Phase transition of the contact process on random regular graphs

Overview & Synopsis

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.