Localization and delocalization of eigenvectors for heavy-tailed random matrices

Charles Bordenave; Alice Guionnet

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arXiv · 1201.1862

Localization and delocalization of eigenvectors for heavy-tailed random matrices

Charles Bordenave·Alice Guionnet

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Abstract

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.

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Charles Bordenave, Alice Guionnet. 2012-01-31. Localization and delocalization of eigenvectors for heavy-tailed random matrices. https://arxiv.org/abs/1201.1862

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