Ewens measures on compact groups and hypergeometric kernels

TL;DR

This paper explores the spectral properties of Ewens measures on compact groups, extending known results to the symplectic case and analyzing the associated orthogonal polynomials and kernels.

Contribution

It extends the analysis of Ewens measures to symplectic groups and studies the resulting spectral properties and kernels.

Findings

Explicit distributions for factors in the characteristic polynomial decomposition.

Introduction of a family of probability measures analogous to Ewens sampling formula.

Asymptotic analysis of orthogonal polynomials leading to a limit kernel.

Abstract

On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as $n$ tends to infinity to a limit kernel at the singularity.