A new approach to derive Pfaffian structures for random matrix ensembles

TL;DR

This paper extends a new method for deriving Pfaffian structures to orthogonal and symplectic ensembles, simplifying the calculation of correlation functions in these complex matrix models.

Contribution

It introduces a unified approach to derive Pfaffian structures for a broad class of orthogonal and symplectic ensembles, including previously uncharacterized cases.

Findings

Pfaffian structures derived for new ensembles like real Ginibre and real chiral ensembles.

Unified method simplifies calculations of correlation functions.

Extension of the supersymmetry approach to broader symmetry classes.

Abstract

Correlation functions for matrix ensembles with orthogonal and unitarysymplectic rotation symmetry are more complicated to calculate than in the unitary case. The supersymmetry method and the orthogonal polynomials are two techniques to tackle this task. Recently, we presented a new method to average ratios of characteristic polynomials over matrix ensembles invariant under the unitary group. Here, we extend this approach to ensembles with orthogonal and unitary-symplectic rotation symmetry. We show that Pfaffian structures can be derived for a wide class of orthogonal and unitary-symplectic rotation invariant ensembles in a unifying way. This includes also those for which this structure was not known previously, as the real Ginibre ensemble and the Gaussian real chiral ensemble with two independent matrices as well.