Conformal Approach to Modelling Random Aggregation

My research will make a major contribution to solving a long-standing problem at the interface of probability, complex analysis and mathematical physics. The focus is on planar random growth processes which grow by successive aggregation of particles. Of specific interest are Laplacian models: models for which the rate of growth of the cluster boundary is determined by its harmonic measure. These arise in a variety of physical and industrial settings, from cancer to polymer creation. Examples include:
- diffusion-limited aggregation (DLA);
- the Eden model for biological cell growth;
- dielectric-breakdown models for the discharge of lightning.

Many random growth models were originally formulated as discrete sets on a lattice. However, progress is sparse in this setting owing to the lack of available mathematical techniques. Indeed, the question of whether there exists a universal scaling limit for DLA has been an important open problem in both mathematics and physics for almost 40 years. I have recently introduced a family of Laplacian random growth models called Aggregate Loewner Evolution (ALE) in which growing clusters are constructed using compositions of conformal mappings. This family includes versions of the physically occurring models above; but also models which I have shown to be analytically tractable.

The main aim of this proposal is to establish scaling limits across all parameter ranges for the family of growth processes described by the ALE construction. Specific objectives include:
- identifying phase transitions in the large-scale geometry of the clusters;
- proving that the fluctuations lie in the Kardar-Parisi-Zhang (KPZ) universality class for certain parameter values;
- establishing the relationship between random growth and Schramm-Loewner Evolution (SLE).

My proposed methodology involves combining techniques arising in the theory of regularity structures with Loewner evolution. The development of this methodology has the potential to make significant impacts in probability and analysis.

The non-academic economic and societal impact of this proposal falls into three areas: provision of highly trained researchers with an ability to engage with different sectors; public engagement activities to inspire the next generation of researchers; and long term potential benefits to the industrial, health and environmental sectors.

The proposed project will be of considerable benefit to the group involved in the research. Tackling the problems outlined in this proposal will establish us at the forefront of the interface of probability, analysis and mathematical physics. Setting up and running this research project will be a major advance in my career and reinforce my research independence. The RA will have the experience of carrying out deep and technical research on the boundary of three areas, of being joint author on high quality publications, and of interacting and collaborating with leading researchers in all of these communities. This will help them to build a strong independent academic career and enhance their long-term employability.

There is an ongoing need to inspire the next generation to consider careers as researchers in science and mathematics. This project is an ideal means by which to do this. It provides an example of how several areas of mathematics can be combined to give insight into the behaviour of physical processes such as lightning, cell growth and mineral deposition. Furthermore, the models themselves can be explained to a general audience and the resulting images are visually appealing. We plan to engage actively with the public at a local and national level, through presentations at school and university visit days and PhD taster sessions, at science fairs such as the British Science Association's British Science Festival, and through the popular science outlets in the national media.

The Laplacian random growth models that I will study as part of this project occur in a wide variety of physical and industrial settings. The Eden model describes clusters of cells that are constrained from moving such as lichen growing on a rock, or cancer cells grown on a slide; DLA describes the aggregation of soot in a diesel engine; and dielectric-breakdown models describe the deposition of minerals such as oil in sandy substrates. In addition, these models are closely related to Hele-Shaw flow which has a number of industrial applications. Although the precise questions that I am proposing to research are focussed on expanding our fundamental understanding of these processes, it is highly plausible that once this work is exposed to appropriate researchers in the industrial, health and environmental sectors, significant applications may be developed in the longer term. Indeed, two applied problems have been identified as pathways to impact. The first is to investigate whether ALE clusters can be used to differentiate between benign and malignant melanoma. The second is to relate the parameter values of the model to properties of physical clusters grown using crystallisation techniques.