Logarithmic Universality in Random Matrix Theory

TL;DR

This paper demonstrates a new form of universality, called logarithmic universality, in certain random matrix ensembles relevant to quantum chromodynamics, showing invariance of spectral correlators under specific matrix modifications.

Contribution

It introduces and proves the concept of logarithmic universality in unitary invariant random matrix ensembles with determinant factors.

Findings

Microscopic spectral correlators are invariant under matrix expansions in the determinant.

Logarithmic universality applies to ensembles relevant for QCD spectra.

A simple random matrix model with Ginsparg-Wilson symmetry shares spectral properties with chiral random matrix theory.

Abstract

Universality in unitary invariant random matrix ensembles with complex matrix elements is considered. We treat two general ensembles which have a determinant factor in the weight. These ensembles are relevant, e.g., for spectra of the Dirac operator in QCD. In addition to the well established universality with respect to the choice of potential, we prove that microscopic spectral correlators are unaffected when the matrix in the determinant is replaced by an expansion in powers of the matrix. We refer to this invariance as logarithmic universality. The result is used in proving that a simple random matrix model with Ginsparg-Wilson symmetry has the same microscopic spectral correlators as chiral random matrix theory.