A covariance formula for the number of excursion set components of Gaussian fields and applications

TL;DR

This work addresses second-order fluctuations of Gaussian-field excursion/level-set component counts by deriving an exact covariance formula via an interpolation between a field and an independent copy. The core method introduces pivotal-point densities and a two-point framework that links covariance to integrals of the covariance kernel against pivotal measures. The authors then leverage this formula to obtain variance results: for short-range fields with $K\in L^1$, ${\rm Var}[N_*(R)]=\sigma^2 R^d+o(R^d)$ with an explicit $\sigma^2$, and they provide general upper bounds in broader settings, including oscillatory long-range cases. The results clarify how variance scales with domain size and correlation structure, offering improved bounds for challenging cases such as monochromatic random waves and guiding expectations for limit theorems in Gaussian topology.

Abstract

We derive a covariance formula for the number of excursion or level set components of a smooth stationary Gaussian field on $\mathbb{R}^d$ contained in compact domains. We also present two applications of this formula: (1) for fields whose correlations are integrable we prove that the variance of the component count in large domains is of volume order and give an expression for the leading constant, and (2) for fields with slower decay of correlation we give an upper bound on the variance which is of optimal order if correlations are regularly varying, and improves on best-known bounds if correlations are oscillating (e.g.\ monochromatic random waves).

Figures (2)

• Figure 1: Top panel: An illustration of a critical point which is 'negatively pivotal' for both the excursion and level set component counts at level $\ell$. We consider a small perturbation $\pm\delta h$ of the excursion set on a neighbourhood $W$ of the critical point $x$, where $h$ is a local positive perturbation (precise definitions will be given in Section \ref{['s:vub']}). In this case both excursion and level component counts decrease by one as we pass from the negative perturbation (left) to the positive perturbation (right). Middle panel: A critical point which is 'negatively pivotal' for the level set and 'not pivotal' for the excursion set. Bottom panel: A critical point of a boundary stratum which is 'negatively pivotal' for both the level set and the excursion set. Note that other configurations are also possible.
• Figure 2: Stratified critical points on the boundary and the effect of local perturbations on excursion/level sets. The excursion sets $\{f\geq\ell\}$ in the top and middle panels intersect $\mathbb{R}^d\setminus B$ and so they do not contribute to the (interior) component count regardless of the local perturbation around $x$. The excursion set $\{f\geq\ell\}$ in the bottom panel is interior but touches the boundary at $x$, therefore the component count is changed by a local perturbation.

Theorems & Definitions (67)

• Theorem 1: Covariance formula for component counts
• Proposition 1: Classical Gaussian interpolation formula
• proof
• Remark 1
• Theorem 2: Variance formula for the component count
• Remark 2
• Theorem 3
• Corollary 1
• Theorem 4: Variance upper bound
• Remark 3