Week 07.04.2025 – 13.04.2025

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. An intriguing property of their model is a conjectured phase transition between models that converge to growing disks, and 'turbulent' non-disk like models. In previous work with Norris and Silvestri, we have shown that the global fluctuations present in these models exhibit behaviour that can be interpreted as the beginnings of a macroscopic phase transition from disks to non-disks. In this talk I will discuss work in progress with Larissa Richards in which we explore how the correlation structure of local fluctuations near the cluster boundary changes at the point of phase transition.