On One-Dimensional Cluster-cluster Model

The Cluster-cluster model was introduced by Meakin et al. in 1984. Each x∈Zd starts with a cluster of size 1 with probability p∈(0,1] independently. Each cluster performs a continuous-time SRW with rate. 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. Focusing on dimension d=1, we show that for α>-2, at time t, the cluster size is of order t1α+2, and for α<-2 we get an infinite cluster in finite time a.s. Additionally, for α=0 we show convergence in distribution of the scaling limit.