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Anchored burning bijections on finite and infinite graphs. (English) Zbl 1341.60126

Summary: 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 \(\mathbb Z^d\), \(d \geq 2\), 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 \(\mathbb Z^d\). We discuss some open problems related to these findings.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
05C05 Trees
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