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The abelian sandpile model on randomly rooted graphs and self-similar groups

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Abstract

The Abelian sandpile model is an archetypical model of the physical phenomenon of self-organized criticality. It is also well studied in combinatorics under the name of chip-firing games on graphs. One of the main open problems about this model is to provide rigorous mathematical explication for predictions about the values of its critical exponents, originating in physics. The model was initially defined on the cubic lattices ℤd, but the only case where the value of some critical exponent has been established so far is the case of the infinite regular tree—the Bethe lattice.

This paper is devoted to the study of the abelian sandpile model on a large class of graphs that serve as approximations to Julia sets of postcritically finite polynomials and occur naturally in the study of automorphism group actions on infinite rooted trees. While different from the square lattice, these graphs share many of its geometric properties: they are of polynomial growth, have one end, and random walks on them are recurrent. This ensures that the behaviour of sandpiles on them is quite different from that observed on the infinite tree. We compute the critical exponent for the decay of mass of sand avalanches on these graphs and prove that it is inversely proportional to the rate of polynomial growth of the graph, thus providing the first rigorous derivation of the critical exponent different from the mean-field (the tree) value.

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References

  1. D. Aldous and R. Lyons, Processes on unimodular random networks, Electronic Journal of Probability 12 (2007), 1454–1508.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Ali and D. Dhar, Structure of avalanches and breakdown of simple scaling in the Abelian sandpile model in one dimension, Physical Review E. Statistical, Nonlinear and Soft Matter Physics 52 (1995), 4804–4816.

    Article  Google Scholar 

  3. S. R. Athreya and A. A. Jarai, Infinite volume limit for the stationary distribution of abelian sandpile models, Communications in Mathematical Physics 249 (2004), 197–213.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Bacher, P. de la Harpe and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bulletin de la Société Mathématique de France 125 (1997), 167–198.

    MATH  Google Scholar 

  5. P. Bak, K. Tang and K. Wiesenfeld, Self-organized criticality, Physical Review A 38 (1988), 364–374.

    Article  MATH  MathSciNet  Google Scholar 

  6. L. Bartholdi and B. Virág, Amenability via random walks, Duke Mathematical Journal 130 (2005), 39–56.

  7. I. Benjamini and O. Schramm, Recurrence of distributional limits of finite planar graphs, Electronic Journal of Probability 6 (2001), 1–23.

    Article  MathSciNet  Google Scholar 

  8. I. Benjamini, R. Lyons, Y. Peres and O. Schramm, Uniform spanning forests, The Annals of Probability 29 (2001), 1–65.

    MATH  MathSciNet  Google Scholar 

  9. N. L. Biggs, Chip-firing and the critical group of a graph, Journal of Algebraic Combinatorics 9 (1999), 25–45.

  10. A. Björner L. Lovasz and P. Shor, Chip-firing games on graphs, European Journal of Combinatorics 12 (1991), 283–291.

    Article  MATH  MathSciNet  Google Scholar 

  11. I. Bondarenko, Groups generated by bounded automata and their Schreier graphs, Ph.D. Thesis, Texas A&M University, 2007.

  12. I. Bondarenko, D. D’Angeli and T. Nagnibeda, Ends of Schreier graphs of self-similar groups, preprint.

  13. W. Chen and T. Schedler, Concrete and abstract structure of the sandpile group for thick trees with loops, arXiv:math/0701381, (2007).

  14. F. Chung and R. B. Ellis, A chip-firing game and Dirichlet eigenvalues, Discrete Mathematics 257 (2002), 341–355.

  15. F. Daerden and C. Vanderzande, Sandpiles on a Sierpiński gasket, Physica A 256 (1998), 533–546.

    Article  Google Scholar 

  16. D. D’Angeli, A. Donno, M. Matter and T. Nagnibeda, Schreier graphs of the Basilica group, Journal of Modern Dynamics 4 (2010), 167–205.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Dhar, Self-organized critical state of sandpile automaton models, Physical Review Letters 64 (1990), 1613–1616.

    Article  MATH  MathSciNet  Google Scholar 

  18. D. Dhar, Theoretical studies of self-organized criticality, Physica A 369 (2006), 29–70.

    Article  MathSciNet  Google Scholar 

  19. D. Dhar and S. N. Majumdar, Abelian sandpile model on the Bethe lattice, Journal of Physics A. Mathematical and General 23 (1990), 4333–4350.

    Article  MathSciNet  Google Scholar 

  20. D. Dhar and S. N. Majumdar, Height correlations in the abelian sandpile model, Journal of Physics A. Mathematical and General 24 (1991), 357–362.

    Article  Google Scholar 

  21. R. Diestel, Graph Theory, 3rd edn., Springer, Berlin, 2006.

    Google Scholar 

  22. R. Grigorchuk, Solved and unsolved problems around one group, in Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, (L. Bartholdi, T. Ceccherini-Silberstein, T. Smirnova-Nagnibeda and A. Żuk, eds.), Progress in Mathematics, Vol. 248, Birkhäuser, Basel, 2005, pp. 117–218.

