Skip to main content
SpringerLink
Log in
Book cover

Probabilistic Cellular Automata pp 79–88Cite as

The Sandpile Cellular Automaton

  • Chapter
  • First Online:

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 27))

Abstract

This is a short introduction to the sandpile cellular automaton. It is aimed at non-specialist, who may not be familiar with statistical physics models. Technicalities are kept to a minimum and the emphasis is on motivation, clear definitions, key properties and some of the challenges.

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

References

  1. Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364–374 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bollobás, B.: Modern graph theory. Graduate Texts in Mathematics, vol. 184. Springer, New York (1998)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Dhar, D.: Theoretical studies of self-organized criticality. Phys. A 369, 29–70 (2006)

    Article  MathSciNet  Google Scholar 

  5. Dhar, D., Majumdar, S.N.: Abelian sandpile model on the Bethe lattice. J. Phys. A 23, 4333–4350 (1990)

    Article  MathSciNet  Google Scholar 

  6. Fey, A., Levine, L., Peres, Y.: Growth rates and explosions in sandpiles. J. Stat. Phys. 138, 143–159 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fey, A., Levine, L., Wilson, D.B.: Approach to criticality in sandpiles. Phys. Rev. E 82, 031121 (2010). https://doi.org/10.1103/PhysRevE.82.031121

  8. 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. Progress in Probability, vol. 60. Birkhäuser, Basel (2008)

    Chapter  Google Scholar 

  9. Járai, A.A.: Sandpile Models. Lecture Notes for the 9th Cornell Probability Summer School (2014). arXiv:1401.0354

  10. Járai, A.A., Redig, F.: Infinite volume limit of the abelian sandpile model in dimensions \(d\ge 3\). Probab. Theory Relat. Fields 141, 181–212 (2008)

    Article  MATH  Google Scholar 

  11. Járai, A.A., Werning, N.: Minimal configurations and sandpile measures. J. Theor. Probab. 27, 153–167 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Járai, A.A., Ruszel, W., Saada, E.: Sandpiles on Galton-Watson trees. In preparation (2014)

    Google Scholar 

  13. Járai, A.A., Redig, F., Saada, E.: Approaching criticality via the zero dissipation limit in the abelian avalanche model. J. Stat. Phys. 159(6), 1369–1407 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jeng, M., Piroux, G., Ruelle, P.: Height variables in the Abelian sandpile model: scaling fields and correlations. J. Stat. Mech. Theory Exp. P10015+63 (2006). https://doi.org/10.1088/1742-5468/2006/10/P10015

  15. Jensen, H.J.: Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems. Cambridge Lecture Notes in Physics, vol. 10. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  16. Kassel, A., Wilson, D.B.: Looping rate and sandpile density of planar graphs. Am. Math. Mon. 123(1), 19–39 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kenyon, R., Wilson, D.B.: Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs. J. Am. Math. Soc. 28(4), 985–1030 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Levine, L.: Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle. Comm. Math. Phys. 335(2), 1003–1017 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Levine, L., Peres, Y.: Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile. Potential Anal. 30, 1–27 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lyons, R., Morris, B.J., Schramm, O.: Ends in uniform spanning forests. Electron. J. Probab. 13, 1702–1725 (2008). https://doi.org/10.1214/EJP.v13-566

  21. Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge University Press, New York (2016)

    Book  MATH  Google Scholar 

  22. Majumdar, S.N., Dhar, D.: Height correlations in the Abelian sandpile model. J. Phys. A 24, L357–L362 (1991)

    Article  Google Scholar 

  23. Majumdar, S.N., Dhar, D.: Equivalence between the Abelian sandpile model and the \(q \rightarrow 0\) limit of the Potts model. Phys. A 185, 129–145 (1992)

    Article  Google Scholar 

  24. Pegden, W., Smart, C.K.: Convergence of the Abelian sandpile. Duke Math. J. 162, 627–642 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Poghosyan, V.S., Priezzhev, V.B., Ruelle, P.: Return probability for the loop-erased random walk and mean height in the Abelian sandpile model: a proof. J. Stat. Mech. Theory Exp. P10004+12 (2011). https://doi.org/10.1088/1742-5468/2011/10/P10004

  26. Priezzhev, V.B.: Structure of two-dimensional sandpile I. height probabilities. J. Stat. Phys. 74, 955–979 (1994)

    Article  Google Scholar 

  27. Priezzhev, V.B.: The upper critical dimension of the abelian sandpile model. J. Stat. Phys. 98, 667–684 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Redig, F.: Mathematical aspects of the abelian sandpile model. Mathematical Statistical Physics. Elsevier, Amsterdam (2006)

    Google Scholar 

  29. Spitzer, F.: Principles of Random Walk. Graduate Texts in Mathematics, vol. 34, 2nd edn. Springer, New York (1976)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    Antal A. Járai

Authors
  1. Antal A. Járai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antal A. Járai .

Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques et Applications UMR 7348, Université de Poitiers, CNRS, Poitiers, France

    Pierre-Yves Louis

  2. University of Technology, Eindhoven, Noord-Brabant, The Netherlands

    Francesca R. Nardi

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Járai, A.A. (2018). The Sandpile Cellular Automaton. In: Louis, PY., Nardi, F. (eds) Probabilistic Cellular Automata. Emergence, Complexity and Computation, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-65558-1_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-65558-1_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-65556-7

  • Online ISBN: 978-3-319-65558-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation