A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains
type: Item
instance of: scholarly article
title: A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains
zbMATH Open document ID: 1154.60075
DOI: 10.1214/07-AOP379
author: Harry Kesten
author: Vladas Sidoravicius
published in: The Annals of Probability
publication date: 2008-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/0809.4175
review text: The authors depart from their previous [Ann. Probab. 33, No.~6, 2402--2462 (2005; Zbl 1111.60074)], apparently unrelated paper on the time rate of the spread of infection in a moving population. Once interpreted as a specific growth model, based on the concept of continuous-time symmetric random walks, one may justifiably transcribe some of its results to the one-dimensional diffusion-limited aggregation. If compared to the standard DLA considerations, where one adds one particle at a time, in the present analysis one has at disposal an inifinity of the like from the start. All particles perform independent continuous-time random walks with a fixed jump rate, until being absorbed by the aggregate. An explicit construction of the growth process in terms of color-chainging particles, allows to give estimates on the asymptotic behavior of the growth process (Theorems 2 and 3). Open problems are mentioned. In view of difficulties with a rigorous proof of the existence of a nontrivial invariant measure in some cases, suitable caricature (toy) models were proposed to support the positive recurrence property validity.
reviewed by: Piotr Garbaczewski
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82C41
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 5353658
zbMATH Keywords: diffusion-limited aggregation
zbMATH Keywords: growth model
zbMATH Keywords: time rate of aggregation
zbMATH Keywords: Markov chains
zbMATH Keywords: continuous-time random walk
zbMATH Keywords: positive recurrence
zbMATH Keywords: Poisson random variables
MaRDI profile type: Publication
arXiv ID: 0809.4175
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OpenAlex ID: W3100653518
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