Survival of contact processes on the hierarchical group. (English) Zbl 1191.82028

Summary: We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.


82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C28 Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics
92D30 Epidemiology
60J25 Continuous-time Markov processes on general state spaces
Full Text: DOI arXiv


[1] Bleher, P. M.; Major, P., Critical phenomena and universal exponents in statistical physics. On Dyson’s hierarchical model, Ann. Probab., 15, 2, 431-477 (1987) · Zbl 0628.60101
[2] Brydges, D.; Evans, S. N.; Imbrie, J. Z., Self-avoiding walk on a hierarchical lattice in four dimensions, Ann. Probab., 20, 1, 82-124 (1992) · Zbl 0742.60067
[3] Dawson, D.A.: Stochastic models of evolving information systems. In: CMS Conference Proceedings, vol. 26, pp. 1-14 (Ottawa, Canada, 1998). AMS, Providence (2000) · Zbl 0951.60090
[4] Dawson, D. A.; Greven, A., Hierarchical models of interacting diffusions: multiple time scale phenomena, phase transition and pattern of cluster-formation, Probab. Theory Relat. Fields, 96, 435-473 (1993) · Zbl 0794.60101
[5] Dawson, D. A.; Gorostiza, L. G., Percolation in a hierarchical random graph, Commun. Stoch. Anal., 1, 1, 29-47 (2007) · Zbl 1328.60210
[6] Donnelly, P.; Kurtz, T. G., A countable representation of the Fleming-Viot measure-valued diffusion, Ann. Probab., 24, 2, 698-742 (1996) · Zbl 0869.60074
[7] Donnelly, P.; Kurtz, T. G., Genealogical processes for Fleming-Viot models with selection and recombination, Ann. Appl. Probab., 9, 4, 1091-1148 (1999) · Zbl 0964.60075
[8] Durrett, R.: Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks/Cole, Pacific Grove (1988) · Zbl 0659.60129
[9] Dyson, F. J., Existence of a phase transition in a one-dimensional Ising ferromagnet, Commun. Math. Phys., 12, 91-107 (1969) · Zbl 1306.47082
[10] Fleischmann, K.; Swart, J. M., Trimmed trees and embedded particle systems, Ann. Probab., 32, 3, 2179-2221 (2004) · Zbl 1048.60063
[11] Hara, T.; Hattori, T.; Watanabe, H., Triviality of hierarchical Ising model in four dimensions, Commun. Math. Phys., 220, 1, 13-40 (2001) · Zbl 1001.82044
[12] Holley, R. A.; Liggett, T. M., The survival of the contact process, Ann. Probab., 6, 198-206 (1978) · Zbl 0375.60111
[13] Kurtz, T. G., Martingale problems for conditional distributions of Markov processes, Electron. J. Probab., 3, 9, 1-29 (1998) · Zbl 0907.60065
[14] Liggett, T. M., Interacting Particle Systems (1985), New York: Springer, New York · Zbl 0559.60078
[15] Liggett, T. M., The survival of one-dimensional contact processes in random environments, Ann. Probab., 20, 696-723 (1992) · Zbl 0754.60126
[16] Liggett, T. M., Improved upper bounds for the contact process critical value, Ann. Probab., 23, 697-723 (1995) · Zbl 0832.60093
[17] Liggett, T. M., Stochastic Interacting Systems: Contact, Voter and Exclusion Process (1999), Berlin: Springer, Berlin · Zbl 0949.60006
[18] Rogers, L. C.G.; Pitman, J. W., Markov functions, Ann. Probab., 9, 4, 573-582 (1981) · Zbl 0466.60070
[19] Sawyer, S.; Felsenstein, J., Isolation by distance in a hierarchically clustered population, J. Appl. Probab., 20, 1-10 (1983) · Zbl 0514.92013
[20] Swart, J.M.: Extinction versus unbounded growth. Habilitation Thesis of the University Erlangen-Nürnberg, ArXiv:math/0702095v1 (2007)
[21] Swart, J.M.: The contact process seen from a typical infected site. J. Theor. Probab. (2008). doi:doi:10.1007/s10959-008-0184-4 (ArXiv:math.PR/0507578v5) · Zbl 1175.60085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.