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, of 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 points, 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-It\^o chaos decomposition of Stecconi and Todino (2025), and a sharp analysis of the error in the pointwise Weyl law associated to an 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.