Log-Gases and Random Matrices (LMS-34) [Course Book ed.] 9781400835416

Citation preview

Log-Gases and Random Matrices

London Mathematical Society Monographs Editors: Martin Bridson, Terry Lyons, and Peter Sarnak Editorial Advisers: Mikhail Gromov, Jean-Francois Le Gall, and Richard Taylor The London Mathematical Society Monographs Series was established in 1968. Since that time it has published outstanding volumes that have been critically acclaimed by the mathematics community. The aim of this series is to publish authoritative accounts of current research in mathematics and high-quality expository works bringing the reader to the frontiers of research. Of particular interest are topics that have developed rapidly in the last ten years but that have reached a certain level of maturity. Clarity of exposition is important and each book should be accessible to those commencing work in its field. The original series was founded in 1968 by the Society and Academic Press; the second series was launched by the Society and Oxford University Press in 1983. In January 2003, the Society and Princeton University Press united to expand the number of books published annually and to make the series more international in scope.

LMS-34. Log-Gases and Random Matrices, by P. J. Forrester LMS-33. Prime-Detecting Sieves, by Glyn Harman LMS-32. The Geometry and Topology of Coxeter Groups, by Michael W. Davis LMS-31. Analysis of Heat Equations on Domains, by El Maati Ouhabaz

Log-Gases and Random Matrices

P.J. Forrester

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

Copyright © 2010 by Princeton University Press Requests for permission to reproduce material from this work should be sent to Permissions, Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved

Library of Congress Cataloging-in-Publication Data Forrester, Peter (Peter John) Log-gases and random matrices / P.J. Forrester. p. cm. -- (London Mathematical Society monographs) ISBN 978-0-691-12829-0 (hardcover : alk. paper) 1. Random matrices. 2. Jacobi polynomials. 3. Integral theorems. I. Title. QA188.F656 2010 519.2--dc22 2009053314

British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed Printed on acid-free paper. f Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Preface

Often it is asked what makes a mathematical topic interesting. Some qualities which come to mind are usefulness, beauty, depth and fertility. Usefulness is usually measured by the utility of the topic outside mathematics. Beauty is an alluring quality of much of mathematics, with the caveat that it is often something only a trained eye can see. Depth comes via the linking together of multiple ideas and topics, often seemingly removed from the original context. And fertility means that with a reasonable effort there are new results, some useful, some with beauty, and a few maybe with depth, still awaiting to be found. More than fifteen years ago I embarked on a project to write in monograph form a development of the theory of solvable log-gas systems in statistical mechanics. As a researcher in the field, I had personally witnessed and experienced some of the interesting qualities of this topic, and I was keen that these be recorded in a form which could serve as a reference for researchers in related fields. Little did I realize that in the ensuing years these related fields would be the subject of intense research activity, requiring a revision of both the focus of the book, and my own research directions, to properly reflect these developments. Although my focus thus evolved away from the statistical mechanics of log-gas systems, this subject still proved to be a unifying theme in the presentation of the subject matter. And as a further give away as to my own research origins, there is a fairly strong flavor of the language of classical equilibrium statistical mechanics throughout, although a similar background of the reader can hardly be expected. More likely the motivation of the reader will come from the topics of random matrices, Painlev´e systems, stochastic growth processes, or Jack polynomials. These are some of the the related fields referred to above, which have been the subject of much recent activity, and which promise to remain interesting topics into the future. Of these it is random matrices which appears along side log-gases as the unifying theme of the book. This marriage of topics has a fine historical pedigree, with the log-gas picture of eigenvalues of random matrices being used to great advantage in the pioneering work of Dyson [147]. While providing a directed logical framework, a development of the intersection between log-gases and random matrices necessarily excludes substantial portions of each of the topics taken separately. However, the latter is necessary in order to achieve a mostly self-contained presentation. Seeking the common intersection of two topics can then be seen as a way of achieving this in a fairly democratic manner. In addition there is intersection with a third topic at work, keeping a further bound on the content, but also being responsible for much of the richness of the mathematics. This third topic is integrable systems. In general the exact calculation of correlations and probability distributions for interacting statistical mechanical systems is an intractable problem; however, underlying integrable structures make log-gases and random matrices an exception. The development of this topic leads to the study of determinantal and Pfaffian processes and the corresponding orthogonal polynomials, as well as Painlev´e systems and Jack polynomials. The quality of usefulness marked the beginning of the study of random matrices and log-gases in mathematical physics. As already mentioned, log-gases were introduced as a tool by Dyson to study random matrices, or as expressed in [201], to liberate the mathematics where none yet exists. Random matrices themselves were introduced by Wigner as a model for the statistical properties of the highly excited energy levels of heavy nuclei. Many of the early works on this theme (up to 1965) are conveniently collected together in the work of Porter [447], along with an introductory review. Long before their occurrence in physics, random matrices appeared in mathematics, especially in relation to the Haar measure on classical groups. Perhaps the first work of this type is due to Hurwitz, who computed the

