Critical value asymptotics for the contact process on random graphs

Recent progress in the study of the contact process (see Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly [Ann. Probab. 49 (2021), pp. 244–286]) has verified that the extinction-survival threshold \lambda _1 on a Galton-Watson tree is strictly positive if and only if the offspring distribution \xi has an exponential tail. In this paper, we derive the first-order asymptotics of \lambda _1 for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if \xi is appropriately concentrated around its mean, we demonstrate that \lambda _1(\xi ) \sim 1/\mathbb {E} \xi as \mathbb {E}\xi \rightarrow \infty, which matches with the known asymptotics on d-regular trees. The same results for the short-long survival threshold on the Erdős-Rényi and other random graphs are shown as well.