On Diffusion Limited Deposition
Résumé
We propose a simple model of columnar growth through {\it diffusion limited aggregation} (DLA). Consider a graph $G_N\times\N$, where the basis has N vertices GN:={1,…,N}, and two vertices (x,h) and (x′,h′) are adjacent if |h−h′|≤1. Consider there a simple random walk {\it coming from infinity} which {\it 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 α<β such that the maximal pile height at time αN/log(N) is of order log(N), while at time βN/log(N) is larger than Nχ. This suggests that a \emph{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 \emph{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.