Random Fractals
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
The proposed research is in Pure Mathematics, namely on the border of Complex Analysis, Probability, and Mathematical Physics. I will mostly study random fractal structures and use them to study important models of Statistical Physics. Fractals are self-similar structures that look the same at all places and on all scales. They appear everywhere in nature, notable examples are Romanesco broccoli, shorelines, lightnings, and sea shells. They also appear in many models of statistical physics that describe such phenomena as magnetism, flow of fluid through porous matter, spread of epidemics, robustness of networks and many others. One of the fundamental parameters of fractal structures is the dimension. Roughly speaking it describes the growth rate of the fractal structures. For example a straight interval of diameter N, which is made of the particles of unit size, contains N particles. We say that the interval has dimension 1. The square of size N contains N squared particles and has dimension 2. The characteristic property of fractal structures is that the structure of size N contains approximately N^d particles where 1
Planned Impact
The proposed research lies on the border of several areas of fundamental research: complex analysis, probability, and mathematical physics. As this is the case for most of the research in pure mathematics, the main impact will be of academic nature and the main beneficiaries will be mathematicians and physicists working in neighbouring areas of research. Besides this pure academic impact there are two other ways how the proposed project could have societal and economical impact. During the project I will train two post-docs, one or two DPhil students and several M.Sc and undergraduate students who will contribute to the highly skilled workforce. The second impact is a long-term one. Most of the research is related to the models of mathematical physics. In the long term the pure research on these problems might lead to the technological advances based on the better understanding of the physics behind these models.
People |
ORCID iD |
| Dmitry Belyaev (Principal Investigator / Fellow) |
Publications
Beliaev
(2019)
Conformal Maps And Geometry
Beliaev D
(2021)
Mean conservation of nodal volume and connectivity measures for Gaussian ensembles
in Advances in Mathematics
Beliaev D
(2018)
Volume distribution of nodal domains of random band-limited functions.
in Probability theory and related fields
Beliaev D
(2020)
On the number of excursion sets of planar Gaussian fields
in Probability Theory and Related Fields
Beliaev D
(2021)
Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials
in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Beliaev D
(2020)
A covariance formula for topological events of smooth Gaussian fields
in The Annals of Probability
Beliaev D
(2016)
Integral means spectrum of whole-plane SLE
Beliaev D
(2017)
Integral Means Spectrum of Whole-Plane SLE
in Communications in Mathematical Physics
Beliaev D
(2017)
Discretisation schemes for level sets of planar Gaussian fields
Beliaev D
(2022)
Intermediate and small scale limiting theorems for random fields
in Communications in Number Theory and Physics
| Description | Complicated random structures appear in many areas of mathematics, physics, engineering, chemistry, biology and many other areas of science. The main goal of the project was to study several important classes of random clusters. The main progress happened in the study of level sets of Gaussian fields. A lot of spatial data that appears as the result of the interaction of many forces can be effectively modelled by these fields. For example, these fields were used starting from the 1940s to model ocean waves. We made great progress in understanding the large scale behaviour of level sets of such field. In particular, we now understand much better the connection between the level sets and percolation. We know that the number of the level sets depends in a nice way on the level and we managed to estimate the variance of the number of level sets, which is the first result of this kind. We also know that the large connectivity properties exhibit a sharp phase transition at a certain level. In dimension 2 this is the zero level. We also know that at the critical level the probability that there is a level set which crosses a given domain is uniformly bounded independently of the size of the domain (but depends on its shape). The second main direction of research was the study of the growing particle clusters where the main mean of transport is diffusion. Clusters of this type appear in electrodeposition, mineral deposition, bacterial growth etc. One of the main mathematical models describing such clusters is the so-called Diffusion Limited Aggregation (DLA) model. This model is notoriously hard to analyse analytically and there are virtually no theoretical results about its growth or fractal structure. On the other hand, there are many numerical results about its behaviour. Almost all of these results used the growth rate to estimate the fractal dimension. We have shown numerically that on the square lattice this assumption is not valid since the clusters have very strong anisotropic growth which dominates the fractal growth. This is a new paradigm in the study of DLA and shed new light on the difference between lattice and off-lattice versions of DLA. The expertise gained while working on the main objectives turned out to be useful in seemingly unrelated areas of research. In particular, it led to a successful collaboration with biochemists and mathematical biologists. |
| Exploitation Route | The main beneficiaries of the project are other researchers working on these and related problems. We envision that many of our results will be used by other mathematicians to achieve further progress in the study of the geometry of Gaussian fields, Diffusion Limited Aggregation and Schramm-Loewner Evolution. Some of the results are already actively used and cited by other researchers. We are actively trying to engage with scientists that use either Gaussian fields or Aggregation models in their research. Some of this engagement has already lead to a collaboration with end-users. But we expect that there will be further progress in this direction. In particular, we envision that our results will be used in the analysis of the geometry of the Cosmic Microwave Background Radiation (CMBR) and in data analysis. |
| Sectors | Other |
| Description | Biochemistry |
| Organisation | University of Oxford |
| Department | Wolfson College |
| Country | United Kingdom |
| Sector | Charity/Non Profit |
| PI Contribution | This was a collaboration with a group of Prof Sansom from the Department of Biochemistry, University of Oxford. My contribution was in the development of the mesoscale model which captures the principle behaviour of clustering proteins but is more amenable to simulation than the previously used model. |
| Collaborator Contribution | Below is the list of contributions: M.C., A.L.D. and M.S.P.S. designed the simulation studies. M.C. and A.L.D. conducted the molecular dynamics simulations and analysis. M.C., A.L.D., P.R., J.H. and T.R. developed methods for analysis of clustering and diffusion of membrane proteins in simulations. M.C., A.L.D., D.B. and B.H. developed and implemented the mesoscale model. P.R., O.B., J.P. and C.K. designed the experiments. O.B. and P.R. conducted all experiments and data analysis. P.R. purified the proteins used in the study and labelled colicins with fluorophores. M.C., A.L.D. and M.S.P.S. drafted the paper with assistance from C.K., P.R. and T.R. |
| Impact | This collaboration was multi-disciplinary: Ben Hambly and I are from Mathematics, the rest of the team are from the Biochemistry. This collaboration led to a publication in the Nature Communications Chavent M, Duncan AL, Rassam P, Birkholz O, Hélie J, Reddy T, Beliaev D... Sansom MSP. (2018). How nanoscale protein interactions determine the mesoscale dynamic organisation of bacterial outer membrane proteins.. Nature communications, 9 (1), pp. 2846 |
| Start Year | 2015 |
| Description | Case study |
| Form Of Engagement Activity | A magazine, newsletter or online publication |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Media (as a channel to the public) |
| Results and Impact | A newsletter highlighting the research of D. Belyaev in collaboration with S. Muirhead and I. Wigman |
| Year(s) Of Engagement Activity | 2017 |
| URL | https://www.maths.ox.ac.uk/node/26411 |
| Description | Nodal structures |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | Organized a workshop where the world-leading experts interested in interactions between probability, number theory and random fields could meet and exchange ideas. |
| Year(s) Of Engagement Activity | 2020 |
| Description | Random Waves at Oxford |
| Form Of Engagement Activity | Participation in an activity, workshop or similar |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Other audiences |
| Results and Impact | Organised an international conference that brought together world-leading researchers in the area. |
| Year(s) Of Engagement Activity | 2018 |
| URL | http://people.maths.ox.ac.uk/belyaev/RandomWaveOxford/workshop.html |