Convergence of the Abelian sandpile
Convergence of the Abelian sandpile
复制标题
阿贝尔沙堆的收敛
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Charles K. Smart
中科院分区：
文献类型：
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作者：
W. Pegden;Charles K. Smart
阿贝尔沙堆增长模型是一个扩散过程，用于放置在整数格$mathbb{Z}^d$的顶点上的芯片配置，其中具有至少2d芯片{em topple}的站点将1个芯片分配给格中的每个邻居，直到没有更多的topplings是可能的。从一个初始配置组成的$n$芯片放在一个单一的顶点，重新缩放的稳定配置似乎收敛到一个特定的分形图案为$n 不需要钱。然而，很少有人已经证明的外观稳定的配置。我们使用偏微分方程技术证明了重新标度的稳定配置确实收敛到一个唯一的极限为n 不需要钱。我们刻画的极限作为一个椭圆障碍问题的解的拉普拉斯算子。
The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $mathbb{Z}^d$, in which sites with at least 2d chips {em topple}, distributing 1 chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of $n$ chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as $n o infty$. However, little has been proved about the appearance of the stable configurations. We use PDE techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as $n o infty$. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.