Kadanoff Sand Piles, following the snowball

TL;DR

This paper analyzes the Kadanoff Sand Pile Model, revealing that fixed points exhibit a nearly periodic wavy pattern emerging from disordered regions, with the pattern covering almost the entire configuration as the number of grains grows.

Contribution

It proves the emergence of a regular wavy pattern in fixed points of the Kadanoff Sand Pile Model, clarifying the structure of stable configurations.

Findings

Fixed points display a nearly periodic wavy pattern.

The disordered segment's size becomes negligible as grains increase.

Pattern emergence occurs from initially disordered regions.

Abstract

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.