The Cluster-cluster model was defined by Meakin in 1984. Consider a stochastic process on the graph Z^d.
Each x in 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|^{-\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>= 1, there is no spontaneous creation of an infinite cluster in a finite time a.s.
Focusing on dimension d=1, we show that for \alpha>-2, at time t, the cluster size is of order t^\frac{1}{\alpha + 2}, and for \alpha < -2 we get an infinite cluster in finite time a.s.
Additionally, for \alpha = 0 we show convergence in distribution of the scaling limit.
Joint work with Noam Berger (TUM) and Daniel Sharon (Technion)
