Large deviations for extreme eigenvalues of deformed Wigner random matrices

TL;DR

This paper establishes a large deviation principle for the largest eigenvalue of deformed Wigner matrices, covering Gaussian and certain non-Gaussian cases, with results depending on the type of ensemble and deformation.

Contribution

It provides the first large deviation principles for the extreme eigenvalues of deformed Wigner matrices, including non-Gaussian cases with diagonal deformation.

Findings

Large deviation principle at speed N for Gaussian ensembles with full-rank deformation.

Restricted large deviation results for non-Gaussian ensembles with diagonal deformation.

Dependence of the large deviation behavior on the type of ensemble and deformation.

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 $(-\infty, x_c)$, where $x_c$ depends on the deformation only and can be infinite.