On the relation between orthogonal, symplectic and unitary matrix ensembles

TL;DR

This paper explores the relationships between matrix kernels in orthogonal, symplectic, and unitary ensembles of Hermitian matrices, providing formulas that connect these kernels without relying on skew-orthogonal polynomials.

Contribution

It derives explicit formulas linking matrix kernels of orthogonal and symplectic ensembles to scalar kernels of unitary ensembles, avoiding skew-orthogonal polynomials.

Findings

Formulas expressing matrix kernel entries in terms of scalar kernels

Extra terms appear when the weight function derivative ratio is rational

General formulas for additional terms are provided

Abstract

For the unitary ensembles of $N\times N$ Hermitian matrices associated with a weight function $w$ there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are $2\times2$ matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever $w'/w$ is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of $w'/w$. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations.