Abstract
In the Abelian sandpile models introduced by Dhar, long-time behavior is determined by an invariant measure supported uniformly on a set of implicitly defined recurrent configurations of the system. Dhar proposed a simple procedure, theburning algorithm, as a possible test of whether a configuration is recurrent, and later with Majumdar verified the correctness of this test when the toppling rules of the sandpile are symmetric. We observe that the test is not valid in general and give a new algorithm which yields a test correct for all sandpiles; we also obtain necessary and sufficient conditions for the validity of the original test. The results are applied to a family of deterministic one-dimensional sandpile models originally studied by Lee, Liang, and Tzeng.
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References
Per Bak, Chao Tang, and Kurt Wiesenfeld, Self-organized criticality: An explanation of 1/f noise,Phys. Rev. Lett. 59:381–384 (1987).
Per Bak, Chao Tang, and Kurt Wiesenfeld, Self-organized criticality,Phys. Rev. A 38:364–374 (1988).
Leo P. Kadanoff, Sidney R. Nagel, Lei Wu, and Su-min Zhou, Scaling and universality in avalanches,Phys. Rev. A 39:6524–6537 (1989).
Deepak Dhar, Self-organized critical state of sandpile automaton models,Phys. Rev. Lett. 64:1613–1616 (1990).
Deepak Dhar and S. N. Majumdar, Abelian sandpile model on the Bethe lattice,J. Phys. A 23:4333–4350 (1990).
S. N. Majumdar and Deepak Dhar, Height correlations in the Abelian sandpile model,J. Phys. A 24:L357-L362 (1991).
S. N. Majumdar and Deepak Dhar, Equivalence of the Abelian sandpile model and theq→0 limit of the Potts model,Physica A 185:129–145 (1991).
S.-C. Lee, N. Y. Liang, and W.-J. Tzeng, Exact solution of a deterministic sandpile model in one dimension,Phys. Rev. Lett. 67:1479–1481 (1991).
S.-C. Lee and W.-J. Tzeng, Hidden conservation law for sandpile models,Phys. Rev. A 45:1253–1254 (1992).
G. Bergman, The diamond lemma for ring theory,Adv. Math. 29:178–218 (1978).
M. H. A. Newmann, On theories with a combinatorial definition of “Equivalence,”Ann. Math. 43:211–264 (1942).
W. T. Tutte,Graph Theory Encyclopedia of Mathematics and its Applications, Vol. 21, Gian-Carlo Rota, ed.; Addison Wesley, Reading, Massachusetts, 1984).
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Speer, E.R. Asymmetric abeiian sandpile models. J Stat Phys 71, 61–74 (1993). https://doi.org/10.1007/BF01048088
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DOI: https://doi.org/10.1007/BF01048088