Alan Sola: One dimensional scaling limits in a Laplacian random growth model

Abstract

In joint work with A. Turner and F. Viklund, we consider growth models defined using conformal maps in which local growth 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: for \(\eta>1\), aggregating particles attach to their immediate predecessors with high probability, while for \(\eta<1\) there is a positive probability that this does not happen.