Universality for zeros of random analytic functions

Abstract

Let ${\xi }_0,{\xi }_1,...$ 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 $$G_n(z)=\sum _{k=0}^{\infty }{\xi }_k{f}_{k,n}{z}^k,$$ where $f_{k,n}$ are deterministic complex coefficients. Let ${\nu }_n$ be the random measure assigning the same weight $1/n$ to each complex zero of $G_n$. Assuming essentially that $-\frac{1}{n}\log f_{[tn],n}\to u(t)$ as $n\to \infty$, where $u(t)$ is some function, we show that the measure ${\nu }_n$ 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 ${\xi }_k$'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.

26 pages, 8 figures, 1 table

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