The contact process on finite homogeneous trees revisited

We consider the contact process with infection rate λ on T_n^d, the d-ary tree of height n. We study the extinction time τ_Tn^d, 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 τ_Tn^d. Let λ₂ denote the upper critical value for the contact process on the infinite d-ary tree. First, if λ < λ₂, then τT^d _n divided by the height of the tree converges in probability, as n → ∞, to a positive constant. Second, if λ > λ₂, then log E[τ_Tn^d] divided by the volume of the tree converges in probability to a positive constant, and τ_Tn^d/E[τ_Tn^d] converges in distribution to the exponential distribution of mean 1.