EPSRC Reference: 
EP/T027940/2 
Title: 
Conformal Approach to Modelling Random Aggregation 
Principal Investigator: 
Turner, Professor AG 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Statistics 
Organisation: 
University of Leeds 
Scheme: 
Standard Research 
Starts: 
01 June 2022 
Ends: 
31 March 2024 
Value (£): 
267,943

EPSRC Research Topic Classifications: 
Mathematical Analysis 
Statistics & Appl. Probability 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
My research will make a major contribution to solving a longstanding 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:
 diffusionlimited aggregation (DLA);
 the Eden model for biological cell growth;
 dielectricbreakdown 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 largescale geometry of the clusters;
 proving that the fluctuations lie in the KardarParisiZhang (KPZ) universality class for certain parameter values;
 establishing the relationship between random growth and SchrammLoewner 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.

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Organisation Website: 
http://www.leeds.ac.uk 