Delocalization at small energy for heavy-tailed random matrices

Abstract : We prove that the eigenvectors associated to small enough eigenvalues of an heavy-tailed symmetric random matrix are delocalized with probability tending to one as the size of the matrix grows to infinity. The delocalization is measured thanks to a simple criterion related to the inverse participation ratio which computes an average ratio of L 4 and L 2-norms of vectors. In contrast, as a consequence of a previous result, for random matrices with sufficiently heavy tails, the eigenvectors associated to large enough eigenvalues are localized according to the same criterion. The proof is based on a new analysis of the fixed point equation satisfied asymptotically by the law of a diagonal entry of the resolvent of this matrix.
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Communications in Mathematical Physics, Springer Verlag, 2017, 354, pp.115-159
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-01655012
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Soumis le : lundi 4 décembre 2017 - 15:26:38
Dernière modification le : mercredi 23 mai 2018 - 17:58:04

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  • HAL Id : ensl-01655012, version 1

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Charles Bordenave, Alice Guionnet. Delocalization at small energy for heavy-tailed random matrices. Communications in Mathematical Physics, Springer Verlag, 2017, 354, pp.115-159. 〈ensl-01655012〉

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