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

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 $1 \leq i,j \leq N$, where the $X_{i,j}^{N}$ for $1 \leq i \leq j \leq N$ are independent and centered. We then denote $\Sigma_{i,j} ^N = \text{Var} (X_{i,j}^{N}) ( 1 + \textbf{1}_{ i =j})^{-1}$ 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.