Stabilizability and percolation in the infinite volume sandpile model

Abstract

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 $μ$, are $μ$-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In $d=1$ and $μ$ a product measure with density $ρ=1$ (the known critical value for stabilizability in $d=1$) with a positive density of empty sites, we prove that $μ$ is not stabilizable. Furthermore, we study, for values of $ρ$ such that $μ$ is stabilizable, percolation of toppled sites. We find that for $ρ>0$ 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.

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