Classes of lattices induced by chip firing (and sandpile) dynamics

In this paper we study three classes of models widely used in physics, computer science and social science: the chip firing game (CFG), the Abelian sandpile model (ASM) and the CFG on a mutating graph (or mutating CFG (MCFG)). These three models are variations of a game on a graph, in the vertices of which are stored chips, and where vertices can be fired, sending one chip along each outgoing edge. We study the set of configurations reachable from a given initial configuration, called the configuration space of a model. It is known that the order induced over the configurations by the evolution rule is a lattice, a special kind of partially ordered set. Our aim is to compare for inclusion the classes of lattices induced by these models. We show that the MCFG and the CFG induce exactly the same class of lattices, and we give new results towards the characterization of the class of lattices induced by the ASM.