Motors and Impossible Firing Patterns in the Parallel Chip-Firing Game

Abstract

The parallel chip-firing game is an automaton on graphs in which vertices “fire” chips to their neighbors. This simple model contains much emergent complexity and has many connections to different areas of mathematics. In this work, we study firing sequences, which describe each vertex's interaction with its neighbors in this game. First, we introduce the concepts of motors and motorized games. Motors both generalize the game and allow us to isolate local behavior of the (ordinary) game. We study the effects of motors connected to a tree and show that motorized games can be transformed into ordinary games if the motor's firing sequence occurs in some ordinary game. Then, we completely characterize the periodic firing sequences that can occur in an ordinary game, which have a surprisingly simple combinatorial description.