On Diffusion Limited Deposition

We propose a simple model of columnar growth through {\it diffusion limited aggregation} (DLA). Consider a graph $G_N\times\realmathbb{N}$, 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 {\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 $\alpha<\beta$ 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 \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.