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Hinsen, Alexander; Jahnel, Benedikt; Cali, Elie; Wary, Jean-Philippe

Phase transitions for chase-escape models on Poisson-Gilbert graphs. (English) Zbl 1434.60194

Electron. Commun. Probab. 25, Paper No. 25, 14 p. (2020).
Summary: We present results on phase transitions of local and global survival in a two-species model on Poisson-Gilbert graphs. Initially, there is an infection at the origin that propagates on the graph according to a continuous-time nearest-neighbor interacting particle system. The graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.

Cited in 4 Documents

MSC:

60J25 Continuous-time Markov processes on general state spaces
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments

Keywords:

interacting particle systems; random graphs; survival; extinction; percolation; Boolean model
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References:

[1] [And92] E.D. Andjel. Survival of multidimensional contact process in random environments. Bol. Soc. Brasil. Mat. (N.S.), 23(1-2):109-119, 1992. · Zbl 0768.60094 · doi:10.1007/BF02584814
[2] [CD09] S. Chatterjee and R. Durrett. Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Probab., 37(6):2332-2356, 2009. · Zbl 1205.60168 · doi:10.1214/09-AOP471
[3] [dALRR15] G.F. de Arruda, E. Lebensztayn, F.A. Rodrigues, and P.M. Rodríguez. A process of rumour scotching on finite populations. R. Soc. Open Sci., 2(9), 2015.
[4] [DJT18] R. Durrett, M. Junge, and S. Tang. Coexistence in chase-escape. Electron. Commun. Probab., 25(22):1-14, 2020. · Zbl 1434.60285 · doi:10.1214/20-ECP302
[5] [Dur10a] R. Durrett. Random Graph Dynamics, Volume 20 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. · Zbl 1223.05002
[6] [Dur10b] R. Durrett. Some features of the spread of epidemics and information on a random graph. Proc. Natl. Acad. Sci. USA, 107(10):4491-4498, 2010.
[7] [Kor05] G. Kordzakhia. The escape model on a homogeneous tree. Electron. Commun. Probab., 10(12):113-124, 2005. · Zbl 1111.60075 · doi:10.1214/ECP.v10-1140
[8] [Kor15] I. Kortchemski. A predator-prey sir type dynamics on large complete graphs with three phase transitions. Stochastic Process. Appl., 125(3):886-917, 2015. · Zbl 1334.60181 · doi:10.1016/j.spa.2014.10.005
[9] [Kor16] I. Kortchemski. Predator-prey dynamics on infinite trees: a branching random walk approach. J. Theoret. Probab., 29(3):1027-1046, 2016. · Zbl 1349.92125 · doi:10.1007/s10959-015-0603-2
[10] [Lig85] T.M. Liggett. Interacting Particle Systems. Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 1985. · Zbl 0559.60078
[11] [Lig92] T.M. Liggett. The survival of one-dimensional contact processes in random environments. Ann. Probab., 20(2):696-723, 1992. · Zbl 0754.60126 · doi:10.1214/aop/1176989801
[12] [LSS97] T.M. Liggett, R.H. Schonmann, and A.M. Stacey. Domination by product measures. Ann. Probab., 25(1):71-95, 1997. · Zbl 0882.60046 · doi:10.1214/aop/1024404279
[13] [MMVY16] T. Mountford, J.-C. Mourrat, D. Valesin, and Q. Yao. Exponential extinction time of the contact process on finite graphs. Stochastic Process. Appl., 126(7):1974-2013, 2016. · Zbl 1346.82023 · doi:10.1016/j.spa.2016.01.001
[14] [MR96] R. Meester and R. Roy. Continuum Percolation. Cambridge University Press, Cambridge, 1996. · Zbl 0858.60092
[15] [MS16] L. Ménard and A. Singh. Percolation by cumulative merging and phase transition for the contact process on random graphs. Ann. Sci. Éc. Norm. Supér. (4), 49(5):1189-1238, 2016. · Zbl 1364.60137
[16] [MV16] J.-C. Mourrat and D. Valesin. Phase transition of the contact process on random regular graphs. Electron. J. Probab., 21(31):1-17, 2016. · Zbl 1342.82104 · doi:10.1214/16-EJP4476
[17] [MVY13] T. Mountford, D. Valesin, and Q. Yao. Metastable densities for the contact process on power law random graphs. Electron. J. Probab., 18(103):1-36, 2013. · Zbl 1281.82018 · doi:10.1214/EJP.v18-2512
[18] [Pen91] M.D. Penrose. On a continuum percolation model. Adv. in Appl. Probab., 23(3):536-556, 1991. · Zbl 0729.60106 · doi:10.2307/1427621
[19] [PP96] M.D. Penrose and A. Pisztora. Large deviations for discrete and continuous percolation. Adv. in Appl. Probab., 28(1):29-52, 1996. · Zbl 0853.60085
[20] [TKL18] S.
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