Three central limit theorems for the unbounded excursion component of a Gaussian field

TL;DR

The work analyzes the geometry of the unbounded excursion component $\{f\ge\ell\}_{\infty}$ for smooth, stationary Gaussian fields $f$ with fast-decaying covariance, proving law of large numbers and central limit theorems for three global statistics—volume, surface area, and Euler characteristic—restricted to growing boxes. The authors develop a white-noise-based CLT for nonlocal functionals, combine it with a Morse-theoretic stability framework to control perturbations, and derive explicit (semi-explicit) formulas for limiting variances, establishing positivity under suitable conditions. In the planar case, the results hold for all supercritical levels $\ell<\ell_c$, with higher dimensions requiring stronger decay and a truncated-connection decay assumption; a finitary CLT is also obtained for practical observations in finite domains. The methods highlight the efficacy of martingale approaches for nonlocal geometric functionals of Gaussian fields and lay groundwork for extensions to broader classes of fields and topological statistics.

Abstract

For a smooth, stationary Gaussian field $f$ on Euclidean space with fast correlation decay, there is a critical level $\ell_c$ such that the excursion set $\{f\geq\ell\}$ contains a (unique) unbounded component if and only if $\ell<\ell_c$. We prove central limit theorems for the volume, surface area and Euler characteristic of this unbounded component restricted to a growing box. For planar fields, the results hold at all supercritical levels (i.e. all $\ell<\ell_c$). In higher dimensions the results hold at all sufficiently low levels (all $\ell<-\ell_c<\ell_c$) but could be extended to all supercritical levels by proving the decay of truncated connection probabilities. Our proof is based on the martingale central limit theorem.

Figures (7)

• Figure 1: The excursion sets $\{f\geq\ell\}$ are shown in white for the Matérn Gaussian field $f:\mathbb{R}^2\to\mathbb{R}$ with parameter $\nu=10$ (see Example \ref{['ex:fields']} for a description of this field). The largest connected component of the excursion set is highlighted in green. The critical level $\ell_c$ for this model is known to be zero.
• Figure 2: The excursion sets $\{f\geq\ell\}$ are shown in white for the Bargmann-Fock field $f:\mathbb{R}^2\to\mathbb{R}$ (this model is described in Example \ref{['ex:fields']}). The boundary of the largest connected component of the excursion set is highlighted in orange.
• Figure 3: The largest component (highlighted in green) of the excursion set $\{f\geq\ell\}\cap\Lambda_n$ surrounds three 'holes' and hence has an Euler characteristic of minus two. The Euler characteristic is well defined for the class of 'basic complexes' which includes excursion sets of smooth Gaussian fields (see at07 for details).
• Figure 4: The dashed area shows the excursion set $\{f\geq\ell\}$ restricted to $\Lambda_{(1+\epsilon)n}$. The dark grey area corresponds to $\{f\geq\ell\}_{n,\epsilon}$. In particular, the lower left excursion component in $\Lambda_n$ is not shaded because it is not connected to $\partial\Lambda_{(1+\epsilon)n}$.
• Figure 5: Three ways in which the volume of the unbounded component restricted to a cube can change: in case (i) the topology within the cube changes and we bound $\lvert\Delta_v(B_w)\rvert$ by $1$. In case (ii) there is no topological change in $B_w$ or in any cube which is connected to $B_w$ by a finite excursion component so we bound $\lvert\Delta_v(B_w)\rvert$ by the volume of the solid grey region in the figure. In case (iii) there is a topological change outside $B_w$ affecting which components inside $B_w$ are part of the unbounded component, so we bound $\lvert\Delta_v(B_w)\rvert$ by $1$.

Theorems & Definitions (84)

• Example 1
• Theorem 1.1: mv20riv21 for $d=2$, sev21sev23 for $d\geq 3$
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1
• Corollary 1.6
• Theorem 2.1
• Theorem 2.2