Critical systems in random geometry
Lead Research Organisation:
Durham University
Department Name: Mathematical Sciences
Abstract
Random planar geometry is the study of canonical random geometrical structures arising as scaling limits from 2D statistical physics models. It aims to gain insight into the behaviour of and connections between: random curves (limits of interfaces); random fields (limits of "height functions"); and random metric measure spaces (limits of "random planar maps"). Such objects have been subjects of intense study by physicists for decades. Broadly speaking, it is conjectured that the limits of many discrete models should be essentially independent of their small-scale behaviour; hence, understanding these limits provides scope for describing entire classes of discrete systems simultaneously. On the other hand, proving such universality statements is notoriously challenging. Celebrated results include the identification of Schramm--Loewner evolution curves as scaling limits of percolation interfaces and loop erased random walks, and more recently, the "Brownian map" as the limit of random triangulations of the plane.
This proposal targets similar results in another setting, where the P.I. has recently developed several novel and exciting techniques. This setting corresponds to a particular "universality class" of statistical physics models which display notably different behaviour. This causes standard analytical techniques to break down, meaning that developing a rigorous mathematical theory presents unique challenges. As such, this regime is much less well understood. On the other hand, it is especially relevant from both a physical and mathematical perspective. For example, it is expected to describe universal extreme value behaviour associated with many models; ranging from random matrices to the Riemann-zeta function, a central object in number theory. Developing a deep understanding of the picture here is the focus of this ambitious proposal.
The broad goals of the research are: to rigorously establish conjectural properties of the main mathematical objects; to discover connections between them; and to identify scaling limits. Such results will have direct and significant consequences for open problems in several related fields. As a result, they will provide an exciting platform for the initiation of interdisciplinary collaborations between probability and other mathematical areas (such as complex analysis, number theory) as well as other subjects (such as theoretical physics and computing). Creating a strong collaborative environment between disciplines such as these has been consistently recognised as an area of key strategic importance.
In the longer term, this work will serve to exhibit the United Kingdom as a world-leading centre for research in random geometry. The subsequent expansion of a specialised group in Durham will provide a unique capability for fundamental research in this area, underpinning the UK's ability to develop novel and ground-breaking techniques in the physical sciences, and ultimately, in industry.
This proposal targets similar results in another setting, where the P.I. has recently developed several novel and exciting techniques. This setting corresponds to a particular "universality class" of statistical physics models which display notably different behaviour. This causes standard analytical techniques to break down, meaning that developing a rigorous mathematical theory presents unique challenges. As such, this regime is much less well understood. On the other hand, it is especially relevant from both a physical and mathematical perspective. For example, it is expected to describe universal extreme value behaviour associated with many models; ranging from random matrices to the Riemann-zeta function, a central object in number theory. Developing a deep understanding of the picture here is the focus of this ambitious proposal.
The broad goals of the research are: to rigorously establish conjectural properties of the main mathematical objects; to discover connections between them; and to identify scaling limits. Such results will have direct and significant consequences for open problems in several related fields. As a result, they will provide an exciting platform for the initiation of interdisciplinary collaborations between probability and other mathematical areas (such as complex analysis, number theory) as well as other subjects (such as theoretical physics and computing). Creating a strong collaborative environment between disciplines such as these has been consistently recognised as an area of key strategic importance.
In the longer term, this work will serve to exhibit the United Kingdom as a world-leading centre for research in random geometry. The subsequent expansion of a specialised group in Durham will provide a unique capability for fundamental research in this area, underpinning the UK's ability to develop novel and ground-breaking techniques in the physical sciences, and ultimately, in industry.
Organisations
- Durham University (Fellow, Lead Research Organisation)
- University of Chile (Collaboration)
- University College London (Collaboration)
- University of Bath (Collaboration)
- University of Auckland (Collaboration)
- University of Vienna (Collaboration)
- Kromek (United Kingdom) (Project Partner)
- École Polytechnique Fédérale de Lausanne (Project Partner)
Publications
Aru J
(2022)
A characterisation of the continuum Gaussian free field in arbitrary dimensions
in Journal de l'École polytechnique - Mathématiques
Aru J
(2023)
Thick points of the planar GFF are totally disconnected for all ??0
in Electronic Journal of Probability
Aru J
(2022)
Brownian half-plane excursion and critical Liouville quantum gravity
in Journal of the London Mathematical Society
Harris S
(2024)
Many-to-few for non-local branching Markov process
in Electronic Journal of Probability
Description | Free fields and related topics |
Amount | £0 (GBP) |
Funding ID | 2572437 |
Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
Sector | Public |
Country | United Kingdom |
Start | 09/2021 |
End | 03/2025 |
Description | Research Fellows Scheme |
Amount | € 0 (EUR) |
Organisation | Mathematical Research Institute of Oberwolfach |
Sector | Academic/University |
Country | Germany |
Start | 02/2023 |
End | 02/2023 |
Description | CLE growth fragmentations |
Organisation | University College London |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | Expertise and intellectual input on a project involving random geometry and branching processes. Hosting meetings in Durham. |
Collaborator Contribution | Expertise and intellectual input on a project involving random geometry and branching processes. Hosting meetings in London. |
Impact | Paper in preparation |
Start Year | 2022 |
Description | Chaos on fractals |
Organisation | University of Chile |
Country | Chile |
Sector | Academic/University |
PI Contribution | Expertise and intellectual input |
Collaborator Contribution | Expertise and intellectual input |
Impact | Papers in preparation, lecture series for postgraduate students delivered in Santiago Jan 2023 |
Start Year | 2023 |
Description | Gaussian free field, Gaussian multiplicative chaos and Liouville quantum gravity |
Organisation | University of Vienna |
Country | Austria |
Sector | Academic/University |
PI Contribution | Writing a book on Gaussian free field, Gaussian multiplicative chaos and Liouville quantum gravity |
Collaborator Contribution | Gaussian free field, Gaussian multiplicative chaos and Liouville quantum gravity |
Impact | Book in preparation |
Start Year | 2022 |
Description | Genealogies in critical branching processes |
Organisation | University of Auckland |
Department | Department of Mathematics |
Country | New Zealand |
Sector | Academic/University |
PI Contribution | Expertise and intellectual input |
Collaborator Contribution | Expertise and intellectual input |
Impact | Paper submitted for publication |
Start Year | 2022 |
Description | Genealogies in critical branching processes |
Organisation | University of Bath |
Department | Department of Mathematical Sciences |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | Expertise and intellectual input |
Collaborator Contribution | Expertise and intellectual input |
Impact | Paper submitted for publication |
Start Year | 2022 |
Description | Crash course on the Gaussian free field at University of Helsinki |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | I gave a five lecture mini series on the Gaussian free field at the University of Helsinki |
Year(s) Of Engagement Activity | 2022 |
URL | https://www.helsinki.fi/en/projects/first/events/past-events |
Description | Introductory course on the Gaussian free field at Universidad de Chile |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | Around 50 postgraduate students and academics attended a three lecture mini series I delivered at the Universidad de Chile. This sparked many questions and discussions and the beginning of some potential research projects with attending academics. Students reported benefitting from the exposition and becoming interested in the area. |
Year(s) Of Engagement Activity | 2023 |