Natural boundary and zero distribution of random polynomials in smooth domains

TL;DR

This paper studies how zeros of random polynomials with smooth boundary domains distribute and shows they tend to cluster on the boundary, also establishing conditions for the boundary to be a natural boundary for associated random series.

Contribution

It proves almost sure convergence of zero distributions to equilibrium measures and identifies conditions under which the boundary becomes a natural boundary for random series.

Findings

Zero counting measures converge to equilibrium measure on boundary.

Boundary is almost surely a natural boundary for the random series.

Results apply to polynomials with various standard bases in smooth domains.

Abstract

We consider the zero distribution of random polynomials of the form $P_n(z) = \sum_{k=0}^n a_k B_k(z)$, where $\{a_k\}_{k=0}^{\infty}$ are non-trivial i.i.d. complex random variables with mean $0$ and finite variance. Polynomials $\{B_k\}_{k=0}^{\infty}$ are selected from a standard basis such as Szeg\H{o}, Bergman, or Faber polynomials associated with a Jordan domain $G$ whose boundary is $C^{2, \alpha}$ smooth. We show that the zero counting measures of $P_n$ converge almost surely to the equilibrium measure on the boundary of $G$. We also show that if $\{a_k\}_{k=0}^{\infty}$ are i.i.d. random variables, and the domain $G$ has analytic boundary, then for a random series of the form $f(z) =\sum_{k=0}^{\infty}a_k B_k(z),$ $\partial{G}$ is almost surely a natural boundary for $f(z).$