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2016 On diffusion limited deposition
Amine Asselah, Emilio N.M. Cirillo, Benedetto Scoppola, Elisabetta Scoppola
Electron. J. Probab. 21: 1-29 (2016). DOI: 10.1214/16-EJP4310


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.


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Amine Asselah. Emilio N.M. Cirillo. Benedetto Scoppola. Elisabetta Scoppola. "On diffusion limited deposition." Electron. J. Probab. 21 1 - 29, 2016.


Received: 19 May 2015; Accepted: 26 January 2016; Published: 2016
First available in Project Euclid: 26 February 2016

zbMATH: 1338.60226
MathSciNet: MR3485361
Digital Object Identifier: 10.1214/16-EJP4310

Primary: 60J45, 60K35, 82B24


Vol.21 • 2016
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