Localization and delocalization of eigenvectors for heavy-tailed random matrices

Consider an $n times n$ Hermitian random matrix with, above the diagonal, independent entries with $alpha $ -stable symmetric distribution and $0 < alpha < 2$ . We establish new bounds on the rate of convergence of the empirical spectral distribution of this random matrix as $n$ goes to infinity. When $1 < alpha < 2$ and $ p > 2$ , we give vanishing bounds on the $L^p$ -norm of the eigenvectors normalized to have unit $L^2$ -norm. On the contrary, when $0 < alpha < 2/3$ , we prove that these eigenvectors are localized.