Positive Recurrence of a One-Dimensional Variant of Diffusion Limited Aggregation

Abstract

The present paper studies a variant of the DLA model: At time 0 choose i.i.d. random variables N(x),x∈ℤ⁺={1,2,...}. Each N(x) has a Poisson (μ) distribution. N(x) will be the number of particles at x at time 0. In the model we also put a mark at an integer position. Its position at time t is denoted by M(t). We take M(0)=0. We regard the particles as frozen (i.e., they stay in place) till a certain random time, which generally differs for different particles. At such a random time a particle is “thawed”, that is, it starts to move according to a continuous time simple random walk. At all times there will be i thawed particles in the system. Assume that at time 0 we have i thawed particles in the system at some positions in ℤ⁺, and that the N(x),x∈ℤ⁺ are i.i.d. as described above. The i thawed particles perform independent, continuous time, simple random walks until the first time τ₁ at which one of them jumps from 1 to 0. At this time we move the mark from 0 to 1 (i.e., M(t)=0 for t<τ₁, but $\mathcal{M}\left( {\tau _1 } \right) = 1$). Also at time τ₁ all particles at 1 are “absorbed by the mark”. This includes frozen as well as thawed particles which are at 1 at time τ₁. If r thawed particles are removed at time τ₁, then we thaw another r particles. We take for these the r particles nearest to the mark at time τ₁, with some rule for breaking ties. The particles thawed at time τ₁ start simple random walks at that time.

At any time t there will be i thawed particles strictly to the right of the mark. If M(t)=p, then the mark stays at p till the first time τ≥t at which one of the i thawed particles jumps from p+1 to M(t)=p. We then move the mark to position, p+1 and absorb all particles at p+1. If r thawed particles are absorbed, then we also thaw r new particles. Again we thaw the particles nearest to the mark. We continue this process forever.

We prove that almost surely $lim_{t \to \infty } t^{ - 1} \mathcal{M}\left( t \right)$ exists and is strictly positive, provided μ is large enough.