Thinning and conditioning of the Circular Unitary Ensemble

TL;DR

This paper investigates the effects of independent thinning on eigenvalues of Haar-distributed unitary matrices, analyzing gap probabilities and conditioned eigenvalue statistics through Toeplitz determinants and orthogonal polynomials.

Contribution

It introduces a novel analysis of thinned eigenvalues in the Circular Unitary Ensemble using Toeplitz determinants and asymptotic methods.

Findings

Asymptotic formulas for gap probabilities of thinned eigenvalues

Characterization of eigenvalue statistics conditioned on no eigenvalues in an arc

Connections between thinning operations and orthogonal polynomial techniques

Abstract

We apply the operation of random independent thinning on the eigenvalues of $n\times n$ Haar distributed unitary random matrices. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of random unitary matrices which are conditioned such that there are no thinned eigenvalues on a given arc of the unit circle. Various probabilistic quantities can be expressed in terms of Toeplitz determinants and orthogonal polynomials on the unit circle, and we use these expressions to obtain asymptotics as $n\to\infty$.