A Wigner Surmise for Hermitian and Non-Hermitian Chiral Random Matrices

TL;DR

This paper develops approximate formulas for the distribution of the first eigenvalue in chiral Random Matrix Theory, applicable to both real and complex eigenvalues, with strong validation against known results and practical applications in Lattice Gauge Theory.

Contribution

It introduces a Wigner surmise approach to derive compact eigenvalue distribution formulas for various classes of chiral Random Matrices, including new results for real eigenvalues and intermediate non-Hermiticity.

Findings

Excellent agreement with known large-N results

New compact expressions for real eigenvalues in orthogonal and symplectic classes

Effective description of lattice QCD data with chemical potential

Abstract

We use the idea of a Wigner surmise to compute approximate distributions of the first eigenvalue in chiral Random Matrix Theory, for both real and complex eigenvalues. Testing against known results for zero and maximal non-Hermiticity in the microscopic large-N limit we find an excellent agreement, valid for a small number of exact zero-eigenvalues. New compact expressions are derived for real eigenvalues in the orthogonal and symplectic classes, and at intermediate non-Hermiticity for the unitary and symplectic classes. Such individual Dirac eigenvalue distributions are a useful tool in Lattice Gauge Theory and we illustrate this by showing that our new results can describe data from two-colour QCD simulations with chemical potential in the symplectic class.