Mathematical Surveys

and Monographs

Volume 171

American Mathematical Society

Eigenvalue Distribution of Large Random Matrices

Leonid PasturMariya Shcherbina

Eigenvalue Distribution of Large Random Matrices

http://dx.doi.org/10.1090/surv/171

Mathematical Surveys

and Monographs

Volume 171

Eigenvalue Distribution of Large Random Matrices

Leonid Pastur Mariya Shcherbina

American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEE

Ralph L. Cohen, ChairEric M. Friedlander

Michael A. SingerBenjamin Sudakov

Michael I. Weinstein

2010 Mathematics Subject Classification. Primary 60F05, 60B20, 15B52, 15B57.

For additional information and updates on this book, visitwww.ams.org/bookpages/surv-171

Library of Congress Cataloging-in-Publication Data

Pastur, L. A. (Leonid Andreevich)Eigenvalue distribution of large random matrices / Leonid Pastur, Mariya Shcherbina.

p. cm. — (Mathematical surveys and monographs ; v. 171)Includes bibliographical references and index.ISBN 978-0-8218-5285-9 (alk. paper)1. Distribution (Probability theory) 2. Random matrices. I. Shcherbina, Mariya, 1958–

II. Title.

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10 9 8 7 6 5 4 3 2 1 16 15 14 13 12 11

Contents

Preface ix

Chapter 1. Introduction 11.1. Objectives and Problems 11.2. Example 131.3. Comments and Problems 21

Part 1. Classical Ensembles 33

Chapter 2. Gaussian Ensembles: Semicircle Law 352.1. Technical Means 352.2. Deformed Semicircle Law 432.3. The Case of Random H(0) 542.4. Problems 59

Chapter 3. Gaussian Ensembles: Central Limit Theorem for LinearEigenvalue Statistics 69

3.1. Covariance for Traces of the Resolvent 693.2. Central Limit Theorem for Linear Eigenvalue Statistics of

Differentiable Test Functions 743.3. Central Limit Theorem for (ϕ(M))jj 903.4. Problems 94

Chapter 4. Gaussian Ensembles: Joint Eigenvalue Distribution and RelatedResults 101

4.1. Joint Eigenvalue Probability Density 1014.2. Orthogonal Polynomial Techniques 1074.3. Simplest Applications 1134.4. Comments and Problems 118

Chapter 5. Gaussian Unitary Ensemble 1295.1. Hermite Polynomials 1295.2. Bulk of the Spectrum 1315.3. Edges of the Spectrum 1475.4. Problems 152

Chapter 6. Gaussian Orthogonal Ensemble 1596.1. Correlation and Cluster Functions 1596.2. Bulk of the Spectrum 1666.3. Edges of the Spectrum 1716.4. Problems 175

v

vi CONTENTS

Chapter 7. Wishart and Laguerre Ensembles 1777.1. Generalities 1777.2. Normalized Counting Measure of Eigenvalues 1827.3. Central Limit Theorem for Linear Eigenvalue Statistics 1897.4. Joint Eigenvalue Distribution 1927.5. Local Regimes 1957.6. Comments and Problems 203

Chapter 8. Classical Compact Group Ensembles: Global Regime 2118.1. Classical Compact Groups as Random Matrix Ensembles 2118.2. Limiting Normalized Counting Measure and Central Limit Theorem

for Linear Eigenvalue Statistics of Unitary Matrices 2188.3. Moments of Traces of Matrices of Classical Compact Groups 2228.4. More Central Limit Theorems for Linear Eigenvalue Statistics of

Matrices of Classical Groups 2348.5. Problems 244

Chapter 9. Classical Compact Group Ensembles: Further Results 2499.1. Joint Eigenvalue Distribution and Related Results 2499.2. Circular Ensembles 2649.3. Problems 273

Chapter 10. Law of Addition of Random Matrices 27510.1. Normalized Counting Measure of Eigenvalues 27610.2. Fluctuations of the Traces of Resolvents 29010.3. The Case of Many Summands 30410.4. Problems 306

Part 2. Matrix Models 315

Chapter 11. Matrix Models: Global Regime 31711.1. Convergence of the Normalized Counting Measure of Eigenvalues 31711.2. Properties of the Limiting Measure 33811.3. Fluctuations of the Normalized Counting Measure of Eigenvalues 35211.4. Comments and Problems 359

Chapter 12. Bulk Universality for Hermitian Matrix Models 36912.1. Basic Results 36912.2. Proof of Basic Results 37012.3. Problems 383

