Scaling limits of anisotropic Hastings-Levitov clusters

TL;DR

This paper investigates the scaling limits of anisotropic Hastings-Levitov clusters, revealing how anisotropy influences shape evolution and harmonic measure, with results expressed via Loewner hulls and differential equations.

Contribution

It introduces a variation of the HL(0) model with anisotropic growth and characterizes its scaling limits and fluctuations, linking them to Loewner theory.

Findings

Limit shapes as Loewner hulls

Harmonic measure evolution via ODE

Characterization of stochastic fluctuations

Abstract

We consider a variation of the standard Hastings-Levitov model HL(0), in which growth is anisotropic. Two natural scaling limits are established and we give precise descriptions of the effects of the anisotropy. We show that the limit shapes can be realised as Loewner hulls and that the evolution of harmonic measure on the cluster boundary can be described by the solution to a deterministic ordinary differential equation related to the Loewner equation. We also characterise the stochastic fluctuations around the deterministic limit flow.