Large deviations theory aims to estimate the probability of rare events and to describe entropy. For instance it allows to estimate the probability that after throwing a fair coin 100 times, we observe heads twenty times, whereas the law of large numbers would have predicted to see heads approximately half the times.
Large random matrices first appeared in the analysis of large arrays of data in the work of Wishart in the thirthy’s, in theoretical physics since the work of Wigner, and since then in many scientific domains, including for instance recently in statistical learning. They are also connected with many other domains such as random tilings or quantum chaos. In particular, a large deviations theory for large random matrices would have groundbreaking applications in many domains of mathematics, physics and statistics.
The classical results from large deviations theory do not apply to most relevant quantities in random matrix theory, such as the spectrum of a random matrix, because they are complicated functions of independent variables. During the last twenty years, important advances allowed to analyze the large deviations for very particular models of random matrices, but a full theory is still lacking. The goal of this project is to develop such a theory.
The first part of the project concerns the large deviations for the spectrum of large random matrices. The second considers precise large deviations for Coulomb Gases and more general mean field strongly interacting particles systems, including the so-called Sinh models from quantum physics, integrable quantum field theories or discrete models such as random tilings. The next objectives focus on multi-matrix models which arise in physics and combinatorics, including spherical integrals and unitary models, and reaches its climax with long standing open questions about Voiculescu’s entropy. The last objective concentrates on applications of large random matrices and large deviations in machine and statistical learning.
