• Guglielmo PaolettiEmail author
Part of the Springer Theses book series (Springer Theses)


In the last twenty years, after the first article by Bak, Tang and Wiesenfeld on Self-Organized criticality (SOC), a large amount of work has been done trying to better understand different features of this class of models. The most studied among them is the Abelian Sandpile Model (ASM), that was actually proposed in that seminal paper as first archetype of SOC. The attention has been focused mainly on the comprehension of the critical properties, in particular the determination of the critical exponents of the avalanches. During my PhD I worked on the Abelian Sandpile Model using unconventional approaches and focusing on not-standard features of the model, related with the pattern formation that can be seen in the evolution of particularly chosen configurations under deterministic conditions, that happened to catch the attention of the scientific community only in the last few years.


Cellular Automaton Conformal Field Theory Deterministic Dynamic Tutte Polynomial Smith Normal Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    D. Thompson, On Growth and Form, 2nd edn. (Dover, New York, 1917)Google Scholar
  3. 3.
    R. Thom, Structural Stability and Morphogenesis (W.A. Benjamin, Reading, 1975)Google Scholar
  4. 4.
    T.A. Witten, L.M. Sander, Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47, 1400–1403 (1981)ADSCrossRefGoogle Scholar
  5. 5.
    T.A. Witten, L.M. Sander, Diffusion-limited aggregation. Phys. Rev. B 27, 5686–5697 (1983)Google Scholar
  6. 6.
    P. Meakin, Diffusion-controlled cluster formation in 2-6-dimensional space. Phys. Rev. A 27, 1495–1507 (1983)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    T. Vicsek, Fractal Growth Phenomena (World Scientific, Singapore, 1989)Google Scholar
  8. 8.
    B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1982)Google Scholar
  9. 9.
    B. Chopard, M. Droz, Cellular Automata Modeling of Physical Systems (Cambridge University Press, Cambridge, 1998)Google Scholar
  10. 10.
    S. Wolfram, Cellular Automata and Complexity (Addison-Wesley, Reading, 1994)Google Scholar
  11. 11.
    S. Wolfram, Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601–644 (1983)Google Scholar
  12. 12.
    M. Kardar, G. Parisi, Y.-C. Zhang, Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    H. Herrmann, Geometrical cluster growth models and kinetic gelation. Phys. Rep. 136, 153–224 (1986)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    L. Barabasi, H. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995)Google Scholar
  15. 15.
    D. Dhar, T. Sadhu, S. Chandra, Pattern formation in growing sandpiles. EPL (Europhys. Lett.) 85, 48002 (2009)ADSCrossRefGoogle Scholar
  16. 16.
    P. Bak, How Nature Works (Copernicus, Secaucus, 1996)Google Scholar
  17. 17.
    S.S. Manna, Large-scale simulation of avalanche cluster distribution in sand pile model. J. Stat. Phys. 59, 509–521 (1990). doi: 10.1007/BF01015580 ADSCrossRefGoogle Scholar
  18. 18.
    D. Dhar, Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64, 1613–1616 (1990)MathSciNetADSCrossRefzbMATHGoogle Scholar
  19. 19.
    D. Dhar, Sandpiles and self-organized criticality. Phys. A: Stat. Mech. Its Appl. 186, 82–87 (1992)Google Scholar
  20. 20.
    D. Dhar, P. Ruelle, S. Sen, D.N. Verma, Algebraic aspects of abelian sandpile models. J. Phys. A: Math. Gen. 28, 805 (1995)MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. 21.
    M. Creutz, abelian sandpiles. Nucl. Phys. B (Proc. Suppl.) 20, 758–761 (1991)Google Scholar
  22. 22.
    M. Creutz, abelian sandpiles. Comput. Phys. 5, 198–203 (1991)Google Scholar
  23. 23.
    M. Creutz, P. Bak, Fractals and self-organized criticality, in Fractals in Science, ed. by A. Bunde, S. Havlin (Springer, Berlin, 1994), pp. 26–47Google Scholar
  24. 24.
