Rotor-routing and spanning trees on planar graphs

Understand

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.