Anchored burning bijections on finite and infinite graphs

作者：Gamlin Samuel L^*; Jarai Antal A

来源：Electronic Journal of Probability, 2014, 19(0): 117.

DOI：10.1214/EJP.v19-3542

• abelian sandpile
• uniform spanning tree
• loop-erased random walk
• Wilson%26apos
• s algorithm
• burning algorithm
• wired spanning forest

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 %26gt;= 2, we show how the anchored bijection, combined with Wilson%26apos;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 Z(d). We discuss some open problems related to these findings.

• 出版日期2014-12-16