One-dimensional scaling limits in a planar Laplacian random growth model

We consider a family of growth models defined using conformal maps in which the local growth rate is determined by \(|\Phi _n'|^{-\eta }\), where \(\Phi _n\) is the aggregate map for n particles. We establish a scaling limit result in which strong feedback in the growth rule leads to one-dimensional limits in the form of straight slits. More precisely, we exhibit a phase transition in the ancestral structure of the growing clusters: for \(\eta >1\), aggregating particles attach to their immediate predecessors with high probability, while for \(\eta <1\) almost surely this does not happen.