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Large Deviations in Random Matrix Theory

Project description

Developing a theory of large devaiations to include complex functions of independent variables

The theory of large deviations is concerned with estimating the probability of rare events. The theory can address the question of how quickly the probability of observing a behaviour different than the one predicted by the law of large numbers decays to zero. Despite progress in the field, classical results from large deviations do not apply to complex functions of independent variables such as the eigenvalues of random matrices. The aim of the EU-funded LDRaM project is to develop a theory that describes a broad range of random matrix models. The project's results will have important implications not only for probability theory and operator algebra but also for theoretical physics, statistics and statistical learning.

Objective

Large deviations theory develops the art of estimating the probability of rare events. The classical theory concentrates on the study of the probability of deviating from the behavior predicted by the law of large numbers, namely the probability that the empirical mean of independent variables differs from its expectation. Such a classical framework does not apply in random matrix theory where one deals with complicated functions of independent variables or strongly interacting random variables, for instance the eigenvalues of a matrix with independent entries. During the last twenty years, important advances allowed to analyze large deviations for a few specific models of random matrices, but a full understanding of these questions is still missing. The object of this project is to develop such a theory. Two notable examples motivate this project. The first is to understand how the distribution of the entries of a random matrix affects the probability of the rare events of its spectrum as its dimension goes to infinity. The second is to prove in great generality the convergence of matrix integrals and Voiculescu's microstates entropy, as well as analyze their limit. The impact of this project would go beyond probability and operator algebra as it would apply to other fields such as theoretical physics, statistics and statistical learning.

Keywords

Host institution

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
Net EU contribution
€ 2 384 537,50
Address
RUE MICHEL ANGE 3
75794 Paris
France

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Region
Ile-de-France Ile-de-France Paris
Activity type
Research Organisations
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Total cost
€ 2 384 537,50

Beneficiaries (1)