Mass equidistribution for random polynomials

TL;DR

This paper investigates the asymptotic distribution of zeros of multivariate random polynomials with orthogonal bases and subgaussian coefficients, proving they become equidistributed with respect to a deterministic current as degree increases.

Contribution

It establishes mass equidistribution of zeros for a broad class of random polynomials with subgaussian coefficients, extending previous results to multivariate cases.

Findings

Zeros become equidistributed with a deterministic current

Results hold for a wide class of subgaussian coefficients

Uses Bergman kernel asymptotics and concentration inequalities

Abstract

The purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of associated orthogonal polynomials with subgaussian coefficients. This class of probability distributions contains a wide range of random variables including standard Gaussian and all bounded random variables. We prove that for almost every sequence of random polynomials their normalized zero currents become equidistributed with respect to a deterministic extremal current. The main ingredients of the proof are Bergman kernel asymptotics, mass equidistribution of random polynomials and concentration inequalities for subgaussian quadratic forms.