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Emergence on Decreasing Sandpile Models

Author

Listed:
  • Kévin Perrot

    (LIF - Laboratoire d'informatique Fondamentale de Marseille - AMU - Aix Marseille Université - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche Scientifique)

  • Éric Rémila

    (GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne - ENS de Lyon - École normale supérieure de Lyon - UL2 - Université Lumière - Lyon 2 - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique)

Abstract

Sand is a proper instance for the study of natural algorithmic phenomena. Idealized square/cubic sand grains moving according to ``simple'' local toppling rules may exhibit surprisingly ``complex'' global behaviors. In this paper we explore the language made by words corresponding to fixed points reached by iterating a toppling rule starting from a finite stack of sand grains in one dimension. Using arguments from linear algebra, we give a constructive proof that for all decreasing sandpile rules the language of fixed points is accepted by a finite (Muller) automaton. The analysis is completed with a combinatorial study of cases where the {\em emergence} of precise regular patterns is formally proven. It extends earlier works, and asks how far can we understand and explain emergence following this track?

Suggested Citation

  • Kévin Perrot & Éric Rémila, 2015. "Emergence on Decreasing Sandpile Models," Post-Print halshs-01212069, HAL.
    • Handle: RePEc:hal:journl:halshs-01212069
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-01212069
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    References listed on IDEAS

    as
    1. Dhar, Deepak, 2006. "Theoretical studies of self-organized criticality," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 369(1), pages 29-70.
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