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Infinite volume limit of the Abelian sandpile model in dimensions d ≥  3
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  • Published: 17 July 2007

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

  • Antal A. Járai1 &
  • Frank Redig2 

Probability Theory and Related Fields volume 141, pages 181–212 (2008)Cite this article

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Abstract

We study the Abelian sandpile model on \({\mathbb{Z}}^{d}\) . In d ≥  3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit μ of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure μ, 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 forest measure on \({\mathbb{Z}}^{d}\) .

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Authors and Affiliations

  1. Carleton University, School of Mathematics and Statistics, 1125 Colonel By Drive, Room 4302, Herzberg Building, Ottawa, ON, K1S 5B6, Canada

    Antal A. Járai

  2. Mathematisch Instituut, Universiteit Leiden, Snellius, Niels Bohrweg 1, 2333 CA, Leiden, The Netherlands

    Frank Redig

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  1. Antal A. Járai
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  2. Frank Redig
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Correspondence to Antal A. Járai.

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Járai, A.A., Redig, F. Infinite volume limit of the Abelian sandpile model in dimensions d ≥  3. Probab. Theory Relat. Fields 141, 181–212 (2008). https://doi.org/10.1007/s00440-007-0083-0

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  • Received: 08 August 2004

  • Revised: 21 May 2007

  • Published: 17 July 2007

  • Issue Date: May 2008

  • DOI: https://doi.org/10.1007/s00440-007-0083-0

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Keywords

  • Abelian sandpile model
  • Wave
  • Addition operator
  • Uniform spanning tree
  • Two-component spanning tree
  • Loop-erased random walk
  • Tail triviality

Mathematics Subject Classification (2000)

  • 60K35
  • 82C22
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