Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble

Abstract

The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both real and complex eigenvalues. For a fixed number of real eigenvalues, an earlier work has given the explicit form of the joint eigenvalue probability density function. We use this to derive a Pfaffian formula for the corresponding summed up generalized partition function. This Pfaffian formula allows the probability that there are exactly k eigenvalues to be written as a determinant with explicit entries. It can be used too to give the explicit form of the correlation functions, provided certain ske..