Chapter 13. Universality for Special Points of Hermitian Matrix Models 38513.1. Generic Soft Edges 38613.2. Internal Points 40813.3. Some Properties of Jacobi Matrices 43113.4. Problems 435

Chapter 14. Jacobi Matrices and Limiting Laws for Linear EigenvalueStatistics 437

14.1. Asymptotics of Orthogonal Polynomials and Quasiperiodic JacobiMatrices 437

CONTENTS vii

14.2. Fluctuations of Linear Eigenvalue Statistics 45114.3. Intermediate and Local Regimes 46414.4. Problems 466

Chapter 15. Universality for Real Symmetric Matrix Models 46915.1. Generalities 46915.2. Invertibility of M(0,n) 47215.3. Universality for Real Symmetric Matrix Models 47815.4. Problems 483

Chapter 16. Unitary Matrix Models 48516.1. Global Regime 48516.2. Bulk Universality for Unitary Matrix Models 49416.3. Problems 498

Part 3. Ensembles with Independent and Weakly DependentEntries 499

Chapter 17. Matrices with Gaussian Correlated Entries 50117.1. Definition and Finite-n Results 50117.2. Limiting Equations 50417.3. Parametric Limits for Certain Ergodic Operators 50917.4. Problems 522

Chapter 18. Wigner Ensembles 52518.1. Generalities 52518.2. Martingale Bounds for Moments of Spectral Characteristics 53118.3. Deformed Semicircle Law 53518.4. Central Limit Theorem for Linear Eigenvalue Statistics 53918.5. Further Asymptotic Results on Linear Eigenvalue Statistics 55618.6. Limits of Extreme Eigenvalues 55918.7. Other Results 56518.8. Problems 580

Chapter 19. Sample Covariance and Related Matrices 58319.1. Limiting Normalized Counting Measure of Eigenvalues 58319.2. Central Limit Theorem for Linear Eigenvalue Statistics 59719.3. Other Results 607

Bibliography 611

Index 631

Preface

Random matrices is an active field of mathematics and physics. Initiated inthe 1920s–1930s by statisticians and introduced in physics in the 1950s–1960s byWigner and Dyson, the field, after about two decades of the "normal science" de-velopment restricted mainly to nuclear physics, has became very active since theend of the 1970s under the flow of accelerating impulses from quantum field theory,quantum mechanics (quantum chaos), statistical mechanics, and condensed mat-ter theory in physics, probability theory, statistics, combinatorics, operator theory,number theory, and theoretical computer science in mathematics, and also telecom-munication theory, qualitative finances, structural mechanics, etc. In addition toits mathematical richness random matrix theory was successful in describing vari-ous phenomena of these fields, providing them with new concepts, techniques, andresults.

Random matrices in statistics have arisen as sample covariance matrices andhave provided unbiased estimators for the population covariance matrices. Abouttwenty years later physicists began to use random matrices in order to model theenergy spectra of complex quantum systems and later the systems with complexdynamics. These, probabilistic and spectral, aspects have been widely representedand quite important in random matrix theory until the present flourishing state ofthe theory and its applications to a wide variety of seemingly unrelated domains,ranging from room acoustics and financial markets to zeros of the Riemann ζ-function.

One more aspect of the theory concerns integrals over matrix measures de-fined on various sets of matrices of an arbitrary (mostly large) dimension. Matrixintegrals proved to be partition functions of models of quantum field theory andstatistical mechanics and generating functions of numerical characteristics of com-binatorial and topological objects; they satisfy certain finite-difference and differ-ential identities connected to many important integrable systems. However, thematrix integrals themselves, their dependence on parameters, etc., can often beinterpreted in spectral terms related to random matrices whose probability law isa matrix measure in the integral.

Thus, random matrix theory can be viewed as a branch of random spectraltheory, dealing with situations where operators involved are rather complex andone has to resort to their probabilistic description. It is worth noting that ap-proximately at the same time as Wigner and Dyson, i.e., in the 1950s, Anderson,Dyson, and Lifshitz proposed to use finite-difference and differential operators withrandom coefficients, i.e., again certain random matrices, to describe the dynamicsof elementary excitations in disordered media (crystals with impurities, amorphoussubstances), thereby creating another branch of random spectral theory, known

ix

x PREFACE

now as random operator theory (see e.g. [396]) and its theoretical physics counter-part, the theory of disordered systems (see e.g. [345]). The statistical approach inboth cases goes a step further from that of quantum statistical mechanics, wheretraditionally the operators (Hamiltonians and observables) are not random but thequantum states are random and their probability law (Gibbs measure) is determinedby the corresponding Hamiltonian. Note that even this tradition was broken in the1970s, when the intensive studies of disordered magnets, spin glasses in particular,began and the random statistical mechanics Hamiltonians, hence the randomizedGibbs measures, were introduced; see e.g. [93, 357].

