## The Annals of Probability

- Ann. Probab.
- Volume 37, Number 2 (2009), 654-675.

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

Anne Fey, Ronald Meester, and Frank Redig

#### Abstract

We study the sandpile model in infinite volume on 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.

#### Article information

**Source**

Ann. Probab., Volume 37, Number 2 (2009), 654-675.

**Dates**

First available in Project Euclid: 30 April 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1241099924

**Digital Object Identifier**

doi:10.1214/08-AOP415

**Mathematical Reviews number (MathSciNet)**

MR2510019

**Zentralblatt MATH identifier**

1165.60033

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J25: Continuous-time Markov processes on general state spaces 60G99: None of the above, but in this section

**Keywords**

Abelian sandpile stabilizability percolation phase transition toppling procedure

#### Citation

Fey, Anne; Meester, Ronald; Redig, Frank. Stabilizability and percolation in the infinite volume sandpile model. Ann. Probab. 37 (2009), no. 2, 654--675. doi:10.1214/08-AOP415. https://projecteuclid.org/euclid.aop/1241099924