vi

PREFACE

volume form of a general unitary matrix parametrized in terms of Euler angles [301]. The book of Weyl [540] contains the Haar volume form written in terms of eigenvalues and eigenvectors for the classical groups, and the book of Hua [300] inter-relates these forms to similar measures relating to spaces of Hermitian matrices. In mathematical statistics Wishart [547] gave the volume form of a rectangular matrix X in terms of the volume of the corresponding positive definite matrix XT X. Two other early mathematical works of lasting importance to the field are those of Dixon [137] and Selberg [483], both of which relate to multidimensional integrals with integrands which can be interpreted as probability measures associated with random matrices. The historical development of random matrices is well documented. Two recent informative accounts are [226], [75]. However, as already stated, the present work addresses only the intersection of the topics of log-gases, random matrices and integrable systems, and so a more extensive historical introduction beyond that already given does not serve as as an informative introduction to the content. Instead it is perhaps worth isolating some of our major topics, giving them some context and providing commentary on how they are to be developed. Jacobians All the works referenced above in relation to how random matrices appear in mathematics relate to Jacobians. To gain insight into the prevalance of Jacobians throughout random matrix theory, consider, for example, the problem of studying the eigenvalues of an N × N real symmetric random matrix, in the situation that the joint distribution on the space of the independent elements is given. The dimension of this space is N (N + 1)/2. The eigenvalue/eigenvector decomposition provides a change of variables from the independent elements of the matrix to its N eigenvalues and N (N − 1)/2 variables associated with its eigenvectors. A strategy then to study the eigenvalues is to perform this change of variables, and an essential ingredient for this task is the computation of the corresponding Jacobian. In the case of real symmetric matrices, complex Hermitian and quaternion real Hermitian matrices, these Jacobians are computed in Chapter 1. Chapter 1 also contains the computation of the Jacobian for a change of variables from the independent elements of an N ×N real symmetric tridiagonal matrix to its eigenvalues and a further N − 1 independent variables relating to its eigenvectors, and a Jacobian relating to the Householder transformation. In Chapter 2 Jacobians are computed in relation to spaces of unitary matrices, including orthogonal and symplectic unitary matrices, which have dimensions O(N 2 ). Jacobians are also computed for the change of variables from the elements to the eigenvalues and variables relating to the eigenvectors for certain unitary and real orthogonal Hessenberg matrices. In these latter circumstances the underlying spaces are of dimension O(N ). The singular value decomposition of rectangular matrices (or equivalently certain decompositions of positive definite matrices), the block decomposition of unitary matrices, and positive definite matrices formed from bidiagonal matrices are some of the settings which give rise to calculations of Jacobians undertaken in Chapter 3. Jacobians of a different sought appear in Chapter 4. Here rational functions with random coefficients in their partial fraction expansion are encountered, and we seek to change variables from a description in terms of these coefficients to one in terms of the zeros. For this purpose use is make of tools already known from the computation of Jacobians in Chapters 1–3, in particular the calculus of wedge products, and also the classical Vandermonde and Cauchy determinants. In Chapter 11 Jacobians are encountered in the change of variables of differential operators given in terms of the elements of parameter-dependent random matrices, to the differential operators given in terms of corresponding eigenvalues and variables relating to the eigenvectors. Finally, in Chapter 15, a task similar to that addressed in Chapter 1 is undertaken, namely the change of variables from the description of N × N real, complex, or quaternion real matrices in terms of the independent elements, to one in terms of the eigenvalues (which are typically complex) and an appropriate number of other variables. Also computed are some Jacobians relating to the change of variables of a random polynomial from its coefficients to its zeros. Determinantal point processes and orthogonal polynomials of one variable A determinantal point process is a statistical system of many particles (points) in which the k-point corre-

PREFACE

vii

lation function is a k × k determinant for each k. The study of eigenvalues of random matrices with complex entries, and also of log-gas systems at the special coupling β = 2 (in the cases considered in this work, the former are mostly special cases of the latter, due to our subject matter being typically restricted to the intersection of the two fields) gives rise to determinantal point processes. Furthermore, the corresponding determinants are determined by just one quantity, referred to as the correlation kernel. To exhibit this fact an essential role is played by orthogonal polynomials. It turns out that in the cases of interest it is the classical orthogonal polynomials which are required. Because full information on the asymptotic properties of these polynomials is known in the existing literature, it is possible to proceed and calculate scaling limits. A generalization of a determinant point process is a Pfaffian point process, in which the k-point correlation function is a 2k × 2k Pfaffian (or equivalently a k × k quaternion determinant) for each k. The eigenvalues of matrix ensembles studied in Chapters 1–3 in which the matrices are diagonalized by real orthogonal or symplectic unitary matrices are examples of Pfaffian point processes. These eigenvalues can be interpreted in terms of log-gas systems at the particular coupling β = 1 and β = 4 respectively. In the theory of Pfaffian processes skew orthogonal polynomials play a role analogous to that played by orthogonal polynomials in the theory of determinantal processes. For the particular skew inner products encountered from the random matrix problems of Chapters 1–3, the required skew orthogonal polynomials can be expressed in terms of classical orthogonal polynomials, and moreover the elements of the Pfaffian are determined by a single 2 × 2 block, the elements of which can be expressed in a summed form suitable for asymptotic analysis. In Chapter 15 non-Hermitian Gaussian random matrices are studied, with real, complex, and real quaternion entries. The eigenvalues in the complex case form a determinantal point process, while in the other two cases a Pfaffian point process results. The Selberg integral, Jack polynomials and generalized hypergeometric functions Familiar in the theory of the Gauss hypergeometric function is the Euler integral, which has the feature that it can be evaluated in terms of gamma functions. The Selberg integral can be considered as an N -dimensional generalization of the Euler integral. In a random matrix context, it appears as the normalization of various ensembles considered in Chapters 1–3. In a log-gas context, it gives the partition function for general β > 0. When written in a trigonometric form, extra parameters can be interpreted as providing the full distribution of certain linear statistics in the circular β-ensemble. In the case β = 2, and in the limit N → ∞, this ties in with the Fisher-Hartwig asymptotic formula from the theory of Toeplitz determinants, covered in Chapter 14. One of the structures underlying the Selberg integral is a further multidimensional integral referred to as the Dixon-Anderson integral. Like the Selberg integral, it can arrived at by the consideration of a problem in random matrix theory, and it too can be evaluated in terms of gamma functions. The many free parameters in the Dixon-Anderson integral allow for an interpretation giving an inter-relation between the distribution of every second eigenvalue in classical matrix ensembles at β = 1, and the joint distribution of the eigenvalues for a related classical matrix ensemble at β = 4. The integrand of the Selberg integral and its various limits is, up to normalization, the eigenvalue probability density function of the various classical β-ensembles given in Chapters 1–3. Theory linking the Selberg integral with the Dixon-Anderson integral can also be used to provide stochastic three-term recurrences (in the degree N ) for the corresponding characteristic polynomials. The integrand of the Euler integral is the weight function for the classical Jacobi polynomials (when defined on the interval [0, 1]). Likewise, the integrand of the Selberg integral, and its various limiting forms, can be used to define inner products which permit complete sets of orthogonal polynomials with special properties. The most fundamental are the Jack polynomials, which relate to the integrand of the Selberg integral in trigonometric form, specialized to correspond to the eigenvalue probability density function for the circular β-ensemble. Using the Jack polynomials as a basis, generalized classical Hermite, Laguerre and Jacobi polynomials, which are multivariable counterparts of the one-variable classical orthogonal polynomials of the same name, can be studied. Another viewpoint of the Jack polynomials is as the polynomial (in complex exponential variables) portion of the eigenfunctions for the Fokker-Planck operator of the Dyson Brownian motion model of the log-gas on a circle. This topic is developed in Chapter 11. A crucial feature is an alge-

