Scaling Limits And Fluctuations For Random Growth Under Capacity Rescaling

We evaluate a strongly regularised version of the Hastings-Levitov model HL(alpha) for 0 <= 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.