Large deviations for the largest eigenvalue of matrices with variance profiles

Abstract

In this article we consider Wigner matrices (XN)N∈N with variance profiles which are of the form XN(i,j)=σ(i∕N,j∕N)ai,j∕ N where σ is a symmetric real positive function of [0,1]2, either continuous or piecewise constant and where the ai,j are independent, centered of variance one above the diagonal. We prove a large deviation principle for the largest eigenvalue of those matrices under the condition that they have sharp sub-Gaussian tails and under some additional assumptions on σ. These sub-Gaussian bounds are verified for example for Gaussian variables, Rademacher variables or uniform variables on [− 3, 3]. This result is new even for Gaussian entries.