However, as in statistical mechanics, the infinite size limit and related asymp-totic regimes play a quite important role in random spectral theory, random matrixtheory in particular. This is also in agreement with principal settings of probabilitytheory, since, according to the classics, "... the epistemological value of the theoryof probability is revealed only by limit theorems. Moreover, without limit theoremsit is impossible to understand the real content of the primary concept of all oursciences – the concept of probability" [238, Preface].

By the way, the large size asymptotic regimes, which are used almost every-where in this book, can also be applied to draw a borderline between randomoperators and random matrices. In our opinion, this can be inferred from the large-n behavior of the number νn of the entries of the same order of magnitude of ann× n matrix on its principal and adjacent diagonals (these matrices are known asthe band matrices). If νn/n has a finite integer limit as n→∞, then there exists alimiting object, a random operator in l2(Z). In particular, in the case of hermitiann×n matrices, if the limit is an odd positive integer 2p+1, then we have a hermit-ian finite-difference operator of order 2p with random coefficients and the spectralproperties of this "limiting" operator are strongly related to those of its "finite box"restriction. This approach to the spectral analysis of selfadjoint operators in l2(Zd)and L2(Rd), d ≥ 1, dates back to the work by H. Weyl of the 1910s and has provedto be quite efficient since then. If, however, νn/n → ∞, n → ∞, then we have a"genuine" random matrix and have to deal with various asymptotic regimes or justestimates, despite the fact that many of them can be used to characterize certaininfinite-dimensional operators, as, for example, in the quantum chaos studies sincethe 1970s or in recent studies of asymptotic eigenvalue spacing as possible toolsto distinguish between the pure point and the absolutely continuous spectrum ofrandom operators. Besides, there exists a variety of results in both theories whichallow one to say, by using the terminology of statistical mechanics, that randommatrix theory can often be viewed as the mean field version of random operatortheory.

We now comment on basic terminology, conventions, and the contents of thebook. Since random matrix theory is largely asymptotic theory, it deals not withrandom matrices of a fixed size n, but rather with sequences of random matricesdefined for all positive integer n’s, despite the fact that we write quite often, togetherwith the random matrix community, "random matrix", i.e., the singular form ofour main object. Moreover, to make our formulas, often long, more readable, wedo not write the subindex n in matrices to indicate their size, excepting the caseswhere it can lead to misunderstandings. It is always understood that we deal withn × n matrices and that we are interested mostly in the large-n behavior of theirspectral characteristics.

PREFACE xi

Next is the term ensemble or random matrix ensemble, whose meaning is some-times a bit vague in random matrix texts. The term seems to be borrowed from theearly days of probability theory and statistical mechanics (where it is widely useduntil now). We use the term just to designate the sequence of matrix probabilitylaws determining the random matrix in the above meaning. We also use quite oftenthe term spectrum, while discussing the large size limit of random matrices, despitethe fact that according to the above, there is no limiting operator, as is the case inrandom operator theory. In this book the term is just a synonym for the supportof the limiting Normalized Counting (or empirical) Measure of eigenvalues of therandom matrix in question. This has to be compared with random operator the-ory that deals with differential and finite-difference operators with random ergodiccoefficients and where also there exists the limiting Normalized Counting Measureof eigenvalues of the finite box restrictions of corresponding operators. Here thespectrum of the "limiting" operator does indeed coincide with probability 1 withthe support of the limiting measure [396, Sections 4.C and 5.C]. Moreover, thereexist certain families of random ergodic operators interpolating between the twocases; see e.g. Section 17.3 of the book.

Now we comment on the contents of the book. The detailed contents canbe seen from the Contents and from the introductions to the chapters. We willtherefore restrict ourselves to general remarks. The book treats three main themes:the existence and the properties of the nonrandom limiting Normalized CountingMeasure of eigenvalues, the fluctuation laws of linear eigenvalue statistics, andthe local regimes. The first two themes are often referred to as the global (ormacroscopic) regime and require the scaling of matrix entries (or the spectral axis)guaranteeing the existence of the well-defined limit of the Normalized CountingMeasure of eigenvalues in question. The themes are similar to those in probabilitytheory on the Law of Large Numbers and the Central Limit Theorem for the sumsof independent or weakly dependent random variables. The main difference here isthat the eigenvalues of a random matrix are strongly dependent even if its entriesare independent; thus one needs new techniques or, at least, appropriately extendedand treated versions of existing probabilistic techniques.

The third theme is entirely the random matrix theme. It is on the local regimes,i.e., on the statistics of eigenvalues falling into intervals, whose length is of order ofmagnitude of typical spacing between eigenvalues and thus tends to zero with anappropriate rate as the size of the matrix tends to infinity.

In treating the above themes, we confine ourselves to the normal random matri-ces, more precisely, to real symmetric, hermitian, orthogonal, and unitary matrices.Random matrix theory studies also quaternion real and symplectic matrices, whichare, roughly speaking, the hermitian and unitary matrices with quaternion (2 × 2matrix) entries. They possess a number of interesting properties that can be foundin [217] and [356] and references therein.

We do not consider complex matrices, the random matrix jargon term for real,but not real symmetric or orthogonal, and complex, but not hermitian or unitary,matrices. This is a big and fast developing field with a lot of interesting recentresults and it deserves a separate book.

The book consists of an introduction and three parts. In the introduction wediscuss first the archetype Gaussian Ensembles of random matrix theory, deriving

xii PREFACE

them from the requirements of orthogonal or unitary invariance (for the real sym-metric and hermitian matrices, respectively). We also discuss briefly other widelyused (but not all) ensembles. We then introduce certain notions, objects, and set-tings of random matrix theory by using an elementary example of diagonal matriceswith i.i.d. random diagonal entries, i.e., in fact, the standard probabilistic set up.In particular, we introduce the main asymptotic regimes of the theory.

Part 1 is devoted to classical ensembles, i.e., the Gaussian Ensembles for realsymmetric and hermitian matrices, introduced by Wigner in the 1950s, the WishartEnsemble for the real symmetric matrices, well known in statistics since the late1920s, its hermitian analog, known as the Laguerre Ensemble, and the ensemblesof real symmetric, hermitian, orthogonal, and unitary matrices whose randomnessis due to the classical groups (orthogonal, unitary) and related symmetric spaces,seen as the matrix probability spaces, with the normalized to unity Haar measureor its restrictions.

We first study in detail the global regime. This is carried out by using basicallytwo technical tools: certain versions of integration by parts, which we call the dif-ferential formulas, and the Poincaré-type inequalities, providing an efficient boundfor the variance of relevant random objects. In particular, the inequalities lead al-most immediately to the bound of the variance of linear eigenvalue statistics, whichis of the order O(1) as n → ∞, unlike O(n) for i.i.d. Gaussian random variables.This is a first manifestation of strong statistical dependence of eigenvalues, one ofthe principal sources of new and often highly nontrivial results of random matrixtheory.

We then pass to the local (bulk and edge) regimes and establish basic factsabout them, thereby presenting a considerable part of the random matrix "arsenal"both for random matrix theory itself and for numerous applications.

This part of the book is rather traditional. We only mention that our pre-sentation of the global regime is based on the systematic use of the Stieltjes andthe Fourier transforms of the Normalized Counting Measure, providing the linkswith the resolvent and the unitary group for the matrices in question, and is ratherefficient in the context. The main technical tool for the local regimes here is theorthogonal polynomial techniques, introduced in random matrix theory by Gaudin,Mehta, Wigner, and Dyson, based in fact on observations of analysts of the nine-teenth century.

Part 2 is on the Matrix Models (known also as the invariant ensembles) of her-mitian and real symmetric matrices. This class of random matrix ensembles shareswith the Gaussian Ensembles the property of invariance with respect to orthogonalor unitary transformations; however their entries are strongly dependent, unlikethose for the Gaussian Ensembles. The main technical tools here are variationalmethods and so-called determinantal formulas for marginal densities (correlationfunctions in statistical mechanics) of the joint eigenvalue distribution whose es-sential ingredients are special orthogonal polynomials, known as the polynomialsorthogonal with respect to varying weights. This leads to important representa-tions for relevant spectral characteristics, quite convenient for the large-n asymp-totic analysis of local regimes. One can then use the asymptotics of orthogonalpolynomials to complete the analysis. This strategy is used in Part 1, where onedeals with the classical polynomials whose asymptotics are well known. To use the

PREFACE xiii

strategy in the case of Matrix Models, one needs the asymptotics for the polyno-mials orthogonal with respect to varying weights. They are obtained and appliedto the study of the local regime of the Matrix Models in a series of recent works(see e.g. [152, 154, 162] and Chapter 14). We use these new asymptotics tostudy the fluctuations of linear eigenvalue statistics of Matrix Models in Chapter14. As for their local regimes, we carry out a direct analysis of determinantal for-mulas based essentially on spectral properties of Jacobi matrices, associated withthe corresponding orthogonal polynomials, rather than on their asymptotics.

