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

Gideon Amir

doi:10.1214/15-AIHP731

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

Gideon Amir

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	1513-1527
Number of pages	15
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	53
Issue number	4
DOIs	https://doi.org/10.1214/15-AIHP731
State	Published - Nov 2017

Bibliographical note

Publisher Copyright:
© Association des Publications de l'Institut Henri Poincaré, 2017.

Funding

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.

Funders	Funder number
German-Israeli Foundation for Scientific Research and Development
Israel Science Foundation	1471

Keywords

DLA
Diffusion limited aggregation
Harmonic measure
Phase transition
Random walk
Renewal structure

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10.1214/15-AIHP731

Cite this

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KW - Phase transition

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KW - Renewal structure

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

Abstract

Bibliographical note

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Keywords

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