viii

PREFACE

braic theory of the Fokker-Planck operator, in which it is decomposed into fundamental commuting operators relating to the degenerate Hecke algebra of type A, and involving exchange operators. The presentation of the theory of Jack polynomials given in Chapter 12 begins from a study of the simultaneous nonsymmetric polynomial eigenfunctions of these commuting operators. The underlying degenerate Hecke algebra allows these polynomials to be constructed inductively using two fundamental operations (transposition and raising), and these operations allow for the explicit evaluation of associated scalar quantities such as various normalizations. The operations of symmetrization and antisymmetrization also play an important role. It is well known that the Euler integral can be extended to provide the solutions of the Gauss hypergeometric differential equation. Likewise, weighting the Selberg integrand by an appropriate factor gives rise to multidimensional integrals which relate to multidimensional hypergeometric functions based on Jack polynomials. These are studied in Chapter 13. With the parameters specialized, these integrals can be interpreted as correlations for log-gas systems. Duality formulas, in which multidimensional integrals of this type are expressed as other multidimensional integrals, this time of dimension independent of N , provide the basis for the asymptotic analysis of the corresponding correlations for all even β at least. Furthermore, Jack polynomial theory can be used to compute the bulk dynamical two-point density-density correlation for the Dyson Brownian motion model perturbed from its equilibrium state for all values of rational values of β. Painlev´e transcendents The Painlev´e differential equations are a distinguished family of second order nonlinear equations. In applied mathematics they are perhaps best known for their role in soliton theory, and thus the study of integrable partial differential equations. Certain solutions of the Painlev´e differential equations — the Painlev´e transcendents — appear in the calculation of gap probabilities for classical random matrix systems corresponding to log-gas systems with β = 1, 2 and 4 (although the latter two are restricted to those instances in which their is an inter-relation with a β = 2 log-gas system; one way the latter comes about is by superimposing two β = 1 ensembles, and integrating over every second eigenvalue, while in the bulk the β = 1 gap probability is transformed by making use of an evenness symmetry). The viewpoint taken in Chapter 8 on these calculations is an algebraic theory of Painlev´e systems based on a Hamiltonian formulation, due mainly to Okamoto, which has the feature of using the Toda lattice equation to inductively construct determinant solutions from a seed solution (the latter relating to an underlying linear second order equation). These determinants can be identified with the gap probabilities of certain log-gas systems at β = 2. Moreover (formal) scaling of the differential operators gives analogous characterization of the gap probabilities in various scaling limits. As a consequence of these characterizations, high precision calculation of the gap probabilities can be undertaken. In Chapter 9 additional viewpoints on these results are considered. One is the study of function theoretic properties of the gap probabilities expressed as Fredholm determinants. Indeed, starting with the Fredholm determinant form seems necessary to account for the scaled limit rigorously. Instead of using function theoretic properties, this starting point can also be developed from a Riemann-Hilbert viewpoint, which in turn is closely related to studying isomonodromic deformations of linear second order differential equations. The Fredholm determinant evaluations allow the high precision calculations of the gap probabilities initiated from the Painlev´e evaluations to be extended. Macroscopic electrostatics and asymptotic formulas Averaging a linear statistic against the eigenvalue spectrum of a random matrix gives a mean value proportional to N (the number of eigenvalues), but a variance of order unity. In applications, this effect shows itself in the study of the statistical properties of the conductance of a quantum wire, noted in Chapter 3. It can be anticipated, and a precise formula for the variance formulated, by hypothesizing that for large length scales the log-gas behaves like a macroscopic conductor, and then using linear response arguments based on the predictions of two-dimensional electrostatics. Moreover, this hypothesis leads to the prediction that the full distribution of the linear statistic will be a Gaussian. For some log-gas systems this has been rigorously established, one of these being that corresponding to the Szeg¨o asymptotic formula from the theory of Toeplitz determinants.

PREFACE

ix

One of the most basic predictions from macroscopic electrostatics is the leading form of the density profile for random matrix ensembles. It can be used too to predict the O(1/N ) correction to this form. Another application is to gap probabilities, for which the large gap size asymptotics can be predicted to the leading two orders, and at the soft edge the large deviation forms of the left and right tails can be computed. When available, the exact results agree with these predictions. Non-intersecting paths and models in statistical mechanics The generating function for non-intersecting paths on an acyclic directed graph is well known to be given in the form of a determinant. In a number of cases of interest this determinant can be evaluated, revealing that the joint probability density function for the paths possessing prescribed coordinates is of a log-gas form, with β = 1 or 2. Non-intersecting paths underly a number of statistical mechanical models, in particular the polynuclear growth model and the Hammersely model of directed percolation. To understand how this comes about requires a study of the Robinson-Schensted-Knuth correspondence from bijective combinatorics. This in turn leads naturally to the study of Schur polynomials, which are in fact examples of Jack polynomials. It is shown that fluctuations of the primary observable quantities in the polynuclear growth model and the Hammersely model of directed percolation (the height of the profile and length of the path, respectively) can be expressed as random matrix averages over the unitary group, and that these matrix averages can be rigorously analyzed in the appropriate scaling limits. Various symmetrizations of the Hammersely model of directed percolation are particularly natural. Examples of these relate to averages over random matrices from the orthogonal and symplectic groups. Transformations of these averages to relate to gap probabilities in Laguerre random matrix ensembles with β = 1 and 4 allows the the rigorous analysis of the scaling limits. Applications of random matrix theory All the random matrix ensembles introduced in Chapters 1–3, for β = 1, 2 and 4 at least, can be associated with problems in quantum physics. The work of Wigner and Dyson relates the Gaussian ensembles to quantum Hamiltonians; the circular ensembles relate to scattering from a disordered cavity; Verbaarschot has given an interpretation of chiral random matrices in terms of the Dirac equation as it relates to QCD; and quantum transport problems lead to the Jacobi ensemble. For general values of β > 0 the eigenvalue p.d.f.’s of the β-ensembles appear as the ground state wave function of a class of quantum many-body problems with the 1/r2 pair potential. The eigenvalue p.d.f. for the complex random matrices of Chapter 15 has the interpretation as the absolute value squared of the ground state wave function for spinless fermions confined to a plane in the presence of a perpendicular magnetic field. An application of the GOE to the statistics of high-dimensional random energy landscapes is given in Chapter 1. In Chapter 3 features of Wishart matrices and the Jacobi ensemble relating to multivariate statistics are discussed, as is the application of Wishart matrices to wireless communication, numerical analysis and quantum entanglement (the latter requires a further constraint on the trace). In Chapters 5 and 14 an account is given too of the application of both the GUE and CUE to the study of statistical properties of the zeros of the Riemann zeta function. The applications to statistical mechanics, as summarized under the previous heading, is given in Chapter 10. It is clear from the above descriptions that the chapters have not been organized according to these headings. Instead the ordering has been determined by the desire to first define and motivate the various classical random matrix ensembles and their generalizations (for example, β extensions, minor processes), to give the mathematics leading to the determination of the corresponding eigenvalue p.d.f.’s, and to relate the latter to log-gases. This accounts for Chapters 1–4. Chapters 5–7 are about the calculation of correlations for the p.d.f’s encountered in Chapters 1–4 when the former can be expressed in terms of determinants (Chapter 5) or Pfaffians (Chapter 6). Chapters 8 and 9 give the theory leading to the computation of gap probabilities and spacing distributions in some of the systems for which the correlations were computed in the previous three chapters. With knowledge of the evaluation of gap probabilities and related random matrix averages in terms

