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Equivalence between the Abelian sandpile model and the q→0 limit of the Potts model

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  • Majumdar, S.N.
  • Dhar, Deepak

Abstract

We establish an equivalence between the undirected Abelian sandpile model and the q→0 limit of the q-state Potts model. The equivalence is valid for arbitrary finite graphs. Two-dimensional Abelian sandpile models, thus, correspond to a conformal field theory with central charge c = −2. The equivalence also gives a Monte Carlo algorithm to generate random spanning trees. We study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the Abelian sandpile model. The average number of sites burnt upto time t varies at ta. In two dimensions our numerically determined value of a agrees with the theoretical prediction a = 85. We relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes.

Suggested Citation

  • Majumdar, S.N. & Dhar, Deepak, 1992. "Equivalence between the Abelian sandpile model and the q→0 limit of the Potts model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 129-145.
    • Handle: RePEc:eee:phsmap:v:185:y:1992:i:1:p:129-145
    DOI: 10.1016/0378-4371(92)90447-X
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    1. Tzeng, Gwo-Hshiung & Chen, June-Jye & Lan, Chang-Jen, 1991. "The influence of modal choice on energy conservation strategies in transport : Application of a logit model," Energy Economics, Elsevier, vol. 13(4), pages 290-299, October.
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    Cited by:

    1. Wu, Xiaoxia & Zhang, Lianzhu & Chen, Haiyan, 2017. "Spanning trees and recurrent configurations of a graph," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 25-30.
    2. Antal A. Járai & Nicolás Werning, 2014. "Minimal Configurations and Sandpile Measures," Journal of Theoretical Probability, Springer, vol. 27(1), pages 153-167, March.
    3. Liao, Yunhua & Fang, Aixiang & Hou, Yaoping, 2013. "The Tutte polynomial of an infinite family of outerplanar, small-world and self-similar graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(19), pages 4584-4593.
    4. Antal A. Járai & Minwei Sun, 2021. "Asymptotic Height Distribution in High-Dimensional Sandpiles," Journal of Theoretical Probability, Springer, vol. 34(1), pages 349-362, March.

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