Asymptotic density in a threshold coalescing and annihilating random walk

Abstract

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 $$P\{\text{the origin is occupied at time $t$}\}\sim C(d, n) t^{−\frac{1}{n-1}},\\ E\{\text{number of particles at the origin at time $t$}\} \sim C(d, n) t^{−\frac{1}{n-1}}.$$ 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 $dE(t)/dt$.