Statistics on Yau's conjecture: Variance asymptotics

Abstract

We investigate the probabilistic counterpart of Yau's conjecture on the nodal volume of Laplace eigenfunctions on compact manifolds, by studying the high-frequency variance asymptotics of Riemannian random waves. We establish (Theorem A) a quantitative bound for the fluctuations of their nodal volumes, depending on different regimes of spectral windows, including the monochromatic one: with spectral size 1. Notably, our bounds improve, by more than a power 2, the existing results in the literature, cf. Canzani and Hanin (2020), in the case of manifolds without conjugate pairs, in particular negatively curved ones. As a corollary, we prove that Berry's cancellation phenomenon occurs for monochromatic Riemannian Random Waves on such chaotic manifolds. Our proofs rely on a local and global analysis combining the Kac-Rice formula, the new Wiener-Itô chaos decompositions of Stecconi and Todino (2025), and a sharp analysis of the error in the pointwise Weyl law associated to arbitrary spectral window (Theorem B). We introduce a general machinery (Theorem C), which ensures variance decay under broad geometric conditions, subject to correlation decay assumptions.