Universality for zeros of random analytic functions

Abstract

Let $ξ_0,ξ_1,...$ be independent identically distributed (i.i.d.) random variables such that $\E \log (1+|ξ_0|)<\infty$. We consider random analytic functions of the form $$ G_n(z)=\sum_{k=0}^{\infty} ξ_k f_{k,n} z^k, $$ where $f_{k,n}$ are deterministic complex coefficients. Let $ν_n$ be the random measure assigning the same weight $1/n$ to each complex zero of $G_n$. Assuming essentially that $-\frac 1n \log f_{[tn], n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $ν_n$ converges weakly to some deterministic measure which is characterized in terms of the Legendre--Fenchel transform of $u$. The limiting measure is universal, that is it does not depend on the distribution of the $ξ_k$'s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.