Massive partition functions and complex eigenvalue correlations in Matrix Models with symplectic symmetry

TL;DR

This paper derives explicit formulas for massive partition functions and eigenvalue correlations in non-Hermitean symplectic matrix models, revealing simpler structures and connections to Hermitean limits.

Contribution

It provides new Pfaffian-based expressions for complex eigenvalue correlations in non-Hermitean symplectic ensembles, generalizing previous results and clarifying kernel structures.

Findings

Explicit Pfaffian formulas for complex eigenvalue correlations

Simplification of expressions compared to real eigenvalue models

Connection between non-Hermitean and Hermitean matrix models

Abstract

We compute all massive partition functions or characteristic polynomials and their complex eigenvalue correlation functions for non-Hermitean extensions of the symplectic and chiral symplectic ensemble of random matrices. Our results are valid for general weight functions without degeneracies of the mass parameters. The expressions we derive are given in terms of the Pfaffian of skew orthogonal polynomials in the complex plane and their kernel. They are much simpler than the corresponding expressions for symplectic matrix models with real eigenvalues, and we explicitly show how to recover these in the Hermitean limit. This explains the appearance of three different kernels as quaternion matrix elements there in terms of derivatives of a single kernel here. We compare to the Hermitean limit of complex matrix models with unitary symmetry leading to some determinantal identities.