小粒子极限下的Hastings-Levitov凝聚
Hastings-Levitov aggregation in the small-particle limit
摘要：
我们建立了一些平面聚集模型的标度限制。该模型是由一个独立的和相同分布的随机共形映射序列的组成，每个对应于一个粒子的添加。我们研究了小粒径和快速聚集的极限。在Caratheodory的意义上，星系团的增长过程收敛于一个膨胀的圆盘。一个更精细的分析显示，在集群内，分支手指，其径向分量随时间的确定性增加的树结构。向内追踪的任何有限手指样本的变元执行合并布朗运动。手指之间的任何有限间隙样本的变元，向外追踪，也执行合并布朗运动。这些性质与簇边界上的调和测度的演化密切相关，并证明了其收敛于布朗网。
英文摘要：
We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Caratheodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web.
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