Metastable Densities for Contact Processes 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, who showed that for arbitrarily small infection parameter $λ$, 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 $λ$ is kept fixed and $n$ tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.

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