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September 2008 A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains
Harry Kesten, Vladas Sidoravicius
Ann. Probab. 36(5): 1838-1879 (September 2008). DOI: 10.1214/07-AOP379

Abstract

We consider the following problem in one-dimensional diffusion-limited aggregation (DLA). At time t, we have an “aggregate” consisting of ℤ∩[0, R(t)] [with R(t) a positive integer]. We also have N(i, t) particles at i, i>R(t). All these particles perform independent continuous-time symmetric simple random walks until the first time t'>t at which some particle tries to jump from R(t)+1 to R(t). The aggregate is then increased to the integers in [0, R(t')]=[0, R(t)+1] [so that R(t')=R(t)+1] and all particles which were at R(t)+1 at time t' − are removed from the system. The problem is to determine how fast R(t) grows as a function of t if we start at time 0 with R(0)=0 and the N(i, 0) i.i.d. Poisson variables with mean μ>0. It is shown that if μ<1, then R(t) is of order $\sqrt{t}$, in a sense which is made precise. It is conjectured that R(t) will grow linearly in t if μ is large enough.

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Harry Kesten. Vladas Sidoravicius. "A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains." Ann. Probab. 36 (5) 1838 - 1879, September 2008. https://doi.org/10.1214/07-AOP379

Information

Published: September 2008
First available in Project Euclid: 11 September 2008

zbMATH: 1154.60075
MathSciNet: MR2440925
Digital Object Identifier: 10.1214/07-AOP379

Subjects:
Primary: 60K35
Secondary: 60J15, 82C41

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 5 • September 2008
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