我们研究了整数格Z^d上三种不同聚集模型的标度极限：内部DLA，其中粒子执行随机游动，直到到达一个未被占用的站点;转子路由器模型，其中粒子执行确定性类似的随机游动;和可分的沙堆，其中每个站点将其多余的质量均匀地分配给其邻居。当晶格间距趋于零时，我们发现三种模型具有相同的尺度极限，我们将其描述为R^d中某个PDE自由边界问题的解。特别是，内部DLA具有确定性的缩放限制。我们发现标度极限是在位势论和流体力学等许多领域独立出现的求积域。我们的结果既适用于重点源的情况，也适用于Diaconis-Fulton smash和域。在特殊情况下，当所有的粒子开始在一个单一的网站，我们表明，标度限制是一个欧几里德球在R^d，并给出了定量的界限上的收敛速率的球。对于经典阿贝尔沙堆模型的形状，我们还改进了以前最著名的Le Borgne和Rossin在Z^2中的边界以及Fey和Redig在高维中的边界。最后，我们研究了叶子被折叠到单个汇点的正则树的沙堆群，并确定了满沙堆群的分解为循环群的乘积。例如，对于高度为n的正则三元树，沙堆群同构于（Z_3）^{2^{n-3}} x（Z_7）^{2^{n-4}} x. x Z_{2^{n-1}-1} x Z_{2^n-1}。我们用这个结果来证明，转子路由器聚集在定期树产生一个完美的球。
英文摘要：
We study the scaling limits of three different aggregation models on the integer lattice Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains. In the special case when all particles start at a single site, we show that the scaling limit is a Euclidean ball in R^d, and give quantitative bounds on the rate of convergence to a ball. We also improve on the previously best known bounds of Le Borgne and Rossin in Z^2 and Fey and Redig in higher dimensions for the shape of the classical abelian sandpile model. Lastly, we study the sandpile group of a regular tree whose leaves are collapsed to a single sink vertex, and determine the decomposition of the full sandpile group as a product of cyclic groups. For the regular ternary tree of height n, for example, the sandpile group is isomorphic to (Z_3)^{2^{n-3}} x (Z_7)^{2^{n-4}} x ... x Z_{2^{n-1}-1} x Z_{2^n-1}. We use this result to prove that rotor-router aggregation on the regular tree yields a perfect ball.
