多源（汇）发育沙堆的斑图形成
Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks
摘要：
在阿贝尔沙堆模型中，在单个位置添加沙粒会产生美丽但复杂的图案。我们研究了这种模式上的水槽网站的效果。水槽会根据添加的砂粒数$N$更改阵列直径的比例。例如，在二维中，在存在汇位点的情况下，对于大的$N$，图案的直径增长为$\sqrt{(N/\log N)}$，而如果不存在汇位点，则其增长为$\sqrt{N}$。在存在一系列汇位点的情况下，该速率降低至$N^{1/3}$。我们确定这些水槽的几何形状的情况下，当有两条线的水槽网站形成一个楔形，其推广到更高的维度的增长率。我们描述了一个这样的渐近模式的二维F-晶格与一个单一的源相邻的一行的汇网站，在不同的空间特征的位置在图案。对于这个晶格，我们还提供了一个精确的表征模式与两个源，当连接它们的线是沿着其中一个轴。
英文摘要：
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.
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