Power law eigenvalue density, scaling and critical random matrix ensembles

TL;DR

This paper analyzes a class of rotationally invariant random matrix ensembles with power law eigenvalue densities, deriving exact eigenvalue correlation kernels and establishing their classification as critical ensembles.

Contribution

It introduces a new scaling for power law eigenvalue densities and computes the exact two-level kernel, expanding understanding of critical random matrix ensembles.

Findings

Eigenvalue density follows an inverse power law.

Exact two-level kernel derived for these ensembles.

Classified as critical ensembles based on correlation properties.

Abstract

We consider a class of rotationally invariant unitary random matrix ensembles where the eigenvalue density falls off as an inverse power law. Under a new scaling appropriate for such power law densities (different from the scaling required in Gaussian random matrix ensembles), we calculate exactly the two-level kernel that determines all eigenvalue correlations. We show that such ensembles belong to the class of critical ensembles.