EPSRC Reference: 
EP/T004290/1 
Title: 
Dimers and Interaction 
Principal Investigator: 
Chhita, Dr S 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematical Sciences 
Organisation: 
Durham, University of 
Scheme: 
EPSRC Fellowship 
Starts: 
01 January 2020 
Ends: 
31 December 2024 
Value (£): 
770,628

EPSRC Research Topic Classifications: 
Logic & Combinatorics 
Mathematical Analysis 
Mathematical Physics 


EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The overarching theme of this proposal is based on studying local random growth in one and two dimensions. Examples of this random growth include watching the accumulation of ice particles on a car windscreen as snow falls onto it, observing the slow combustion of the burned interface of a piece of paper, or crystal deposition in a corner of a room. A common feature of all these models is that there is a microscopic interface evolving in time in a prescribed random manner. Kardar, Parisi and Zhang, in the 1980's, introduced a mathematical equation to explain the evolution of the microscopic interface, which is now known as the "KPZ equation". For many years, this equation was mathematically illposed. It was only recently that the right mathematical advances were made to understand this equation rigorously in one dimension (recognised by a Fields Medal in 2014). However, very little is known in two or higher dimensions about these random growth models.
A fascinating part of the story is that some onedimensional random growth models can be studied using exact formulas. Acquiring these formulas is a nontrivial task and the usual approach is to exploit the (algebraic) structures inherent in the model.
Once these formulas have been obtained, very delicate computations lead to understanding the limiting random behaviour of these models. This is a difficult problem and has attracted a wide range of mathematicians with various different backgrounds. Amazingly, the random limiting behaviour observed in these types of models have the same random limiting behaviour observed in the KPZ equation, leading to a plethora of different models similar limiting features. These models are grouped under one umbrella known as the "KPZ Universality Class". One of the main challenges of this area is to branch away from models with exact formulas. One expects that the exact formulas, which are extremely powerful leading to deep results, are manifestation of the mathematical model. For instance, very little is known about models having small perturbations away from the models with exact formulas. This is a major challenge for the scientific progression of this field.
The goal of the proposal is based on extending the understanding of this universality class in both one and two dimensions. Indeed, controlling microscopic fluctuations has many important applications, for example chip printing in nanotechnology. The PI has previously been successful in showing a major extension of this universality class by studying the socalled roughsmooth transition and has established dynamics in a particular model in the twodimensional KPZ Universality class. More concretely,
1. the PI plans to establish a robust framework using both algebraic and probabilistic techniques to investigate universality of this transition within the KPZ universality class,
2. make further steps in understanding the twodimensional KPZ Universality class.

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