Universality of Lévy Matrices

Abstract

Motivated by conjectures from physics, we study the eigenvalues and eigenvectors of Lévy matrices, which are symmetric random matrices whose upper triangular entries are independent, identically distributed α-stable distributions. For α < 2, these distributions are heavy-tailed, with infinite second moment. For α ∈ (1, 2), we show that at all finite non- zero energies, Lévy matrices exhibit completely delocalized eigenvectors and local eigenvalue statistics that asymptotically match those of the Gaussian Orthogonal Ensemble. For almost all α ∈ (0, 2), we prove the same result for small energies, including zero. Additionally, for almost all α ∈ (0, 2), we analyze the statistics of eigenvector entries of Lévy matrices at small energies and show that the limiting distribution of any such entry is non-Gaussian. For entries of the eigenvector corresponding to the median eigenvalue, we identify this distribution explicitly. We also demonstrate the presence of non-trivial correlations between eigenvector entries corresponding to nearby eigenvalues. These findings contrast sharply with the known eigenvector behavior for other random matrix ensembles. Further, our results for both eigenvalues and eigenvectors generalize to a large class of heavy-tailed random matrices.