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Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

Communications in Mathematical Physics volume 249, pages 197–213 (2004)Cite this article

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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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Authors and Affiliations

  1. 7 SJSS marg, Indian Statistical Institute, New Delhi, 110016, India

    Siva R. Athreya

  2. CWI, 94079, 1090, GB Amsterdam, The Netherlands

    Antal A. Járai

Authors
  1. Siva R. Athreya
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  2. Antal A. Járai
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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