Growth of Stationary Hastings-Levitov

TL;DR

This paper introduces a stationary version of the Hastings-Levitov model, demonstrating its potential as a stationary off-lattice DLA model, with geometric and scaling properties aligning with theoretical predictions.

Contribution

The paper constructs and analyzes a stationary Hastings-Levitov model, revealing tight particle size distribution and geometric properties, and establishing convergence to Brownian motion.

Findings

Particle sizes are tight in the stationary model.

Arms in the model converge to Brownian motion with fractal dimension 3/2.

Tree height scales as n^{2/3}, matching theoretical predictions.

Abstract

We construct and study a stationary version of the Hastings-Levitov$(0)$ model. We prove that, unlike in the classical HL$(0)$ model, in the stationary case the size of particles attaching to the aggregate is tight, and therefore SHL$(0)$ is proposed as a potential candidate for a stationary off-lattice variant of Diffusion Limited Aggregation (DLA). The stationary setting, together with a geometric interpretation of the harmonic measure, yields new geometric results such as stabilization, finiteness of arms and arm size distribution. We show that, under appropriate scaling, arms in SHL$(0)$ converge to the graph of Brownian motion which has fractal dimension $3/2$. Moreover we show that trees with $n$ particles reach a height of order $n^{2/3}$, corresponding to a numerical prediction of Meakin from 1983 for the gyration radius of DLA growing on a long line segment.