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Limit theorems for non-local functionals of smooth Gaussian fields via quasi-association

Michael McAuley

TL;DR

The paper develops a unified quasi-association framework to study limit theorems for non-local, approximately additive functionals of smooth Gaussian fields on expanding domains. By leveraging a covariance formula for topological events and Lipschitz control, it proves law of large numbers, variance asymptotics, and central limit theorems (distributional and almost-sure) for topological functionals and the Euler characteristic, under varying decay regimes of the covariance. It also analyzes the volume of the unbounded excursion component, obtaining LLN/LIL-type refinements, CLTs, and quantitative rates via association theory and percolation-type arm decay assumptions. Methodologically, the work combines stratified Morse theory, pivotal measures, and a mesoscopic-additivity decomposition to translate topological features into quasi-independent blocks, enabling explicit rate estimates and potential generalizations to manifolds and non-stationary settings. These results provide a robust, potentially universal approach to non-local Gaussian-field functionals, with implications for percolation-type phenomena and high-dimensional topology on random fields.

Abstract

Many classical objects of study related to the geometry/topology of smooth Gaussian fields (e.g. the volume, surface area or Euler characteristic of excursion sets) have a `locality' property which is crucial to their analysis. More recently, progress has been made in studying `non-local' quantities of such fields (e.g. the component/nodal count or the Betti numbers of excursion sets). In this work we establish limit theorems for non-local, approximately additive functionals of stationary fields evaluated on growing domains. Specifically we show that, for weakly dependent fields, such functionals satisfy a law of large numbers, have variance which is asymptotic to the volume of the domain and satisfy both quantitative and almost-sure central limit theorems. Our approach uses a covariance formula for topological events to establish a form of quasi-association for the functionals.

Limit theorems for non-local functionals of smooth Gaussian fields via quasi-association

TL;DR

The paper develops a unified quasi-association framework to study limit theorems for non-local, approximately additive functionals of smooth Gaussian fields on expanding domains. By leveraging a covariance formula for topological events and Lipschitz control, it proves law of large numbers, variance asymptotics, and central limit theorems (distributional and almost-sure) for topological functionals and the Euler characteristic, under varying decay regimes of the covariance. It also analyzes the volume of the unbounded excursion component, obtaining LLN/LIL-type refinements, CLTs, and quantitative rates via association theory and percolation-type arm decay assumptions. Methodologically, the work combines stratified Morse theory, pivotal measures, and a mesoscopic-additivity decomposition to translate topological features into quasi-independent blocks, enabling explicit rate estimates and potential generalizations to manifolds and non-stationary settings. These results provide a robust, potentially universal approach to non-local Gaussian-field functionals, with implications for percolation-type phenomena and high-dimensional topology on random fields.

Abstract

Many classical objects of study related to the geometry/topology of smooth Gaussian fields (e.g. the volume, surface area or Euler characteristic of excursion sets) have a `locality' property which is crucial to their analysis. More recently, progress has been made in studying `non-local' quantities of such fields (e.g. the component/nodal count or the Betti numbers of excursion sets). In this work we establish limit theorems for non-local, approximately additive functionals of stationary fields evaluated on growing domains. Specifically we show that, for weakly dependent fields, such functionals satisfy a law of large numbers, have variance which is asymptotic to the volume of the domain and satisfy both quantitative and almost-sure central limit theorems. Our approach uses a covariance formula for topological events to establish a form of quasi-association for the functionals.
Paper Structure (17 sections, 29 theorems, 188 equations, 3 figures)

This paper contains 17 sections, 29 theorems, 188 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related literature
  4. Main results
  5. Topological functionals
  6. Volume of the unbounded component
  7. Method of proof
  8. Discussion
  9. Covariance formula and quasi-association
  10. Covariance formula for events
  11. Covariance formula for functionals
  12. Excursion/level-set functionals and the Euler characteristic
  13. Qualitative limit theorems
  14. Quantitative limit theorems
  15. Variance asymptotics
  16. ...and 2 more sections

Key Result

Theorem 2.4

Let $f$ be a Gaussian field satisfying Assumptions a:minimal and a:cov_decay_weak, $\ell\in\mathbb{R}$ and $\Phi$ be a bounded excursion/level-set functional. Then there exists $c=c(\ell)\in\mathbb{R}$ such that, as $R\to\infty$ almost surely and in $L^2$.

Figures (3)

  • Figure 1: The box $\Lambda_R$ subdivided into separated mesoscopic $r$-boxes and stratified by hyperplanes at distance $a$. The shaded regions represent components of the excursion set $\{f\geq\ell\}$, labelled by the term of \ref{['e:top_decomp_outline']} to which they contribute.
  • Figure 2: The box $\Lambda_R$ is subdivided by $a$-planes (denoted by dashed lines). The shaded regions represent excursion set components, which are labelled according to whether they contribute to $\Phi^{(>a)}$ or $\Phi^{(<a)}$.
  • Figure 3: Two possible configurations of arm events illustrating Lemma \ref{['l:arm_event_supercrit']}. The shaded region represents the excursion set $\{g\geq\ell\}$.

Theorems & Definitions (59)

  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.8: Quantitative CLT
  • Theorem 2.11
  • Lemma 3.1
  • proof
  • Theorem 3.2: Theorem 2.13 of bmr20
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • ...and 49 more