Metastable densities for the contact process on power law random graphs

We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett (2009), who showed that for arbitrarily small infection parameter, the survival time of the process is larger than a stretched exponential function of the number of vertices, n. We obtain sharp bounds for the typical density of infected sites in the graph, as is kept fixed and n tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.