.58) The unit in this new algebra is then (0, 1). By standard theory, this corresponds to taking the one-point compactification of G. 

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In rough terms, taking the one-point compactification of G corresponds to conjoining the single multiplicative linear functional “evaluation at ∞” to AE . It is known from Ref. [ALP99] that when Harm = 0, u(x) tends to a common value along P-a.e. path to ∞ for any u ∈ HE . 8.4 Remarks and References The literature regarding this area is vast, and we do not attempt to cite all those pertaining to the subareas. However, some sources with similar context include Refs. [AMV09, KVV06, SVY04, Cho08]; see also Refs. [Die10, BG08, Woe00, Woe09, SCW09] regarding random walk models and the references cited therein. For quantum theory, see Refs. [Sal10, SSS09, Del09, MS09a]. Conjecture 8.42. We conjecture that the converses of Theorem 8.40 and Theorem 8.41 both hold. In other words, we expect that AHarm ∼ = C(bd G), and that if Harm = 0, then the Gel’fand space of AE contains at least two elements that don’t correspond to any vertex of G.

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Chapter 9

Symmetric Pairs Symmetry is what we see at a glance; based on the fact that there is no reason for any difference. ... — Blaise Pascal The most general law in nature is equity-the principle of balance and symmetry which guides the growth of forms along the lines of the greatest structural efficiency. — Herbert Read

This chapter is devoted to the study of symmetric pairs of unbounded operators in Hilbert space. A symmetric pair comprises densely defined operators J : H1 → H2 and K : H2 → H1 , which are compatible in a certain sense: Each is contained in the adjoint of the other. In Section B.3, we provide a streamlined construction of the Friedrichs extension of a densely defined self-adjoint and semibounded operator A on a Hilbert space H by means of a symmetric pair of operators. With the appropriate definitions of H1 and J in terms of A and H, we show that (JJ  )−1 is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A. These results are applied to the study of the graph Laplacian on infinite networks in Section 9.2 in relation to the Hilbert spaces 2 (G0 ) and HE . The study of symmetric pairs arose in part from studying the relationship between the Laplacian on 2 (G0 ) and the Laplacian on HE and in part from the search for a proof of the closability of the energy form E; see Section 9.2. However, it turned out to have applications in other 193

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areas: construction of the Friedrichs (Section B.3) and Krein (Section B.4) extension1 of Hermitian operators, the Malliavin derivative of stochastic calculus (Section 9.3), and the Tomita–Takesaki theory of von Neumann algebras (Section 9.4). 9.1 Symmetric Pairs and Closability In this section, we show by example that symmetric pairs may be used to deduce the closability of operators and sometimes even compute adjoints. See Refs. [DS88, BR79, BR97, Nel69, Rud91, vN32c] for background. Definition 9.1. Suppose H1 and H2 are Hilbert spaces, A, B are operators with dense domains, dom A ⊆ H1 and dom B ⊆ H2 , and A : dom A ⊆ H1 → H2

and B : dom B ⊆ H2 → H1 .

If A and B are linear operators, we say that (A, B) is a symmetric pair iff Aϕ, ψH2 = ϕ, BψH1

for all ϕ ∈ dom A, ψ ∈ dom B.

(9.1)

In other words, (A, B) is a symmetric pair iff A ⊆ B

and B ⊆ A .

(9.2)

In the case when A and B are conjugate linear operators (see Section 9.4), we say that (A, B) is a symmetric pair iff Aϕ, ψH2 = ϕ, BψH1

for all ϕ ∈ dom A, ψ ∈ dom B.

(9.3)

Example 9.2 (Integration by parts). The best understood symmetric pairs arise from some form of integration by parts. For example, consider H1 = H2 = L2 (g), where g is the standard Gaussian measure (i.e., N (0, 1)) 2 on R: dg(x) := (2π)−1/2 ex /2 dx. In this case, one can easily check that (A, B) is a symmetric pair for the following two operators: dom A := {f ∈ L2 (g) .. f  (x) ∈ L2 (g)}, .

A :=

d dx ,

dom B := {f ∈ L2 (g) .. xf (x) − f  (x) ∈ L2 (g)}, .

(9.4)

d B := − dx + Mx , (9.5)

where Mx denotes the operation of multiplying by the independent variable x. Both domains are clearly dense in L2 (g). In the following, we give examples of important symmetric pairs which are not related by integration by parts. 1 The term “extension” here refers to containment of the respective graphs of the operators under consideration.

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195

Definition 9.3. The graph of a linear operator A : H1 → H2 is    x .. Γ(A) := ∈ H1 ⊕ H2 . y = Ax . y Definition 9.4. A densely defined linear operator A : H1 → H2 ,

dom A ⊆ H1 ,

is said to be closable iff Γ(A) (the closure of Γ(A) in the natural Hilbert norm of H1 ⊕ H2 ) is the graph of an operator. In this case, the closure of A is denoted by A, and one has Γ(A) = Γ(A). Example 9.5. The point of this example is to provide an interesting case of an operator that fails to be closable and to illustrate the relationship that can exist between the adjoint of an operator between L2 spaces and the Radon–Nikodym derivative of their respective measures. We return to ¯ this theme in Example B.25; see also Refs. [Ota88, HSdSS07, Jør80]. 2 Let X = [0, 1], and consider L (X, λ) and L2 (X, μ) for measures λ and μ, which are mutually singular. For concreteness, let λ be a Lebesgue measure and let μ be a classical singular continuous Cantor measure. Then, the support of μ is the middle-third Cantor set, which we denote by K, so that μ(K) = 1 and λ(X \ K) = 1. The continuous functions C(X) are a dense subspace of both L2 (X, λ) and L2 (X, μ) (see, for example, Ref. [Rud87, Ch. 2]). Define the “inclusion” operator2 J to be the operator with dense domain C(X) and J : C(X) ⊆ L2 (X, λ) → L2 (X, μ) by Jϕ = ϕ.

(9.6)

We show that dom J  = {0}, so suppose f ∈ dom J  . Without loss of generality, one can assume f ≥ 0 by replacing f with |f | if necessary. By definition, f ∈ dom J  iff there exists g ∈ L2 (X, λ) for which   ϕf dμ = ϕg dλ = ϕ, gλ for all ϕ ∈ C(X). (9.7) Jϕ, f μ = X

X

(ϕn )∞ n=1

⊆ C(X) so that ϕn |K = 1 and One can  choose limn→∞ X ϕn dλ = 0 by considering the appropriate piecewise linear modifications of the constant function 1. For example, see Figure 9.1. 2 As a map between sets, J is the inclusion map C(X) → L2 (X, μ). However, we are considering C(X) ⊆ L2 (X, λ) here, and so, J is not an inclusion map on the level of inner product spaces because the inner products are different. Perhaps, “pseudoinclusion” would be a better term.

Operator Theory and Analysis of Infinite Networks

196

ϕ1

1

ϕ2

1

ϕ3

1

... 0

1

0

0

0

1

0

1

0

Fig. 9.1. A sequence {ϕn } ⊆ C(X) for which ϕn |K = 1 and lim Example 9.5.

Now, we have

 X

ϕn dλ = 0. See



ϕn , J f λ = ϕn , f μ = 1, f μ =

|f | dμ



for any n,

(9.8)

X

 2 but limn→∞ X ϕn g dλ = 0 for any continuous g ∈ L (X, λ).2 Thus, |f | dμ = 0 so that f = 0 μ-a.e. In other words, f = 0 ∈ L (X, μ), X and hence, dom J  = {0}, which is certainly not dense! Thus, one can interpret the adjoint of the inclusion as multiplication by a Radon–Nikodym derivative (“J  f = f dμ dλ ”), which must be trivial when the measures are mutually singular. This comment is made more precise in Example B.25 and Corollary B.26. As a consequence of this extreme situation, the inclusion operator in (9.6) is not closable. Theorem 9.6. If (A, B) is a symmetric pair, then A and B are each closable operators. Moreover : 1. A A is densely defined and self-adjoint with dom A A ⊆ dom A ⊆ H1 , and 2. B  B is densely defined and self-adjoint with dom B  B ⊆ dom B ⊆ H2 . Proof. Since A and B are densely defined and A ⊆ B  and B ⊆ A , it is immediate that A and B  are densely defined; it follows from a theorem by von Neumann that A and B are both closable. By another theorem by von Neumann, A A is self-adjoint; cf. Ref. [Rud91, Thm. 13.13]. The same proof works, mutatis mutandis, for the conjugate linear case (9.3).  Remark 9.7. Whenever (A, B) is a symmetric pair, the above theorem now allows us to assume that A and B are closed operators. Hereinafter, we thus refer to the self-adjoint operators A A and B  B. Remark 9.8. It follows from Theorem 9.6 that (9.2) may be rewritten as A ⊆ B

and B ⊆ A ,

which will be useful for Lemma 9.14.

(9.9)

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197

Lemma 9.9. Let (A, B) be a symmetric pair. Then, A = B  iff B = A . Proof. Due to the symmetry of the situation, it suffices to show only one direction, so assume A = B  . Upon taking adjoints, we have A = (A) =  B  = B, and the proof is complete. Definition 9.10. A symmetric pair (A, B) is called maximal iff equality holds upon taking closures in (9.2), i.e., iff the closure of one operator is the adjoint of the other: A = B

and B = A .

(9.10)

By Lemma 9.9, only one of the equalities in (9.10) needs to be verified in a given instance. Remark 9.11 (Examples of maximal symmetric pairs). The inclusion/Laplacian pair discussed in Section 9.2 (see especially Remark 9.27) is not maximal, in general. The Malliavin derivative in Section 9.3 and the polar decomposition of Tomita–Takesaki theory in Section 9.4 will furnish examples of maximal symmetric pairs. See also Theorem 9.22. Definition 9.12. In Corollaries 9.13 and 9.16 and elsewhere, we use the notation H K to denote the orthogonal complement of K in H: .

H K := {h ∈ H .. h, kH = 0 for all k ∈ K}.

(9.11)

Corollary 9.13. For any symmetric pair (A, B), Γ(B  ) Γ(A) = {0} in H1 ⊕ H2 ⇐⇒ Γ(A ) Γ(B) = {0} in H2 ⊕ H1 . Lemma 9.14. Suppose (A, B) is a symmetric pair. The containment A ⊆ B  of (9.9) is strict if and only if there is a nonzero ψ satisfying ψ ∈ dom(A B  )

and

A B  ψ = −ψ.

(9.12)

Proof. Recall that A = (A) and A = A. The containment A ⊆ B  is strict if and only if there is a nonzero ψ ∈ dom B  such that     u ψ 0= = u, ψH1 + Au, B  ψH2 for all u ∈ dom A, , Au Bψ (9.13) where the first equality follows from the definition of containment of operators and the second by the definition of the usual inner product on H1 ⊕ H2 . Observe that (9.13) implies B  ψ ∈ dom A , and hence,  ψ ∈ dom(A B  ) so that (9.13) is seen to be equivalent to (9.12).

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Remark 9.15. Observe that if the symmetric pair is maximal, then (9.12) is the condition that A A and B  B are essentially self-adjoint.3 Corollary 9.16. If (A, B) is a symmetric pair, then there is a partial isometry V : H1 → H2 such that A = V (A A)1/2 = (B  B)1/2 V . In particular, specH1 (A A) \ {0} = specH2 (B  B) \ {0}. Let P1 denote the orthogonal projection onto H1 ker A, and let P2 denote the projection onto ran A in H2 . Then, V  V = P1 and V V  = P2 . The properties of the partial isometry V in Corollary 9.16 are an important consequence of Theorem 9.6. Remark 9.17. Corollary 9.16 is an immediate consequence of Theorem 9.6 and reappears in the background of the example applications of symmetric pairs given later in this chapter. • In Section 9.3, we apply symmetric pairs to the Malliavin derivative of stochastic calculus, and in this case, the defect of the partial isometry is one-dimensional. More precisely, the partial isometry V is isometric except for one dimension, and the one-dimensional space is spanned by the constant function 1 on the sample space Ω. • In Section 9.4, we apply symmetric pairs to the Tomita–Takesaki theory of von Neumann algebras and show that in the case of modular theory, the defect of the partial isometry is zero (cf. Theorem 9.61) so that V is, in fact, an isometry called the modular conjugation. All of these results follow from our use of the particular symmetric pair at hand. See Refs. [DS88, KR97] for background. 9.1.1 The relation between symmetric operators and symmetric pairs In the case of a symmetric operator, the question of existence of proper symmetric extensions (or possibly even proper self-adjoint extensions) is decided by certain eigenspaces of the adjoint operator A (see Definition 9.21). Lemma 9.14 provides an analogous result for symmetric pairs, and we now extend this fruitful analogy. 3 See

Definition B.8.

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199

The notion of “symmetric pairs of unbounded operators” generalizes the concept from von Neumann of a single symmetric operator as in Definition 9.18. In Theorem 9.19, we make the relation between symmetric operators and symmetric pairs more precise. Theorem 9.19 shows that for every symmetric pair of operators (A, B), there is an associated canonical symmetric operator, denoted by L in the following, with L acting in the direct sum Hilbert space and determined uniquely by the given pair (A, B). We further show in Corollary 9.22 that the notion of “deficiency” for a pair (A, B) is directly connected to the deficiency of the symmetric operator L (in the sense of von Neumann) and hence to von Neumann deficiency indices. Definition 9.18. Let K be a Hilbert space, let D ⊆ K be a dense subspace, and consider a linear operator L on K with dom L = D. We say that L is symmetric (or Hermitian) iff Lw, zK = w, LzK

for all w, z ∈ D,

(9.14)

or equivalently, iff L ⊆ L . Theorem 9.19. Suppose H1 and H2 are Hilbert spaces and A, B are operators with dense domains dom A ⊆ H1 and dom B ⊆ H2 and A : dom A ⊆ H1 → H2

and

B : dom B ⊆ H2 → H1 .

Define K := H1 ⊕ H2 , and let L : K → K be the densely defined linear operator in K given by L(u ⊕ v) := Bv ⊕ Au for u ∈ dom A and v ∈ dom B. In other words,   0 B dom L := dom A ⊕ dom B and L := . (9.15) A 0 Then, (A, B) is a symmetric pair iff L is symmetric. Proof. Let w = u⊕v ∈ dom A⊕dom B, and let z = x⊕y ∈ dom A⊕dom B. On the one hand,     Bv x (9.16) , Lw, zK = = Bv, xH1 + Au, yH2 . Au y On the other hand, w, LzK =

    u By , = v, AxH2 + u, ByH1 . v Ax

(9.17)

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Operator Theory and Analysis of Infinite Networks

Now, from (9.16) and (9.17), it is clear that Lw, zK = w, LzK iff Bv, xH1 + Au, yH2 = v, AxH2 + u, ByH1 , which holds for all x, u ∈ dom A and y, v ∈ dom B iff (9.1) is true.



Corollary 9.20. For L defined as in (9.15), L (x ⊕ y) = A y ⊕ B  x for y ∈ dom A and x ∈ dom B  . In other words,   0 A     dom L = dom B ⊕ dom A and L = . (9.18) B 0 Proof. The proof of (9.18) follows by applying (9.15) to obtain          u x u A y , . ,L =  v B x y v



Definition 9.21. For a densely defined symmetric operator T : K → K, the defect spaces are .

Def ± (T ) := Eig± (T  ) = {u± ∈ dom T  .. T  u± = ± u± }.

(9.19)

Also, the deficiency indices of T are n± (T ) := dim Def ± (T ).

(9.20)

See Definition B.10. It is well known, due to the work of von Neumann, that a densely defined symmetric operator has self-adjoint extensions if and only if its deficiency indices are equal (n+ (T ) = n− (T )) and that it has a unique self-adjoint extension (i.e., essentially self-adjoint ) iff both deficiency indices equal 0. Theorem 9.22. Let (A, B) be a asymmetric pair, and let L be defined as in (9.15). (a) The deficiency indices of L are equal: n+ (L) = n− (L). (b) Furthermore, L is essentially self-adjoint iff (A, B) is a maximal symmetric pair. Proof. (a) If u ⊕ v is an eigenvector of L with eigenvalue , then (9.18) gives          u u A v  u = L , (9.21) = = v Bu v v

Symmetric Pairs

201

which is equivalent to v ∈ dom A , with A v = u,

and u ∈ dom B  , with B  u = v. (9.22)

Now, we can see that   u ∈ Def (L) v

 ⇐⇒

 −u ∈ Def − (L) v

(9.23)

by the following computation:          A v −u u  −u L = , = =− v (i) −B  u (ii) − v v where (i) follows from (9.18) and (ii) follows from (9.22). The other computation is similar. This gives a bijection between Def (L) and Def − (L), whence n+ (L) = n− (L). (b) By the theory by von Neumann, a self-adjoint extension of L corresponds to a partial isometry Q, which maps the defect space Def (L) onto the defect space Def − (L). We denote this extension by LQ and observe that L ⊆ LQ ⊆ L , whence (9.15) and (9.18) give the representation   0 BQ LQ = (9.24) AQ 0 for some operators AQ and BQ satisfying A ⊆ AQ ⊆ B  and B ⊆ BQ ⊆ A .  ⊆ B  , whence Taking adjoints yields B ⊆ AQ ⊆ A and A ⊆ BQ A ⊆ AQ ⊆ B 

and B ⊆ BQ ⊆ A

(9.25)

 = AQ (by because the self-adjointness of LQ implies AQ = BQ and BQ Theorem 9.19). From (9.25), we see that the maximality of the symmetric  pair implies LQ = L and conversely.

Remark 9.23 (Self-adjoint extensions of L). From Corollary 9.22 and its proof, it is clear that every self-adjoint extension of L corresponds to an extension (AQ , BQ ) of (A, B) in the sense that (AQ , BQ ) is a symmetric pair satisfying A ⊆ AQ and B ⊆ BQ . Remark 9.24. Combining (9.21) with (9.23), one can see that u ⊕ v ∈ Def (L ) if and only if        u A v u = = , (9.26) v v − Bu

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where appropriate domain assumptions are made for either direction of the implication. The corresponding identity holds for Def − (L ), and both will be useful hereinafter. According to von Neumann’s theory of self-adjoint extensions of a symmetric operator T , the set of self-adjoint extensions of T is parameterized by partial isometries from Def (T ) = Eig (T  ) to Def − (T ) = Eig− (L ). Denoting such a map by Q (as in the proof of Theorem 9.22(b)) and the Cayley transform by C(T ) := ( − T )( + T )−1 , we apply this to T = L and K = H1 ⊕ H2 to get the following. K = dom C(L) ⊕ Def (L) Q

C

K = ran C(L) ⊕ Def − (L) The structure underlying Theorem 9.22 is a pair of surjective isomorphisms   u Ψ± : Eig−1 (A B  ) → Def ± (L ) by Ψ± (u) = ± Bu for u ∈ Eig−1 (A B  ). Consequently, a partial isometry Q : Def (L) → ˜ as Def − (L) induces an operator on Eig−1 (A B  ) (which we denote by Q) the unique operator making the following diagram commute. Eig−1 (A B  )

˜ Q

Ψ+

Def (L)

Eig−1 (A B  ) Ψ−

Q

Def − (L)

As in the proof of Theorem 9.22(b), we use the notation LQ for the ˜ self-adjoint extension of L corresponding to Q. Let us also denote v := Qu ˜ Then, by (9.26), the action of LQ on a generic element of for u ∈ dom Q.         x x u u .. dom LQ = . + Q + y y − Bu − Bu    u ∈ Def (L) ∈ dom L, − Bu is given by LQ

        x v By + u − v u + = , + Bv Ax + B  u + B  v y − Bu

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where we have used the fact that L ⊆ LQ to compute the action of LQ on x ⊕ y and (9.24) to compute the action of LQ on the rest of the expression. ˜ Note that since Q is a partial isometry, the corresponding restriction on Q is that    2  2 ˜ Qu u 2  2 ˜ 2 ˜ 2 =

u 1 + B u 2 =   ˜ = Qu 1 + B Qu 2 . −B u K − B Qu K 9.2 Applications of Symmetric Pairs to Laplace Operators on Infinite Networks In this section, we use symmetric pairs to investigate the relationship between 2 (G0 ) and HE . Observe that one may consider the Laplace operator in two distinct senses: (i) L : HE → 2 (G0 ) (ii) ΔE : HE → HE

with domain dom L := span{vx }x∈G , (9.27) with domain dom ΔE := span{vx }x∈G . (9.28)

Recall that δx ∈ 2 (G0 ) denotes the Dirac mass at x, i.e., the characteristic function of the singleton {x} and that δx ∈ HE denotes the element of HE which has δx ∈ 2 (G0 ) as a representative. The context will make it clear which meaning is intended. Also, recall that E(δx ) = c(x) < ∞ is immediate from (1.9), and hence, one always has δx ∈ HE . As usual, for functions u, v : G → R, we have the 2 (G0 ) inner product u, v2 :=

u(x)v(x),

(9.29)

x∈G

i.e., the measure on 2 (G0 ) is just the counting measure. In both cases (9.27) and (9.28), the operator has the same domain and is determined on its domain by formula

cxy (u(x) − u(y)). (9.30) u(x) → y∼x

Recall that the sum in (9.30) is finite by the local finiteness assumption on the network, and so, the Laplacian is well defined in either sense. In Theorem 9.25, we apply Theorem 9.6 to the construction laid out in Ref. [Kre47]. This shows how one can recover the closability results described in Sections 8 and 6 in a manner which is both quicker and

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more elegant. More precisely, we show that if the “inclusion” operator is defined by K : 2 (G0 ) → HE by Kϕ = ϕ on the appropriate dense domain in 2 (G0 ), then (K, L) forms a symmetric pair. In contrast to the examples in Sections 9.3 and 9.4, the construction here illustrates a symmetric pair which is not maximal; see Remark 9.27. Theorem 9.25. Define K : span{δx }x∈G → HE by Kδx = δx , and define L : span{vx }x∈G → 2 (G0 ) by L(vx ) = δx − δo , as in (9.27). Then, (K, L) is a symmetric pair, and hence, both operators are closable. Proof. We must establish Kϕ, ψE = ϕ, Lψ2 for all ϕ ∈ span{δx }x∈G and ψ ∈ span{vx }x∈G , for which the following is sufficient: Kδx , vy E = δx , vy E = δx (y) − δx (o) = δx , δy − δo 2 = δx , Lvy 2 , where we have used the reproducing property of vx .



Theorem 9.26. The following are equivalent : (i) For the Laplacian in (9.28), i.e., considered as an operator ΔE : HE → HE with dom ΔE = span{vx }x∈G , ΔE is not essentially self-adjoint. In other words, there is a nonzero ψ ∈ HE satisfying ψ ∈ dom ΔE

and

ΔE ψ = −ψ.

(9.31)

(ii) For the Laplacian in (9.27), i.e., considered as an operator ΔE : HE → 2 (G0 ) with dom Δ = span{vx }x∈G , there is a nonzero ψ ∈ HE satisfying ψ ∈ dom(Δ K  )

and

Δ K  ψ = −ψ.

(9.32)

(iii) The inclusions K ⊆ Δ and Δ ⊆ K  are proper, i.e., there is a nonzero ψ ∈ HE satisfying   ψ ∈ Γ(K  ) Γ(Δ). K ψ Proof. The equivalence (i) ⇐⇒ (ii) follows from the definitions of the respective adjoint operators. The equivalence (ii) ⇐⇒ (iii) is a direct application of Lemma 9.14 to the current situation.  Remark 9.27. In light of Theorem 9.26 and Section 14.3, which establishes that ΔE may not be essentially self-adjoint on HE , it is clear that (K, Δ) is not, in general, a maximal symmetric pair.

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Lemma 9.28. Harm ⊆ dom ΔE and ΔE h = 0 for all h ∈ Harm. Proof. Let u ∈ dom ΔE with ΔE u ∈ Fin; for example, let u = vx for some ⊥ x ∈ G. Then, ΔE u ∈ Fin = {δx }⊥⊥ x∈G iff ΔE u ∈ Harm by Theorem 2.20, so ΔE u, hHE = 0.

(9.33)

It is clear that this also shows that ΔE h = 0.



Theorem 9.25 provides a more effective way of proving a key result of Ref. [Kre47] in the following corollary. Corollary 9.29. In the notation of Theorem 9.25, K  K is a self-adjoint extension of Δ2 and L L is a self-adjoint extension of ΔE . Proof. Self-adjointness of these operators follows from a celebrated theorem by von Neumann, once closability is established (which is given by Theorem 9.25). To establish that ΔE ⊆ L L, note that the definitions give vy , L Lvx E = Lvy , Lvx 2 = δy − δo , δx − δo 2 = (δx − δo )(y) − (δx − δo )(o) = vy , δx − δo E for any vx . This shows that the action of L L agrees with ΔE on  dom ΔE . Remark 9.30. In Theorems 6.1 and 6.8, it is shown that Δ2 is essentially self-adjoint,4 from which it follows that K  K is the unique self-adjoint extension of Δ2 . It is shown in Ref. [Kre47] that L L is the Krein extension of ΔE ; see Section B.4. Example 9.31. If we take H1 = 2 (G0 ), H2 = HE , and D = span{δx }x∈G, then the hypotheses of Theorem B.24 are satisfied. The only detail requiring effort to check is that span{vx }x∈G ⊆ D (whence D is dense in H2 ). To see this, note that the reproducing property of vx gives |ϕ, vx HE | = |ϕ(x) − ϕ(o)| = |ϕ, δx − δo 2 | √ ≤ ϕ 2 δx − δo 2 = 2 ϕ 2 , so one can take C = 2 in (B.31). In this case, the operator Λ is L L, the Krein extension of the energy Laplacian; see Refs. [Kre47, KL10] and Section B.4. 4 See

Definition B.8.

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9.3 Gaussian Fields and the Malliavin Derivative The term “stochastic calculus” refers loosely to an infinite-dimensional theory of integration/differentiation in the context of stochastic processes, in particular, It¯o calculus and its variant, the Malliavin calculus. The Malliavin calculus is a stochastic version of calculus of variations and provides a robust definition of the derivative of a random variable which allows, for example, for integration by parts. In the following, we study the Malliavin derivative and its adjoint, the Skorokhod integral (an extension of the It¯o integral to processes which may not be adapted; cf. Ref. [Gaw99]). The Skorokhod integral may be interpreted as an infinite-dimensional generalization of the divergence operator. As an example application of symmetric pairs, we show how they can be used to provide a streamlined proof that the Malliavin derivative and Skorokhod integral are closable operators and, in fact, are mutually adjoint. Closability of the Malliavin derivative is fundamental to the Malliavin calculus. The context for this section is Gaussian fields, i.e., Gaussian processes indexed by a Hilbert space L; see Definition 9.32. For example, in the special case where the index Hilbert space is chosen to be L = L2 [0, ∞), the resulting Gaussian process is Brownian motion, and then, the sample points in Ω may be understood as paths/trajectories. See Refs. [Hid80, Bel15, PS72, Gaw99] for background material and further details. Consider a probability measure space (Ω , Σ, P). We are interested in operators in the Hilbert spaces of random variables H1 := L2 (Ω , P)

and H2 := L2 (Ω , P) ⊗ L.

(9.34)

Note that an element of H1 is a random variable taking values in R, and H2 is a random variable taking values in L. We see in Definition 9.37 that the Malliavin derivative is a densely defined operator on H1 with range in H2 . It will be enough for us to work with real Hilbert spaces, in which case we refer to (9.1). We use the notation   F dP and E[F G] = F G dP = F, GH1 for all F, G ∈ H1 . E[F ] = Ω

Ω

Since H2 is a tensor product of Hilbert spaces, it inherits a natural inner product from the tensor structure: F1 ⊗ k1 , F2 ⊗ k2 H2 = F1 , F2 H1 k1 , k2 L = E[F1 F2 ]k1 , k2 L .

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207

Suppose that for i = 1, 2, we have ψi ∈ L2 (Ω , P) ⊗ L. In other words, suppose ψi : Ω → L are measurable functions on Ω with 

ψi (ω) 2 dP(ω) < ∞. Ω

Then,

 ψ1 , ψ2 H2 =

Ω

ψ1 (ω), ψ2 (ω)L dP(ω) = E[ψ1 (·), ψ2 (·)L ].

Definition 9.32. A Gaussian field or Gaussian Hilbert space is (Ω , Σ, P, Φ), where Φ : L → L2 (Ω , P) is interpreted as a Wiener process indexed by L satisfying the following properties: (i) E[Φ(h)] = 0 for all h ∈ L, and (ii) for any finite subset {h1 , . . . , hn } ⊆ L, the random variables {Φ(hi )}ni=1 are jointly Gaussian. That is, the density of the joint distribution of {Φ(hi )}ni=1 is given by 1

−1

gGn (x) := (2π)−n/2 (det Gn )−1/2 e− 2 x,Gn

x

,

x ∈ Rn ,

where Gn is the Gramian matrix with entries [Gn ]i,j = hi , hj L . Theorem 9.33. Given any real Hilbert space H, there always exist associated Gaussian fields, i.e., an associated probability space and Gaussian process Φ, with properties as specified in Definition 9.32. A proof of Theorem 9.33 may be found in Ref. [Hid80] or Ref. [PS72]. Remark 9.34. In the special case of the Itˆ o–Wiener integral  ∞ h(t) dΦt , h ∈ L, Φ(h) = 0

we recover from Definition 9.32 the usual Brownian motion, as described in Ref. [Hid80]. Definition 9.35. The symmetric Fock space is the Hilbert completion of ∞ n Sym(L⊗n ), where L⊗n := L (9.35) Γsym (L) := n=0

j=1

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208

and “Sym” denotes the operator that symmetrizes a tensor. See Ref. [BR79] for details. The symmetric Fock space provides a useful model for H1 , as is seen from Lemma 9.36; it is also used in Lemma 9.38 to give insight on the domain of the Malliavin derivative. Lemma 9.36. The symmetric Fock space is isometrically isomorphic to H1 = L2 (Ω , Σ, P). ∞ ⊗n Proof. For k ∈ L, define ek := n=0 k√n! . Then, for ki ∈ L, i = 1, 2, the vectors eki are the unique elements of Γsym (L) such that ek1 , ek2 Γsym (L) =

∞

k1 , k2 nL = ek1 ,k2 L . n! n=0

(9.36)

Moreover, the mapping W0 : {ek }k∈L → L2 (Ω , P) by

1

2

W0 (ek ) = eΦ(k)− 2 k 2

(9.37)

extends by linearity and closure to a unitary isomorphism W : Γsym (L) →  L2 (Ω , P). Definition 9.37. For n ∈ N, let p ∈ R[x1 , . . . , xn ] be a polynomial in n real variables, so p : Rn → R. Define the Malliavin derivative by .

dom T := spanp∈R[x1 ,...,xn ] {p(Φ(h1 ), . . . , Φ(hn )) .. (hi )ni=1 ⊆ L, and n ∈ N}, (9.38) T (F ) :=

n

j=1

∂p ∂xj

 (Φ(h1 ), . . . , Φ(hn )) ⊗ hj

for F ∈ dom T.

