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.
|Number of pages||15|
|Journal||Annales de l'institut Henri Poincare (B) Probability and Statistics|
|State||Published - Nov 2017|
Bibliographical noteFunding Information:
1Supported by the Israel Science Foundation (Grant No. 1471) and by a Grant from the GIF, the German-Israeli Foundation for Scientific Research and Development.
© Association des Publications de l'Institut Henri Poincaré, 2017.
- Diffusion limited aggregation
- Harmonic measure
- Phase transition
- Random walk
- Renewal structure