Kévin Perrot ; Thi Ha Duong Phan ; Trung Van Pham
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On the set of Fixed Points of the Parallel Symmetric Sand Pile Model
dmtcs:2974 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2011,
DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems
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https://doi.org/10.46298/dmtcs.2974
On the set of Fixed Points of the Parallel Symmetric Sand Pile ModelArticle
Authors: Kévin Perrot 1,2,3; Thi Ha Duong Phan 4; Trung Van Pham 4
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Kévin Perrot;Thi Ha Duong Phan;Trung Van Pham
1 Laboratoire de l'Informatique du Parallélisme
2 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe MC3
3 Modèles de calcul, Complexité, Combinatoire
4 Institut de Mathématiques [Hanoi]
Sand Pile Models are discrete dynamical systems emphasizing the phenomenon of $\textit{Self-Organized Criticality}$. From a configuration composed of a finite number of stacked grains, we apply on every possible positions (in parallel) two grain moving transition rules. The transition rules permit one grain to fall to its right or left (symmetric) neighboring column if the difference of height between those columns is larger than 2. The model is nondeterministic and grains always fall downward. We propose a study of the set of fixed points reachable in the Parallel Symmetric Sand Pile Model (PSSPM). Using a comparison with the Symmetric Sand Pile Model (SSPM) on which rules are applied once at each iteration, we get a continuity property. This property states that within PSSPM we can't reach every fixed points of SSPM, but a continuous subset according to the lexicographic order. Moreover we define a successor relation to browse exhaustively the sets of fixed points of those models.
Thi Ha Duong Phan, Emergence, complexity and computation, A Survey on the Stability of (Extended) Linear Sand Pile Model, pp. 253-281, 2022, 10.1007/978-3-030-92551-2_16.
Kévin Perrot;Éric Rémila, arXiv (Cornell University), Emergence of Wave Patterns on Kadanoff Sandpiles, pp. 634-647, 2014, 10.1007/978-3-642-54423-1_55.