Part 3 deals with ensembles determined by independent but not necessarilyGaussian random variables, mostly with real symmetric and hermitian Wigner ma-trices, whose upper triangular entries are independent, and with sample covariancematrices, for which the corresponding data matrices have independent entries. Westudy in detail the existence and properties of the limiting Normalized CountingMeasure of eigenvalues and the fluctuation of linear eigenvalue statistics by usingthe differentiation formulas, martingale-type bounds (instead of Poincaré-type in-equalities of Part 1), and an "interpolation trick", allowing us to use results on theclassical ensembles. As for the local regime, where considerable progress has beenachieved recently, we present a brief review of results obtained and methods usedin Sections 18.7 and 19.3.

Random matrix theory is the result of a nontrivial synthesis of ideas and con-structions from several branches of mathematics and physics. Therefore it employsa wide range of often specialized concepts and methods belonging to various fieldsthat have been traditionally only very tenuously related. For the same reason, itattracts the interest of scientists from a number of branches of mathematics andrelated sciences. Finally, the theory has accumulated a good deal of profound factsand interrelations between them, some of which have not yet been rigorously provedin the generality in which they are believed to be true. Because of the above andthe wide variety of recent developments, it seems hardly possible to present theessentials of the theory in a book of reasonable size by using the traditional styleof mathematical writing, where everything is proved in detail, thus comprising areasonably complete and self-contained text. We therefore depart sometimes fromthis style, basically in two cases. The first case is where we need certain resultsof analysis, probability, operator theory, etc. They are formulated without proofor with just the sketch of a proof, however with the appropriate references. Suchstatements are called propositions, in contrast to theorems and lemmas, which areproved in full. Other results, especially those obtained quite recently, are also justformulated or described, and their proofs, which are as a rule cumbersome and tech-nically complicated, are replaced by discussions of the main ideas involved. Resultsof this type are either presented as remarks, comments, sometimes problems andspecial sections that are more survey-like: for example, Comments 1.3.1 and 7.6.1,Section 18.7, and Problem 2.4.13. Note also that the importance and driving forceof random matrix theory are mostly due to its numerous and diverse applications.Their sufficiently comprehensive description requires much more space and exper-tise than we possess. This is why we mention this or that application and/or linkand provide a selection of references (mostly recent) after the presentation of thecorresponding result.

xiv PREFACE

We are aware that this type of presentation may not satisfy everyone, but wehope that our intention of giving a comprehensive impression of the subject willserve as at least a partial justification.

Likewise, we did our best to write a book that is of interest to a sufficiently wideaudience, but we could not avoid being subjective in the choice of results and refer-ences, determined to a large extent by our points of view and our works (mostly dueto spectral theory and mathematical physics) and the lack of space. We apologizestrongly for not including or/and mentioning many important contributions.

Our final remark concerns notation: throughout the book we write the integralwithout limits for the integral in the whole line and C, C1, etc., and c, c1, etc., forgeneric quantities which do not depend on the matrix size, special parameters, etc.,but whose values may be different in different formulas.

We would also like to thank the coauthors of our joint papers and numerouscolleagues with whom many ideas and results were obtained and discussed.

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Index

Airy kernel, 149, 199Anderson model, 511, 522

band matrices, 64Bessel kernel, 201Bogolyubov inequality, 322Boltzmann-Gibbs distribution, 122bulk, 26bulk of spectrum, 15Burkholder inequality, 528

