# On the spectrum of the QCD Dirac operator

## TL;DR

This paper argues that the microscopic fluctuations of Dirac eigenvalues in QCD are universal and can be accurately modeled by chiral random matrix theory, supported by multiple lines of evidence.

## Contribution

It provides evidence that the microscopic spectral correlations of lattice QCD Dirac spectra are universally described by chiral random matrix theory, regardless of underlying dynamics.

## Key findings

- Spectral correlations obey Dyson-Mehta-Wigner statistics.
- Valence quark mass dependence matches chiral random matrix predictions.
- Microscopic spectral density is universal and insensitive to distribution details.

## Abstract

In this lecture we argue that the fluctuations of Dirac eigenvalues on the finest scale, i.e. on the scale of the average level spacing do not depend on the underlying dynamics and can be obtained from a chiral random matrix theory with the same low energy effective theory. We present three pieces of evidence supporting that such microscopic correlations of lattice QCD Dirac spectra are given by chiral random matrix theory. First, we find that the spectral correlations of eigenvalues in the bulk of the spectrum obey the Dyson-Mehta-Wigner statistics. Second, we show that the valence quark mass dependence for sufficiently small quark masses, as calculated by the Columbia group, can be obtained from the microscopic spectral density of chiral random matrix theory. Third, in the framework of chiral random matrix models, we present results showing that the microscopic spectral density is strongly universal, i.e. is insensitive to the details of the probability distribution.