In many applications of random matrices, knowing when the extremal eigenvalues of such matrices are atypical is of paramount importance. In other words, what can we say about its large deviations. In the last decade, there has been numerous advances in this area. This talk will be about such an advance for the large deviations of the largest eigenvalue of random matrices with variance profiles, that is symmetric matrices with independent centered entries on and above the diagonal but with variances that can vary from entry to entry. This work is a part of a line of research that tackles random matrices with so-called sharp sub-Gaussian entries (such as Rademacher entries) and uses spherical integrals as proxies for the largest eigenvalue. However, a notable fact for this model is that the large deviations were previously unknown even for Gaussian models. This talk is based on a joint work with Raphaël Ducatez and Alice Guionnet.
