Scaling limits of random growth processes

Liddle, George and Turner, Amanda (2021) Scaling limits of random growth processes. PhD thesis, Lancaster University.

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Abstract

The topic of this thesis is random growth processes. These occur naturally in many real world settings such as in the growth of tumours and lightning strikes. As such we would like to model the processes so that we can effectively study their properties. In particular, we are interested in what the shape of the process is as it grows and so we wish to evaluate the scaling limits of the random processes. In Chapter 1, we will provide the background material needed in order to study the random growth models. We will give examples of real world processes that we would like to study before describing the models used to study them. We then provide some previous results in the area to provide context for the independent research that follows. Chapter 2 will follow [LT21a] closely. In this paper we evaluate a strongly regularised version of the Hastings-Levitov model HL(α) for 0≤α<2. We consider the scaling limit of the model under capacity rescaling. We first consider the case where α=0 and show that the limiting structure of the cluster is not a disk, unlike in the small-particle limit. Then when 0<α<2 we show that under the same rescaling the cluster approaches a disk and we analyse the fluctuations. In Chapter 3, we present results from a second paper [LT21b]. In this paper we study the anisotropic version of the Hastings-Levitov model AHL(ν). We consider the evolution of the harmonic measure on logarithmic timescales and show that there exists a logarithmic time window on which the harmonic measure flow, started from the unstable fixed point, moves stochastically from the unstable point towards a stable point. Finally, in Chapter 4, we give the conclusions of this thesis and the scope for future work.