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Guionnet A. Large Random Matrices: Lectures on Macroscopic Asymptotics

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Guionnet A. Large Random Matrices: Lectures on Macroscopic Asymptotics
Berlin: Springer, 2009. — 284 p.
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.
Contents :

Front Matter
Introduction
Front Matter
Wigner’s theorem
Wigner's matrices; more moments estimates
Words in several independent Wigner matrices
Front Matter
Concentration inequalities and logarithmic Sobolev inequalities
Generalizations
Concentration inequalities for random matrices
Front Matter
Maps and Gaussian calculus
First-order expansion
Second-order expansion for the free energy
Front Matter
Large deviations for the law of the spectral measure of Gaussian Wigner's matrices
Large Deviations of the Maximum Eigenvalue
Front Matter
Stochastic analysis for random matrices
Large deviation principle for the law of the spectral measure of shifted Wigner matrices
Asymptotics of Harish-Chandra-Itzykson-Zuber integrals and of Schur polynomials
Asymptotics of some matrix integrals
Front Matter
Free probability setting
Freeness
Free entropy
Front Matter
Basics of matrices
Front Matter
Basics of probability theory
Back Matter
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