## 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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