Almost sure asymptotics for Riemannian random waves

TL;DR

This paper proves that Riemannian random waves on compact manifolds converge to a universal Gaussian field almost surely, extending classical results and providing new almost-sure asymptotics for nodal volumes.

Contribution

It establishes almost-sure convergence of Riemannian random waves to a universal Gaussian field and derives almost-sure asymptotics for nodal volumes, addressing a longstanding open question.

Findings

Convergence of rescaled random waves to a Gaussian field

Almost-sure asymptotics for nodal volume

Extension of Salem--Zygmund CLT to Riemannian manifolds

Abstract

We consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact Riemannian manifold. With probability one with respect to the Gaussian coefficients, we establish that, both for large band and monochromatic models, the process properly rescaled and evaluated at an independently and uniformly chosen point $X$ on the manifold, converges in distribution under the sole randomness of $X$ towards an universal Gaussian field as the frequency tends to infinity. This result extends the celebrated central limit Theorem of Salem--Zygmund for trigonometric polynomials series to the more general framework of compact Riemannian manifolds. We then deduce from the above convergence the almost-sure asymptotics of the nodal volume associated with the random wave. To the best of our knowledge, in the real Riemannian case, these asymptotics were only known in expectation and not in the almost sure sense due to the lack of sufficiently accurate variance estimates. This in particular addresses a question of S. Zelditch regarding the almost sure equidistribution of nodal volume.