Emergence on Decreasing Sandpile Models

Kévin Perrot (1) , Éric Rémila (2)

1 LIF - Laboratoire d'informatique Fondamentale de Marseille
2 GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne

Sand is a proper instance for the study of natural algorithmic phenomena. Idealized square/cubic sand grains moving according to ``simple'' local toppling rules may exhibit surprisingly ``complex'' global behaviors. In this paper we explore the language made by words corresponding to fixed points reached by iterating a toppling rule starting from a finite stack of sand grains in one dimension. Using arguments from linear algebra, we give a constructive proof that for all decreasing sandpile rules the language of fixed points is accepted by a finite (Muller) automaton. The analysis is completed with a combinatorial study of cases where the {\em emergence} of precise regular patterns is formally proven. It extends earlier works, and asks how far can we understand and explain emergence following this track?

Mots clés: sandpile models; fixed points; emergence

Domaines: Economies et finances; Mathématique discrète [cs.DM]

Kévin Perrot, Éric Rémila. Emergence on Decreasing Sandpile Models. MFCS 2015 40th International Symposium on Mathematical Foundations of Computer Science, Aug 2015, Milan, Italy. ⟨halshs-01212069⟩