2013/01/06 by Kévin Perrot, Perrot, Kévin, Éric Rémila +2 · 1 citation
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Algorithm #Combinatorics (math.CO) #Cover (algebra) #Dimension (graph theory) #Discrete Mathematics (cs.DM) #Engineering #FOS: Computer and information sciences #FOS: Mathematics #Fixed point #Geometry #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Physics #Pile #Point (geometry) #Pure mathematics #Simple (philosophy) #Simplicity #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cs.DM #math.CO
paper · pdf · doi:10.48550/arxiv.1301.0997
published in arXiv (Cornell University) (Cornell University) · Research report. 26 pages, 9 figures
arxiv created 2013/01/06 · openalex publication_date 2013/01/06 · arxiv updated 2013/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is about cubic sand grains moving around on nicely packed columns in one dimension (the physical sand pile is two dimensional, but the support of sand columns is one dimensional). The Kadanoff Sand Pile Model is a discrete dynamical system describing the evolution of a finite number of stacked grains --as they would fall from an hourglass-- to a stable configuration. Grains move according to the repeated application of a simple local rule until reaching a stable configuration from which no rule can be applied, namely a fixed point. The main interest of the model relies in the difficulty of understanding its behavior, despite the simplicity of the rule. We are interested in describing the shape of fixed point configurations according to the number of initially stacked sand grains. In this paper, we prove the emergence of a wavy shape on fixed points, i.e., a regular pattern is (nearly) periodically repeated on fixed points. Interestingly, the regular pattern does not cover the entire fixed point, but eventually emerges from a seemingly highly disordered segment. Fortunately, the relative size of the part of fixed points non-covered by the pattern repetition is asymptotically null.