Natural boundary and zero distribution of random polynomials 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 Szegő, 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 $f(z).$

9 pages