Skip to main content
SpringerLink
Log in

Directed Nonabelian Sandpile Models on Trees

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We define two general classes of nonabelian sandpile models on directed trees (or arborescences), as models of nonequilibrium statistical physics. Unlike usual applications of the well-known abelian sandpile model, these models have the property that sand grains can enter only through specified reservoirs.

In the Trickle-down sandpile model, sand grains are allowed to move one at a time. For this model, we show that the stationary distribution is of product form. In the Landslide sandpile model, all the grains at a vertex topple at once, and here we prove formulas for all eigenvalues, their multiplicities, and the rate of convergence to stationarity. The proofs use wreath products and the representation theory of monoids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ayyer A., Klee S., Schilling A.: Combinatorial Markov chains on linear extensions. J. Algebr. Combin. 39(4), 853–881 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  2. Almeida, J.: Finite semigroups and universal algebra, Series in Algebra, vol. 3. World Scientific Publishing Co. Inc., River Edge (1994) (Translated from the 1992 Portuguese original and revised by the author)

  3. Ayyer, A., Strehl, V.: The spectrum of an asymmetric annihilation process. In: 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), Discrete Math. Theor. Comput. Sci. Proc., AN, pp. 461–472. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2010)

  4. Ayyer, A., Strehl, V.: Stationary distribution and eigenvalues for a de Bruijn process. In: Kotsireas, I.S., Zima, E.V. (eds.) Advances in Combinatorics, pp. 101–120. Springer, Berlin (2013)

  5. Ayyer, A., Schilling, A., Steinberg, B., Thiéry, N.M.: \({{\fancyscript{R}}}\)-trivial monoids and Markov chains. Int. J. Algebra Comput. (2014, to appear). arXiv:1401.4250

  6. Ayyer A.: Algebraic properties of a disordered asymmetric Glauber model. J. Stat. Mech. Theory Exp. 2011(02), P02034 (2011)

    Article  MathSciNet  Google Scholar 

  7. Brown K.S., Diaconis P.: Random walks and hyperplane arrangements. Ann. Probab. 26(4), 1813–1854 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brzozowski J.A., Fich F.E.: Languages of \({{\mathcal{R}}}\) -trivial monoids. J. Comput. Syst. Sci. 20(1), 32–49 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bidigare P., Hanlon P., Rockmore D.: A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements. Duke Math. J. 99(1), 135–174 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Biggs N.L.: Chip-firing and the critical group of a graph. J. Algebr. Combin. 9(1), 25–45 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Björner, A.: Random walks, arrangements, cell complexes, greedoids, and self-organizing libraries. In: Building Bridges, Bolyai Soc. Math. Stud., vol. 19, pp. 165–203. Springer, Berlin (2008)

  12. Björner A.: Note: Random-to-front shuffles on trees. Electron. Commun. Probab. 14, 36–41 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Björner A., Lovász L., Shor P.W.: Chip-firing games on graphs. Eur. J. Combin 12(4), 283–291 (1991)

    Article  MATH  Google Scholar 

  14. Brown K.S.: Semigroups, rings, and Markov chains. J. Theor. Probab. 13(3), 871–938 (2000)

    Article  MATH  Google Scholar 

  15. Bak P., Tang C., Wiesenfeld K.: Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59(4), 381–384 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  16. Chung F., Graham R.: Edge flipping in graphs. Adv. Appl. Math. 48(1), 37–63 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cori R., Le Borgne Y.: The sand-pile model and Tutte polynomials. Adv. Appl. Math. 30(1), 44–52 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. I. Mathematical Surveys, No. 7. American Mathematical Society, Providence (1961)

  19. Derrida B., Evans M.R., Hakim V., Pasquier V.: Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A 26(7), 1493–1517 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Deo, N,: Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall Series in Automatic Computation. Prentice-Hall Inc., Englewood Cliffs (1974)

