Abstract
Stochastic search plays a crucial role in many natural and artificial phenomena across scales and disciplines. Examples range from the spread of misinformation in online social networks to resource collection by animals. Quantifying the efficiency with which the search occurs has a long and rich history dating back at least as far as the start of the 20th century. Since then, the theoretical efforts have provided insights and tools to analyse search processes. However, they mostly have been restricted to one-dimensional space or highly symmetric scenarios with coarse-grained descriptions of the environment in which the search takes place. As such, these approximations or approaches are no longer sufficient to analyse highly resolved empirical observations that are coming out of the sciences in recent times. In order to take advantage of the minute details brought about by these observations, a new approach is needed.Here, we lay the foundations for that approach by constructing a modular mathematical framework based on lattice randoms walk (LRWs) to study stochastic processes. We report results ranging from fundamental problems in random walk theory to complex search processes. Starting with the former, we obtain time-dependent Green's function for two distinct LRWs: the biased lattice random walk in arbitrary dimensions and boundary conditions; and the random walk on the hexagonal lattice for periodic boundary conditions and the cylindrical boundary condition with absorbing ends. We present a unified approach to model any particle-environment interactions that affect only the movement of agents, i.e. inert interactions, ranging from permeable barriers to regions with decreased mobility. We employ our findings to study transmission events between pairwise agents. Lastly, we build on the previous results to provide general expressions for the visitation statistics and the cover times of random walks.
| Date of Award | 24 Jan 2023 |
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| Original language | English |
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| Supervisor | Luca Giuggioli (Supervisor) & Filippo Simini (Supervisor) |