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The contact process on finite homogeneous trees revisited

We consider the contact process with infection rate $λ$ on $\mathbb{T}_n^d$, the $d$-ary tree of height $n$. We study the extinction time $τ_{\mathbb{T}_n^d}$, 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 $τ_{\mathbb{T}_n^d}$. Let $λ_2$ denote the upper critical value for the contact process on the infinite $d$-ary tree. First, if $λ< λ_2$, then $τ_{\mathbb{T}_n^d}$ divided by the height of the tree converges in probability, as $n \to \infty$, to a positive constant. Second, if $λ> λ_2$, then $\log \mathbb{E}[τ_{\mathbb{T}_n^d}]$ divided by the volume of the tree converges in probability to a positive constant, and $τ_{\mathbb{T}_n^d}/\mathbb{E}[τ_{\mathbb{T}_n^d}]$ converges in distribution to the exponential distribution of mean 1.