GOE statistics for Levy matrices

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