Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

An Erratum to this article was published on 03 April 2006

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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Correspondence to Antal A. Járai.

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An erratum to this article is available at http://dx.doi.org/10.1007/s00220-006-1557-0.

Communicated by M. Aizenman

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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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Keywords

  • Neural Network
  • Complex System
  • Nonlinear Dynamics
  • Span Tree
  • Volume Version