Sorting via chip-firing

TL;DR

This paper studies a labeled chip-firing process on an infinite path graph, demonstrating a sorting property where chips end up ordered when starting from the origin with an even number of chips.

Contribution

It introduces a novel labeled chip-firing variant with a confluence property leading to sorting, extending understanding of chip-firing dynamics and configurations.

Findings

Labeled chip-firing on an infinite path sorts chips when starting from the origin with even count.

Stabilization preserves a natural partial order on configurations.

Extensions to other graphs and initial configurations are discussed.

Abstract

We investigate a variant of the chip-firing process on the infinite path graph: rather than treating the chips as indistinguishable, we label them with positive integers. To fire an unstable vertex, i.e. a vertex with more than one chip, we choose any two chips at that vertex and move the lesser-labeled chip to the left and the greater-labeled chip to the right. This labeled version of the chip-firing process exhibits a remarkable confluence property, similar to but subtler than the confluence that prevails for unlabeled chip-firing: when all chips start at the origin and the number of chips is even, the chips always end up in sorted order. Our proof of sorting relies upon an independently interesting lemma concerning unlabeled chip- firing which says that stabilization preserves a natural partial order on configurations. We also discuss some extensions of this sorting phenomenon to other graphs (variants of the infinite path), to other initial configurations, and to other Cartan-Killing types.