    M. Creutz, Xtoys: cellular automata on xwindows. Nucl. Phys. B (Proc. Suppl.) 47, pp. 846–849 (1996).
  25. 25.
    S.N. Majumdar, D. Dhar, Height correlations in the abelian sandpile model. J. Phys. A: Math. Gen. 24, L357 (1991)ADSCrossRefGoogle Scholar
  26. 26.
    V.B. Priezzhev, Exact height probabilities in the abelian sandpile model. Phys. Scripta 1993, 663 (1993)CrossRefGoogle Scholar
  27. 27.
    V.B. Priezzhev, Structure of two-dimensional sandpile. I. Height probabilities. J. Stat. Phys. 74, 955–979 (1994). doi: 10.1007/BF02188212 ADSCrossRefGoogle Scholar
  28. 28.
    S.N. Majumdar, D. Dhar, Equivalence between the abelian sandpile model and the \(q \rightarrow 0\) limit of the Potts model. Phys. A: Stat. Mech. Its Appl. 185, 129–145 (1992)ADSCrossRefGoogle Scholar
  29. 29.
    S. Mahieu, P. Ruelle, Scaling fields in the two-dimensional abelian sandpile model. Phys. Rev. E 64, 066130 (2001), arXiv:0107150Google Scholar
  30. 30.
    P. Ruelle, A \(c=-2\) boundary changing operator for the abelian sandpile model. Phys. Lett. B 539, 172–177 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  31. 31.
    G. Piroux, P. Ruelle, Logarithmic scaling for height variables in the abelian sandpile model. Phys. Lett. B 607, 188–196 (2005)ADSCrossRefGoogle Scholar
  32. 32.
    G. Piroux, P. Ruelle, Boundary height fields in the abelian sandpile model. J. Phys. A: Math. Gen. 38, 1451 (2005)Google Scholar
  33. 33.
    G. Piroux, P. Ruelle, Pre-logarithmic and logarithmic fields in a sandpile model. J. Stat. Mech.: Theory Exp. 2004, 10005 (2004)Google Scholar
  34. 34.
    M. Jeng, Conformal field theory correlations in the abelian sandpile model. Phys. Rev. E 71, 016140 (2005)ADSCrossRefGoogle Scholar
  35. 35.
    S. Moghimi-Araghi, M. Rajabpour, S. Rouhani, abelian sandpile model: a conformal field theory point of view. Nucl. Phys. B 718, 362–370 (2005)MathSciNetADSCrossRefzbMATHGoogle Scholar
  36. 36.
    M. Jeng, G. Piroux, P. Ruelle, Height variables in the abelian sandpile model: scaling fields and correlations. J. Stat. Mech.: Theory Exp. 2006, P10015 (2006)CrossRefGoogle Scholar
  37. 37.
    V. Poghosyan, S. Grigorev, V. Priezzhev, P. Ruelle, Pair correlations in sandpile model: a check of logarithmic conformal field theory. Phys. Lett. B 659, 768–772 (2008)MathSciNetADSCrossRefzbMATHGoogle Scholar
  38. 38.
    A.D. Sokal, The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, in Surveys in Combinatorics, ed. by B. Webb (Cambridge University Press, Cambridge, 2005)Google Scholar
  39. 39.
    S. Caracciolo, J.L. Jacobsen, H. Saleur, A.D. Sokal, A. Sportiello, Fermionic field theory for trees and forests. Phys. Rev. Lett. 93, 080601 (2004)MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    S. Caracciolo, C. De Grandi, A. Sportiello, Renormalization flow for unrooted forests on a triangular lattice. Nucl. Phys. B 787, 260–282 (2007)ADSCrossRefzbMATHGoogle Scholar
  41. 41.
    A. Björner, L. Lovász, P. Shor, Chip-fring games on graphs. Eur. J. Combin. 12 (1991)Google Scholar
  42. 42.
    A. Björner, L. Lovász, Chip-firing games on directed graphs. J. Algebraic Comb. 1, 305–328 (1992). doi: 10.1023/A:1022467132614 CrossRefzbMATHGoogle Scholar
  43. 43.