(9.39)

Note that T (F ) ∈ H2 , as in (9.34). Lemma 9.38. dom T is dense in H1 . .

Proof. Since span{ek .. k ∈ L} is dense in Γsym (L), it follows that dom T  is dense in L2 (Ω , P) by Lemma 9.36. Definition 9.39. For dom T as in (9.38), define the stochastic integral operator dom S := dom T ⊗ L, S(F ⊗ k) := F · Φ(k) − T (F ), k.

(9.40) (9.41)

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209

Note that (9.40) is an algebraic tensor product, not a Hilbert tensor product. It is immediate that S is densely defined because the span of the elementary tensors is dense in the tensor product. Remark 9.40. We see in Theorem 9.44 that (S, T ) is a symmetric pair in the sense of Definition 9.1. T

+ H2 = L2 (Ω , P) ⊗ L

H1 = L2 (Ω , P) k S

Recall that H2 is the Hilbert space of vector-valued functions (taking values in L) corresponding to H1 ; see (9.34) and the following remarks. Definition 9.41. We denote the standard Gaussian density by 1

2

gn (x) = (2π)−n/2 e− 2 x ,

x ∈ Rn ,

(9.42)

and recall that the corresponding expectation is given by  p(x1 , . . . , xn )gn (x1 , . . . , xn )dx1 . . . dxn , E [p (Φ(h1 ), . . . , Φ(hn ))] := Rn

(9.43) where the vectors {hi }ni=1 ⊆ L are chosen to be orthogonal, i.e., with hi , hj H = δij . Without this restriction, the joint distribution of the vectorvalued random variable (Φ(h1 ), . . . , Φ(hn )) ∈ Rn is the n-dimensional Gaussian whose covariance matrix is the Gramian with entries hi , hj H . Lemma 9.42. For all k ∈ L and F ∈ dom T , we have T (F ), 1 ⊗ kH2 = F, Φ(k)H1 .

(9.44)

Proof. Note that the tensor structure of H2 = L2 (Ω , P) ⊗ L means T (F ), 1 ⊗ kH2 = E [T (F ), kL ] ,

(9.45)

where the expectation on the right-hand side is with respect to the Gaussian density as in (9.43). Consequently, we may prove (9.44) in the form E[T (F ), kL ] = E[F · Φ(k)] for all k ∈ L, F ∈ dom T

(9.46)

since E[F · Φ(k)] = F, Φ(k)H1 . Choosing an F includes a choice of (i )ni=1 . The calculations which follow will simplify greatly when k is orthogonal to i . We can effect this by

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applying the Gram–Schmidt procedure to this set, which will only result in a different polynomial in the representation of T (and we are free to range over all polynomials). Thus, we may pass from (i )ni=1 to (hi )ni=1 , where hi , hj L = δij . In other words, applying the spectral theorem to the Gramian of (i )ni=1 allows us to diagonalize this matrix so that we can work instead with the standard Gaussian covariance matrix I. By choosing h1 = k in the Gram–Schmidt procedure, we have k, hi L = 1 for i = 1 and 0 otherwise. Consequently, applying orthogonality and integration by parts to (9.39) yields 

  ∂p  E [T (F ), kL ] = E Φ(h1 ), . . . , Φ(hn ) ∂x1  ∂p = (x1 , . . . , xn )gn (x1 , . . . , xn )dxn . . . dxn ∂x n 1 R  ∂gn p(x1 , . . . , xn ) (x1 , . . . , xn )dx1 . . . dxn =− ∂x1 Rn  x1 p(x1 , . . . , xn )gn (x1 , . . . , xn )dx1 . . . dxn = Rn

k, hi L = δ1i by (9.43) IBP by (9.42)

    = E p Φ(h1 ), . . . , Φ(hn ) Φ(h1 )

by (9.43)

= E[F · Φ(k)]

Φ(k) = Φ(h1 ),



which is (9.46).

While using Lemma 9.42 to derive the main result in this section (Theorem 9.44), we need the following simple result about the Leibniz rule in this context. Lemma 9.43. The Malliavin derivative T is a module derivation. In other words, T (HK) = T (H)K + HT (K)

for any H, K ∈ dom T.

(9.47)

Proof. Since the vector space of all polynomials of degree of at most n is invariant under changes in the coordinates, it suffices to note that we can find n and h1 , . . . , hn ∈ L, which can be used to represent both H and K simultaneously: H = p(Φ(h1 ), . . . , Φ(hn )) and K = q(Φ(h1 ), . . . , Φ(hn )). Now, (9.47) follows from the fact that

∂(pq) ∂xi

=

∂p ∂xi q

∂q + p ∂x . i



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211

Theorem 9.44. The Malliavin derivative T of Definition 9.37 and the stochastic integral operator S of Definition 9.39 form a symmetric pair : T (F ), G ⊗ kH2 = F, S(G ⊗ k)H1

for any F, G ∈ dom T, k ∈ L. (9.48)

Consequently, T and S are closable linear operators. Furthermore, (T, S) is a maximal symmetric pair (see Definition 9.10), whence T  = S and S = T . Proof. We compute (9.48) first: T (F ), G ⊗ kH2 = E [T (F ), kL G]

by (9.45)

= E [T (F G), kL ] − E [F T (G), kL ]

by (9.47)

= E [F G · Φ(k)] − E [F T (G), kL ]

by (9.46)

= E [F (G · Φ(k) − T (G), kL )]

by linearity

= E [F · S(G ⊗ k)]

by (9.41)

= F, S(G ⊗ k)H1 . The closability of T and S now follows from Theorem 9.44. To see maximality, we show that no element of Γ(S  ) is orthogonal to Γ(T ). More precisely, since T (eΦ(k) ) = eΦ(k) ⊗k by Corollary 9.48, we show that if F ∈ dom S  and  Φ(k)    e F , = 0 for all k ∈ L, (9.49) eΦ(k) ⊗k S  (F ) then F = 0. Observe that (9.49) is equivalent to   E eΦ(k) (F + S  (F ), kL ) = 0 for all k ∈ L, and since span{eΦ(k) }k∈L is dense in H1 , (9.50) implies F = 0.

(9.50) 

Corollary 9.45. For Tk (F ) := T (F ), kL , we have that Tk is a derivation and that Tk + Tk = MΦ(k) , where Mf denotes the multiplication operator (by the function f ).

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Proof. By (9.41), we have S(F ⊗ k) = MΦ(k) F − Tk (F ) so that Theorem 9.44 gives (Tk + Tk )(F ) = Tk (F ) + MΦ(k) F − Tk (F ) = MΦ(k) F

for all F ∈ dom T.

It is immediate from (9.39) that Tk is a derivation.



Remark 9.46. The operator S of Definition 9.39 is essentially integration by parts (compare with Example 9.2), and this is highlighted by the proof of Theorem 9.44, which is essentially the same as the calculus proof of integration by parts based on the Leibniz rule. Example 9.47 extends this parallel. Example 9.47 (Vector bundles). The construction above is an infinitedimensional generalization of vector bundles. For a vector field X on a manifold M and f ∈ C ∞ (M ), let f X(ϕ)(m) = f (m)X(ϕ)(m),

∀m ∈ M, ϕ ∈ C ∞ (M ).

Then, (f X) = −f X + MX(f ) . Corollary 9.48. Let (Ω , Σ, P, Φ) be a Gaussian field, and let T : H1 → H2 be the closure of T as in Definition 9.37. For k ∈ L, the exponential random variables eΦ(k) lie in dom(T  T ), and   T  T (eΦ(k) ) = Φ(k) − k 2L eΦ(k) . See Remark 9.17 for a discussion of the significance of the following result on ker(T ). Corollary 9.49. Let T , L, and (Ω , Σ, P, Φ) be as above. Then, ker(T ) is the one-dimensional subspace of H1 = L2 (Ω , P) spanned by the constant function 1 on Ω . Proof. Let pH n denote the Hermite polynomial of degree n so that t2

etx− 2 =

∞

n=0

pH n (x)

tn . n!

By (9.37), the isomorphism W : Γsym (L) → L2 (Ω , P) of Lemma 9.38 maps Hn (as defined in (9.35)) into span{pH n (Φ(k)}k∈L . For F ∈ dom T with

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213

T (F ) = 0, one has T  T (F ) = 0 so that 0 = T  T (F ) =

∞

nFn ,

n=0

where Fn ∈ W (Hn ). Therefore, nFn = 0 for n = 1, 2, . . . , and hence, Fn = 0 for n = 1, 2, . . . . It follows that Fn = 0 can only hold for n = 0.  Since W (H0 ) = span 1, the conclusion follows. 9.4 Tomita–Takesaki Theory Tomita–Takesaki theory was devised in the 1970s as a tool for understanding von Neumann algebras, which are type III factors; cf. Refs. [Tak70, Tak03, BR79, KR97, KR86]. A factor is a von Neumann algebra whose center consists only of multiples of the identity operator. Given a factor M, a rough outline of the classification is as follows: Type I: B(H), the algebra of all bounded linear operators on some Hilbert space H. It is called type In if dim H = n. Type II1 : M is of type II1 iff M is not type I but has a finite and faithful trace. Type II∞ : M is of type II∞ iff M is not type II1 but has an infinite but semifinite faithful trace. Here, semifinite means that the domain is dense in M in the w∗ topology. Type III: Everything else, so no trace of a trace. The study of type III von Neumann algebras M and associated commutant M goes back to the early days. Among the big advances from Tomita–Takesaki theory is the following result. Theorem 9.50. Given M type III and with a choice of a faithful normal state, we can make explicit the assertion that the commutant M is realized as an anti-isomorphic copy of M, even spatially implemented (see Theorem 9.61 for a more precise statement). Here, spatially refers to an operator in H. In this case, the operator is called the modular conjugation and denoted by J, and so, one has JMJ = M ; cf. Remarks 9.57 and 9.60. We refer to Refs. [KR86, BR79, Kad98, Tak70, Tak03, vD76, HV13] for details; see Ref. [DS88] for more general background.

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In this context, we prove that the basic involutions of Tomita– Takesaki theory are closable and compute the adjoints of their closures (cf. Ref. [KR86, Section 9.2]). Closability of the involutions of Tomita– Takesaki theory are fundamental for constructing the modular automorphisms necessary for the study of von Neumann algebras, which are type III factors. While closability of the pertinent operators is previously known in both of these example scenarios, we found in each case that the method of symmetric pairs provides for shorter, simpler, and more elegant proofs. It yields closability and other conclusions in a simple and unified manner and serves to unify two otherwise disparate areas of infinite-dimensional analysis. We invite the reader to compare the relatively brief and straightforward proofs of Theorems 9.56 and 9.62 with the machinery of Ref. [KR86, Thm. 2.7.8, Rem. 5.6.3, Lem. 9.2.1, Lem. 9.2.28, Cor. 9.2.30, Cor. 9.2.31]. Part of von Neumann’s classification of rings of operators (now called von Neumann algebras) included the classification of factors; of these, the type III factors were considered intractable up to the 1970s. The primary reason for this is the nonavailability of even semifinite traces; see the discussion in Section 9.4. The Tomita–Takesaki theory was a response to this obstacle, and it has proved successful, leading to a structure theory for these previously intractable cases (although it applies equally well to von Neumann algebras of all types). The part of the theory of relevance to us here is the modular conjugation (and automorphisms) and its connection to a certain closable operator as well as its associated polar decomposition. See Refs. [Kad98, Tak70, Tak03, KR86, BR79, HV13] for further background. Let H be a Hilbert space, and consider the bounded linear operators on H, denoted by B(H). We use the usual bracket notation to denote the commutator of two operators: [x, y] := xy − yx,

x, y ∈ B(H).

Definition 9.51. The commutator of a von Neumann algebra M ⊆ B(H) is M := {x ∈ B(H) .. [x, m] = 0, ∀m ∈ M}. .

(9.51)

Definition 9.52. For ξ ∈ H with ξ = 1, we say that ξ is M-cyclic iff .

Mξ := {mξ .. m ∈ M}

(9.52)

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215

is dense in H, and we say that ξ is M-separating iff M ξ := {m ξ .. m ∈ M } .

(9.53)

is dense in H. Remark 9.53. Throughout the rest of this section, we assume that a cyclic and separating vector ξ ∈ H has been fixed. It can be shown that M ξ is dense in H iff mξ = 0, m ∈ M

=⇒

m = 0,

(9.54)

and this is the motivation for the terminology separating in Definition 9.52. Given M and a fixed faithful normal state, one may construct a pair (H, ξ) via the GNS construction such that ξ will be a cyclic and separating vector in H for M. See Refs. [KR97, BR79, vD76] for background material and further details. However, in general, a von Neumann algebra M ⊆ B(H) with dim H = ℵ0 cannot be realized with cyclic and separating vectors. For example, with M = B(H) and M = CIH , there is no ξ ∈ H for which M ξ is dense in H. We mention this just to emphasize the importance of the assumption of a cyclic and separating vector ξ ∈ H. Definition 9.54. Define (unbounded) conjugate-linear operators Sξ and Fξ by dom Sξ := Mξ,

Sξ (mξ) := m ξ,

for all mξ ∈ Mξ

(9.55)

and dom Fξ := M ξ,

Fξ (m ξ) := (m ) ξ,

for all m ξ ∈ M ξ.

(9.56)

Note that by our choice of ξ, both Sξ and Fξ are densely defined; see Remark 9.53 and Definition 9.52. We show that (Sξ , Fξ ) is a symmetric pair as follows. Sξ

Mξ ⊆ Hi

) M ξ ⊆ H

Fξ

Note that since Sξ and Fξ are conjugate-linear operators, we refer to (9.3). Remark 9.55. It is immediate from (9.52)–(9.54) that ker Sξ = 0 and ran Sξ = H.

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Theorem 9.56. (Sξ , Fξ ) is a symmetric pair, and thus, Sξ and Fξ are closable with Sξ ⊆ Fξ

and

Fξ ⊆ Sξ .

(9.57)

Proof. Take u = mξ ∈ Mξ and v = m ξ ∈ M ξ. Then, (9.55) gives Sξ u, vH = Sξ mξ, m ξH = m ξ, m ξH = ξ, mm ξH , = ξ, m mξH since m ∈ M implies mm = m m. Then, Sξ u, vH = ξ, m mξH = ξ, (m m) ξH = ξ, m (m ) ξH . Using (9.56), we continue from the previous line as follows: Sξ u, vH = mξ, Fξ (m ξ)H = u, Fξ vH , so (Sξ , Fξ ) is a symmetric pair; the rest follows from Theorem 9.6.



Remark 9.57. The significance of Theorem 9.56 is that it allows for the polar decomposition of Sξ (and Fξ ), as in Definition 9.59, which is essential for Tomita–Takesaki theory. See Ref. [DS88] for details on the polar decomposition in the unbounded case. Remark 9.58. The Laplace operator Δ studied elsewhere in this work is not related to the modular operator defined as follows, which is unfortunately traditionally also denoted by the same symbol. Here, we use the slightly different (non-italicized) symbol  for the modular conjugation, in a mild attempt to avoid confusion. Definition 9.59. In view of Theorem 9.56, the modular operator

:H→H

by

 := Sξ S ξ

(9.58)

is well defined; it is well known to be self-adjoint and positive (see Ref. [BR79, Prop. 2.5.11]). Also, the modular conjugation is the conjugate linear partial isometry J :H→H

defined by S ξ = J 1/2 .

(9.59)

Remark 9.60. As mentioned in Section 9.4, the foundational results of Tomita–Takesaki is Theorem 9.61. These formulas are used to develop a robust structure theory for von Neumann algebras, even those which are type III factors (the most difficult case). A factor is a von Neumann algebra whose center consists only of multiples of the identity operator. Loosely, a factor M is said to be type III iff M does not have any

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217

(even semifinite) trace; cf. Section 9.4. However, as a precursor to (9.60), it must be shown that Sξ is closable so that (9.59) makes sense; symmetric pairs provide a streamlined method for carrying out this verification, as shown in Theorem 9.56. See Refs. [BR79, HV13] for details. Theorem 9.61 (Tomita–Takesaki [Tak70]). With J as defined in (9.59), JMJ = M

and

 t M− t = M,

for all t ∈ R.

(9.60)

Observe that this result implies that the defect of J is zero (it is an isometry with a dense domain whose adjoint also has a dense domain) because of the existence of a cyclic and separating vector; see Remark 9.17. A first step in the lengthy proof of Theorem 9.61 is verification of the fact that Sξ MSξ ⊆ M , and this is where the symmetric pair (Sξ , Fξ ) of Theorem 9.56 may be applied. Consider that on dom Sξ , we have the identity (Sξ xSξ )y = y(Sξ xSξ ) for all x, y ∈ M,

(9.61)

which can be verified directly from the definition of Sξ . Note that the operators x, y ∈ M appearing in (9.61) are understood to be acting on their natural dense domains in H so that each side of the equality makes sense; see the last part of Remark 9.53. Equation (9.61) indicates that for any x ∈ M, one has Sξ xSξ ∈ M , whence Sξ MSξ ⊆ M . The remainder of the proof of Theorem 9.61 may now be followed along the usual lines: (9.58) and (9.59) and von Neumann’s double commutant theorem M = M eventually imply that JxJ ∈ M

for all x ∈ M.

(9.62)

It follows from Sξ2 = I that J  J = I, and thus, (9.62) implies JMJ = M . See Ref. [BR79] for details. Theorem 9.62. The pair (Sξ , Fξ ) is a maximal symmetric pair so that Sξ = Fξ and Fξ = Sξ . Proof. From Theorem 9.56, we have the graph containment Γ(Sξ ) ⊆ Γ(Fξ ). To prove that this is actually an equality, we show that     ζ ζ  ) and =⇒ ζ = 0. (9.63) ∈ Γ(F ⊥ Γ(Sξ ) ξ Fξ ζ Fξ ζ

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From the hypothesis in (9.63), we have the identity mξ, ζH + m ξ, Fξ ζH = 0

for all m ∈ M.

(9.64)

If we take m = m ∈ M in (9.64), it implies Fξ ζ = −ζ

(9.65)

because ξ is cyclic. By way of contradicting (9.63), suppose that ζ = 0. Then, for εk satisfying 0 < εk < 12 ζ 2 , we can choose mk ∈ M satisfying   (i) mk = (mk ) and (ii) mk ξ, ζ − ζ 2  < εk (9.66) because ξ is M-separating, and hence, M -cyclic. But then, mk ξ, ζH = −mk ξ, Fξ ζH

by (9.65)

= −(mk ) ξ, Fξ ζH

by (9.66i)

=

−(mk ) ξ, ζH

by (9.56)

=

−mk ξ, ζH

by (9.66i) again.

Now, (9.66ii) implies ζ 2 = − ζ 2 , whence ζ = 0.



9.5 Remarks and References The development of symmetric pairs was originally motivated by the study of Laplace operators on infinite networks as an operator on 2 (G0 ) and on HE and considering span{δx } as a common linear subspace which is dense in one Hilbert space (2 (G0 )) but not in the other. It is known from Theorems 6.1 and 6.8 (see also Refs. [Woj07, KL12, KL10]) that Δ is essentially self-adjoint5 on its natural domain in 2 (G0 ), but in Theorem 9.26, it is shown that Δ may not be essentially self-adjoint on its natural domain in HE (see Section 14.3 for an example).

5 See

Definition B.8.

Chapter 10

The Dissipation Space HD Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone. — A. Einstein

While the vectors in HE represent voltage differences, there is a second Hilbert space HD which serves to complete our understanding of metric geometry for resistance networks. This section is about an isometric embedding d of the Hilbert space HE of voltage functions into the Hilbert space HD of current functions and the projection Pd that relates d to its adjoint. This dissipation space HD will be needed for several purposes, including the resolution of the compatibility problem discussed in Section 1.5 and the solution to the Dirichlet problem in the energy space via its solution in the (ostensibly simpler) dissipation space; see Figure 10.1. The geometry of the embedding d is a key feature of our solution in Theorem 10.8 to a structure problem regarding current functions on graphs. Section A.3 contains definitions of the terms isometry, coisometry, projection, initial projection, final projection, and other notions used in this section. A good reference is Ref. [LP16], especially Chapter 2. In this section, we find it helpful to use the notation Ω (x, y) = c−1 xy . Definition 10.1. Considering Ω as a measure on G1 , the currents comprise the Hilbert space .

HD := {I : G1 → C .. I(y, x) = −I(x, y), ID < ∞},

219

(10.1)

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where the norm and inner product are given, respectively, by ID := D(I)1/2

and

I1 , I2 D := D(I1 , I2 ) =

1 2

(10.2) 

Ω (x, y)I1 (x, y)I2 (x, y).

(x,y)∈G1

Observe that HD is clearly a closed subspace of 2 (G1 , Ω ), but it is not clear that HE can be represented as an 2 space in any clear or easy manner (but see Theorem 7.20). As an 2 space, HD is obviously complete. This last statement is made precise in Theorem 10.12, where it is shown that HD is larger than HE by precisely the space of currents supported on cycles. The fundamental relationship between HE and HD is given by the following operator which implements Ohm’s law. It can also be considered as a coboundary operator in the sense of homology. Further motivation for the choice of symbology is explained in Section 10.3. Definition 10.2. The drop operator d = dc : HE → HD is defined by (dv)(x, y) := cxy (v(x) − v(y))

(10.3)

and converts potential functions into currents (that is, weighted voltage drops) by implementing Ohm’s law. In particular, for v ∈ P(α, ω), we get dv ∈ F (α, ω) in the notation of Definitions 1.23 and 1.21. As Lyons comments in Ref. [LP16, Section 9.3], thinking of the resistance Ω (x, y) as the length of the edge (xy), d is a discrete version of the directional derivative ∂v v(x) − v(y) ≈ (x), (dv)(x, y) = Ω (x, y) ∂y and the elements of HD are akin to discrete vector fields. Lemma 10.3. d is an isometry. Proof. Lemma 1.30 may be restated as follows: dv2D = u2E .



10.1 The Structure of HD In Theorem 10.8, we are now able to characterize HD by using Lemma 10.3 to extend Lemma 2.20 (the decomposition HE = Fin ⊕ Harm). First, however, we need some terminology. Whenever we consider the closed span of a set of vectors S in HE or HD , we continue to use the notation [S]E or [S]D to denote the closure of the span in E or D, respectively.

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221

Fig. 10.1. The action of d and d on the orthogonal components of HE and HD . See Theorem 10.8 and Definition 10.9.

Definition 10.4. Define the weighted edge neighborhoods to be ηz := dδz , i.e.,  cxy , x = z ∼ y, ηz (x, y) = (10.4) 0, else. Then, denote the space of all such currents by N bd := [dFin]D = [ηz ]D . This space is called  in Ref. [LP16, Chapters 2 and 9]. Lemma 10.5. D(ηz ) = c(z). Proof. Computing directly,   D(ηz ) = Ω (x, y)ηz (x, y)2 = czy = c(z). (x,y)∈G1

y∼x



Definition 10.6. For each h ∈ Harm, we have div(dh) = 0 so that dh satisfies the homogeneous Kirchhoff law by Corollary 1.39. Therefore, we denote Kir := dHarm = ker div and call them Kirchhoff currents. Definition 10.7. Recall that a cycle (of length n) is a sequence of n edges ϑ = {(x1 , x2 ), (x2 , x3 ), . . . , (xn−1 , xn ), (xn , x1 )}, where no edge appears twice, nor does any edge (x, y) appear with its reversal (y, x) (observe that in such cases, one has a sum of cycles). Denote the closed span of the characteristic functions of cycles ϑ ∈ L by Cyc := [χϑ ]D . The space Cyc is called 3 in Ref. [LP16, Chapters 2 and 9]. We are now able to describe the structure of HD . See Ref. [LP16, (9.6)] for a different proof. Theorem 10.8. HD = N bd ⊕ Kir ⊕ Cyc. Proof. Lemma 1.37 expresses the fact that dHE is orthogonal to Cyc. Since d is an isometry, dHE = dFin ⊕ dHarm, and the result follows from Theorem 2.20 and the definitions just above. See Figure 10.1. 

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222

Definition 10.9. Recall that a projection on a Hilbert space is by definition an operator satisfying P = P  = P 2 . The following notation will be used for projection operators: PFin : HE → Fin,

PHarm : HE → Harm,

Pd : HD → dHE ,

Pd⊥ = PCyc : HD → Cyc,

PN bd : HD → N bd,

PKir : HD → Kir.

Figure 10.1 may assist the reader with seeing how these operators relate. Lemma 10.10. The adjoint of the drop operator d : HD → HE is given by (d I)(x) − (d I)(y) = Ω (x, y)Pd I(x, y).

(10.5)

Proof. Since Pd d = d and Pd = Pd by definition, dv, ID = Pd dv, ID = dv, Pd ID =

=

1 2



Ω (x, y)cxy (v(x) − v(y))Pd I(x, y)

by (10.3)

(x,y)∈G1

1  cxy (v(x) − v(y))(d I)(x) − (d I)(y)) 2 0 x,y∈G

by (10.5). 

Remark 10.11. Observe that (10.5) only defines the function d I up to the addition of a constant, but elements of HE are equivalence classes, so this is sufficient. Also, (d I)(x) − (d I)(y) = Ω (x, y)I(x, y) satisfies the same calculation as in the proof of Lemma 10.10. However, the compatibility problem described in Section 1.5 prevents this from being a well-defined operator on all of HD . One can think of d as a weighted boundary operator and d as the corresponding coboundary operator; this approach is carried out extensively in Ref. [Soa94], although the author does not include the weight as part of his definition.

The Dissipation Space HD

223

Theorem 10.12. d and d are partial isometries with initial and final projections d d = IHE

and

dd = Pd ,

(10.6)

respectively. Furthermore, d : HE → N bd ⊕ Kir is unitary. Proof. Lemma 10.3 states that d is an isometry; the first identity of (10.6) follows immediately. The second identity of (10.6) follows from the computation dd I(x, y) = d (d I(x) − d I(y)) = d (Ω (x, y)Pd I(x, y))

by (10.5)

= cxy (Ω (x, y)Pd I(x, y) − Ω (y, y)Pd I(y, y)) by (10.3) = Pd I(x, y).

Definition 1.7.

The last claim is also immediate from the previous computation.



We are now able to give a proof of the completeness of HE that is independent of Section 4.1; see also Remark 4.6. Lemma 10.13. dom E/{constants} is complete in the energy norm. Proof. Let {vj } be a Cauchy sequence. Then, {dvj } is Cauchy in HD by Theorem 10.12, so it converges to some I ∈ HD (completeness of HD is just the Riesz–Fischer theorem). We now show that vj → d I ∈ HE : E(vj − d I) = E(d (dvj − I)) ≤ D(dvj − I) → 0 

again by Theorem 10.12. 10.1.1 An orthonormal basis for HD

Recall from Remark 1.16 that we may always choose an orientation on G1 . We use the notation e = (x, y) ∈ G1 to indicate that e is in the orientation ← − and e = (y, x) is not. For example, there is a term in the sum  D(I) = Ω ( e)I( e)2 (10.7)  e∈G1

← − for e, but there is no term for e (and hence no leading coefficient of 12 , as in (10.2)).

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Definition 10.14. For e = (x, y) ∈ G1 , denote by ϕe the normalized Dirac mass on an edge: 1 δe . ϕe :=  Ω ( e)

(10.8)

Lemma 10.15. The weighted edge masses {ϕe } form an orthonormal basis for HD . Proof. It is immediate that every function in Fin(G1 ) can be written as a (finite) linear combination of such functions. Since HD is just a weighted 2 space (as noted in Definition 10.1), it is clear that Fin(G1 ) is dense in HD . To check orthonormality, ϕe1 , ϕe2 D =



Ω ( e)ϕe1 ( e)ϕe2 ( e) = δe1 ,e2 ,

 e∈G1

where δe1 ,e2 is the Kronecker delta since ϕe1 ( e)ϕe2 ( e) = ce iff e1 = e2 , and zero otherwise.  Remark 10.16. Incidentally, the same calculation from the proof of Lemma 10.15 verifies ϕe ∈ HD . It would be nice if ϕe ∈ dHE , as this would allow us to “pull back” ϕe to obtain a localized generating set for HE , i.e., a collection of functions with finite support. Unfortunately, this is not the case whenever e is contained in a cycle, and the easiest explanation is probabilistic. If x ∼ y, then the Dirac mass on the edge (x, y) corresponds to the experiment of passing 1 A from x to its neighbor y. However, there is always some positive probability that current will flow from x to y around the other part of the cycle and hence the minimal current will not be ϕe ; see Lemma 3.52 for a more precise statement. Of course, we can apply d to obtain a nice result as in Lemma 10.17; however, d ϕe will generally not have finite support and may be difficult to compute. In light of Theorem 10.12 and the previous paragraph, it is clear that any element d ϕe is an element of P(x, y) for some x ∈ G0 and some y ∼ x. Lemma 10.17. The collection {d ϕe } is a Parseval frame for HE . Proof. The image of an orthonormal basis under a partial isometry is always a frame. That we have a Parseval frame (i.e., a tight frame with

The Dissipation Space HD

225

bounds A = B = 1) follows from the fact that d is an isometry:   |d ϕe , vE |2 = |ϕe , dvD |2 = dv2D = v2E .  e∈G1

 e∈G1

We used Lemma 10.15 for the second equality and Theorem 10.12 for the third.  10.2 The Divergence Operator In Section 10.4, we see how Pd allows one to solve certain potential-theoretic problems, but first, we need an operator which enables us to study Δ with respect to HD rather than HE . While the term “divergence” is standard in mathematics, the physics literature sometimes uses “activity” to connote the same idea, e.g., Refs. [Pow75]– [Pow79]. We like the term “divergence” as it corresponds to the intuition that the elements of HD are (discrete) vector fields. Definition 10.18. The divergence operator is div : HD → HE given by  I(x, y). (10.9) div(I)(x) := y∼x

To see that div is densely defined, note that div(δ(x,y) )(z) = δx (z) − δy (z), and the space of finitely supported edge functions Fin(G1 ) is dense in 2 (G1 , Ω ) = HD . Theorem 10.19. Δd = div and Δ = div d. Proof. To compute Δd I for a finitely supported current I ∈ Fin(G1 ), let v := d I, so  cxy (v(x) − v(y)) defn Δ Δ(d I)(x) = Δv(x) = y∼x

=



cxy Ω (x, y)Pd I(x, y)

defn d

y∼x

= div(Pd I)(x)

cxy = Ω (x, y)−1 .

