Equidistribution of zeros of random polynomials

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 Szegő, 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 $\mathbb{E}[{\log }^{+}|A_0|]<\infty$.