| 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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