Scaling limits for planar aggregation with subcritical fluctuations

TL;DR

This paper investigates the scaling limits of planar aggregation models with harmonic measure-based growth, revealing a transition from stable disk shapes to critical fluctuations as parameters vary.

Contribution

It introduces a framework for analyzing conformal map-based growth models and identifies the critical fluctuation behavior near stability boundaries.

Findings

Clusters converge to a disk shape under certain conditions.

Fluctuations become critical as parameters approach stability limits.

Methodology applies to broader classes of random growth models.

Abstract

We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. We show that, when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. The methodology developed in this paper provides a blueprint for analysing more general random growth models, such as the Hastings-Levitov family.