New directions arising from a special diffusion process on the integer lattice

所谓的阿贝尔沙堆代表了一个惊人的例子，从一个简单的环境中产生了令人惊讶的数学。沙堆的规则很简单：给出一个无限棋盘(即整数格子)上的“筹码”配置，一个人可以“推翻”任何至少有4个筹码的正方形，方法是将一个筹码从这个方格发送到它相邻的四个方格中的每一个。当从一大堆筹码开始，一直到没有一个正方形的筹码超过3个时，结果是一个惊人的分形图。研究这一过程导致发现了数学中看似不同的领域之间令人惊讶的联系，本项目旨在利用这些新的联系，以便更深入地了解沙堆过程，并丰富我们对这些其他领域的理解。这个数学领域与相变和物理学中的保形场理论也有联系。更具体地说，这个项目涉及Bak，Tang和Wiesenfeld的所谓阿贝尔沙堆模型。通过识别整数超调和函数、阿波罗圆填充和欧几里得平面的某些规则拼接之间的新联系，我们现在能够表征沙堆的标度极限并分析其局部分形结构。这个项目有两个主要目标。首先是扩展我们对沙堆的知识：例如，我们想要加强我们可以证明沙堆过程所允许的那种收敛，表征沙堆的Dirichlet问题的长期研究实例的解，并将我们对沙堆过程的尺度极限的知识扩展到正方形格子的宇宙之外。另一方面，我们希望将为我们的沙堆分析开发的新视角应用于其他领域，有可能证实关于阿波罗圆填充的数论猜想，或表征具有某些对称性质的平面平铺。

The so-called Abelian sandpile represents a striking example of surprising mathematics arising from a simple setting. The rules of the sandpile are simple: given a configuration of "chips" on an infinite chessboard (i.e., the integer lattice), one can "topple" any square with at least 4 chips, by sending one chip from this square to each of its four neighboring squares. When begun from a single large stack of chips and continued until no square has more than 3 chips, the result is a striking fractal configuration. Studying this process has lead to the discovery of surprising connections between seemingly disparate areas of mathematics, and this project aims to leverage these new connections both for the sake of a deeper understanding of the sandpile process, and to enrich our understandings of these other areas as well. This area of mathematics has connections with phase transitions and conformal field theory in physics as well.More specifically, this project concerns the so-called Abelian sandpile model of Bak, Tang, and Wiesenfeld. By identifying new connections between integer superharmonic functions, Apollonian circle packings, and certain regular tilings of the Euclidean plane, we are now able to characterize the scaling limit of the sandpile and analyze its local fractal structure. This project has two primary goals. The first is to extend our knowledge of the sandpile: for example, we would like to strengthen the kind of convergence we can prove the sandpile process admits, characterize solutions to long-studied instances of the Dirichlet problem for the sandpile, and extend our knowledge of the scaling limit of the sandpile process beyond the universe of the square lattice. On the other hand, we would like to bring new perspectives developed for our analysis of the sandpile to bear on other areas, with the possibility of confirming number-theoretic conjectures regarding Apollonian circle packings, or characterizing tilings of the plane with certain symmetry properties.