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.
Keywords
Cite
@article{arxiv.math/0408060,
title = {Infinite volume limit of the Abelian sandpile model in dimensions d >= 3},
author = {Antal A. Jarai and Frank Redig},
journal= {arXiv preprint arXiv:math/0408060},
year = {2011}
}
Comments
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