Variance linearity for real Gaussian zeros

TL;DR

This paper studies the zero set of stationary Gaussian processes on the real line, establishing that their variance is at least linear and exploring conditions for linear variance, with examples of super rigidity and mixing.

Contribution

It provides lower bounds for the variance of zeros, proves non-hyperuniformity, and characterizes conditions for linear variance in Gaussian zero processes.

Findings

Zero set variance is at least linear

The process is never hyperuniform

Examples of super rigid, weakly mixing zero sets

Abstract

We investigate the zero set of a stationary Gaussian process on the real line, and in particular give lower bounds for the variance of the number of points on a large interval, in all generality. We prove that this point process is never hyperuniform, i.e. the variance is at least linear, and give necessary conditions to have linear variance, which are close to be sharp. We study the class of symmetric Bernoulli convolutions and give an example where the zero set is super rigid, weakly mixing, and not hyperuniform.