Stabilizability and percolation in the infinite volume sandpile model
| Autor: | Fey, Anne, Meester, Ronald, Redig, Frank |
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| Rok vydání: | 2007 |
| Předmět: | |
| Zdroj: | Annals of Probability 2009, Vol. 37, No. 2, 654-675 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1214/08-AOP415 |
| Popis: | 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. Comment: 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) |
| Databáze: | arXiv |
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