Spectral Statistics of Dirac Ensembles

TL;DR

This paper investigates the spectral properties of Dirac operators from random noncommutative geometries, revealing a universal eigenvalue distribution in the large N limit, characterized by a convolution of the semicircle law.

Contribution

It demonstrates the universality of the eigenvalue spectrum for Dirac ensembles and establishes a genus expansion for a broad class of multi-trace multimatrix models.

Findings

Eigenvalue spectrum converges to a convolution of semicircle laws

Universality holds across different geometries for Gaussian potentials

Genus expansion proven for multi-trace multimatrix models

Abstract

In this paper we find spectral properties in the large $N$ limit of Dirac operators that come from random finite noncommutative geometries. In particular for a Gaussian potential the limiting eigenvalue spectrum is shown to be universal regardless of the geometry and is given by the convolution of the semicircle law with itself. For simple non-Gaussian models this convolution property is also evident. In order to prove these results we show that a wide class of multi-trace multimatrix models have a genus expansion.