TY - JOUR

T1 - CRITICAL VALUE ASYMPTOTICS FOR THE CONTACT PROCESS ON RANDOM GRAPHS

AU - Nam, Danny

AU - Nguyen, Oanh

AU - Sly, Allan

N1 - Funding Information:
Received by the editors October 31, 2019, and, in revised form, January 26, 2021. 2020 Mathematics Subject Classification. Primary 60K35, 05C80. The first author is supported by a Samsung scholarship. The third author is supported by NSF grant DMS-1352013, Simons Investigator grant and a MacArthur Fellowship.
Publisher Copyright:
© 2022 American Mathematical Society

PY - 2022/6/1

Y1 - 2022/6/1

N2 - 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.

AB - 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.

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U2 - 10.1090/tran/8399

DO - 10.1090/tran/8399

M3 - Article

AN - SCOPUS:85130184309

SN - 0002-9947

VL - 375

SP - 3899

EP - 3967

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 6

ER -