Chip-Firing and Fractional Bases

TL;DR

This paper analyzes a specific chip-firing process on an infinite path graph, detailing the final configuration starting from an initial number of chips at the origin, with firing rules based on fractional bases.

Contribution

It introduces and characterizes a novel chip-firing process involving fractional bases on an infinite path graph, providing explicit descriptions of the final states.

Findings

Explicit description of the final chip configuration

Connection to fractional base representations

Analysis of the process dynamics and stability

Abstract

We study a particular chip-firing process on an infinite path graph. At any time when there are at least $a+b$ chips at a vertex, $a$ chips fire to the left and $b$ chips fire to the right. We describe the final state of this process when we start with $n$ chips at the origin.