Large deviations for extreme eigenvalues of deformed Wigner random matrices

Abstract

We present a large deviation principle at speed N for the largest eigenvalue of some additively deformed Wigner matrices. In particular this includes Gaussian ensembles with full-rank general deformation. For the non-Gaussian ensembles, the deformation should be diagonal, and we assume that the laws of the entries have sharp sub-Gaussian Laplace transforms and satisfy certain concentration properties. For these latter ensembles we establish the large deviation principle in a restricted range $(-\mathrm{\infty },x_c)$, where $x_c$ depends on the deformation only and can be infinite.