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A covariance formula for topological events of smooth Gaussian fields

Dmitry Beliaev, Stephen Muirhead, Alejandro Rivera

TL;DR

This work derives an exact covariance formula for topological events of smooth Gaussian fields on manifolds, expressing Cov(A1,A2) as an integral of the covariance kernel K against signed pivotal measures dπ^±. The method hinges on a finite-dimensional core established via Piterbarg’s formula and discriminant geometry, then extends to infinite dimensions through careful approximation and limiting arguments. The formula yields broad applications, including strong mixing bounds for topological events, decorrelation and lower concentration for topological counts, and an alternative perspective on the Harris criterion in level-set percolation. The approach unifies prior results (e.g., Rivera–Vanneuville, Piterbarg) and provides a versatile framework for connectivity and percolation-type questions in Gaussian fields on general manifolds.

Abstract

We derive a covariance formula for the class of `topological events' of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class, and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g. the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by a recent paper by Rivera and Vanneuville, in which a correlation inequality was derived for certain topological events on the plane, as well as by an old result of Piterbarg, in which a similar covariance formula was established for finite-dimensional Gaussian vectors.

A covariance formula for topological events of smooth Gaussian fields

TL;DR

This work derives an exact covariance formula for topological events of smooth Gaussian fields on manifolds, expressing Cov(A1,A2) as an integral of the covariance kernel K against signed pivotal measures dπ^±. The method hinges on a finite-dimensional core established via Piterbarg’s formula and discriminant geometry, then extends to infinite dimensions through careful approximation and limiting arguments. The formula yields broad applications, including strong mixing bounds for topological events, decorrelation and lower concentration for topological counts, and an alternative perspective on the Harris criterion in level-set percolation. The approach unifies prior results (e.g., Rivera–Vanneuville, Piterbarg) and provides a versatile framework for connectivity and percolation-type questions in Gaussian fields on general manifolds.

Abstract

We derive a covariance formula for the class of `topological events' of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class, and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g. the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by a recent paper by Rivera and Vanneuville, in which a correlation inequality was derived for certain topological events on the plane, as well as by an old result of Piterbarg, in which a similar covariance formula was established for finite-dimensional Gaussian vectors.

Paper Structure

This paper contains 31 sections, 32 theorems, 181 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Topological events
  3. Strong mixing in the Euclidean setting
  4. Application to lower concentration for topological counts
  5. A covariance formula for topological events
  6. The covariance formula
  7. Applications
  8. Strong mixing for topological events
  9. Sequences of fields: The Kostlan ensemble
  10. Decorrelation for topological counts
  11. Positive association for increasing topological events
  12. The Harris criterion
  13. Proof sketch
  14. Heart of the proof: the finite-dimensional case
  15. Restating the formula in terms of the discriminant
  16. ...and 16 more sections

Key Result

Corollary 1.1

The quantity $\mathrm{Cov}(A_1,A_2)$ is equal to where:

Figures (3)

  • Figure 1: An illustration of the crossing events $A_i$ and the pivotal events $\textup{Piv}_x(A_i)$ that appear in the covariance formula in Corollary \ref{['c:crossings']}. Left panels: Two realisations of a field $f$ which exhibit the left-right crossing event for $\{f > 0\}$ in the rectangle $B$ (shown in grey). Right panels: After a small perturbation of $f$ (compared to the left panel), the left-right crossing event no longer occurs. Central panels: The 'pivotal event' at which the crossing event first fails in this perturbation; this event can be of three possible types, either involving a level-$0$ critical point $x$ of $f$ in the interior of $B$ (top figure), or involving a level-$0$ critical point $x$ of $f$ restricted to the top side of $B$ (bottom figure), or involving a level-$0$ point $x$ on the corner of $B$ (not shown).
  • Figure 2: Left: An example of a tame stratification $\mathcal{F} = \{F_1, F_2\}$ of a compact set $B$. Here the generalised tangent bundle $T_xF_2|_{F_1}$ is well-defined since, as the points $x_k$ converge to $x\in F_1$, the respective tangent planes also converge. Right: A rough depiction of the 'rapid spiral sheet', which is an example of a set that cannot be tamely stratified (see Example \ref{['ex:stratified_sets']}); here tangent planes do not converge, and so the generalised tangent bundle is not well-defined.
  • Figure 3: Outside of a null set, the boundary of $\hat{A}$ is a hypersurface $\widetilde{V}_{F_{{}}}'$ which is covered by the disjoint union over $x\in F$ of the $\widetilde{V}_{x_{{}}}'$. Left (functional view): A small neighbourhood $U$ of $u$ in $V$ is split by $\widetilde{V}_{F_{{}}}'$ into two parts, $\mathcal{C}_1$ and $\mathcal{C}_2$, which are inside two different topological classes (one of them belongs to $\hat{A}$ and one does not). Right (spatial view): When $u$ changes continuously within $U\cap \widetilde{V}_{F_{{}}}'$, the corresponding level-$0$ stratified critical point $x$ changes continuously within $F$. Central panels shows three functions in $\widetilde{V}_{F_{{}}}'$ and their critical points. Small perturbations of these functions all belong to the same topological class, for perturbations positive near the critical point they belong to $\mathcal{C}_1$ (right panels) and for negative perturbations to $\mathcal{C}_2$ (left panels).

Theorems & Definitions (89)

  • Corollary 1.1
  • Corollary 1.2: Strong mixing for topological events
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6: Lower concentration for topological counts
  • Remark 1.7
  • Definition 2.1: Stratified set
  • Remark 2.2
  • Example 2.3: Trivial stratification
  • ...and 79 more