On diffusion limited deposition. (English) Zbl 1338.60226

Summary: 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 \(|h-h'|\leq 1\). 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.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
60J45 Probabilistic potential theory
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