On one-dimensional Cluster cluster model

Abstract

The Cluster-cluster model was introduced by Meakin et al in 1984. Each $x\in {\mathbb{Z}}^d$ starts with a cluster of size 1 with probability $p\in (0,1]$ independently. Each cluster $C$ performs a continuous-time SRW with rate $|C{|}^{-\hat{I}\pm }$. 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 $\hat{I}\pm >-2$, at time $t$, the cluster size is of order $t^{\frac{1}{\hat{I}\pm +2}}$, and for $\hat{I}\pm <-2$ we get an infinite cluster in finite time a.s. Additionally, for $\hat{I}\pm =0$ we show convergence in distribution of the scaling limit.

20 pages, 6 figures