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

TL;DR

The paper investigates the fluctuations of the number of sign and level-set clusters for the Gaussian free field on $\mathbb{Z}^d$ with $d\ge3$ inside large domains. It develops a comprehensive Wiener-Itô chaos framework for nonlocal level-set functionals, introducing pivotal intensities and semi-local localisation to control the chaos components. Depending on the dimension and level, the authors establish Gaussian limits or Hermite-type non-Gaussian limits, and they uncover Berry cancellation phenomena that suppress fluctuations for sign clusters. The results significantly advance the understanding of fluctuations for strongly correlated Gaussian fields and open the door to analyzing other nonlocal geometric functionals via chaos expansions coupled with percolation-type inputs.

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-Itô 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.

Figures (2)

• Figure 1: Two complete Feynman diagrams on the vertices $\{X_{i,j}\}$ in the case $I=2,k=4$; only the first diagram contributes to the diagram formula.
• Figure 2: Illustration of pivotal configurations for the cluster count in $\Lambda_R$ (i.e. $\Xi(E)$ is the sum of the number of components of $E$ and $\Lambda_R \setminus E$ that do not intersect $\partial \Lambda_R$, where $E$ are the black vertices). Left: The configuration is $(-1)$-pivotal at $y_1$, $1$-pivotal at $y_2$, and not pivotal at $y_3$. Right: The configuration is $1$-pivotal at $(y_1,y_2)$.

Theorems & Definitions (102)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Theorem 1.6
• Theorem 1.10
• Theorem 2.1: Diagram formula jan97
• Proposition 2.2: Chaos expansion for smooth functionals
• proof
• Remark 2.3
• Definition 2.4