GOE statistics for Lévy matrices

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