Asymptotic normality of linear statistics of zeros of random polynomials

TL;DR

This paper proves a central limit theorem for linear statistics of zeros of certain random polynomials, using Bergman kernel asymptotics to analyze the behavior of these zeros under varying weights.

Contribution

It introduces a CLT for zeros of random orthogonal polynomials with Gaussian coefficients and derives Bergman kernel asymptotics for weighted polynomial spaces.

Findings

Establishment of a CLT for linear statistics of zeros

Derivation of Bergman kernel asymptotics for weighted polynomial spaces

Application to random orthogonal polynomials with Gaussian coefficients

Abstract

In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted $L^2$-space of polynomials endowed with varying measures of the form $e^{-2n\varphi_n(z)}dz$ under suitable assumptions on the weight functions $\varphi_n$.