### The contact process on random d-regular graphs, static and dynamic

Probability Seminar

4th May 2022, 4:00 pm – 5:00 pm

Online, Zoom

We consider the contact process on random d−regular graphs, briefly presenting earlier work on the case where the graph is static, and focusing on more recent work where the graph evolves simultaneously with (and independently of) the contact process. In both cases, the analysis involves fixing the infection rate of the contact process and the graph parameters, taking the number of vertices N to infinity and studying the asymptotic behaviour of τ_N, the time it takes for the infection to disappear. Concerning the static graph, we prove that this behavior undergoes a phase transition: there is a value λ_c such that τ_N is of order log(N) if λ < λ_c, whereas τ_N grows exponentially with N if λ > λ_c. The latter situation is called the metastable regime. Turning to the dynamic graph setting, our choice of graph evolution is a Markovian edge−switching mechanism, whose rate is chosen so that the evolving local landscape seen by a fixed vertex approaches a limiting dynamic graph process. We again show the existence of a metastable regime for the contact process on these graphs, and notably, we show that this regime occurs for values of λ that would be subcritical in the static graph. Joint work with Jean−Christophe Mourrat (static graphs) and with Gabriel Baptista da Silva and Roberto Imbuzeiro Oliveira (dynamic graphs).

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