  21. Dhar D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613–1616 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Dhar D.: The abelian sandpile and related models. Phys. A Stat. Mech. Appl. 263(1), 4–25 (1999)

    Article  MathSciNet  Google Scholar 

  23. Dhar, D.: Some results and a conjecture for Manna’s stochastic sandpile model. Phys. A Stat. Mech. Appl. 270(1), 69–81 (1999)

  24. Dhar, D.: Theoretical studies of self-organized criticality. Phys. A Stat. Mech. Appl. 369(1), 29–70 (2006). Fundamental Problems in Statistical Physics Proceedings of the 11th International Summerschool on ‘Fundamental problems in statistical physics’, Leuven, Belgium 11th International Summerschool on ‘Fundamental problems in statistical physics’

  25. Denton, T., Hivert, F., Schilling, A., Thiéry, N.M.: On the representation theory of finite \({{\mathcal{J}}}\)-trivial monoids. Sém. Lothar. Combin. 64, Art. B64d, 44 (2010/2011)

  26. Diaconis, P.: Group representations in probability and statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 11. Institute of Mathematical Statistics, Hayward (1988)

  27. Diaconis, P.: From shuffling cards to walking around the building: an introduction to modern Markov chain theory. In: Proceedings of the International Congress of Mathematicians, vol. I (Berlin, 1998), pp. 187–204 (1998)

  28. Devroye L., Lugosi G.: Combinatorial methods in density estimation, Springer Series in Statistics Series. Springer, Berlin (2001)

    Google Scholar 

  29. Eilenberg, S.: Automata, languages, and machines, vol. B. Academic Press, Harcourt Brace Jovanovich Publishers, New York (1976) [with two chapters (“Depth decomposition theorem” and “Complexity of semigroups and morphisms”) by Bret Tilson, Pure and Applied Mathematics, vol. 59 (1976)]

  30. Goles E., Morvan M., Phan H.D.: The structure of a linear chip firing game and related models. Theor. Comput. Sci. 270(1-2), 827–841 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  31. Grigorchuk R.I., Nekrashevich V.V., Sushchanskiĭ V.I.: Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova 231(Din. Sist. Avtom. i Beskon. Gruppy), 134–214 (2000)

    Google Scholar 

  32. Grigorchuk R.I.,  Zuk A.: The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedicata 87(1–3), 209–244 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Higgins, P.M.: Techniques of semigroup theory. Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1992) (with a foreword by G. B. Preston)

  34. Holroyd, A.E., Levine, L., Mészáros, K., Peres, Y., Propp, J., Wilson, D.B.: Chip-firing and rotor-routing on directed graphs. In and Out of Equilibrium 2, pp. 331–364 (2008)

  35. Howie, J.M.: Fundamentals of semigroup theory, London Mathematical Society Monographs. New Series, vol. 12. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, (1995)

  36. Ivashkevich E.V., Priezzhev V.B.: Introduction to the sandpile model. Phys. A Stat. Mech. Appl. 254(1-2), 97–116 (1998)

    Article  Google Scholar 

  37. Krohn, K., Rhodes, J., Tilson, B.: Algebraic theory of machines, languages, and semigroups, Chapters 1, 5–9. In: Arbib, M.A. (ed.) With a major contribution by Kenneth Krohn and John L. Rhodes. Academic Press, New York (1968)

  38. Kambites M., Silva P.V., Steinberg B.: The spectra of lamplighter groups and Cayley machines. Geom. Dedicata 120, 193–227 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  39. Levin, D.A., Peres, Y., Wilmer, E.L.: Markov chains and mixing times. American Mathematical Society, Providence (2009) (with a chapter by James G. Propp and David B. Wilson)

  40. Manna S.S.: Two-state model of self-organized criticality. J. Phys. A Math. Gen. 24(7), L363 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  41. Meldrum J.D.P.: Wreath products of groups and semigroups, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 74. Longman, Harlow (1995)

