Electronic Journal of Probability

On diffusion limited deposition

Amine Asselah, Emilio N.M. Cirillo, Benedetto Scoppola, and Elisabetta Scoppola

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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.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 19, 29 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B24: Interface problems; diffusion-limited aggregation 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

diffusion limited aggregation cluster growth random walk

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Asselah, Amine; Cirillo, Emilio N.M.; Scoppola, Benedetto; Scoppola, Elisabetta. On diffusion limited deposition. Electron. J. Probab. 21 (2016), paper no. 19, 29 pp. doi:10.1214/16-EJP4310. https://projecteuclid.org/euclid.ejp/1456499641

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