A Structure Theory of the Sandpile Monoid for Directed Graphs

The Abelian Sandpile Model is a diffusion process on (directed) graphs, studied, under various names, in statistical physics, discrete dynamical systems, theoretical computer science, algebraic graph theory, and other fields. The model takes a directed multigraph X with a sink accessible from all nodes; associates a configuration space with X and defines transition rules between the configurations; and finally, defines a a finite commutative monoid M (the sandpile monoid) on the “stable configurations” and a finite abelian group G (the sandpile group) on the “recurrent configurations.” We add the sandpile semigroup S and the Rees quotient S/G, the sandpile quotient to the list. We study the structure of these algebraic objects and their connection to the combinatorial structure of the underlying directed graphs. We demonstrate that the basic theory follows from elementary facts about commutative monoids. In particular, we point out that G is both the unique minimal ideal and the universal group quotient of M. We also note that the semilattice of idempotents of a finite commutative monoid M is also the universal semilattice quotient of M and that this semilattice arises as the meet semilattice of a lattice which, in the context of sandpiles, we call the sandpile lattice. Our main result establishes a dual isomorphism between the sandpile lattice and the lattice of ideals of the accessibility poset of cyclic strong components (strong components which contain a cycle) of the underlying digraph. As a consequence, we characterize the sandpile lattices up to isomorphism as finite distributive lattices. Finally we introduce the notion of transience class of a sandpile and relate it to the nilpotence class of sandpile quotient. The “transience class” concept offers a new direction of study of the Abelian Sandpile Model.