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

作者：Kesten Harry^*; Sidoravicius Vladas

来源：Annals of Probability, 2008, 36(5): 1838-1879.

DOI：10.1214/07-AOP379

• diffusion-limited aggregation
• positive recurrence
• Lyapounov function
• growth model

We consider the following, problem in one-dimensional diffusion-limited agregation (DLA). At time t, we have an "aggregate" consisting of Z boolean AND [0, R(t)] [with R(t) a positive integer]. We also have N(i, t) particles at i, i > R(t). All these particle, perform independent continuous-time symmetric simple random Walks until the first time t' > i at Which some particle tries to jump from R(t) + l to R(t). The aggreate is then increased to the integers in [0, R(t')] = [0, R(t) + 1] [so that R(t') = R(i) + l] and all particles which Were at R(t) + l 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 mu > 0. It is shown that if mu < 1. then R(t) is of order root t, in a sense which is made precise. It is conjectured that R(t) will grow linearly in t if mu is large enough.

• 出版日期2008-9