This establishes div Pd = Δd , from which the result follows from Lemma 10.21. The second identity follows from the first by rightmultiplying by d and applying (10.6). 

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Remark 10.20. Theorem 10.19 may be reformulated as follows: Let u, v ∈ HE and I := dv. Then, Δv = u if and only if div I = u. This result will help us solve div I = w for the general initial condition w in Section 10.4. Also, it is shown in Theorems 6.1 and 6.8 that Δ is essentially self-adjoint on 2 (1). In that context, the results of Theorem 10.19 have a more succinct form. Corollary 10.21. The kernel of div is Kir ⊕ Cyc, whence div PN bd = div, div PN⊥bd = 0 and div(HD ) ⊆ Fin. Proof. If I ∈ Kir so that I = dh for h ∈ Harm, then div I(x) = Δh = 0 follows from Theorem 10.19. If I = χϑ for ϑ ∈ L, then    χϑ (x, y) = χϑ (x, y) + χϑ (x, y) div I(x) = y∼x

(x,y)∈ϑ

(x,y)∈ϑ /

= (−1 + 1) + 0 = 0. To show div(HD ) ⊆ Fin, it now suffices to consider I ∈ N bd. Since div ηz =  Δδz by Theorem 10.19, the result follows from closing the span. In particular, Corollary 10.21 shows that the range of div lies in HE , as stated in Definition 10.18. The identity div PN bd = div implies that the solution space F (α, ω) is invariant under minimization; see Theorem 10.30. Remark 10.22. Since div is defined without reference to c, d “hides” the measure c from the Laplacian. To highlight the similarities with the Laplacian, recall from Definition 1.21 that a current I satisfies the homogeneous or nonhomogeneous Kirchoff laws iff div I = 0 or div I = δα − δω , respectively. In Section 10.3, we consider an interesting analogy between the previous two results and the complex function theory. Corollary 10.23. Δ ⊇ d div∗ and div div∗ = ΔΔ∗ . Proof. These follow from Theorem 10.19 by taking adjoints. The inclusion is for the case when Δ may be unbounded, in which case we must be careful about domains. When T is any bounded operator, dom T  S  ⊆ dom(ST ) ; cf. Ref. [Rud91]. To see this, observe that v ∈ dom T  S  if and only if v ∈ dom S  , so assume this. Then, |ST u, v| = |T u, S  v| ≤ T  · S  v · u = Kv u, for Kv = T  · S  v. This shows that v ∈ dom(ST ) .

∀u ∈ dom ST 

The Dissipation Space HD

227

Lemma 10.24. For fixed x ∈ G0 , div is norm continuous in I: (10.10) | div(I)(x)| ≤ |c(x)|1/2 ID .  Proof. Using c(x) := y∼x cxy as in (1.3), direct computation yields  2  2     √      2 | div(I)(x)| =  I(x, y) =  cxy Ω (x, y)I(x, y) y∼x  y∼x        ≤ cxy  Ω (x, y)|I(x, y)|2   y∼x

y∼x

≤ |c(x)|D(I), where we have used the Schwarz inequality and the definitions of c, D, and div.  Corollary 10.25. For v ∈ HE , |Δv(x)| ≤ c(x)1/2 vE . Proof. Apply Theorem 10.24 to I = dv, and use the second claim of Theorem 10.19.  One consequence of the previous lemma is that the space of functions satisfying the nonhomogeneous Kirchhoff condition (1.18) is also closed, as we show in Theorem 10.30. In Remark 2.12, we discussed some reproducing kernels for operators on HE ; we now introduce one for the divergence operator div, using the weighted edge neighborhoods {ηz } of Definition 10.4. Lemma 10.26. The currents {ηz } form a reproducing kernel for div. Proof. By Lemma 10.24, the existence of a reproducing kernel follows from Riesz’s theorem. Since it must be of the form div(I)(z) = kz , ID , we verify that   Ω (x, y)ηz (x, y)I(x, y) = I(z, y) = div(I)(z).  ηz , ID = (x,y)∈G1

y∼z

10.3 Analogy with Calculus and Complex Variables The material in this book bears many analogies with vector calculus and complex function theory. Several points are obvious, such as the existence and uniqueness of harmonic functions and the discrete Gauss– Green formula of Lemma 1.13. In this section, we point out a couple more subtle comparisons.

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The drop operator d is analogous to the complex derivative   ∂ 1 ∂ 1 ∂ d = := + ∂= , dz ∂z 2 ∂x ∂y as may be seen from the discussion of the compatibility problem in Section 1.5. Recall from the proof of Theorem 10.8 that Lemma 1.37 expresses the fact that I, χϑ D = 0, ∀ϑ ∈ L

⇐⇒

∃v ∈ HE

such that dv = I.

This result is analogous to Cauchy’s theorem: If v is a complex function on an open set, then v = f  (that is, v has an antiderivative) if and only if every closed contour integral of v is 0. Indeed, even the proofs of the two results follow similar methods. The divergence operator div may be compared to the Cauchy–Riemann operator   1 ∂ 1 ∂ ∂ := − . (10.11) ∂¯ = ∂ z¯ 2 ∂x ∂y Indeed, in Theorem 10.19, we found that div d = cΔ, which may be ¯ = 1 Δ. The Cauchy–Riemann compared with the classical identity ∂∂ 4 ¯ equation ∂f = 0 characterizes the analytic functions, and div I = 0 characterizes the currents satisfying the homogeneous Kirchhoff law; see Definition 1.20. In Section 10.4, we give a solution to the inhomogeneous equation div I = w when w is given and satisfies certain conditions. The analogous problems in complex variables are as follows: Let W ⊆ C be a domain with a smooth boundary bd W , and let ∂¯ be the Cauchy–Riemann operator (10.11). Suppose that ν is a compactly supported (0, 1) form in W . We consider the boundary value problem ¯ = ν, ∂f

¯ = 0. with ∂ν

The Bochner–Martinelli theorem states that the solution f is given by the following integral representation: f (ζ)ω(dζ, z) − ν(ζ) ∧ ω(dζ, z), (10.12) f (z) = bd W

W

The Dissipation Space HD

229

where ω is the Cauchy kernel. In fact, this theorem continues to hold when W is a domain in Cn if one uses the Bochner–Martinelli kernel: ω(ζ, z) =

n (n − 1)!  ¯ (ζk − z¯k )dζ¯1 ∧ dζ1 ∧ · · · (ˆj) · · · ∧ dζ¯n ∧ dζn , 2π |ζ − z|2n k=1

(10.13) where ˆj means that the term dζ¯j ∧ dζj has been omitted and where ∂ϕ =

 ∂ϕ dzk k ∂zk

¯ = and ∂ϕ

 ∂ϕ d¯ zk . k ∂z ¯k

Indeed, in Lemma 10.26, we obtained a reproducing kernel for div; this is analogous to the Bochner–Martinelli kernel K(z, w); see Ref. [Kyt95] for more on the Bochner–Martinelli kernel. Theorem 2.10 shows that vx is analogous to the Bergman kernel, which reproduces the holomorphic functions within L2 (Ω ), where Ω ⊆ C is a domain. Indeed, the Bergman kernel is also associated with a metric, the Bergman metric, which is defined by    ∂γ   dB (x, y) := inf  (t) , (10.14) γ γ ∂t where the infimum is taken over all piecewise C 1 paths γ from x to y; cf. Ref. [Kra01]. 10.4 Solving Potential-Theoretic Problems with Operators We begin by discussing the minimizing nature of the projections PFin and PN bd . Theorem 10.27 shows how d solves the compatibility problem of 1.5: Given a current flow I ∈ HD , there does not necessarily exist a potential function v ∈ HE for which dv = I. Nonetheless, there is a potential function associated with I which satisfies dv = PN bd I, and it can be found via the minimizing projection. Consequently, Theorem 10.27 can be seen as an analogue of Theorem 1.40. Theorem 10.30 shows that the solution space F (α, ω) is invariant under PN bd . Coupled with the results of Theorem 10.27, this shows that if one can find any solution I ∈ F (α, ω), one can obtain another solution to the same Dirichlet problem with minimal dissipation, namely, PN bd I.

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10.4.1 Resolution of the compatibility problem In this section, we relate the projections PFin : HE −→ Fin

and PN bd : HD −→ N bd = dFin

of Definition 10.9 to some questions which arose in Section 1.3. The operators PFin and PN bd are minimizing projections because they strip away excess energy/dissipation due to harmonic or cyclic functions: • If v ∈ P(x, y), then PFin v is the unique minimizer of E in P(x, y). • If I ∈ F (x, y), then PN bd I is the unique minimizer of D in F (x, y). In a similar sense, Pd is also a minimizing projection. Probability notions will play a key role in our solution to questions about divergence in electrical networks (Definition 10.18) as well as our solution to a potential equation. The divergence will be important again in Section 11.2, where we use it to provide a foundation for a probabilistic model which is dynamic (in contrast to other related ideas in the literature) in the sense that the Markov chain is a function of a current I, which may vary. Theorem 10.27. Given v ∈ HE , there is a unique I ∈ HD which satisfies d I = v and minimizes ID . Moreover, it is given by Pd I, where I is any solution to d I = v. Proof. Given v ∈ HE , we can find some I ∈ HD for which d I = v by Theorem 10.12. Then, the orthogonal decomposition I = Pd I + Pd⊥ I gives I2D = Pd I2D + Pd⊥ I2D ≥ Pd I2D

(10.15)

so that ID ≥ Pd ID shows that Pd I minimizes the dissipation norm.  Finally, note that d Pd I = d dd I = d I by Corollary 10.10. Corollary 10.28. d is a solution operator in the sense that if I is any element of HD , then d I is the unique element v ∈ HE for which dv = Pd I. Corollary 10.29. DPN bd = EPFin . Hence, for I = d(vx − vy ), RF (x, y) = E(d I)1/2 = D(Pd I)1/2 .

and

RW (x, y) = min{D(I)1/2 .. I ∈ F (x, y)}.

(10.16)

The Dissipation Space HD

231

Proof. Given I ∈ HD , let I0 = PN bd I. Then, define v by v := d PN bd I0 = d PN bd I. Applying d to both sides gives dv = PN bd I by (10.6) (since PN bd ≤ Pd ) so that taking dissipations and applying Lemma 10.3 gives D(PN bd I) = D(dv) = E(v) = E(PFin v) because ran d PN bd ⊆ Fin by Theorems 10.8 and 10.12.  Theorem 10.30. For any α, ω ∈ G0 , the subset F (α, ω) is closed with respect to  · D and invariant under PN bd . Proof. From (1.18) and (10.9), we have that I ∈ F (α, ω) if and only div I = δα − δω . Suppose that {In } ⊆ F (α, ω) is a sequence of currents for D which In −−→ I. Then, div In = δα −δω for every n, and from Lemma 10.24, the inequality |(div In )(x) − (div I)(x)| ≤ |c(x)|1/2 In − ID gives div I(x) = δα − δω . Note that x is fixed, and so, c(x) is just a constant in the inequality above. For invariance, note that div PN bd = div by Corollary 10.21. Then, I ∈ F (α, ω) implies div PN bd I = div I = δα − δω

=⇒

PN bd I ∈ F (α, ω).



⊥ ⊥ Since PN bd is a subprojection of PCyc and PKir , we have an easy corollary.

Corollary 10.31. For any α, ω ∈ G0 , F (α, ω) is invariant under Pd = ⊥ ⊥ and PKir . PCyc Remark 10.32. Putting these tools together, we have obtained an extremely simple method for solving the equation Δv = δα − δω : 1. Find any current I ∈ F (α, ω). This is trivial; one can simply take the characteristic function of a path from α to ω. 2. Apply PN bd to I to “project away” harmonic currents and cycles. 3. Apply d to PN bd I. Since PN bd I ∈ dFin, this only requires an application of Ohm’s law in reverse as in (10.5). Then, v = d PN bd I is the desired energy-minimizing solution (since any harmonic component is removed). As a bonus, we already obtained the current Pd I induced by v. The only nontrivial part of the process described above is the computation of PN bd . For further analysis, one must understand the cycle space Cyc of G and the space Kir of harmonic currents. We hope to make progress on this problem in the future, see Remark 17.5.

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10.4.2 Generalization of the current problem We have already seen how d and PN bd solve certain problems on the resistance network. Given any current I on the network, d produces the unique associated potential function v (i.e., such that Pd I is induced by v). Moreover, Pd I is the unique current flow associated with I, which is physically realistic. In this section, we show how to solve resistance network problems with more general conditions by applying the tools from Hilbert space theory developed in the previous section in this fashion. To be specific, let X = {x1 , x2 , . . . , xn , . . . } ⊆ G0 be a specified subset of vertices,  and consider f = x∈X f (x)δx as prescribing weights on this set. Then, div I = f means that the current I satisfies the Kirchhoff laws:   f (x), x ∈ X, I(x, y) = (10.17) 0, else y∼x in a direct generalization of (1.18). A solution I to (10.17), that is, to div I = f , will represent the current flow induced by imposing the specified voltages at the appropriate vertices, see the first paragraph of Section 1.3. Definition 10.33. Let X = {x1 , x2 , . . . , xn } ⊆ G0 . For f = denote .

Ff := {I : G1 → R .. div I = f }.

 X

f (x)δx , (10.18)

When X is finite, the hypotheses of Lemma 10.34 are automatically met, and so, Ff is nonempty. Theorem 10.36 shows that Ff gives solutions to a certain inhomogeneous Dirichlet problem. If X = G0 , then the solution to div I = f is given by I = df , as is shown in Lemma 1.30 (in fact, this is precisely the content of (1.23)). However, if X  G0 , the following solution must be used. Lemma 10.34. For each x ∈ G0 , let wx be a solution to Δw = δx .   Suppose X ⊆ G0 , and let f = x∈X f (x)δx . If u = x∈X f (x)wx ∈ HE , then I = du solves the initial value problem div I = f.

(10.19)

  Proof. Assume initially that X is finite. For I := d x∈X f (x)wx ,    div I = div d f (x)wx = Δ f (x)wx = f (x)δx = f. x∈X

x∈X

x∈X

The Dissipation Space HD

233

The second equality comes from Theorem 10.19; the third comes from the linearity of Δ and the definition of wx . If X is infinite, then let {Xn } be a nested sequence of sets increasing  to X, i.e., Xn ⊆ Xn+1 and X = n Xn . Define fn := x∈Xn f (x)wx so that f − fn E → 0, and let In be the solution obtained by the method just described so that div In = fn . With n > m, we show that {In } is Cauchy: 

  2 d is linear f (x)wx − f (x)wx In − Im D = Dd ⎛ =E⎝

Xn

Xm

⎞



f (x)wx ⎠

x∈Xn \Xm

⎛ ≤E⎝



Correlation 10.29

⎞

f (x)wx ⎠ = E(f − fm ) −−−−−→ 0. n→∞

x∈X\Xm

Thus, I := lim In is well defined and div I = f .



Lemma 10.35. For f ∈ HE , Ff ⊆ HD is invariant under PN bd and closed. The proof of Lemma 10.35 is identical to that of Theorem 10.30. Its use lies in the fact that if I solves div I = w, then PN bd I is also a solution, and thus, one has a solution which lies in the range of d. Theorem 10.36. The Dirichlet problem Δv = w is solved by v = d I, where I ∈ Fw . Proof. The set Fw is nonempty by Lemma 10.34, from which the result is  immediate by Lemma 10.19: Δv = Δd I = div I = w. Remark 10.37. The significance of Theorem 10.36 is that it allows us to solve a problem in HE by working entirely in HD , which is just an 2 space and much better understood.

10.5 Remarks and References After completing a first draft of this book, we discovered several of the results of this section in Refs. [LP16, Soa94]. Both of these texts are excellent; Lyons emphasizes the connections with probability, and Chapters 2 and 9 are most pertinent to the current discussion, and

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Soardi emphasizes the (co)homological perspective and parallels with vector calculus. The subspace of currents spanned by edge neighborhoods N bd = [dFin]D = [ηz ]D is called  in Ref. [LP16, Chapters 2 and 9], and the subspace of cycles Cyc := [χϑ ]D is called 3. The reader may also wish to consult Refs. [DR08, Sto08, DEIK07, BB05, Chu07, CR06, Tho90, Zem97, Zem91] with regard to the material in this chapter.

Chapter 11

Probabilistic Interpretations

From its shady beginnings devising gambling strategies and counting corpses in medieval London, probability theory and statistical inference now emerge as better foundations for scientific models, especially those of the process of thinking and as essential ingredients of theoretical mathematics, even the foundations of mathematics itself. — David Mumford God not only plays dice. He also sometimes throws the dice where they cannot be seen. — S. Hawking

In Chapter 7, we constructed a measure P on SG where SG ⊆ HE ⊆ SG is a certain Gel’fand triple. In this chapter, we develop a different but analogous measure on the space of infinite paths in bd G. We carry out this construction for harmonic functions on (G, c) in Section 11.1, where the cxy of the measure is defined in terms of transition probabilities p(x, y) = c(x) random walk and the associated cylinder sets. When the random walk on (G, c) is transient, the current induced by a monopole gives a unit flow to infinity; such a current induces an orientation on the edges G1 and a new, naturally adapted Markov chain. The state space of this new process is also G0 , but the transition probabilities are now defined by the induced current I(x,y) p(x, y) = div . We call the fixed points of the corresponding transition I (x) operator the “forward-harmonic” functions and carry out the analogous construction for them in Section 11.2. The authors are currently working to determine whether or not these measures can be readily related to each other or the measure P of Section 7.2. 235

236

Operator Theory and Analysis of Infinite Networks

Some excellent references for probability on graphs include Refs. [LP16, Woe09, Woe00]. 11.1 Path Space of a General Random Walk We begin by recalling some terms from Section 3.6 and providing some more details. Let γ = (x0 , x1 , x2 , . . . , xn ) be any finite path starting at x = x0 . The probability of a random walk started at x traversing this path is P(γ) :=

n 

p(xk−1 , xk ),

(11.1)

k=1 c

xy where p(x, y) := c(x) is the probability that the walk moves from x to y as in (3.41). This intuitive notion can be extended via Kolmogorov consistency to the space of all infinite paths starting at x. Let Xn (γ) denote the nth coordinate of γ; one can think of γ as an event and Xn as the random walk (a random variable), in which case

Xn = Location of the random walk at time n.

(11.2)

Definition 11.1. Let Γ denote the space of all infinite paths γ in (G, c). Then, a cylinder set in Γ is specified by fixing the first n coordinates: .

Γ(x1 ,x2 ,...,xn ) := {γ ∈ Γ .. Xk (γ) = xk , k = 1, . . . , n}.

(11.3)

Define P(c) on cylinder sets by P(c) (Γ(x1 ,x2 ,...,xn ) ) :=

n 

p(xi−1 , xi ).

(11.4)

i=1

Remark 11.2. It is clear from Definition 11.1 that the probability of a random walk following the finite path γ = (x0 , x1 , x2 , . . . , xn ) is equal to the measure of the set of all infinite walks which agree with γ for the first n steps: Combining (11.1) and (11.4) gives P(c) (Γ(x1 ,x2 ,...,xn ) (x)) = P(γ). Observe that (11.4) is a conditional probability: P(c) (Γ(x1 ,x2 ,...,xn ) (x)) = P(c) {γ ∈ Γ(x) | Xk (γ) = xk , k = 1, . . . , n}. (11.5) Remark 11.3 (Kolmogorov consistency). We use Kolmogorov’s consistency theorem to construct a measure on the space of paths beginning at vertex x ∈ G0 , see Ref. [Jor06, Lem. 2.5.1] for a precise statement of this extension principle in its function theoretic form and Ref. [Jor06, Exc. 2.4

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Probabilistic Interpretations

and 2.5] for the method we follow here. The idea is that we consider a sequence of functionals {μ(n) }, where μ(n) is defined on An := span{χΓ(x ,...,x 0

n)

.. .

xi ∼ xi−1 , i = 1, . . . , n}.

(11.6)

Alternatively, .

An := {f : Γ → R .. f (γ1 ) = f (γ2 ) whenever Xk (γ1 ) = Xk (γ2 ) for k ≤ n}. (11.7) That is, an element of An cannot distinguish between two paths which agree for the first n steps. This means that μ(n) is a “simple functional” in the sense that it is constant on each cylinder set of level n:  μ(n) [f ] = a(x0 ,...,xn ) μ(n) [χΓ(x ,...,x ) ]. (11.8) 0

n

x0 ,...,xn

If the functionals μ(n) are mutually consistent in the sense that μ(n+1) [f ] = μ(n) [f ], then Kolmogorov’s consistency theorem gives a unique Borel probability measure on the space of all paths. More precisely, Kolmogorov’s theorem gives the existence of a limit functional which is defined for functions on paths of infinite length, and this corresponds to a measure by Riesz’s theorem; see Refs. [Jor06, Kol56]. In the following, 1 denotes the constant function with value equal to 1. Theorem 11.4 (Kolmogorov). If each μ(n) : An → C is a positive linear functional satisfying the consistency condition μ(n+1) [f ] = μ(n) [f ]

for all f ∈ An ,

(11.9)

then there exists a positive linear functional μ defined on the space of functions on infinite paths such that μ[f ] = μ(n) [f ],

f ∈ An ,

(11.10)

where f is considered as a function on an infinite path which is zero after the first n edges. Moreover, if we require the normalization μ(n) [1] = 1, then μ is determined uniquely. We now show that P(c) extends to a natural probability measure on the space of infinite paths Γ(x).

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Operator Theory and Analysis of Infinite Networks

Theorem 11.5. For (G, c), there is a unique measure P(c) defined on Γ which satisfies   (c) f dP = f dP(c,n) = E(n) [f ], ∀f ∈ An . (11.11) E[f ] = Γ

Γ

Proof. We must check condition (11.9) for μ(n) = P(c,n) , defined by P(c,n) (χΓ(x ,...,x ) ) := 0

n 

n

p(xi−1 , xi ),

i=1

with (11.4) in mind. Think of f ∈ An as an element of An+1 and apply P(c,n+1) to it:  P(c,n+1) [f ] = a(x0 ,...,xn+1 ) P(c,n+1) (χΓ(x ,...,x ) ) 0

n+1

x0 ,...,xn+1

=





a(x0 ,...,xn )

x0 ,...,xn xn+1

=



p(xi−1 , xi )p(xn , xn+1 )

i=1

a(x0 ,...,xn )

x0 ,...,xn

n 

n 

p(xi−1 , xi )

i=1



p(xn , xn+1 )

xn+1 ∼xn

= P(c,n) [f ]

 since xn+1 ∼xn p(xn , xn+1 ) = 1. For the second equality, note that f ∈ An , so we can use the same constant a for each (n + 2)-tuple that begins with  (x0 , . . . , xn ). 11.1.1 A boundary representation of the bounded harmonic functions of finite energy Definition 11.6. A cocycle ζ : Γ → R is a measurable function on the infinite path space which is independent of the first finitely many vertices in the path: ζ(γ) = ζ(σγ),

(11.12)

where σ is the shift operator, i.e., if γ = (x0 , x1 , x2 , . . . ), then σγ = (x1 , x2 , x3 , . . . ). Intuitively, a cocycle is a function on the boundary bd G; it depends only on the asymptotic trajectory of a path or random walk. A cocycle does not see where the random walk began, only where it went; it is thus a special kind of martingale, as we will see in the following.

Probabilistic Interpretations

239

The goal of this section is to show that the formula h(x) = Ex [ζ]

(11.13)

spells out a bijective correspondence between certain harmonic functions h and cocycles ζ on the space of infinite paths; see Theorem 11.9. A good reference for this section is Ref. [Jor06, Thm. 2.7.1]. Note that the left-hand side of (11.13) involves no measure theory, in contrast to the right-hand side, where the expectation refers to the integration of cocycles ζ against the probability measure P(c) . The underlying Borel probability space of P(c) is the σ-algebra of measurable sets generated by the cylinder sets in Γ, i.e., by the subsets in Γ, which fix only a finite number of places (in the infinite paths). The condition on a measurable function ζ on Γ which accounts for h defined by (11.13) being harmonic is that ζ is invariant under a finite left shift; cf. (11.12). It turns out that in making the integrals Ex (ζ) precise, the requirement that ζ be measurable is a critical assumption. In fact, there is a variety of nonmeasurable candidates for such functions, ζ, on Γ. Definition 11.7. For any measurable function f : Γ → R, we write  f (γ) dP(c) (11.14) Ex [f ] := E[f | X0 = x] = Γ(x)

for the expected value of f , conditioned on the path starting at x. Lemma 11.8. If h is harmonic, then for any n = 1, 2, . . . ,  h ◦ Xn dP(c) . h(x) =

(11.15)

Γx

Proof. By the definition of the cylinder measure dP(c) (Definition 11.1),    h ◦ X1 dP(c) = p(x, y) h ◦ X0 dP(c) Γx

Γy

y∼x

=

 y∼x

so iteration and Ph = h gives

p(x, y)h(y)  Γx

 Γy

dP(c) = Ph(x),

(11.16)

h◦Xn dP(c) = h(x) for every n = 1, 2, . . . . 

Theorem 11.9. The bounded harmonic functions are in bijective correspondence with the bounded and measurable cocycles. More precisely, if ζ is

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Operator Theory and Analysis of Infinite Networks

such a cocycle, then it defines a bounded harmonic function via hζ (x) := Ex [ζ].

(11.17)

Conversely, if h is bounded and harmonic, then it defines a bounded and measurable cocycle via for P(c) − a.e. γ ∈ Γ(x).

ζh (γ) := lim h(Xn (γ)) n→∞

(11.18)

Proof. (⇒) Recall that Δ = c − T; we show that chζ = T hζ whenever ζ  . is a cocycle. If Γ(x,y) := {γ ∈ Γ(x) .. X1 (γ) = y}, then Γ(x) = y∼x Γ(x,y) is a disjoint union and   (c) ζ(γ) dP = ζ(γ) dP(c) . hζ (x) = Ex [ζ] = Γ(x)

y∼x

Γ(x,y)

For each γ ∈ Γ(x,y) , one has P(γ) = P(x, y)P(σγ) = p(x, y)P(σγ) by (11.1), whence  c(x)hζ (x) = c(x) p(x, y)ζ(σγ) dP(c) Γ(x,y)

y∼x

=





cxy

Γ(y)

y∼x

ζ(γ) dP(c) = T hζ (x),

where the cocycle property (11.29) is used for the second equality. (⇐) Now, let h be a bounded harmonic function. Since lim h(Xn (γ)) = lim h(Xn+1 (γ)) = lim h(Xn (σγ)),

n→∞

n→∞

n→∞

the cocycle property (11.29) is obviously satisfied whenever the limit exists. Let Σn denote the σ-algebra generated by the cylinder sets of level n, and denote Xn (γ) = xn . Then, Xn+1 (γ) is a neighbor y ∼ xn , and  E[h(Xn+1 ) | Σn ] = E[h(Xn+1 ) | Σn ] p(xn , y) =



y∼xn

p(xn , y)E[h(Xn+1 ) | Σn ]

y∼xn



=E



p(xn , y)h(Xn+1 ) | Σn

y∼xn

= E[h(Xn ) | Σn ] = h(Xn ).

(11.19)

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Probabilistic Interpretations

Since h is bounded, this shows that h(Xn ) is a bounded martingale, whence by Doob’s theorem (cf. Ref. [Doo53]), it converges pointwise P(c) -a.e. on Γ, and (11.18) makes ζh well defined P(c) -a.e. on Γ. (↔) We conclude with a proof that these two constructions correspond to inverse operations. If ζ is a cocycle, we must show that limn→∞ EXn (γ) [ζ] = ζ(γ). To this end, for A ⊆ Γ, define the conditioned (c) measure PA := PP(c)(A∩·) so that dPA = P(c)1(A) χA dP(c) . Now, for a fixed (A) γ ∈ Γ, let An = Γ(x,X1 (γ),...,Xn (γ)) be the cylinder set whose first n + 1 coordinates agree with γ. Applying the measure identity lim μ(An ) =

μ(An ) for nested sets, we obtain limn→∞ PAn = δγ as a weak limit of measures. Now,    n→∞ ζ(ξ) dP(c) (ξ) = ζ(ξ) dPAn (ξ) −−−−−→ ζ(ξ) dδγ EXn (γ) [ζ] = ΓXn (γ)

Γ

Γ

= ζ(γ).

