The Bernardi Process and Torsor Structures on Spanning Trees

Abstract

Abstract. Let G be a ribbon graph, i.e., a connected finite graph G together with a cyclic ordering of the edges around each vertex. By adapting a construction due to Olivier Bernardi, we associate to any pair (v, e) consisting of a vertex v and an edge e adjacent to v a bijection β (v,e) between spanning trees of G and elements of the set Pic g (G) of degree g divisor classes on G, where g is the genus of G in the sense of Baker-Norine. We give a new proof that the map β (v,e) is bijective by explicitly constructing an inverse. Using the natural action of the Picard group Pic 0 (G) on Pic g (G), we show that the Bernardi bijection β (v,e) gives rise to a simply transitive action β v of Pic 0 (G) on the set of spanning trees which does not depend on the choice of e. A plane graph has a natural ribbon structure (coming from the counterclockwise orientation of the plane), and in this case we show that β v is independent of v as well. Thus for plane graphs, the set of spanning trees is naturally a torsor for the Picard group. Conversely, we show that if β v is independent of v then G together with its ribbon structure is planar. We also show that the natural action of Pic 0 (G) on spanning trees of a plane graph is compatible with planar duality.These findings are formally quite similar to results of Holroyd et al. and ChanChurch-Grochow, who used rotor-routing to construct an action r v of Pic 0 (G) on the spanning trees of a ribbon graph G, which they show is independent of v if and only if G is planar. It is therefore natural to ask how the two constructions are related. We prove that β v = r v for all vertices v of G when G is a planar ribbon graph, i.e. the two torsor structures (Bernardi and rotor-routing) on the set of spanning trees coincide. In particular, it follows that the rotor-routing torsor is compatible with planar duality. We conjecture that for every non-planar ribbon graph G, there exists a vertex v with β v = r v .We thank Spencer Backman, Melody Chan, and Dan Margalit for helpful discussions and feedback on an earlier draft. We also thank Chi Ho Yuen for catching a sign error in Theorem 6.1, and the anonymous referees for their extraordinarily careful proofreading and numerous useful suggestions. This work began at the American Insitute of Mathematics workshop "Generalizations of chip firing and the critical group" in July 2013, and we would like to thank AIM as well as the organizers of that conference (L. Levine, J. Martin, D. Perkinson, and J. Propp) for providing a stimulating environment.

Hypergraph polynomials and the Bernardi process

“…In the case of graphs, hypertrees on the vertex set are easily seen to be the duals of the so called break divisors. In this special case, the above bijection-by-dissection agrees with the bijection between spanning trees and break divisors, defined by Bernardi [3] and studied further by Baker and Wang [1]. In [1] the inverse of Bernardi's bijection is described in a way that is formally different from the original Bernardi algorithm.…”

Geometric bijections between spanning trees and break divisors

“…Now we give an alternative proof of Theorem 22, the idea is to prove the commutativity of two finer diagrams separately. In particular, our proof produces a stronger assertion than the proof in [9].…”

“…In fact, Baker and Wang showed that the torsors induced by Bernardi bijections are isomorphic to the torsors induced by rotor routing if the ribbon structure is planar. Based on these facts as well as some computational evidence, Baker asked whether planar Bernardi bijections come from the above geometric picture [9,Remark 5.2]. In this paper, we answer Baker's question in the affirmative, and we give alternative proofs and sharpenings of Baker and Wang's results.…”

“…Baker and Wang proved several propositions about Bernardi torsors of plane graphs in their paper [9]. (They also use the convention that the ribbon structures on planar duals are induced by the clockwise orientation of the plane.)…”

Sandpiles, Spanning Trees, and Plane Duality

“…We would also like to refer the reader to the recent preprint [3], which arrives at another proof of Theorem 3.1 via a completely different route. In that paper, Baker and Wang prove that the bijections obtained by Bernardi in [4,Theorem 45] give rise to another simply transitive action of the sandpile group on the spanning trees of a ribbon graph G with a fixed root vertex.…”