Matrix averages relating to the Ginibre ensembles

Research paper by Peter J. Forrester, Eric M. Rains

Indexed on: 02 Jul '09Published on: 02 Jul '09Published in: Mathematical Physics


The theory of zonal polynomials is used to compute the average of a Schur polynomial of argument $AX$, where $A$ is a fixed matrix and $X$ is from the real Ginibre ensemble. This generalizes a recent result of Sommers and Khorozhenko [J. Phys. A {\bf 42} (2009), 222002], and furthermore allows analogous results to be obtained for the complex and real quaternion Ginibre ensembles. As applications, the positive integer moments of the general variance Ginibre ensembles are computed in terms of generalized hypergeometric functions, these are written in terms of averages over matrices of the same size as the moment to give duality formulas, and the averages of the power sums of the eigenvalues are expressed as finite sums of zonal polynomials.