Abstract
We study the stationary distribution of the standard Abelian sandpile model in the box Λn = [-n, n]d ∩ ℤd for d≥ 2. We show that as n→ ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in ℤd. This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on ℤd. In the case d > 4, we also make use of Wilson’s method.
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Communicated by M. Aizenman
An erratum to this article is available at http://dx.doi.org/10.1007/s00220-006-1557-0.
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Athreya, S., Járai, A. Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models. Commun. Math. Phys. 249, 197–213 (2004). https://doi.org/10.1007/s00220-004-1080-0
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DOI: https://doi.org/10.1007/s00220-004-1080-0