Abstract
We study the rotor router model and two deterministic sandpile models. For the rotor router model in ℤd, Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter h (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension d≥1.
For the rotor router model, the limiting shape is a sphere for all values of h. For one of the sandpile models, and h=2d−2 (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a sphere in the limit h→−∞. Finally, we prove that the rotor router shape contains a diamond.
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Fey-den Boer, A., Redig, F. Limiting Shapes for Deterministic Centrally Seeded Growth Models. J Stat Phys 130, 579–597 (2008). https://doi.org/10.1007/s10955-007-9450-6
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DOI: https://doi.org/10.1007/s10955-007-9450-6