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

Abstract

We consider a family of growth models defined using conformal maps in which the local growth rate is determined by $|{\Phi }_n^{\prime }{|}^{-\eta }$, 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.

In version 3, the order of sections has been rearranged and a few minor typos have been corrected