    Google Scholar 

  23. R. Grigorchuk, V. Nekrashevych and V. I. Sushchanskii, Automata, dynamical systems and groups, Trudy Matematicheskogo Instituta Imeni V. A. Steklova 231 (2000), 134–214.

    MathSciNet  Google Scholar 

  24. R. Grigorchuk and A. Żuk, On a torsion-free weakly branch group defined by a three-state automaton, International Journal of Algebra and Computation 12 (2002), 223–246.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Gromov, Structures métriques pour les variétés riemanniennes, (J. Lafontaine and P. Pansu, eds.), Textes mathématiques, Vol. 1, CEDIC, Paris, 1981.

    Google Scholar 

  26. B. Kutnjak-Urbanc, S. Zapperi, S. Milosevic and H. Eugene Stanley, Sandpile model on the Sierpiński gasket fractal, Physical Review E. Statistical, Nonlinear and Soft Matter Physics 54 (1996), 272–277.

    Article  Google Scholar 

  27. Y. Le Borgne and D. Rossin, On the identity of the sandpile group, Discrete Mathematics 256 (2002), 775–790.

  28. L. Levine, The sandpile group of a tree, European Journal of Combinatorics 30 (2009), 1026–1035.

    Article  MATH  MathSciNet  Google Scholar 

  29. D. Lorenzini, Smith normal form and laplacians, Journal of Combinatorial Theory. Series B 98 (2008), 1271–1300.

    Article  MATH  MathSciNet  Google Scholar 

  30. C. Maes, F. Redig and E. Saada, The abelian sandpile model on an infinite tree, The Annals of Probability 30 (2002), 2081–2107.

    Article  MATH  MathSciNet  Google Scholar 

  31. C. Maes, F. Redig, E. Saada and A. Van Moffaert, On the thermodynamic limit for a one-dimensional sandpile process, Markov Processes and Related Fields 6 (2000), 1–21.

    MATH  MathSciNet  Google Scholar 

  32. M. Matter, Abelian Sandpile Model on Randomly Rooted Graphs, Ph.D thesis, Universit é de Genève, 2012. Available at http://archive-ouverte.unige.ch/unige:21849.

  33. R. Meester, F. Redig and D. Znamenski, The abelian Sandpile: a mathematical introduction, Markov Processes and Related Fields 7 (2001), 509–523.

    MATH  MathSciNet  Google Scholar 

  34. V. Nekrashevych, Combinatorics of polynomial iterations, in Complex Dynamics-Families and Friends, (D. Schleicher, ed.), A. K. Peters, 2009, Wellesly, MA, pp. 169–214.

  35. V. Nekrashevych, Iterated monodromy groups, in Groups St Andrews 2009 in Bath, Vol. 1, London Mathematical Society Lecture Note Series, Vol. 387, Cambridge University Press, Cambridge, 2011, pp. 41–93.

    Chapter  Google Scholar 

  36. V. Nekrashevych, Self-Similar Groups, American Mathematical Society, Providence, RI, 2005.

    Book  MATH  Google Scholar 

  37. V. B. Priezzhev, The upper critical dimension of the abelian sandpile model, Journal of Statistical Physics 98 (2000), 667–684.

    Article  MATH  MathSciNet  Google Scholar 

  38. V. B. Priezzhev, D. V. Ktitarev and E. V. Ivashkevich, Formation of avalanches and critical exponents in an abelian sandpile model, Physical Review Letters 76 (1996), 2093–2096.

    Article  Google Scholar 

  39. F. Redig, Mathematical aspects of the abelian sandpile model, Lecture notes, Les Houches, 2005.

  40. L. G. Rogers and A. Teplyaev, Laplacians on the Basilica Julia set, Communications on Pure and Applied Analysis 9 (2010), 211–231.

    Article  MATH  MathSciNet  Google Scholar 

  41. P. Ruelle and S. Sen, Toppling distributions in one-dimensional abelian sandpiles, Journal of Physics A. Mathematical and General 25 (1992), 1257–1264.

    Article  MathSciNet  Google Scholar 

  42. J.-P. Serre, Trees, Translated from the French Original by John Stillwell, Corrected 2nd printing of the 1980 English Translation, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.

    Google Scholar 

  43. I. Spakulova, Percolation and Ising model on tree-like graphs, Ph.D thesis, Vanderbilt University, 2008.

  44. J. Tannery, Introduction à la théorie des fonctions d’une variable. I Chapitre 3, Librairie scientifique A. Hermann, 1904.

  45. E. Toumpakari, On the sandpile group of regular trees, European Journal of Combinatorics 28 (2007), 822–842.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Section de mathématiques, Université de Genève, 2-4 Rue du Livère CP 64, 1211, Genève 4, Switzerland

    M. Matter & T. Nagnibeda

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  1. M. Matter
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  2. T. Nagnibeda
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Correspondence to M. Matter.

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The authors acknowledge the support of the Swiss National Science Foundation Grant PP0022-118946.

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Matter, M., Nagnibeda, T. The abelian sandpile model on randomly rooted graphs and self-similar groups. Isr. J. Math. 199, 363–420 (2014). https://doi.org/10.1007/s11856-013-0068-x

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  • DOI: https://doi.org/10.1007/s11856-013-0068-x

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