Open Access
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

Abstract

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.

Citation

Download Citation

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

Information

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

Subjects:
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
Back to Top