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

Abstract

We consider the following problem in one-dimensional diffusion-limited aggregation (DLA). At time t, we have an "aggregate" consisting of ℤ ∩n [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 √t, in a sense which is made precise. It is conjectured that A(t) will grow linearly in t if μ is large enough. © 2008 Institute of Mathematical Statistics.