x

PREFACE

of Painlev´e transcendents thus established, we proceed in Chapter 10 to show how this can be put to use in the analysis of certain models in statistical mechanics relating to non-intersecting paths. The generalization of the Gaussian ensembles to be parameter dependent, or equivalently to have Brownian-motion valued entries, is introduced in Chapter 11, leading to the Calogero-Sutherland quantum many-body system and families of commuting operators. The polynomial eigenfunctions of these commuting operators are studied in Chapters 12 and 13, culminating in the computation of correlation functions for general β. Theory from Chapter 4 on the Dixon-Anderson integral again appears in Chapter 13, for its relevance to the computation of correlations for general β (or more precisely, for the inter-relations it provides), while theory from Chapters 1–3 relating to the β-ensembles is developed to give characterizations of the general β bulk and edge states in terms of stochastic differential equations. Continuing the general β theme, the study of fluctuation formulas is taken up in Chapter 14. The topic of the log-gas in a two-dimensional domain (which is in fact where my own studies began), and the corresponding random matrix ensembles in which the eigenvalues are complex, is the theme of final chapter of the book. After this introduction to the content and organization, a few words about the presentation are appropriate. As already remarked it has been my desire to give enough detail so that the development is self-contained. A large portion of the necessary working is carried out in the body of the text, but use too has been made of an exercises format which is both more space efficient and less laborious. I have aimed to structure the exercises with sufficient intermediate results so that they can reasonably be worked through, without the need to consult the original references. Generally it has been my intention to keep the subject matter moving. Consequently, there are a small number of results requiring a technical working beyond the main stream of the book, which necessarily have been omitted. I have been most fortunate to have had research fellowships from the Australian Research Council for the duration of this project. This has freed up time and energy for me to follow, and to be part of, many of the developments which have taken place since I began writing. Both being an active researcher in the field and following the developments have been necessary for writing this monograph. While rewarding, studying the research literature is often difficult and inefficient. My own learning was most efficient when studying instead monographs, in particular those of Gupta and Nagar [279], Haake [284], Hua [300], Mehta [398], Macdonald [376] and Muirhead [410]. Similarly it is my hope that this work will prove itself to be an efficient learning resource in preparation for future researches. There are a number of individuals who have over the years lent their assistance to this project, both directly and indirectly. My wife Gail places value on the worth of such academic pursuits, and provided a home environment to make it possible. For getting me started in research, and teaching me some fundamentals, I thank R.J. Baxter, B. Jancovici and (the late) E.R. Smith. Collaborations with K. Aomoto, T.H. Baker, P. Desrosiers, N.E. Frankel, T. Nagao, E.M. Rains and N.S. Witte have been of great value. E. Due˜nez provided some critical comments on my earlier writing on the circular ensembles which were of much help, and P. Sarnak saw enough potential in these earlier writings to recommend the work to Princeton University Press. Most recently F. Bornemann has provided me with high precision numerical data calculated from Fredholm determinants for use in Chapters 8 and 9, and A. Mays provided some help in relation to the proofreading.

Peter Forrester Melbourne, Australia January 2010

Contents

Preface

v

Chapter 1. Gaussian matrix ensembles 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Random real symmetric matrices The eigenvalue p.d.f. for the GOE Random complex Hermitian and quaternion real Hermitian matrices Coulomb gas analogy High-dimensional random energy landscapes Matrix integrals and combinatorics Convergence The shifted mean Gaussian ensembles Gaussian β -ensemble

Chapter 2. Circular ensembles 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

Scattering matrices and Floquet operators Definitions and basic properties The elements of a random unitary matrix Poisson kernel Cauchy ensemble Orthogonal and symplectic unitary random matrices Log-gas systems with periodic boundary conditions Circular β -ensemble Real orthogonal β -ensemble

Chapter 3. Laguerre and Jacobi ensembles 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12

Chiral random matrices Wishart matrices Further examples of the Laguerre ensemble in quantum mechanics The eigenvalue density Correlated Wishart matrices Jacobi ensemble and Wishart matrices Jacobi ensemble and symmetric spaces Jacobi ensemble and quantum conductance A circular Jacobi ensemble Laguerre β -ensemble Jacobi β -ensemble Circular Jacobi β -ensemble

Chapter 4. The Selberg integral 4.1 4.2 4.3

Selberg’s derivation Anderson’s derivation Consequences for the β -ensembles

1 1 5 11 20 30 33 41 42 43

53 53 56 61 66 68 71 73 76 81

85 85 90 98 106 110 111 115 118 125 127 129 130

133 133 137 145

xii

4.4 4.5 4.6 4.7 4.8

CONTENTS

Generalization of the Dixon-Anderson integral Dotsenko and Fateev’s derivation Aomoto’s derivation Normalization of the eigenvalue p.d.f.’s Free energy

Chapter 5. Correlation functions at β = 2 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Successive integrations Functional differentiation and integral equation approaches Ratios of characteristic polynomials The classical weights Circular ensembles and the classical groups Log-gas systems with periodic boundary conditions Partition function in the case of a general potential Biorthogonal structures Determinantal k-component systems

Chapter 6. Correlation functions at β = 1 and 4 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Correlation functions at β = 4 Construction of the skew orthogonal polynomials at β = 4 Correlation functions at β = 1 Construction of the skew orthogonal polynomials and summation formulas Alternate correlations at β = 1 Superimposed β = 1 systems A two-component log-gas with charge ratio 1:2

Chapter 7. Scaled limits at β = 1, 2 and 4 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

Scaled limits at β = 2 — Gaussian ensembles Scaled limits at β = 2 — Laguerre and Jacobi ensembles Log-gas systems with periodic boundary conditions Asymptotic behavior of the one- and two-point functions at β = 2 Bulk scaling and the zeros of the Riemann zeta function Scaled limits at β = 4 — Gaussian ensemble Scaled limits at β = 4 — Laguerre and Jacobi ensembles Scaled limits at β = 1 — Gaussian ensemble Scaled limits at β = 1 — Laguerre and Jacobi ensembles Two-component log-gas with charge ratio 1:2

Chapter 8. Eigenvalue probabilities — Painleve´ systems approach 8.1 8.2 8.3 8.4 8.5 8.6

Definitions Hamiltonian formulation of the Painlev´e theory σ -form Painlev´e equation characterizations The cases β = 1 and 4 — circular ensembles and bulk Discrete Painlev´e equations Orthogonal polynomial approach

Chapter 9. Eigenvalue probabilities — Fredholm determinant approach 9.1 9.2 9.3 9.4 9.5

Fredholm determinants Numerical computations using Fredholm determinants The sine kernel The Airy kernel Bessel kernels

