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
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
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
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