Singular scaling limits in a planar random growth model
26th January 2018, 3:30 pm – 4:30 pm
Main Maths Building, SM4
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 and conjectured that this family exhibits a phase transition from clusters that converge to disks to clusters that converge to random irregular shapes. In this talk we will introduce a natural extension of the Hastings-Levitov family. For this model, we are able to prove that both singular and absolutely continuous scaling limits can occur. Specifically, we can show that for certain parameter values, the resulting cluster can be shown to converge to a randomly oriented one-dimensional slit, whereas for other parameter values the scaling limit is a deterministically growing disk.
This is based on work in progress with Alan Sola (Stockholm) and Fredrik Viklund (KTH).