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) .