Stabilizability and percolation in the infinite volume sandpile model. (English) Zbl 1165.60033

Summary: 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.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
60G99 Stochastic processes
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