Abstract

The random waves are random linear combinations of eigenfunctions of the Laplace–Beltrami operator on Riemannian manifolds. Our first main result shows that the critical radius of an embedding, defined via these eigenfunctions, of Riemannian manifolds into higher-dimensional Euclidean spaces has a uniform lower bound. The proof relies on the local Weyl law for the kernel of the spectral projection. This result allows the application of Weyl's tube formula to derive tail probabilities for the suprema of random waves. The talk is elementary which is accessible to graduate students, focusing mostly on geometric analysis, with only a few lines of probability appearing.