Electronic Journal of Probability

Internal aggregation models on comb lattices

Wilfried Huss and Ecaterina Sava

Full-text: Open access

Abstract

The two-dimensional comb lattice $\mathcal{C}_2$ is a natural spanning tree of the Euclidean lattice  $\mathbb{Z}^2$. We study three related cluster growth models on $\mathcal{C}_2$: internal diffusion limited aggregation (IDLA), in which random walkers move on the vertices of $\mathcal{C}_2$ until reaching an unoccupied  site where they stop; rotor-router aggregation in which particles perform deterministic walks, and stop when reaching a site previously unoccupied; and the divisible sandpile model where at  each vertex there is a pile of sand, for which, at each step, the mass exceeding $1$ is distributed equally among the neighbours. We describe the shape of the divisible sandpile cluster on $\mathcal{C}_2$,  which is then used to give inner bounds for IDLA and rotor-router aggregation.

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 30, 21 pp.

Dates
Accepted: 12 April 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062352

Digital Object Identifier
doi:10.1214/EJP.v17-1940

Mathematical Reviews number (MathSciNet)
MR2915666

Zentralblatt MATH identifier
1244.82047

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 05C81: Random walks on graphs

Keywords
growth model comb lattice internal diffusion limited aggregation rotor-router aggregation divisible sandpile asymptotic shape random walk rotor-router walk

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Huss, Wilfried; Sava, Ecaterina. Internal aggregation models on comb lattices. Electron. J. Probab. 17 (2012), paper no. 30, 21 pp. doi:10.1214/EJP.v17-1940. https://projecteuclid.org/euclid.ejp/1465062352


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