Rotationally invariant family of L\'evy like random matrix ensembles

TL;DR

This paper introduces a new family of rotationally invariant random matrix ensembles parameterized by , which generalize critical ensembles to include Le9vy-like distributions with power law eigenvalue densities, and analyzes their eigenvalue correlations.

Contribution

The paper presents a novel family of random matrix ensembles characterized by a parameter or Le9vy-like distributions, extending known ensembles and deriving their eigenvalue correlations using new orthogonal polynomials.

Findings

Eigenvalue densities exhibit power law tails for <1.

Eigenvalue correlations differ qualitatively from Gaussian and critical ensembles.

The model provides explicit formulas for correlations in Le9vy-like ensembles.

Abstract

We introduce a family of rotationally invariant random matrix ensembles characterized by a parameter $\lambda$. While $\lambda=1$ corresponds to well-known critical ensembles, we show that $\lambda \ne 1$ describes "L\'evy like" ensembles, characterized by power law eigenvalue densities. For $\lambda > 1$ the density is bounded, as in Gaussian ensembles, but $\lambda <1$ describes ensembles characterized by densities with long tails. In particular, the model allows us to evaluate, in terms of a novel family of orthogonal polynomials, the eigenvalue correlations for L\'evy like ensembles. These correlations differ qualitatively from those in either the Gaussian or the critical ensembles.