Randomness on the Lattice

TL;DR

This paper reviews lattice QCD studies on the eigenvalues of the Dirac operator, showing their fluctuations align with chiral Random Matrix Theory and explaining deviations with chiral perturbation theory.

Contribution

It demonstrates the applicability of chiral Random Matrix Theory to lattice QCD eigenvalue fluctuations and analytically accounts for deviations beyond the Thouless energy.

Findings

Eigenvalue fluctuations match chiral Random Matrix Theory predictions.

Deviations from RMT are explained by partially quenched chiral perturbation theory.

Provides analytical understanding of eigenvalue behavior beyond the Thouless energy.

Abstract

In this lecture we review recent lattice QCD studies of the statistical properties of the eigenvalues of the QCD Dirac operator. We find that the fluctuations of the smallest Dirac eigenvalues are described by chiral Random Matrix Theories with the global symmetries of the QCD partition function. Deviations from chiral Random Matrix Theory beyond the Thouless energy can be understood analytically by means of partially quenched chiral perturbation theory.