Phase transition of the contact process on random regular graphs

Abstract

We consider the contact process with infection rate $\lambda$ on a random $(d+1)$-regular graph with $n$ vertices, $G_n$. We study the extinction time ${\tau }_{G_n}$ (that is, the random amount of time until the infection disappears) as $n$ is taken to infinity. We establish a phase transition depending on whether $\lambda$ is smaller or larger than ${\lambda }_1({\mathbb{T}}^d)$, the lower critical value for the contact process on the infinite, $(d+1)$-regular tree: if $\lambda <{\lambda }_1({\mathbb{T}}^d)$, ${\tau }_{G_n}$ grows logarithmically with $n$, while if $\lambda >{\lambda }_1({\mathbb{T}}^d)$, it grows exponentially with $n$. This result differs from the situation where, instead of $G_n$, the contact process is considered on the $d$-ary tree of finite height, since in this case, the transition is known to happen instead at the _upper_ critical value for the contact process on ${\mathbb{T}}^d$.

16 pages

References in corpus (1)

• Exponential extinction time of the contact process on finite graphs

Cited by in corpus (1)

• A Metastability Result for the Contact Process on a Random Regular Graph