One-dimensional long-range diffusion-limited aggregation I

Gideon Amir ^[1] ; Omer Angel ^[2] ; Itai Benjamini ^[3] ; Gady Kozma ^[3]
1. [1] Bar-Ilan University
  
  Bar-Ilan University
  
  Israel
2. [2] University of British Columbia
  
  University of British Columbia
  
  Canadá
3. [3] Weizmann Institut
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Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 5, 2016, págs. 3546-3579
Idioma: inglés
DOI: 10.1214/15-AOP1058
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Resumen
- We examine diffusion-limited aggregation generated by a random walk on ZZ 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 nβ+o(1)nβ+o(1), with an explicitly given ββ. 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.