    C.M. Lopez, Chip firing and the Tutte polynomial. Ann. Comb. 1, 253–259 (1997). doi: 10.1007/BF02558479 MathSciNetADSCrossRefzbMATHGoogle Scholar
  44. 44.
    T. Sadhu, D. Dhar, Pattern formation in growing sandpiles with multiple sources or sinks. J. Stat. Phys. 138, 815–837 (2010). doi: 10.1007/s10955-009-9901-3 MathSciNetADSCrossRefzbMATHGoogle Scholar
  45. 45.
    D. Dhar, T. Sadhu, Pattern formation in fast-growing sandpiles, ArXiv e-prints (2011), arXiv:1109.2908v1Google Scholar
  46. 46.
    T. Sadhu, D. Dhar, The effect of noise on patterns formed by growing sandpiles, J. Stat. Mech.: Theory Exp. 2011, P03001 (2011), arXiv:1012.4809Google Scholar
  47. 47.
    S.H. Liu, T. Kaplan, L.J. Gray, Geometry and dynamics of deterministic sand piles. Phys. Rev. A 42, 3207–3212 (1990)MathSciNetADSCrossRefGoogle Scholar
  48. 48.
    D. Dhar, The abelian sandpile and related models. Phys. A: Stat. Mech. Its Appl. 263, 4–25 (1999). Proceedings of the 20th IUPAP International Conference on Statistical PhysicsGoogle Scholar
  49. 49.
    Y. Le-Borgne, D. Rossin, On the identity of the sandpile group. Discrete Math. 256, 775–790 (2002). LaCIM 2000 Conference on Combinatorics, Computer Science and Applications.Google Scholar
  50. 50.
    A. Fey-den Boer, F. Redig, Limiting shapes for deterministic centrally seeded growth models. J. Stat. Phys. 130, 579–597 (2008). doi: 10.1007/s10955-007-9450-6 MathSciNetADSCrossRefzbMATHGoogle Scholar
  51. 51.
    L. Levine, Y. Peres, Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile. Potential Analysis 30, 1–27 (2009). doi: 10.1007/s11118-008-9104-6 MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    S. Ostojic, Patterns formed by addition of grains to only one site of an abelian sandpile. Phys. A: Stat. Mech. Its Appl. 318, 187–199 (2003)ADSCrossRefzbMATHGoogle Scholar
  53. 53.
    D.B. Wilson, Sandpile Aggregation Pictures on Various Lattices.
  54. 54.
    G.F. Lawler, M. Bramson, D. Griffeath, Internal diffusion limited aggregation. Ann. Probab. 20, 2117–2140 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    V.B. Priezzhev, D. Dhar, A. Dhar, S. Krishnamurthy, Eulerian walkers as a model of self-organized criticality. Phys. Rev. Lett. 77, 5079–5082 (1996)ADSCrossRefGoogle Scholar
  56. 56.
    Y.P. Lionel Levine, Spherical asymptotics for the rotor-router model in \({\mathbb{Z}}^d\). Indiana Univ. Math. J. 57, 431–450 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    A.E. Holroyd, L. Levine, K. Mészáros, Y. Peres, J. Propp, D.B. Wilson, Chip-firing and rotor-routing on directed graphs, in In and Out of Equilibrium 2, ed. by V. Sidoravicius, M.E. Vares, Progress in Probability, vol. 60 (Birkhäuser, Basel, 2008), pp. 331–364. doi:10.1007/978-3-7643-8786-0_17 Google Scholar
  58. 58.
    J. Gravner, J. Quastel, Internal DLA and the Stefan problem. Ann. Probab. 28, 1528–1562 (2000)Google Scholar
  59. 59.
    L. Levine, Y. Peres, Scaling limits for internal aggregation models with multiple sources. J. Anal. Math. 111, 151–219 (2010). doi: 10.1007/s11854-010-0015-2 MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    A. Fey-den Boer, S.H. Liu, Limiting shapes for a non-abelian sandpile growth model and related cellular automata. Preprint (2010), arXiv:1006.4928v2Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.LIP6 - (Université Paris 6) UPMCParis Cedex 05France

Personalised recommendations