Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks

Abstract

Adding sand grains at a single site in the 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 the 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 the presence of a line of sink sites, this rate reduces to N 1/3. We determine the growth rates for various sink geometries along with the case when there are two lines of sink sites forming a wedge, and generalizations to higher dimensions. We characterize the asymptotic pattern in the large N limit for one such case, the two-dimensional F-lattice with a single source adjacent to a line of sink sites. The characterization is done in terms of the positions 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 of the lattice.