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Scaling limits and fluctuations for random growth under capacity rescaling

Random growth occurs in many real world settings, for example, we see it exhibited in the growth of tumours, bacterial growth and lightning patterns. As such we would like to be able to model such processes to determine their behaviour in their scaling limits. Well studied models to describe these different processes include the Eden model and DLA. In a 1998 paper, Hastings and Levitov introduced a one parameter family of conformal maps HL(alpha) which can be used to model Laplacian growth processes and allows us to vary between the previous models by varying the parameter alpha. We will consider a regularised version of this model and analyse its scaling limits and fluctuations for different values of alpha under capacity rescaling.