  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 $G_N:=\{1,\dots,N\}$, 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 $N^\chi$. 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.
