  Consider an n x 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 we
give vanishing bounds on the Lp-norm of the eigenvectors normalized to have
unit L2-norm goes to 0. On the contrary, when 0 < alpha < 2/3, we prove that
these eigenvectors are localized.
