Tightness of Stationary Nodal Measures

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.