Dimers and Interaction

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 ill-posed. 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 one-dimensional 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 so-called rough-smooth transition and has established dynamics in a particular model in the two-dimensional 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 two-dimensional KPZ Universality class.

This proposal has both short and long term impacts on society and the economy. These impacts include the training of highly specialised personnel, knowledge generation, outreach and developing a toolkit that has importance to other disciplines based on historical connections of applied probability to other scientific disciplines.

1. Training of highly specialised personnel:
The proposal is centred around hiring and further training of two postdoctoral researchers. Further development of these individuals in a state of the art discipline of mathematics (applied probability, algebraic combinatorics, integrable systems) means that these individuals will be highly sought after in a wide range of industries. In particular, these individuals will gain the required professional skills that are transferable to many sectors of industry, including education, technology and financial industries.

2. Knowledge:
As part of the proposal, the PI will continue his work in knowledge generating and disseminating activities. These include publishing research outputs in the top journals of probability and mathematics, giving talks in international and national conferences, and strengthening the PI's existing collaboration network as well as forging new collaborations. During the period of funding, the PI will organise a scientific workshop bringing many key international mathematicians to Durham University. These efforts are important in safeguarding the health of probabilistic research in the UK, which has had many important historical contributions to the financial and technological industries, as well as maintaining the UK's competitiveness in this area of research.

3. Presentation to the wider community:
An important part of research is to foster and encourage dialogue between probabilistic research and the general public. Such a conversation is hugely important as it ensures that the public is up to date with mathematical research and understands the huge role it plays in today's world. The PI will aim to achieve this using public lectures and look for new opportunities to engage the public.

4. Applications to other fields:
Research in probability has enjoyed numerous applications to other fields that have had considerable impacts on the economy, society and general well-being of people. These applications have also influenced research in probability by giving new lines of research problems and challenges for researchers to explore. Two applications of probability that are closely related to the PI's area of expertise are the use of "determinantal point processes" in machine learning, and "statistical mechanics" in its role in understanding complex systems. The latter has had some success in understanding the brain. The PI will look to explore and exploit these types of connections to other fields throughout the grant funding period.