CRITICAL VALUE ASYMPTOTICS FOR THE CONTACT PROCESS ON RANDOM GRAPHS

Danny Nam; Oanh Nguyen; Allan Sly

doi:10.1090/tran/8399

CRITICAL VALUE ASYMPTOTICS FOR THE CONTACT PROCESS ON RANDOM GRAPHS

Danny Nam, Oanh Nguyen, Allan Sly

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

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.

Original language	English (US)
Pages (from-to)	3899-3967
Number of pages	69
Journal	Transactions of the American Mathematical Society
Volume	375
Issue number	6
DOIs	https://doi.org/10.1090/tran/8399
State	Published - Jun 1 2022

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1090/tran/8399

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title = "CRITICAL VALUE ASYMPTOTICS FOR THE CONTACT PROCESS ON RANDOM GRAPHS",

abstract = "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-R{\'e}nyi and other random graphs are shown as well.",

author = "Danny Nam and Oanh Nguyen and Allan Sly",

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AU - Nguyen, Oanh

AU - Sly, Allan

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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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CRITICAL VALUE ASYMPTOTICS FOR THE CONTACT PROCESS ON RANDOM GRAPHS

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