One-Dimensional Scaling Limits in a Planar Laplacian Random Growth Model

Alan Sola,Fredrik Viklund,Amanda Turner

doi:10.1007/s00220-019-03460-1

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.

Highlights

Discrete versions can be formulated on a lattice in all dimensions: some famous examples of this type of growth process include diffusion-limited aggregation (DLA) [28], the Eden model [4], or the more general dielectric breakdown model (DBM) [22]
We study a particular instance of a conformal growth model, focusing instead on the concentration aspect of Laplacian growth and showing that anisotropic scaling limits arise in the presence of strong feedback in the growth rule
Hastings [7], and subsequently Mathiesen and Jensen [19], study a model that essentially corresponds to Aggregate Loewner evolution (ALE)(2, η) modulo a slightly different parametrization in η. (an alternative name for the growth model in this paper could have been DBM(α, η) or HL(α, η), but we have opted for a different terminology to avoid confusion with lattice models, and to emphasize connections with the Loewner equation, see below.) Hastings argues that for large enough exponents, more precisely, for η 3 in our parametrization, the corresponding clusters become one-dimensional; he points out that the behavior of the models depends strongly on the choice of regularization

Summary

Introduction

(an alternative name for the growth model in this paper could have been DBM(α, η) or HL(α, η), but we have opted for a different terminology to avoid confusion with lattice models, and to emphasize connections with the Loewner equation, see below.) Hastings argues that for large enough exponents, more precisely, for η 3 in our parametrization, the corresponding clusters become one-dimensional; he points out that the behavior of the models depends strongly on the choice of regularization Another model that fits into this general framework is the Quantum Loewner Evolution model (QLE(γ , η)) of Miller and Sheffield [20,21] which is proposed as a scaling limit of DBM(η) on a γ -Liouville quantum gravity surface. A similar difficulty arises from the dependence of the particle sizes on the derivatives of the conformal mappings In this case the model is well-defined without the need for a regularization parameter in (5), it is no longer possible to guarantee that the resulting clusters have total capacity bounded above and below. The feedback mechanism in (4) is sensitive so that a single “bad” angle can destroy the genealogical structure of the growing slit by leading to the creation of a new, competing tip further down the slit, which could lead to a splitting of growth into two branches

Overview of Results

Loewner Flows

Analysis of the Slit Map

Ancestral Lines and Convergence for ALE

Estimates on Conformal Maps via Loewner’s Equation