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