    Google Scholar 

  42. Meldrum J.D.P.: Wreath products of groups and semigroups, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 74. Longman, Harlow (1995)

    Google Scholar 

  43. MacDonald C.T., Gibbs J.H.: Concerning the kinetics of polypeptide synthesis on polyribosomes. Biopolymers 7(5), 707–725 (1969)

    Article  Google Scholar 

  44. MacDonald C.T., Gibbs J.H., Pipkin A.C.: Kinetics of biopolymerization on nucleic acid templates. Biopolymers 6(1), 1–25 (1968)

    Article  Google Scholar 

  45. Margolis S., Steinberg B.: Quivers of monoids with basic algebras. Compos. Math. 148(5), 1516–1560 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  46. Nekrashevych, V.: Self-similar groups, Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence (2005)

  47. Pin, J.-E.: Varieties of formal languages. Foundations of Computer Science. Plenum Publishing Corp., New York (1986) (with a preface by M.-P. Schützenberger, Translated from the French by A. Howie)

  48. Postnikov A., Shapiro B.: Trees, parking functions, syzygies, and deformations of monomial ideals. Trans. Am. Math. Soc. 356(8), 3109–3142 (2004). (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  49. Rhodes J., Steinberg B.: The q-theory of finite semigroups, Springer Monographs in Mathematics. Springer, New York (2009)

    Book  Google Scholar 

  50. Stein W.A., et al. Sage Mathematics Software (Version 5.9). The Sage Development Team (2013). http://www.sagemath.org

  51. The Sage-Combinat community. Sage-Combinat: enhancing Sage as a toolbox for computer exploration in algebraic combinatorics (2008). http://combinat.sagemath.org

  52. Sadhu T., Dhar D.: Steady state of stochastic sandpile models. J. Stat. Phys. 134(3), 427–441 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. Schütz G.M., Ramaswamy R., Barma M.: Pairwise balance and invariant measures for generalized exclusion processes. J. Phys. A Math. Gen. 29(4), 837 (1996)

    Article  ADS  MATH  Google Scholar 

  54. Stanley, R.P.: Enumerative combinatorics, vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999) (with a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin)

  55. Steinberg B.: Möbius functions and semigroup representation theory. J. Combin. Theory Ser. A 113(5), 866–881 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  56. Steinberg B.: Möbius functions and semigroup representation theory. II. Character formulas and multiplicities. Adv. Math. 217(4), 1521–1557 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  57. Sasamoto T., Wadati M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A Math. Gen. 31(28), 6057 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  58. Tripathy G., Barma M.: Steady state and dynamics of driven diffusive systems with quenched disorder. Phys. Rev. Lett. 78, 3039–3042 (1997)

    Article  ADS  Google Scholar 

  59. Toumpakari, E.C.: On the Abelian Sandpile Model. Ph.D. thesis, University of Chicago, Department of Mathematics (2005)

Download references

Author information

Author notes

  1. Arvind Ayyer

    Present address: Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

Authors and Affiliations

  1. Department of Mathematics, UC Davis, One Shields Ave., Davis, CA, 95616-8633, USA

    Arvind Ayyer & Anne Schilling

  2. Department of Mathematics, City College of New York, Convent Avenue at 138th Street, New York, NY, 10031, USA

    Benjamin Steinberg

  3. Laboratoire de Recherche en Informatique - CNRS: UMR 8623, Université Paris-Sud, 91405, Orsay, France

    Nicolas M. Thiéry

Authors
  1. Arvind Ayyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anne Schilling
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Benjamin Steinberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nicolas M. Thiéry
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arvind Ayyer.

Additional information

Communicated by N. Reshetikhin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ayyer, A., Schilling, A., Steinberg, B. et al. Directed Nonabelian Sandpile Models on Trees. Commun. Math. Phys. 335, 1065–1098 (2015). https://doi.org/10.1007/s00220-015-2343-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-015-2343-7

Keywords

Search

Navigation