Random complex zeroes, I. Asymptotic normality

TL;DR

This paper proves that smooth functionals of zeroes in three models of Gaussian random analytic functions exhibit asymptotic normality, enhancing understanding of their statistical behavior.

Contribution

It establishes asymptotic normality for linear statistics of zeroes across elliptic, flat, and hyperbolic Gaussian models, a novel unification.

Findings

Asymptotic normality proven for all three models

Applicable to smooth linear functionals of zeroes

Advances understanding of zero distribution statistics

Abstract

We consider three models (elliptic, flat and hyperbolic) of Gaussian random analytic functions distinguished by invariance of their zeroes distribution. Asymptotic normality is proven for smooth functionals (linear statistics) of the set of zeroes.