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

We review an elementary derivation of the Borodin-Sinclair-Forrester-Nagao Pfaffian point process, which characterizes the law of real eigenvalues for the real Ginibre ensemble in the large matrix size limit, in terms of averages of products of characteristic polynomials. This derivation relies on a number of interesting structures associated with the real Ginibre ensemble such as the hidden symplectic symmetry of the statistics of real eigenvalues. It leads to a representation for the K-point correlation function for any K is an element of N in terms of an integral over the symmetric space U(2K)/USp(2K) and this paper gives the proof of asymptotic exactness for this integral.