Diffusion Limited Aggregation on a Cylinder

We consider the DLA process on a cylinder G × N. It is shown that this process “grows arms”, provided that the base graph G has small enough mixing time. Specifically, if the mixing time of G is at most log(2−ε) |G|, the time it takes the cluster to reach the m-th layer of the cylinder is at most of order m · |G| log log|G| . In particular we get examples of infinite Cayley graphs of degree 5, for which the DLA cluster on these graphs has arbitrarily small density. In addition, we provide an upper bound on the rate at which the “arms” grow. This bound is valid for a large class of base graphs G, including discrete tori of dimension at least 3. It is also shown that for any base graph G, the density of the DLA process on a G-cylinder is related to the rate at which the arms of the cluster grow. This implies, that for any vertex transitive G, the density of DLA on a G-cylinder is bounded by 2/3.