Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT

TL;DR

This paper explores the interpolation between different random matrix ensembles using orthogonal and skew-orthogonal polynomials, deriving formulas and correlation functions relevant for lattice QCD applications.

Contribution

It introduces a unifying framework for Wilson random matrix ensembles, deriving new polynomial relations and correlation functions that simplify numerical and theoretical analyses.

Findings

Derived recursion relations and formulas for Wilson polynomials

Expressed correlation functions in terms of two-flavor partition functions

Simplified expressions using Pfaffian factorization for numerical applications

Abstract

The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble as well as the Gaussian unitary ensemble are associated to the Dyson index $\beta=2$ the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian as well as the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel-Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With help of these results we derive the unquenched $k$-point correlation function of the Hermitian and then non-Hermitian Wilson random matrix in terms of two flavour partition functions only. This representation is due to a Pfaffian factorization drastically simplifying the expressions for numerical applications. It also serves as a good starting point for studying the Wilson-Dirac operator in the $\epsilon$-regime of lattice quantum chromodynamics.