156 160 165 172 180

186 186 193 197 200 207 212 217 223 229

236 236 246 251 263 269 274 278

283 283 290 297 298 301 308 312 316 319 323

328 328 333 349 363 372 375

380 380 385 386 393 399

CONTENTS

9.6 9.7 9.8 9.9 9.10

Eigenvalue expansions for gap probabilities The probabilities Eβsoft (n; (s, ∞)) for β = 1, 4 The probabilities Eβhard (n; (0, s); a) for β = 1, 4 Riemann-Hilbert viewpoint Nonlinear equations from the Virasoro constraints

Chapter 10. Lattice paths and growth models 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Counting formulas for directed nonintersecting paths Dimers and tilings Discrete polynuclear growth model Further interpretations and variants of the RSK correspondence Symmetrized growth models The Hammersley process Symmetrized permutation matrices Gap probabilities and scaled limits Hammersley process with sources on the boundary

Chapter 11. The Calogero–Sutherland model 11.1 11.2 11.3 11.4 11.5 11.6 11.7

Shifted mean parameter-dependent Gaussian random matrices Other parameter-dependent ensembles The Calogero-Sutherland quantum systems The Schr¨odinger operators with exchange terms The operators H (H,Ex) , H (L,Ex) and H (J,Ex) Dynamical correlations for β = 2 Scaled limits

Chapter 12. Jack polynomials 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Nonsymmetric Jack polynomials Recurrence relations Application of the recurrences A generalized binomial theorem and an integration formula Interpolation nonsymmetric Jack polynomials The symmetric Jack polynomials Interpolation symmetric Jack polynomials Pieri formulas

Chapter 13. Correlations for general β 13.1 13.2 13.3 13.4 13.5 13.6 13.7

Hypergeometric functions and Selberg correlation integrals Correlations at even β Generalized classical polynomials Green functions and zonal polynomials Inter-relations for spacing distributions Stochastic differential equations Dynamical correlations in the circular β ensemble

Chapter 14. Fluctuation formulas and universal behavior of correlations 14.1 14.2 14.3 14.4 14.5 14.6

Perfect screening Macroscopic balance and density Variance of a linear statistic Gaussian fluctuations of a linear statistic Charge and potential fluctuations Asymptotic properties of Eβ (n; J) and Pβ (n; J)

xiii

403 416 421 426 435

440 440 456 463 471 480 487 492 495 500

505 505 512 516 521 524 530 540

543 543 550 553 555 558 564 579 583

592 592 601 613 627 633 634 640

658 658 663 665 672 680 688

xiv

14.7

CONTENTS

Dynamical correlations

Chapter 15. The two-dimensional one-component plasma 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 15.11

Complex random matrices and polynomials Quantum particles in a magnetic field Correlation functions General properties of the correlations and fluctuation formulas Spacing distributions The sphere The pseudosphere Metallic boundary conditions Antimetallic boundary conditions Eigenvalues of real random matrices Classification of non-Hermitian random matrices

698

701 701 706 711 718 725 729 738 744 747 752 760

Bibliography

765

Index

785

Chapter One Gaussian matrix ensembles The Gaussian ensembles are introduced as Hermitian matrices with independent elements distributed as Gaussians, and joint distribution of all independent elements invariant under conjugation by appropriate unitary matrices. The Hermitian matrices are divided into classes according to the elements being real, complex or real quaternion, and their invariance under conjugation by orthogonal, unitary, and unitary symplectic matrices, respectively. These invariances are intimately related to time reversal symmetry in quantum physics, and this in turn leads to the eigenvalues of the Gaussian ensembles being good models of the highly excited spectra of certain quantum systems. Calculation of the eigenvalue p.d.f.’s is essentially an exercise in change of variables, and to calculate the corresponding Jacobians both wedge products and metric forms are used. The p.d.f.’s coincide with the Boltzmann factor for a log-gas system at three special values of the inverse temperature β = 1, 2 and 4. Thus the eigenvalues behave as charged particles, all of like sign, which are in equilibrium. The Coulomb gas analogy, through the study of various integral equations, allows for the prediction of the leading asymptotic form of the eigenvalue density. After scaling, this leading asymptotic form is referred to as the Wigner semicircle law. The Wigner semicircle law is applied to the study of the statistics of critical points for a model of high-dimensional energy landscapes, and to relating matrix integrals to some combinatorial problems on the enumeration of maps. Conversely, the latter considerations also lead to the proof of the Wigner semicircle law in the case of the GUE. The shifted mean Gaussian ensembles are introduced, and it is shown how the Wigner semicircle law can be used to predict the condition for the separation of the largest eigenvalue. In the last section a family of random tridiagonal matrices, referred to as the Gaussian β-ensemble, are presented. These interpolate continuously between the eigenvalue p.d.f.’s of the Gaussian ensembles studied previously.

1.1 RANDOM REAL SYMMETRIC MATRICES Quantum mechanics singles out three classes of random Hermitian matrices. We will begin our study by specifying one of these—Hermitian matrices with all entries real, or equivalently real symmetric matrices. The independent elements are taken to be distributed as independent Gaussians, but with the variance different for the diagonal and off-diagonal entries. D EFINITION 1.1.1 A random real symmetric N × N matrix X is said to belong to the Gaussian orthogonal ensemble (GOE) if the diagonal and upper triangular elements are independently chosen with p.d.f.’s 2 1 √ e−xjj /2 2π

and

2 1 √ e−xjk , π

respectively. The p.d.f.’s of Definition 1.1.1 are examples of the normal (or Gaussian) distribution 2 2 1 √ e−(x−μ) /2σ , 2 2πσ

denoted N[μ, σ]. With this notation, note that an equivalent construction of GOE matrices is to let Y be an N × N random matrix of independent standard Gaussians N[0, 1] and to form X = 12 (Y + YT ).

2

CHAPTER 1

The joint p.d.f. of all the independent elements is P (X) :=

N  j=1

N  2 1 −x2jj /2  1 −x2jk √ e √ e = AN e−xjk /2 π 2π 1≤j J of

J2 . 4c

c D X |yj |2 E , N j=1 λ − λj N

1=

(1.138)

where each |yj |2 has mean unity, and {λj } are the eigenvalues of a member of the specified Gaussian ensemble but with H0 = 0. We know that the density of the {λj } is then given by the semicircle law (1.129). Hence for N large Z J N “ DX 2N λ “ J 2 ”1/2 ” |yj |2 E ρb (y) dy = ∼ 1− 1− 2 , 2 λ − λj J λ −J λ − y j=1 where the first relation follows from the fact that the eigenvalues and eigenvectors are independently distributed and |yj |2  = 1, while the equality, which requires that λ > J, follows from (1.132). Substituting this in (1.138) and solving for λ gives the stated result.

