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September 2016 One-dimensional long-range diffusion-limited aggregation I
Gideon Amir, Omer Angel, Itai Benjamini, Gady Kozma
Ann. Probab. 44(5): 3546-3579 (September 2016). DOI: 10.1214/15-AOP1058


We examine diffusion-limited aggregation generated by a random walk on $\mathbb{Z}$ with long jumps. We derive upper and lower bounds on the growth rate of the aggregate as a function of the number of moments a single step of the walk has. Under various regularity conditions on the tail of the step distribution, we prove that the diameter grows as $n^{\beta+o(1)}$, with an explicitly given $\beta$. The growth rate of the aggregate is shown to have three phase transitions, when the walk steps have finite third moment, finite variance, and conjecturally, finite half moment.


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Gideon Amir. Omer Angel. Itai Benjamini. Gady Kozma. "One-dimensional long-range diffusion-limited aggregation I." Ann. Probab. 44 (5) 3546 - 3579, September 2016.


Received: 1 December 2013; Revised: 1 July 2015; Published: September 2016
First available in Project Euclid: 21 September 2016

zbMATH: 1353.82051
MathSciNet: MR3551204
Digital Object Identifier: 10.1214/15-AOP1058

Primary: 82C24
Secondary: 60K35 , 97K50 , 97K60

Keywords: Diffusion limited aggregation , DLA , Green’s function , harmonic measure , phase transition , Random walk , stable Green’s function , Stable process

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 5 • September 2016
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