The Stochastic Sandpile Model on Complete Graphs

Abstract

The stochastic sandpile model (SSM) is a generalisation of the standard Abelian sandpile model (ASM), in which topplings of unstable vertices are made random. When unstable, a vertex sends one grain to each of its neighbours independently with probability $p \in (0,1)$. We study the SSM on complete graphs. Our main result is a description of the recurrent states of the model. We show that these are given by convex sums of recurrent states of the ASM. This allows us to recover a well-known result: that the number of integer lattice points in the $n$-dimensional permutation polytope is equal to the number of labelled spanning trees on $n$ vertices. We also provide a stochastic version of Dhar's burning algorithm to check if a given (stable) state is recurrent or not, which runs in linear time. Finally, we study a family of so-called partial SSMs in which some vertices topple randomly while others topple deterministically (as in the ASM, sending one grain to all neighbours). We show that this distinction is meaningful, yielding sets of recurrent states that are in general different from those of both the ASM and SSM. We also show that to get all recurrent states of the SSM, we can allow up to two vertices to topple deterministically.