1.9 GAUSSIAN β-ENSEMBLE The p.d.f. (1.28) is realized by the eigenvalues of the GOE, GUE and GSE for the values of β equal to 1, 2 and 4 respectively. In this section a family of random tridiagonal matrices, referred to as the Gaussian βensemble, with (1.28) as their eigenvalue p.d.f. for general β > 0, will be studied. They can be motivated by the reduction of GOE or GUE matrices to tridiagonal form. 1.9.1 Householder transformations A familiar technique in numerical linear algebra is the similarity transformation of a real symmetric matrix to tridiagonal form using a sequence of reflection matrices, referred to as Householder transformations. Explicitly, let A be a real symmetric matrix [aij ]i,j=1,...,N . Then one can construct a sequence of symmetric real orthogonal matrices U(1) , U(2) , . . . , U(N −2) such that the transformed matrix U(N −2) U(N −3) · · · U(1) AU(1) U(2) · · · U(N −2) =: B(N −2) is a symmetric tridiagonal matrix. These matrices have the structure   0j×N −j 1j (j) (j) (j)T , = U = 1N − 2u u 0N −j×j VN −j×N −j

(1.139)

(1.140)

44

CHAPTER 1

where u(j)T u(j) = 1 and VN −j×N −j is symmetric real orthogonal. Geometrically U(j) corresponds to a reflection in the hyperplane orthogonal to u(j) . (1) Consider first the construction of U(1) . Choosing the components ul of u(1) as 1 a12 1/2 a1l (1) (1) (1) 1− u1 = 0, u2 = , ul = − (l ≥ 3), (1.141) (1) 2 α 2αu2 (1)

where α = (a212 + · · · + a21N )1/2 , we then have u(1)T [al1 ]l=1,...,N = (a12 − α)/2u2 . This in turn implies that B(1) := U(1) AU(1)

(1.142)

has b11 = a11 ,

b1k = bk1 = 0 (k ≥ 3)

b12 = b21 = α,

and is thus tridiagonal with respect to the first row and column. The matrices U(j) , j = 2, 3, . . . in order are (j) (j) (j) now defined by the formulas (1.141), but with u1 = u2 = · · · = uj = 0, and the analogue of the entries a1l replaced by the elements in the first row of the bottom right (N − j + 1) × (N − j + 1) submatrix of B(j−1) . A number of works (see [157] and references therein) posed the question as to the form of B(N −2) when A is a member of the GOE. It was found that like A itself, the elements of B(N −2) are all independent (apart from the requirement that B(N −2) be symmetric) with a distribution that can be calculated explicitly. P ROPOSITION 1.9.1 Let N[0, 1] refer to the standard normal distribution as defined below Definition 1.1.1, and let χ ˜k denote the square root of the gamma distribution Γ[k/2, 1], the latter being specified by the p.d.f. (1/Γ(k/2))uk/2−1 e−u , u > 0, and realized by the sum of the squares of k independent Gaussian √ 2 distributions N[0, 1/ 2]. (The p.d.f. of χ ˜k is thus equal to (2/Γ(k/2))uk−1 e−u , u > 0.) For A a member of the GOE, the tridiagonal matrix B(N −2) obtained by successive Householder transformations is given by ⎤ ⎡ N[0, 1] χ ˜N −1 ⎥ ⎢ χ ˜N −2 ⎥ ⎢ ˜N −1 N[0, 1] χ ⎥ ⎢ χ ˜ N[0, 1] χ ˜ N −2 N −3 ⎥ ⎢ ⎥. ⎢ . . . .. .. .. ⎥ ⎢ ⎥ ⎢ ⎣ χ ˜2 N[0, 1] χ ˜1 ⎦ χ ˜1 N[0, 1] Proof. Let GOEn denote the ensemble of n × n GOE matrices. From the Householder algorithm, the first row and column of B(N−2) are he same as those of B(1) in (1.142), and thus from (1.141) we have (N−2)

b11

= N[0, 1],

(N−2)

b12

=χ ˜N−1 ,

where use has been made of the assumption that A is a member of GOEN , and the definition of χ ˜2N−1 as a sum of squares of Gaussians. To proceed further we must compute the distribution of the bottom N −1×N −1 block of B(1) . In general, (1) denoting such a block of the matrix X by XN−1 , it follows from (1.140) that BN−1 = VN−1 AN−1 VN−1 . Since the elements of the real orthogonal matrix VN−1 are independent of the elements of AN−1 , which itself is a member of (1) GOEN−1 , it follows immediately from the general invariance of the GOE under orthogonal transformations that BN−1 (1) is also a member of GOEN−1 . Applying the Householder transformation to BN−1 , we thus get (N−2)

b22

Continuing inductively gives the stated result.

= N [0, 1],

(N−2)

b23

=χ ˜N−2 .

45

GAUSSIAN MATRIX ENSEMBLES

1.9.2 Tridiagonal matrices The result of Proposition 1.9.1 suggests investigating the Jacobian for the change of variables from a general real symmetric tridiagonal matrix ⎡ ⎤ an bn−1 ⎢ bn−1 an−1 bn−2 ⎥ ⎢ ⎥ ⎢ ⎥ bn−2 an−2 bn−3 ⎢ ⎥ (1.143) T=⎢ ⎥, .. .. .. ⎢ ⎥ . . . ⎢ ⎥ ⎣ b2 a2 b1 ⎦ b1 a1 to its eigenvalues and variables relating to its eigenvectors. First, for each eigenvalue λk and correspond(1) ing eigenvector vk , it is easy to see by direct substitution that once the first component vk =: qk of vk is specified, all other components can be expressed in terms of λk and the elements of T. To make the eigendecomposition unique we specify that qk > 0, and furthermore note that T, being symmetric, can be orthogonally diagonalized, and so doing this we have n

qk2 = 1.

(1.144)

k=1

The Jacobian for the change of variables from b := (bn−1 , . . . , b1 ),

a := (an , an−1 , . . . , a1 ),

(1.145)

to λ := (λ1 , . . . , λn ),

q := (q1 , . . . , qn−1 )

(1.146)

can be calculated using the method of wedge products. However, one must first establish some auxiliary results. P ROPOSITION 1.9.2 Let (X)11 denote the top-left hand entry of the matrix X. We have n

qj2 . λj − λ

(1.147)

n−1 2i bi (λi − λj ) = i=1 n 2 . i=1 qi

(1.148)

((T − λ1)−1 )11 =

j=1

Also 

2

1≤i 0, the number of positive values in {si } equals N (μ). This can equivalently be stated in terms of {xi }. P ROPOSITION 1.9.8 The number of sign changes in the shooting eigenvector x equals n − N (μ), which is the number of eigenvalues of A greater than μ. 1.9.4 Prufer ¨ phases There is a parametrization, in terms of Pr¨ ufer phases and amplitudes, of the shooting vectors well suited to analysis of the large n limit of the bulk eigenvalues (see Section 13.6). To introduce the parametrization, first observe that the three-term recurrence satisfied by the shooting vector bj xj+1 + aj xj + bj−1 xj−1 = μxj

(j = 1, . . . , n; b0 := 0, bn := −1)

is equivalent to the matrix equation      uj uj+1 (μ − aj )/bj −1/bj = bj 0 vj vj+1 where



uj vj



 =

1 0 0 bj−1



xj xj−1

(j = 1, . . . , n),

(1.166)

(1.167)

 (1.168)

50

CHAPTER 1

(note that the matrix in (1.167) has unit determinant and so as a transformation is volume preserving). Choosing the initial condition u1 = 1, v1 = 0 we see that     1 uj , = Tj 0 vj where Tj := Vj−1 · · · V1 is referred to as a transfer matrix D EFINITION 1.9.9 The Pr¨ufer phases θjμ and amplitudes Rjμ > 0 are such that    μ  Rj cos θjμ uj = , vj Rjμ sin θjμ

(1.169)

μ where −π/2 < θj+1 − θjμ < 3π/2.

