The contact process on finite homogeneous trees revisited

Abstract

We consider the contact process with infection rate $\lambda$ on ${\mathbb{T}}_n^d$, the $d$-ary tree of height $n$. We study the extinction time ${\tau }_{{\mathbb{T}}_n^d}$, 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 ${\tau }_{{\mathbb{T}}_n^d}$. Let ${\lambda }_2$ denote the upper critical value for the contact process on the infinite $d$-ary tree. First, if $\lambda <{\lambda }_2$, then ${\tau }_{{\mathbb{T}}_n^d}$ divided by the height of the tree converges in probability, as $n\to \infty$, to a positive constant. Second, if $\lambda >{\lambda }_2$, then $\log \mathbb{E}[{\tau }_{{\mathbb{T}}_n^d}]$ divided by the volume of the tree converges in probability to a positive constant, and ${\tau }_{{\mathbb{T}}_n^d}/\mathbb{E}[{\tau }_{{\mathbb{T}}_n^d}]$ converges in distribution to the exponential distribution of mean 1.

22 pages, 1 figure

References in corpus (1)

• Exponential extinction time of the contact process on finite graphs