On the emergence of regularities on one-dimensional decreasing sandpiles

Abstract

Decreasing sandpiles model the dynamics of configurations where each position i∈N contains a finite number of stacked grains hi, such that hi≥hi+1 (decrease property). Grains move according to a decreasing local rule R=(r1,r2,…,rp) such that rj≥rj+1, meaning that rj grains move from columns i to i+j for all 1≤j≤p, if it does not contradict the decrease property. We are interested in the fixed point reached starting from a finite number of grains on a unique column.In [21], we proved the emergence of wave patterns periodically covering fixed points, for rules of the form (1,…,1) (Kadanoff sandpile models). The present work is a significative extension: for large classes of decreasing sandpile model instances, we prove the emergence of patterns of various shapes periodically covering fixed points. We introduce new automata to analyze their asymptotic structure, and use the least action principle. The difficulty of understanding the behavior of sandpile models, despite the simplicity of the rules, is what makes the problem challenging.