Limit theorems for the number of sign and level-set clusters of the Gaussian free field

TL;DR

This paper investigates the asymptotic behavior of sign and level-set clusters in the Gaussian free field on high-dimensional integer lattices, revealing Gaussian fluctuations in higher dimensions and complex behaviors at critical levels.

Contribution

It introduces the first application of chaos expansion techniques to analyze non-local functionals of strongly correlated Gaussian fields, providing new insights into cluster fluctuation behaviors.

Findings

Gaussian fluctuations in dimensions d ≥ 4 at non-critical levels

Potential non-Gaussian fluctuations in dimension d=3 depending on level

Sign clusters exhibit Berry cancellation, reducing fluctuation magnitude

Abstract

We study the limiting fluctuations of the number of sign and level-set clusters of the Gaussian free field on $\mathbb{Z}^d$, $d \ge 3$, that are contained in a large domain. In dimension $d \ge 4$ we prove that the fluctuations are Gaussian at all non-critical levels, while in dimension $d=3$ we show that fluctuations may be Gaussian or non-Gaussian depending on the level. We also show that the sign clusters experience a form of Berry cancellation in all dimensions, that is, the fluctuations of the sign cluster count is suppressed compared to generic levels. Our proof is based on controlling the Weiner-It\^{o} chaos expansion of the cluster count using percolation theoretic inputs; to our knowledge this is the first time that chaos expansion techniques have been applied to analyse a non-local functional of a strongly correlated Gaussian field.