Small particle limits in a regularized Laplacian random growth model

TL;DR

This paper investigates a regularized Laplacian random growth model, analyzing the limiting shapes of clusters and the scaling limits of harmonic measure flow, revealing connections to the Brownian web and internal branching structures.

Contribution

It introduces a regularized version of Hastings-Levitov growth, establishes convergence to growing disks, and characterizes the harmonic measure flow limits under various parameter regimes.

Findings

Clusters converge to growing disks as capacity c approaches 0.

Harmonic measure flow converges to Brownian web, stopped Brownian web, or identity map depending on parameters.

Simulation results support theoretical findings and explore uncovered parameter regimes.

Abstract

We study a regularized version of Hastings-Levitov planar random growth that models clusters formed by the aggregation of diffusing particles. In this model, the growing clusters are defined in terms of iterated slit maps whose capacities are given by c_n=c|\Phi_{n-1}'(e^{\sigma+i\theta_n})|^{-\alpha}, \alpha \geq 0, where c>0 is the capacity of the first particle, {\Phi_n}_n are the composed conformal maps defining the clusters of the evolution, {\theta_n}_n are independent uniform angles determining the positions at which particles are attached, and \sigma>0 is a regularization parameter which we take to depend on c. We prove that under an appropriate rescaling of time, in the limit as c converges to 0, the clusters converge to growing disks with deterministic capacities, provided that \sigma does not converge to 0 too fast. We then establish scaling limits for the harmonic measure flow, showing that by letting \alpha tend to 0 at different rates it converges to either the Brownian web on the circle, a stopped version of the Brownian web on the circle, or the identity map. As the harmonic measure flow is closely related to the internal branching structure within the cluster, the above three cases intuitively correspond to the number of infinite branches in the model being either 1, a random number whose distribution we obtain, or unbounded, in the limit as c converges to 0. We also present several findings based on simulations of the model with parameter choices not covered by our rigorous analysis.