Equivalence of local weak and Benjamini--Schramm formulations of Berry's random wave model

Abstract

Berry's random wave model predicts that high-energy eigenfunctions of classically chaotic systems can be modeled locally, at the wavelength scale, by the Gaussian monochromatic random wave. Several rigorous formulations of this prediction have been proposed. We prove the equivalence of two such formulations on a closed manifold: local weak limits and Benjamini--Schramm convergence of manifolds decorated by eigenfunctions. The latter considers pointed manifolds equipped with a smooth function, up to isometry, without recording local orthonormal coordinates. The key step is to recover the unaveraged local function law from its orthogonal average using translation invariance and the ergodicity of the random wave. Finally, either formulation implies strong inverse localization, allowing high-energy eigenfunctions to approximate arbitrary solutions of the Euclidean Helmholtz equation, with applications to their nodal sets.

arXiv comment34 pages. Revised version with simplified proofs of several lemmas and improved exposition