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Sorting via chip-firing. (English) Zbl 1369.05148
Summary: We investigate a variant of the chip-firing process on the infinite path graph $$\mathbb{Z}$$: 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 some extensions of this sorting phenomenon to other graphs (variants of the infinite path), to other initial configurations, and to other Cartan-Killing types.

##### MSC:
 05C57 Games on graphs (graph-theoretic aspects) 05C82 Small world graphs, complex networks (graph-theoretic aspects) 68R10 Graph theory (including graph drawing) in computer science 91A43 Games involving graphs
##### Keywords:
chip-firing; abelian sandpile model; confluence
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