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Preface | p. xiii |
Introduction | p. 1 |
Real and complex Wigner matrices | p. 6 |
Real Wigner matrices: traces, moments and combinatorics | p. 6 |
The semicircle distribution, Catalan numbers and Dyck paths | p. 7 |
Proof #1 of Wigner's Theorem 2.1.1 | p. 10 |
Proof of Lemma 2.1.6: words and graphs | p. 11 |
Proof of Lemma 2.1.7: sentences and graphs | p. 17 |
Some useful approximations | p. 21 |
Maximal eigenvalues and Füredi-Komlós enumeration | p. 23 |
Central limit theorems for moments | p. 29 |
Complex Wigner matrices | p. 35 |
Concentration for functionals of random matrices and logarithmic Sobolev inequalities | p. 38 |
Smoothness properties of linear functions of the empirical measure | p. 38 |
Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities | p. 39 |
Concentration for Wigner-type matrices | p. 42 |
Stieltjes transforms and recursions | p. 43 |
Gaussian Wigner matrices | p. 45 |
General Wigner matrices | p. 47 |
Joint distribution of eigenvalues in the GOE and the GUE | p. 50 |
Definition and preliminary discussion of the GOE and the GUE | p. 51 |
Proof of the joint distribution of eigenvalues | p. 54 |
Selberg's integral formula and proof of (2.5.4) | p. 58 |
Joint distribution of eigenvalues: alternative formulation | p. 65 |
Superposition and decimation relations | p. 66 |
Large deviations for random matrices | p. 70 |
Large deviations for the empirical measure | p. 71 |
Large deviations for the top eigenvalue | p. 81 |
Bibliographical notes | p. 85 |
Hermite polynomials, spacings and limit distributions for the Gaussian ensembles | p. 90 |
Summary of main results: spacing distributions in the bulk and edge of the spectrum for the Gaussian ensembles | p. 90 |
Limit results for the GUE | p. 90 |
Generalizations: limit formulas for the GOE and GSE | p. 93 |
Hermite polynomials and the GUE | p. 94 |
The GUE and determinantal laws | p. 94 |
Properties of the Hermite polynomials and oscillator wave-functions | p. 99 |
The semicircle law revisited | p. 101 |
Calculation of moments of LN | p. 102 |
The Harer-Zagier recursion and Ledoux's argument | p. 103 |
Quick introduction to Fredholm determinants | p. 107 |
The setting, fundamental estimates and definition of the Fredholm determinant | p. 107 |
Definition of the Fredholm adjugant, Fredholm resolvent and a fundamental identity | p. 110 |
Gap probabilities at 0 and proof of Theorem 3.1.1 | p. 114 |
The method of Laplace | p. 115 |
Evaluation of the scaling limit: proof of Lemma 3.5.1 | p. 117 |
A complement: determinantal relations | p. 120 |
Analysis of the sine-kernel | p. 121 |
General differentiation formulas | p. 121 |
Derivation of the differential equations: proof of Theorem 3.6.1 | p. 126 |
Reduction to Painlevé V | p. 128 |
Edge-scaling: proof of Theorem 3.1.4 | p. 132 |
Vague convergence of the largest eigenvalue: proof of Theorem 3.1.4 | p. 133 |
Steepest descent: proof of Lemma 3.7.2 | p. 134 |
Properties of the Airy functions and proof of Lemma 3.7.1 | p. 139 |
Analysis of the Tracy-Widom distribution and proof of Theorem 3.1.5 | p. 142 |
The first standard moves of the game | p. 144 |
The wrinkle in the carpet | p. 144 |
Linkage to Painlevé II | p. 146 |
Limiting behavior of the GOE and the GSE | p. 148 |
Pfaffians and gap probabilities | p. 148 |
Fredholm representation of gap probabilities | p. 155 |
Limit calculations | p. 160 |
Differential equations | p. 170 |
Bibliographical notes | p. 181 |
Some generalities | p. 186 |
Joint distribution of eigenvalues in the classical matrix ensembles | p. 187 |
Integration formulas for classical ensembles | p. 187 |
Manifolds, volume measures and the coarea formula | p. 193 |
An integration formula of Weyl type | p. 199 |
Applications of Weyl's formula | p. 206 |
Determinantal point processes | p. 214 |
Point processes: basic definitions | p. 215 |
Determinantal processes | p. 220 |
Determinantal projections | p. 222 |
The CLT for determinantal processes | p. 227 |
Determinantal processes associated with eigenvalues | p. 228 |
Translation invariant determinantal processes | p. 232 |
One-dimensional translation invariant determinantal processes | p. 237 |
Convergence issues | p. 241 |
Examples | p. 243 |
Stochastic analysis for random matrices | p. 248 |
Dyson's Brownian motion | p. 249 |
A dynamical version of Wigner's Theorem | p. 262 |
Dynamical central limit theorems | p. 273 |
Large deviation bounds | p. 277 |
Concentration of measure and random matrices | p. 281 |
Concentration inequalities for Hermitian matrices with independent entries | p. 282 |
Concentration inequalities for matrices with dependent entries | p. 287 |
Tridiagonal matrix models and the ß ensembles | p. 302 |
Tridiagonal representation of ß ensembles | p. 303 |
Scaling limits at the edge of the spectrum | p. 306 |
Bibliographical notes | p. 318 |
Free probability | p. 322 |
Introduction and main results | p. 323 |
Noncommutative laws and noncommutative probability spaces | p. 325 |
Algebraic noncommutative probability spaces and laws | p. 325 |
C*-probability spaces and the weak*-topology | p. 329 |
W*-probability spaces | p. 339 |
Free independence | p. 348 |
Independence and free independence | p. 348 |
Free independence and combinatorics | p. 354 |
Consequence of free independence: free convolution | p. 359 |
Free central limit theorem | p. 368 |
Freeness for unbounded variables | p. 369 |
Link with random matrices | p. 374 |
Convergence of the operator norm of polynomials of independent GUE matrices | p. 394 |
Bibliographical notes | p. 410 |
Appendices | p. 414 |
Linear algebra preliminaries | p. 414 |
Identities and bounds | p. 414 |
Perturbations for normal and Hermitian matrices | p. 415 |
Noncommutative matrix Lp-norms | p. 416 |
Brief review of resultants and discriminants | p. 417 |
Topological preliminaries | p. 418 |
Generalities | p. 418 |
Topological vector spaces and weak topologies | p. 420 |
Banach and Polish spaces | p. 422 |
Some elements of analysis | p. 423 |
Probability measures on Polish spaces | p. 423 |
Generalities | p. 423 |
Weak topology | p. 425 |
Basic notions of large deviations | p. 427 |
The skew field H of quaternions and matrix theory over F | p. 430 |
Matrix terminology over F and factorization theorems | p. 431 |
The spectral theorem and key corollaries | p. 433 |
A specialized result on projectors | p. 434 |
Algebra for curvature computations | p. 435 |
Manifolds | p. 437 |
Manifolds embedded in Euclidean space | p. 438 |
Proof of the coarea formula | p. 442 |
Metrics, connections, curvature, Hessians, and the Laplace-Beltrami operator | p. 445 |
Appendix on operator algebras | p. 450 |
Basic definitions | p. 450 |
Spectral properties | p. 452 |
States and positivity | p. 454 |
von Neumann algebras | p. 455 |
Noncommutative functional calculus | p. 457 |
Stochastic calculus notions | p. 459 |
References | p. 465 |
General conventions and notation | p. 481 |
Index | p. 484 |
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