Large deviations for the largest eigenvalue of matrices with variance profiles

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 [0,1]2, either continuous or piecewise constant and ai,j independent, centered one above diagonal. We prove large deviation principle for largest eigenvalue those under condition that they have sharp sub-Gaussian tails some additional assumptions on σ. These bounds verified example Gaussian variables, Rademacher variables uniform [− 3, 3]. This result new even entries.