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The partly symmetric real Ginibre ensemble consists of matrices formed aslinear combinations of real symmetric and real anti-symmetric Gaussian randommatrices. Such matrices typically have both real and complex eigenvalues. For afixed number of real eigenvalues, an earlier work has given the explicit formof the joint eigenvalue probability density function. We use this to derive aPfaffian formula for the corresponding summed up generalized partitionfunction. This Pfaffian formula allows the probability that there are exactly$k$ eigenvalues to be written as a determinant with explicit entries. It can beused too to give the explicit form of the correlation functions, providedcertain skew orthogonal polynomials are computed. This task is accomplished interms of Hermite polynomials, and allows us to proceed to analyze variousscaling limits of the correlations, including that in which the matrices areonly weakly non-symmetric.

Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble