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Large deviations and stochastic calculus for large random matrices. (English) Zbl 1189.60059

Summary: 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 a lot of interest, in particular due to a series 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 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 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 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.

MSC:

60F10 Large deviations
15B52 Random matrices (algebraic aspects)
46L54 Free probability and free operator algebras
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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