Delocalization at small energy for heavy-tailed random matrices

TL;DR

This paper demonstrates that eigenvectors corresponding to small eigenvalues of heavy-tailed symmetric random matrices are delocalized as the matrix size increases, contrasting with localization at large eigenvalues.

Contribution

It introduces a novel analysis of the resolvent's fixed point equation to establish delocalization of small eigenvalue eigenvectors in heavy-tailed matrices.

Findings

Eigenvectors for small eigenvalues are delocalized with high probability.

Eigenvectors for large eigenvalues are localized.

The analysis uses a new approach to the resolvent's fixed point equation.

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 L4 and L2-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.