Dimers and Interaction
Lead Research Organisation:
Durham University
Department Name: Mathematical Sciences
Abstract
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.
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.
Planned Impact
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.
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.
People |
ORCID iD |
Sunil Chhita (Principal Investigator / Fellow) |
Publications
Ayyer A
(2021)
Correlations in totally symmetric self-complementary plane partitions
in Transactions of the London Mathematical Society
Ayyer A
(2023)
GOE fluctuations for the maximum of the top path in alternating sign matrices
in Duke Mathematical Journal
Beffara V
(2022)
Local geometry of the rough-smooth interface in the two-periodic Aztec diamond
in The Annals of Applied Probability
Chhita S
(2023)
On the Domino Shuffle and Matrix Refactorizations
in Communications in Mathematical Physics
Chhita S
(2021)
The domino shuffling algorithm and Anisotropic KPZ stochastic growth
in Annales Henri Lebesgue
Macera D
(2024)
Non-Lyapunov annealed decay for 1d Anderson eigenfunctions
in Journal of Mathematical Physics
Description | Correlations in TSSCPPs |
Organisation | Indian Institute of Science Bangalore |
Country | India |
Sector | Academic/University |
PI Contribution | Collaboration with Arvind Ayyer working on totally self-complimentary plane partitions (TSSCPPs) and alternating sign matrices (ASMs). |
Collaborator Contribution | The TSSCPPs are boxed plane partitions with maximal degrees of symmetry. These are known to be equinumerous with ASMs (alternating sign matrices which are equivalent to the six vertex model with domain wall boundary conditions at the ice point). Here, we find a formula for the correlations of the TSSCPPs which means that any probabilities of any local configuration can be computed. This project is hopefully a first step in a wider project. |
Impact | One paper: "Correlations in totally symmetric self-complementary plan partitions", which is in the area of Combinatorics, Mathematical physics and probability. The paper is published in Transactions of the London Mathematical Society. |
Start Year | 2020 |
Description | Fluctuations for ASMs |
Organisation | Indian Institute of Science Bangalore |
Country | India |
Sector | Academic/University |
PI Contribution | One submitted preprint with Arvind Ayyer (IISc) and Kurt Johansson (KTH) |
Collaborator Contribution | The contributions we made in this paper were to find the first results for the fluctuations at the interface for a six-vertex model with domain wall boundary conditions away from the free-fermion point. The six-vertex model is a simplified model of ice and has a deep history in statistical mechanics. The difficulty with studying the six-vertex model is that the exact formulas are too complicated. We bypassed these complications by using a "probabilistic bijection" with the uniformly random six-vertex model with domain wall boundary conditions and another model which has good formulas, providing rigorous evidence of Tracy-Widom type fluctuations of these particular models, thus reaffirming universality. |
Impact | GOE fluctuations for the maximum of the top path in alternating sign matrices ArXiv preprint: https://arxiv.org/abs/2109.02422 |
Start Year | 2021 |
Description | Fluctuations for ASMs |
Organisation | Royal Institute of Technology |
Country | Sweden |
Sector | Academic/University |
PI Contribution | One submitted preprint with Arvind Ayyer (IISc) and Kurt Johansson (KTH) |
Collaborator Contribution | The contributions we made in this paper were to find the first results for the fluctuations at the interface for a six-vertex model with domain wall boundary conditions away from the free-fermion point. The six-vertex model is a simplified model of ice and has a deep history in statistical mechanics. The difficulty with studying the six-vertex model is that the exact formulas are too complicated. We bypassed these complications by using a "probabilistic bijection" with the uniformly random six-vertex model with domain wall boundary conditions and another model which has good formulas, providing rigorous evidence of Tracy-Widom type fluctuations of these particular models, thus reaffirming universality. |
Impact | GOE fluctuations for the maximum of the top path in alternating sign matrices ArXiv preprint: https://arxiv.org/abs/2109.02422 |
Start Year | 2021 |
Description | On the domino shuffle and matrix refactorizations |
Organisation | Kunliga Tekniska Hoegskolan |
Country | Sweden |
Sector | Academic/University |
PI Contribution | We worked with Maurice Duits to find a general approach for correlations in general weighted Aztec diamonds. We found that the method that I used previously, is equivalent to the one used by Duits in a 2019. |
Collaborator Contribution | A joint collaboration; see above |
Impact | https://arxiv.org/pdf/2208.01344.pdf |
Start Year | 2021 |
Description | Rough-Smooth Interface |
Organisation | Kunliga Tekniska Hoegskolan |
Country | Sweden |
Sector | Academic/University |
PI Contribution | One submitted preprint with Kurt Johansson (KTH) and Vincent Beffara (Grenoble) |
Collaborator Contribution | The rough smooth interface appears in many 2d discrete probabilistic models such as random tiling models. This interface sits between two macroscopic regions, one of which having exponential decay and the other having polynomial decay. Previously, a specific averaging had given the presence of random matrix theory statistics at this region in the two-periodic Aztec diamond. Here, we study (for some values of the natural parameter associated to the two-periodic Aztec diamond) the local geometry at this interface. |
Impact | One paper: "Local Geometry of the rough-smooth interface in the two-periodic Aztec diamond" which is accepted into Annals of Applied Probability |
Start Year | 2018 |
Description | Rough-Smooth Interface |
Organisation | University of Grenoble |
Country | France |
Sector | Academic/University |
PI Contribution | One submitted preprint with Kurt Johansson (KTH) and Vincent Beffara (Grenoble) |
Collaborator Contribution | The rough smooth interface appears in many 2d discrete probabilistic models such as random tiling models. This interface sits between two macroscopic regions, one of which having exponential decay and the other having polynomial decay. Previously, a specific averaging had given the presence of random matrix theory statistics at this region in the two-periodic Aztec diamond. Here, we study (for some values of the natural parameter associated to the two-periodic Aztec diamond) the local geometry at this interface. |
Impact | One paper: "Local Geometry of the rough-smooth interface in the two-periodic Aztec diamond" which is accepted into Annals of Applied Probability |
Start Year | 2018 |
Description | Massolit videos on probability |
Form Of Engagement Activity | A broadcast e.g. TV/radio/film/podcast (other than news/press) |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Schools |
Results and Impact | Provided a set of mini courses for GSCE probability that is hosted by Massolit for the Edexcel curriculum. This is hosted on Massolit and subscribed to by schools, who have access to videos and educational content, as well as providing important resources for GCSE revision. |
Year(s) Of Engagement Activity | 2022 |
URL | https://www.massolit.io/courses/probability-i-pearson-edexcel-gcse-mathematics-9-1-foundation-tier |