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 LG 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.