Randomly coalescing random walk in dimension $ geq $ 3

Suppose at time $0$ each site of $Z^d$ contains one particle, which starts to perform a continuous time random walk. The particles interact only at times when a particle jumps to an already occupied site: if there are $j$ particles present, then the jumping particle is removed from the system with probability $p_j$. We assume that $p_j$ is increasing in $j$. In an earlier paper we proved that if the dimension $d$ is at least 6, then $p(t) := P{$there is at least one particle at the origin at time $t sim C(d)/t$, with $C(d)$ an explicitly identified constant. We also conjectured that the result holds for $d ge 3$. In the present paper we show that, under the quite natural condition that the number of particles per site is bounded, this is indeed the case. The key step in the proof is to improve a certain variance bound, which is needed to estimate the error terms in an approximate differential equation for $p(t)$. We do this by making more refined use of coupling methods and(correlation) inequalities.