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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 infectiondisappears) as $n$ is taken to infinity. We establish a phase transitiondepending on whether $\lambda$ is smaller or larger than$\lambda_1(\mathbb{T}^d)$, the lower critical value for the contact process onthe 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 differsfrom the situation where, instead of $G_n$, the contact process is consideredon the $d$-ary tree of finite height, since in this case, the transition isknown to happen instead at the _upper_ critical value for the contact processon $\mathbb{T}^d$.

Phase transition of the contact process on random regular graphs