Localization and delocalization of eigenvectors for heavy-tailed random matrices

By Charles Bordenave, Alice Guionnet. 2012

91 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c0bcf8f55374dc1978e6d6280c05b4c9dccdd52c

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

No-gaps delocalization for general random matrices (2015, background): “Showing that the eigenvectors of general random matrices have these properties, too, became a focus of attention in the recent years [16, 17, 11, 12, 6, 13, 14, 15, 36, 39, 5, 7, 28].”
Extremal eigenvalues and eigenvectors of deformed Wigner matrices (2013, background): “[2] Anderson, G. W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices, Cambridge University Press (2010).”
Delocalization of eigenvectors of random matrices with independent entries (2013, background): “Delocalization properties with varying degrees of strength and generality were then established for several other symmetric and Hermitian ensembles – band matrices [5, 6, 9], sparse matrices (adjacency matrices of Erdös-Renyi graphs) [7, 8], heavy-tailed matrices [2, 1], and sample covariance matric”