Journal of Applied Probability

The entirely coupled region of supercritical contact processes

Achillefs Tzioufas

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We consider translation-invariant, finite-range, supercritical contact processes. We show the existence of unbounded space-time cones within which the descendancy of the process from full occupancy may with positive probability be identical to that of the process from the single site at its apex. The proof comprises an argument that leans upon refinements of a successful coupling among these two processes, and is valid in d-dimensions.

Article information

J. Appl. Probab., Volume 53, Number 3 (2016), 925-929.

First available in Project Euclid: 13 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82843 82C43: Time-dependent percolation [See also 60K35]

Contact processes asymptotic shape theorem coupling finite range coupling event coupling time


Tzioufas, Achillefs. The entirely coupled region of supercritical contact processes. J. Appl. Probab. 53 (2016), no. 3, 925--929.

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  • Andjel, E., Mountford, T., Pimentel, L. P. R. and Valesin, D. (2010). Tightness for the interface of the one-dimensional contact process. Bernoulli 16, 909–925.
  • Bezuidenhout, C. and Gray, L. (1994). Critical attractive spin systems. Ann. Prob. 22, 1160–1194.
  • Bezuidenhout, C. and Grimmett, G. (1990). The critical contact process dies out. Ann. Prob. 18, 1462–1482.
  • Durrett, R. (1980). On the growth of one-dimensional contact processes. Ann. Prob. 8, 890–907.
  • Durrett, R. (1995). Ten lectures on particle systems. In Lectures on Probability Theory (Lecture Notes Math. 1608), Springer, Berlin, pp. 97–201.
  • Durrett, R. and Griffeath, D. (1983). Supercritical contact processes on $\Z$. Ann. Prob. 11, 1–15.
  • Durrett, R. and Schonmann, R. H. (1987). Stochastic growth models. In Percolation Theory and Ergodic Theory of Infinite Particle Systems, Springer, New York, pp. 85–119.
  • Garet, O. and Marchand, R. (2012). Asymptotic shape for the contact process in random environment. Ann. Appl. Prob. 22, 1362–1410.
  • Garet, O. and Marchand, R. (2014). Large deviations for the contact process in random environment. Ann. Prob. 42, 1438–1479.
  • Harris, T. E. (1974). Contact interactions on a lattice. Ann. Prob. 2, 969–988.
  • Harris, T. E. (1978). Additive set valued Markov processes and graphical methods. Ann. Prob. 6, 355–378.
  • Kingman, J. F. C. (1968). The ergodic theory of subadditive stochastic processes. J. R. Statist. Soc. B 30, 499–510.
  • Liggett, T. M. (1985). An improved subadditive ergodic theorem. Ann. Prob. 13, 1279–1285.
  • Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.
  • Mollison, D. (1972). Possible velocities for a simple epidemic. Adv. Appl. Prob. 4, 233–257.
  • Mollison, D. (1977). Spatial contact models for ecological and epidemic spread. J. R. Statist. Soc. B 39, 283–326.
  • Tzioufas, A. (2011). Contact processes on the integers. Doctoral Thesis, Heriot-Watt University.
  • Tzioufas, A. (2016). Highly supercritical oriented percolation in two dimensions revisited. Preprint. Available at