Large deviation principle for the largest eigenvalue of random matrices with a variance profile

Abstract

We establish large deviation principles for the largest eigenvalue of large random matrices with variance profiles. For $N\in \mathbb{N}$, we consider random $N\times N$ symmetric matrices $H^N$ which are such that $H_{ij}^N=\frac{1}{\sqrt{N}}{X}_{i,j}^N$ for $1\le i,j\le N$, where the $X_{i,j}^N$ for $1\le i\le j\le N$ are independent and centered. We then denote ${\Sigma }_{i,j}^N=\text{Var}(X_{i,j}^N)(1+{\text{1}}_{i=j}{)}^{-1}$ the variance profile of $H^N$. Our large deviation principle is then stated under the assumption that the ${\Sigma }^N$ converge in a certain sense toward a real continuous function $\sigma$ of $[0,1{]}^2$ and that the entries of $H^N$ are sharp sub-Gaussian. Our rate function is expressed in terms of the solution of a Dyson equation involving $\sigma$. This result is a generalization of a previous work by the third author and is new even in the case of Gaussian entries.