Index ∗-algebra, 160 2-Schatten norm, 114 R-transform, 180 S-transform, 181 k-point correlation function, 196 absolutely continuous, 12 absolutely integrable, 13, 16 Airy function, 206, 260 Airy kernel, 206 almost sure convergence, 32, 135 almost surely, 6 amalgamated free product, 176 antichain, 215 approximation to the identity, 144 asymptotic free independence, 174 asymptotic notation, 5 asymptotically almost surely, 6 Atiyah convexity theorem, 44 augmented matrix, 133 Azuma’s inequality, 63 Bai-Yin theorem, 129, 131 Baker-Campbell-Hausdorff formula, 252 Bernoulli distribution, 11 Bernoulli ensemble, 105 Berry-Ess´ een theorem, 87, 95, 100, 101, 213 bijective proof, 122 Bochner’s theorem, 257 Borel-Cantelli lemma, 14, 34 Brown measure, 232, 233 Burgers’ equation, 153 Carleman continuity theorem, 90 Catalan number, 122 Cauchy transform, 142 Cauchy-Binet formula, 253 Cauchy-Schwarz inequality, 164 central limit theorem, 79, 86 characteristic function, 15 Chebyshev’s inequality, 17 Chernoff bound, 79 Chernoff inequality, 61 Christoffel-Darboux formula, 197 circular law, 193, 209, 224 circular unitary ensemble, 268 classical probability theory, 160 compressible, 213, 215, 218 concentration of measure, 56 condition number, 209 conditional expectation, 27 conditional probability, 27 conditional probability measure, 24 conditioning, 24, 29 convergence in distribution, 32 convergence in expectation, 135 convergence in probability, 32, 135 convergence in the sense of ∗-moments, 172 convergence in the sense of moments, 172 correlation, 58 Courant-Fischer minimax theorem, 42 covariance matrix, 86 cumulant, 180 cumulative distribution function, 12 279
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