Surprising Pfaffian factorizations in Random Matrix Theory with Dyson index $\beta=2$

TL;DR

This paper reveals a surprising Pfaffian structure in Dyson index =2 random matrix ensembles, demonstrating a hidden universality and connecting orthogonal and skew-orthogonal polynomials.

Contribution

It derives a non-trivial Pfaffian determinant for =2 ensembles, previously thought to be exclusive to =1,4, and relates orthogonal and skew-orthogonal polynomials.

Findings

Unveiled Pfaffian structures in =2 ensembles

Established relation between orthogonal and skew-orthogonal polynomials

Detailed analysis of chiral unitary ensembles

Abstract

In the past decades, determinants and Pfaffians were found for eigenvalue correlations of various random matrix ensembles. These structures simplify the average over a large number of ratios of characteristic polynomials to integrations over one and two characteristic polynomials only. Up to now it was thought that determinants occur for ensembles with Dyson index $\beta=2$ whereas Pfaffians only for ensembles with $\beta=1,4$. We derive a non-trivial Pfaffian determinant for $\beta=2$ random matrix ensembles which is similar to the one for $\beta=1,4$. Thus, it unveils a hidden universality of this structure. We also give a general relation between the orthogonal polynomials related to the determinantal structure and the skew-orthogonal polynomials corresponding to the Pfaffian. As a particular example we consider the chiral unitary ensembles in great detail.