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An Introduction to Random Matrices.pdf

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    Cover
    Half-title
    Series-title
    Title
    Copyright
    Dedication
    Contents
    Preface
    1 Introduction
    2 Real and complex Wigner matrices
    2.1 Real Wigner matrices: traces, moments and combinatorics
    2.1.1 The semicircle distribution, Catalan numbers and Dyck paths
    2.1.2 Proof #1 of Wigner’s Theorem 2.1.1
    2.1.3 Proof of Lemma 2.1.6: words and graphs
    2.1.4 Proof of Lemma 2.1.7: sentences and graphs
    2.1.5 Some useful approximations
    2.1.6 Maximal eigenvalues and F¨uredi–Koml´os enumeration
    2.1.7 Central limit theorems for moments
    2.2 Complex Wigner matrices
    2.3 Concentration for functionals of random matrices and logarithmic Sobolev inequalities
    2.3.1 Smoothness properties of linear functions of the empirical measure
    2.3.2 Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities
    2.3.3 Concentration for Wigner-type matrices
    2.4 Stieltjes transforms and recursions
    2.4.1 Gaussian Wigner matrices
    2.4.2 General Wigner matrices
    2.5 Joint distribution of eigenvalues in the GOE and the GUE
    2.5.1 Definition and preliminary discussion of the GOE and the GUE
    2.5.2 Proof of the joint distribution of eigenvalues
    2.5.3 Selberg’s integral formula and proof of (2.5.4)
    2.5.4 Joint distribution of eigenvalues: alternative formulation
    2.5.5 Superposition and decimation relations
    2.6 Large deviations for random matrices
    2.6.1 Large deviations for the empirical measure
    Exponential tightness
    A large deviation upper bound
    A large deviation lower bound
    2.6.2 Large deviations for the top eigenvalue
    2.7 Bibliographical notes
    3 Hermite polynomials, spacings and limit distributions for the Gaussian ensembles
    3.1 Summary of main results: spacing distributions in the bulk and edge of the spectrum for the Gaussian ensembles
    3.1.1 Limit results for the GUE
    3.1.2 Generalizations: limit formulas for the GOE and GSE
    3.2 Hermite polynomials and the GUE
    3.2.1 The GUE and determinantal laws
    3.2.2 Properties of the Hermite polynomials and oscillator wave-functions
    3.3 The semicircle law revisited
    3.3.1 Calculation of moments of LN
    3.3.2 The Harer–Zagier recursion and Ledoux’s argument
    3.4 Quick introduction to Fredholm determinants
    3.4.1 The setting, fundamental estimates and definition of the Fredholm determinant
    3.4.2 Definition of the Fredholm adjugant, Fredholm resolvent and a fundamental identity
    Multiplicativity of Fredholm determinants
    3.5 Gap probabilities at 0 and proof of Theorem 3.1.1
    3.5.1 The method of Laplace
    3.5.2 Evaluation of the scaling limit: proof of Lemma 3.5.1
    3.5.3 A complement: determinantal relations
    3.6 Analysis of the sine-kernel
    3.6.1 General differentiation formulas
    3.6.2 Derivation of the differential equations: proof of Theorem 3.6.1
    3.6.3 Reduction to Painlev´e V
    3.7 Edge-scaling: proof of Theorem 3.1.4
    3.7.1 Vague convergence of the largest eigenvalue: proof of Theorem 3.1.4
    3.7.2 Steepest descent: proof of Lemma 3.7.2
    3.7.3 Properties of the Airy functions and proof of Lemma 3.7.1
    3.8 Analysis of the Tracy–Widom distribution and proof of Theorem 3.1.5
    3.8.1 The first standard moves of the game
    3.8.2 The wrinkle in the carpet
    3.8.3 Linkage to Painlev´e II
    3.9 Limiting behavior of the GOE and the GSE
    3.9.1 Pfaffians and gap probabilities
    Pfaffian integration formulas
    Determinant formulas for squared gap probabilities
    3.9.2 Fredholm representation of gap probabilities
    Matrix kernels and a revision of the Fredholm setup
    Main results
    3.9.3 Limit calculations
    Statements of main results
    Proofs of bulk results
    3.9.4 Differential equations
    Block matrix calculations
    Proof of Theorem 3.1.6
    Proof of Theorem 3.1.7
    3.10 Bibliographical notes
    4 Some generalities
    4.1 Joint distribution of eigenvalues in the classical matrix ensembles
    4.1.1 Integration formulas for classical ensembles
    The Gaussian ensembles
    Laguerre ensembles and Wishart matrices
    Jacobi ensembles and random projectors
    The classical compact Lie groups
    4.1.2 Manifolds, volume measures and the coarea formula
    4.1.3 An integration formula of Weyl type
    4.1.4 Applications of Weyl’s formula
    4.2 Determinantal point processes
    4.2.1 Point processes: basic definitions
    4.2.2 Determinantal processes
    4.2.3 Determinantal projections
    4.2.4 The CLT for determinantal processes
    4.2.5 Determinantal processes associated with eigenvalues
    The sine process
    The Airy process
    4.2.6 Translation invariant determinantal processes
    4.2.7 One-dimensional translation invariant determinantal processes
    4.2.8 Convergence issues
    4.2.9 Examples
    The biorthogonal ensembles
    Birth–death processes conditioned not to intersect
    4.3 Stochastic analysis for random matrices
    4.3.1 Dyson’s Brownian motion
    4.3.2 A dynamical version of Wigner’s Theorem
    4.3.3 Dynamical central limit theorems
    4.3.4 Large deviation bounds
    4.4 Concentration of measure and random matrices
    4.4.1 Concentration inequalities for Hermitian matrices with independent entries
    Entries satisfying Poincare’s inequality
    Matrices with bounded entries and Talagrand’s method
    4.4.2 Concentration inequalities for matrices with dependent entries
    The setup with M = Rm and µ=Lebesgue measure
    The setup with M a compact Riemannian manifold
    Applications to random matrices
    4.5 Tridiagonal matrix models and the β ensembles
    4.5.1 Tridiagonal representation of β ensembles
    4.5.2 Scaling limits at the edge of the spectrum
    4.6 Bibliographical notes
    5 Free probability
    5.1 Introduction and main results
    5.2 Noncommutative laws and noncommutative probability spaces
    5.2.1 Algebraic noncommutative probability spaces and laws
    5.2.2 C-probability spaces and the weak*-topology
    C-probability spaces
    Weak*-topology
    5.2.3 W-probability spaces
    Laws of self-adjoint operators
    5.3 Free independence
    5.3.1 Independence and free independence
    5.3.2 Free independence and combinatorics
    Basic properties of non-crossing partitions
    Free cumulants and freeness
    5.3.3 Consequence of free independence: free convolution
    Multiplicative free convolution
    5.3.4 Free central limit theorem
    5.3.5 Freeness for unbounded variables
    5.4 Link with random matrices
    5.5 Convergence of the operator norm of polynomials of independent GUE matrices
    5.6 Bibliographical notes
    Appendices
    A Linear algebra preliminaries
    A.1 Identities and bounds
    A.2 Perturbations for normal and Hermitian matrices
    A.3 Noncommutative matrix Lp-norms
    A.4 Brief review of resultants and discriminants
    B Topological preliminaries
    B.1 Generalities
    B.2 Topological vector spaces and weak topologies
    B.3 Banach and Polish spaces
    B.4 Some elements of analysis
    C Probability measures on Polish spaces
    C.1 Generalities
    C.2 Weak topology
    D Basic notions of large deviations
    E The skew field H of quaternions and matrix theory over F
    E.1 Matrix terminology over F and factorization theorems
    E.2 The spectral theorem and key corollaries
    E.3 A specialized result on projectors
    E.4 Algebra for curvature computations
    F Manifolds
    F.1 Manifolds embedded in Euclidean space
    “Critical” vocabulary
    Lie groups and Haar measure
    F.2 Proof of the coarea formula
    F.3 Metrics, connections, curvature, Hessians, and the Laplace–Beltrami operator
    Curvature of classical compact Lie groups
    G Appendix on operator algebras
    G.1 Basic definitions
    G.2 Spectral properties
    G.3 States and positivity
    G.4 von Neumann algebras
    G.5 Noncommutative functional calculus
    H Stochastic calculus notions
    References
    General conventions and notation
    Index
    This page intentionally left blank
    CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 118 Editorial Board B. BOLLOB ´AS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO An Introduction to Random Matrices The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader’s understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence. greg w. anderson is Professor of Mathematics at the University of Minnesota. alice guionnet is CNRS Research Director at the Ecole Normale Sup´erieure in Lyon (ENS-Lyon). ofer zeitouni is Professor of Mathematics at both the University of Minnesota and the Weizmann Institute of Science in Rehovot, Israel.
    CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS Editorial Board: B. Bollob´as, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: www.cambridge.org/series/sSeries.asp?code=CSAM Already published 65 A. J. Berrick & M. E. Keating An introduction to rings and modules with K-theory in view 66 S. Morosawa et al. Holomorphic dynamics 67 A. J. Berrick & M. E. Keating Categories and modules with K-theory in view 68 K. Sato L´evy processes and infinitely divisible distributions 69 H. Hida Modular forms and Galois cohomology 70 R. Iorio & V. Iorio Fourier analysis and partial differential equations 71 R. Blei Analysis in integer and fractional dimensions 72 F. Borceux & G. Janelidze Galois theories 73 B. Bollob´as Random graphs (2nd Edition) 74 R. M. Dudley Real analysis and probability (2nd Edition) 75 T. Sheil-Small Complex polynomials 76 C. Voisin Hodge theory and complex algebraic geometry, I 77 C. Voisin Hodge theory and complex algebraic geometry, II 78 V. Paulsen Completely bounded maps and operator algebras 79 F. Gesztesy & H. Holden Soliton equations and their algebro-geometric solutions, I 81 S. Mukai An introduction to invariants and moduli 82 G. Tourlakis Lectures in logic and set theory, I 83 G. Tourlakis Lectures in logic and set theory, II 84 R. A. Bailey Association schemes 85 J. Carlson, S. M¨uller-Stach & C. Peters Period mappings and period domains 86 J. J. Duistermaat & J. A. C. Kolk Multidimensional real analysis, I 87 J. J. Duistermaat & J. A. C. Kolk Multidimensional real analysis, II 89 M. C. Golumbic & A. N. Trenk Tolerance graphs 90 L. H. Harper Global methods for combinatorial isoperimetric problems 91 I. Moerdijk & J. Mrˇcun Introduction to foliations and Lie groupoids 92 J. Koll´ar, K. E. Smith & A. CortiRational and nearly rational varieties 93 D. Applebaum L´evy processes and stochastic calculus (1st Edition) 94 B. Conrad Modular forms and the Ramanujan conjecture 95 M. Schechter An introduction to nonlinear analysis 96 R. Carter Lie algebras of finite and affine type 97 H. L. Montgomery & R. C. Vaughan Multiplicative number theory, I 98 I. Chavel Riemannian geometry (2nd Edition) 99 D. Goldfeld Automorphic forms and L-functions for the group GL(n,R) 100 M. B. Marcus & J. Rosen Markov processes, Gaussian processes, and local times 101 P. Gille & T. Szamuely Central simple algebras and Galois cohomology 102 J. Bertoin Random fragmentation and coagulation processes 103 E. Frenkel Langlands correspondence for loop groups 104 A. Ambrosetti & A. Malchiodi Nonlinear analysis and semilinear elliptic problems 105 T. Tao & V. H. Vu Additive combinatorics 106 E. B. Davies Linear operators and their spectra 107 K. Kodaira Complex analysis 108 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Harmonic analysis on finite groups 109 H. Geiges An introduction to contact topology 110 J. Faraut Analysis on Lie groups: An Introduction 111 E. Park Complex topological K-theory 112 D. W. Stroock Partial differential equations for probabilists 113 A. Kirillov, Jr An introduction to Lie groups and Lie algebras 114 F. Gesztesy et al. Soliton equations and their algebro-geometric solutions, II 115 E. de Faria & W. de Melo Mathematical tools for one-dimensional dynamics 116 D. Applebaum L´evy processes and stochastic calculus (2nd Edition) 117 T. Szamuely Galois groups and fundamental groups
    An Introduction to Random Matrices GREG W. ANDERSON University of Minnesota ALICE GUIONNET Ecole Normale Sup´erieure de Lyon OFER ZEITOUNI University of Minnesota and Weizmann Institute of Science
    CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521194525 © G. W. Anderson, A. Guionnet and O. Zeitouni 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2009 ISBN-13 978-0-511-78780-5 eBook (EBL) ISBN-13 978-0-521-19452-5 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
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