Infinite volume limit of the Abelian sandpile model in dimensions d >= 3

Abstract

We study the Abelian sandpile model on Z^d. In dimensions at least 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit mu of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure mu, 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 tree measure on Z^d.

First version: LaTeX; 29 pages. Revised version: LaTeX; 29 pages. The main result of the paper has been extended to all dimensions at least 3, with a new and simplyfied proof of finiteness of the two-component spanning tree. Second revision: LaTeX; 32 pages

References in corpus (4)

• The abelian sandpile and related models
• Introduction to the Sandpile Model
• The Abelian sandpile; a mathematical introduction
• On the thermodynamic limit for a one-dimensional sandpile process

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• Logarithmic corrections to scaling in the four-dimensional uniform spanning tree
• Abelian sandpiles: an overview and results on certain transitive graphs
• Anchored burning bijections on finite and infinite graphs
• A planar Ising model of self-organized criticality