Cluster-Cluster model in ${\mathbb{Z}}^d$

Abstract

We consider a stochastic process on ${\mathbb{Z}}^d$ for $d\ge 1$. Given a translation invariant and ergodic starting configuration of finite clusters, each cluster $C$ performs a continuous time simple random walk with rate $|C{|}^{-\alpha }$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. In all dimensions, we show that if $\alpha \ge 0$, there is almost surely no spontaneous creation of an infinite cluster within finite time. Moreover, for any $\alpha \le -1-2/d$ there is a finite-time blowup almost surely. In the regime $\alpha \in (-1,0)$ we show that the behavior greatly depends on the initial configuration. In addition, in dimension one, we establish the exact phase diagram.