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 skew orthogonal polynomials are computed. This task is accomplished in terms of Hermite polynomials, and allows us to proceed to analyze various scaling limits of the correlations, including that in which the matrices are only weakly non-symmetric.