Perturbed Toeplitz operators and radial determinantal processes

TL;DR

This paper investigates rotation invariant determinantal point processes in the complex plane, establishing criteria for central limit theorems for angular statistics using perturbed Toeplitz matrices.

Contribution

It introduces a new criterion linking perturbed Toeplitz determinants to CLTs for angular statistics in determinantal ensembles.

Findings

Criteria for CLT validity in these ensembles

Connection between generating functions and perturbed Toeplitz determinants

Applicable to eigenvalues of Gaussian matrices and roots of random polynomials

Abstract

We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criteria for a central limit theorem to hold for angular statistics of the points. The proof exploits an exact formula relating the generating function of such statistics to the determinant of a perturbed Toeplitz matrix.