The Annals of Probability

A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains

Harry Kesten and Vladas Sidoravicius

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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.

Article information

Ann. Probab., Volume 36, Number 5 (2008), 1838-1879.

First available in Project Euclid: 11 September 2008

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J15 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Diffusion-limited aggregation positive recurrence Lyapounov function growth model


Kesten, Harry; Sidoravicius, Vladas. A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains. Ann. Probab. 36 (2008), no. 5, 1838--1879. doi:10.1214/07-AOP379.

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