Equidistribution of zeros of random polynomials

TL;DR

This paper proves that the zeros of certain random polynomials become uniformly distributed along the boundary of a domain, given a mild condition on the distribution of the random coefficients.

Contribution

It establishes the almost sure convergence of zero distributions of random polynomials with deterministic bases to the equilibrium measure, under a finite logarithmic moment condition.

Findings

Zeros of random polynomials concentrate on boundary of domain

Convergence holds if and only if the coefficients have finite log-plus expectation

Results apply to various polynomial bases like Szegő, Bergman, and Faber

Abstract

We study the asymptotic distribution of zeros for the random polynomials $P_n(z) = \sum_{k=0}^n A_k B_k(z)$, where $\{A_k\}_{k=0}^{\infty}$ are non-trivial i.i.d. complex random variables. Polynomials $\{B_k\}_{k=0}^{\infty}$ are deterministic, and are selected from a standard basis such as Szeg\H{o}, Bergman, or Faber polynomials associated with a Jordan domain $G$ bounded by an analytic curve. We show that the zero counting measures of $P_n$ converge almost surely to the equilibrium measure on the boundary of $G$ if and only if $\mathbb{E}[\log^+|A_0|]<\infty$.