On diffusion limited deposition

Abstract

We propose a simple model of columnar growth through diffusion limited aggregation (DLA). Consider a graph $G_N\times \mathbb{N} $, where the basis has $N$ vertices $G_N:=\{1,\dots ,N\}$, and two vertices $(x,h)$ and $(x',h')$ are adjacent if $|h-h'|\le 1$. 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.