280 Index

determinantal integration formula, 195

Dirac distribution, 11

discrete uniform distribution, 11

disintegration, 28

distance between random vector and

subspace, 78

dual basis, 219

Duhamel formula, 256

Dyck word, 122

Dyson operator, 248

empirical spectral distribution, 135, 162

entropy formula, 38

entropy function, 38

epsilon net argument, 110, 212

ESD, 135

Ess´ een concentration inequality, 85

essential range, 29

event, 2

expectation, 13

expected empirical spectral

distribution, 162

exponential moment, 15

exponential moment method, 62, 64, 71

extension (probability space), 3

faithfulness, 163

Fekete points, 266

first moment method, 14, 57

Fock space, 176

Fokker-Planck equation, 104

Fourier moment, 15

free central limit theorem, 182

free convolution, 178

free cumulant, 181

free independence, 157, 174

free probability, 157

free product, 176

Frobenius inner product, 254

Frobenius norm, 47, 114

Fubini-Tonelli theorem, 31

functional calculus, 168

Gamma function, 36

Gaudin-Mehta formula, 196, 259

Gaussian, 12

Gaussian concentration inequality, 71

Gaussian ensemble, 105

Gaussian Orthogonal Ensemble (GOE),

105

Gaussian symmetrisation inequality, 112

Gaussian Unitary Ensemble (GUE),

106, 259

Gelfand-Naimark theorem, 170

Gelfand-Tsetlin pattern, 50

generic chaining, 111

geometric distribution, 11

Ginibre formula, 183, 191, 251

GNS construction, 169

Golden-Thompson inequality, 253

H¨ older inequality, 54

Hadamard product, 111

Hadamard variation formula, 49

harmonic oscillator, 200, 259

heat equation, 243

Herbst’s argument, 77

Herglotz function, 144

Herglotz representation theorem, 144

Hermite differential equation, 200

Hermite polynomial, 194, 259, 267

Hermite recurrence relation, 199

high probability, 6

Hilbert-Schmidt inner product, 254

Hilbert-Schmidt norm, 47, 114

hodograph transform, 154

Hoeffding’s inequality, 63

Hoeffding’s lemma, 61

Horn’s conjecture, 39

iid, 23

iid matrices, 105

incompressible, 213, 215

independence, 19

indicator function, 8

inner product, 163

inverse Littlewood-Offord problem, 216

inverse moment problem, 91

Ito calculus, 242

Ito’s formula, 242

Jensen’s inequality, 18

Johansson formula, 250

joint distribution, 19

joint independence, 19

Ky Fan inequality, 40

L´ evy’s continuity theorem, 83, 84

Lagrange multiplier, 41

Laplace’s method, 38

large deviation inequality, 17, 56, 60

law, 10

law of large numbers (strong), 66