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Zeros and critical points of Gaussian fields: cumulants asymptotics and limit theorems

Michele Ancona, Louis Gass, Thomas Letendre, Michele Stecconi

TL;DR

This work develops a general cumulant framework for the geometry of zeros and critical points of smooth centered stationary Gaussian fields in Euclidean space, focusing on the large-scale asymptotics of the (d−k)-volume of the zero set and, for k=1, the count of critical points. The authors derive explicit cumulant asymptotics for linear statistics supported on growing domains, and show Law of Large Numbers and Central Limit Theorems under mild decay and non-degeneracy hypotheses; they further extend to one-parameter families with local scaling limits, including Kostlan polynomials, where local limits converge to the Bargmann–Fock field. The strategy combines Kac–Rice formulas with a refined interpolation (Kergin) and a Hadamard-type analysis, organized via partitions and edge-connected graphs to manage the combinatorics. The results yield both moment and distributional limit theorems, with corollaries such as hole probabilities and functional CLTs, and provide a robust framework for understanding equidistribution of random submanifolds arising as vanishing loci in Gaussian models on manifolds.

Abstract

Let $f:\mathbb{R}^d \to \mathbb{R}^k$ be a smooth centered stationary Gaussian field and $\mathcal{B} \subset \mathbb{R}^d$ be a bounded Borel set. In this paper, we determine the asymptotics as $R \to \infty$ of all the cumulants of the $(d-k)$-dimensional volume of $f^{-1}(0) \cap R\mathcal{B}$. When $k=1$, we obtain similar asymptotics for the number of critical points of $f$ in $R\mathcal{B}$. Our main hypotheses are some regularity and non-degeneracy of the field, as well as mild integrability conditions on the first derivatives of its covariance kernel. As corollaries of these cumulants estimates, we deduce a strong Law of Large Numbers and a Central Limit Theorem for the nodal volume (resp.~the number of critical points) of a regular and non-degenerate enough field whose covariance decays fast enough at infinity. Our results hold more generally for a one-parameter family $(f_R)$ of Gaussian fields admitting a stationary local scaling limit as $R \to \infty$, for example Kostlan polynomials in the large degree limit. They also hold for the random measures of integration over the vanishing locus of $f_R$ as $R \to +\infty$.

Zeros and critical points of Gaussian fields: cumulants asymptotics and limit theorems

TL;DR

This work develops a general cumulant framework for the geometry of zeros and critical points of smooth centered stationary Gaussian fields in Euclidean space, focusing on the large-scale asymptotics of the (d−k)-volume of the zero set and, for k=1, the count of critical points. The authors derive explicit cumulant asymptotics for linear statistics supported on growing domains, and show Law of Large Numbers and Central Limit Theorems under mild decay and non-degeneracy hypotheses; they further extend to one-parameter families with local scaling limits, including Kostlan polynomials, where local limits converge to the Bargmann–Fock field. The strategy combines Kac–Rice formulas with a refined interpolation (Kergin) and a Hadamard-type analysis, organized via partitions and edge-connected graphs to manage the combinatorics. The results yield both moment and distributional limit theorems, with corollaries such as hole probabilities and functional CLTs, and provide a robust framework for understanding equidistribution of random submanifolds arising as vanishing loci in Gaussian models on manifolds.

Abstract

Let be a smooth centered stationary Gaussian field and be a bounded Borel set. In this paper, we determine the asymptotics as of all the cumulants of the -dimensional volume of . When , we obtain similar asymptotics for the number of critical points of in . Our main hypotheses are some regularity and non-degeneracy of the field, as well as mild integrability conditions on the first derivatives of its covariance kernel. As corollaries of these cumulants estimates, we deduce a strong Law of Large Numbers and a Central Limit Theorem for the nodal volume (resp.~the number of critical points) of a regular and non-degenerate enough field whose covariance decays fast enough at infinity. Our results hold more generally for a one-parameter family of Gaussian fields admitting a stationary local scaling limit as , for example Kostlan polynomials in the large degree limit. They also hold for the random measures of integration over the vanishing locus of as .
Paper Structure (48 sections, 99 theorems, 312 equations)

This paper contains 48 sections, 99 theorems, 312 equations.

Table of Contents

  1. Keywords:
  2. MSC 2020:
  3. Introduction
  4. Zero set of a stationary Gaussian field in a growing set
  5. Cumulants asymptotics
  6. Limit theorems and other corollaries
  7. Discussion of the main hypotheses and examples
  8. Kostlan polynomials
  9. Strategy of proof
  10. Organization of the paper
  11. Partitions, cumulants and diagonals
  12. Partitions of a finite set
  13. Cumulants
  14. Product spaces and diagonals
  15. Thick diagonals
  16. ...and 33 more sections

Key Result

Theorem 1.1

Let $p \geqslant 2$ be an integer and $Z$ be the zero set of a centered stationary Gaussian field $f:\mathbb{R}^d \to \mathbb{R}^k$ of class $\mathcal{C}^{2p-1}$. We assume that (hyp: non-degeneracy) and (hyp: decay of correlations limit) hold. Then, there exists $\gamma_2(f) \geqslant 0$, depending

Theorems & Definitions (275)

  • Theorem 1.1: Central moments asymptotics for volumes
  • Remark 1.2
  • Theorem 1.3: Central moments asymptotics for critical points
  • Remark 1.4
  • Theorem 1.5: Central Limit Theorem, stationary case
  • Theorem 1.6: Law of Large Numbers, stationary case
  • Theorem 1.7: Cumulants asymptotics for zero sets
  • Theorem 1.8: Cumulants asymptotics for critical points
  • Remark 1.9
  • Corollary 1.10: Moment asymptotics
  • ...and 265 more