Characteristic polynomials in real Ginibre ensembles

Akemann G, Phillips MJ, Sommers H-J. Characteristic polynomials in real Ginibre ensembles. J. Phys. A: Math. Theor. 2009;42(1):012001.We calculate the average of two characteristic polynomials for the realGinibre ensemble of asymmetric random matrices, and its chiral counterpart.Considered as quadratic forms they determine a skew-symmetric kernel from whichall complex eigenvalue correlations can be derived. Our results are obtained ina very simple fashion without going to an eigenvalue representation, and arecompletely new in the chiral case. They hold for Gaussian ensembles which arepartly symmetric, with kernels given in terms of Hermite and Laguerrepolynomials respectively, depending on an asymmetry parameter. This allows usto interpolate between the maximally asymmetric real Ginibre and the GaussianOrthogonal Ensemble, as well as their chiral counterparts