Interactions and Information Transmission in Diffusing Agents

This project falls within the EPSRC 'Mathematical Sciences' research area.
Transmission events occur on the microscopic scale through short-range interaction events between a pair of individuals. With many individuals in a complex system, these transmission events lead to the emergence of patterns on the macroscopic scale. Examples include the sharing of food locations in socially living animals, the passing of information between search robots or the spreading of infectious pathogens through a population. The locations and frequency of these interactions are inherently stochastic processes. As such, the field of statistical physics lends itself perfectly to the study of such systems. By exploiting the most recent developments in the field - the resolution of a 100-year old problem in lattice random walks (LRW) [1] - it is now possible to determine the transmission probability between pairs of randomly moving individuals [2]. Through these advances, much is now known about the behaviour of randomly moving and interacting agents in d-dimensional, hypercubic, bounded lattices. However, there is still much to learn about both movement and interactions dynamics on other, less standard lattices structures. These include lattices with multiple heterogenous internal states, directed lattices and other 2-dimensionally tessellating lattices - triangular and hexagonal.
Currently, many studies of interacting agents, especially those in non-standard lattice structures rely on stochastic simulations [3], which are often highly computationally expensive. Through advancements in analytic solutions of occupation probabilities of a random walker in non-standard bounded lattices, this PhD aims to broaden the scope of domains in which studies of interaction transmission dynamics are able to be obtained analytically. This is likely to be achieved through methods such as the deployment of a technique to deal with inhomogeneous sites in a lattice, the defect technique [4] and the method of images [5], a technique borrowed from electrostatics used to bound spatial dynamics.
Currently my attention lies in obtaining analytical solutions in hexagonal and triangular lattices. By extending techniques deployed to obtain state-of-the-art results for analytic formulas representing the Lattice Random Walk occupation probability, the propagator or Lattice Green's Function, in bounded Euclidean Lattices [1], I am able to obtain previously unknown analytic propagators in Hexagonal domains. I do this by modelling the dynamics using the so called Her's Co-ordinate system [6]. Not only is this a significant theoretical advance - known analytical results modelling hexagonal LRWs all rely on mappings to standard Euclidean lattices [7] - propagators in true bounded hexagonal space are of interest as previous computational studies [8] show that the territorial structure of a population of scent-marking territorial animals naturally forms a tessellated hexagonal lattice structure. This is pertinent due to the spread of Mycobacterium bovis, the causative bacterium of bovine Tuberculosis (bTB) within the European Badger (Meles meles) population, an issue costing the UK taxpayer millions of pounds every year [9]. Hence, with an analytic framework that elucidates interaction and transmission dynamics in non-standard lattices, one is able to tackle such problems from a brand new, computationally inexpensive angle.
[1] L. Giuggioli; Phys. Rev. X; 2020
[2] L. Giuggioli, S. Sarvaharman; Submitted; 2022
[3] T. Stutz, et. al.; PLoS One; 2021
[4] V. M. Kenkre; In Memory Functions, Projection Operators, and the Defect Technique; 2021
[5] E. Montroll, B. J. West; In Studies in Statistical Mechanics: Fluctuation Phenomena, 1979
[6] I. Her; Proc ASME DTC; 1992
[7] E. Montroll; Math. Phys.; 1968
[8] A. Heiblum Robles, L. Giuggioli; Phys. Rev. E; 2018
[9] United Kingdom Government; 2018