部分均衡与可分沙堆模型的推广
Partial Balayage and a Generalization of the Divisible Sandpile Model
摘要：
在最近的工作L。Levine和Y. Peres，观察到晶格上颗粒聚集的三种模型-可分沙堆，转子-路由器聚集和内部扩散限制聚集-共享一个共同的缩放极限，因为晶格间距趋于零，如果它们以相同的初始质量配置开始。很容易观察到，这个标度极限与将这个初始质量组态的部分平衡量与勒贝格测度的势论运算完全相同。然而，从部分平衡运算的理论中可以清楚地看出，人们可以将质量构形的部分平衡取为比勒贝格测度更一般的度量，而对于Levine和Peres描述的三种聚集模型，人们不能这样做。因此，在本文中，我们推广这些模型之一，可分沙堆模型，主要是在一个有界的设置，并表明，这种推广的自然缩放限制是由一个一般的部分平衡操作。
英文摘要：
In recent work by L. Levine and Y. Peres, it was observed that three models for particle aggregation on the lattice - the divisible sandpile, rotor-router aggregation, and internal diffusion limited aggregation - share a common scaling limit as the lattice spacing tends to zero, if they are started with the same initial mass configuration. It is straightforward to observe that this scaling limit is precisely the same as the potential-theoretic operation of taking the partial balayage of this initial mass configuration to the Lebesgue measure. However, from the theory of the partial balayage operation it is clear that one may take the partial balayage of a mass configuration to a more general measure than the Lebesgue measure, which one cannot do for the three aggregation models described by Levine and Peres. In this paper we therefore generalize one of these models, the divisible sandpile model, in mainly a bounded setting, and show that a natural scaling limit of this generalization is given by a general partial balayage operation.
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