Index 281

law of large numbers (weak), 66

least singular value, 209

Leibniz formula for determinants, 250

Lidskii inequality, 46

Lindeberg condition, 86

Lindeberg trick, 93

Lindskii inequality, 45

linearity of convergence, 80

linearity of expectation, 14

log-Sobolev inequality, 75

logarithmic potential, 228

logarithmic potential continuity

theorem, 228

Markov inequality, 14

McDiarmid’s inequality, 69

mean field approximation, 192

measurable space, 7

median, 18

method of stationary phase, 38

method of steepest descent, 203

moment, 15

moment map, 44

moment method, 89, 115

negative moment, 15

negative second moment identity, 217

neighbourhood recurrence, 245

Neumann series, 164, 225

Newton potential, 193

Newtonian potential, 228

non-commutative H¨ older inequality, 47,

255

non-commutative probability space,

161, 169

non-negativity, 163

normal distribution, 12

normal matrix, 52

nuclear norm, 47

operator norm, 106

Ornstein-Uhlenbeck equation, 245

Ornstein-Uhlenbeck operator, 103

Ornstein-Uhlenbeck process, 104, 244

orthogonal polynomials, 194

overwhelming probability, 6

pairwise independence, 19

Paley-Zygmund inequality, 18

partial flag, 44

partial trace, 43

permutahedron, 44

Poisson distribution, 11

Poisson kernel, 144

polynomial rate, 151

predecessor comparison, 145

principal minor, 50

probabilistic way of thinking, 4

probability density function, 12

probability distribution, 10

probability measure, 2

probability space, 2

Prokhorov’s theorem, 33

pseudospectrum, 225

pushforward, 10

random matrices, 10

random sign matrices, 105

random variable, 7

Rayleigh quotient, 43

recurrence, 245

regular conditional probability, 29

resolvent, 15, 143

Riemann-Lebesgue lemma, 82

sample space, 2

scalar random variable, 8

scale invariance, 80

Schatten norm, 46, 53, 254

Schubert variety, 44

Schur complement, 147

Schur-Horn inequality, 44

Schur-Horn theorem, 44

second moment method, 17, 58

self-consistent equation, 150

semi-classical analysis, 201

semicircular element, 173

singular value decomposition, 51

singular vector, 51

singularity probability, 214

spectral instability, 224

spectral measure, 167

spectral projection, 41, 48

spectral theorem, 39, 41, 167

spectral theory, 160

spectrum, 39

Sperner’s theorem, 215

Stein continuity theorem, 98

Stein transform, 99

Stieltjes continuity theorem, 144

Stieltjes transform, 143, 164, 178

Stirling’s formula, 35

stochastic calculus, 236, 244

stochastic process, 237

sub-exponential tail, 16