Matrix averages relating to Ginibre ensembles
Peter J Forrester, Eric M Rains
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL | IOP PUBLISHING LTD | Published : 2009
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 Khoruzhenko (2009 J. Phys. A: Math. Theor. 42 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..View full abstract
The work of PJF was supported by the Australian Research Council.