Equilibrium distribution of zeros of random polynomials

Abstract

We consider ensembles of random polynomials of the form $p(z)=\sum_{j = 1}^N a_j P_j$ where $\{a_j\}$ are independent complex normal random variables and where $\{P_j\}$ are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain $Ω\subset C$ relative to an analytic weight $ρ(z) |dz|$. In the simplest case where $Ω$ is the unit disk and $ρ=1$, so that $P_j(z) = z^j$, it is known that the average distribution of zeros is the uniform measure on $S^1$. We show that for any analytic $(Ω, ρ)$, the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on $\partialΩ$ as $N\to\infty$. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of $(Ω, ρ)$.