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

TL;DR

This paper analyzes the asymptotic behavior of cumulants and limit theorems for the volume of zero sets and critical points of smooth Gaussian fields over large domains, under regularity and decay conditions.

Contribution

It provides the first detailed asymptotic analysis of cumulants for zero sets and critical points of Gaussian fields, leading to LLN and CLT results.

Findings

Asymptotics of cumulants for zero sets and critical points as domain size grows

Establishment of strong Law of Large Numbers for nodal volume and critical points

Derivation of Central Limit Theorem under decay and regularity conditions

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 $R \to +\infty$.