GOE statistics for Lévy matrices

We establish eigenvector delocalization and bulk universality for Lévy matrices, which are real, symmetric, N × N random matrices H whose upper triangular entries are independent, identically distributed α -stable laws. First, if α∈(1,2) and E 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 ∞ . Second, we show for almost all α∈(0,2) , there exists a constant c(α)>0 such that the same statements hold if |E|<c(α) .