(11.20)

On the other hand, if h is harmonic, we must show that Ex [ζh ] = h. Then, for ζh (γ) := limn→∞ h(Xn (γ)), boundedness allows us to apply the dominated convergence theorem and compute   lim (h ◦ Xn (γ)) dP(c) = lim h ◦ Xn (γ) dP(c) . (11.21) Ex [ζh ] = Γx n→∞

n→∞

Γx

Now, the sequence on the right-hand side of (11.21) is constant by Lemma 11.8, so Ex [ζh ] = h(x). Finally, observe that the conclusion of boundedness results, in either direction, from the hypothesis of boundedness and the formulas (11.17) and (11.18). Also, measurability of the cocycle follows from the pointwise convergence of measurable functions in (11.18).  Remark 11.10. The (⇒) direction of the proof of Theorem 11.9 may also be computed:   p(x, y)Ex [ζ | X1 = y] = cxy Ey [ζ] hζ (x) = c(x)Ex [ζ] = c(x) =



y∼x

y∼x

cxy hζ (y),

y∼x

where Ex [ζ | X1 = y] = Ey [ζ] because the random walk is a Markov process. See, for example, Ref. [LPW08, Prop. 9.1].

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Operator Theory and Analysis of Infinite Networks

11.2 Forward-Harmonic Functions The current passing through a given edge may be interpreted as the expected value of the number of times that a given unit of charge passes through it. This perspective is studied extensively in the literature; see cxy Refs. [DS84, LP16] for excellent treatments. In this case, p(x, y) = c(x) helps one construct a current which is harmonic or dissipation-minimizing. However, that is not what we do here; we are interested in studying current functions whose dissipation is finite but not necessarily minimal. In Theorem 1.48, we show that the experiment always induces a “downstream” current flow between the selected two points, that is, a path along which the potential is strictly decreasing. These probability notions will play a key role in our solution to questions about activity; cf. Definition 11.11. We use the forward path measure again in our representation formula (Theorem 11.21) for the class of forward-harmonic functions on G. The corresponding Markov process is dynamically adapted to the network (and the charge on it). This representation is dynamic and nonisotropic, which sets it apart from other related representation formulas in the literature. 11.2.1 Activity of a current and the probability of a path Given a (fixed) current, we are interested in computing “how much of the current” takes any specified path from x to some other (possibly distant) vertex y. This will allow us to answer certain existence questions (see Theorem 1.48) and provide the basis for the study of the forward-harmonic functions studied in Section 11.2. Note that, in contrast to (11.1), the probabilistic interpretation given in Definition 11.13 (and the discussion preceding it) does not make any reference to c. In this section, we follow Ref. [Pow76b] closely. Definition 11.11. The divergence of a current I : G1 → R is the function on x ∈ G0 defined by 1  y∼x |I(x, y)|, x = α, ω divI (x) := 2 (11.22) 1, x = α, ω, which describes the total “current traffic” passing through x ∈ G0 . Thus, div is an operator mapping functions on G1 to functions on G0 ; see Section 10.2 for details.

Probabilistic Interpretations

243

For convenience, we restate Definition 1.46. Definition 11.12. Let v : G0 → R be given, and suppose we fix α and ω for which v(α) > v(ω). Then, a current path γ (or simply, a path) is an edge path from α to ω with the extra stipulation that v(xk ) < v(xk−1 ) for each k = 1, 2, . . . , n. Denote the set of all current paths by Γ = Γα,ω (dependence on the initial and terminal vertices is suppressed when context precludes confusion). Also, define Γα,ω (x, y) to be the subset of current paths from α to ω which pass through the edge (x, y) ∈ G1 . Suppose we fix a source α and sink ω and consider a single current path I(x,y) γ from α to ω. With divI defined as in (11.22), one can consider div I (x) as the probability that a unit of charge at x will pass to a “downstream” neighbor y. Note that I(x, y) > 0 and divI = 0 since we are considering an edge of our path γ. This allows us to define a probability measure on the path space Γα,ω . Definition 11.13. If γ ∈ Γ follows the vertex path (α = x0 , x1 , x2 , . . . , xn = ω), then define the probability of γ by P(γ) :=

n  I(xk−1 , xk ) . divI (xk−1 )

(11.23)

k=1

This quantity gives the probability that a unit of charge at α will pass to ω by traversing the path γ. 11.2.2 Forward-harmonic transfer operator In this section, we consider the functions h : G0 → C, which are forwardharmonic, that is, functions which are harmonic with respect to a current I. We make the standing assumption that the network is transient; this guarantees the existence of a monopole at every vertex, and the induced current will be a unit flow to infinity; cf. Corollary 1.42. We orient the edges by a fixed unit current flow I to infinity, as in Definition 1.16. The forward-harmonic functions are fixed points of a transfer operator induced by the flow which gives the value of h at one vertex as a convex combination of its values at its downstream neighbors. The main idea is to construct a measure on the space of paths beginning at vertex x ∈ G0 and then use this measure to define forward-harmonic functions. In fact, we are able to produce all forward-harmonic functions

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Operator Theory and Analysis of Infinite Networks

from the class of functions which satisfy a certain cocycle condition, see Definition 11.18. In Theorem 11.21, we give an integral representation for the harmonic functions, and in Corollary 11.22, we show that if I has a universal sink, then the only forward harmonic functions are the constants. Definition 11.14. Given a fixed minimal current I = Pd I, we denote the set of all current paths in the resistance network (G, c) by ΓI := {γ = (x0 , x1 , . . . .. (xi , xi+1 ) ∈ G1 , xi ≺ xi+1 }. .

(11.24)

For n = 1, 2, . . . , we denote the set of all current paths of length n by (n)

ΓI

:= {γ = (x0 , x1 , . . . , xn ) .. (xi , xi+1 ) ∈ G1 , xi ≺ xi+1 }, .

(11.25)

and denote the collection of paths starting at x by ΓI (x) := {γ ∈ ΓI (n) x0 = x}, and likewise for ΓI (x).

.. .

Here, the orientation is determined by I, and if I(x, y) = 0 for some (x, y) ∈ G1 , then this edge will not appear in any current path in ΓI , and for all practical purposes, it may be considered as having been removed from G for the moment. Definition 11.15. When a minimal current I = Pd I is fixed, the set of forward neighbors of x ∈ G0 is 0 . nbr+ I (x) := {y ∈ G . x ≺ y, x ∼ y}. .

(11.26)

Definition 11.16. If v : G0 → R, define the forward Laplacian of v by 

Δv(x) := cxy (v(x) − v(y)). (11.27) y∈nbr+ (x)

= 0. A function h is forward-harmonic iff Δh For Definitions 11.14–11.16, the dependence on I may be suppressed when context precludes confusion. Theorem 11.17. For I ∈ HD and x ∈ G0 , there is a unique measure Px defined on ΓI (x) which satisfies Px [f ] = P(n) x [f ],

f ∈ An .

(11.28)

Proof. We only need to check Kolmogorov’s consistency condition (11.9); see Refs. [Jor06, Kol56]. For n < m, consider An ⊆ Am by assuming that

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Probabilistic Interpretations

f depends only on the first n edges of γ. (For brevity, we denote a function on n edges as a function on n + 1 vertices.) Then,  (m) f (γ) dP(m) Px [f ] = x (γ) ΓI (x)

 =

ΓI (x)

 =

ΓI (x)

f (x0 , x1 , x2 , . . . , xn ) dP(m) x (γ) f (x0 , x1 , x2 , . . . , xn ) dP(n) x (γ)

= P(n) x [f ].



11.2.3 A boundary representation for the forward-harmonic functions We now show that the forward-harmonic functions are in bijective correspondence with the cocycles when defined as follows. Definition 11.18. A cocycle is a function f : ΓI → R which is compatible with the probabilities on current paths in the sense that it satisfies f (γ) = f (x0 , x1 , x2 , x3 . . . ) =

cx0 x1 divI (x0 ) f (x1 , x2 , x3 . . . ) c+ (x0 )I(x0 , x1 )

(11.29)

whenever γ = (x0 , x1 , x2 , x3 . . . ) ∈ ΓI is a current path as in Defini tion 11.14 and (x0 , x1 ) is the first edge in γ. Also, c+ (x) := y∈nbr+ (x) cxy is the sum of conductances of edges leaving x. If the operator m is given by multiplication by m(x, y) =

cxy divI (x) c+ (x)I(x, y)

(11.30)

and σ denotes the shift operator, the cocycle condition can be rewritten as f = mf σ. Using ek = (xk−1 , xk ) to denote the edges, this gives f (e1 , e2 , . . . ) = m(e1 ) . . . m(en )f (en+1 , en+2 , . . . ) =

∞ 

m(ek ).

(11.31)

k=1

Definition 11.19. Define the forward transfer operator TI induced by I by  1 cxy f (y). (11.32) (TI f )(x) := + c (x) + y∈nbr (x)

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Operator Theory and Analysis of Infinite Networks

Lemma 11.20. If f : Γ(x) → R is a cocycle and one defines hf (x) := Px [f ], it follows that hf (x) is a fixed point of the forward transfer operator TI . Proof. With hf so defined, we conflate the linear functional Px with the measure associated to it via Riesz’s theorem and compute  1 cxy Py [f ] def T, hf (TI hf )(x) = + c (x) + y∈nbr (x)

=

 y∈nbr+ (x)

=

 y∈nbr+ (x)

=

 y∈nbr+ (x)

cxy C + (x) 



 y∈nbr+ (x)

=

 y∈nbr+ (x)

f (γ) dPy (γ)

cxy f (σγ) dPx (γ) c+ (x)

ΓI (x)

I(x, y) divI (x)

× f (σγ) dPx (γ) =

ΓI (y)

I(x, y) divI (x)

 ΓI (x)

cxy + c (x)

Py as a measure

change of vars

divI (x) I(x, y) just algebra

 ΓI (x)

f (γ) dPx (γ)

I(x, y) Px (f ) divI (x)

by (11.29)

Px as a functional 

= Px (f )

I(x,y) divI (x)

= 1.

To justify the change in variables, note that if γ is a path starting at x whose first edge is (x, y), then σ γ is a path starting at y. Moreover, since y is a downstream neighbor of x, every path γ starting at y corresponds to exactly one path starting at x, namely, ((x, y), γ).  Theorem 11.21. The forward-harmonic functions are in bijective correspondence with the bounded and measurable cocycles. More precisely, if ζ is such a cocycle, then hζ (x) := Px [ζ]

(11.33)

is harmonic and bounded. Conversely, if h is harmonic and bounded, then ζh (γ) := limn→∞ h(Xn (γ)),

γ ∈ Γx

(11.34)

247

Probabilistic Interpretations

is a bounded and measurable cocycle, where Xn (γ) is the nth vertex from x along the path γ. Proof. (⇒) Let ζ be a cocycle and define hζ as in (11.33) with C + (x) as in Definition 11.19, then compute 

ζ (x) = cxy (Px [ζ] − Py [ζ]) Δh y∈nbr+ (x)



= Px [ζ]

y∈nbr+ (x)

= Px [ζ]C + (x) −



cxy − 

cxy Py [ζ]

y∈nbr+ (x)

cxy Py [ζ],

y∈nbr+ (x)

which is 0 by Lemma 11.20.

= 0. Observe that Xn is a Markov chain with a (⇐) Let h satisfy Δh transition probability of Px at x. The above computations show that h is then a fixed point of TI , and hence, h(Xn ) is a bounded martingale. By Doob’s theorem (cf. Ref. [Doo53]), it converges pointwise Px -ae on Γ, and (11.34) makes ζh well defined. One can see that ζh is a cocycle by the same arguments as in the proof of Ref. [Jor06, Thm. 2.7.1]. The conclusions regarding boundedness and measurability follow as in the proof of Theorem 11.9.  Corollary 11.22. If I has a universal sink, or in other words, if all current paths γ end at some common point ω, then the only forwardharmonic function is the zero function. Proof. Every harmonic function comes from a cocycle, which in turn comes from a harmonic function as a martingale limit by the previous theorem. However, formula (11.34) yields fh (γ) := lim h(Xn (γ)) = h(ω), n→∞

∀γ ∈ Γ.

Thus, every cocycle is constant, and hence, (11.29) implies fh ≡ 0. Then, (11.33) gives h ≡ 0.  11.3 Remarks and References The probability literature is among the largest of all the subdisciplines of mathematics, and so, the following list of suggested references barely begins to scratch the surface: Refs. [IR08, AHL06, Aik05, BW05, CK04, ˇ BZ09, IKW09, Sil09, YZ05, OP19].

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Operator Theory and Analysis of Infinite Networks

Of these references, some are more specialized. However for prerequisite material (if needed), the reader may find the following sources especially relevant: Refs. [LP16, LPW08, AF09, Per99]. Remark 11.23 (Open problem). Is it true that if I minimizes D on F (α, ω), then all current paths flow from α to ω? If so, this would fill another hole in the proof of distΔ (x, y) = distD (x, y) (recall (3.1) and (3.3)) in Ref. [Pow76b]. One may be able to approach this by reasoning along the following lines: If I ∈ F (α, ω) is minimal, then Theorem 1.40 implies I is induced by a potential v which minimizes E over P(α, ω). Since orthogonality gives E(v) = E(PFin v) + E(PHarm v), such a minimal v must lie in Fin. In fact, v = fα − fω , where fx = PFin vx .

Chapter 12

Spectral Comparisons

Motivated by a recent work on the analysis of large networks, we consider the natural (discrete) Laplace operator on the network. If the weights on the edges of the network are given by a (conductance) function c, we denote this operator by Δc . In this chapter, we study the dependence of the spectrum of Δc on c. In addition to operator theory and spectral theory, we use tools from metric geometry and variational calculus. The main result in this section (Theorem 12.20) deals with monotonicity of the spectrum of Δc . We illustrate this with explicit models on the binary tree and on a one-dimensional lattice. Most of the material in this chapter originally appeared in Ref. [JP14]. 12.1 Comparing Graphs with Different Conductance Functions In this chapter, we use the framework developed in earlier chapters to compute certain spectral theoretic information as well as resistance metrics on the underlying vertex set. In particular, we explore how certain quantities depend on the choice of c in comparison to another conductance function, which we denote by b. It will be assumed that both b and c yield a connected weighted graph, although we allow for the case when cxy > 0 and bxy = 0 (so that x and y are neighbors in (G, c) but not in (G, b)). The data, defined from b and c, to be compared are as follows: 1. the energy forms E (b) and E (c) and the respective energy Hilbert spaces HE (b) and HE (c) that they define; 2. the systems of dipole vectors that form reproducing kernels for the two Hilbert spaces, see Definition 2.11; 249

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Operator Theory and Analysis of Infinite Networks

3. the respective Laplace operators Δ(b) and Δ(c) and their spectra; 4. the spaces of finite-energy harmonic functions on HE (b) and HE (c) ; and 5. the effective resistance metrics on HE (b) and HE (c) . We focus our study on the case when one of the two energy-Hilbert spaces is contractively contained in the other, which corresponds to the inequality b ≤ c. In this case, we believe that our results have applications to percolation theory and the study of random walks in random environments as well as to dilation theory (see Ref. [Arv10]) and the contractive inclusion of Hilbert spaces (see Ref. [Sar94]). Of special operator theoretic significance is the adjoint of the contractive inclusion mapping. The issues involved with the adjoint operator are subtle, as the computation of the adjoint operator depends on the Hilbert inner products used. It is the adjoint operator that allows one to compute the respective energy kernels for the two Hilbert spaces; see Definition 2.11. We further derive an invariant (involving induced linear maps between the respective spaces of finite-energy harmonic functions) which distinguishes two networks when G is fixed and the conductance functions vary. We also give a necessary and sufficient condition on a fixed conductance function c having its energy Hilbert space E (c) boundedly contained in HE (b) (b = 1), i.e., contractive containment in the “flat” energy Hilbert space corresponding to constant conductance b. The significance of this is that computations in HE (b) are typically much easier and that the conclusions obtained there may then be transferred to HE (c) . 12.2 Comparing Different Conductance Functions In our proofs, we make use of tools from the theory of unbounded operators, and readers may find Ref. [Kat95] helpful. Similarly, we refer to Ref. [ST81] for graphs and networks. Given a network (G, c), we are interested in comparing its energy space HE = HE (c) and Laplace operator Δ = Δ(c) with those corresponding to a different conductance function b. To clarify the dependence on the conductance functions, we use scripts to distinguish between objects corresponding to different underlying conductance functions. For example, Δ(c) = Δ in (1.4) and E (c) = E in (1.9), as opposed to (Δ(b) v)(x) :=

 y∼x

bxy (v(x) − v(y))

(12.1)

Spectral Comparisons

251

and E (b) (u, v) = u, vE (b) =

1  bxy (u(x) − u(y))(v(x) − v(y)), 2

(12.2)

x,y∈G

.

with domain dom E (b) = {u : G → C .. E (b) (u) < ∞}. It is clear that HE (b) (b) also depends on b and so too does the energy kernel {vx }x∈G . We take the domains to be dom Δ(b) = span{vx(b) }x∈G

and

dom Δ(c) = span{vx(c) }x∈G .

(12.3)

Remark 12.1. Given a network (G, c) and a new conductance function b ≤ c, it may be that bxy = 0 even though cxy > 0, and consequently, the edge structure of (G, b) may be very different from (G, c). However, we always make the assumption that (G, b) is connected so that Lemma 12.5 may be applied. Definition 12.2. Let b : G0 × G0 → [0, ∞) be a symmetric function satisfying bxy ≤ cxy

for all x, y ∈ G0 .

In this case, we write b ≤ c. Note that we always assume that (G, b) is connected; see Remark 12.1. Lemma 12.3. For b ≤ c, inclusion gives a natural contractive embedding I : HE (c) → HE (b) . Proof. Since b ≤ c, one has 1  1  E (b) (u) = bxy |u(x) − u(y)|2 ≤ cxy |u(x) − u(y)|2 = E (c) (u) 2 2 x,y∈G

x,y∈G

(12.4) for any function u : G → R, and hence, IuE (b) ≤ uE (c) .



Lemma 12.4. I(Fin(c) ) → Fin(b) and I  (Harm(b) ) → Harm(c) . Proof. The first follows from the fact that I(δx ) = δx , whence the second follows because adjoints preserve the orthocomplements (see Theorem 2.20), i.e.,      ⊥ = Harm(c) . I  Harm(b) = I  (Fin(b) )⊥ ⊆ Fin(c) 

Operator Theory and Analysis of Infinite Networks

252

Lemma 12.5 clarifies the nature of the blanket assumption that (G, b) is connected; see Remark 12.1. Lemma 12.5. If (G, c) is a network and b ≤ c, then the following are equivalent : (i) (G, b) is connected. (ii) ker E (b) = ker E (c) = C1. (iii) ker I = 0. Proof. To see (i) ⇐⇒ (ii), observe that E (b) (u) is given by a sum of nonnegative terms and hence vanishes if and only if each summand does. Thus, E (b) (u) = 0 iff u is locally constant. For (ii) =⇒ (iii), note that I(u) = 0 implies uE (b) = 0 and hence that u is a constant function, whence u = 0 in HE (b) . For (iii) =⇒ (ii), suppose (G, b) is not connected, and define u = 1 on one component and u = 0 on the complement. Then,  I(u)E (b) = 0 but u = 0 in HE (c) . (b)

(c)

Lemma 12.6. I  vx = vx , and for general u ∈ HE (b) , one can compute I  via bxy (u(x) − u(y)). (12.5) (I  u)(x) − (I  u)(y) = cxy Proof. For u ∈ HE (c) ⊆ HE (b) , I  vx(b) , uE (c) = vx(b) , IuE (b) = u(x) − u(o) = vx(c) , uE (c) . Now, for u ∈ HE (b) and v ∈ HE (c) , the latter claim follows from the fact that 1  u, IvE (b) = bxy (u(x) − u(y))(v(x) − v(y)) 2 x,y∈G

is equal to I  u, vE (c) =

1  cxy ((I  u)(x) − (I  u)(y))(v(x) − v(y)). 2



x,y∈G

Corollary 12.7. I is injective. (c)

Proof. ker I = {0} because span{vx } = ran I  is dense in HE (c) .



Remark 12.8. Corollary 12.7 may appear trivial, but it is not. Suppose H1 and H2 are two Hilbert spaces with the same underlying vector space V but different inner products for which v2 ≤ v1 for all v ∈ V . Then,

Spectral Comparisons

253

the identity map ι : V → V induces an embedding H1 → H2 , which can fail to be injective. For example, take H2 to be the Hardy space H+ (D) on the unit disk and take H1 to be u(z)H+ (D), the image of H2 under the operation of multiplication by the function u ∈ H ∞ (D). That is, .

H1 = {uh .. h ∈ H2 },

uh1 := h2 .

There are functions u ∈ H ∞ (D) for which uh1 = 0 and uh2 = 0 even when h is a nonzero element of H2 ; see Ref. [Sar94] for details. Lemma 12.9. If δxy is the Kronecker delta, then vx(b) , Δ(b) vy(b) E (b) = δxy + 1 = vx(c) , Δ(c) vy(c) E (c) ,

∀x, y ∈ G \ {o}. (12.6)

Proof. Note that vx(b) , Δ(b) vy(b) E (b) = (Δ(b) vy(b) )(x) − (Δ(b) vy(b) )(o) = δx , vy(b) E (b) − δo , vy(b) E (b) because δx ∈ HE (b) and δx , uE (b) = Δ(b) u(x). Now, the result follows via δx , vy(b) E (b) − δo , vy(b) E (b) = (δx (y) − δx (o)) − (δo (y) − δo (o)) = δxy + 1 since x, y = o.



Lemma 12.10. For 1 < b ≤ c, one has Δ(b) = IΔ(c) I  . Proof. Applying Lemmas 12.9 and 12.6, vx(b) , Δ(b) vy(b) E (b) = vx(c) , Δ(c) vy(c) E (c) = I  vx(b) , Δ(c) I  vy(b) E (c) = vx(b) , IΔ(c) I  vy(b) E (c) .



Thus, we have a commuting square as follows. HE (c) o Δ(c)

I



HE (c)

HE (b) 

I

(12.7)

Δ(b) =IΔ(c) I 

/ HE (b) (b)

Note that one can recover the dipole property of vx from Lemmas 12.6 (b) (b) (c) and 12.10: Δ(b) vx = IΔ(c) I  vx = IΔ(c) vx = I(δx − δo ) = δx − δo .

Operator Theory and Analysis of Infinite Networks

254

Corollary 12.11. I  ∈ Hom(Harm(b) , Harm(c) ) is a spectral invariant. Proof. This is basically a restatement of Lemma 12.4.



This spectral invariance is also apparent from the formula Δ(b) = IΔ I of Lemma 12.10. While I is not a norm-preserving map, it is standard from spectral theory that one can write I in terms of its polar decomposition as I = U P , and then, Δ(b) = IΔ(c) I  implies that a unitary equivalence is given by Δ(b) = U Δ(c) U  . In the case when dim Harm(b) = dim Harm(c) = 1, the spectral invariant of Corollary 12.11 is just a number. This is computed explicitly for the geometric integers in Example 12.24. (c) 

Corollary 12.12. If b ≤ c and Δ(c) is bounded on HE (c) , then Δ(b) is bounded on HE (b) . Proof. Lemma 12.10 immediately gives Δ(b) HE (b) →HE (b) ≤ Δ(c) HE (c) →HE (c) .



Corollary 12.13. If c = 1 and Δ(c) is bounded on HE (c) , then Δ(b) is bounded on HE (b) for any bounded conductance function b. Proof. Writing b∞ for the supremum of b, we have bxy ≤ b∞ cxy = b∞ , so Corollary 12.12 applies to the network with conductances b∞ · 1.



Theorem 12.14. Let c be an arbitrary conductance function, and let 1 = χG1 , as in Definition 1.2. Then, HE (c) is contained in HE (1) if and only if there is an ε > 0 such that cxy ≥ ε for all x, y ∈ G with cxy > 0. Proof. For the forward direction, suppose K < ∞ satisfies u2E (1) ≤ Ku2E (c) for all u ∈ HE (c) . Note that E (c) (δx ) = c(x) follows directly from (1.9), so c(x) = δx 2E (c) ≥

1 1 δx 2E (1) ≥ K K

since δx E (1) ≥ 1 by the connectedness of the network.

Spectral Comparisons

255

For the converse, u2E (1) =

1  1  cxy 1 (u(x) − u(y))2 = u2E (c) , (u(x) − u(y))2 ≤ 2 2 ε ε x,y∈G

x,y∈G

so I : HE (c) → HE (1) is a bounded operator with IHE (c) →HE (1) ≤

√1 . ε



Example 12.15 (Horizontally connected binary tree). This example shows that the boundedness of the conductance function is not sufficient to imply the boundedness of the Laplacian and illustrates the interplay between spectral reciprocity and effective resistance (see also Chapter 6). To begin, let (G, b) be the binary tree where every edge x, y) ∈ G1 has conductance cxy = 1. Now, let (G, c) be the network obtained by connecting all vertices at level k with an edge of conductance ck , as in Figure 12.1. The resulting network is no longer a tree, but we call it a horizontally connected binary tree for lack of a better name (or perhaps, following inspiration from Ref. [Kai03]). Note that b ≤ c. Suppose that ck = 1 for each k, so cxy is globally constant on G1 . However, c(x) = 2k + 2 for x in level k, so c(x) is clearly unbounded on G0 . (As usual, level k consists of all vertices in (G, b) for which the shortest path to o contains exactly k edges.) Let Kn be the complete graph on n vertices. Using Schur complements (for example, as in Section 3.6 o

o G1 c1

level 1

G2 G3

c2 level 2

c3 level 3

...

...

...

... Fig. 12.1.

Construction of a “horizontally connected binary tree” of Example 12.15.

Operator Theory and Analysis of Infinite Networks

256

or Refs. [Kig01, Kig03]), one can compute RKn (x, y) = 21−n for any F (x, y) can be made x, y ∈ Kn . Consequently, it is easy to see that R(G,c) arbitrarily small by choosing x, y in level k for sufficiently large k. By spectral reciprocity (see Chapter 6), this implies that Δ(c) is unbounded on HE (c) . Thus, this network provides an example of how boundedness of cxy does not imply boundedness of Δ(c) . For an example of how boundedness of cxy does not imply boundedness of Δ on other spaces, see Ref. [Woj07]. Suppose that we choose ck so as to make c(x) bounded on G0 . Then, we must have ck = O(2−k ) as k → ∞, so we define ck = 2−k . Using this, one can compute that RGk (x, y) = 1 for x, y in level k of Gk for every k. Lemma 12.16. Suppose b ≤ c. If Δ(c) is self-adjoint, then Δ(b) is also self-adjoint. Proof. Take adjoints on both sides of Δ(b) = IΔ(c) I  (see Lemma 12.10). Note that the domains are as in (12.3).  Example 12.17 (Geometric integers). For a fixed constant c > 1, let (Z, cn ) denote the network with integers for vertices and with geometrically increasing conductances defined by cn−1,n = cmax{|n|,|n−1|} so that the network under consideration is ···

c3

•

−2

c2

•

−1

c

• 0

c

•

c2

1

•

c3

2

• 3

c4

··· ,

as in Example 12.24, and fix o = 0. In Example 14.3, it is shown that Δ(c) is not typically self-adjoint by exhibiting a defect vector ϕ ∈ HE (c) , which satisfies Δ(c) ϕ = −ϕ.

(12.8)

However, for b = 1, Δ(b) is bounded and Hermitian and thus clearly selfadjoint. This example shows that the converse of Lemma   12.16 does not hold. Using Fourier theory, one can show that HE (b) ∼ = L2 (−π, π), sin2 ( 2t ) ; see Section B.3. So, Lemma 12.10 gives Δ(b) = IΔ(c) I  , where Δ(b) is bounded and Δ(c) is unbounded and not self-adjoint. The inclusion I : HE (c) → HE (b) indicates that HE (b) = HE (c) ⊕ HE⊥(c) ,

Spectral Comparisons

257

where HE⊥(c) = HE (b)  HE (c) , and that Δ(c) is a matrix corner of Δ(b) : (b)

Δ



Δ(c) = A

 A . B

(12.9)

This gives another way to relate the operators Δ(b) and Δ(c) . 12.2.1 The adjoint of Δ(b) with respect to E (c) For the results in this section, we consider the adjoint of Δ(b) with respect c to E (c) and denote it by Δ(b) ; in other words, we are interested in c

Δ(b) u, vE (c) = u, Δ(b) vE (c) . It will be helpful to know the action of I  on Fin, as given in Lemma 12.18; this result also generalizes the dipole property Δv = δx − δy of Definition 1.23. (c)

Lemma 12.18. For 1 < b ≤ c, one has span{vx } ⊆ dom Δ(b)

c

c

Δ(b) vx(c) = I  (δx − δo ).

and (12.10)

Proof. For any fixed x ∈ G and u ∈ HE (c) , we have the estimate vx(c) , Δ(b) uE (c) = Δ(b) u(x) − Δ(b) u(o) = δx − δo , uE (b) ≤ δx − δo E (b) · uE (b) (c)

c

by Lemma 2.16 followed by (2.6). This shows that span{vx } ⊆ dom Δ(b) (c) and vx , Δ(b) uE (c) = δx − δo , uE (b) , which gives (12.10).  −1

For Theorem 12.20, we need to define Δ(c) via the spectral theorem. To this end, we introduce the following blanket assumption (which remains in place for the remainder of this section). Assumption 12.19. Suppose a conductance function c has been fixed. If the corresponding Laplace operator Δ(c) is not self-adjoint, then we replace it with the Friedrichs extension, as described in Section B.3. With Assumption 12.19 in place, we can work with Δ(c) as a self-adjoint operator. Then, by the spectral theorem, for any u ∈ HE (c) , there is a Borel

Operator Theory and Analysis of Infinite Networks

258 (c)

measure μu on [0, ∞) such that ∞ (c) (c) ψ(λ) dμu (λ) = u, ψ(Δ )uE (c) = 0

0

∞

ψ(u)P (dλ)u2E (c) , (12.11)

where P is the projection-valued measure in the spectral resolution of Δ(c) . This will be useful for Theorem 12.22. Furthermore, we also have that ∞ (c) −1 e−λΔ dλ. (12.12) Δ(c) := 0

This definition of the inverse is a standard of the spectral

∞ −λt application 1 dλ = t . theorem and is based on the fact that 0 e −1

c

Theorem 12.20. For 1 < b ≤ c, one has Δ(b) = Δ(c) Δ(b) Δ(c) , where −1 Δ(c) is the inverse of the Friedrichs extension defined as in (12.12). c

Proof. We first show Δ(c) Δ(b) = Δ(b) Δ(c) , which is equivalent to (c) (b)c (b) (c) − Δ Δ ) = 0 by Corollary 12.7. Applying Lemmas 12.18 I(Δ Δ and 12.10, one has c

c

Δ(c) Δ(b) vx(c) = IΔ(c) Δ(b) vx(c) = IΔ(c) I  (δx − δo ) = Δ(b) (δx − δo ). (c)

Then, using the dipole property Δ(c) vx = δx − δo yields Δ(b) (δx − δo ) = Δ(b) (Δ(c) vx(c) ) = Δ(b) (Δ(c) vx(c) ) = Δ(b) Δ(c) (vx(c) ). c

(c)

(c)

Now, we have Δ(c) Δ(b) (vx ) = Δ(b) Δ(c) (vx ) for any x, whence c (c) Δ(c) Δ(b) = Δ(b) Δ(c) follows from the density of span{vx } in HE (c) . (c) (b) (c) It follows from the preceding argument that Δ Δ (span{vx }) ⊆ −1 dom Δ(c) , and so, the proof is complete.  12.3 Moments of Δ(c) and Monotonicity of Spectral Measures Note that we continue to assume Δ(c) is a self-adjoint operator, as discussed in Assumption 12.19. (c)

(c)

Lemma 12.21. For u = vx − vy

and ψ(λ) = λk , k = 0, 1, 2, we have

k = 0 : u, uE (c) = RF (x, y), k = 1 : u, Δ(c) uE (c) = 2 − 2δxy , 2

k = 2 : u, Δ(c) uE (c) = c(x) + 2cxy + c(y).

