26 research outputs found
Limiting shapes for deterministic centrally seeded growth models
We study the rotor router model and two deterministic sandpile models. For
the rotor router model in , 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 (the initial number of particles at each site), allows to
prove a number of new limiting shape results in any dimension .
For the rotor router model, the limiting shape is a sphere for all values of
. For one of the sandpile models, and (the maximal value), the
limiting shape is a cube. For both sandpile models, the limiting shape is a
sphere in the limit . Finally, we prove that the rotor router
shape contains a diamond.Comment: 18 pages, 3 figures, some errors corrected and more explanation
added, to appear in Journal of Statistical Physic
Chip-Firing and Rotor-Routing on Directed Graphs
We give a rigorous and self-contained survey of the abelian sandpile model
and rotor-router model on finite directed graphs, highlighting the connections
between them. We present several intriguing open problems.Comment: 34 pages, 11 figures. v2 has additional references, v3 corrects
figure 9, v4 corrects several typo