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.