How universal is the Wigner distribution?

TL;DR

This paper investigates the universality of the Wigner distribution in Gaussian ensembles of complex matrices, revealing it only applies for a specific case and deriving new eigenvalue densities for others.

Contribution

It identifies an enhanced symmetry in matrix ensembles and derives the eigenvalue density for cases where the Wigner distribution does not hold.

Findings

Wigner distribution applies only for m=1.

New eigenvalue density forms are derived for m ≥ 2.

Enhanced symmetry simplifies the analysis of the radial sector.

Abstract

We consider Gaussian ensembles of m N x N complex matrices. We identify an enhanced symmetry in the system and the resultant closed subsector, which is naturally associated with the radial sector of the theory. The density of radial eigenvalues is obtained in the large N limit. It is of the Wigner form only for m=1. For m \ge 2, the new form of the density is obtained.