Random walk on sphere packings and Delaunay triangulations in arbitrary dimension

TL;DR

This paper proves that random walks on various tilings of Euclidean space, including Delaunay triangulations and sphere packings, converge to Brownian motion, with applications to irregular point distributions and new convergence proofs.

Contribution

It establishes convergence of random walks on a broad class of tilings to Brownian motion, including irregular and random point configurations, with new uniform convergence results.

Findings

Random walks on Delaunay triangulations converge to Brownian motion.

Uniform convergence of finite volume schemes for the Laplace equation is proven.

A new, concise proof of a 2020 result in 2D is provided.

Abstract

We prove that random walks on a family of tilings of d-dimensional Euclidean space, with a canonical choice of conductances, converge to Brownian motion modulo time parameterization. This class of tilings includes Delaunay triangulations (the dual of Voronoi tesselations) and sphere packings. Our regularity assumptions are deterministic and mild. For example, our results apply to Delaunay triangulations with vertices sampled from a d-dimensional Gaussian multiplicative chaos measure. As part of our proof, we establish the uniform convergence of certain finite volume schemes for the Laplace equation, with quantitative bounds on the rate of convergence. In the special case of two dimensions, we give a new, short proof of the main result of Gurel-Gurevich--Jerison--Nachmias (2020).