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 $\mathbb{Z}^d$ . 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 $p_j$. We show that if $p_j$ is increasing in $j$ and if the dimension $d$ is at least 6 and if we start with one particle at each site of $\mathbb{Z}^d$, then $p(t):= P\{there is at least one particle at the origin at time t\}\sim C(d)/t$. The constant $C(d)$ is explicitly identified. We think the result holds for every dimension $d \geq 3$ and we briefly discuss which steps in our proof need to be sharpened to weaken our assumption $d \geq 6$.

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.