This book focuses on the behaviour of large random matrices. Standard results are covered, and the presentation emphasizes elementary operator theory and differential equations, so as to be accessible to graduate students and other non-experts. The introductory chapters review material on Lie groups and probability measures in a style suitable for applications in random matrix theory. Later chapters use modern convexity theory to establish subtle results about the convergence of eigenvalue distributions as the size of the matrices increases. Random matrices are viewed as geometrical objects with large dimension. The book analyzes the concentration of measure phenomenon, which describes how measures behave on geometrical objects with large dimension. To prove such results for random matrices, the book develops the modern theory of optimal transportation and proves the associated functional inequalities involving entropy and information. These include the logarithmic Sobolev inequality, which measures how fast some physical systems converge to equilibrium.

Introduction	p. 1
Metric measure spaces	p. 4
Weak convergence on compact metric spaces	p. 4
Invariant measure on a compact metric group	p. 10
Measures on non-compact Polish spaces	p. 16
The Brunn-Minkowski inequality	p. 22
Gaussian measures	p. 25
Surface area measure on the spheres	p. 27
Lipschitz functions and the Hausdorff metric	p. 31
Characteristic functions and Cauchy transforms	p. 33
Lie groups and matrix ensembles	p. 42
The classical groups, their eigenvalues and norms	p. 42
Determinants and functional calculus	p. 49
Linear Lie groups	p. 56
Connections and curvature	p. 63
Generalized ensembles	p. 66
The Weyl integration formula	p. 72
Dyson's circular ensembles	p. 78
Circular orthogonal ensemble	p. 81
Circular symplectic ensemble	p. 83
Entropy and concentration of measure	p. 84
Relative entropy	p. 84
Concentration of measure	p. 93
Transportation	p. 99
Transportation inequalities	p. 103
Transportation inequalities for uniformly convex potentials	p. 106
Concentration of measure in matrix ensembles	p. 109
Concentration for rectangular Gaussian matrices	p. 114
Concentration on the sphere	p. 123
Concentration for compact Lie groups	p. 126
Free entropy and equilibrium	p. 132
Logarithmic energy and equilibrium measure	p. 132
Energy spaces on the disc	p. 134
Free versus classical entropy on the spheres	p. 142
Equilibrium measures for potentials on the real line	p. 147
Equilibrium densities for convex potentials	p. 154
The quartic model with positive leading term	p. 159
Quartic models with negative leading term	p. 164
Displacement convexity and relative free entropy	p. 169
Toeplitz determinants	p. 172
Convergence to equilibrium	p. 177
Convergence to arclength	p. 177
Convergence of ensembles	p. 179
Mean field convergence	p. 183
Almost sure weak convergence for uniformly convex potentials	p. 189
Convergence for the singular numbers from the Wishart distribution	p. 193
Gradient flows and functional inequalities	p. 196
Variation of functionals and gradient flows	p. 196
Logarithmic Sobolev inequalities	p. 203
Logarithmic Sobolev inequalities for uniformly convex potentials	p. 206
Fisher's information and Shannon's entropy	p. 210
Free information and entropy	p. 213
Free logarithmic Sobolev inequality	p. 218
Logarithmic Sobolev and spectral gap inequalities	p. 221
Inequalities for Gibbs measures on Riemannian manifolds	p. 223
Young tableaux	p. 227
Group representations	p. 227
Young diagrams	p. 229
The Vershik ¿ distribution	p. 237
Distribution of the longest increasing subsequence	p. 243
Inclusion-exclusion principle	p. 250
Random point fields and random matrices	p. 253
Deterrninantal random point fields	p. 253
Deterrninantal random point fields on the real line	p. 261
Deterrninantal random point fields and orthogonal polynomials	p. 270
De Branges's spaces	p. 274
Limits of kernels	p. 278
Integrable operators and differential equations	p. 281
Integrable operators and Hankel integral operators	p. 281
Hankel integral operators that commute with second order differential operators	p. 289
Spectral bulk and the sine kernel	p. 293
Soft edges and the Airy kernel	p. 299
Hard edges and the Bessel kernel	p. 304
The spectra of Hankel operators and rational approximation	p. 310
The Tracy-Widom distribution	p. 315
Fluctuations and the Tracy-Widom distribution	p. 321
The Costin-Lebowitz central limit theorem	p. 321
Discrete Tracy-Widom systems	p. 327
The discrete Bessel kernel	p. 328
Plancherel measure on the partitions	p. 334
Fluctuations of the longest increasing subsequence	p. 343
Fluctuations of linear statistics over unitary ensembles	p. 345
Limit groups and Gaussian measures	p. 352
Some inductive limit groups	p. 352
Hua-Pickrell measure on the infinite unitary group	p. 357
Gaussian Hilbert space	p. 365
Gaussian measures and fluctuations	p. 369
Hermite polynomials	p. 373
Tensor products of Hilbert space	p. 373
Hermite polynomials and Mehler's formula	p. 375
The Ornstein-Uhlenbeck semigroup	p. 381
Hermite polynomials in higher dimensions	p. 384
From the Ornstein-Uhlenbeck process to the Burgers equation	p. 392
The Ornstein-Uhlenbeck process	p. 392
The logarithmic Sobolev inequality for the Ornstein-Uhlenbeck generator	p. 396
The matrix Ornstein-Uhlenbeck process	p. 398
Solutions for matrix stochastic differential equations	p. 401
The Burgers equation	p. 408
Noncommutative probability spaces	p. 411
Noncommutative probability spaces	p. 411
Tracial probability spaces	p. 414
The semicircular distribution	p. 418
References	p. 424
Index	p. 433
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