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) similar to 1/E xi as E xi -> infinity which matches with the known asymptotics on d-regular trees. The same results for the short-long survival threshold on the Erdos-Renyi and other random graphs are shown as well.