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 |Φ _n'|^-η , where Φ _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 η >1 , aggregating particles attach to their immediate predecessors with high probability, while for η <1 almost surely this does not happen.