A natural stochastic extension of the sandpile model on a graph

TL;DR

This paper introduces a stochastic extension of the sandpile model on graphs, where grains are sent with probability p, and characterizes its recurrent configurations using graph orientations, linking it to the Tutte polynomial.

Contribution

It defines a new stochastic sandpile model, characterizes its recurrent configurations via graph orientations, and relates its polynomial to the Tutte polynomial.

Findings

Recurrent configurations differ from the classical model for p<1.

The set of recurrent configurations is characterized by graph orientations.

The lacking polynomial satisfies a recurrence similar to the Tutte polynomial.

Abstract

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 $p \in (0,1]$. For $p=1$, this coincides with the standard Abelian sandpile model. In general, for $p\in(0,1)$, 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.