Convergence of random walks to Brownian motion on cubical complexes

Cubical complexes are metric spaces constructed by gluing together unit cubes in an analogous way to the construction of simplicial complexes. We construct Brownian motion on such spaces, define random walks, and prove that the transition kernels of the random walks converge to that for Brownian motion. The proof involves pulling back onto the complex the distribution of Brownian sample paths on the standard cube, and combining this with a distribution on walks between cubes in the complex. The main application lies in analysing sets of evolutionary trees: several tree spaces are cubical complexes and we briefly describe our results and some applications in this context. Our results extend readily to a class of polyhedral complex in which every cell of maximal dimension is isometric to a given fixed polyhedron.