Rotorrouting and spanning trees on planar graphs
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, selforganized criticality, random walks, and algorithms. Given a ribbon graph G, Holroyd et al. used the "rotorrouting" 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2013
 arXiv:
 arXiv:1308.2677
 Bibcode:
 2013arXiv1308.2677C
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 16 pages. v2: final version, to appear in IMRN