Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics

A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance property is developed. This involves isolating a class of invariant unitary matrices, to be referred to as cyclic Pólya ensembles, and examining their properties with respect to the spherical transform on U(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm U(N)$$\end{document}. Included in the cyclic Pólya ensemble class are Haar invariant unitary matrices, 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, and give a uniqueness statement for the corresponding weights. The natural appearance of bilateral hypergeometric series is highlighted, with this special function playing the role of the Meijer G-function in the transform theory of unitary invariant product of positive definite matrices. We construct a family of functions forming bi-orthonormal pairs which underly the correlation kernel of the corresponding determinantal point processes, and furthermore obtain an integral formula for the correlation kernel involving just two of these functions.