Strong Emergence of Wave Patterns on Kadanoff Sandpiles

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.