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 $p \in (0,1]$. 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.

💡 Research Summary

The paper introduces a stochastic generalisation of the Abelian sandpile model (ASM) on a finite graph G that contains a distinguished sink vertex. In the classical ASM each unstable vertex topples by sending exactly one grain to every neighbour. The authors replace this deterministic rule by a probabilistic one: when a vertex v is unstable it attempts to send a grain to each neighbour independently with probability p∈(0,1]. When p=1 the model reduces to the ordinary ASM, while for any 0<p<1 the dynamics become a Markov chain with a richer set of recurrent (steady‑state) configurations.

The central theoretical contribution is a complete characterisation of the recurrent configurations for the stochastic model. To achieve this the authors define the “lack” of a vertex v as the difference between the number of grains currently present at v and the minimum number required for stability (which equals the degree of v). A configuration is recurrent if and only if there exists an orientation of the underlying undirected graph such that, for every vertex v, the out‑degree in that orientation is at least the lack of v. This condition generalises the well‑known orientation characterisation of ASM recurrent states (where the out‑degree must equal the degree) and reduces to it when the lack of every vertex is zero (i.e., when p=1).

Having identified the combinatorial object that encodes recurrence, the authors introduce the “lacking polynomial”
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