Gaussian complex zeroes are not always normal: limit theorems on the disc

TL;DR

This paper investigates the zeroes of hyperbolic-invariant random holomorphic functions on the unit disc, revealing a phase transition in their distribution depending on covariance decay, with implications for Gaussian chaos.

Contribution

It establishes a new limit theorem describing the transition from normal to skewed distributions of zeroes based on covariance decay rates.

Findings

Normal distribution of zeroes when covariance decays rapidly

Skewed distribution in long-range dependence regime

Connection to Gaussian multiplicative chaos

Abstract

We study the zeroes of a family of random holomorphic functions on the unit disc, distinguished by their invariance with respect to the hyperbolic geometry. Our main finding is a transition in the limiting behaviour of the number of zeroes in a large hyperbolic disc. We find a normal distribution if the covariance decays faster than a certain critical value. In contrast, in the regime of 'long-range dependence' when the covariance decays slowly, the limiting distribution is skewed. For a closely related model we emphasise a link with Gaussian multiplicative chaos.