Critical value asymptotics for the contact process on random graphs

Nam, Danny; Nguyen, Oanh; Sly, Allan

doi:10.1090/tran/8399

Critical value asymptotics for the contact process on random graphs
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by Danny Nam, Oanh Nguyen and Allan Sly PDF

Trans. Amer. Math. Soc. 375 (2022), 3899-3967 Request permission

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 $\lambda _1$ 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.

References

D. Aldous and J. A. Fill, Reversible Markov chains and random walks on graphs, Unfinished monograph, recompiled 2014, available at http://www.stat.berkeley.edu/$\sim $aldous/RWG/book.html, 2002.
Carol Bezuidenhout and Geoffrey Grimmett, The critical contact process dies out, Ann. Probab. 18 (1990), no. 4, 1462–1482. MR 1071804
Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly, Survival and extinction of epidemics on random graphs with general degree, Ann. Probab. 49 (2021), no. 1, 244–286. MR 4203338, DOI 10.1214/20-AOP1451
Amir Dembo and Andrea Montanari, Gibbs measures and phase transitions on sparse random graphs, Braz. J. Probab. Stat. 24 (2010), no. 2, 137–211. MR 2643563, DOI 10.1214/09-BJPS027
Rick Durrett, Random graph dynamics, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 20, Cambridge University Press, Cambridge, 2007. MR 2271734
Rick Durrett, Probability: theory and examples, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, vol. 31, Cambridge University Press, Cambridge, 2010. MR 2722836, DOI 10.1017/CBO9780511779398
T. E. Harris, Contact interactions on a lattice, Ann. Probability 2 (1974), 969–988. MR 356292, DOI 10.1214/aop/1176996493
Xiangying Huang and Rick Durrett, The contact process on random graphs and Galton Watson trees, ALEA Lat. Am. J. Probab. Math. Stat. 17 (2020), no. 1, 159–182. MR 4057187, DOI 10.30757/alea.v17-07
Svante Janson, The probability that a random multigraph is simple, Combin. Probab. Comput. 18 (2009), no. 1-2, 205–225. MR 2497380, DOI 10.1017/S0963548308009644
Jeong Han Kim, Poisson cloning model for random graphs, International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 873–897. MR 2275710
Steven Lalley and Wei Su, Contact processes on random regular graphs, Ann. Appl. Probab. 27 (2017), no. 4, 2061–2097. MR 3693520, DOI 10.1214/16-AAP1249
Thomas M. Liggett, Multiple transition points for the contact process on the binary tree, Ann. Probab. 24 (1996), no. 4, 1675–1710. MR 1415225, DOI 10.1214/aop/1041903202
Thomas M. Liggett, Stochastic interacting systems: contact, voter and exclusion processes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 324, Springer-Verlag, Berlin, 1999. MR 1717346, DOI 10.1007/978-3-662-03990-8
Thomas M. Liggett, Interacting particle systems, Classics in Mathematics, Springer-Verlag, Berlin, 2005. Reprint of the 1985 original. MR 2108619, DOI 10.1007/b138374
Eyal Lubetzky and Allan Sly, Cutoff phenomena for random walks on random regular graphs, Duke Math. J. 153 (2010), no. 3, 475–510. MR 2667423, DOI 10.1215/00127094-2010-029
Michael Molloy and Bruce Reed, A critical point for random graphs with a given degree sequence, Proceedings of the Sixth International Seminar on Random Graphs and Probabilistic Methods in Combinatorics and Computer Science, “Random Graphs ’93” (Poznań, 1993), 1995, pp. 161–179. MR 1370952, DOI 10.1002/rsa.3240060204
Jean-Christophe Mourrat and Daniel Valesin, Phase transition of the contact process on random regular graphs, Electron. J. Probab. 21 (2016), Paper No. 31, 17. MR 3492935, DOI 10.1214/16-EJP4476
Robin Pemantle, The contact process on trees, Ann. Probab. 20 (1992), no. 4, 2089–2116. MR 1188054
Jonathon Peterson, The contact process on the complete graph with random vertex-dependent infection rates, Stochastic Process. Appl. 121 (2011), no. 3, 609–629. MR 2763098, DOI 10.1016/j.spa.2010.11.003
A. M. Stacey, The existence of an intermediate phase for the contact process on trees, Ann. Probab. 24 (1996), no. 4, 1711–1726. MR 1415226, DOI 10.1214/aop/1041903203
Remco van der Hofstad, Random graphs and complex networks. Vol. 1, Cambridge Series in Statistical and Probabilistic Mathematics, [43], Cambridge University Press, Cambridge, 2017. MR 3617364, DOI 10.1017/9781316779422
Xiaofeng Xue, Contact processes with random vertex weights on oriented lattices, ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015), no. 1, 245–259. MR 3343484
Xiaofeng Xue, Critical value for the contact process with random recovery rates and edge weights on regular tree, Phys. A 462 (2016), 793–806. MR 3531601, DOI 10.1016/j.physa.2016.06.001
Xiaofeng Xue and Yu Pan, Contact processes with random recovery rates and edge weights on complete graphs, J. Stat. Phys. 169 (2017), no. 5, 951–971. MR 3719634, DOI 10.1007/s10955-017-1898-4