The Sandpile Cellular Automaton

  • Antal A. JáraiEmail author
Part of the Emergence, Complexity and Computation book series (ECC, volume 27)


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.


  1. 1.
    Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364–374 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bollobás, B.: Modern graph theory. Graduate Texts in Mathematics, vol. 184. Springer, New York (1998)Google Scholar
  3. 3.
    Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64, 1613–1616 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dhar, D.: Theoretical studies of self-organized criticality. Phys. A 369, 29–70 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dhar, D., Majumdar, S.N.: Abelian sandpile model on the Bethe lattice. J. Phys. A 23, 4333–4350 (1990)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fey, A., Levine, L., Peres, Y.: Growth rates and explosions in sandpiles. J. Stat. Phys. 138, 143–159 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fey, A., Levine, L., Wilson, D.B.: Approach to criticality in sandpiles. Phys. Rev. E 82, 031121 (2010).
  8. 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)CrossRefGoogle Scholar
  9. 9.
    Járai, A.A.: Sandpile Models. Lecture Notes for the 9th Cornell Probability Summer School (2014). arXiv:1401.0354
  10. 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)CrossRefzbMATHGoogle Scholar
  11. 11.
    Járai, A.A., Werning, N.: Minimal configurations and sandpile measures. J. Theor. Probab. 27, 153–167 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Járai, A.A., Ruszel, W., Saada, E.: Sandpiles on Galton-Watson trees. In preparation (2014)Google Scholar
  13. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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).
  15. 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)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kassel, A., Wilson, D.B.: Looping rate and sandpile density of planar graphs. Am. Math. Mon. 123(1), 19–39 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Levine, L.: Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle. Comm. Math. Phys. 335(2), 1003–1017 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Levine, L., Peres, Y.: Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile. Potential Anal. 30, 1–27 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lyons, R., Morris, B.J., Schramm, O.: Ends in uniform spanning forests. Electron. J. Probab. 13, 1702–1725 (2008).
  21. 21.
    Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge University Press, New York (2016)CrossRefzbMATHGoogle Scholar
  22. 22.
    Majumdar, S.N., Dhar, D.: Height correlations in the Abelian sandpile model. J. Phys. A 24, L357–L362 (1991)CrossRefGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. 24.
    Pegden, W., Smart, C.K.: Convergence of the Abelian sandpile. Duke Math. J. 162, 627–642 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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).
  26. 26.
    Priezzhev, V.B.: Structure of two-dimensional sandpile I. height probabilities. J. Stat. Phys. 74, 955–979 (1994)CrossRefGoogle Scholar
  27. 27.
    Priezzhev, V.B.: The upper critical dimension of the abelian sandpile model. J. Stat. Phys. 98, 667–684 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Redig, F.: Mathematical aspects of the abelian sandpile model. Mathematical Statistical Physics. Elsevier, Amsterdam (2006)Google Scholar
  29. 29.
    Spitzer, F.: Principles of Random Walk. Graduate Texts in Mathematics, vol. 34, 2nd edn. Springer, New York (1976)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK

Personalised recommendations