一维长程扩散限制凝聚III --极限凝聚
One-dimensional long-range diffusion-limited aggregation III -- The limit aggregate
摘要：
本文研究了文献[AABK 09]中定义的一维长程扩散极限凝聚过程的极限凝聚$A_\infty = \bigcup_{n\geq 0} A_n$的结构。我们证明了（在一定的正则性条件下）对于有限三阶矩的游动$A_\infty$具有更新结构和正密度，而对于有限方差的游动更新结构不再存在且$A_\infty$具有0密度.我们定义了一个树结构的聚集，并显示这些随机树的度和数量的结果。我们引入了一个新的“调和竞争”模型，不同的颜色竞争调和测度，并显示树结构是如何在这个模型中共存。
英文摘要：
In this paper we study the structure of the limit aggregate $A_\infty = \bigcup_{n\geq 0} A_n$ of the one-dimensional long range diffusion limited aggregation process defined in [AABK09]. We show (under some regularity conditions) that for walks with finite third moment $A_\infty$ has renewal structure and positive density, while for walks with finite variance the renewal structure no longer exists and $A_\infty$ has 0 density. We define a tree structure on the aggregates and show some results on the degrees and number of ends of these random trees. We introduce a new "harmonic competition" model where different colours compete for harmonic measure, and show how the tree structure is related to coexistence in this model.
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