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Tightness of Stationary Nodal Measures

Louis Gass, Giovanni Peccati

TL;DR

The paper proves a functional central limit theorem for the nodal volume field of a smooth, stationary Gaussian field, showing that the rescaled nodal measure converges to a Brownian sheet in the Skorokhod space over the unit cube. The authors develop a robust method combining Kac–Rice formulas with cumulant analysis and new moment bounds, enabling tightness via a multidimensional Kolmogorov–Chentsov criterion. This functional convergence strengthens previous finite-dimensional CLTs and yields implications for non-local functionals in specific 2D cases, such as overcrowding-type limits. The results clarify the contrast with Berry’s random wave model, where integrability conditions fail and tightness remains unresolved, thus advancing the probabilistic understanding of nodal geometry in Gaussian fields.

Abstract

We study the rescaled nodal volume field $ξ_R$ associated with a smooth, stationary Gaussian field on $[0,R]^d$, whose covariance satisfies adequate integrability conditions. Our main theorem shows that, as $R \to \infty$, the process $ξ_R$ converges in distribution, in an appropriate space of càdlàg mappings, to a standard Brownian sheet. The proof relies on a recent finite-dimensional CLT by Ancona, Gass, Letendre, and Stecconi (2025), as well as on a multidimensional Kolmogorov--Chentsov criterion for tightness due to Bickel and Wichura (1971). The application of the latter requires new moment estimates that are of independent interest. Our results stand in sharp contrast with Berry's random wave model, where the required integrability conditions fail and the question of tightness remains open.

Tightness of Stationary Nodal Measures

TL;DR

The paper proves a functional central limit theorem for the nodal volume field of a smooth, stationary Gaussian field, showing that the rescaled nodal measure converges to a Brownian sheet in the Skorokhod space over the unit cube. The authors develop a robust method combining Kac–Rice formulas with cumulant analysis and new moment bounds, enabling tightness via a multidimensional Kolmogorov–Chentsov criterion. This functional convergence strengthens previous finite-dimensional CLTs and yields implications for non-local functionals in specific 2D cases, such as overcrowding-type limits. The results clarify the contrast with Berry’s random wave model, where integrability conditions fail and tightness remains unresolved, thus advancing the probabilistic understanding of nodal geometry in Gaussian fields.

Abstract

We study the rescaled nodal volume field associated with a smooth, stationary Gaussian field on , whose covariance satisfies adequate integrability conditions. Our main theorem shows that, as , the process converges in distribution, in an appropriate space of càdlàg mappings, to a standard Brownian sheet. The proof relies on a recent finite-dimensional CLT by Ancona, Gass, Letendre, and Stecconi (2025), as well as on a multidimensional Kolmogorov--Chentsov criterion for tightness due to Bickel and Wichura (1971). The application of the latter requires new moment estimates that are of independent interest. Our results stand in sharp contrast with Berry's random wave model, where the required integrability conditions fail and the question of tightness remains open.

Paper Structure

This paper contains 10 sections, 10 theorems, 60 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview and motivation
  3. Notation
  4. Setting
  5. Main results
  6. Proof
  7. Kac--Rice formula and cumulants
  8. Second moment estimate
  9. Fourth moment estimate
  10. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1.1

Let $f$ be a random process satisfying assumptions $(H1)$ (resp. $(\widetilde{H}1)$), $(H2)$ and $(H3)$. Then, as $R\to+\infty$, the random field $\xi_R$ defined in e:xi converges in distribution in $\mathcal{D}([0,1]^d)$ towards a $d$-dimensional standard Brownian sheet on $[0,1]^d$.

Figures (4)

  • Figure 1: Heatmap and 3D plot of a standard Brownian sheet $W$ on $[0,1]^2$.
  • Figure 2: Simulations of the Bargmann--Fock field on the square $[0,R]^2$, and of the associated nodal-length field $\xi_R$ on $[0,1]^2$. From left to right, the columns display the Bargmann--Fock field ((a), (d), (g)), the nodal-length field $\xi_R$ shown as a heatmap ((b), (e), (h)), and a three-dimensional plot of $\xi_R$((c), (f), (i)). From top to bottom, the rows correspond to $R=20$, $R=50$, and $R=80$.
  • Figure 3: The set $L(a,b)$ for $a=2, \, b=3$ (red curve), $a=\infty,\, b=3$ (blue curve), and $a=2,\, b=\infty$ (orange curve). The breaking point in each curve has coordinates $(k, 1-k/a)$.
  • Figure 4: Two pairs of adjacent rectangles $(A,B)$ contained in $[0,1]^2$. In the background, a realization of $x\mapsto f(Rx)$, where $f$ is the Bargmann-Fock field and $R=10$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 2.1: Kac--Rice
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 8 more