Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A central limit theorem for the number of excursion set components of Gaussian fields

Dmitry Beliaev, Michael McAuley, Stephen Muirhead

TL;DR

The paper establishes a central limit theorem for the number of excursion/level-set components of smooth, stationary Gaussian fields in large domains under short-range correlations, with fluctuations of volume order. It introduces a robust martingale framework based on a moving-average representation $f=q*W$, proving a CLT and showing that the limiting variance is strictly positive under mild integral conditions, with the Bargmann–Fock field as a primary example. A key technical advance is a third-moment bound for critical points, which underpins the CLT and extends to stability results for a broad class of topological functionals. The work contrasts with Hermite-expansion approaches by providing a more flexible method that can handle non-additive topological functionals and yields a semi-explicit form for the limiting variance; these results have potential implications for fields like cosmology, percolation theory, and Morse-theoretic analyses of Gaussian random fields.

Abstract

For a smooth stationary Gaussian field on $\mathbb{R}^d$ and level $\ell \in \mathbb{R}$, we consider the number of connected components of the excursion set $\{f \ge \ell\}$ (or level set $\{f = \ell\}$) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g. the Bargmann-Fock field), a central limit theorem holds with volume-order scaling. Previously such a result had only been established for `additive' geometric functionals of the excursion/level sets (e.g. the volume or Euler characteristic) using Hermite expansions. Our approach, based on a martingale analysis, is more robust and can be generalised to a wider class of topological functionals. A major ingredient in the proof is a third moment bound on critical points, which is of independent interest.

A central limit theorem for the number of excursion set components of Gaussian fields

TL;DR

The paper establishes a central limit theorem for the number of excursion/level-set components of smooth, stationary Gaussian fields in large domains under short-range correlations, with fluctuations of volume order. It introduces a robust martingale framework based on a moving-average representation , proving a CLT and showing that the limiting variance is strictly positive under mild integral conditions, with the Bargmann–Fock field as a primary example. A key technical advance is a third-moment bound for critical points, which underpins the CLT and extends to stability results for a broad class of topological functionals. The work contrasts with Hermite-expansion approaches by providing a more flexible method that can handle non-additive topological functionals and yields a semi-explicit form for the limiting variance; these results have potential implications for fields like cosmology, percolation theory, and Morse-theoretic analyses of Gaussian random fields.

Abstract

For a smooth stationary Gaussian field on and level , we consider the number of connected components of the excursion set (or level set ) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g. the Bargmann-Fock field), a central limit theorem holds with volume-order scaling. Previously such a result had only been established for `additive' geometric functionals of the excursion/level sets (e.g. the volume or Euler characteristic) using Hermite expansions. Our approach, based on a martingale analysis, is more robust and can be generalised to a wider class of topological functionals. A major ingredient in the proof is a third moment bound on critical points, which is of independent interest.
Paper Structure (21 sections, 29 theorems, 215 equations, 1 figure)

This paper contains 21 sections, 29 theorems, 215 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Existing results on the component count
  3. CLT for the component count
  4. Positivity of the limiting variance
  5. Third moment bounds
  6. Outline of the paper
  7. Acknowledgements
  8. Stability of the component count
  9. Stratified domains and critical points
  10. Stability of the component count
  11. Application to Gaussian fields
  12. Proof of the CLT
  13. Martingale CLT for lexicographic arrays
  14. Application to the component count
  15. Positivity of the limiting variance
  16. ...and 6 more sections

Key Result

Theorem 1.2

Suppose Assumption a:clt holds. Let $\ell \in \mathbb{R}$ and $\star \in \{\mathrm{ES},\mathrm{LS}\}$. Then there exists $\sigma = \sigma_\star(\ell) \ge 0$ such that, as $R \to \infty$, where $Z$ is a standard normal random variable.

Figures (1)

  • Figure 1: An example of a stratified domain in $d=2$; the dashed lines show the boundary of $\mathcal{R}$ while the shaded region, thick lines and circles show the stratification of a domain $D$.

Theorems & Definitions (67)

  • Theorem 1.2: CLT for the component count
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.6: Third moment bound for critical points
  • Corollary 1.7: Third moment bound for the component count
  • Remark 1.8
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 57 more