We consider an interacting random walk on $\mathbb{Z}^d$ where
particles interact only at times when a particle jumps to a site at which there
are at least $n - 1$ other particles present. If there are $i \ge n - 1$
particles present, then the particle coalesces (is removed from the system)
with probability $c_i$ and annihilates (is removed along with another particle)
with probability $a_i$. We call this process the $n$-threshold randomly
coalescing and annihilating random walk. We show that, for $n \ge 3$, if both
$a_i$ and $a_i + c_i$ are increasing in $i$ and if the dimension $d$ is at
least $2n + 4$, then
$P\{\text{the origin is occupied at time $t$}\}\sim
C(d, n) t^{1/(n-1)}$
$E\{\text{ number of particles at the origin at time
$t$}\} C(d, n) t^{1/(n-1)}$
¶ The constants $C(d, n)$ are explicitly identified. The proof is an
extension of a result obtained by Kesten and van den Berg for the 2-threshold
coalescing random walk and is based on an approximation for $dE(t)/dt$.