Variance of the Number of Zeroes of Shift-Invariant Gaussian Analytic Functions

TL;DR

This paper investigates the variance in the number of zeros of shift-invariant Gaussian analytic functions in a strip, revealing asymptotic bounds and conditions for different growth behaviors of the variance.

Contribution

It provides new asymptotic bounds for the variance of zero counts and characterizes conditions affecting its growth rate based on spectral measures.

Findings

Variance grows between linear and quadratic in T

Conditions identified for linear, quadratic, or intermediate variance growth

Provides bounds and spectral measure criteria for variance behavior

Abstract

Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle $[0,T]\times [a,b]$ is asymptotically between $cT$ and $CT^2$, with positive constants $c$ and $C$. We also supply with conditions (in terms of the spectral measure) under which the variance asymptotically grows linearly, as a quadratic function of $T$, or has intermediate growth.