Inhomogeneous contact processes on trees

Abstract

We consider an inhomogeneous contact process on a tree$\mathbb{T}_k$ of degreek, where the infection rate at any site isλ, the death rate at any site in$S \subset \mathbb{T}_k$ isδ (with 0 <δ ⩽ 1) and that at any site in$\mathbb{T}_k - S$ is 1. Denote by$\lambda _c (\mathbb{T}_k )$ the critical value for thehomogeneous model (i.e.,δ=1) on$\mathbb{T}_k$ and byϑ(δ, λ) the survival probability of the inhomogeneous model on$\mathbb{T}_k$. We prove that whenk > 4, if$S = \mathbb{T}_\sigma$, a subtree embedded in$\mathbb{T}_k$, with 1 ⩽σ ⩽ √k, then three existsδ ^σ_c strictly between ($\lambda _c (\mathbb{T}_k )/\lambda _c (\mathbb{T}_\sigma )$) and 1 such that ($\theta (\delta ,\lambda _c (\mathbb{T}_k )) = 0$) whenδ >δ ^σ_c andϑ(δ, λ _c($\theta (\delta , \lambda _c (\mathbb{T}_k )) > 0$) > 0 whenδ <δ ^σ_c ; ifS={o}, the origin of$\theta (\delta , \lambda _c (\mathbb{T}_k )) = 0$, then$\theta (\delta , \lambda _c (\mathbb{T}_k )) = 0$ for anyδ ε (0, 1).