Asymptotic distribution of complex zeros of random analytic functions

TL;DR

This paper studies the asymptotic distribution of zeros of certain random analytic functions, showing convergence to a deterministic measure independent of the distribution of coefficients, with applications to random matrices and geometric ensembles.

Contribution

It establishes a general law for the zero distribution of random analytic functions under broad conditions, linking it to the Legendre-Fenchel transform of a function u(t).

Findings

The zero counting measure converges to a deterministic measure.

The limiting measure is independent of the distribution of the coefficients.

Applications include geometric ensembles and a circular law analogue.

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 \[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}\xi_kf_{k,n}z^k,\] 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.