Ladder Sandpiles

Details Ladder Sandpiles Ladder Sandpiles

2007-04-23

arxiv.org/abs/0704.2913v2

Markov Proc. Relat. Fields 13 (2007), 493-518.

Mathematics

Probability

30 pp; v2 has only minor changes

Scientific paper

We study Abelian sandpiles on graphs of the form $G \times I$, where $G$ is an arbitrary finite connected graph, and $I \subset \Z$ is a finite interval. We show that for any fixed $G$ with at least two vertices, the stationary measures $\mu_I = \mu_{G \times I}$ have two extremal weak limit points as $I \uparrow \Z$. The extremal limits are the only ergodic measures of maximum entropy on the set of infinite recurrent configurations. We show that under any of the limiting measures, one can add finitely many grains in such a way that almost surely all sites topple infinitely often. We also show that the extremal limiting measures admit a Markovian coding.

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