Firing Patterns in the Parallel Chip-Firing Game

Abstract

The $\textit{parallel chip-firing game}$ is an automaton on graphs in which vertices "fire'' chips to their neighbors. This simple model, analogous to sandpiles forming and collapsing, contains much emergent complexity and has connections to different areas of mathematics including self-organized criticality and the study of the sandpile group. In this work, we study $\textit{firing sequences}$, which describe each vertex's interaction with its neighbors in this game. Our main contribution is a complete characterization of the periodic firing sequences that can occur in a game, which have a surprisingly simple combinatorial description. We also obtain other results about local behavior of the game after introducing the concept of $\textit{motors}$.

Keywords

• chip-firing
• combinatorics
• finite automata
• graph theory
• sandpile model
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
• [math.math-co] mathematics [math]/combinatorics [math.co]