Measures for Transient Configurations of the Sandpile-Model

  • Matthias Schulz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4173)


The Abelian Sandpile-Model (ASM) is a well-studied model for Self-Organized Criticality (SOC), for which many interesting algebraic properties have been proved. This paper deals with the process of starting with the empty configuration and adding grains of sand, until a recurrent configuration is reached.

The notion studied in this paper is that the configurations at the beginning of the process are in a sense very far from being recurrent, while the configurations near the end of the process are quite close to being recurrent; this leads to the idea of ordering the transient configurations, such that configurations closer to being recurrent generally are greater than configurations far from recurrent. Then measures are defined which increase monotonically with respect to these orderings and can be interpreted as “degrees of recurrence”. Diagrams for these measures are shown and briefly discussed.


Neutral Element Connected Subset Calculation Rule Span Forest Dirichlet Eigenvalue 
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  1. 1.
    Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Dhar, D., Ruelle, P., Sen, S., Verma, D.N.: Algebraic aspects of abelian sandpile models. J. PHYS. A 28, 805 (1995)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chung, F., Ellis, R.: A chip-firing game and dirichlet eigenvalues. Discrete Mathematics 257, 341–355 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Creutz, M.: Cellular automata and self-organized criticality. In: Bhanot, G., Seiden, P.E., Chen, S.Y. (eds.) Some new directions in science on computers, pp. 147–169. World Scientific, Singapore (1996)Google Scholar
  5. 5.
    Ivashkevich, E.V., Ktitarev, D.V., Priezzhev, V.B.: Waves of topplings in an abelian sandpile. Physica A: Statistical and Theoretical Physics 209, 347–360 (1994)CrossRefGoogle Scholar
  6. 6.
    Dartois, A., Magnien, C.: Results and conjectures on the sandpile identity on a lattice. In: Morvan, M., Rémila, É. (eds.) Discrete Models for Complex Systems, DMCS 2003. DMTCS Proceedings., Discrete Mathematics and Theoretical Computer Science, vol. AB, pp. 89–102 (2003)Google Scholar
  7. 7.
    Majumdar, S.N., Dhar, D.: Equivalence between the abelian sandpile model and the q − − − > 0 limit of the potts model. Physica A: Statistical and Theoretical Physics 185, 129–145 (1992)Google Scholar
  8. 8.
    Schulz, M.: Untersuchungen am sandhaufen-modell. Master’s thesis, University of Karlsruhe (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Matthias Schulz
    • 1
  1. 1.Department for Computer SciencesUniversity of KarlsruheKarlsruheGermany

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