阿贝尔沙堆的收敛性
Convergence of the Abelian sandpile
摘要：
阿贝尔沙堆增长模型是一个扩散过程的芯片配置放置在整数格$\mathbb{Z}^d$的顶点，其中网站至少有2d芯片{ \em倾倒}，分配1芯片的晶格中的每个邻居，直到没有更多的倾倒是可能的。从一个初始配置的$n$芯片放置在一个单一的顶点，重新缩放的稳定配置似乎收敛到一个特定的分形图案为$n\to \infty$。然而，很少有人已经证明的外观稳定的配置。我们使用PDE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$n \to \infty$。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。
英文摘要：
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 $n\to \infty$. 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 $n \to \infty$. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.
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