Confluence in Labeled Chip-Firing

TL;DR

This paper provides a new proof that labeled chip-firing on a line results in sorted chips for even counts and applies these methods to resolve related conjectures about confluence in such systems.

Contribution

It introduces a novel proof technique for a known result and extends the analysis to settle several open conjectures on confluence.

Findings

Labeled chip-firing on a line with even chips always sorts chips.

The new proof technique simplifies understanding of confluence.

Several related conjectures on confluence are resolved.

Abstract

In 2016, Hopkins, McConville, and Propp proved that labeled chip-firing on a line always leaves the chips in sorted order if the number of chips is even. We present a novel proof of this result. We then apply our methods to resolve a number of related conjectures concerning the confluence of labeled chip-firing systems.