Limiting Point Processes for Rescalings of Coalescing and Annihilating Random Walks on $Z^d$

Abstract

Let $p(x, y)$ be an arbitrary random walk on $Z^d$. Let $\xi_t$ be the system of coalescing random walks based on $p$, starting with all sites occupied, and let $\eta_t$ be the corresponding system of annihilating random walks. The spatial rescalings $P(0 \in \xi_t)^{1/d}\xi_t$ for $t \geqq 0$ form a tight family of point processes on $R^d$. Any limiting point process as $t \rightarrow\infty$ has Lesbegue measure as its intensity, and has no multiple points. When $p$ is simple random walk on $Z^d$ these rescalings converge in distribution, to the simple Poisson point process for $d \geq 2$, and to a non-Poisson limit for $d = 1$. For a large class of $p$, we prove that $P(0 \in \eta_t)/P(0 \in \xi_t) \rightarrow 1/2$ as $t \rightarrow\infty$. A generalization of this result, proved for nearest neighbor random walks on $Z^1$, and for all multidimensional $p$, implies that the limiting point process for rescalings $P(0 \in \xi_t)^{1/d}\eta_t$ of the system of annihilating random walks is the one half thinning of the limiting point process for the corresponding coalescing system.