Sandpiles, spanning trees, and plane duality

TL;DR

This paper proves that the rotor-routing actions of sandpile groups on spanning trees are compatible under planar duality, confirming a conjecture by Baker, and deepening the understanding of graph duality and sandpile dynamics.

Contribution

It establishes the compatibility of rotor-routing actions with plane duality for planar graphs, confirming a conjecture and linking sandpile groups with graph duality.

Findings

Rotor-routing actions are compatible under plane duality.

Sandpile groups of dual graphs are related through these actions.

Confirms Baker's conjecture on duality compatibility.

Abstract

Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well-known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G*, and furthermore that the sandpile groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G* on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.