Matrix averages relating to the Ginibre ensembles

Abstract

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.

13 pages

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Cited by in corpus (6)

• Raney distributions and random matrix theory
• On the moments of the characteristic polynomial of a Ginibre random matrix
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• Maximum of the characteristic polynomial of the Ginibre ensemble
• Wilson loops and free energies in $3d$ $\mathcal{N}=4$ SYM: exact results, exponential asymptotics and duality
• Schur expansion of random-matrix reproducing kernels