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Metastable densities for the contact process on power law random graphs. (English) Zbl 1281.82018
Summary: We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of S. Chatterjee and R. Durrett [Ann. Probab. 37, No. 6, 2332–2356 (2009; Zbl 1205.60168)], who showed that for arbitrarily small infection parameter \(\lambda\), 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 \(\lambda\) is kept fixed and \(n\) tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.

MSC:
82C22 Interacting particle systems in time-dependent statistical mechanics
05C80 Random graphs (graph-theoretic aspects)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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