Laplacian growth as one-dimensional turbulence

TL;DR

This paper introduces a conformal mapping-based model of Laplacian stochastic growth that reveals a phase transition between stable and turbulent regimes, providing insights into large-scale cluster dynamics and scaling laws.

Contribution

It presents a novel conformal mapping model for Laplacian growth, identifying a phase transition and analyzing large-scale cluster features and scaling behaviors.

Findings

Identification of stable and turbulent growth regimes

Derivation of scaling laws for cluster size

Analysis of the transition dynamics using Fourier components

Abstract

A new model of Laplacian stochastic growth is formulated using conformal mappings. The model describes two growth regimes, stable and turbulent, separated by a sharp phase transition. The first few Fourier components of the mapping define the web, an envelope of the cluster. The web is used to study the transition and the dynamics of large-scale features of the cluster characterized by evolution from macro- to micro-scales. Also, we derive scaling laws for the cluster size.