On the set of Fixed Points of the Parallel Symmetric Sand Pile Model

TL;DR

This paper investigates the fixed points of the Parallel Symmetric Sand Pile Model, revealing a continuous subset of fixed points compared to the Symmetric Sand Pile Model, and introduces a successor relation for systematic exploration.

Contribution

It introduces a comparison between PSSPM and SSPM, establishing a continuity property and defining a successor relation for fixed point enumeration.

Findings

PSSPM cannot reach all SSPM fixed points, only a continuous subset.

A continuity property links the fixed points of PSSPM and SSPM.

A successor relation enables exhaustive browsing of fixed points.

Abstract

Sand Pile Models are discrete dynamical systems emphasizing the phenomenon of 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.