Abstract
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.
Highlights
Sand Pile Models were introduced in 1988 ([BTW88]) to highlight Self-Organized Criticality (SOC)
We provide a deterministic procedure to reach the extremal fixed points of Parallel Symmetric Sand Pile Model (PSSPM) according to the total lexicographic order, and prove that any fixed point between these two extremal fixed points reachable in Symmetric Sand Pile Model (SSPM) is reachable in PSSPM
We have studied the set of fixed points of PSSPM(n) and compared it to the set of fixed points of SSPM(n) using the natural lexicographic order
Summary
Sand Pile Models were introduced in 1988 ([BTW88]) to highlight Self-Organized Criticality (SOC). SOC characterizes dynamical systems having critical attractors, i.e., systems that evolve toward a stable state from which small perturbations have uncontrolled consequences on the system This property is straightforward to figure out in the scope of sand pile models : consider a flat table on which we add grains one by one. This example illustrates the SOC of sand pile models. We can add one more rule, symmetric to the previous one : if the difference of height between columns i and i − 1 is larger than two, one grain can fall from column i to column i − 1 This leads to SSPM (symmetric sand pile model), studied in [Pha08] and [FMP07].
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