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Inhomogeneous contact processes on trees

Authors:	C Chris Wu

Institution:	(1) Department of Mathematics, Penn State University, Beaver Campus, 15061 Monaca, Pennsylvania

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 _c ^σ 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 _c ^σ andϑ(δ, λ _c( $\theta (\delta , \lambda _c (\mathbb{T}_k )) > 0$ ) > 0 whenδ <δ _c ^σ ; 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).

Keywords:	Contact process inhomogeneity trees
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