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

TL;DR

This paper studies a family of planar Laplacian growth models using conformal maps, revealing a phase transition in the cluster structure where growth becomes one-dimensional for certain parameters.

Contribution

It establishes a scaling limit showing one-dimensional growth in a conformal map-based model and identifies a phase transition in the cluster's ancestral structure.

Findings

For ta>1, particles attach to immediate predecessors with high probability.

For ta<1, this attachment does not occur almost surely.

The model exhibits a phase transition in the growth pattern based on ta.

Abstract

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.