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.