An introduction to the Abelian sandpile model on a finite graph

The Sandpile model was introduced in the late eighties by three physicists, Bak, Tang and Wiesenfeld, with the aim of providing a model with features of self-organized criticality. Self-organized criticality appears in Nature through a variety of phenomena (earthquakes, forest fires, etc.) and is characterized by the fact that the internal dynamics of a given system leads it into a non-equilibrium state without external tuning of any parameter.
After its introduction, the Sandpile model was widely studied by physicists, among them by Dhar, who established several algebraic and combinatorial properties of it. In particular, he observed that the model is abelian, a crucial property making the model mathematically tractable.
In this talk, we review basic properties of the Abelian sandpile model. Given a finite connected graph, the model consists in configurations on the vertices of the graph, and transformation rules between successive configurations. As we shall see, there is a group structure which naturally arises from the dynamics of the model. The elements of the corresponding finite abelian group are recurrent configurations (in a probabilistic meaning), but they can also be characterized deterministically as “allowed configurations” for a certain “burning” algorithm. An interesting fact is that recurrent configurations are in bijection with spanning trees of the underlying graph.