Metastability for the contact process on the configuration model with infinite mean degree

We study the contact process on the configuration model with a power law degree distribution, when the exponent is smaller than or equal to two. We prove that the extinction time grows exponentially fast with the size of the graph and prove two metastability results. First the extinction time divided by its mean converges in distribution toward an exponential random variable with mean one, when the size of the graph tends to infinity. Moreover, the density of infected sites taken at exponential times converges in probability to a constant. This extends previous results in the case of an exponent larger than $2$ obtained in \cite{CD,MMVY,MVY}.
Proposition 6.2 replaced by a weaker version (after a gap in its proof was mentioned to us by Daniel Valesin). Does not affect the two main theorems of the paper