Delocalization at small energy for heavy-tailed random matrices

Charles Bordenave (1) , Alice Guionnet (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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.

Domains: Mathematics [math]; Probability [math.PR]

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