Strong Emergence of Wave Patterns on Kadanoff Sandpiles

Abstract : Emergence is easy to exhibit, but very hard to formally explain. This paper deals with square sand grains moving around on nicely stacked columns in one dimension (the physical sandpile is two dimensional, but the support of sand columns is one dimensional). The Kadanoff sandpile model is a discrete dynamical system describing the evolution of finitely many sand grains falling from an hourglass (or equivalently from a finite stack of sand grains) to a stable configuration. The repeated application of a simple local rule let grains move until reaching a fixed point. The difficulty of understanding its behavior, despite the simplicity of its rule, is the main interest of the model. In this paper we prove the emergence of exact wave patterns periodically repeated on fixed points. Remarkably, those regular patterns do not cover the entire fixed point, but eventually emerge from a seemingly disordered segment: grains are added on the left, triggering avalanches that become regular as they fall down the sandpile. The proof technique we set up associated arguments of linear algebra and combinatorics, which interestingly allow to formally demonstrate the emergence of regular patterns without requiring a precise understanding of the non-regular initial segment's dynamic.
Type de document :
Article dans une revue
The Electronic Journal of Combinatorics, Open Journal Systems, 2017, 24 (2), pp. 2-4
Liste complète des métadonnées

Littérature citée [34 références]  Voir  Masquer  Télécharger

https://halshs.archives-ouvertes.fr/halshs-01417254
Contributeur : Nelly Wirth <>
Soumis le : vendredi 4 mai 2018 - 14:31:37
Dernière modification le : mercredi 31 octobre 2018 - 12:24:13
Document(s) archivé(s) le : lundi 24 septembre 2018 - 23:16:30

Fichier

5619-20523-3-PB.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : halshs-01417254, version 1

Citation

Kévin Perrot, Eric Rémila. Strong Emergence of Wave Patterns on Kadanoff Sandpiles. The Electronic Journal of Combinatorics, Open Journal Systems, 2017, 24 (2), pp. 2-4. 〈halshs-01417254〉

Partager

Métriques

Consultations de la notice

247

Téléchargements de fichiers

16