Scaling limits of random growth models

Summary

Random growth processes occur widely in the physical world. One of the most well-known, yet notoriously difficult, examples is diffusion-limited aggregation (DLA) which models e.g. mineral deposition. This process is usually initiated from a cluster containing a single "seed" particle, to which successive particles then attach themselves. One approach to mathematically modelling planar random growth seeded from a single particle is to take the seed particle to be the unit disk and to represent the randomly growing clusters as compositions of conformal mappings of the exterior disk. In 1998, Hastings and Levitov proposed a family of models using this approach, which includes a version of DLA. These models appear to exhibit a phase transition from disk-like clusters to random fractal clusters. This project will explore variants of the Hastings-Levitov family with the aim of establishing a mechanism for the breakdown of isotropic behaviour.

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