Averages of products of characteristic polynomials and the law of real eigenvalues for the real Ginibre ensemble

An elementary derivation of the Borodin-Sinclair-Forrester-Nagao Pfaffian point process, which characterises the law of real eigenvalues for the real Ginibre ensemble in the large matrix size limit, uses the averages of products of characteristic polynomials. This derivation reveals a number of interesting structures associated with the real Ginibre ensemble such as the hidden symplectic symmetry of the statistics of real eigenvalues and an integral representation for the $K$-point correlation function for any $K\in \mathbb{N}$ in terms of an asymptotically exact integral over the symmetric space $U(2K)/USp(2K)$.