Critical radii and suprema of random waves over Riemannian manifolds

TL;DR

The paper studies random waves on smooth, compact Riemannian manifolds under the spherical ensemble and establishes a positive universal limit for the critical radius of a deterministic embedding $i_{\lambda}$ into $\mathbb{R}^{k_{\lambda}}$, derived via a local Weyl law. This universal limit enables Weyl's tube formula to yield explicit tail probabilities for the supremum of the random waves and to derive the asymptotics of the excursion probability and the expected Euler characteristic of excursion sets. A detailed analysis of the local geometry of the embedding and a kernel-based representation of the critical radius underpin the results, including the behavior of the induced metric $g_{\lambda}$ and the principal curvatures. The findings provide precise large-deviation formulas for excursion probabilities, linking spectral geometry with probabilistic geometry on manifolds and offering universal constants dependent only on the dimension $d$ of $M$. Overall, the work advances understanding of how high-frequency eigenfunctions govern extreme values and topology of random waves on manifolds through a deterministic-embedding tube-geometry framework.

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.

Theorems & Definitions (8)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Corollary 2.1
• Lemma 4.1
• proof
• Proposition 5.1
• proof