Asymptotic distribution of complex zeros of random analytic functions

Abstract

Let $\xi_{0},\xi_{1},\ldots$ be independent identically distributed complex-valued random variables such that $\mathbb{E}\log(1+|\xi_{0}|)<\infty$. We consider random analytic functions of the form \[\mathbf{G}_{n}(z)=\sum_{k=0}^{\infty}\xi_{k}f_{k,n}z^{k},\] where $f_{k,n}$ are deterministic complex coefficients. Let $\mu_{n}$ be the random measure counting the complex zeros of $\mathbf{G}_{n}$ according to their multiplicities. 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 $\frac{1}{n}\mu_{n}$ converges in probability to some deterministic measure $\mu$ which is characterized in terms of the Legendre–Fenchel transform of $u$. The limiting measure $\mu$ 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.