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.
Type de document :
Article dans une revue
Communications in Mathematical Physics, Springer Verlag, 2017, 354, pp.115-159
https://hal-ens-lyon.archives-ouvertes.fr/ensl-01655012
Contributeur : Alice Guionnet
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Soumis le : lundi 4 décembre 2017 - 15:26:38
Dernière modification le : mercredi 28 février 2018 - 10:22:56
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〉
