GOE Statistics for Levy Matrices

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