Heavy-tailed random matrices

TL;DR

This paper explores the impact of heavy-tailed distributions on the eigenvalues and eigenvectors of non-Gaussian random matrices, revealing how rare, extreme events alter their universal properties.

Contribution

It analyzes the universal macroscopic properties of heavy-tailed random matrices, including Levy matrices, stable free random variable matrices, and deformed standard ensembles.

Findings

Heavy tails significantly alter eigenvalue distributions.

Universal properties differ from Gaussian ensembles.

Heavy-tailed matrices exhibit unique spectral behaviors.

Abstract

We discuss non-Gaussian random matrices whose elements are random variables with heavy-tailed probability distributions. In probability theory heavy tails of the distributions describe rare but violent events which usually have dominant influence on the statistics. They also completely change universal properties of eigenvalues and eigenvectors of random matrices. We concentrate here on the universal macroscopic properties of (1) Wigner matrices belonging to the Levy basin of attraction, (2) matrices representing stable free random variables and (3) a class of heavy-tailed matrices obtained by parametric deformations of standard ensembles.