Overcrowding estimates for zero count and nodal length of stationary Gaussian processes

TL;DR

This paper establishes that for certain stationary Gaussian processes, the probability of observing an unusually high number of zeros or large nodal length significantly exceeds the expected value is exponentially small, under specific spectral conditions.

Contribution

It provides new exponential bounds on the probability of large deviations for zero counts and nodal lengths in stationary Gaussian processes under spectral measure assumptions.

Findings

Probability of large zero count deviations is exponentially small.

Probability of large nodal length deviations is exponentially small.

Results apply to processes on  and  with spectral measure conditions.

Abstract

Assuming certain conditions on the spectral measures of centered stationary Gaussian processes on $\mathbb{R}$ (or ${\mathbb{R}}^2$), we show that the probability of the event that their zero count in an interval (resp., nodal length in a square domain) is larger than $n$, where $n$ is much larger than the expected value of the zero count in that interval (resp., nodal length in that square domain), is exponentially small in $n$.