阿贝尔沙堆中的阿波罗结构
Apollonian structure in the Abelian sandpile
摘要：
阿贝尔沙堆过程通过将任何顶点与至少4个芯片一起倾倒，将其芯片分配给其4个邻居中的每一个，来演化整数格上的芯片配置。当从一大堆芯片开始时，沙堆的终态具有一种奇怪的分形结构，这种结构一直无法解释。使用整数-上调和函数可达到的二次增长的表征，我们证明了沙堆PDE最近表现出的沙堆的标度极限的特征承认某些分形解决方案，给出了一个精确的数学角度的沙堆的分形性质。
英文摘要：
The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on the fractal nature of the sandpile.
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