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The Infinite Volume Limit of Dissipative Abelian Sandpiles

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Abstract

We construct the thermodynamic limit of the stationary measures of the Bak-Tang-Wiesenfeld sandpile model with a dissipative toppling matrix (sand grains may disappear at each toppling). We prove uniqueness and mixing properties of this measure and we obtain an infinite volume ergodic Markov process leaving it invariant. We show how to extend the Dhar formalism of the ‘abelian group of toppling operators’ to infinite volume in order to obtain a compact abelian group with a unique Haar measure representing the uniform distribution over the recurrent configurations that create finite avalanches

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Authors and Affiliations

  1. Instituut voor Theoretische Fysica, K.U. Leuven, Celestijnenlaan 200D, 3001, Leuven, Belgium

    C. Maes

  2. Faculteit Wiskunde en Informatica, Technische Universiteit Eindhoven, The Netherlands and EURANDOM, 513, 5600 MB, Eindhoven, The Netherlands

    F. Redig

  3. Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, site Colbert, CNRS, UMR 6085, 76821, Mont-Saint-Aignan Cedex, France

    E. Saada

Authors
  1. C. Maes
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  2. F. Redig
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  3. E. Saada
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Additional information

Communicated by H. Spohn

Work partially supported by Tournesol project – nr. T2001.11/03016VF

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Maes, C., Redig, F. & Saada, E. The Infinite Volume Limit of Dissipative Abelian Sandpiles. Commun. Math. Phys. 244, 395–417 (2004). https://doi.org/10.1007/s00220-003-1000-8

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  • DOI: https://doi.org/10.1007/s00220-003-1000-8

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