Sandpiles, Spanning Trees, and Plane Duality
Melody Chan, Darren Glass, Matthew Macauley, David Perkinson, Caryn Werner, Qiaoyu Yang · SIAM Journal on Discrete Mathematics · 2015
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.