# Scaling limits and fluctuations for random growth under capacity rescaling

## TL;DR

This paper investigates the behavior of a regularized Hastings-Levitov growth model under capacity rescaling, revealing different limiting shapes and fluctuation behaviors depending on the parameter lpha, and identifying potential phase transitions at lpha=0 and 2.

## Contribution

It introduces a new capacity-rescaled analysis of HL(lpha), showing non-disk limits for lpha=0 and disk limits for 0<lpha<2, and characterizes the Gaussian fluctuations and phase transitions.

## Key findings

- Limit shape for lpha=0 is not a disk.
- For 0<lpha<2, the cluster approaches a disk.
- Fluctuations behave like a Gaussian field depending on lpha.

## Abstract

We evaluate a strongly regularised version of the Hastings-Levitov model HL$(\alpha)$ for $0\leq \alpha<2$. Previous results have concentrated on the small-particle limit where the size of the attaching particle approaches zero in the limit. However, we consider the case where we rescale the whole cluster by its capacity before taking limits, whilst keeping the particle size fixed. We first consider the case where $\alpha=0$ and show that under capacity rescaling, the limiting structure of the cluster is not a disk, unlike in the small-particle limit. Then we consider the case where $0<\alpha<2$ and show that under the same rescaling the cluster approaches a disk. We also evaluate the fluctuations and show that, when represented as a holomorphic function, they behave like a Gaussian field dependent on $\alpha$. Furthermore, this field becomes degenerate as $\alpha$ approaches 0 and 2, suggesting the existence of phase transitions at these values.