A natural stochastic extension of the sandpile model on a graph

We introduce a new model of a stochastic sandpile on a graph G containing a sink. When unstable, a site sends one grain to each of its neighbours independently with probability p ∈ (0, 1). The case p = 1 coincides with the standard Abelian sandpile model. In general, for p ∈ (0, 1), the set of recurrent configurations of this sandpile model is different from that of the Abelian sandpile model. We give a characterisation of this set in terms of orientations of the graph G. We also define the lacking polynomial _{L G} as the generating function counting this set according to the number of grains, and show that this polynomial satisfies a recurrence which resembles that of the Tutte polynomial.