Correlation functions for random complex zeroes: strong clustering and local universality

TL;DR

This paper establishes strong clustering of correlation functions for zeros of Gaussian Entire Functions, provides universal bounds, and demonstrates asymptotic normality of zero count fluctuations in large domains.

Contribution

It proves strong clustering of correlation functions and universal bounds for zeros of Gaussian analytic functions, linking these to normal fluctuations of zero counts.

Findings

Strong clustering of k-point correlation functions.

Universal local bounds for zero correlation functions.

Asymptotic normality of zero count fluctuations.

Abstract

We prove strong clustering of k-point correlation functions of zeroes of Gaussian Entire Functions. In the course of the proof, we also obtain universal local bounds for k-point functions of zeroes of arbitrary nondegenerate Gaussian analytic functions. In the second part of the paper, we show that strong clustering yields the asymptotic normality of fluctuations of some linear statistics of zeroes of Gaussian Entire Functions, in particular, of the number of zeroes in measurable domains of large area. This complements our recent results from the paper "Fluctuations in random complex zeroes" (arXiv:1003.4251v1).