Structural properties of Stochastic Abelian Sandpile

Abstract

We present some combinatorial results on the stochastic abelian sandpile model. These models are characterized by nondeterministic toppling rules. The recurrence checking for the deterministic case can be performed using the well known burning test which detects presence of forbidden sub-configurations (FSC) in strongly polynomial time. In the stochastic case, however, even for Manna's model, which is perhaps the simplest non-trivial example, no such procedure is known. In this paper, we address the decision problem of the existence of any FSC in a general stochastic sandpile. We demonstrate a polynomial time algorithm which, given the sandpile graph and toppling rules, decides if there exists an FSC. In the event of a positive answer, it generates at least one FSC for the given sandpile. Repeated application of the algorithm can be used to find many distinct FSCs. We also demonstrate a procedure for creating larger FSCs from smaller ones and use this to create FSCs for the Manna's model. We hope that the structural analysis of stochastic sandpile we perform in this paper, will prove useful in the eventual formulation of a deterministic procedure to decide recurrence.