Casimir operator, 214Cauchy distribution, 59, 218Cauchy Ensemble, 218Cayley transform, 217centered linear statistics, 17chiral random matrix, 181Christoffel-Darboux formula, 108Circular Ensembles, 211circular law, 24cluster functions, 8Combes-Thomas estimates, 431concentration inequalities, 215condition number, 181contracted zero distribution, 349convergence in distribution, 13correlation functions, xii, 8Counting Measure of eigenvalues, 6cumulants, 528

deformed GUE, 43deformed semicircle law, 43, 44, 535determinantal formulas, 110differentiation formula, 39, 212, 528distribution function, 12double scaling limit, 29Duhamel formula, 75Dyson Brownian Motion, 121

eigenvalue repulsion, 145empirical eigenvalue distribution, 54ensemble, xiequilibrium measure, 318ergodic operators, 442, 512

finite band Jacobi matrices, 440fractile, 66Fredholm determinant, 109free convolution, 275free probability, 275

gap probability, 7, 112Gaussian Ensembles, xiGaussian Orthogonal Ensemble (GOE), 2Gaussian Symplectic Ensemble (GSE), 2Gaussian Unitary Ensemble (GUE), 2Gaussian universality classes, 22generating functional, 8Glivenko-Cantelli theorem, 54global (or macroscopic) regime, 16global regime, 12Gram theorem, 110, 111Gumbell distribution, 582

Hadamard inequality, 145hard edge, 202Hastings-McLeod solution, 408, 409heavy tails, 569Heine formulas, 456Hermite polynomials, 129

Integrated Density of States, 442, 513intermediate regime, 18, 139interpolating random matrix, 535interpolation trick, xiii, 42invariant ensembles, 101isotropic vectors, 585Itsykson-Zuber formula, 127Itzykson–Zuber integral, 131

Jacobi Ensemble, 63, 195Jacobi matrix, 109, 438, 440, 444

Kirkwood-Salzburg equations, 226

Laguerre Ensemble, 6, 181Laguerre polynomials, 196Langevin equation, 118Laplace characteristic function, 455Laplace characteristic functional, 8

631

632 INDEX

large deviations, 15linear statistic, 6local bulk (or microscopic) regime, 16local hard edge regime, 151, 202local relaxation flow, 575local semicircle law, 158, 570local soft edge regime, 19, 150log-concave measure, 585logarithmic energy, 319

MANOVA, 180Marchenko-Pastur law, 188marginals, 7martingale bounds, 526Matrix Models, 5, 101microscopic limits, 22minimum energy problem, 318modified Robin constant, 347

Nevanlinna functions, 36noncommutative additive convolution, 287noncommutative multiplicative

convolution, 288Normalized Counting Measure of

eigenvalues, 6null or white Wishart distribution, 178

operator norm, 37Ornstein-Uhlenbeck process, 118

Painlevé equation, 29, 146, 152, 202, 409Pfaffian, 165Plancherel-Rotah formulas, 130Poincaré-Nash inequality, 40Poisson law, 15population covariance matrix, 179

quarter-circle law, 188quasi-Bloch generalized eigenfunctions, 441

R-transform, 287random matrix ensemble, xi, 1regular, 445reproducing kernel, 108resolvent, 37Riemann theta function, 440rigidity of spectrum, 147

sample, 178sample covariance matrix, 178–180sample mean, 180scaling limits, 22selfenergy, 523semicircle law, 48semicircle universality class, 23sin-kernel, 143sine-kernel, 142singular values, 178, 607special points, 15, 18spectrum, xi, 15

spiked population covariance matrix, 200Stieltjes transform, 35Stieltjes-Perron inversion formula, 35strong universality, 28subexponential decay, 571

tight sequence, 13tightness, 13Toeplitz determinant, 26, 152, 222

unitary Matrix Models, 485universality, 23, 140universality classes, 21, 23universality conjecture, 26

vague convergence, 13Vandermonde determinant, 110varying weight, 109

weak convergence, 12weak universality, 577Weibull distribution, 582weight, 108Wigner Ensembles, 5Wishart distribution, 179Wishart Ensemble, 5, 178, 179

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Random matrix theory is a wide and growing fi eld with a variety of concepts, results, and techniques and a vast range of applications in mathematics and the related sciences. The book, written by well-known experts, offers beginners a fairly balanced collection of basic facts and methods (Part 1 on classical ensembles) and presents experts with an exposition of recent advances in the subject (Parts 2 and 3 on invariant ensembles and ensembles with independent entries).

The text includes many of the authors’ results and methods on several main aspects of the theory, thus allowing them to present a unique and personal perspective on the subject and to cover many topics using a unifi ed approach essentially based on the Stieltjes transform and orthogonal polynomials. The exposition is supplemented by numerous comments, remarks, and problems. This results in a book that presents a detailed and self-contained treatment of the basic random matrix ensembles and asymptotic regimes.

This book will be an important reference for researchers in a variety of areas of mathematics and mathematical physics. Various chapters of the book can be used for graduate courses; the main prerequisite is a basic knowledge of calculus, linear algebra, and probability theory.