Stabilizability and percolation in the infinite volume sandpile model

Details Stabilizability and percolation in the infinite volume sandpile model Stabilizability and percolation in the infinite volume sandpile model

2007-10-04

arxiv.org/abs/0710.0939v2

Annals of Probability 2009, Vol. 37, No. 2, 654-675

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AOP415 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/08-AOP415

We study the sandpile model in infinite volume on $\mathbb{Z}^d$. In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure $\mu$, are $\mu$-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In $d=1$ and $\mu$ a product measure with density $\rho=1$ (the known critical value for stabilizability in $d=1$) with a positive density of empty sites, we prove that $\mu$ is not stabilizable. Furthermore, we study, for values of $\rho$ such that $\mu$ is stabilizable, percolation of toppled sites. We find that for $\rho>0$ small enough, there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail, as is the case in the subcritical regime for ordinary percolation.

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