Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks

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.

27 pages, 17 figures. Figures with better resolution is available at http://www.theory.tifr.res.in/~tridib/pss.html

References in corpus (4)

• Discrete Riemann Surfaces and the Ising model
• Limiting shapes for deterministic centrally seeded growth models
• Discrete Holomorphicity at Two-Dimensional Critical Points
• Explicit characterization of the identity configuration in an Abelian Sandpile Model

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• Modelling proportionate growth
• Proportionate growth in patterns formed in the rotor-router model
• The spectrum of the abelian sandpile model
• Notes on identical configurations in Abelian Sandpile Model with initial height
• Sandpiles and Dominos