One-dimensional long-range diffusion-limited aggregation III - The limit aggregate

摘要

In this paper we study the structure of the limit aggregate A(infinity) = boolean OR(n >= 0) A(n) of the one-dimensional long range diffusion limited aggregation process defined in (Ann. Probab. 44 (2016) 3546-3579). We show (under some regularity conditions) that for walks with finite third moment A(infinity) has renewal structure and positive density, while for walks with finite variance the renewal structure no longer exists and A(infinity) 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.