Rescaled Levy-Loewner hulls and random growth

TL;DR

This paper investigates the behavior of rescaled hulls generated by radial Loewner evolution driven by unimodular Lévy processes, analyzing their limits, growth localization, and connections to models like SLE and Hastings-Levitov.

Contribution

It introduces a new framework for analyzing rescaled Lévy-Loewner hulls, establishing their weak limits and exploring their relation to known growth models and stochastic processes.

Findings

Rescaled hulls have a Hausdorff dimension of 1.

Growth tends to localize as jump parameters increase.

Connections to SLE and Hastings-Levitov models are established.

Abstract

We consider radial Loewner evolution driven by unimodular L\'evy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process. The process involves two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov $\rm{HL}(0)$ model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1. Using a different type of compound Poisson process, where the Poisson kernel is replaced by the heat kernel, as driving function, we recover one case of the aforementioned model and $\rm{SLE}(\kappa)$ as limits.