Cyclic P\'olya Ensembles on the Unitary Matrices and their Spectral Statistics

TL;DR

This paper extends the spherical transform framework to analyze products of non-Haar distributed unitary matrices, introducing cyclic Pólya ensembles and deriving their eigenvalue statistics with potential applications in spectral analysis.

Contribution

It introduces cyclic Pólya frequency functions and ensembles for unitary matrices, providing a new analytic framework for their spectral properties.

Findings

Defined cyclic Pólya frequency functions and ensembles.

Derived determinantal point processes for eigenvalues.

Discussed challenges in local spectral statistics analysis.

Abstract

The framework of spherical transforms and P\'olya 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 P\'olya frequency functions and show their relation to the cyclic P\'olya 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.