A natural stochastic extension of the sandpile model on a graph

2012/09/10 by Yao-ban Chan, Chan, Yao-ban, Jean‐François Marckert +3
Physics and Astronomy · Computer Science · Mathematics · #Theoretical and Computational Physics #Topological and Geometric Data Analysis #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1209.2038

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 ∈ (0,1]. For p=1, this 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.

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