Asymptotic density in a coalescing random walk model

Abstract

We consider a system of particles, each of which performs a continuous time random walk on ℤd. The particles interact, only at times when a particle jumps to a site at which there are a number of other particles present. If there are j particles present, then the particle which just jumped is removed from the system with probability Pj. We show that if Pj is increasing in j and if the dimension d is at least 6 and if we start with one particle at each site of ℤd, then p(t) := P{there is at least one particle at the origin at time t} ∼ C(d)/t. The constant C(d) is explicitly identified. We think the result holds for every dimension d ≥ 3 and we briefly discuss which steps in our proof need to be sharpened to weaken our assumption d ≥ 6. The proof is based on a justification of a certain mean field approximation for dp(t)/dt. The method seems applicable to many more models of coalescing and annihilating particles. Key words and phrases. Coalescing random walk, method of bounded difference, asymptotic particle density.