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

Scaling limits and fluctuations for random growth under capacity rescaling