Phase transition of the contact process on random regular graphs

We consider the contact process with infection rate on a random (d+1) -regular graph with n vertices, Gₙ. We study the extinction time ₆䂸 (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 is smaller or larger than ₁ (T ᵈ), the lower critical value for the contact process on the infinite, (d+1) -regular tree: if 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.