Pattern formation in growing sandpiles

TL;DR

This paper analyzes the complex patterns formed in a directed Abelian sandpile model on a square lattice, providing an exact characterization of the asymptotic pattern as grains are added.

Contribution

It offers a detailed analysis and exact description of the asymptotic pattern formation in a directed sandpile model with a specific initial configuration.

Findings

Pattern size scales as the square root of the number of grains added

Exact characterization of the asymptotic pattern features

Identification of the shape and position of pattern features

Abstract

Adding grains at a single site on a flat substrate in the Abelian sandpile models produce beautiful complex patterns. We study in detail the pattern produced by adding grains on a two-dimensional square lattice with directed edges (each site has two arrows directed inward and two outward), starting with a periodic background with half the sites occupied. The size of the pattern formed scales with the number of grains added $N$ as $\sqrt{N}$. We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features of the pattern.