GOE Statistics for Levy Matrices

In this paper 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 ∈ℝ is any energy 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 (GOE) 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).