Sorting via Chip-Firing

Sam Hopkins ; Thomas McConville ; Jim Propp

Séminaire lotharingien de combinatoire, 78B (2017)

Résumé

We investigate a variant of the chip-firing process on the infinite path graph Z: 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 extensions to other variants of the infinite path, an intriguing empirical observation on random firing of labeled chips, and a possible generalization to other types.