Fluctuations of the number of excursion sets of planar Gaussian fields

Dmitry Beliaev, Michael McAuley, Stephen Muirhead

Published 2019-08-28Version 1

The number of connected components of the excursion set above a level $\ell$ (or level set at $\ell$) of a smooth planar Gaussian field in the ball of radius $R$ is known to have mean of order $R^2$ for any $\ell$. We show that for certain fields with positive spectral density near the origin (including the Bargmann-Fock field), and for certain values of $\ell$, 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.