Large deviations for the largest eigenvalue of matrices with variance profiles

Abstract

In this article we consider Wigner matrices $X_N$ with variance profiles (also called Wigner-type matrices) which are of the form $X_N(i,j)=\sigma (i/N,j/N)a_{i,j}/\sqrt{N}$ where $\sigma$ is a symmetric real positive function of $[0,1{]}^2$ and $\sigma$ will be taken either continuous or piecewise constant. We prove a large deviation principle for the largest eigenvalue of those matrices under the same condition of sharp sub-Gaussian bound and for some other assumptions on $\sigma$. These sub-Gaussian bounds are verified for example for Gaussian variables, Rademacher variables or uniform variables on $[-\sqrt{3},\sqrt{3}]$.

arXiv comment43 pages, 3 figures

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