TY - JOUR

T1 - Anchored burning bijections on finite and infinite graphs

AU - Gamlin, Samuel L.

AU - Járai, Antal A.

PY - 2014/12/16

Y1 - 2014/12/16

N2 - 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 Zd, 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.

AB - 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 Zd, 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.

KW - Abelian sandpile

KW - Burning algorithm

KW - Loop-erased random walk

KW - Uniform spanning tree

KW - Wilson’s algorithm

KW - Wired spanning forest

UR - http://www.scopus.com/inward/record.url?scp=84919827935&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1214/EJP.v19-3542

U2 - 10.1214/EJP.v19-3542

DO - 10.1214/EJP.v19-3542

M3 - Article

AN - SCOPUS:84919827935

SN - 1083-6489

VL - 19

SP - 1

EP - 23

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

M1 - 117

ER -