Spectral Comparisons

259

Proof. The case k = 0 follows immediately from (3.9). For k = 1, (12.6) gives vx(c) , Δ(c) vx(c) E (c) − vx(c) , Δ(c) vy(c) E (c) − vy(c) , Δ(c) vx(c) E (c) + vy(c) , Δ(c) vy(c) E (c) = 2 − (δxy + 1) − (δxy + 1) + 2. For k = 2, we use the fact that the Friedrichs extension is self-adjoint and the dipole property Lemma 2.16 to compute 2

u, Δ(c) uE (c) = Δ(c) u, Δ(c) uE (c) = δx − δy , δx − δy E (c) = c(x) + 2cxy + c(y). 

For the last step, we used (1.11). (c)

In Theorem 12.22, we use μu as given by (12.11). Also, let ∞ (c) mk (u) := λk dμ(c) u

(12.13)

0

(c)

(b)

be the kth moment of μu , and similarly for μu . We now consider the moments of Δ via spectral theory. Theorem 12.22 (Monotonicity of spectral measures). Let (G, c) be a given network, and let b ≤ c. Then, (b)

(c)

m1 (u) = m1 (I  u)

and

(b)

(c)

m2 (u) ≤ m2 (I  u).

(12.14)

Proof. First, note that Lemma 12.10 gives (b)

(c)

m1 = u, Δ(b) uE (b) = u, IΔ(c) I  uE (b) = I  u, Δ(c) I  uE (c) = m1 . For the second moments, using Lemma 12.10 again gives (b)

m2 = u, (Δ(b) )2 uE (b) = u, IΔ(c) I  IΔ(c) I  uE (b) 

= Δ(c) I  u, I  IΔ(c) I  uE (c) . Since I  I is contractive by Lemma 12.3, 



Δ(c) I  u, I  IΔ(c) I  uE (c) ≤ I  I · Δ(c) I  u, Δ(c) I  uE (c) ≤ u, I(Δ(c) )2 I  uE (c) , (b)

(c)

whence m2 ≤ m2 .



Operator Theory and Analysis of Infinite Networks

260

(b)

(b)

Remark 12.23. If bxy < cxy for some edge (xy), then m2 (vx ) < (c) (b) m2 (I  vx ). 12.4 Examples Example 12.24 (Geometric integers). Let (Z, cn ) be the network whose vertices are the integers with conductances given by  cmax{|m|,|n|} , |m − n| = 1, cm,n = 0, else, as in the following diagram. ···

c4

•

c3

−3

•

c2

−2

•

−1

c

•

c

0

• 1

c2

• 2

c3

•

c4

3

···

It is shown in Theorem 14.29 that Harm is one-dimensional for this network, and it is shown in Section 14.3 that Δ is not essentially self-adjoint (as an operator on HE ) for this network. We compare (Z, bn ) and (Z, cn ), where 1 < b ≤ c. In this case, dim Harm(b) = dim Harm(c) = 1, and we can compute the (numerical) spectral invariant of Corollary 12.11. Choose unit vectors hb ∈ Harm(b) and hc ∈ Harm(c) :

1 1 sgn(n) sgn(n) hb (n) = √ 1 − |n| , hc (n) = √ 1 − |n| . (12.15) b c 2 c−1 2 b−1 Now, since I  hb , uE (c) = hb , uE (b) for all u ∈ HE (c) , we have hb , vn(c) E (b) = I  hb , vn(c) E (c) = Khc , vn(c) E (c) = Khc , vn(c) E (c) , (12.16) following the ansatz that I  should be just a numerical constant (scaling factor). Suppose for simplicity that n > 0, as the other computation is similar. On the left-hand side of (12.16), we can compute directly from (1.9): 

hb , vn(c)

 E (b)

=2

∞  j=1

=

√

b

j

1 − b−j 1 − b1−j √ − √ 2 b−1 2 b−1

b − 1vn(c) (n) =



 vn(c) (j) − vn(b) ( j − 1)

n  √ √ 1 1 − c−n . (12.17) b−1 = b−1 n c c−1 j=1

Spectral Comparisons

261

Meanwhile, on the right-hand side of (12.16), the reproducing property gives

1 1 (c) (12.18) 1− n . hc , vn E (c) = hc (n) − hc (o) = √ c 2 c−1 Substituting (12.17) and (12.18) into (12.16) gives

√ 1 1 − c−n 1 =K √ b−1 1− n , c−1 c 2 c−1 and so, the corresponding spectral invariant is  1 − b     , K =  I |Harm(b)  = 1−c and this is the factor by which I  scales the basis vector hb ; see Corollary 12.11. 12.5 Remarks and References The use of analysis on infinite discrete systems and spectral theory of associated Laplace operators is of relevance to operator algebras and to the mathematics of computation: sampling, approximation, learning theory, and more; see, for example, Refs. [CJ12, SG10, SU12, GMP11, ZCX11, Des02, SZ07]. Remark 12.25 (Open problem). For a fixed conductance function b : G0 × G0 → [0, ∞), what are the closed subspaces K ⊆ HE (b) such that K∼ = HE (c) for some conductance function c with b ≤ c?

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Chapter 13

Examples and Applications

The art of doing mathematics consists in finding that special case which contains all the germs of generality. — D. Hilbert

13.1 Finite Graphs Example 13.1. Consider a “linear” electrical resistance network consistas ing of several resistors connected in series with resistances Ω i = c−1 i indicated: α = x0

Ω1

/ x1

Ω2

/ x2

Ω3

/ x3

Ω4

/ ···

Ωn

/ xn = ω

Construct a dipole v ∈ P(α, ω) on this network as follows. Let v(x0 ) = V be fixed. Then determine v(x1 ) via (1.19): Δv(x0 ) = Δv(x1 ) =

2 

1 Ω1 (V

1 Ωk (v(xk−1 )

− v(x1 )) = 1 =⇒ v(x1 ) = V − Ω1 , − v(xk )) = 0 =⇒ v(x2 ) = V − Ω1 − Ω2 ,

k=1

and so forth. Three things to notice about this extremely elementary example are (i) v is fixed by its value at one point and any other dipole on this graph can differ only by a constant, (ii) we recover the basic fact of electrical theory that the voltage drop across resistors in series is just the sum of the resistances, and (iii) all current flows are induced (this is not true of more general graphs).

263

264

Operator Theory and Analysis of Infinite Networks

Consider the basis {e0 , e1 , . . . , eN }, where ek = δxk , the unit Dirac mass at k. The Laplace operator for this model has the matrix ⎡

1

−1

0

...

⎢ 2 −1 ... ⎢−1 ⎢ ⎢ 0 −1 2 ... ⎢ Δ=⎢ . ⎢ . .. ⎢ . . ⎢ ⎢ . . . −1 ⎣ 0 0

...

0

⎤

⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ 2 −1⎦

0 −1

(13.1)

1

One may obtain a unitary representation on 2 (ZN ) by using the diagonal matrix U (ζ) = diag(1, ζ, ζ 2 , . . . , ζ N ), where ζ := e2π /(N +1) is a primitive ¯ It is easy to check that for any matrix (N + 1)th root of 1, so that ζ −1 = ζ. M ∈ MN +1 (C), one has [U (ζ)M U (ζ) ]j,k = ζ j−k [M ]j,k . Then define Δ(ζ) :=U (ζ)ΔU (ζ) , and see that Δ(ζ) = C − U (ζ) T U (ζ) . It is clear that spec Δ = spec Δ(ζ), because Δv = λv

⇐⇒

Δ(ζ)[U (ζ)v] = λ[U (ζ)v].

Decompose the transfer operator into the sum of two shifts, so that T = M+ + M− , where M+ has ones below the main diagonal and zeros elsewhere, and M− has ones above the diagonal and zeros elsewhere. Then ¯ − and M− = M  . we have U (ζ)M+ U (ζ) = ζM+ and U (ζ)M− U (ζ) = ζM + By induction, the characteristic polynomial can be written pn (x) = det(xI − Tn ) = xpn−1 (x) − pn−2 (x), 2 + 1, and corresponding with p1 = x, p2 = x2 − 1, p3 = x3 − 2x, p4 = x4 − 3x√ √ Perron–Frobenius eigenvalues λ1 = 0, λ2 = 1, λ3 = 2, λ4 = φ = 12 (1+ 5).

spec Δ2 = {0, 1},

spec Δ3 = {0, 1, 3},

spec Δ4 = {0, 2, 2 ±

√ 2}.

Example 13.2. The correspondence P(α, ω) → F (α, ω) described in Lemma 1.30 is not bijective, i.e., the converse to the theorem is false, as can be seen from the following example. Consider the following electrical

Examples and Applications

265

network with resistances Ωi = c−1 i . Ω1

α = x0

/ x1 Ω3

Ω2

 x2

 / x3 = ω

Ω4

One can verify that the following gives a current flow I = It on the graph for any t ∈ [0, 1]: / x1 x0 t

1−t

 x2

t

1−t

/ x 3

In fact, there are many flows on this network; let χϑ be the characteristic function of the cycle ϑ = (x0 , x1 , x3 , x2 ) / x1 xO 0 ϑ=  x2 o x3 so that χϑ (e) = 1 for each e ∈ {e1 = (x0 , x1 ), e2 = (x1 , x2 ), e3 = (x2 , x3 ), e4 = (x3 , x0 )}. Then It +εχϑ will be a flow for any ε ∈ R. (Although this formulation seems more awkward than simply allowing t to take any value in R, it is easier to work with characteristic functions of cycles when there are many cycles in the network.) However, there will be only one value of t and ε for which the above flow corresponds to a potential function, and that potential function is the following: V

V −



Ω1 +Ω3  Ω2 4 k=1 Ωk

/V −

/V −

Ω2 +Ω4  Ω1 4 k=1 Ωk



(Ω1 +Ω3 )(Ω2 +Ω4 ) 4 k=1 Ωk

This is the potential function which “balances” the flow around both sides of the square; it can be computed as in the previous example. These ideas are given formally in Theorem 1.40. Example 13.3 (Finite cyclic model). In this case, let GN have vertices given by xk = e2π k/N for k = 1, 2, . . . , N , with edges connecting each vertex to its two nearest neighbours. For example, when N = 9,

266

Operator Theory and Analysis of Infinite Networks

x x3 ff 2 NN x1,    ,,

x4

G9

x9 = x0  

x5< 0. Define the horizontal conductances between nearest neighbours by cxn ,xn−1 = cyn ,yn−1 = αn , and define the vertical conductance of the “rungs” of the ladder by cxn ,yn = β n , as in Figure 13.4. This network was suggested to us by Agelos Georgakopoulos as an example of a one-ended network with nontrivial Harm. The function u constructed below is the first example of an explicitly computed nonconstant harmonic function of finite energy on a graph with one end (existence of such a phenomenon was first proved in [CW92]). Numerical experiments indicate that this function is also bounded (and even that the sequences ∞ {u(xn )}∞ n=0 and {u(yn )}n=0 actually converge very quickly), but we have not yet been able to prove this. Numerical evidence also suggests that Δ is not essentially self-adjoint on this network, but we have not yet proved this, either. However, compare with the defect on the geometric integers discussed in Section 14.3. This graph clearly has one end. We will show that such a network has nontrivial resistance boundary if and only if α > 1 and in this case, the boundary consists of one point for β = 1, and two points for β such that (1 + α1 )2 < α/β 2 . It will be made clear that the paths γx = (x1 , x2 , x3 , . . . ) and γy = (y1 , y2 , y3 , . . . ) are equivalent in the sense of Definition 7.38 if and only if β = 1. For presenting the construction of u, choose β < 1 satisfying 4β 2 < α (at the end of the construction, we explain how to adapt the proof for the less restrictive condition (1+ α1 )2 < α/β 2 ). We now construct a nonconstant

Fig. 13.4. One-sided ladder network, geometric conductances. Edge labels here indicate conductances, and α > 1 > β > 0.

272

Operator Theory and Analysis of Infinite Networks

u ∈ Harm with u(x0 ) = 0 and u(y0 ) = −1. If we consider the flow induced by u, the amount of current flowing through one edge determines u completely (up to a constant). Once it is clear that there are two boundary points in this case, it is clear that specifying the value of u at one (and grounding the other) determines u completely. Due to the symmetry of the graph, we may abuse notation and write n ˇ for the vertex “across the rung” from n. For a function u for xn or yn , and n on the ladder, denote the horizontal increments and the vertical increments by δu(n) := u(n + 1) − u(n)

and

σu(n) := u(n) − u(ˇ n),

respectively. Thus, for n ≥ 1, we can express the equation Δu(n) = 0 by Δu(n) = αn δu(n − 1) − αn+1 δu(n) + β n σu(n) = 0, which is equivalent to δu(n) =

1 βn δu(n − 1) + n+1 σu(n). α α

Since symmetry allows one to assume that u(ˇ n) = 1 − u(n), we may replace σu(n) by 2u(n) + 1 and obtain that any u satisfying  n  n β β 2 1 + u(n) + (13.8) u(n + 1) = u(n) + u(n)−u(n−1) α α α α α is harmonic. It remains to see that u has finite energy. Our estimate for E(u) < ∞ requires the assumption that α > 4β 2 , but numerical computations indicate that u defined by (13.8) will be both bounded and of finite energy, for any β < 1 < α. First, note that u(1) = α1 and so an immediate induction using (13.8) shows that δu(n) = u(n + 1) − u(n) > 0 for all n ≥ 1, and so u is strictly increasing. Since β < 1 < α, we may choose N so that  n β α−1 . < n ≥ N =⇒ α 2 Then n ≥ N implies u(n + 1) ≤ 2u(n) +

1 , α

(13.9)

Examples and Applications

273 β α

by using (13.8) and the fact that u(n) is increasing and (13.8) to write

 β n δu(n) = α1 (δu)(n − 1) + α2 u(n) + α1 α =

1 αn (δu)(0)

+

n−1 

1 αk

2

α u(n − k) +

1 α

< 1. Now use

 β n−k α

k=0

=

1

αn+1

+

β(1−β n ) αn+1 (1−β)

+

2

n 

αn+1

β k u(k),

k=1

where the second line comes by iterating the first, and the third by algebraic simplification. Applying the estimate (13.9) gives 2

n 

β k u(k) ≤ 22

k=1

n 

β k u(k − 1) +

2 α

k=1

= 22

n 

n 

βk

k=1

β k u(k − 1) + 2 αβ ·

1−β n 1−β ,

k=2

and iterating gives   n−1 1 β  k βk − βn β(1 − β n ) (2β)n δu(n) ≤ n+1 1 + + +2 . 2 α 1−β α α 1−β

(13.10)

k=0

Now the energy E(u) = (13.10) as follows: E(u) ≤

∞  n=0

1 αn+1

∞ n=0

2

αn+1 (δu(n)) can be estimated by using

 β(1 − β n ) (2β)n 1+ + 1−β α

2β + 2β n+1 − 2n+2 β n+1 − 22 β n+2 + (2β)n+2 + α(1 − β)(2β − 1)

2

and the condition α > 4β 2 ensures convergence. Note that this computations above can be slightly refined: instead of α > 4β 2 , one need only assume that α > (1 + α1 )2 β 2 . Then, fix ε > 0 for which α/β 2 > (1 + α1 )2 + ε and choose N so that n ≥ N implies n (β/α) < 1 + α1 + ε(1 + 2α + αε). Then the calculations can be repeated, with most occurrences of 2 replaced by 1 + α1 + ε.

Operator Theory and Analysis of Infinite Networks

274

Remark 13.14 (Comparison of Example 13.13 to the 1-dimensional integer lattice). Example 14.33 shows that for α > 1, the “geometric half-integers” network 0

α

1

α2

2

α3

3

α4

···

supports a monopole but not a harmonic function of finite energy. These conductances correspond to the biased random walk where, at each vertex, the walker has transition probabilities ⎧ ⎨ 1 , m = n − 1, 1+α p(n, m) = ⎩ α , m = n + 1. 1+α In particular, this is a spatially homogeneous distribution. In contrast, the random walk corresponding to Example 13.13 has transition probabilities ⎧ 1 ⎪ m = n − 1, β n, ⎪ ⎪ ⎨ 1+α+( α ) p(n, m) = 1+α+α( β )n , m = n + 1, α ⎪ ⎪ n ⎪ ⎩ (β/α)β n , m = n ˇ. 1+α+( α ) Thus, Example 13.13 is geometrically asymptotic to the geometric halfintegers. One can even think of Example 13.13 as describing the scattering theory of the geometric half-integer model, in the sense of [LP89]. In this theory, a wave (described by a function) travels towards an obstacle. After the wave collides with the obstacle, the original function is transformed (via the “scattering operator”) and the resulting wave travels away from the obstacle. The scattering is typically localized in some sense, corresponding to the location of the collision. To see the analogy with the present scenario, consider the current flow defined by the harmonic function u constructed in Example 13.13, i.e., induced by Ohm’s law: I(x, y) = cxy (u(x) − u(y)). With div|I| (x) := 1  |I(x, z)|, this current defines a Markov process with tran.. 2 {z . I(x,z)>0}

sition probabilities P (x, y) =

I(x, y) , div|I| (x)

if I(x, y) > 0,

and P (x, y) = 0 otherwise; see Chapter 11. This describes a random walk where a walker started on the bottom edge of the ladder will tend to step leftwards, but with a geometrically increasing probability of stepping to the upper edge, and then walking rightwards off towards infinity.

Examples and Applications

275

The walker corresponds to the wave, which is scattered as it approaches the geometrically localized obstacle at the origin. 13.3 Remarks and References The material of this chapter is an assortment of examples, some finite weighted graphs and others infinite. The infinite models are understood with the use of limit considerations. A good background reference is [LPW08]. Additionally, the reader may find the sources [Woe00, Woe03, KW02, Kig03, Kig01, Lyo03, LPS03, LP03, HL03, HJL02, Pem09, FHS09, MS09b, Bil13, Chr08] to be useful.

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Chapter 14

Lattice Networks Observe also (what modern writers almost forgot, but some older writers, such as Euler and Laplace, clearly perceived) that the role of inductive evidence in mathematical investigation is similar to its role in physical research. — G. P´ olya

The integer lattices Zd ⊆ Rd are some of the most widely studied infinite graphs and have an extensive literature; see Refs. [DS84, Woe00, LP16, Tel06a], for example. We begin with some results for the simple lattices; in Section 14.2, we consider the case when c is nonconstant. Because the case when c = 1 is amenable to Fourier analysis, we are able to compute many explicit formulas for many expressions, including vx and R(x, y). For d ≥ 3, we even compute R(x, ∞) = limy→∞ R(x, y) in Theorem 14.9 and give a formula for the monopole w. There is a small amount of overlap here with the results of Ref. [Soa94, Section V.2], where the focus is more on solving Poisson’s equation Δu = f . In Section 16.3, we employ our formulas in the refinement of an application to the isotropic Heisenberg model of ferromagnetism. In the current context, we may choose canonical representatives when working pointwise: Given u ∈ HE , we use the representative which tends to 0 at infinity. We take this as a standing assumption for this section, as it allows us to use the Fourier transform without ambiguity or unnecessary technical details. To see that this is justified, note that 2 (c) is dense1 in 1 Technically,

the embedded image of J : 2 (c) → Fin is dense in Fin; see Definition 6.84. 277

278

Operator Theory and Analysis of Infinite Networks

Fin by Theorem 6.85 and hence dense in HE for these examples, as it is well known that there are no nonconstant harmonic functions of finite energy on the integer lattices (we provide a proof in Theorem 14.17 for completeness). Clearly, all elements of 2 (c) vanish at ∞ for c = 1. Remark 14.1. As mentioned in Remark 0.1, one of the applications of the current investigation is in numerical analysis. Discretization of the real line amounts to considering a graph which is a scaled copy of the integers . Gε = (εZ, 1ε 1), where εZ = {εn .. n ∈ Z}. After finding the solution to a given problem as a function of the parameter ε, one lets ε → 0. Let xn denote the vertex at n.

The difference operator D acts on a function on this network by Df (xn ) := f (xn ) − f (xn+1 ). The adjoint of D with respect to 2 is D f (xn ) = f (xn ) − f (xn−1 ). Then, D D = Δ. 14.1 Lattice Networks with Constant Conductances The lattice network (Zd , c) with an edge between any two vertices which are one unit apart is called simple or translation-invariant when c = 1. The term “simple” originates in the literature on random walks. Example 14.2. One may compute the energy kernel directly using (3.1), that is, by finding a solution vx to Δv = δx − δ0 , as depicted in Figure 14.1. Then, R(o, x) = vx (x) − vx (o) = x − 0 = x, which is unbounded as x → ∞. This also provides an example of a function vx ∈ HE for which vx ∈ / 2 (c), as discussed in Section 6.5.2 and elsewhere. In Lemma 14.4, we obtain a general formula for vx on (Zd , 1). Figure 14.4 of Example 14.16 shows how this compares with that in Figure 14.5.

Fig. 14.1.

The function vx , a solution to Δv = δx − δ0 in (Z, 1).

Lattice Networks

279

To see how the function v = vx1 = χ[1, ∞) may be approximated by elements of Fin, define ⎧ ⎨1 − k , 1 ≤ k ≤ n, n (14.1) un (xk ) = ⎩0, else. The reader can verify that un minimizes E(v − u) over the set of u for which spt(u) ⊆ [1, n − 1] and that E(v − un ) = (1 − (1 − n1 ))2 +

n−1 

 (1 − nk ) − (1 −

 k−1 2 n )

+ (1 − (1 − 0))2

k=1

1 n − 1 n→∞ + −−−−−→ 0. n2 n2 The fact that v ∈ [Fin] but limk→∞ v(xk ) = limk→∞ v(x−k ) reflects that (Z, 1) has two graph ends, unlike the other integer lattices; cf. Ref. [PW90]. Therefore, (Z, 1) also provides an example of a network with more than one end which does not support nontrivial harmonic functions. An explicit formula is given for the potential configuration functions {vx } on the simple d-dimensional lattice in Lemma 14.4. By combining this formula with the dipole formulation of resistance distance =

R(x, y) = v(x) − v(y) for v = vx − vy from (3.1), we are able to compute an explicit formula for resistance distance on the translation-invariant lattice network Zd in Theorem 14.7. Results in this section exploit the Fourier duality Zd Td ; Ref. [Rud62] is a good reference. We are using Pontryagin duality of locally compact abelian groups; as an additive group of rank d, the discrete lattice Zd is the dual of the d-torus Td . Conversely, Td is the compact group of unitary characters on Zd (the operation in Td is complex multiplication). This duality is the basis for our Fourier analysis in this context. For convenience, we view Td as a d-cube, i.e., the Cartesian product of d period intervals of length 2π. In this form, the group operation in Td is written additively, and the Haar measure on Td is normalized with the familiar factor of (2π)−d . In Ref. [P´ol21], P´olya proved that the random walk on the simple integer lattice is transient if and only if d ≥ 3; see Ref. [DS84] for a nice introduction and a proof using electrical resistance networks. In the current context, this can be reformulated as the statement that there exist monopoles on Zd if and only if d ≥ 3. We offer a new characterization of this dichotomy, which we recover in Theorem 14.5 via a new (and completely constructive) proof.

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Operator Theory and Analysis of Infinite Networks

In Remark 14.18, we show that in the infinite integer lattices, functions in HE may be approximated by functions of finite support. Sometimes P´olya’s result is restated: The resistance to infinity is finite if and only if d ≥ 3. There is an ambiguity in this statement which is specific to the nature of resistance metric. One interpretation is that one can construct a unit flow to infinity; this is the terminology of Ref. [DS84] for a current with div(I) = δx , and it is clear that this is the induced current of a monopole. Probabilistically, this definition may be rephrased: For a random walk beginning at x ∈ G0 , the expected hitting time of the sphere of (shortest-path) radius n remains bounded as n → ∞. This approach interprets “infinity” as the “set of all points at infinity.” By contrast, we prove a much stronger result for the simple lattice networks (Zd , 1) in Theorem 14.9, where we show that limy→∞ R(x, y) is bounded as y → ∞ for any x ∈ G0 . To see the strength of this result, note that the simple (c = 1) homogeneous trees of degree d ≥ 3 have finite resistance to infinity, even though limy→∞ R(x, y) = ∞ for any x ∈ G0 and any choice of y → ∞. This is discussed further in Example 15.2 of the previous section. The heuristic explanation is that the resistance distance between two places is much smaller when there is high connectivity between them; there is much more connectivity between x and the “set of all points at infinity” than between x and a single “point at infinity.” In the next result, we obtain the Fourier transform of the Laplacian; we recently noted that this corresponds almost identically to the inverse Fourier transform H of the “potential kernel” of Ref. [Soa94, Section V.2]. Lemma 14.3. On the electrical resistance network (Zd , 1), the spectral (Fourier) transform of Δ is multiplication by S(t) = S(t1 , . . . , td ) = 4

d  k=1

 sin2

tk . 2

(14.2)

Proof. Each point x in the lattice Zd has 2d neighbors, so we need to find the L2 (Td ) Fourier representation of Δv(x) = (2dI − T )v(x) = 2dv(x) −

d 

v(x1 , . . . , xk ± 1, . . . , xd ). (14.3)

k=1

Here, t = (t1 , . . . , td ) ∈ Td and x = (x1 , . . . , xd ) ∈ Zd . The kth entry of t can be written as tk = t · εk , where εk = [0, . . . , 0, 1, 0, . . . , 0] has 1 in the k th slot. Then, moving one step in the lattice by x → x + εk corresponds

Lattice Networks

to e

x·t

→ e

tk

e

x·t

281

under the Fourier transform, and  d  t − t

Δv(t) = 2d − (e k + e k ) vˆ(t)  =2

k=1 d 



(1 − cos(tk )) vˆ(t)

k=1

=4

d  k=1

 sin2

tk 2

vˆ(t).



Lemma 14.4. Let {vx }x∈Zd be the potential configuration on the integer lattice (Zd , 1). Then, for y ∈ Zd ,

cos((x − y) · t) − cos(y · t) 1 dt. (14.4) vx (y) = (2π)d Td S(t) Proof. Under the Fourier transform, Lemma 14.3 indicates that the vx = e x·t − 1, whence equation Δvx = δx − δo becomes S(t)ˆ

x·t −1 1 − y·t e dt. (14.5) e vx (y) = d (2π) Td S(t) Since we may assume vx is R-valued, the result follows.



