Abstract
We study the Abelian sandpile model on \({\mathbb{Z}}^{d}\) . In d ≥ 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit μ of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure μ, and study ergodic properties of this process. The main techniques we use are a connection between the statistics of waves and uniform two-component spanning trees and results on the uniform spanning forest measure on \({\mathbb{Z}}^{d}\) .
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Járai, A.A., Redig, F. Infinite volume limit of the Abelian sandpile model in dimensions d ≥ 3. Probab. Theory Relat. Fields 141, 181–212 (2008). https://doi.org/10.1007/s00440-007-0083-0
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DOI: https://doi.org/10.1007/s00440-007-0083-0
Keywords
- Abelian sandpile model
- Wave
- Addition operator
- Uniform spanning tree
- Two-component spanning tree
- Loop-erased random walk
- Tail triviality