Tightness of Stationary Nodal Measures

TL;DR

This paper proves that the rescaled nodal volume field of a smooth stationary Gaussian field converges to a Brownian sheet as the domain size grows, under certain covariance conditions, contrasting with Berry's random wave model.

Contribution

It establishes the convergence in distribution of the rescaled nodal volume field to a Brownian sheet, providing new moment estimates and tightness criteria for such Gaussian fields.

Findings

Convergence of the rescaled nodal volume field to a Brownian sheet.

Development of new moment estimates for tightness.

Contrast with Berry's random wave model where conditions fail.

Abstract

We study the rescaled nodal volume field $\xi_R$ 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 $\xi_R$ converges in distribution, in an appropriate space of c\`adl\`ag 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.