The following result is well known in the literature (cf. Refs. [DS84, NW59], for example) but usually stated in terms of current flow induced by the monopole. Theorem 14.5. With S(t) as in (14.2), the network (Zd , 1) has a monopole

cos(x · t) 1 dt (14.6) w(x) = − d (2π) Td S(t) if and only if d ≥ 3, in which case the monopole is unique. Proof. As in the proof of Lemma 14.4, we use the Fourier transform to solve Δw = −δo by converting it into S(t)w(t) ˆ = −1. This gives (14.6), t 1 1 d and since cos ≈ ∈ L (T ) for t ≈ 0, the integral is finite iff d ≥ 3 by S(t) S(t) the same argument as in the proof of Theorem 14.9; see (14.12). It remains to check that w ∈ HE . Note that it follows from Theorem 14.17 that the boundary term of (2.19) vanishes, and hence, we may compute the energy

Operator Theory and Analysis of Infinite Networks

282

Fig. 14.2. “Nerd Sniping” from xkcd by Randall Munroe (all rights reserved by the author), whom we gratefully acknowledge for allowing us to reproduce his comic. See Theorem 14.7 for the solution, and see Section 16.3 for implications with regard to magnetism in R2 and R3 .

for d ≥ 3 via w E = Δ1/2 w 2 =

Td

S(t)w(t) ˆ 2 dt =

Td

1 dt < ∞. S(t)

(14.7)

Uniqueness is an immediate corollary of the previous theorem and Theorem 14.17; if w were another, then Δ(w − w ) = δo − δo = 0 and w − w is constant.  Remark 14.6. Upon comparing (14.6) with (14.4), it is easy to see why all networks support finite-energy dipoles: The numerator in the integral for the monopole is of the order of 1 for t ≈ 0, while the corresponding numerator for the dipole is o(t) for t ≈ 0. Figure 14.2 serves to illustrate the relationships outlined above. Theorem 14.7. Resistance distance on the integer lattice (Zd , 1) is given by

sin2 ((x − y) · 2t ) 1 (14.8) R(x, y) =   dt. d 2 tk (2π)d Td k=1 sin 2

Proof. Let {vx }x∈Zd be the potential configuration on Zd . Then, vx − vy ∈ P(x, y), so by (3.2), we may use (3.9) to compute the resistance distance

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via R(x, y) = vx (x) + vy (y) − vx (y) − vy (x) since RF = RW on Zd . Using ex = e x·t , substitute in the terms from (14.5) of Lemma 14.4:

ex (ex − 1) + ey (ey − 1) − ex (ey − 1) − ey (ex − 1) 1 dt R(x, y) = d (2π) Td S(t)

1 1 − ex + 1 − ey − ey−x + ex − ey−x + ey = dt (2π)d Td S(t)

1 2 − 2 cos((x − y) · t) = dt, (14.9) (2π)d Td S(t) and the formula follows from the half-angle identity. Corollary 14.8. If y ∼ x, then R(x, y) =

1 d



on (Zd , 1).

Proof. The symmetry of (Zd , 1) indicates that the distance from x to its neighbor will not depend on which of the 2d neighbors is chosen. For k = 1, 2, . . . , d, let yk be a neighbor of x in the k th direction. Then, (14.4) gives

sin2 ( t2k ) 1 R(x, yk ) = (14.10)   dt.  (2π)d Td dk=1 sin2 tk 2 Thus,

d k=1

R(x, yk ) = 1, and R(x, yk ) = R(x, yj ) gives the result.



Theorem 14.9. With S(t) as in (14.2), the metric space ((Zd , 1), R) is bounded if and only if d ≥ 3, in which case

1 2 dt for d ≥ 3. (14.11) lim R(x, y) = y→∞ (2π)d Td S(t) Proof. This result hinges upon the convergence properties of the integrand for R(x, y), as computed in Lemma 14.7. In particular, to see that 1/S(t) ∈ L1 (Td ), one only needs to check for t ≈ 0, where  1 1 = O  2 , as t → 0. S(t) tk   Switching to spherical coordinates, 1/S(ρ) = O ρ−2 , as ρ → 0, and one requires

1 |ρ−2 |ρd−1 dSd−1 < ∞, (14.12) 0

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where dSd−1 is the usual (d − 1)-dimensional spherical measure. Of course, (14.12) holds precisely when −2 + d − 1 > −1, i.e., when d > 2. Similarly, the function cos((x − y) · t)/S(t) ∈ L1 (Td ) iff d ≥ 3. Therefore, (14.9) gives

2 2 1 cos((x − y) · t) dt − dt, for d ≥ 3. R(x, y) = d d (2π) Td S(t) (2π) Td S(t) Now, replace y with a sequence of vertices tending to infinity as in Definition 2.62. By the Riemann–Lebesgue lemma, the second integral vanishes, and for any such y → ∞, we have (14.11). Note that this is independent of x ∈ G0 , as one would expect from the translational invariance of the network since c = 1.  Definition 14.10. Denote R∞ := limy→∞ R(o, y), as it is clear from the previous result that the limit does not depend on the choice of y. Corollary 14.11. For d ≥ 3, there exists x ∈ Zd for which R(o, x) > R∞ . Proof. From (14.8), it is clear that R(o, x) ≤ R(o, ∞) if and only if

1 1 1 − cos(x · t) 1 dt ≤ dt, (2π)d Td S(t) (2π)d Td S(t) which is equivalent to 1 (2π)d

Td

cos(x · t) dt ≥ 0. S(t)

(14.13)

However, (14.13) cannot hold for all x ∈ Zd , as such an inequality would mean that all Fourier coefficients of w are nonnegative, in violation of Heisenberg’s uncertainty principle.  Remark 14.12. Corollary 14.11 leads to the paradoxical conclusion that given x ∈ G0 , there may be a y which is “further from x than infinity.” This is the case for d = 3; numerical computation of (14.11) gives lim R(x, y) ≈ 0.5054620038965394 in Z3 ,

y→∞

(14.14)

and for y = (1, 1, 1), R (o, y) ≈ 0.5334159062457338.

(14.15)

In fact, numerical computations indicate the following extremely bizarre situation: R(x, y2k ) < R(x, ∞) < R(x, y2k+1 ) for yn := (n, n, 0).

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285

x

y

Z3

∞ Fig. 14.3. In Z3 , it may happen that R(x, y) > R(x, ∞), where R(x, ∞) = limz→∞ R(x, z). This phenomenon is represented here schematically as a “black hole.”

We now turn to a graphical expression of this point. Indeed, Figure 14.3 serves to illustrate the relationships outlined above. Remark 14.13. An application of Bochner’s theorem (see Theorem 7.4) yields a unique Radon probability measure P on Td such that

1 e t·x dP(t) = e− 2 R(o,x) , ∀x ∈ Zd . Td

See Corollary 7.25 for an interpretation of this result in terms of stochastic integrals (via the Wiener transform). Corollary 14.14. For (Zd , 1), vx ∈ 2 (Zd ) if and only if d ≥ 3. Proof. By computations similar to those in the 14.9, one   proof of Theorem can see that in absolute values, the integrand (e x·t − 1)/S(t) of (14.5) is in L2 (Td ) if and only if d ≥ 3, in which case Parseval’s theorem applies.  Corollary 14.15. For (Zd , 1), the monopole w ∈ 2 (Zd ) if and only if d ≥ 5. Proof. The proof is almost identical to that of Corollary 14.14, except  that the integrand is 1/S(t), which is in L2 (Td ) if and only if d ≥ 5. Example 14.16. To see why R is not bounded on (Z, 1), one can evaluate (14.4) explicitly via the Fej´er kernel:

sin2 ((x − y) 2t ) 1   R(x, y) = dt (2π) T sin2 2t

sin2 ((x − y) 2t ) 1   dt |x − y| = (2π) T |x − y| sin2 2t

1 FN (t) dt = |x − y| (2π) T = |x − y|,

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3.0 2.5

-4

-2

2.0

2.0

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

2

4

6

8

10

-4

-2

2

4

6

8

10

-4

-2

2

4

6

8

10

Fig. 14.4. The function vx for x = 1, 2, 3 in Z, as obtained from the Fej´ er kernel. See Example 14.16.

where FN (t) is the Fej´er kernel with N = |x− y|; see Figure 14.4. Of course, this was to be expected because R coincides with the shortest path metric on trees. The following result is well known; we include it for completeness and the novelty of the proof. Theorem 14.17. h is a harmonic function on (Zd , 1) if and only if h is linear (or affine). Consequently, HE = Fin for Zd . Proof. From Δh = 0, the Fourier transform gives S(t)h(t) = 0. By the formula of Lemma 14.3 for S(t), this means ˆ h can only be supported at t = 0 and hence that ˆ h is a distribution which is a linear combination of derivatives of the Dirac mass at t = 0; see Ref. [Rud91] for this structure theorem from the theory of distributions. ˆ = P (δo ), where δo is the Dirac mass at t = 0 and Denoting this by h(t) P is some polynomial. The inverse Fourier transform gives h(x) = P (x). If the degree of P is 2 or higher, then Δh will have a constant term of −2d (cf. (14.3)) and hence cannot vanish identically. It is clear that a linear function on Zd has infinite energy; consequently, Harm is empty on this network, and the second conclusion follows.  Remark 14.18. For (Zd , 1), it is instructive to work out directly why Fin is dense in HE . That is, let us suppose that the boundary term vanishes for every v ∈ HE and use this to prove that every function which is orthogonal to Fin must be constant (and hence 0 in HE ). This shows that Fin is dense in HE in the energy norm.

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287

Proof. If v ∈ HE , then v E = v, Δvc < ∞; the Fourier transform sends  v → vˆ(t) = Z vn e n·t and

vˆ(t)S(t)ˆ v (t) dt < ∞, (14.16) v, Δvc → (2π)−d Td

  where S(t) = 4 k=1 sin2 t2k , as in Lemma 14.3. Then, note that the Schwarz inequality gives 2 

S(t)ˆ v (t) dt ≤ S(t) dt S(t)ˆ v (t)2 dt d

Td

Td

Td

so that S(t)ˆ v (t) ∈ L1 (Td ). From the other hypothesis, v ⊥ Fin means that δx , vE = 0 for each x ∈ G0 , whence Parseval’s equation gives

−d e m·t S(t)ˆ v (t) dt = 0, ∀m. 0 = δxm , vE = δxm , Δvc → (2π) Td

1

This implies that S(t)ˆ v (t) = 0 in L (Td ), and hence, vˆ can only be supported at t = 0. From Schwartz’s theory of distributions, this means vˆ(t) = f0 (t) + c0 δ0 (t) + c1 Dδ0 (t) + c2 D(2) δ0 (t) + · · · , where f0 is an L1 function and all the other terms are derivatives of the Dirac mass at t = 0 (D(2) is a differential operator of rank 2, etc.). If vˆ is just a function, then it is 0 a.e., and we are done. If the distribution δ0 (t) is a component of vˆ, then F −1 (δ0 ) = 1, which is zero in HE . In one dimension, the distribution δ0 (t) cannot be a component of vˆ because F −1 (δ0 )(xm ) = m, and this function does not have finite energy (the computation of the energy picks up a term of 1 on every edge of the lattice Zd ). The computation is similar for higher derivatives of δ0 , but they diverge even faster. For higher dimensions, note that D1 δ(0,0) = Dδ0 ⊗ δ0 and E(Dδ0 ⊗ δ0 ) = E(Dδ0 )E(δ0 ) (this is a basic fact about quadratic forms on a Hilbert space), and so, this devolves into the same argument as for the one-dimensional case.  Remark 14.18 does not hold for general graphs; see Example 15.2. Also, ˜ 1/2 vˆ, as mentioned in ˜ 1/2 (v + k) = Δ the end of the proof shows why Δ Remark 6.14; the addition of a constant corresponds on the Fourier side to the addition of a Dirac mass outside the support of χ. Example 14.19 (Nontrivial harmonic functions on Zd ∪ Zd ). Consider the disjoint union of two copies of Zd with c = 1 and d ≥ 3.

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Operator Theory and Analysis of Infinite Networks

Now, identify the origins o1 , o2 of the two lattices with a single edge of conductance co1 o2 = 1. Let w1 ∈ HE be a monopole on the first copy of Zd , as ensured by Theorem 14.5. We may assume w1 is normalized so that w(o1 ) = 1 and then extend w1 to the rest of the network by letting w ˜1 (x) = 1 for all x in the second copy of Zd . Similarly, let w2 be a function which is a monopole on the second Zd , satisfying w(o2 ) = 1, and extend ˜2 (x) = 1 for x in the first copy of Zd . Now, one can it to w ˜2 by defining w ˜2 = −δo2 . Note that the unit drop in w ˜1 across the check that Δw ˜1 = Δw edge co1 o2 moves the Dirac mass of Δw1 to the second copy of Zd . Now, define h := w1 − w2 .

(14.17)

It is easy to check that h ∈ HE and that h ∈ Harm. Corollary 14.20. If w is the monopole on (Zd , 1), d ≥ 3, then w(x) = 12 (R(o, x) − R(o, ∞)),

(14.18)

and consequently, limx→∞ w(x) = 0. Proof. Subtract (14.11) from (14.9) and compare with (14.6). For the latter statement, one can take the limit of (14.18) as x → ∞ directly or apply the Riemann–Lebesgue lemma to (14.6).  Corollary 14.21. If w is the monopole on (Zd , 1), then E(w) =

1 lim R(x, y). 2 y→∞

(14.19)

Proof. Compare (14.19) with (14.11) and note that Harm = {0}, so w is unique.  The case of (Zd , 1) for d = 1 is a tree and hence very simple with R(x, y) = |x−y| and for d ≥ 3 may be fairly well understood by the formulas given above. However, the case d = 2 seems to remain a bit mysterious. It appears that R(x, y) ≈ log(1 + |x − y|); we now give two results in this direction. Remark 14.22. From Theorem 14.9, it is clear that for d ≥ 3, if yn ∼ zn and both tend to ∞, one has limn→∞ (R(x, yn ) − R(x, zn )) = 0. In fact, this

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289

remains true in Z2 but not Z. For Z, yn ∼ zn

=⇒

n→∞

|R(x, yn ) − R(x, zn )| = 1 −−−−−→ 1 = 0.

A little more work is required for Z2 , where we work with x = o for simplicity:

cos(y · t) − cos(y · t + tk ) 1 dt R(o, z) − R(o, y) = (2π)2 T2 S(t)

1 cos(y · t)(1 − cos tk ) + sin(y · t) sin tk = dt. 2 (2π) T2 S(t) tk sin tk 1 2 One can check that 1−cos S(t) , S(t) ∈ L (T ) by converting to spherical coordinates and making the estimate

ρ ρ dρ dθ < ∞. 2 T2 ρ

Now, the Riemann–Lebesgue Lemma shows that R(o, y) − R(o, z) tends to 0 as y (and hence also z) tends to ∞. We now turn to a graphical expression of this point. Indeed, Figure 14.5 serves to illustrate the relationships outlined as follows. Example 14.23. The following example shows that even though vx is a bounded function on any network (Lemma 7.41), the corresponding multiplication operator may not be bounded; see Remark 8.2. This highlights the disparity between C  (HE ) from Definition 8.13 and AE from Definition 8.33. Consider the integer network with unit conductances (Z, 1) as follows.

We label the vertex xn by “n” to simplify the notation. Then, if (8.43) holds, Corollary 8.28 would give Mvn = |δ1 v1 | + 2|δ2 v2 | + · · · + n|δn vn | + n|δn+1 vn+1 | + · · ·

Fig. 14.5.

The energy kernel element vn on the integer network (Z, 1).

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290

for each fixed n. The operator norm corresponding to one of these terms is   √ √ k→∞   n|δn+k vn+k | = n δn+k E vn+k E = n 2 n + k −−−−−→ ∞, so clearly (8.43) cannot hold. Checking Theorem 8.21 directly is harder; one must compute 1/2

−1/2

Mvn HE →HE = sup VF DF VF F

2 →2 ,

where the latter is the operator norm on 2 (F ) and F ranges over all finite subsets of X. For our purposes, it will suffice to consider sets F of the form F = {1, 2, . . . , n}. The matrix for VF is then ⎤ ⎡ 1 1 1 1 ··· ⎥ ⎢1 2 2 2 ··· ⎥ ⎢ ⎥ ⎢1 2 3 3 ··· ⎥ ⎢ ⎥ ⎢ ⎥ ⎢1 2 3 4 ··· ⎥, ⎢. . . . .. .. .. ⎥ ⎢ . . . . .. . ⎥ ⎢. . . . . . . ⎥ ⎢ ⎥ ⎢ · · · n − 2 n − 2 n − 2 ⎥ ⎢ ⎣ ··· n − 2 n− 1 n− 1 ⎦ ··· n − 2 n− 1 n −1/2 but V 1/2 and can be complicated even for small F . For example, for  V 111 VF = 1 2 2 , one has 123

⎡ 1/2

VF

⎢ =⎣

[[−8 − 28γ + 49γ 3 , 3]]

[[8 − 28γ + 49γ 3 , 2]]

[[8 − 28γ + 49γ 3 , 2]]

[[13 − 21γ − 49γ 2 + 49γ 3 , 3]]

[[1 + 7γ − 49γ 2 + 49γ 3 , 2]] [[41 − 49γ − 49γ 2 + 49γ 3 , 2]] ⎤ [[1 + 7γ − 49γ 2 + 49γ 3 , 2]] 1 + [[−8 + 98γ 2 + 49γ 3 , 2]] ⎦ , [[97 − 105γ − 49γ 2 + 49γ 3 , 3]] where [[p(γ), k]] is the root of the polynomial p(γ) closest to the number k, and the formula for V −1/2 is similar. Example 14.24 (Noncompactness of the transfer operator). Let G be the integers Z with edges only between vertices of distance one apart (as in Example 14.2 with d = 1), with c ≡ 1. Then, the transfer operator T := σ + + σ − consists of the sum of two unilateral shifts, for which the

Lattice Networks

finite truncations (as described ⎡ 0 1 ⎢1 0 ⎢ ⎢0 1 ⎢ ⎢ ⎢0 0 TN = ⎢ ⎢ .. .. ⎢. . ⎢ ⎢0 0 ⎢ ⎣0 0 0 0

291

just above) are the banded matrices ⎤ 0 0 ··· 0 0 0 1 0 ··· 0 0 0 ⎥ ⎥ 0 1 ··· 0 0 0 ⎥ ⎥ ⎥ 1 0 ··· 0 0 0 ⎥ (14.20) .. .. . . . . . ⎥ ⎥. . .. .. .. ⎥ . . ⎥ 0 0 ··· 0 1 0 ⎥ ⎥ 0 0 ··· 1 0 1 ⎦ 0 0 ··· 0 1 0

Then, consider the vectors ξn 2 = 2

ξn := (0, . . . , 0, 1, 12 , 13 , 14 , . . . ),   

∞  1 π2 . = k2 3

(14.21)

k=1

n zeros

Then, TN does not converge to T uniformly because for n = N ,    1 1 ξn , (T − Tn )ξn c = = ξk ξk+1 = (k − n)(k − n + 1) k(k + 1) |k|>n

≥

  |k|≥1

|k|>n

1 k+1

2 ≈

|k|≥1

π2 , 6

which is bounded away from 0 as n → ∞. 14.2 Non-Simple Integer Lattice Networks In this section, we illustrate some of the phenomena that may occur on integer lattices when the conductances are allowed to vary. Many of these examples serve to demonstrate certain definitions or general properties discussed in previous sections. Example 14.25 (Symmetry of the graph vs. symmetry of the network). Consider a two-dimensional integer lattice, the case d = 2 in Example 14.2, and think of these points as living in the complex plane, √ so each vertex is m + n , where m and n are integers and = −1. It is possible to define the conductances in such a way that a function v(z) has finite energy, but v( z) does not (this is just precomposing with a symmetry of the graph: rotation by π2 ). However, v(z) is in 2 (1) if and only if v( z) is in 2 (1). Thus, 2 (1) does not see the graph.

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Define the conductances by  cxy =

1, 2

| Im(y)|

y = x + 1, , y =x+

so that the conductances of horizontal edges are all 1 and the conductances of vertical edges grow by 2k . Now, consider the function  2−|Re(x)| , y = 0, v(z) = v(x + y) = 0, y = 0. When computing the energy E(v), the only contributing terms are the edges along the real axis, and the edges immediately adjacent to the real axis: E(v(z)) = horizontal + vertical = 2(1/2 + 1/4 + 1/8 + · · · ) + 4(1/2 + 1/4 + 1/8 + · · · ) = 6, which is finite. However, E(v( z)) = 2(1 + 1 + 1 + · · · ) + 4(1/2 + 1/4 + 1/8 + · · · ) = ∞. Example 14.26 (An example where 2  HE ). Let Z have cn−1,n = n. Consider 2 (G0 , ν), where ν is the counting measure. The Dirac functions δxk satisfy δxk = 1, so {δxk } is a bounded sequence in 2 (G0 ). However, the Laplacian is ⎡ ⎤ .. . ⎢ ⎥ ⎥ Δ=⎢ ⎣ −n 2n + 1 −(n + 1) ⎦, .. . k→∞

and E(δxk ) = δxk , Δδxk  = 2k + 1 −−−−−→ ∞. So, we cannot have the bound v E ≤ K v for any constant K. This is “corrected” by using the measure c instead. In this case, δk c = 2k + 1 so that {δk } is not bounded, and we must use {δk / c(k)}. But then, the Laplacian is ⎤ ⎡ .. . ⎥ ⎢ n n+1 ⎥ Δc = ⎢ ⎣ − 2n+1 1 − 2n+1 ⎦ , .. .  δ and E √ xk = c(x1k ) E(δx ) = 1. c(xk )

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Example 14.27. It is quite possible to have unbounded functions of finite energy. Consider the network (G, c) = (Z, 1) with vertices at each integer and unit conductances to the nearest nearest neighbors. Then, it is simple n to show that u(n) = i=1 n1 and v(n) = log |1+n| are unbounded and have ∞ 1 π2 finite energy—use the identity n=1 n2 = 6 . For v, note that log |1 + n| −  log |1 + (n − 1)| = log 1 + n1  ≤ n1 . Example 14.28 (An unbounded function with finite energy). Let Z have cn−1,n = n12 . Then, the function f (n) = n is clearly unbounded, but E(f ) =

 1  1 π2 2 < ∞. (f (n) − f (n − 1)) = = n2 n2 6

Conclusion: It is possible to have unbounded functions of finite energy if c decays sufficiently fast. See Example 15.9 (and Figure 15.4) for an example of a unbounded harmonic function of finite energy. We’ve seen that there are no nontrivial harmonic functions of finite energy on (Zd , c) when c = 1. However, the situation is very different when c is not bounded. Theorem 14.29. Harm = 0 for (Z, c) iff is spanned by a single bounded function. Proof. (⇒) Fix u(0) = 0, define u(1) = u(n) − u(n − 1) =



1 c01 ,

c−1 xy < ∞. In this case, Harm and let u(n) be such that

1 , cn−1,n

∀n.

(14.22)

Now, u is harmonic: Δu(n) = cn−1,n (u(n) − u(n − 1)) − cn,n+1 (u(n + 1) − u(n)) = cn−1,n

1 1 − cn,n+1 = 0, cn−1,n cn,n+1

and u is of finite energy E(u) =

 n∈Z

cn−1,n (u(n) − u(n − 1))2 =

 n∈Z

1 < ∞. cn−1,n

Note that once the values of u(0) and u(1) are fixed, all the other values of u(n) are determined by (14.22). Therefore, Harm is one-dimensional.

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294

(⇐) If Δu(n) = cn−1,n (u(n) − u(n − 1)) − cn,n+1 (u(n + 1) − u(n)) = 0 for every n, then cn−1,n (u(n) − u(n − 1)) = cn,n+1 (u(n + 1) − u(n)) = a for some fixed a (the amperage of a sourceless current). Then,   1 E(u) = cn−1,n (u(n) − u(n − 1))2 = a2 1, let (Z, cn ) denote the network with integers for vertices and with geometrically increasing conductances defined by cn−1,n = cmax{|n|,|n−1|} so that the network under consideration is ···

c3

−2

c2

−1

c

0

c

1

c2

2

c3

3

c4

···

We fix o = 0. Lemma 14.31. On (Z, cn ), the energy kernel is given by ⎧ k ≤ 0, ⎪ ⎪0, ⎨ k+1 1−r vn (k) = 1 ≤ k ≤ n, n > 0, 1−r , ⎪ ⎪ ⎩ 1−rn+1 1−r , k ≥ n, and similarly for n < 0. Furthermore, the function wo (n) = ar|n| , a := r 2(1−r) , defines a monopole, and h(n) = sgn(n)(1 − wo (n)) defines an element of Harm. Proof. It is easy to check that Δwo (0) = 2c(a−ar) = 1 and that Δwo (n) = cn (arn − arn−1 ) + cn+1 (arn − arn+1 ) = 0 for n = 0. The reader may check r so that wo ∈ HE . The computations for vx and h are that E(wo ) = 2(1−r) essentially the same. See Figure 14.6. 

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Fig. 14.6. The functions v1 , v2 , and v3 on (Z, cn ). Also, the monopole wo and the projection f1 = PFin v1 . See Lemma 14.31.

Remark 14.32. In Figure 14.6, one can also see that f1 = PFin v1 induces a current flow of 1 A from 1 to 0, with 1+r 2 A flowing down the one-edge A flowing down the path from 1 to 0 and the remaining current of 1−r 2 “pseudo-path” from 1 to +∞ and then from −∞ to 0. See Remark 3.20. Example 14.33 (Geometric half-integer model). It is also interesting to consider (Z+ , cn ), as this network supports a monopole but has Harm = 0. 0

c

1

c2

2

c3

3

c4

···

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Operator Theory and Analysis of Infinite Networks

As in Lemma 14.31, it is straightforward to check that wo (n) = ar|n| , r , defines a monopole on the geometric half-integer model (Z+ , cn ). a := (1−r) However, it is also easy to check by induction that Harm = 0 for this model. For k = 2, 3, . . . , the network (Z+ , k n ) can be considered as the “projection” of the homogeneous tree of degree k (k , k1 1) under a map which sends x to n ∈ Z iff there are n edges between x and o. Example 14.34 (Decomposition in D). In Remark 2.58, we discussed the Hilbert space D and its inner product u, vo := u(o)v(o) + u, vE . Since (Z+ , cn ) and (Z, cn ) are both transient for c > 1 (but only the latter contains harmonic functions), it is interesting to consider PD0 1 for these ⊥ 1 models (see Remark 2.57). The projections v = PD0 1 and u = 1−v = PD 0 n on (Z, c ) are given by v(x) = 2 − 2a + ar|x|

and u(x) = 2a − 1 − ar|x| ,

(14.24)

1 − where with a = 3−2c and r = c−1 , one can check that v ∈ M+ o and u ∈ Mo ; see Definition 2.55 and Lemma 2.56. In particular, Δv = (1 − vo )δ0 and Δu = −uo δ0 (as usual, o = 0). Now, consider the representative of w ∈ Mo given by ! w(x) = (2 − a)χ[−∞, 0] + 1 + 2a(r|x| − c) χ[1, ∞) . (14.25)

A straightforward computation shows that w = v + h with h ∈ HDo . The function v = PD0 1 was computed for (Z, cn ) in (14.24) by using the ⊥ 1 = 1 − v and formula E(u) = uo − u2o from Lemma 2.59, where u := PD 0 uo = u(o). For a general network (G, c), this formula implies that (uo , E(u)) lies on a parabola with uo ∈ [0, 1) and maximum at ( 12 , 14 ). From (14.24), it 1 is clear that the network (Z, cn ) provides an example of how uo = 1 − 2c−1 can take any value in [0, 1). Note that c = 1 corresponds to E(u) = 0, which is the recurrent case. Example 14.35 (Star networks). Let (Sm , cn ) be a network constructed by conjoining m copies of (Z+ , cn ) by identifying the origins of each; let o be the common origin. Example 14.36 (Fin2 not dense in Fin). On (Z, 2n ), Fin2 = cl span{δx − δo } is not dense in Fin (see Definition 5.5). To illustrate this, we compute the projection of −δo to Fin2 . This may be accomplished by

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297

Fig. 14.7. The projection of the Dirac mass −δo onto Fin2 ; see Example 14.36 and also Lemmas 14.31 and 14.31.

computing

  . un := projection of − δo to span{δx − δo .. |x| ≤ n}

and then taking the limit as n → ∞. The result is depicted in Figure 14.7. We leave the computation of the case of general geometric conductance (Z, cn ) as an exercise. 14.3 An Example of When Δ is Not Essentially Self-Adjoint on HE We construct a function u ∈ dom E satisfying Δu = −u on (Z, cn ), c > 1, the one-dimensional integer lattice with geometrically growing conductances. We do this in two stages: (i) Construct a defect vector on (Z+ , cn ), and (ii) combine two copies of this defect vector to obtain an example on (Z, cn ). Example 14.37 (Defect on the positive integers). (Z+ , c) where

We consider

cn−1,n = cn , n ≥ 1, for some fixed c > 1. Thus, the network under consideration is as follows. 0

c

1

c2

2

c3

3

c4

···

Now, recursively define a system of polynomials in r = 1/c by " # " # " #" #" # pn 1 1 1 1 1 1 0 = n · · · . qn 1 + rn r r2 1 + r2 r 1+r 1

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We show that u(n) := qn satisfies Δu = −u and has finite energy. It will be helpful to note that pn = cn (u(n) − u(n − 1)),

(14.26)

and hence, pn+1 = pn + qn

and qn+1 = qn + rn+1 pn+1 .

Now, Δu = −u because Δu(n) = pn − pn+1 = −qn = −u(n). We need the following lemma to show that u ∈ HE . Lemma 14.38. There is an m such that pn ≤ nm and qn ≤ (n+ 1)m − nm for n ∈ Z+ . Proof. We prove both bounds simultaneously by induction, so assume both bounds hold for n and prove pn+1 ≤ (n + 1)m

and

qn+1 ≤ (n + 2) − (n + 1)m . m

The estimate for pn+1 = pn +qn is immediate from the inductive hypotheses. For the qn+1 estimate, choose an integer m so that  2 2 2 t .. m(m − 1) ≥ max{t r . t ≥ 0} = . e log c Then, (n + 1)2 rn+1 ≤ m(m − 1) for all n, so  m  m m(m − 1) n n+2 n+1 2+r ≤2+ ≤ + (n + 1)2 n+1 n+1 !m !m n 1 by using the binomial theorem to expand n+1 = 1 − n+1 and !m !m n+2 1 = 1 + n+1 . Multiplying by (n + 1)m gives n+1 ((n + 1)m − nm ) + rn+1 (n + 1)m ≤ (n + 2)m − (n + 1)m , which is sufficient because the left-hand side is an upper bound for qn+1 =  qn + rn+1 pn+1 . Lemma 14.39. The defect vector u(n) := qn has finite energy and is bounded.

