Open Access
March 2009 Stabilizability and percolation in the infinite volume sandpile model
Anne Fey, Ronald Meester, Frank Redig
Ann. Probab. 37(2): 654-675 (March 2009). DOI: 10.1214/08-AOP415


We study the sandpile model in infinite volume on ℤd. 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.


Download Citation

Anne Fey. Ronald Meester. Frank Redig. "Stabilizability and percolation in the infinite volume sandpile model." Ann. Probab. 37 (2) 654 - 675, March 2009.


Published: March 2009
First available in Project Euclid: 30 April 2009

zbMATH: 1165.60033
MathSciNet: MR2510019
Digital Object Identifier: 10.1214/08-AOP415

Primary: 60G99 , 60J25 , 60K35

Keywords: Abelian sandpile , percolation , phase transition , stabilizability , toppling procedure

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 2 • March 2009
Back to Top