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

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 and the Gaussian unitary ensemble are associated with the Dyson index β = 2, the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian and 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 the help of these results, we derive the unquenched k-point correlation function of the Hermitian and the non-Hermitian Wilson random matrix in terms of two-flavor partition functions only. This representation is due to a Pfaffian factorization. It drastically simplifies the expressions, which can be easily numerically evaluated. It also serves as a good starting point for studying the Wilson–Dirac operator in the ϵ-regime of lattice quantum chromodynamics.