Anchored burning bijections on finite and infinite graphs

Samuel L. Gamlin, Antal A. Járai

OAI: oai:purehost.bath.ac.uk:openaire_cris_publications/75900423-08b0-44b0-95c7-74b292d09115 • DOI: https://doi.org/10.1214/EJP.v19-3542

Let G be an infinite graph such that each tree in the wired uniform spanning forest on G has one end almost surely. On such graphs G, we give a family of continuous, measure preserving, almost one-to-one mappings from the wired spanning forest on G to recurrent sandpiles on G, that we call anchored burning bijections. In the special case of Z^d, d≥2, we show how the anchored bijection, combined with Wilson’s stacks of arrows construction, as well as other known results on spanning trees, yields a power law upper bound on the rate of convergence to the sandpile measure along any exhaustion of Zd. We discuss some open problems related to these findings.

Abelian sandpile Burning algorithm Loop-erased random walk Uniform spanning tree Wilson’s algorithm Wired spanning forest