Phase transition of the contact process on random regular graphs

Abstract : We consider the contact process with infection rate λ on a random (d + 1)-regular graph with n vertices, Gn. We study the extinction time τ Gn (that is, the random amount of time until the infection disappears) as n is taken to infinity. We establish a phase transition depending on whether λ is smaller or larger than λ 1 (T d), the lower critical value for the contact process on the infinite, (d + 1)-regular tree: if λ < λ 1 (T d), τ Gn grows logarithmically with n, while if λ > λ 1 (T d), it grows exponentially with n. This result differs from the situation where, instead of Gn, the contact process is considered on the d-ary tree of finite height, since in this case, the transition is known to happen instead at the upper critical value for the contact process on T d.
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Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2016, 21, 〈10.1214/16-EJP4476〉
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Jean-Christophe Mourrat, Daniel Valesin. Phase transition of the contact process on random regular graphs. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2016, 21, 〈10.1214/16-EJP4476〉. 〈ensl-01401885〉

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