Rotor-Routing Induces the Only Consistent Sandpile Torsor Structure on Plane Graphs

TL;DR

This paper proves that rotor-routing is the unique consistent sandpile torsor algorithm on plane graphs, establishing its special role in the structure of spanning trees and extending the concept to regular matroids.

Contribution

It rigorously defines and proves the uniqueness of rotor-routing as the only consistent sandpile torsor algorithm on plane graphs, and extends the framework to regular matroids.

Findings

Rotor-routing is the only consistent sandpile torsor algorithm on plane graphs.

There are exactly three other consistent algorithms with similar structure.

Conjecture that the Backman-Baker-Yuen algorithm is the unique consistent algorithm on regular matroids.

Abstract

We make precise and prove a conjecture of Klivans about actions of the sandpile group on spanning trees. More specifically, the conjecture states that there exists a unique ``suitably nice'' sandpile torsor structure on plane graphs which is induced by rotor-routing. First, we rigorously define a sandpile torsor algorithm (on plane graphs) to be a map which associates each plane graph (i.e., planar graph with an appropriate ribbon structure) with a free transitive action of its sandpile group on its spanning trees. Then, we define a notion of consistency, which requires a torsor algorithm to be preserved with respect to a certain class of contractions and deletions. Using these definitions, we show that the rotor-routing sandpile torsor algorithm is consistent. Furthermore, we demonstrate that there are only three other consistent algorithms on plane graphs, which all have the same structure as rotor-routing. We also define sandpile torsor algorithms on regular matroids and suggest a notion of consistency in this context. We conjecture that the Backman-Baker-Yuen algorithm is consistent, and that there are only three other consistent sandpile torsor algorithms on regular matroids, all with the same structure.