Scaling limits of random growth processes
Lead Research Organisation:
Lancaster University
Department Name: Mathematics and Statistics
Abstract
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. In earlier work, Norris and Turner showed that the scaling limit of the simplest of the Hastings-Levitov models is a growing disk in the small-particle limit. This is the limit in which the capacity of each particle tends to zero while the number of particles is tuned so that the total capacity of the cluster remains constant. This PhD project will instead aim to obtain scaling limits in the limit where the whole cluster is rescaled by its total capacity.
Organisations
Publications
Liddle G
(2021)
Scaling limits and fluctuations for random growth under capacity rescaling
in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509504/1 | 30/09/2016 | 29/09/2021 | |||
| 1943420 | Studentship | EP/N509504/1 | 30/09/2017 | 30/05/2021 | George Liddle |
| Description | Random growth occurs in many real world situations. Examples of random growth of this include tumoral growth, bacterial colony growth and lightning strikes. We build mathematical models that describe such growth so that we can further understand the behaviour of the growth process. Well studied growth models include Diffusion Limited Aggregation(DLA) for mineral aggregation and the Eden model for biological cell growth. I have been researching the scaling limits of the Hastings-Levitov model which is used to describe Laplacian growth processes and allows us to vary between previous models such as DLA and the Eden model by varying a parameter alpha. The model is formed by attaching particles one by one onto the boundary of a disk to form a growing cluster. The parameter alpha allows us to vary the size of the particle being added. This model has previously been well studied in the "small particle limit" where the side of the attached particle gets smaller as we send the number of particles to infinity. In my first paper, I studied a different scaling limit where the particle size is not sent to zero but instead we rescale the whole cluster by its total size at each point. I then look at what the shape of the cluster is as the number of particles attached tends to infinity. I first looked at the case where the parameter alpha was equal to zero. I expected that in this case the scaling limit would be a disk. However, I managed to show that in fact the scaling limit isn't a disk when alpha equals zero. This is particularly interesting because it is in contrast to the small particle limit where we do get a disk in this case. I then looked at the case where the parameter alpha was between zero and two strictly. In order to study this process I needed to use a regularisation on our model because without the regularisation the cluster becomes very difficult to analyse due to its random nature. I've managed to show that the scaling limit of the regularised model does approach a disk for alpha between zero and two strictly. Furthermore, I've managed to calculate the fluctuations on this result to show how far we are away from a disk. We show the fluctuations are given by a Gaussian field. I formed a paper from these results which has been accepted for publication in Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques. My current research is focused on a variation of the model called the Anisotropic Hastings-Levitov model. In this model the particles are not attached uniformly onto the boundary but according to some other probability measure. This model has previously been studied in the small-particle limit up to compact timescales. However, I have studied the model on longer logarithmic timescales and have established the behaviour of the model on these timescales. I have written a second paper which details this research which I expect to submit for publication soon. |
| Exploitation Route | There has been a large amount of research into random growth processes. From an academic perspective one might be interested in how to study the model Hastings-Levitov without the need for a regularisation for alpha greater than zero or how the model behaves for different values of alpha that haven't been studied. From a non-academic perspective these models are formed to model real world processes and as such results on the model may tell us something about how a particular growth process behaves. |
| Sectors | Other |
| URL | https://www.lancaster.ac.uk/maths/people/george-liddle |
| Description | Institute of Mathematical Statistics Hannan Graduate Student Travel Award |
| Amount | $500 (USD) |
| Organisation | Institute of Mathematical Statistics |
| Sector | Charity/Non Profit |
| Country | United States |
| Start | 04/2020 |
| End | 12/2022 |