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December 2018 Diffusion limited aggregation on the Boolean lattice
Alan Frieze, Wesley Pegden
Ann. Appl. Probab. 28(6): 3528-3557 (December 2018). DOI: 10.1214/18-AAP1392

Abstract

In the Diffusion Limited Aggregation (DLA) process on $\mathbb{Z}^{2}$, or more generally $\mathbb{Z}^{d}$, particles aggregate to an initially occupied origin by arrivals on a random walk. The scaling limit of the result, empirically, is a fractal with dimension strictly less than $d$. Very little has been shown rigorously about the process, however.

We study an analogous process on the Boolean lattice $\{0,1\}^{n}$, in which particles take random decreasing walks from $(1,\dots,1)$, and stick at the last vertex before they encounter an occupied site for the first time; the vertex $(0,\dots,0)$ is initially occupied. In this model, we can rigorously prove that lower levels of the lattice become full, and that the process ends by producing an isolated path of unbounded length reaching $(1,\dots,1)$.

Citation

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Alan Frieze. Wesley Pegden. "Diffusion limited aggregation on the Boolean lattice." Ann. Appl. Probab. 28 (6) 3528 - 3557, December 2018. https://doi.org/10.1214/18-AAP1392

Information

Received: 1 May 2017; Revised: 1 December 2017; Published: December 2018
First available in Project Euclid: 8 October 2018

zbMATH: 06994399
MathSciNet: MR3861819
Digital Object Identifier: 10.1214/18-AAP1392

Subjects:
Primary: 60C05
Secondary: 82C24

Keywords: Boolean cube, DLA

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 6 • December 2018
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