New chaos decomposition of Gaussian nodal volumes

Abstract

We investigate the random variable defined by the volume of the zero set of a smooth Gaussian field, on a general Riemannian manifold possibly with boundary, a fundamental object in probability and geometry. We prove a new explicit formula for its Wiener-It\^o chaos decomposition that is notably simpler than existing alternatives and which holds in greater generality, without requiring the field to be compatible with the geometry of the manifold. A key advantage of our formulation is a significant reduction in the complexity of computing the variance of the nodal volume. Unlike the standard Hermite expansion, which requires evaluating the expectation of products of $2+2n$ Hermite polynomials, our approach reduces this task--in any dimension $n$--to computing the expectation of a product of just four Hermite polynomials. As a consequence, we establish a new exact formula for the variance, together with lower and upper bounds. Importantly, in contrast to previous results, our approach applies to highly non-isotropic situations, allowing the study of Riemannian random waves on arbitrary manifolds. By introducing two parameters associated to any Gaussian field: the frequency and the eccentricity, we quantify the deviation from the standard settings (e.g., spheres) and establish a quantitative version of Berry's cancellation phenomenon valid on all manifolds.