Connectivity functions for random walk percolation models

Historically, it has been of great interest and practical relevance to study spreading phenomena of a fluid (water, pollutants,...) through a medium, and many Mathematical models have been introduced for this purpose. M. Sahimi (1994) assigns two possible categories to such models: diffusion processes and percolation processes. In the first, the assumption is that the source of randomness lies with the fluid, whereas in the second, the motion of the fluid is wholly determined by the structure of the medium, which itself is considered to be randomly generated. The point of view taken by percolation processes was initially studied and contextualised by S. R. Broadbent and J. M. Hammersley (1957), and has since led to further exploration of various random geometries such as the Bernoulli percolation model or Gaussian Free Field (GFF) excursion sets. The latter give rise to new and exciting universality classes, characterized by the presence of long-range correlations between local observables, which decay polynomially with the distance.

All these models exhibit what is called a phase transition, presumably of second order and characterized by critical exponents, which takes place as the density changes from a subcritical phase, consisting of small connected islands only, to a supercritical phase comprising an infinite cluster.

The goal of this research project is to further develop our mathematical understanding of long-range correlated percolation models. One such model on the integer lattice is called the random interlacements, and was first introduced by A.-S. Sznitman (2010) to deal with probabilistic covering and fragmentation problems attached to random walk traces. Such models were considered e.g. by M.J. Brummelhuis and H.J. Hillhorst (1991) as models of corrosion.

By further delving into this area, this project aims to solve several open problems relating to the interlacements and will progress understanding of large deviation properties of the simple random walk. Of particular interest is the behaviour of the truncated two-point function for the vacant set of the walk on the torus at suitable timescales. In determining the precise leading asymptotic behaviour for this observable, both in sub- and supercritical regime, this project aims to tackle a long-lasting and central question in this area. It is likely that the methodology and results obtained for this purpose will be adaptable to other scenarios involving long-range dependence.

This project falls within the EPSRC Mathematical Analysis, Probability and Mathematical Physics research areas, in particular relating to random structures, stochastic analysis and critical phenomena.

Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors

MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY

The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.

FINANCIAL SERVICES and GOVERNMENT

The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.

DATA SCIENCE:

Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs