In this paper, we establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold @math for a Galton-Watson tree is strictly positive if and only if its offspring distribution @math has an exponential tail, i.e., @math , settling a conjecture by Huang and Durrett hd18 . On the random graph with degree distribution @math , we show that if @math has an exponential tail, then for small enough @math the contact process with the all-infected initial condition survives for @math -time w.h.p. (short survival), while for large enough @math it runs over @math -time w.h.p. (long survival). When @math is subexponential, we prove that the contact process w.h.p. displays long survival for any fixed @math .
@cite_1: The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival @math and (ii) when it is geometric( @math ) we have @math , where the @math are much smaller than previous estimates. We also study the critical value @math for "prolonged persistence" on graphs with @math vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known @math we give estimates on the rate of convergence. Physicists tell us that @math where @math is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.
Less is known for the contact process on general Galton-Watson trees. Recently, Huang and Durrett @cite_1 proved that on Galton-Watson trees, @math if the offspring distribution @math is subexponential. Along with Theorem , we now have the complete characterization of the existence of extinction in the contact process on Galton-Watson trees.