AbstractThis thesis is concerned with the voter model and the contact process, two interacting particle systems in the sense of [Liggett, 1985]. For both systems, we are interested in the termination time of the process, called either consensus or extinction, and to bound this time in each case we shift our attention from the original process which is irreversible to a related reversible process. A reversible Markov chain is a simple random walk on a weighted graph with site-dependent stepping rates, and so there are many standard techniques for their analysis which we will apply.
The voter model is a classical interacting particle system modelling how global consensus is formed across a network, by local imitation. We analyse the time to consensus for the voter model when the underlying structure is a subcritical scale-free inhomogeneous random graph (in the sense of [Bollob´as et al., 2007]). The reason that we focus on subcritical random graphs is that, as we will see below, the behaviour observed here cannot be captured by mean-field methods. Moreover, we generalise the model to include a ‘temperature’ parameter. The interplay between the temperature and the structure of the random graph leads to a very rich phase diagram, where in the different phases different parts of the underlying geometry dominate the time to consensus. We also consider a discursive voter model, where voters discuss their opinions with their neighbours. We find a different phase diagram for this discursive model in the subcritical case, and then begin to explore discursive voter model consensus and mixing on the supercritical network. Our proofs rely on the well-known duality to coalescing random walks and a detailed understanding of the structure of the random graphs.
Finally, we prove a phase transition for the contact process (a simple model for infection without immunity) on a homogeneous random graph that is initially Erd˝osR´enyi, but reacts dynamically to the infection to try to prevent an epidemic via updating in only the infected neighbourhoods, at constant rate. Under this graph dynamic, the presence of infection can help to prevent the spread and so many monotonicity-based techniques fail but analysis is made possible nonetheless via a forest construction.
|Date of Award||20 Jan 2021|
|Supervisor||Marcel Ortgiese (Supervisor) & Peter Morters (Supervisor)|