Asymptotic distribution of complex zeros of random analytic functions

Let ${\xi }_0,{\xi }_1,\dots$ be independent identically distributed complex- valued random variables such that $\mathbb{E}\log (1+\mathrm{\mid }{\xi }_0\mathrm{\mid })<\mathrm{\infty }$. We consider random analytic functions of the form $${\mathbf{G}}_n(z)=\sum _{k=0}^{\mathrm{\infty }}{\xi }_k{f}_{k,n}{z}^k,$$ 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 $n\to \mathrm{\infty }$, 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.