The Annals of Applied Probability

The contact process in a dynamic random environment

Daniel Remenik

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We study a contact process running in a random environment in ℤd where sites flip, independently of each other, between blocking and nonblocking states, and the contact process is restricted to live in the space given by nonblocked sites. We give a partial description of the phase diagram of the process, showing in particular that, depending on the flip rates of the environment, survival of the contact process may or may not be possible for large values of the birth rate. We prove block conditions for the process that parallel the ones for the ordinary contact process and use these to conclude that the critical process dies out and that the complete convergence theorem holds in the supercritical case.

Article information

Ann. Appl. Probab., Volume 18, Number 6 (2008), 2392-2420.

First available in Project Euclid: 26 November 2008

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Contact process random environment complete convergence block construction interacting particle system


Remenik, Daniel. The contact process in a dynamic random environment. Ann. Appl. Probab. 18 (2008), no. 6, 2392--2420. doi:10.1214/08-AAP528.

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  • Andjel, E. D. (1992). Survival of multidimensional contact process in random environments. Bol. Soc. Brasil. Mat. (N.S.) 23 109–119.
  • Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Die Grundlehren der mathematischen Wissenschaften 196. Springer, New York.
  • Bezuidenhout, C. and Grimmett, G. (1990). The critical contact process dies out. Ann. Probab. 18 1462–1482.
  • Bramson, M., Durrett, R. and Schonmann, R. H. (1991). The contact process in a random environment. Ann. Probab. 19 960–983.
  • Broman, E. I. (2007). Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment. Ann. Probab. 35 2263–2293.
  • Durrett, R. and Møller, A. M. (1991). Complete convergence theorem for a competition model. Probab. Theory Related Fields 88 121–136.
  • Durrett, R. and Schinazi, R. (1993). Asymptotic critical value for a competition model. Ann. Appl. Probab. 3 1047–1066.
  • Durrett, R. and Schonmann, R. H. (1987). Stochastic growth models. In Percolation Theory and Ergodic Theory of Infinite Particle Systems (Minneapolis, Minn., 1984–1985). The IMA Volumes in Mathematics and Its Applications 8 85–119. Springer, New York.
  • Durrett, R. and Swindle, G. (1991). Are there bushes in a forest? Stochastic Process. Appl. 37 19–31.
  • Harris, T. E. (1972). Nearest-neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9 66–89.
  • Harris, T. E. (1974). Contact interactions on a lattice. Ann. Probab. 2 969–988.
  • Klein, A. (1994). Extinction of contact and percolation processes in a random environment. Ann. Probab. 22 1227–1251.
  • Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • Liggett, T. M. (1992). The survival of one-dimensional contact processes in random environments. Ann. Probab. 20 696–723.
  • Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 324 Springer, Berlin.
  • Luo, X. (1992). The Richardson model in a random environment. Stochastic Process. Appl. 42 283–289.
  • Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics 119. Cambridge Univ. Press, Cambridge.