Universality for zeros of random analytic functions

TL;DR

This paper proves a universal limiting distribution for zeros of certain random analytic functions, independent of the distribution of coefficients, and applies the results to various geometric ensembles and a polynomial version of the circular law.

Contribution

It establishes a universal weak limit for zero measures of random analytic functions under broad conditions, linking it to the Legendre--Fenchel transform of a deterministic function.

Findings

The zero measures converge to a deterministic limit characterized by a Legendre--Fenchel transform.

The universality of the limiting measure does not depend on the distribution of the coefficients.

Applications include ensembles in constant curvature geometries and a polynomial analogue of the circular law.

Abstract

Let $\xi_0,\xi_1,...$ be independent identically distributed (i.i.d.) random variables such that $\E \log (1+|\xi_0|)<\infty$. 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 $-\frac 1n \log f_{[tn], n}\to u(t)$ 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.