Note that it follows from (1.167) and (1.168) that {θjμ } satisfies the first order recurrence μ b2j cot θj+1 = − tan θjμ + (μ − aj ),

θ1μ = 0.

(1.170)

A consequence is an identity which tells us that θjμ is a decreasing function of μ (see also Exercises 1.9 q.5). P ROPOSITION 1.9.10 We have ∂ μ θj = − u2l . ∂μ j−1

(Rjμ )2

(1.171)

l=1

Proof. Differentiating (1.170) with respect to μ and making use of (1.168) and (1.169) gives the recurrence μ )2 (Rj+1

μ ∂θj+1 ∂θjμ = (Rjμ )2 − u2j . ∂μ ∂μ

This together with the initial condition ∂θ1μ /∂μ = 0 implies (1.171).

We are now in a position to relate θnμ to N (μ) for the tridiagonal matrix (1.143) [326] . First note from the recurrence (1.166) that for μ → ∞, xj is positive while xj−1 /xj → 0. Recalling (1.169), this implies limμ→∞ θjμ = 0. But it has just been shown that θjμ is a decreasing function of μ. The facts that xn+1 = μ μ un+1 = Rn+1 cos θn+1 and that xn+1 = 0 if and only if μ is an eigenvalue then imply the kth largest μ λk eigenvalue λk of T is such that θn+1 = (π/2) + π(k − 1), and moreover that θn+1 relates to the number of eigenvalues of T greater that μ, n − N (μ), according to 1  1  μ  (1.172)  θn+1 − (n − N (μ)) ≤ . π 2 E XERCISES 1.9 1. The objective of this exercise is to derive the Vandermonde determinant evaluation

]j,k=1,...,N det[xk−1 j

˛ ˛ ˛ ˛ ˛ := ˛ ˛ ˛ ˛

1 1 .. . 1

x1 x2 .. . xN

x21 x22 .. . x2N

··· ··· .. . ···

xN−1 1 xN−1 2 .. . xN−1 N

˛ ˛ ˛ ˛ Y ˛ (xk − xj ). ˛= ˛ ˛ 1≤j 0, a > −1

and (J)

w2 (x) = (1 − x)a (1 + x)b ,

−1 < x < 1, a, b > −1

respectively (in the latter two cases w2 (x) = 0 outside the specified intervals). The monic orthogonal polynomials associated with these weight functions are proportional to the Hermite, Laguerre and Jacobi classical polynomials respectively (see, e.g., [508]). Explicitly −n p(G) Hn (x), n (x) = 2 n a p(L) n (x) = (−1) n!Ln (x), Γ(a + b + n + 1), (a,b) n P (x) p(J) n (x) = 2 n! Γ(a + b + 2n + 1) n

(5.46)

where  n (2m)! n−2m x (−1) 2 , Hn (x) = 2m 2m m! m=0   n n + a xm , Lan (x) = (−1)m n − m m! m=0 m     n x−1 n + a (n + a + b + 1)m n (a,b) Pn (x) = (a + 1)m 2 m n m=0



[n/2]

m n−m

(5.47)

(the notation (u)p is defined in (5.83) below). The corresponding normalizations are (G)

(pn , pn )2

= π 1/2 2−n n!,

(L)

(pn , pn )2 = Γ(n + 1)Γ(a + n + 1), Γ(n + 1)Γ(a + b + 1 + n)Γ(a + 1 + n)Γ(b + 1 + n) (J) . (pn , pn )2 = 2a+b+1+2n Γ(a + b + 2n + 1)Γ(a + b + 2n + 2)

(5.48)

Using the above formulas in Proposition 5.1.2, with KN (x, y) evaluated according to Proposition 5.1.3 and (5.13), gives an explicit expression for the n-particle correlation in each of the ensembles.

201

CORRELATION FUNCTIONS AT β = 2

5.4.2 Circular ensembles and the Cauchy weight By making the transformation (2.50) we know from (2.51) that N  j=1



wβ (eiθj )

|eiθk − eiθj |β dθ1 · · · dθN

1≤j xy+1

(y+1) > · · · > x(y) y > xy

>

(y) x2

> ··· >

(y+1) xb+c−y−1

(y = 1, . . . , b − 1),

(y = b, . . . , c − 1), (y)

> xb+c−y

(y = c, . . . , b + c − 1).

(10.64)

Furthermore, they must stay within the bounds of the hexagon, and are restricted to odd (even) numbered sites for y odd (even).

460

CHAPTER 10

Figure 10.10 Interlacing particle system which results by associating particles with the center point of the vertical-axis parallelograms of a tiling.

Suppose now that the parallelograms are rescaled so that the leftmost extremity (which occurs on the line y = b) is at x = 0, while the rightmost extremity (which occurs on the line y = 0) is at x = 1. By taking the number of lattice paths and thus rhombi to infinity, a continuous multi-species particle system is obtained (the different lines are the different species), which is specified by the interlacings (10.64), supplemented by the requirement that all coordinates are bound between 0 and 1. This multi-species system is referred to as a bead process [94], [177]. One point of interest is the distribution of particles on a given line r. P ROPOSITION 10.2.3 The p.d.f. on line r is proportional to ⎧ r  (r)  ⎪ (r) (r) (r) ⎪ ⎪ (xj )c−r (1 − xj )b−r (xj − xk )2 , 1 ≤ r ≤ b, ⎪ ⎪ ⎪ j=1 ⎪ 1≤j0 , and for η a composition, let η +

643

CORRELATIONS FOR GENERAL β

denote the associated partition, as in Definition 12.1.1. We have N 

 (C)  δ(xj )z −l Pκ (z; 2/β)

j=1

= LN −1 N ×

N 



1/2

−1/2

dx2 e−2πilx2 · · ·

j=2

=



|1 − e2πixj |β



1/2 −1/2

dxN e−2πilxN Pκ (1, e2πix2 , . . . , e2πixN )

|e2πixk − e2πixj |β

2≤j l, we proceed as in the proof of Proposition 12.3.2, this time setting a = −b = −l +  in (12.142). According to Proposition 12.6.10, (4.4) and (12.46) we then have N DX

˛ E(C) ˛ δ(xj )˛z −l Pκ (z; 2/β) = LN−1 N (|κ| − N l)Pκ ((1)N ; 2/β)

j=1

× lim

→0

N 1Y Γ(1 + β(j − 1)/2)Γ(βj/2 + 1) .  j=1 Γ(1 − l +  + β(N − j)/2 + κj )Γ(1 + l −  + β(N − j)/2 − κj )Γ(1 + λ)

The product contains the terms

1 . Γ(1 − l +  + κN )Γ(1 + l −  − κ1 )

With κN = 0 and l ∈ Z>0 , 1/Γ(1−l++κN ) is proportional to  for small , and similarly for κ1 > l, 1/Γ(1+l−−κ1 ) is also proportional to  for small , thus implying the result.

