Universality of random matrices in the microscopic limit and the Dirac operator spectrum

TL;DR

This paper proves the universality of correlation functions in certain random matrix ensembles at the microscopic scale, linking it to the spectral properties of the Dirac operator.

Contribution

It demonstrates the universality of the microscopic spectral correlations for chiral and unitary random matrix ensembles using orthogonal polynomial asymptotics.

Findings

Correlation functions are universal in the microscopic limit.

Reduction of recursion relations to Bessel equations explains local spectral behavior.

Potential implications for the universality of the Dirac operator spectrum.

Abstract

We prove the universality of correlation functions of chiral unitary and unitary ensembles of random matrices in the microscopic limit. The essence of the proof consists in reducing the three-term recursion relation for the relevant orthogonal polynomials into a Bessel equation governing the local asymptotics around the origin. The possible physical interpretation as the universality of soft spectrum of the Dirac operator is briefly discussed.