Fermionic mapping for eigenvalue correlation functions of weakly non-Hermitian symplectic ensemble

Abstract

The eigenvalues of an arbitrary quaternionic matrix have a joint probability distribution function first derived by Ginibre. We derive the j.p.d. for the weakly non-Hermitian version of this problem and then show that there exists a mapping of this system onto a fermionic field theory. This mapping is used 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.