The result of Proposition 13.7.1 implies the sum in (13.201) can be reduced to a sum over partitions only, provided twice the real part is taken. Thus ρT(1,1) (x, y; τ ) = 2Re

κ κ=0N

N 1  (C)

Nκ

 (C) (1)  δ(xj )Pκ (z (1) ; 2/β)

j=1

N   (C) (C)  (0) × Pκ (z (0) ; 2/β)p0 eβW δ(xj ) e2πi|κ|(y−x)/L e−τ e(κ;2/β)/β . (13.205) j=1

It remains to evaluate the inner product Pκ (z (0) ; 2/β)|p0 eβW p0 = e−βW 

(C)

(C)

N j=1

(0)

δ(xj )(C) in (13.205). In the case

(C)

/N0N (initial state equals equilibrium state),

N  (C) (C)  (0) δ(xj ) = Pκ (z (0) ; 2/β)p0 eβW j=1

N 1  (C) N0N

j=1

 (C) (0)  δ(xj )Pκ (1/z (0); 2/β)

(13.206)

644

CHAPTER 13

and thus is evaluated according to (13.204). In the case p0 = 1/LN (initial state a perfect gas)  1/2 N   (C) N  1/2  βW (C) (0) (0) (0) (0) Pκ (z ; 2/β)p0 e δ(xj ) = dx2 · · · dxN Pκ (1/z (0) ; 2/β) L −1/2 −1/2 j=1 N (0) Pκ (1/z1 , 0, . . . , 0) L  (0) −κ1 N , κ = (κ1 , 0, . . . , 0), L (z1 ) = 0 otherwise. =

(13.207)

Consideration of these facts gives the following formulas for ρT(1,1) in the finite system [212]. P ROPOSITION 13.7.2 For the perfect gas initial condition ρT(1,1) (x, y; τ ) =

∞ 2 2 2 2N Re e2πiκ1 (y−x)/Le−(2π) τ (κ1 +β(N −1)κ1 /2)/βL , 2 L κ =1

(13.208)

1

while for the case that the initial state is equal to the equilibrium state ρT(1,1) (x, y; τ ) =

N  (Γ(1 + β(j − 1)/2))4 2 2/β 2 2 ¯2/β Re |κ| (κ − 1)! (κ)f (κ) f 1 N N L2 (Γ(1 + β(N − j)/2 + κj ))2 j=1 κ=0

×

  1 exp 2πi(y − x)|κ|/L − e(κ; 2/β)τ /β (.13.209) (Γ(1 + β(j − 1)/2 − κj ))2 j=2 N 

Proof. In deriving (13.208) we use the result (13.207) in (13.205), together with the facts that for (κ1 , 0, . . . , 0),

(β/2)|κ| dκ = 1, which allows the ratio of (13.204) and (13.202) to be simplified. In deriving (13.209), we substitute (13.204), (13.206) and (13.202) in (13.205), and use the results „ l(κ) Y“ j=2

−

” β (j − 1) 2 κj

«2 =

Pκ(2/β) ((1)N ) =

N “Y j=2

Γ(1 + β(j − 1)/2) ”2 , Γ(1 + β(j − 1)/2 − κj )

˛ bκ ˛ β/2 = fN (κ), ˛ hκ α→2/β

(13.210)

where fN (κ) is specified by (12.59), as well as the formula (12.60) for dκ . β/2

13.7.2 Bulk limit The formula (13.208) for ρT(1,1) in the case of perfect gas initial conditions is in the form of a Riemann sum approximation to an integral, and the following result is immediate. P ROPOSITION 13.7.3 In the bulk limit N, L → ∞, N/L = ρ (constant),  ∞ 2 2 bulk (x, y; τ ) = 2ρ2 e−(2πρ) (u +βu/2)τ /β cos(2πρ(x − y)u) du. ρT(1,1)

(13.211)

0

In contrast, the formula (13.209) for ρT(1,1) in the case that the initial state is equal to the equilibrium state requires further analysis before its limiting value can be computed. One essential difficulty is that in (13.209) the sum is over all partitions κ, and thus N independent quantities, whereas in (13.208) only the largest part κ1 enters into the summation. This feature would appear to make the problem of computing the thermodynamic limit intractable. However, if we restrict attention to the case β rational a significant simplification takes place.

645

CORRELATIONS FOR GENERAL β

P ROPOSITION 13.7.4 Let β be rational and write β/2 = p/q where p and q are relatively prime. Then the summand in (13.209) is nonzero if and only if κ is of the form κ = (α1 , . . . , αq , p, . . . , p, . . . , 1, . . . , 1, 0, . . . , 0), 7 89 : 7 89 : where α1 ≥ α2 ≥ · · · ≥ αq and q +

β1 p s

p j=1

(13.212)

βp 1 s

βj ≤ N .

QN

Γ(1+β(j−1)/2−κj ))2 in (13.209) is non-zero if and only if β(j−1)/2−κj ∈ {−1, −2, . . . } for each j = 1, 2, . . . , N . With β/2 = p/q and j = q + 1 this means p − κq+1 ∈ {−1, −2, . . . } and thus κq+1 ≤ p as required.

Proof. The factor (

j=2

As a consequence of (13.7.4), for β rational the summation in (13.209) can be taken over the p + q coordinates αi (i = 1, . . . , q) and βj (j = 1, . . . , p). In fact it is more convenient to replace the coordinates βj by γj (j = 1, . . . , p) defined so that κq+γa +k = p − a, k = 1, . . . , γa+1 − γa (γ0 := 0, γp+1 := N − q). β/2

P ROPOSITION 13.7.5 For β/2 = p/q and κ as specified above and large N, {αj }, {γj } we have fN (κ) = ABC/D, where     A := (j − i)β/2 + αi − αj ∼ (αi − αj )β/2 ; β/2

1≤i