LDPs for the largest eigenvalue of sub-Gaussian Wigner matrices

I will discuss large deviation principles for the right-most eigenvalue of Wigner matrices with sub-Gaussian entries. Previous work of Guionnet and Husson established a universal rate function for the light-tailed "sharp sub-Gaussian" case, where large deviations result from "delocalized" tilting strategies. On the other hand, large deviations for heavy-tailed matrices and sparse graphs and networks have been shown to be governed by localization phenomena. The general sub-Gaussian case is in some sense critical, with non-universal rate functions determined by a mixture of competing localized and delocalized tilts. Our approach also leads to information on the conditional structure of the associated eigenvector.

This is based on joint work with Raphael Ducatez and Alice Guionnet.