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299

Proof. Applying Lemma 14.38 to the formula for E yields E(u) =

∞ 

cn (u(n) − u(n − 1))2 =

n=1

∞  n=1

rn p2n ≤

∞ 

rn n2m = Li−2m (r) < ∞

n=1

since a polylogarithm indexed by a negative integer is continuous on R, except for a single pole at 1 (but recall that r ∈ (0, 1)).  Lemma 14.39 ensures that the defect vector is bounded; in the example in Figure 14.8, the defect vector has a limiting value of ≈ 4.04468281, although the function value does not exceed 4 until x = 10. The first few values of the function are  u=

3 17 173 3237 114325 7774837 1032268341 270040381877 140010315667637 , , , , , , , , ,... 2 8 64 1024 32768 2097152 268435456 68719476736 35184372088832



≈ [1.5, 2.125, 2.7031, 3.1611, 3.4889, 3.7073, 3.8455, 3.9296, 3.9793, 4.0080, . . . ] .

While we are unable to provide a nice closed-form formula for the defect vector, we can provide generating functions for it using pn = pn (r) and qn = qn (r) obtained just above. Define P (x) =

∞  n=0

pn (r)xn

∞ 

and Q(x) =

qn (r)xn .

n=0

Multiplying both sides of pn+1 = pn + qn by xn+1 and summing up from n = 0 to ∞, P (x) = xP (x) + xQ(x),

(14.27)

Fig. 14.8. A Mathematica plot of the defect vector u on (Z+ , 2n ); see Example 14.37 and Lemma 14.39. The left plot shows u(x) for x = 0, 1, . . . , 10, and the plot on the right shows data points for u(x), x = 10, 11, 12, . . . .

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300

where we have used the fact that p0 = 0. Meanwhile, multiplying both sides of qn+1 = qn + rn+1 pn+1 by xn+1 and summing up from n = 0 to ∞, Q(x) − 1 = xQ(x) + P (rx).

(14.28)

Write (14.27) in the form (1 − x)P (x) = xQ(x) and substitute in (1 − x)Q(x) = 1 + P (rx) from (14.28) to get 1 + P (rx) = (1 − x)Q(x) = (1−x)2 P (x) or x P (x) =

x (1−x)2

+

x (1−x)2 P (rx)

=

x (1−x)2

x(rx) (1−x)2 (1−rx)2

x(rx) 2 (1−x)2 (1−rx)2 P (r x)

+ ··· =

+

∞ $ n  n=0 k=0

∞  rk x rn(n+1)/2 xn % = . n k 2 (1 − rk x)2 k=0 (1 − r x) n=0

k→∞

Note that r ∈ (0, 1), so P (rk x) −−−−−→ P (0) = 0 again since p0 = 0. Now, (14.27) gives Q(x) = 1−x x P (x), whence Q(x) =

∞  rn(n+1)/2 xn−1 %n . k 2 k=1 (1 − r x) n=0

Example 14.40 (Defect on the integers). We consider (Z, c) as in Definition 14.30: ···

c3

−2

c2

−1

c

0

c

1

c2

2

c3

3

c4

···

Proceeding as in Example 14.37, one uses Δu(0) = −u(0) to compute   1 2c(u(0) − u(1)) = −u(0) =⇒ u(1) = 1 + 2c u(0), and obtain the initial values p1 = 12 and q1 = 1 + r2 . Therefore, for Z, we instead use the polynomials defined by # " #" " # " #" # 1 1 1 1 1 1 pn 0 = n · · · 1 . qn 1 + rn r r2 1 + r2 r 2+r 2 The other computations are essentially identical to those for (Z+ , cn ). 14.4 Remarks and References The infinite lattices offer a second attractive family of examples; and they are especially relevant for lattice-spin models in physics, as discussed in Chapter 16. The book by Soardi [Soa94] is a nice introduction to the subject, and a classical introductory reference is Ref. [Spi76]. Of the results in the

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301

literature of relevance to this chapter, Refs. [GvN51, GS06, SZ09, Mar99, CL07, Lig99, Lig95, Lig93] are especially relevant. The geometric integers of Example 14.37 came about from our desire to apply von Neumann’s theory of unbounded operators and their deficiency indices [vN32a, vN32b, vN32c, DS88] to the metric geometry of infinite weighted graphs (G, c). Starting with (G, c), there are two natural Hilbert spaces 2 (G0 ) (where G0 is the vertex set) and the energy Hilbert space HE . An intriguing aspect of Section 14.3 is that the boundary features of (G, c) deriving from deficiency indices cannot be accounted for by using the more naive of the two Hilbert spaces 2 (G0 ); HE is forced upon us. The geometric integers discussed in Example 14.37 is called a weighted linear graph in Ref. [AF09] and is studied in conjunction with birth and death processes; see the references cited therein.

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Chapter 15

Infinite Trees Great fleas have little fleas upon their backs to bite ’em,And little fleas have lesser fleas, and so ad infinitum. And the great fleas themselves, in turn, have greater fleas to go on,While these again have greater still, and greater still, and so on. — A. De Morgan

The n-ary trees play an important role in symbolic dynamics, and they support a rich family of nontrivial harmonic functions of finite energy. These graphs are essentially homogeneous trees; the only difference is that the homogeneous tree has one more branch at the root, as can be seen from Figure 15.1. We use the latter examples as they are simpler yet still sufficient for our purposes and because of our independent interest in symbolic dynamics. However, almost all remarks extend to the homogeneous trees without effort; these examples are well studied because of their close relationship with group theory (especially free groups). Also, they provide an excellent test bed for studying the effects of varying c and for illustrating several of our theorems. A network whose underlying graph is a homogeneous tree always allows for the construction of a nontrivial harmonic function. In particular, Fin is not dense in HE by Lemma 2.49 that these are equivalent. Remark 15.1. If the origin were removed from the binary tree, we adopt the convention that vertices in one component are “positive” and indices in the other are “negative.” If the vertices are indexed with binary numbers (using the empty string ∅ to denote the origin o = x∅ ), then indices beginning with 1 are positive and indices beginning with 0 are negative. 303

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304

xo

xo

Fig. 15.1. The homogeneous tree of degree 3 (left) and the binary tree from symbolic dynamics (right). The root of the tree is labeled xo . If the gray branch is pruned from the homogeneous tree, the two become isomorphic.

1

1

1 x

vx

RF(x,o) = (vx) = 1 0 o 1 4

1 8

x

0

0

0

1 16

fx 1 2

3 4

RH(x,o) = (hx) =

1 4

1 16

1 8

1 4

RW(x,o) = (fx) =

o 3 4

1 2

hx o

15 16

7 8

x 1 4

1 8

1 16

Fig. 15.2. The reproducing kernel on the tree with c = 1. For a vertex x which is adjacent to the origin o, this figure illustrates the elements vx , fx = PFin vx , and hx = PHarm vx , as discussed in Example 15.2. This figure should be compared with Figure 14.6 (with c = 2).

. Example 15.2 (The reproducing kernel on the tree). Let ( , 1) be the binary tree network in Figure 15.1 with constant conductances. Figure 15.2 depicts vx as well as its decomposition in terms of Fin and Harm. We have chosen x to be adjacent to the origin o; the binary label of this vertex would be x1 .

Infinite Trees

1

1

1

305

1

1

x

(k)

vx

0

0

0

2k 1 2k+2  2 x

0

0

2k-j 1 2k+2  2

0

j = 0,1, ... , k

2k-1 1 2k+2  2

fx(k)

1 2k-j 1 2k+2 2

2k 1 2k+2  2

1 2

1

j = 0,1, ... , k

1

1 22k+21 2

0

1

k

(k)

hx

x

1 22k+2 1 2 k-1

1

2k-j 1 2k+2  2

j = 0,1, ... , k

1 2

2k 1 2k+2  2

2k-j 1 2k+2  2

j = 0,1, ... , k 0

0

y Fig. 15.3. Approximants to the reproducing kernel on the tree with c = 1; see Example 15.2.

In Figure 15.2, the numbers indicate the value of the function at that vertex; artistic liberties have been taken. If vertices s and t are the same distance from o, then |fx (s)| = |fx (t)|, and similarly for hx . Note that hx provides an example of a nonconstant harmonic function in HE . Another / 2 , see Corollary 2.73. It is easy to see that key point is that hx ∈ 1 1 limz→±∞ hx (z) = 2 ± 2 , whence hx is bounded. (k) For fx = PFin vx in Figure 15.2, the illustration of fx in Figure 15.3 is .. the projection of vx (or fx ) to span{δx . x ∈ Gk }, where Gk consists of all vertices within k steps of o. The lines on the right-hand side of each figure just indicate that the function is constant in the remainder of the graph

Operator Theory and Analysis of Infinite Networks

306

(k)

(at a value of 0 or 1); in particular, note that fx (y) = 0 for every vertex y, which is at least k + 1 steps from the origin. Also, observe that   δs δt − , Δfx(k) = δx − δy + k+2 k+2 2 −2 2 −2 + − s∈bd Gk+1

t∈bd Gk+1

bd G+ k+1

where is the subset of bd Gk+1 that lies on the positive branch, etc. It is interesting to note that if one were to identify all the vertices of (k) + would become harmonic at this bd Gk+1 = bd G+ k+1 ∪ bd Gk+1 , then fx (k)

(k)

new vertex. Observe also that hx = vx − fx is its orthogonal complement and is harmonic everywhere except on bd Gk+1 . Note that for x further from the origin, one can see that vx will appear similar to vx in Figure 14.1. Namely, vx will increase by 1 with each step along the path from o to x, and along each branch, the value will be extended by a constant (equal to the distance from o). Example 15.3 (A function of finite energy which is not approx(k) imable by Fin). We continue to refer to Figure 15.3. Since fx ∈ Fin (k) and it is easy to see that fx − fx E → 0 and that Δfx = δx − δy , this approximation verifies that fx = PFin vx . It also shows that minf ∈Fin vx − f E ≥ hx E = 12 . Example 15.4 (A monopole which is not a “dipole at infinity”). Let ( , 1) be the binary tree network in Figure 15.1 with constant conductances c = 1. Let |x| be the number of edges in the path connecting x to o. Define a function 1 wo (x) = 1 − |x| (15.1) 2 so that, essentially, wo = 2|hx − 12 | for hx of Example 15.2. It is easy to check that Δwo = −δo so that wo is a monopole at the root/origin o. To see that wo ∈ HE ,    2 ∞  1 1 n E(wo ) = 2 2 1 − n − 1 − n+1 2 2 n=0   ∞ 2  1 =2 2n n+1 2 n=0 =

∞ 1 1 2 n=0 2n

= 1.

Infinite Trees

307

However, wo is not a “dipole at infinity” in the sense that there is no sequence {xn } of distinct and successively adjacent vertices for which {vxn } converges to wo (this is in contrast to the integer lattices Zd , d ≥ 3). Observe that RF (x, y) coincides with the shortest-path distance on this network (as it does on any tree). If {xn } is a sequence tending to ∞ (i.e., for any N , there is an n such that xn is more than n steps from o for all n ≥ N ), then E(vxn ) = RF (xn , o) = n so that wo is not a limit of a sequence of dipoles. Of course, since {vx } is dense in HE , wo is the limit of the linear . combinations of dipoles. In fact, let bd Gk = {x ∈ G0 .. R(x, o) = k} as before. Then, 

wo (x) = lim

k→∞

z∈bd Gk

vz . 2k

Example 15.5 (A function with nonvanishing boundary sum). In Theorem 2.40, we showed that   u(x)Δv(x) + lim u(x) ∂∂v (x). u, v E = lim k→∞

Let wo (x) = 1 − .. .

1 2|x|

k→∞

x∈int Gk

x∈bd Gk

be the monopole from Example 15.4. With Gk :=

{x |x| ≤ k}, we have ∂wo ∂  (x)

=

    1 1 1 1 − k − 1 − k−1 = k 2 2 2

Since Δw = −δo , we have energy of wo is

 Gk

.

for x ∈ bd Gk = {x .. |x| = k}.

wo (x)Δwo (x) = −w(o) for each k, and the

  1 E(wo ) = wo , wo E = − 1 − 0 + lim k→∞ 2

   1 1 = 1. 1− k 2 2k

x∈bd Gk

For hx , the harmonic function with E(hx ) = becomes E(hx , hx ) = lim

k→∞

 x∈bd Gk

1 4

in Example 15.2, this

x hx (x) ∂h ∂  (x) =

1 . 4

In fact, one can obtain this by computing the boundary term directly: Each of the 2k−1 vertices in bd G+ k is connected by a single edge to Gk , and

Operator Theory and Analysis of Infinite Networks

308

similarly for the 2k−1 vertices in bd G− k , so  y∈bd Gk

2k+1 − 1 1 1 −1 · k+1 + 2k−1 k+1 · k+1 2k+1 2 2 2   1 1 = 1− k . 4 2

k−1 x hx (y) ∂h ∂  (y) = 2

Example 15.6 (The tree supports many nontrivial harmonic functions). We can use hx of Example 15.2 to describe an infinite forest of mutually orthogonal harmonic functions on the binary tree. Let z ∈ G be represented by a finite binary sequence, as discussed in Remark 15.1. Define a morphism ϕz : G → G by prepending, i.e., ϕz (x) = zx. This has the effect of “rigidly” translating the tree so that the image lies on the subtree with root z. Then, hz := hx ◦ϕz is harmonic and is supported only on the subtree with root z. The supports of hz1 and hz2 intersect if and only if Im(ϕzi ) ⊆ Im(ϕzj ). For concreteness, suppose it is Im(ϕz1 ) ⊆ Im(ϕz2 ). If they are equal, it is because z1 = z2 , and we don’t care. Otherwise, compute the dissipation of the induced currents:  Ω (x, y) dhz1 (x, y) dhz2 (x, y). dhz1 , dhz2 D = 12 (x,y)∈ϕz1 (G1 )

Note that dhz2 (x, y) always has the same sign on the subtree with root z1 = o, but dhz1 (x, y) appears in the dissipation sum positively signed with the same multiplicity as it appears negatively signed. Consequently, all the terms cancel, and 0 = dhz1 , dhz2 D = hz1 , hz2 E shows that hz1 ⊥ hz2 . Example 15.7 (Haar wavelets). Example 15.6 can be heuristically described in terms of Haar wavelets. Consider the boundary of the tree as a copy of the unit interval with hx as the basic Haar mother wavelet via the “shadow” cast by limn→±∞ hx (xn ) = ±1. Then, hz is a Haar wavelet localized to the subinterval of the support of its shadow, etc. Of course, this heuristic is a bit misleading since the boundary is actually isomorphic to {0, 1}N with its natural cylinder-set topology. Example 15.8. On the binary tree with c = 1, the monopole wo is given by wo (x) = 2−|x| and can be written as v + Δv for v ∈ dom ΔV : v(x) := 2−|x| −

√  2 1+

√1 2

n+1

.

Infinite Trees

309

We leave it to the reader to check that this v satisfies the above equation, and also, E(v) =  G0

√ 2(46 2−65) √ √ , (2− 2)2 ( 2−1)

E(Δv) =

√ 4(99−70 2) √ √ , (2− 2)4 ( 2−1)

and

√ vΔv = 27 (−23 + 17 2).

Example 15.9 (An unbounded harmonic function of finite energy). Figure 15.4 is a sketch of an unbounded harmonic function of finite energy on the binary tree with c = 1. To construct it, pick one ray from o to ∞, and let |x|  1 h(x) = k k=1

for x along this ray. Then, if x is in the ray and y ∼ x, fix h(y) so that h 1 ), and define h along the rest is harmonic at x (i.e., h(y) = h(x) + |x|(|x|+1) of this branch by h(z) =

h(y) − h(x) . 2|y−z|

If w denotes the monopole at o defined by w(x) = 2−|x| , as discussed in Example 15.8 and previously, then we are essentially attaching a scaled copy of w to each neighbor of the chosen ray. See Figure 15.4.

n

+ 12

11 6

1 + 12

25 12

+ 1n

1 Σ k =1 k

1 + n+1

1 + n(n+1)

+ 16

1 +1

3 2

+ 13

+ 14

+ 12

0 −1 Fig. 15.4.

An unbounded harmonic function of finite energy. See Example 15.9.

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310

It is clear that h(x) → ∞ logarithmically along the chosen ray; the energy coming from h(x) on this ray is ∞  2  π2 1 . = E(h)|ray = n 6 n=1 The energy from each branch incident upon the ray is  2 1 1 1 1 E(h)|branch(n) = E(w) = 2 . + + n(n + 1) n(n + 1) n (n + 1)2 n(n + 1)  π2 Summing up, E(h) = E(h)|ray + ∞ n=0 E(h)|branch(n) = 2 − 2. We leave it to the reader to check that h is harmonic. Example 15.10. On the binary tree with c = 1, the function uξ (x) = ξ −|x| has energy E(uξ ) = 2

(1 − ξ)2 −∞ for which ϕ, Sϕ ≥ cϕ, ϕ

for all ϕ ∈ dom S.

(B.2)

The spectrum of a semibounded operator lies in some halfline [κ, ∞), and the defect indices of a semibounded operator always agree (see Definition B.10). The graph Laplacian Δ considered in much of this book falls into this class.

339

340

Operator Theory and Analysis of Infinite Networks

Definition B.4. An operator S on H is called bounded iff there exists k ∈ R such that |v, Sv| ≤ kv2

for every v ∈ D.

(B.3)

The spectrum of a bounded operator lies in a compact subinterval of R. Bounded Hermitian operators are automatically self-adjoint. When (G, Ω ) satisfies the Powers bound (6.68), the transfer operator falls into this class. Definition B.5. For an operator S on the Hilbert space H, the graph of S is .

v ] .. v ∈ H} ⊆ H ⊕ H, G(S) := {[ Av

(B.4)

v [ Sv ]2Graph := v2H + Sv2H

(B.5)

with the norm and the corresponding inner product. The operator S is closed iff G(S) is closed in H ⊕ H or closable if the closure of G(S) is the graph of an operator. In this case, the corresponding operator is S clo , the closure of S. The domain of S clo is therefore defined as dom S clo := {u .. lim u − un H = lim v − Sun H = 0} .

n→∞

n→∞

(B.6)

for some v ∈ H and Cauchy sequence {un } ⊆ dom S. Then, one defines S clo u := v. If S is Hermitian, then S clo will also be Hermitian, but it will not be self-adjoint in general. Remark B.6. It is important to observe that an operator S is closable if and only if S  has a dense domain. However, this is clearly satisfied when S is Hermitian with a dense domain since then dom S ⊆ dom S  . Definition B.7. Suppose that S is a linear operator on H with a dense domain dom S. Define the graph rotation operator G : H ⊕ H → H ⊕ H by G(u, v) := (−v, u). It is easy to show that the graph of S  is G(S  ) = (G(G(S)))⊥ .

(B.7)

For any semibounded operator S on a Hilbert space, there are unique self-adjoint extensions Smin (the Friedrichs extension) and Smax (the Krein extension) such that S ⊆ S clo ⊆ Smin ⊆ S˜ ⊆ Smax , (B.8) where S˜ is any nonnegative self-adjoint extension of S. For general unbounded operators, these inclusions may all be strict. In (B.8), A ⊆ B

341

Some Operator Theories

means graph containment, i.e., it means G(A) ⊆ G(B), where G is as in Definition B.5. The case when Smin = Smax is particularly important. Definition B.8. An operator is defined to be essentially self-adjoint iff it has a unique self-adjoint extension. An operator is essentially self-adjoint if and only if it has defect indices 0,0 (see Definition B.10). A self-adjoint operator is trivially essentially self-adjoint. Theorem B.9 ([vN32a, Rud91, DS88]). Let S be a Hermitian operator. 1. S is closable, its closure S clo is Hermitian, and S  = (S clo ) . 2. Every closed Hermitian extension T of S satisfies S ⊆ T ⊆ T  ⊆ S. 3. S is essentially self-adjoint if and only if dom(S clo ) = dom S  . 4. S is essentially self-adjoint precisely when both its defect indices are 0. 5. S has self-adjoint extensions iff S has equal defect indices. The Hermitian operator S := QP Q of Example B.18 has defect indices 1,1 and yet is not even semibounded. Definition B.10. Let S be an operator with adjoint S  . For λ ∈ C, define .

Def λ (S) := ker(λ − S  ) = {v ∈ dom S  .. S  v = λv}.

(B.9)

Then, Def λ (S) is the defect space of S corresponding to λ. Elements of Def λ (S) are called defect vectors. The number dim Def λ (S) is constant in the connected components of the resolvent set C \ σ(S) and is called the defect index of the component containing λ. Note that if S is Hermitian, then its resolvent set can have at most two connected components. Furthermore, if S is semibounded, then its resolvent set can have only one connected component, and we have only one defect index to compute: the dimension of Def(S) = Def −1 (S). These facts explain the two consequences of the following theorem, which can be found in Refs. [vN32a, Rud91, DS88]. Theorem B.11 (von Neumann). For a Hermitian operator S on H, one has .

.

dom S  = dom S clo ⊕ {v ∈ H .. S  v = v} ⊕ {v ∈ H .. S  v = − v}, where the orthogonality of the direct sum on the right-hand side is with respect to the graph inner product (B.5) (not the inner product of H).

342

Operator Theory and Analysis of Infinite Networks

Consequently, S is essentially self-adjoint if and only if Sv = ± v

=⇒

v ∈ H.

v = 0,

(B.10)

For a semibounded Hermitian operator S on H, one has .

dom S  = dom S clo ⊕ {v ∈ H .. S  v = −v}, where again the orthogonality of the direct sum on the right-hand side is with respect to the graph inner product (B.5). Consequently, S is essentially self-adjoint if and only if S  v = −v

=⇒

v = 0,

v ∈ H.

(B.11)

A solution v to (B.10) or (B.11) is called a defect vector (as in Definition B.10) or an vector at ∞. The idea of the proof in von Neumann’s theorem is to obtain the essential self-adjointness of a Hermitian operator S by using the following stratagem: An unbounded function applied to a bounded self-adjoint operator is an unbounded self-adjoint operator. In this case, the function is f (x) = λ − x−1 . If we can see that (λ − Smin )−1 is bounded and self-adjoint, then f ((λ − Smin )−1 ) = Smin is an unbounded self-adjoint operator. First, note that ¯ − S)] = ran(λ ¯ − Smin ) = ker(λ − S  )⊥ = Def λ (S)⊥ . [ran(λ  Note that if λ ∈ res(S) and (λ − Smin )−1 = R x E(dx) with projectionvalued measures E : B(R) → P roj(H), then  Smin = (λ − x−1 ) E(dx). R

This will show that Smin is self-adjoint; if Def λ (S) = 0 for “enough” λ, then Smin is self-adjoint and hence S is essentially self-adjoint. Lemma B.12. If S is bounded and Hermitian, then it is essentially selfadjoint. Proof. S is bounded iff it is defined everywhere by the Hellinger–Toeplitz theorem. Since S  is also everywhere defined in this case, it is clear the two operators have the same domain. 

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Some Operator Theories

Lemma B.13. For an operator S which is semibounded but not necessarily closed, Def(S)⊥ = ran(1 + S clo ).

(B.12)

Proof. Recall that Def(S) = Def −1 (S). General theory gives Def(S)⊥ = (ker 1 + S  )⊥ = (ran(1 + S))⊥⊥ = (ran(1 + S))clo . It remains to check that ran(1 + S clo ) = (ran(1 + S))clo . If S is not semibounded, one may have only the containment (⊆): Note that u ∈ ran(1 + S clo ) iff u = v + Sv for v ∈ dom S clo . Then, v = lim vn for some clo vn ∈ dom S, and the containment is clear. For (⊇), let u ∈ ran(1 + S) so that u = lim un , where un = vn Svn for vn ∈ dom S, and note that vn 2 ≤ vn 2 + Svn , vn  + vn , Svn  + Svn 2 = vn + Svn 2 = un 2 , (B.13) where the inequality uses the fact that S is semibounded. By passing to a subsequence if necessary, (B.13) implies vn − vm  ≤ un − um , whence {un } Cauchy implies {vn } is also Cauchy and hence has a limit v. Therefore, n→∞

Svn = un − vn −−−−−→ u − v, which allows one to define S clo v = u − v and see u ∈ ran(1 + S clo ).



Example B.14 (The defect of the Laplacian on (0, ∞)). Probably the most basic example of defect vectors (and how an Hermitian operator can fail to be essentially self-adjoint) is provided by the Laplace operator d2 2 Δ = − dx 2 on the Hilbert space H = L (0, ∞). Example B.18 gives an even more striking (though less simple) example. We take Δ as having the dense domain D = {f ∈ C0∞ (0, ∞) .. f (k) (0) = lim f (k) (x) = 0, k = 0, 1, 2, . . . }. .

x→∞

We always have u, Δu ≥ 0 for u ∈ D. However, Δ u = −u is satisfied by e−x ∈ H \ D. To see this, take any test function ϕ ∈ D, and compute the weak derivative via integration by parts (applied twice):     2 −x d2 d = −(−1)2 dx , ϕ = −e−x , ϕ. e−x , Δϕ = e−x , − dx 2ϕ 2e ˜ and so, Δ ˜ fails to be Thus, the domain of Δ is strictly larger than Δ, self-adjoint. One might try the approximation argument used to prove essential selfadjointness of the Laplacian on 2 (c) in Theorem 6.1: Let {vn } ⊆ D be a

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sequence with vn − e−x c → 0. Since Δ agrees with Δ when restricted to D, vn , Δvn c = vn , Δ vn c → e−x , Δ e−x c . However, there are two mistakes here. First, one does not have convergence unless the original sequence is chosen so as to approximate v in the nonsingular quadratic form u, vΔ := u, v + u, Δ v. Second, one cannot approximate e−x with respect to this nonsingular quadratic form by elements of D. In fact, e−x is orthogonal to D in this sense: ϕ, e−x Δ = ϕ, e−x c + ϕ, Δe−x c = ϕ, e−x c − ϕ, e−x c = 0. Alternatively, observe that by von Neumann’s theorem (Theorem B.11), a general element in the domain of Δ is v + ϕ∞ , where v ∈ dom Δ and ϕ∞ is in the defect space. B.2 Banded Matrices Definition B.15. Consider the matrix MS corresponding to an operator S in some orthonormal basis {bx }, so the entries of MS are given by MS (x, y) := bx , Sby .

(B.14)

We say MS is a banded matrix iff every row and column contains only finitely many nonzero entries. A fortiori, MS is uniformly banded if no row or column has more than N nonzero entries for some N ∈ N. With MS defined as in (B.14), it is immediate that MS is Hermitian whenever S is MS (x, y) = bx , Sby  = Sbx , by  = by , Sbx  = MS (y, x).

(B.15)

Banded matrices are of interest in the current context because the graph Laplacian is always a banded matrix when there is a uniform bound deg(x) ≤ N ; recall the form of MΔ given in (6.8). Since Δ = c − T, the transfer operator T is also banded. In general, the bandedness of an operator does not imply the operator is self-adjoint. In fact, see Example B.18 for a Hermitian operator on 2 which is not self-adjoint, despite having a uniformly banded matrix. However, this property does make it much easier to compute the adjoint. Lemma B.16. Let S be an unbounded Hermitian operator on H with a dense domain of definition D = dom S ⊆ H. Suppose that the matrix

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MS defined as in (B.14) is banded with respect to the orthonormal basis {bx }x∈X , and define vˆ(x) := bx , v. Then, v ∈ dom S  and S  v = w if and ˆ is only if vˆ, w ˆ ∈ 2 (X), and w(x)  MS (x, y)ˆ v (y). (B.16) w(x) ˆ = y∈X 

Thus, S is represented by the banded matrix MS . Proof. (⇒) To see the form of w ˆ in (B.16),    MS (x, y)ˆ v (y) = bx , Sby by , v = Sbx , by by , v y∈X

y∈X

Hermitian

y∈X

= Sbx , v

Parseval

= bx , w

S  v = w,

where the last equality is possible since v ∈ dom S  . It is the hypothesis of bandedness that guarantees all these sums are finite and hence meaningful. Conversely, first note that it is the hypothesis of bandedness which makes the sum in (B.16) finite, ensuring w ˆ is well defined. Suppose (B.16) holds and that vˆ, w ˆ ∈ 2 (X). To show v ∈ dom S  , we must find a constant K < ∞ for which |v, Su| ≤ Ku for every u ∈ D:  v, bx bx , Su Parseval v, Su = x∈X

=



v, bx Sbx , u

x∈X

=



vˆ(x)

x∈X

=

 



 y

MS (y, x)by , u

vˆ(x)MS (y, x)ˆ u(y)

by (B.15)

y∈X x∈X

 2       vˆ(x)MS (x, y) |ˆ u(y)|2 |v, Su|2 ≤    y∈X x∈X

by Schwarz,

y∈X

and see that we can take K = w ˆ 2.



Lemma B.17. Let A be an operator on 2 (Z) whose matrix MA is uniformly banded, with all bands having no more than β nonzero entries. Then, A ≤ β sup |axy |. x,y

(B.17)

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Proof. The Schwarz inequality gives  

1/2 A ≤ max supx |axy |2 , y

supy

 x

|axy |2

1/2 

.

