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 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
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