Nodal Geometry, Heat Diffusion and Brownian Motion

TL;DR

This paper combines Brownian motion and heat diffusion tools to analyze the shape and size of nodal domains on Riemannian manifolds, providing bounds on their inradius and spatial containment.

Contribution

It extends Lieb's theorem on nodal domain inradius and refines Mangoubi's results, offering new geometric bounds using probabilistic methods.

Findings

Any nodal domain almost fully contains a ball of radius ~1/√λ.

Nodal domains cannot be contained in thin tubular neighborhoods of finite surface unions.

Provides probabilistic bounds on the geometry of nodal domains.

Abstract

We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain $\Omega_\lambda$ almost fully contains a ball of radius $\sim \frac{1}{\sqrt{\lambda}}$. This also gives a slight refinement of a result by Mangoubi, concerning the inradius of nodal domains (\cite{Man2}). On the other hand, we also prove that no nodal domain can be contained in a reasonably thin tubular neighbourhood of unions of finitely many surfaces inside $M$.