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2014 Anchored burning bijections on finite and infinite graphs
Samuel Gamlin, Antal Járai
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Electron. J. Probab. 19: 1-23 (2014). DOI: 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 $Z^d$, $d \ge 2$, 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 $Z^d$. We discuss some open problems related to these findings.


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Samuel Gamlin. Antal Járai. "Anchored burning bijections on finite and infinite graphs." Electron. J. Probab. 19 1 - 23, 2014.


Accepted: 16 December 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1341.60126
MathSciNet: MR3296533
Digital Object Identifier: 10.1214/EJP.v19-3542

Primary: 60K35

Keywords: Abelian sandpile , burning algorithm , Loop-erased random walk , Uniform spanning tree , Wilson's algorithm , wired spanning forest

Vol.19 • 2014
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