GOE Statistics for Levy Matrices

TL;DR

This paper proves eigenvector delocalization and GOE universality for Levy matrices with stable law entries, showing that their spectral statistics resemble those of classical Gaussian ensembles under certain conditions.

Contribution

It establishes delocalization and universality results for Levy matrices with alpha-stable entries, extending random matrix theory to heavy-tailed distributions.

Findings

Eigenvectors are delocalized for energies away from zero.

Spectral statistics near certain energies converge to GOE predictions.

Results hold for a wide range of stability parameters alpha.

Abstract

In this paper we establish eigenvector delocalization and bulk universality for L\'{e}vy matrices, which are real, symmetric, $N \times N$ random matrices $\textbf{H}$ whose upper triangular entries are independent, identically distributed $\alpha$-stable laws. First, if $\alpha \in (1, 2)$ and $E \in \mathbb{R}$ is any energy bounded away from $0$, we show that every eigenvector of $\textbf{H}$ corresponding to an eigenvalue near $E$ is completely delocalized and that the local spectral statistics of $\textbf{H}$ around $E$ converge to those of the Gaussian Orthogonal Ensemble (GOE) as $N$ tends to $\infty$. Second, we show for almost all $\alpha \in (0, 2)$, there exists a constant $c(\alpha) > 0$ such that the same statements hold if $|E| < c (\alpha)$.