Large deviations for the largest eigenvalue of random matrices

In large deviation theory, we consider sequences of random variables that converge toward a limit and we try to evaluate how the probability that they take other values decays. Aside from Gaussian matrices for which explicit formulas are known to describe the spectrum, little is known of the large deviations for the empirical measure or the largest eigenvalue in the general case. In this talk, I will consider sub-Gaussian random matrix models and I will explain how to use spherical integrals to obtain large deviation principles for the largest eigenvalue of those matrices.