Rotor-routing and spanning trees on planar graphs

TL;DR

This paper investigates the rotor-routing model's action of the sandpile group on spanning trees of ribbon graphs, proving it is independent of the basepoint precisely when the graph is planar.

Contribution

It establishes a necessary and sufficient condition for the rotor-routing action to be basepoint-independent, linking planarity of the graph to this property.

Findings

Rotor-routing action is independent of basepoint if and only if the graph is planar.

The result connects graph planarity with algebraic properties of the sandpile group.

Provides a characterization of when the rotor-routing model's action is well-defined without basepoint dependence.

Abstract

The sandpile group Pic^0(G) of a finite graph G is a discrete analogue of the Jacobian of a Riemann surface which was rediscovered several times in the contexts of arithmetic geometry, self-organized criticality, random walks, and algorithms. Given a ribbon graph G, Holroyd et al. used the "rotor-routing" model to define a free and transitive action of Pic^0(G) on the set of spanning trees of G. However, their construction depends a priori on a choice of basepoint vertex. Ellenberg asked whether this action does in fact depend on the choice of basepoint. We answer this question by proving that the action of Pic^0(G) is independent of the basepoint if and only if G is a planar ribbon graph.