2025/05/28 by Michele Stecconi, Stecconi, Michele, Anna Paola Todino +1 · 2 citations
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Quantum chaos and dynamical systems #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2505.22350
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-Itô 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.