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

In this paper we study the structure of the limit aggregate A∞ = ∪ N≥0 An 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∞ has renewal structure and positive density, while for walks with finite variance the renewal structure no longer exists and A∞ 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.