Random walk on sphere packings and Delaunay triangulations in arbitrary dimension

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).