However, uniform bandedness gives

1/2  2 sup |axy | ≤ β sup max |axy |, x

y

and similarly for the other term.

x

y



Example B.18 (Two operators which are each self-adjoint but whose product is not essentially self-adjoint). In Section 7.2, we discussed the Schwartz space S of functions of rapid decay and its dual S  , the space of tempered distributions; cf. (7.4). If we use the orthonormal basis for L2 (R) consisting of the Hermite polynomials, then the operators d ˜ : f (x) → xf (x) have the following matrix form: P˜ : f (x) → 1 dx f (x) and Q ⎤ ⎡ 0 1 √ ⎥ ⎢ ⎥ ⎢1 0 2 ⎥ ⎢ √ √ ⎥ ⎢ ⎥ ⎢ 2 0 3 ⎥ ⎢ ⎥ ⎢ √ .. .. ⎥ ⎢ . . 1⎢ 3 ⎥ (B.18) P = ⎢ ⎥, ⎥ 2⎢ √ . ⎥ ⎢ .. 0 n ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ √ . . ⎢ .⎥ n 0 ⎥ ⎢ ⎦ ⎣ .. .. . . ⎤ ⎡ 0 1 √ ⎥ ⎢ ⎥ ⎢ −1 2 0 ⎥ ⎢ √ √ ⎥ ⎢ ⎥ ⎢ − 2 0 3 ⎥ ⎢ ⎥ ⎢ √ .. .. ⎥ ⎢ 1 ⎢ . . − 3 ⎥ Q= ⎥. ⎢ ⎥ 2 ⎢ √ .. ⎥ ⎢ . 0 n ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ √ . . ⎢ .⎥ − n 0 ⎥ ⎢ ⎦ ⎣ .. .. . . (B.19)

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P and Q are Heisenberg’s matrices, and they satisfy the canonical commutation relation P Q − QP = 2 I. P and Q provide examples of Hermitian operators on 2 (Z), which are each essentially self-adjoint but for which T = QP Q is not essentially self-adjoint. In fact, T has defect indices 1,1 (cf. Definitions B.10 and B.8). These indices are found directly by solving the the defect equation T  f = QP Qf = ± f to obtain the C ∞ solutions ⎧ −1/x ⎨e x , x > 0, f+ (x) = ⎩0, x ≤ 0;

=⇒

x(xf ) = ±f

f− (x) =

⎧ 1/x ⎨ e , x < 0, x ⎩0,

x ≥ 0.

Thus, there is a one-dimensional space of solutions to each defect equation, and the defect indices are 1,1. To see that P and Q are actually self-adjoint, one can observe that P generates the unitary group f (x) → f (x + t) and Q generates the unitary group e tx . Therefore, P and Q are self-adjoint by Stone’s theorem; see Ref. [DS88]. B.3 The Friedrichs Extension For a large class of symmetric operators, there is a canonical choice for a self-adjoint extension, the Friedrichs extension. The importance of the Friedrichs extension of an unbounded Hermitian operator on a Hilbert space stems from its role in the classical theory (and mathematical physics in particular). For example, consider the Laplace operator Δ defined initially on C0∞ (Ω ), where Ω is a regular open subset of Rn for which Ω is compact. Thus, dom Δ consists of smooth functions vanishing at the boundary of Ω . To get a self-adjoint operator in the Hilbert space H = L2 (Ω ) (and an associated spectral resolution), one then assigns boundary conditions; each distinct choice yields a different self-adjoint extension. The two most famous choices of boundary conditions are the Neumann and the Dirichlet conditions. The Friedrichs extension procedure may be described abstractly, as the Hilbert completion of dom Δ with respect to a quadratic form defined in terms of Δ; cf. Refs. [Kat95, DS88]. Nonetheless, in the current example, the Friedrichs extension turns out to correspond to Dirichlet boundary conditions. Neumann conditions turn out to correspond to another self-adjoint extension, the Krein extension, which is discussed further in Section B.4.

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The study of the symmetric pairs introduced in Section 9.1 inspired a novel and concise construction of the Friedrichs extension AF of a semibounded operator A : dom(A) ⊆ H → H: • Use A to define a new and strictly finer topology on H so that Υ : D → H is a contractive (inclusion) embedding. • The key result, Theorem B.20, then gives AF = (ΥΥ )−1 . Consider an operator A : dom A ⊆ H → H whose domain is dense in the Hilbert space H, and assume that A satisfies ϕ, Aϕ ≥ ϕ2

for all ϕ ∈ dom A.

(B.20)

Define HA to be the Hilbert completion of dom A with respect to the norm induced by ψ, ϕA := ψ, Aϕ,

ψ, ϕ ∈ dom A,

(B.21)

and define the inclusion operator Υ : HA → H

by Υϕ = ϕ for ϕ ∈ HA .

(B.22)

Definition B.19. If A is symmetric and nonnegative densely defined operator, then the Friedrichs extension of A is the operator AF with dom AF := dom A ∩ Υ(HA )

and AF ϕ = Aϕ

for ϕ ∈ dom A. (B.23)

Here, as usual, .

dom A := {ψ ∈ H .. ∃C < ∞ with |ψ, Aϕ| ≤ Cϕ, ∀ϕ ∈ dom A}. (B.24) Recall that the symmetric extensions of a symmetric operator A are in bijective correspondence with the subspaces M satisfying dom(A) ⊆ M ⊆ dom(A ),

(B.25)

where the extension of A to M is AM = A |M ; observe that A is densely defined because B ⊆ A in this case (see Remark 9.8). Consequently, AF ϕ is unambiguously determined by (B.23) as a (graph) limit. Theorem B.20. The operator (ΥΥ )−1 is the Friedrichs extension of A. Proof. (1) We first show that the operator (ΥΥ )−1 is a self-adjoint extension of A. The inclusion operator Υ : HA → H is contractive because the estimate (B.20) implies Υf  = f  ≤ f A

for all f ∈ H.

(B.26)

Some Operator Theories

349

From general theory, we know that Υ  = Υ, so both Υ and Υ are contractive with respect to their respective norms, and hence, ΥΥ : H → H is also contractive. We deduce that ΥΥ is a contractive self-adjoint operator in H. Using the self-adjointness of ΥΥ and definitions (B.21) and (B.22), we have that the following holds for any ψ, ϕ ∈ dom A: ψ, JJ  Aϕ = JJ  ψ, Aϕ = J  ψ, Aϕ = J  ψ, ϕA = ψ, Jϕ = ψ, ϕ. (B.27) Since (B.27) holds on the dense subset dom A, we have ΥΥ Aϕ = ϕ for any ϕ ∈ dom A,

(B.28)



and it follows immediately that ΥΥ is invertible on ran A. A fortieri, the identity (B.28) shows that (ΥΥ )−1 is an extension of A. (2) Next, we must show that ran ΥΥ = dom A ∩ Υ(HA ). Let ψ ∈ ran ΥΥ . Then, ψ = ΥΥ ϕ for some ϕ ∈ HA , so y ∈ Υ(HA ) is immediate. To see that ψ ∈ dom A , note that for any ϕ ∈ dom A, part (1) of this proof gives |ψ, Aϕ| = |ΥΥ ξ, Aϕ| = |ξ, ΥΥ Aϕ| = |ξ, ϕ| ≤ ξϕ,

(B.29)

so the bound in (B.24) is satisfied with C = ξ. This shows that ran ΥΥ ⊆ dom A ∩ Υ(HA ). Now, for ψ ∈ dom A ∩ Υ(HA ), we prove the reverse containment. Since  ψ ∈ dom A, we have ψ = ΥΥ Aψ by part (1), so ψ ∈ ran ΥΥ . Definition B.21. If A is semibounded, then A + c + 1 is a symmetric and nonnegative densely defined operator satisfying (B.2), and the Friedrichs extension procedure may be applied to construct (A + c + 1)F as in Theorem B.20. The Friedrichs extension of A is thus defined as AF := (A + c + 1)F − c − 1.

(B.30)

Remark B.22. While several constructions of Friedrichs extension have already been given in the literature, we feel that our Theorem B.20 has the attractive features of both novelty and simplicity. For example, Kato’s approach [Kat95, Section 2.3] depends on first developing a rather elaborate theory of closable forms, while by contrast, our proof is simple and direct. Additionally, the tools developed here are precisely those which we need in our analysis of the network Laplacian as a semibounded Hermitian operator with dense domain in HE , the Hilbert space of functions of finite energy on a graph. For readers interested in earlier approaches to Friedrichs extension,

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we refer to, for example, the books by Dunford–Schwartz [DS88], Kato [Kat95], and Reed & Simon [RS72]. The following corollary shows that part of Kato’s results can be recovered from Theorems 9.6 and B.20. Corollary B.23. For a given Hilbert space H, there is a bijective correspondence between the collection of densely defined closed quadratic forms q, which satisfy q(ϕ, ϕ) ≥ ϕ2 , and the collection of self-adjoint operators A on H, which satisfy A ≥ 1. More precisely: 1. given A, let dom q := dom A1/2 and define q(ϕ, ψ) := A1/2 ϕ, A1/2 ϕ,

∀ϕ, ψ ∈ dom q.

2. given q, let Υ : dom q → H be the inclusion map Υϕ = ϕ and define A := (ΥΥ )−1 . Proof. The proof of (1) is straightforward; the nontrivial direction of the correspondence is (2), but this follows immediately from Theorem B.20.  B.4 A Generalization of the Krein Construction The following result is used to generalize some results of Ref. [Kre47]. It also offers a more streamlined proof; see Corollary 9.29 and Remark 9.30. Theorem B.24. Suppose that H1 and H2 are Hilbert spaces with D ⊆ H1 ∩ H2 and that D is dense in H1 (but not necessarily in H2 ). Define D ⊆ H2 by .

D := {h ∈ H2 .. ∃C ∈ (0, ∞)

for which |ϕ, hH2 | ≤ CϕH1 ,

∀ϕ ∈ D}. (B.31)

Then, D is dense in H2 if and only if there exists a self-adjoint operator Λ in H1 with D ⊆ dom Λ and ϕ, ΛϕH1 = ϕ22

for all ϕ ∈ D.

(B.32)

Proof. Consider the inclusion operator Υ : H1 → H2 given by dom Υ = D,

Υϕ = ϕ,

ϕ ∈ D.

By the definition of dom Υ , we know that h ∈ dom Υ iff there is a finite C = Ch such that |Υϕ, hH2 | ≤ CϕH1 . By (B.31), this means h ∈ dom Υ iff h ∈ D , i.e., dom Υ = D . Consequently, the assumption that (B.31) is dense in H2 is equivalent to Υ being densely defined and hence

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Some Operator Theories

also equivalent to Υ being closable. By a theorem by von Neumann, the operator Λ := Υ Υ is self-adjoint in H1 . Now, for ϕ ∈ D, we have ϕ, ΛϕH1 = ϕ, Υ ΥϕH1 = Υϕ, ΥϕH2 = Υϕ, ΥϕH2 = ϕ, ϕH2 = ϕ22 , which verifies (B.32). For the converse, we need to show that D is dense in H2 . To this end, we exhibit a set V ⊆ D ⊆ H2 , with V dense in H2 . Note that (B.32) implies the existence of a well-defined partial isometry K : H1 → H2 given by KΛ1/2 ϕ = ϕ ∀ϕ ∈ D and satisfying dom K = K  K = ran(Λ1/2 ). We extend K by defining K = 0 on (dom K)⊥ and then defining V := {ψ ∈ . H2 .. K  ψ ∈ dom(Λ1/2 )}. For ψ ∈ V, the definition of K and the Cauchy– Schwarz inequality now yield             |ψ, ϕ2 | = ψ, KΛ1/2 ϕ2  = Λ1/2 K  ψ, ϕ1  ≤ Λ1/2 K  ψ  ϕ1 1

for every ϕ ∈ D, whence V ⊆ D . Since Λ1/2 is densely defined, V is dense  in H2 . Example B.25. Example 9.5 illustrates the relationship that can exist between the adjoint of an operator between L2 spaces and the Radon– Nikodym derivative of their respective measures and how mutual orthogonality of these measures can cause a catastrophic failure of the adjoint. We return to this theme here and show how our main result, Theorem B.24, can be regarded as a noncommutative version of the Lebesgue–Radon–Nikodym ¯ decomposition; see also Refs. [Ota88, HSdSS07, Jør80]. Let (X, A) be a measure space on which two regular, positive, and σ-finite measures μ1 and μ2 are defined. Let Hi := L2 (X, μi ) for i = 1, 2, and let D := Cc (X). Then, the equivalent conditions in the conclusion of Theorem B.24 hold if and only if μ2  μ1 . In this case, Λ corresponds to 2 multiplication by the Radon–Nikodym derivative f := dμ dμ1 , and (B.32) can be written as   dμ2 ϕϕf dμ1 = |ϕ|2 dμ1 ϕ, Λϕ1 = dμ1 X X  = |ϕ|2 dμ2 = ϕ22 , ∀ϕ ∈ C(X). X

The connection is made precise in general by the spectral theorem in the following corollary of Theorem B.24.

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Operator Theory and Analysis of Infinite Networks

Corollary B.26. Assume the hypotheses of Theorem B.24. Then, for every ϕ ∈ D, there is a Borel measure μϕ on [0, ∞) such that  ∞ λ dμϕ (λ). (B.33) ϕ21 = μϕ ([0, ∞)) and ϕ12 = 0

Proof. Following the proof of Theorem B.24, we take Υ : D → H2 by Υϕ = ϕ and obtain the self-adjoint operator Λ = Υ Υ. The spectral theorem yields a spectral resolution  ∞ λEΛ (dλ), Λ= 0

where EΛ is the associated projection-valued measure. If we define μϕ via dμϕ := EΛ ( dλ)ϕ21 , then the conclusions in (B.33) follow from the spectral theorem.  For an additional application of Theorem B.24, see the example of the Laplace operator on the energy space given in Example 9.31. B.5 Remarks and References Representations for unbounded operators by infinite matrices were suggested early in quantum mechanics by Heisenberg. Since then, they have served as sources of other applications as well as the theory of operators in Hilbert space; see especially Ref. [vN55]. The reader may find Refs. [Rud91, KR97, DS88, RS75, Arv02, vN55] to be helpful. B.5.1 Historical context and motivation The importance of the Friedrichs extension of an unbounded Hermitian operator on a Hilbert space stems from its role in the classical theory. The network Laplace operator considered in this chapter is a discrete analogue of the better known Laplace operator associated with a manifold with boundary in harmonic analysis and PDE theory, see for example Refs. [Fri35, Fri39] and the endnotes of Ref. [DS88, Chapter XII]. In classical applications of mathematical physics, this Laplacian is an unbounded operator initially defined on a domain of smooth functions vanishing on the boundary. To get a self-adjoint operator in L2 (and an associated spectral resolution), one then assigns boundary conditions. Each distinct choice yields a different self-adjoint extension (realized in a suitable L2 -space). The two most famous such boundary conditions are the Neumann and the

Some Operator Theories

353

Dirichlet conditions. In the framework of unbounded Hermitian operators in Hilbert space, the Dirichlet boundary conditions correspond to a semibounded self-adjoint extension of Δ called the Friedrichs extension. For boundary value problems on manifolds with boundaries, the Hermitian property comes from a choice of a minimal domain for the given elliptic operator T under consideration, and the semiboundedness then amounts to an a priori coercivity estimate placed as a condition on T . Today, the notion of a Friedrichs extension is typically understood in a more general operator-theoretic context concerning semibounded Hermitian operators with dense domain in Hilbert space; see, for example, Ref. [DS88, p. 1240]. In its abstract Hilbert space formulation, it throws light on a number of classical questions in spectral theory and in quantum mechanics, for example in the study of Sturm–Liouville operators and Schr¨ odinger operators, e.g., Ref. [Kat95]. If a Hermitian operator is known to be semibounded, we know by a theorem by von Neumann that it will automatically have self-adjoint extensions. The selection of appropriate boundary conditions for a given boundary value problem corresponds to choosing a particular self-adjoint extension of the partial differential operator in question. In general, some self-adjoint extensions of a fixed minimal operator T may be semibounded and others not. The Friedrichs extension is both self-adjoint and semibounded and with the same lower bound as the initial operator T (on its minimal domain). We are here concerned with a different context: analysis and spectral theory of problems in discrete context, wherein Δ is the infinitesimal generator of the random walk on (G, c). In this regard, we are motivated by a number of recent papers, some of which are cited above. A desire to quantify the asymptotic behavior of such reversible Markov chains leads to the need for precise and useful notions of the boundaries of infinite graphs. Different conductance functions lead to different Laplacians Δ and also to different boundaries. In the energy Hilbert space HE , this operator Δ will then have a natural dense domain turning it into a semibounded Hermitian operator, and as a result, Friedrichs’ theory applies. As in classical Riemannian geometry, one expects an intimate relationship between metrics and associated Laplace operators. This is comparable to the use of the classical Laplace operator in the study of manifolds with boundary or even just boundaries of open domains in Euclidean space; see, for example, Refs. [Fug91, Fug86].

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Appendix C

Navigation Aids for Operators and Spaces C.1 A Road Map of Spaces Each arrow represents an embedding. span{δx }

/ 2 (c)

/ Fin

span{vx }

/ SG

 / HE O Harm

/

/ SG O  SG Fin

C.2 A Summary of the Operators on Various Hilbert Spaces c unbdd c bdd c=1 Δ on HE unbdd, Herm, poss. defect unbdd, Herm, ess. s.-a. unbdd, Herm, ess. s.-a. bdd, s.-a. Δ on 2 (1) unbdd, Herm, ess. s.-a. unbdd, non-Herm non-Herm bdd, s.-a. Δ on 2 (c) T on HE unbdd, Herm, poss. defect unbdd, Herm, ess. s.-a. unbdd, Herm, ess. s.-a. bdd, s.-a. T on 2 (1) unbdd, Herm, ess. s.-a. unbdd, non-Herm non-Herm bdd, s.-a. T on 2 (c)

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Appendix D

A Guide to the Bibliography

We have endeavored to offer a self-contained presentation, but readers looking to brush up on preliminaries may find a number of terms and ideas used here in Refs. [AF09, DS84, LPW08, YZ05]. These sources have an primarily a probabilistic point of view, but resistance networks (weighted graphs) are essentially equivalent to reversible Markov chains; see the quote by Peres at the start of Chapter 1. Consequently, these books cover the fundamentals on electrical resistance networks as well as related topics, such as reversible Markov chains; hitting and convergence time, and flow rate, parameters; special graphs and trees; symmetric graphs and chains; L2 techniques for bounding mixing times; randomized algorithms; continuous state, infinite state, and random environment; interacting particles; and Markov chain Monte Carlo. The individual chapters inside our book involve a list of diverse areas. In each case, they are discussed and cited. Naturally, for each of the area, pure or applied, the existing literature includes many prior contributions from a number of authors. In the following, we offer a few categories of these interconnected areas; and for each, we list some authors who have contributed. We apologize in advance for omissions. Readers who wish to follow up with more details will be able to locate the relevant papers in our combined references as follows: 1. Analysis on weighted graphs, including resistance networks: 1.1 Metrics: R. Alexander, Y. Bartal, D. Burago, I. Benjamini, Y. Colin de Verdi`ere, G. Sved, C. M. Cramlet, J. Dodziuk, P. G. Doyle, J. L. Snell, J. Kigami, R. Lyons, and M. A. Rieffel.

357

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Operator Theory and Analysis of Infinite Networks

1.2

1.3

1.4

1.5

1.6

1.7

Graph theory: G. Polya, B. Bollobas, Fan Chung, R. Diestel, P. G. Doyle, J. L. Snell, C. Thomassen, G. V. Epifanov, M. L. Lapidus, L. Hartman, I. Raeburn. Potential theory: R. Aharoni, C. Constantinescu, J. Dodziuk, X. W. C. Faber, J. Kigami, V. S. Guliyev, R. Lyons, W. Woess, and P. M. Soardi. Dirichlet forms: N. Bourleau, F. Hirsch, D. I. Cartwright, W. Woess, J. Dodziuk, M. Fukushima, Y. Oshima, V. A. Kaimanovich, Z. Kuramochi, M. Yamasaki, M. Rockner, and B. Schmuland. Boundaries (Martin and others) and discrete analogues of Greens– Gauss–Stokes theorems: H. Aikawa, S. Brofferio, W. Woess, N. Budarina, C. Z. Chen, R. Diestel, Y. Peres, J. L. Doob, I. Ignatiouk-Robert, Z. Kuramochi, H. L. Royden, S. A. Sawyer, M. Silhavy, and P. M. Soardi. Trees: P. Cartier, R. Permantle, Y. Colin de Verdier, M. D. Esposti, S. Isola, R. Diestel, R. Diestel, R. Lyons, Y. Peres, R. Froese, W. Woess, and P. M. Soardi. Fractals, limits, and renormalization: A. F. Beardon, M. Barnsley, J. Hutchinson, O. Bratteli, D. Dutkay, L. Baggett, J. Packer, A. Connes, R. L. Devaney, R. Duits, J. de Graaf, J. Kigami, R. S. Strichartz, D. Guido, T. Isola, M. L. Lapidus, M. Golubitsky, B. M. Hambly, T. Kumagai, P. E. T. Jorgensen, E. Pearse, and S. Pedersen.

2. Tools from the theory of operators in Hilbert space: 2.1 Reproducing kernel Hilbert space: D. Alpay, E. Nelson, N. Aronszajn, D. Larson, and T. Kailath. 2.2 Unbounded Hermitian operators and their extensions: J. von Neumann, M. Krein, K. O. Friedrichs, E. Nelson, P. D. Lax, R. S. Phillips, N. Dunford, J. T. Schwartz, M. Reed, B. Simon, and P. E. T. Jorgensen. 2.3 Shorted operators: W. N. Anderson, G. E. Trapp, C. A. Butler, T. D. Morley, and V. Metz. 2.4 Spectral theory: W. B. Arveson, Y. Colin de Verdier, J. Dodziuk, J. Kigami, R. S. Strichartz, D. Guido, T. Isola, M. L. Lapidus, D. Sarason, and P. M. Soardi.

A Guide to the Bibliography

359

2.5 Transfer operators: V. Baladi, Fan Chung, D. I. Cartwright, W. Woess, L. Saloff-Coste, and D. Dutkay. 2.6 Quadratic forms and perturbation theory: C. Z. Chen, M. Fukushima, Y. Oshima, J. Kigami, and T. Kato. 2.7 Laplacians: F. Chung, R. M. Richardson, J. Dodziuk, J. L. Doob, X. W. C. Faber, W. Woess, L. G. Rogers, A. Teplyaev, P. M. Soardi, and A. Weber. 2.8 Fock space methods: T. Hida, W. B. Arveson, L. Gross, and A. Guichardet. 3. Stochastic integrals: 3.1 Gelfand triples: I. M. Gelfand, N. Wiener, R. A. Minlos, L. Gross, A. Guichardet, and T. Hida. 3.2 Schoenberg–von Neumann embeddings: I. J. Schoenberg, J. von Neumann, B. O. Koopman, J. L. Doob, L. Gross, K. R. Parthasarathy, and K. Schmidt. 4. Probabilistic methods: 4.1 Random walk models and Markov processes: D. Aldous, J. A. Fill, G. Polya, F. Spitzer, D. W. Stroock, C. A. Nash-Williams, A. I. Aptekarev, Fan Chung, Y. Peres, J. L. Doob, R. Lyons, T. Lyons, Y. Peres, A. Telcs, R. Pemantle, I. Ignatiouk-Robert, B. Morris, M. A. Picardello, L. Saloff-Coste, G. G. Yin, and Q. Zhang. 4.2 Potential theory: M. Brelot, R. Lyons, W. Woess, P. Malliavin, and A. N. Kolmogorov. 4.3 Martingale techniques: J. L. Doob, E. Nelson, and W. Woess. 4.4 Diffusion models: R. R. Coifman, I. G. Kevrekidis, M. Maggioni, R. Lyons, Y. Peres, and W. Woess. 5. Tools from other areas of mathematics: 5.1 C ∗ -algebras: E. Andruchow, W. B. Arveson, O. Bratteli, D. W. Robinson, R. V. Kadison, J. A. Ball, I. Cho, A. Connes, and P. S. Muhly. 5.2 Harmonic analysis: C. Berg, P. Cartier, J. Dodziuk, J. L. Doob, R. S. Strichartz, F. P. Greenleaf, G. Olaffson, W. Woess, A. M. Kytmanov, and P. M. Soardi. 5.3 Functional analysis: L. Schwartz, M. A. Al-Gwaiz, H. Aikawa, D. L. Cohn, S. G. Krantz, R. R. Phelps, and W. Rudin.

360

Operator Theory and Analysis of Infinite Networks

5.4 Numerical analysis: K. Atkinson, S. C. Brenner, and H. H. Goldstine. 6. Applications: 6.1 Classical statistical models: G. Polya, T. M. Liggett, Y. Peres, K. Handa, and L. Hartman. 6.2 Quantum statistical mechanics: Equilibrium states and long-range order: R. T. Powers, H. Schwetlick, and E. Nelson. 6.3 Engineering: R. Bott, R. J. Duffin, P. G. Doyle, J. L. Snell, and T. Kailath. 6.4 Physics (other): M. Baake, R. V. Moody, W. Dur, H. J. Briegel, A. Dhar, E. Formenti, O. Guhne, J. Kellendonk, and D. Ruelle.

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Index

A

C

algebra Banach algebra, 187 bounded functions, 187 multiplier C  -algebra, 178 von Neumann, 213 Aronszajn’s theorem, 335

closed operator, 35, 103, 196, 340 closable operator, 195, 204, 211, 216–217, 340–341 cocycle, 238 commutator, 83, 214 compactification, 190–191, 316 compatibility problem, 14, 227, 230, 232 conductance function, 2 current, 8 path, 10 current path, 19, 243 cycle, 221 cycle condition, 15 cyclic vector, 214, 315

B Banach algebra, 187 Banded matrix, 110, 132, 291, 344 uniformly, 110, 138 boundary of a subgraph, 36 representation of harmonic functions, 38, 152 sum, 37, 40, 152 boundary conditions on Δ, 65 boundary form, 105 boundary representation, 38, 152, 165

D defect space, 103–106, 143, 200–201, 298–300 dipole, 10–14, 28, 30, 78 discrete Gauss–Green formula, 37, 46, 104 for deg(x) = ∞, 49 dissipation, 9 389

390

Operator Theory and Analysis of Infinite Networks

divergence, 242 divergence operator, 225 drop operator, 220, 227 E effective resistance, 8, 18, 28, 78 free resistance, 58–63, 82 trace resistance, 68–77 wired resistance, 63–66, 82 energy form, 5–7 energy kernel, 28–35, 89, 174–175, 289 and frames, 118 and probability, 80 extended, 161 for different conductances, 250–254 on Z, 294–296 energy space, 25, 30, 45, 65 compared to 2 (1), 139 evaluation functional, 27, 60 exhaustion, 26, 36, 58 F finitely supported function, 29, 45 Fock space, 163, 166, 208 forward Laplacian, 244 forward-harmonic, 244 frames, 117 dual, 118, 122 Parseval, 224 Friedrichs extension, 258, 340, 347

G Gauss–Green formula, 37, 46, 104 for deg(x) = ∞, 49 Gel’fand space, 188 Gel’fand triple, 155–164 geodesic distance, 81 GNS construction, 315 grounded energy space, 42–45, 92, 173 H harmonic function, 4, 7, 29, 39, 45 and 2 , 49 and boundary form, 105 and monopoles, 40 and self-adjointness, 100, 103, 105 boundary representation, 38, 152, 239 bounded, 239 forward-harmonic, 242–247 Hilbert space and the bounded operators B(H), 214 Fock space, 163, 166, 208 grounded energy space, 43, 92, 173 of random variables, 206 projections to subspaces, 222 reproducing kernel Hilbert space, 27, 335 rigged, 155 Schoenberg–von Neumann construction, 89, 95, 335–338

391

Index

the dissipation space HD , 219–233 the energy space HE , 23–52

N network resistance network, 3

K

O

Kirchhoff’s law, 9, 16 KMS state, 316–322 Krein extension, 205, 340, 350

one-point compactification, 190 orthonormal basis, 108, 111, 344 of V(F ), 125 of VF , 125 of Harm, 102 of HD , 223 of HE , 25, 97, 186

L ladder model, 269–273 Laplace operator, 4 matrix Laplacian, 112–117 on 2 (1), 109–117 on 2 (c), 141–150 on M, 33 on V, 48, 97–106, 118 probabilistic, 144 self-adjoint extension of, 101, 103, 205, 353 self-adjointness of, 109, 113, 144 spectral representation, 116 long-range order, 320, 323 M Markov property, 6 modular operator, 216 monopole, 10, 32, 34, 39–42, 236 and harmonic functions, 40 multiplication operators, 174, 211 boundedness, 177, 181, 184 C  -algebra of bounded, 176 self-adjointness, 174 rank-1, 179

P partial isometry d and d , 223 path current, 10, 19 to ∞, 45 powers bound, 98, 135, 139, 149, 271, 340 projection to V(F ), 124 to Cyc, 222 to Fin, 31, 222, 230 to Harm, 31, 222 to Kir, 222 to N bd, 222, 230 to Hilbert subspaces, 222 projection-valued measure, 41, 116, 159, 258, 342, 352 R random walk, 71, 78, 236 reproducing kernel, 30, 335 for ran ΔM , 35 for Fin, 31

392

Operator Theory and Analysis of Infinite Networks

for Harm, 31 for HE , 28 resistance metric, 28, 53–87 for finite networks, 54 for infinite networks, 57 free resistance, 58–63, 82, 91, 93, 122 on the integer lattice, 282 Powers’ inequality, 268 trace resistance, 68–77 wired resistance, 63–66, 82, 91, 93 rigged Hilbert space, 155 Royden decomposition, 30 S Schoenberg–von Neumann theorem, 90 Schur complement, 69, 74, 77–78 self-adjoint, 97, 101, 257, 339, 346 essentially self-adjoint, 105, 109, 113, 131, 144, 200, 204, 297 self-adjoint extension, 103, 132, 158, 193, 200–205, 340–341, 347 semibounded, 33, 48, 101, 110, 114, 339–342 semibounded operator, 33, 48, 101, 110, 114, 339, 353 separating vector, 215 shorted operator, 77

spectral gap, 122–123, 134, 140, 149 spectral reciprocity, 125–130 theorem, 128 spectral representation, 116, 132, 140, 159, 318 state, 315 subnetwork free, 59 full, 58 trace, 69 wired, 63 symmetric pair, 193–218 and self-adjoint extensions, 201 and self-adjointness, 200 example, 195, 197, 206, 213 maximal, 197–198, 200, 204, 217 with Δ, 204 T trace, 74, 213–217, 323–324 resistance, 68–77 subnetwork, 69 transfer operator, 4, 70, 264, 344 (non)compactness, 290 forward-harmonic, 243–247 on 2 (1), 107, 135–139 transient, 17, 34, 39, 42, 78, 236 U unitalization, 190