Large Random Matrices: Lectures on Macroscopic Asymptotics: École d’Été de Probabilités de Saint-Flour XXXVI ¿ 2006

Alice Guionnet (auth.)9783540698968, 3540698965

Random matrix theory has developed in the last few years, in connection with various fields of mathematics and physics. These notes emphasize the relation with the problem of enumerating complicated graphs, and the related large deviations questions. Such questions are also closely related with the asymptotic distribution of matrices, which is naturally defined in the context of free probability and operator algebra.
The material of this volume is based on a series of nine lectures given at the Saint-Flour Probability Summer School 2006. Lectures were also given by Maury Bramson and Steffen Lauritzen.

Table of contents :
Front Matter….Pages I-XII
Introduction….Pages 1-4
Front Matter….Pages 5-5
Wigner’s theorem….Pages 7-28
Wigner’s matrices; more moments estimates….Pages 29-40
Words in several independent Wigner matrices….Pages 41-46
Front Matter….Pages 47-48
Concentration inequalities and logarithmic Sobolev inequalities….Pages 49-57
Generalizations….Pages 59-64
Concentration inequalities for random matrices….Pages 65-87
Front Matter….Pages 89-91
Maps and Gaussian calculus….Pages 93-107
First-order expansion….Pages 109-118
Second-order expansion for the free energy….Pages 121-145
Front Matter….Pages 147-148
Large deviations for the law of the spectral measure of Gaussian Wigner’s matrices….Pages 149-158
Large Deviations of the Maximum Eigenvalue….Pages 159-162
Front Matter….Pages 166-166
Stochastic analysis for random matrices….Pages 167-181
Large deviation principle for the law of the spectral measure of shifted Wigner matrices….Pages 183-209
Asymptotics of Harish-Chandra-Itzykson-Zuber integrals and of Schur polynomials….Pages 211-216
Asymptotics of some matrix integrals….Pages 217-224
Front Matter….Pages 266-266
Free probability setting….Pages 227-230
Freeness….Pages 231-244
Free entropy….Pages 245-259
Front Matter….Pages 261-261
Basics of matrices….Pages 263-265
Front Matter….Pages 261-261
Basics of probability theory….Pages 265-275
Back Matter….Pages 275-301