Critical radii and suprema of random waves over Riemannian manifolds

TL;DR

This paper investigates the geometric properties of random waves on Riemannian manifolds, establishing universal limits for critical radii and deriving tail probabilities for their suprema using Weyl's tube formula.

Contribution

It introduces a universal limit for the critical radius of eigenfunction-based embeddings of manifolds, enabling new probabilistic estimates for random wave behavior.

Findings

Universal positive limit for critical radius

Application of Weyl's tube formula to tail probabilities

Estimation of Euler characteristic expectations

Abstract

We study random waves on smooth, compact, Riemannian manifolds under the spherical ensemble. Our first main result shows that there is a positive universal limit for the critical radius of a specific deterministic embedding, defined via the eigenfunctions of the Laplace-Beltrami operator, of such manifolds into higher dimensional Euclidean spaces. This result enables the application of Weyl's tube formula to derive the tail probabilities for the suprema of random waves. Consequently, the estimate for the expectation of the Euler characteristic of the excursion set follows directly.