Skew orthogonal polynomials for the real and quaternion real Ginibre ensembles and generalizations (Q2839465)

Considering the real and the quaternion real Ginibre ensembles (which are random matrix models), the author re-calculates the \(k\)-point correlation function (which gives the density of the first \(k\) eigenvalues). The first model is an \(N\times N\) matrix where the entries are independent Gaussian real valued r.v.s., and the second one is a \(2N\times 2N\) matrix formed by independent \(2\times 2\) block matrices of the form \([z \;w;-\overline{w} \;\overline{z}]\), where \(z\) and \(w\) are independent Gaussian complex valued r.v.s. It is known that the correlation functions are completely determined by certain polynomials, which are in terms of the expectation NEWLINE\[NEWLINE x\mapsto E[\det(xI_{2n}-G)], NEWLINE\]NEWLINE where \(I_{2n}\) is the identity matrix of size \(2n\) and \(G\) is \(2n\times 2n\) random matrix whose distribution corresponds to a Ginibre ensemble. The author re-derives the explicit form of such polynomials by direct calculation of the displayed expectation, and he achieves this task by using the so-called Schur and Jack polynomials as well as the Selberg integrals. In addition, the author calculates with this method the corresponding polynomials for other random matrices, called spherical and anti-spherical ensembles.