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.
Reviews / Votes
From the reviews:
"This book is a set of lecture notes on eigenvalues of large random matrices. . useful to all mathematicians and statisticians who are interested in Wigner matrices. . In summary, the book is very much worth perusal." (Vladislav Kargin, Mathematical Reviews, Issue 2010 d)
Series
Edition
Language
Place of publication
Publishing group
Target group
Professional and scholarly
Research
Illustrations
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 18 mm
Weight
ISBN-13
978-3-540-69896-8 (9783540698968)
DOI
10.1007/978-3-540-69897-5
Schweitzer Classification
Gaëtan Borot graduated at ENS Paris in theoretical physics, did his PhD at CEA Saclay, and is now a W2 Group Leader at the Max Planck Institute for Mathematics in Bonn. He was also a visiting scholar at MIT, collaborating with Alice Guionnet on the asymptotic analysis of random matrix models. He is working on the mathematical aspects of geometry and physics, ranging from statistical physics, random matrices, integrable systems, enumerative geometry, topological quantum field theories, etc.
Alice Guionnet is Director of research CNRS at École Normale Supérieure (ENS) Lyon, from MIT where she served as a professor in 2012-2015. She received the MS from ENS Paris in 1993 and the PhD, under the guidance of G. Ben Arous at Université Paris Sud in 1995.A. Guionnet is a world leading probabilist, working on a program related to operator algebra theory and mathematical physics. She has made important contributions in random matrix theory,including large deviations, topological expansions, but also more classical study of their spectrum and eigenvectors. From 2006-2011 she served as Editor-in-Chief of Annales de L'Institut Henri Poincaré (currently on its editorial board), and also serves on the editorial board of Annals of Probability.She has given two Plenary talks and a number of Invited Talks at international meetings, including ICM. Her distinctions include the Miller Institute Fellowship, (2006), the Loève Prize (2009), the Silver Medal of CNRS (2010) and Simon Investigator (2012).
Karol Kajetan Kozlowski is a CNRS Chargé de recherche at the École Normale Supérieure (ENS) Lyon. He graduated from ENS-Lyon in 2005 and did his PhD at the Laboratoire Physique of ENS-Lyon. He was then a post-doctoral fellow at the Deutsches Elektronen-Synchrotron. His main research interest concern quantum integrable models and various aspects of asymptotic analysis.
Wigner matrices and moments estimates.- Wigner#x2019;s theorem.- Wigner's matrices; more moments estimates.- Words in several independent Wigner matrices.- Wigner matrices and concentration inequalities.- Concentration inequalities and logarithmic Sobolev inequalities.- Generalizations.- Concentration inequalities for random matrices.- Matrix models.- Maps and Gaussian calculus.- First-order expansion.- Second-order expansion for the free energy.- Eigenvalues of Gaussian Wigner matrices and large deviations.- Large deviations for the law of the spectral measure of Gaussian Wigner's matrices.- Large Deviations of the Maximum Eigenvalue.- Stochastic calculus.- 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.- Free probability.- Free probability setting.- Freeness.- Free entropy.- Basics of matrices.- Basics of probability theory.