Classes of Lattices Induced by Chip Firing (and Sandpile) Dynamics

Abstract

In this paper we study three classes of models widely used in physics, computer science and social science: the Chip Firing Game, the Abelian Sandpile Model and the Chip Firing Game on a mutating graph. We study the set of configurations reachable from a given initial configuration, called the configuration space of a model, and try to determine the main properties of such sets. We study the order induced over the configurations by the evolution rule. This makes it possible to compare the power of expression of these models. It is known that the configuration spaces we obtain are lattices, a special kind of partially ordered set. Although the Chip Firing Game on a mutating graph is a generalization of the usual Chip Firing Game, we prove that these models generate exactly the same configuration spaces. We also prove that the class of lattices induced by the Abelian Sandpile Model is strictly included in the class of lattices induced by the Chip Firing Game, but contains the class of distributive lattices, a very well known class.