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.
Highlights
Let G be an infinite, locally finite and connected graph with a chosen origin o ∈ G
Internal diffusion limited aggregation (IDLA) is a random walk-based growth model, which was introduced by Diaconis and Fulton [DF91]
In IDLA n particles start at the origin of G, and each particle performs a simple random walk until it reaches a vertex which was not previously occupied
Summary
Let G be an infinite, locally finite and connected graph with a chosen origin o ∈ G. Internal diffusion limited aggregation (IDLA) is a random walk-based growth model, which was introduced by Diaconis and Fulton [DF91]. In IDLA n particles start at the origin of G, and each particle performs a simple random walk until it reaches a vertex which was not previously occupied. There the particle stops, and on occupies this vertex, and a new particle starts its journey at the origin. The resulting random set of n occupied sites in G is called the IDLA cluster, and will be denoted by An. IDLA has received increased attention in the last years. In 1992, Lawler, Bramson and Griffeath [LBG92] showed that for simple random walk on Zd, with d ≥ 2, the limiting shape of IDLA, when properly rescaled, is almost surely an Euclidean ball of radius 1. In 1995, Lawler [Law95] refined this result by giving estimates on the fluctuations. Asselah and Gaudilliere [AG10, AG11a] proved an upper bound of order log(n) for the inner fluctuation δI and of order log2(n)
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