Dualities for characteristic polynomial averages of complex symmetric and self dual non-Hermitian random matrices

Ensembles of complex symmetric, and complex self dual random matrices are known to exhibit local statistical properties distinct from those of the non-Hermitian Ginibre ensembles. On the other hand, in distinction to the latter, the joint eigenvalue probability density function of these two ensembles are not known. Nonetheless, as carried out in the recent works of Liu and Zhang, Akemann et al.~and Kulkarni et al., by considering averages of products of characteristic polynomials, analytic progress can be made. Here we show that an approach based on the theory of zonal polynomials provides an alternative to the diffusion operator or supersymmetric Grassmann integrations methods of these works. It has the advantage of not being restricted to a Gaussian unitary invariant measure on the matrix spaces. To illustrate this, as an extension, we consider averages of products and powers of characteristic polynomials for complex symmetric, and complex self dual random matrices subject to a spherical measure. In the case of powers, when comparing against the corresponding real Ginibre, respectively quaternion Ginibre averages with a spherical measure, one finds the qualitative feature of a decreasing (increasing) profile as the magnitude of the argument of the characteristic polynomial increases. This is analogous to the findings of the second two of the cited works in the Gaussian case.