### Internal aggregation models on comb lattices

**Wilfried Huss**

*(Vienna University of Technology)*

**Ecaterina Sava**

*(Graz University of Technology)*

#### 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.

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Pages: 1-21

Publication Date: April 12, 2012

DOI: 10.1214/EJP.v17-1940

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