Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

Abstract

The framework of spherical transforms and Pólya ensembles is of utility in deriving structured analytic results for sums and products of random matrices in a unified way. In the present work, we will carry over this framework to study products of unitary matrices. Those are not distributed via the Haar measure, but still are drawn from distributions where the eigenvalue and eigenvector statistics factorise. They include the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, give a uniqueness statement for the corresponding weights, and derive the determinantal point processes of the eigenvalue statistics at fixed matrix dimension. An outline is given of problems one may encounter when investigating the local spectral statistics.

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Cited by in corpus (2)

• Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
• Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions