Diffusion limited aggregation on a cylinder (Q934611)

The authors consider the diffusion limited aggregation (DLA) process on a cylinder \(G\times N\), here \(G\) is a finite graph. The main results of the paper are given in Theorems 2.1, 4.2 and 5,1. Theorem 2.1 states that if \(G\) has small enough mixing time, then the time it takes the cluster to reach the \(m\)-th layer of the cylinder is \(o(m \cdot |G|)\), where \(|G|\) is the size of \(G\). Note that for a graph \(G\) with mixing time at most \(\log^{(2-\epsilon)} |G|\) (for any constant \(\epsilon\)), the time to reach the mth layer is at most of order \(m\cdot \frac{|G|}{\log\log|G|}\) . Such a phenomenon is sometimes dubbed as ``the aggregate grows arms'', i.e. grows faster than order \(|G|\) particles per layer. The second result (Thm. 4.2) concerns the density of the limit cluster, the union of all clusters obtained at some finite time. It shows that the expected rate at which the cluster grows bounds this density. From this result it yields that (a) for any vertex transitive graph \(G\), the DLA process on the \(G\)-cylinder has density bounded by 2/3. This includes the cases where \(G\) is a \(d\)-dimensional torus; (b) for \(G\) with small enough mixing time, the density tends to 0 as the size of \(G\) tends to infinity. Finally, the third result is given in Theorem 5.1, which gives a lower bound on the expected time the cluster reaches the \(m\)-th layer, complementing the upper bound in Theorem 2.1. This lower bound implies that the cluster cannot grow too fast, and in fact for many natural graphs it cannot grow faster than \(|G|^c\) for some universal \(0 < c < 1\). The lower bound holds for a wider range of graphs at the base of the cylinder than the upper bound (including \(d\)-dimensional tori for \(d = 3\)).