Electronic Journal of Probability

Internal aggregation models on comb lattices

Wilfried Huss and Ecaterina Sava

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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