Polynomial Ensembles and Recurrence Coefficients

TL;DR

This survey introduces recurrence coefficients as a unified tool to analyze the asymptotic behavior of polynomial ensembles, including convergence, variance, and sampling methods, in both real and complex settings.

Contribution

It provides a simple, unified approach using recurrence coefficients to study the asymptotics of polynomial ensembles, extending classical results to non-orthogonal projections and complex cases.

Findings

Unified approach to asymptotic behavior using recurrence coefficients

Convergence results for real and complex polynomial ensembles

Asymptotic formulas and variance control for linear statistics

Abstract

Polynomial ensembles are determinantal point processes associated with (non necessarily orthogonal) projections onto polynomial subspaces. The aim of this survey article is to put forward the use of recurrence coefficients to obtain the global asymptotic behavior of such ensembles in a rather simple way. We provide a unified approach to recover well-known convergence results for real OP ensembles. We study the mutual convergence of the polynomial ensemble and the zeros of its average characteristic polynomial; we discuss in particular the complex setting. We also control the variance of linear statistics of polynomial ensembles and derive comparison results, as well as asymptotic formulas for real OP ensembles. Finally, we reinterpret the classical algorithm to sample determinantal point processes so as to cover the setting of non-orthogonal projection kernels. A few open problems are also suggested.