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 IntroductionFront Matter Wigner’s theorem Wigner's matrices; more moments estimates Words in several independent Wigner matricesFront Matter Concentration inequalities and logarithmic Sobolev inequalities Generalizations Concentration inequalities for random matricesFront Matter Maps and Gaussian calculus First-order expansion Second-order expansion for the free energyFront Matter Large deviations for the law of the spectral measure of Gaussian Wigner's matrices Large Deviations of the Maximum EigenvalueFront 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 integralsFront Matter Free probability setting Freeness Free entropyFront Matter Basics of matricesFront Matter Basics of probability theory Back Matter
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