Fluctuations in the zero set of the hyperbolic Gaussian analytic function

TL;DR

This paper investigates the variance in the number of zeros of the hyperbolic Gaussian analytic function within expanding discs, revealing a novel change in behavior at a specific intensity level.

Contribution

It uncovers a new phase transition in the variance behavior of the zero set, which was previously unreported in the literature.

Findings

Variance exhibits a phase transition at a critical intensity.

Distribution of zeros remains invariant under automorphisms.

New behavior pattern identified at a certain intensity threshold.

Abstract

The zero set of the hyperbolic Gaussian analytic function is a random point process in the unit disc whose distribution is invariant under automorphisms of the disc. We study the variance of the number of points in a disc of increasing radius. Somewhat surprisingly, we find a change of behaviour at a certain value of the `intensity' of the process, which appears to be novel.