The contact process on finite homogeneous trees revisited

We consider the contact process with infection rate λ on Tdn, the d-ary tree of height n. We study the extinction time τTdn, that is, the random time it takes for the infection to disappear when the process is started from full occupancy. We prove two conjectures of Stacey regarding τTdn. Let λ2 denote the upper critical value for the contact process on the infinite d-ary tree. First, if λ < λ2, then τTdn divided by the height of the tree converges in probability, as n→∞, to a positive constant. Second, if λ> λ2, then logE[τTdn] divided by the volume of the tree converges in probability to a positive constant, and τTdn/E[τTdn] converges in distribution to the exponential distribution of mean 1.