Scaling limits and fluctuations for random growth under capacity rescaling

Liddle, George and Turner, Amanda (2019) Scaling limits and fluctuations for random growth under capacity rescaling. arxiv.org. (Unpublished)

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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.

Item Type:
Journal Article
Journal or Publication Title:
arxiv.org
Additional Information:
39 pages
Subjects:
ID Code:
135885
Deposited By:
Deposited On:
29 Jul 2019 14:55
Refereed?:
No
Published?:
Unpublished
Last Modified:
25 Oct 2020 06:34