Fermionic Mapping For Eigenvalue Correlation Functions Of (Weakly) Non-Hermitian Symplectic Ensemble

TL;DR

This paper introduces a fermionic field theory mapping for eigenvalue correlation functions of quaternionic matrices, enabling analytical computation of densities and correlations in non-Hermitian symplectic ensembles.

Contribution

It presents a novel fermionic mapping approach to analyze eigenvalue correlations in non-Hermitian quaternionic matrices, extending previous probabilistic methods.

Findings

Derived explicit eigenvalue density functions

Computed higher-order correlation functions

Applicable to both strongly and weakly non-Hermitian cases

Abstract

The eigenvalues of an arbitrary quaternionic matrix have a joint probability distribution function first derived by Ginibre. We show that there exists a mapping of this system onto a fermionic field theory and then use this mapping to integrate over the positions of the eigenvalues and obtain eigenvalue density as well as all higher correlation functions for both the strongly and weakly non-Hermitian cases.