On diffusion limited deposition

We propose a simple model of columnar growth through diffusion limited aggregation (DLA). Consider a graph G(N) x N, where the basis has N vertices G(N) : = {1, ..., N}, and two vertices (x, h) and (x', h') are adjacent if vertical bar h - h'vertical bar <= 1. Consider there a simple random walk coming from infinity which deposits on a growing cluster as follows: the cluster is a collection of columns, and the height of the column first hit by the walk immediately grows by one unit. Thus, columns do not grow laterally.We prove that there is a critical time scale N = log (N) for the maximal height of the piles, i.e., there exist constants alpha < beta such that the maximal pile height at time alpha/N = log (N) is of order log (N), while at time beta N/log(N) is larger than N-chi for some positive chi. This suggests that a monopolistic regime starts at such a time and only the highest pile goes on growing. If we rather consider a walk whose height-component goes down deterministically, the resulting ballistic deposition has maximal height of order log (N) at time N.These two deposition models, diffusive and ballistic, are also compared with uniform random allocation and Polya's urn.