Anchored burning bijections on finite and infinite graphs

TL;DR

This paper introduces anchored burning bijections linking uniform spanning forests and recurrent sandpiles on graphs, providing new insights into convergence rates and measure-preserving mappings in infinite graph settings.

Contribution

It constructs a family of measure-preserving, almost one-to-one mappings called anchored burning bijections for infinite graphs with specific spanning forest properties.

Findings

Established anchored burning bijections for certain infinite graphs.

Derived power law bounds on convergence to the sandpile measure in   2.

Connected spanning tree structures with sandpile dynamics using Wilson's construction.

Abstract

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 $\mathbb{Z}^d$, $d \ge 2$, 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 $\mathbb{Z}^d$. We discuss some open problems related to these findings.