Probability Theory and Related Fields

, Volume 147, Issue 3–4, pp 529–563

Survival of contact processes on the hierarchical group



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.


Contact process Survival Hierarchical group Coupling Renormalization group 

Mathematics Subject Classification (2000)

Primary: 82C22 Secondary: 60K35 82C28 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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)MATHCrossRefMathSciNetGoogle Scholar
  2. 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)MATHCrossRefMathSciNetGoogle Scholar
  3. 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)Google Scholar
  4. 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)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dawson D.A., Gorostiza L.G.: Percolation in a hierarchical random graph. Commun. Stoch. Anal. 1(1), 29–47 (2007)MathSciNetGoogle Scholar
  6. 6.
    Donnelly P., Kurtz T.G.: A countable representation of the Fleming–Viot measure-valued diffusion. Ann. Probab. 24(2), 698–742 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Donnelly P., Kurtz T.G.: Genealogical processes for Fleming–Viot models with selection and recombination. Ann. Appl. Probab. 9(4), 1091–1148 (1999)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Durrett, R.: Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks/Cole, Pacific Grove (1988)Google Scholar
  9. 9.
    Dyson F.J.: Existence of a phase transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12, 91–107 (1969)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Fleischmann K., Swart J.M.: Trimmed trees and embedded particle systems. Ann. Probab. 32(3A), 2179–2221 (2004)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hara T., Hattori T., Watanabe H.: Triviality of hierarchical Ising model in four dimensions. Commun. Math. Phys. 220(1), 13–40 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Holley R.A., Liggett T.M.: The survival of the contact process. Ann. Probab. 6, 198–206 (1978)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kurtz T.G.: Martingale problems for conditional distributions of Markov processes. Electron. J. Probab. 3(9), 1–29 (1998)MATHMathSciNetGoogle Scholar
  14. 14.
    Liggett T.M.: Interacting Particle Systems. Springer, New York (1985)MATHGoogle Scholar
  15. 15.
    Liggett T.M.: The survival of one-dimensional contact processes in random environments. Ann. Probab. 20, 696–723 (1992)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Liggett T.M.: Improved upper bounds for the contact process critical value. Ann. Probab. 23, 697–723 (1995)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Liggett T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Process. Springer, Berlin (1999)MATHGoogle Scholar
  18. 18.
    Rogers L.C.G., Pitman J.W.: Markov functions. Ann. Probab. 9(4), 573–582 (1981)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sawyer S., Felsenstein J.: Isolation by distance in a hierarchically clustered population. J. Appl. Probab. 20, 1–10 (1983)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Swart, J.M.: Extinction versus unbounded growth. Habilitation Thesis of the University Erlangen-Nürnberg, ArXiv:math/0702095v1 (2007)Google Scholar
  21. 21.
    Swart, J.M.: The contact process seen from a typical infected site. J. Theor. Probab. (2008). doi:10.1007/s10959-008-0184-4 (ArXiv:math.PR/0507578v5)

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Theoretical Statistics and Mathematics DivisionIndian Statistical InstituteBangaloreIndia
  2. 2.Institute of Information Theory and Automation of the ASCR (ÚTIA)Prague 8Czech Republic

Personalised recommendations