Localization and delocalization of eigenvectors for heavy-tailed random matrices

Abstract

Consider an n × n Hermitian random matrix with, above the diagonal, independent entries with α-stable symmetric distribution and 0 < α < 2. We establish new bounds on the rate of convergence of the empirical spectral distribution of this random matrix as n goes to infinity. When 1 < α < 2 and p > 2, we give vanishing bounds on the Lp-norm of the eigenvectors normalized to have unit L2-norm. On the contrary, when 0 < α < 2/3, we prove that these eigenvectors are localized. © 2013 Springer-Verlag Berlin Heidelberg.