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Research - Sandpiles (1)
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» Sandpiles

1. Introduction

The different models used to simulate sandpiles formation show all the characteristics of Self-Organized Criticality (SOC) [1]. At the end of their evolution, the piles always reach a critical state (or fixed point, or stable state) which does not evolve anymore. Any modification of this state will generate a complete reorganization, until a new critical state is reached, and so on. Sandpiles are a good example of this phenomenon, as sand grains fall until they reach a stable position, but then any addition (or removal) of grains will lead to a full reorganization of the pile.

We are interested in the model SPM (Sand Pile Model [2]) and its generalization, IPM (Ice Pile Model [3]). Their mathematical study is mainly based on algebraic and combinatoric techniques [2, 4], which allow in particular to describe perfectly their dynamics [2, 3]. These good results are interesting but only deal with particular cases, one of our objectives will be to generalize them.

In this perspective, we recall the basic definitions and known results, then we describe what we are interested in : the generalization to arbitrary initial conditions, and to symmetric (multi-directionnal) models.

Bibliography

[1] P. Bak, C. Tang, et K. Wiesenfeld. Self-organized criticality. Physical Review A, 38(1):364−374, 1988.
[2] E. Goles and M. A. Kiwi. Games on line graphs and sand piles. Theoretical Computer Science, 115(2):321−349, 1993.
[3] E. Goles, M. Morvan, and H. D. Phan. Sand piles and order structure of integer partitions. Discrete Applied Mathematics, 117:51−64, 2002.
[4] T. Brylawski. The lattice of integer partitions. Discrete mathematics, 6:201−219, 1973.
[5] E. Formenti and B. Masson. On Computing Fixed Points for Generalized Sandpiles. International Journal of Unconventional Computing, 2(1):13−25, Old City Publishing, 2006.
[6] E. Formenti and B. Masson. A note on fixed points of generalized ice pile models. International Journal of Unconventional Computing, 2(2), Old City Publishing, 2006.
[7] E. Formenti, B. Masson and T. Pisokas. On symmetric sandpiles. In Cellular Automata for Research and Industry (ACRI 2006), Lecture Notes in Computer Sciences, Springer-Verlag, 2006.