Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks

TL;DR

This paper investigates how sink sites affect pattern formation in Abelian sandpile models, revealing altered growth rates and detailed pattern characterizations for various sink geometries in two and higher dimensions.

Contribution

It introduces new growth rate formulas for patterns with sinks and provides detailed asymptotic pattern characterizations for specific geometries.

Findings

Growth rate changes from √N to √(N/log N) with a sink site.

Pattern growth reduces to N^{1/3} with a line of sink sites.

Exact pattern characterization for two sources on a 2D F-lattice.

Abstract

Adding sand grains at a single site in Abelian sandpile models produces beautiful but complex patterns. We study the effect of sink sites on such patterns. Sinks change the scaling of the diameter of the pattern with the number $N$ of sand grains added. For example, in two dimensions, in presence of a sink site, the diameter of the pattern grows as $\sqrt{(N/\log N)}$ for large $N$, whereas it grows as $\sqrt{N}$ if there are no sink sites. In presence of a line of sink sites, this rate reduces to $N^{1/3}$. We determine the growth rates for these sink geometries along with the case when there are two lines of sink sites forming a wedge, and its generalization to higher dimensions. We characterize one such asymptotic patterns on the two-dimensional F-lattice with a single source adjacent to a line of sink sites, in terms of position of different spatial features in the pattern. For this lattice, we also provide an exact characterization of the pattern with two sources, when the line joining them is along one of the axes.