On the zeros of random harmonic polynomials: the Weyl model

TL;DR

This paper extends the analysis of zeros of Gaussian harmonic polynomials to the Weyl ensemble, providing asymptotic formulas for their expected number of zeros, building on prior work related to the Kostlan ensemble.

Contribution

It introduces asymptotic results for the zeros of harmonic polynomials in the Weyl ensemble, expanding the understanding of their zero distribution.

Findings

Derived asymptotics for zeros in the Weyl ensemble

Extended previous Kostlan ensemble results

Provided formulas for expected number of zeros

Abstract

Li and Wei (2009) studied the density of zeros of Gaussian harmonic polynomials with independent Gaussian coefficients. They derived a formula for the expected number of zeros of random harmonic polynomials as well as asymptotics for the case that the polynomials are drawn from the Kostlan ensemble. In this paper we extend their work to cover the case that the polynomials are drawn from the Weyl ensemble by deriving asymptotics for this class of harmonic polynomials.