Large deviations for the largest eigenvalue of matrices with variance profiles

Abstract

In this article we consider Wigner matrices ${(X_N)}_{N\in \mathbb{N}}$ with variance profiles which are of the form $X_N(i,j)=\mathit{\sigma }(i∕N,j∕N)a_{i,j}∕\sqrt{N}$ where σ is a symmetric real positive function of ${[0,1]}^2$, either continuous or piecewise constant and where the $a_{i,j}$ 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 $[-\sqrt{3},\sqrt{3}]$. This result is new even for Gaussian entries.