The cluster-cluster model was introduced by Meakin in the 1980s as a diffusion-driven model for the formation of multiple clusters. A cluster of size m moves at rate m^{-\alpha}. In this talk, we present new results showing that for every \alpha \ge 0, corresponding to the bounded-rates regime, clusters remain finite almost surely for all times. The proof is surprisingly simple: it requires no geometric information and depends only on densities and abstract rate bounds. By contrast, in the speed-up regime \alpha<0, the behavior changes dramatically. There, cluster geometry plays a decisive role. We will exhibit initial configurations that lead to immediate blow-up, alongside others for which blow-up never occurs.

Joint work with Noam Berger, Dominik Shmid and Daniel Sharon.