Local Tail Statistics of Heavy-Tailed Random Matrix Ensembles with Unitary Invariance

TL;DR

This paper investigates the tail behavior of eigenvalues in heavy-tailed, unitarily invariant Hermitian random matrices, revealing unexpected Poisson-like statistics and challenging prior assumptions about stability in such ensembles.

Contribution

It demonstrates that freely stable random matrices do not always have stable eigenvalue statistics, uncovering Poisson-like behavior and proposing new conjectures.

Findings

Heavy-tailed Hermitian matrices can exhibit Poisson eigenvalue statistics.

Eigenvector and eigenvalue statistics are decoupled due to invariance.

Poisson-like statistics observed in heavy-tailed Wigner matrices.

Abstract

We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might exhibit Poisson or Poisson-like statistics. The Poisson statistics have been already observed for heavy-tailed Wigner matrices. We conclude with two conjectures on this peculiar behaviour.