The lattice structure of chip firing games and related models

Abstract

In this paper, we study a classical discrete dynamical system, the chip firing game (CFG), used as a model in physics, economics and computer science. We use order theory and show that the set of reachable states (i.e. the configuration space) of such a system started from any configuration is a lattice, which implies strong structural properties. The lattice structure of the configuration space of a dynamical system is of great interest since it implies convergence (and more) if the configuration space is finite. If it is infinite, this property implies another kind of convergence: all the configurations reachable from two given configurations are reachable from their infimum. In other words, there is a unique first configuration which is reachable from two given configurations. Moreover, the CFG is a very general model, and we show how known models can be encoded as CFGs, and how some results about them can be deduced from this paper. Finally, we introduce a new model, which is a generalisation of the CFG, and about which many interesting questions arise.