Spectral representation of correlation functions for zeros of Gaussian power series with stationary coefficients

TL;DR

This paper investigates the zeros of Gaussian analytic functions with stationary coefficients, providing spectral representations of their correlation functions and revisiting limit theorems with numerical examples.

Contribution

It introduces an integral spectral representation for the correlation functions of GAF zeros, connecting them to the spectral measures of stationary Gaussian coefficients.

Findings

Spectral measures characterize zero correlations of GAFs.

Revisits and extends limit theorems for random analytic functions.

Provides numerical illustrations of GAF examples.

Abstract

We analyze Gaussian analytic functions (GAFs) defined as power series with coefficients modeled by discrete stationary Gaussian processes, utilizing their spectral measures. We revisit some limit theorems for random analytic functions and examine some examples of GAFs through numerical computations. Furthermore, we provide an integral representation of the n-point correlation functions of the zero sets of GAFs in terms of the spectral measures of the underlying coefficient Gaussian processes.