We consider a system of particles, each of which performs a continuous time random walk on ${bf Z^d$. 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 ${bf Z^d$, 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 $d ge 3$ and we briefly discuss which steps in our proof need to be sharpened to weaken our assumption $d ge 6$. 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.

CWI. Probability, Networks and Algorithms [PNA]

Kesten, H., & van den Berg, R. (1998). Asymptotic density in a coalescing random walk model. CWI. Probability, Networks and Algorithms [PNA]. CWI.