Dependence structures in random growth

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The focus of this project is on understanding the long-time behaviour of planar random growth processes. These arise in physical and industrial settings, from cancer to polymer creation. Such models are important, not only due to potential applications arising from their physical prevalence, but also because of connections with new and rapidly evolving areas of mathematics such as Schramm-Loewner Evolution (SLE) and regularity structures. However, as many of these models exhibit complicated long-range dependencies, mathematical analysis has yielded very little progress. One approach to modelling random growth is to represent clusters as compositions of conformal mappings. This enables the generation of clusters using computer simulation and indicates the existence of phase transitions from clusters that converge to disks, to clusters that converge to straight lines. Preliminary simulations suggest that off-critical limits, near the point of transition to straight lines, are simple paths which can be described as stochastic processes. The aim of this project is to undertake a systematic numerical study of clusters near this point of transition. The objective is to understand the dependency structure of these stochastic processes and in particular to identify a relationship between the parameter describing the cluster, and the dependency structure.