# Large deviations of the largest eigenvalue of supercritical sparse Wigner matrices

## Location

Let $X$ be an $n \times n$ sparse random symmetric matrix that is a Hadamard product of $G$ and $\Xi$, where $G$ is a Wigner matrix with centered sub-Gaussian entries (the distribution of the diagonals is allowed to be different) and $\Xi$ is a symmetric matrix with i.i.d. Bernoulli $p$ entries on and above the diagonal positions. In this talk, we will discuss the large deviations of the largest eigenvalue of $X/\sqrt{np}$ under the assumption that $\log n/n \ll p \ll 1$ and that the entries of $G$ satisfy the sub-Gaussian concentration property. As a byproduct, we will also obtain the large deviations of the second-largest eigenvalue of the adjacency matrix of a sparse Erdös–Rényi graph.

This is based on joint work with Fanny Augeri.