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 τGn (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 λ1(Td), the lower critical value for the contact process on the infinite, (d+ 1)-regular tree: if λ < λ1(Td), τGn grows logarithmically with n, while if λ> λ1(Td), 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 Td