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 λ1 on a Galton-Watson tree is strictly positive if and only if the offspring distribution ξ has an exponential tail. In this paper, we derive the first-order asymptotics of λ1 for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if ξ is appropriately concentrated around its mean, we demonstrate that λ1(ξ) ∼ 1/Eξ as Eξ → ∞, which matches with the known asymptotics on d-regular trees. The same results for the short-long survival threshold on the Erdos-Rényi and other random graphs are shown as well.
All Science Journal Classification (ASJC) codes
- Applied Mathematics