Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution

TL;DR

This paper investigates the distribution of zeros of translation-invariant Gaussian analytic functions in a strip, establishing the existence and properties of their limiting zero-counting measures and relating them to the spectral measure.

Contribution

It proves the almost sure existence of the limiting zero-counting measure and characterizes when it is non-random based on the spectral measure, extending Wiener’s work.

Findings

Limiting zero-counting measure exists almost surely.

The measure is non-random if and only if the spectral measure is continuous.

Explicit computation of the mean zero-counting measure in terms of the spectral measure.

Abstract

We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the a limiting horizontal mean counting-measure of the zeroes exists almost surely, and that it is non-random if and only if the spectral measure is continuous (or degenerate). In this case, the mean zero-counting measure is computed in terms of the spectral measure. We compare the behavior with Gaussian analytic function with symmetry around the real axis. These results extend a work by Norbert Wiener.