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2004 Large deviations and stochastic calculus for large random matrices
Alice Guionnet
Probab. Surveys 1: 72-172 (2004). DOI: 10.1214/154957804100000033

Abstract

Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random matrices. We shall describe how such issues are motivated either in physics/combinatorics by the study of the so-called matrix models or in free probability by the definition of a non-commutative entropy. We shall show how classical large deviations techniques can be used in this context.

These lecture notes are supposed to be accessible to non probabilists and non free-probabilists.

Citation

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Alice Guionnet. "Large deviations and stochastic calculus for large random matrices." Probab. Surveys 1 72 - 172, 2004. https://doi.org/10.1214/154957804100000033

Information

Published: 2004
First available in Project Euclid: 8 November 2004

zbMATH: 1189.60059
MathSciNet: MR2095566
Digital Object Identifier: 10.1214/154957804100000033

Subjects:
Primary: 15A52 , 60F10
Secondary: 46L50

Keywords: Integration , large deviations , non-commutative measure , random matrices

Rights: Copyright © 2004 The Institute of Mathematical Statistics and the Bernoulli Society

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