Fluctuations of the number of excursion sets of planar Gaussian fields

TL;DR

This paper investigates the fluctuations in the number of connected components of excursion sets in planar Gaussian fields, revealing that these fluctuations are at least proportional to the boundary length, with stronger bounds under certain spectral conditions.

Contribution

It establishes lower bounds on the fluctuations of excursion set components in planar Gaussian fields, including cases with singular spectral densities and the Random Plane Wave.

Findings

Fluctuations are at least of order R for certain Gaussian fields and levels.

Variance of the number of components is at least of order R^2 in many cases.

For the Random Plane Wave, fluctuations are at least of order R^{3/2}.

Abstract

For a smooth, stationary, planar Gaussian field, we consider the number of connected components of its excursion set (or level set) contained in a large square of area $R^2$. The mean number of components is known to be of order $R^2$ for generic fields and all levels. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain levels $\ell$, these random variables have fluctuations of order at least $R$, and hence variance of order at least $R^2$. In particular, this holds for excursion sets when $\ell$ is in some neighbourhood of zero, and it holds for excursion/level sets when $\ell$ is sufficiently large. We prove stronger fluctuation lower bounds of order $R^\alpha$, $\alpha \in [1,2]$, in the case that the spectral density has a singularity at the origin. Finally, we show that the number of excursion/level sets for the Random Plane Wave at certain levels has fluctuations of order at least $R^{3/2}$, and hence variance of order at least~$R^3$. We expect that these bounds are of the correct order, at least for generic levels.