Limit theorems for internal aggregation models
Limit theorems for internal aggregation models
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内部聚合模型的极限定理
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发表时间：
2007
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通讯作者：
Lionel Levine
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作者：
Lionel Levine
我们研究了三种不同聚集模型在整数晶格Z^d上的缩放极限：内部DLA，粒子在其中随机行走直到到达一个未被占用的位置；转子-路由器模型，其中粒子进行确定性的随机漫步模拟；以及可分割的沙堆，每个场地将其多余的质量均匀地分配给相邻的场地。当晶格间距趋于零时，我们发现所有三种模型都具有相同的缩放极限，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正交域，它在势理论和流体力学等许多领域中都是独立出现的。我们的结果既适用于多点源的情况，也适用于域的Diaconis-Fulton粉碎和。在所有粒子从一个点出发的特殊情况下，我们证明了尺度极限是R^d中的欧几里得球，并给出了收敛到一个球的速度的定量界限。对于经典阿贝尔沙堆模型的形状，我们还改进了先前最著名的Le Borgne和Rossin在Z^2和Fey和Redig在高维的边界。最后，我们研究了树叶被压缩为单个汇顶点的规则树的沙堆群，并确定了满沙堆群分解为循环群的乘积。例如，对于高度为n的正则三叉树，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}。我们用这个结果证明了在规则树上的转子-路由器聚合产生一个完美的球。
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.