GOE statistics for Lévy matrices

Abstract

We establish eigenvector delocalization and bulk universality for Lévy matrices, which are real, symmetric, $N\times N$ random matrices $\mathbf{H}$ whose upper triangular entries are independent, identically distributed $\alpha$-stable laws. First, if $\alpha \in (1,2)$ and $E\in \mathbb{R}$ is bounded away from 0, we show that every eigenvector of $\mathbf{H}$ corresponding to an eigenvalue near $E$ is completely delocalized and that the local spectral statistics of $\mathbf{H}$ around $E$ converge to those of the Gaussian Orthogonal Ensemble as $N$ tends to $\infty$. Second, we show for almost all $\alpha \in (0,2)$, there exists a constant $c(\alpha )>0$ such that the same statements hold if $|E|<c(\alpha )$.