Cylindrical Hastings Levitov

TL;DR

This paper introduces a cylindrical version of the Hastings-Levitov$(0)$ process, proves its convergence to a stationary version under particle size scaling, and demonstrates its relevance as a model for diffusion limited aggregation.

Contribution

It defines a new cylindrical Hastings-Levitov$(0)$ process and establishes its convergence to a stationary process, addressing technical challenges in spatial limits and normalization.

Findings

Process converges to stationary Hastings-Levitov$(0)$

Model captures diffusion limited aggregation phenomena

Handles spatial limit with slit map normalization

Abstract

We define a Hastings-Levitov$(0)$ process on a cylinder and prove that the process converges to Stationary Hastings Levitov$(0)$ under appropriate particle size scaling that depends on the radius of the cylinder. The Stationary Hastings Levitov$(0)$ was shown by Berger, Procaccia and Turner to admit tight particle sizes, without a priori particle size normalization, thus it serves as a good model for the phenomenon of diffusion limited aggregation. Technical challenge, in this paper, is in taking the spatial limit together with the correct slit map normalization. This result also shows that the early life of the Hastings Levitov$(0)$ process in the small particle limit, spatially scaled so the slits have unit length, behaves like the Stationary Hastings Levitov$(0)$.