Hostname: page-component-7479d7b7d-fwgfc Total loading time: 0 Render date: 2024-07-13T00:09:56.606Z Has data issue: false hasContentIssue false
  • English
  • Français

The entirely coupled region of supercritical contact processes

Part of: Special processes Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  24 October 2016

Achillefs Tzioufas
Achillefs Tzioufas*
Affiliation:
Universidad de Buenos Aires
*
* Current address: Instituto de Matemática e Estatística - USP, Rua do Matão, 1010, CEP 05508-900 São Paulo, Brasil. Email address: tzioufas@ime.usp.br
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Contact processesasymptotic shape theoremcouplingfinite rangecoupling eventcoupling time

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82C43: Time-dependent percolation
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 925 - 929
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Andjel, .,Mountford, T.,Pimentel, L. P. R. and Valesin, D. (2010).Tightness for the interface of the one-dimensional contact process.Bernoulli 16,909–925.Google Scholar
[2] Bezuidenhout, C. and Gray, L. (1994).Critical attractive spin systems.Ann. Prob. 22,1160–1194.Google Scholar
[3] Bezuidenhout, C. and Grimmett, G. (1990).The critical contact process dies out.Ann. Prob. 18,1462–1482.CrossRefGoogle Scholar
[4] Durrett, R. (1980).On the growth of one-dimensional contact processes.Ann. Prob. 8,890–907.Google Scholar
[5] Durrett, R. (1995).Ten lectures on particle systems. InLectures on Probability Theory (Lecture Notes Math. 1608),Springer,Berlin, pp.97–201.CrossRefGoogle Scholar
[6] Durrett, R. and Griffeath, D. (1983).Supercritical contact processes on ℤ.Ann. Prob. 11,1–15.Google Scholar
[7] Durrett, R. and Schonmann, R. H. (1987).Stochastic growth models. InPercolation Theory and Ergodic Theory of Infinite Particle Systems,Springer,New York, pp.85–119.Google Scholar
[8] Garet, O. and Marchand, R. (2012).Asymptotic shape for the contact process in random environment.Ann. Appl. Prob. 22,1362–1410.Google Scholar
[9] Garet, O. and Marchand, R. (2014).Large deviations for the contact process in random environment.Ann. Prob. 42,1438–1479.Google Scholar
[10] Harris, T. E. (1974).Contact interactions on a lattice.Ann. Prob. 2,969–988.Google Scholar
[11] Harris, T. E. (1978).Additive set valued Markov processes and graphical methods.Ann. Prob. 6,355–378.CrossRefGoogle Scholar
[12] Kingman, J. F. C. (1968).The ergodic theory of subadditive stochastic processes.J. R. Statist. Soc. B 30,499–510.Google Scholar
[13] Liggett, T. M. (1985).An improved subadditive ergodic theorem.Ann. Prob. 13,1279–1285.Google Scholar
[14] Liggett, T. M. (1999).Stochastic Interacting Systems: Contact, Voter and Exclusion Processes.Springer ,Berlin.CrossRefGoogle Scholar
[15] Mollison, D. (1972).Possible velocities for a simple epidemic.Adv. Appl. Prob. 4,233–257.Google Scholar
[16] Mollison, D. (1977).Spatial contact models for ecological and epidemic spread.J. R. Statist. Soc. B 39,283–326.Google Scholar
[17] Tzioufas, A. (2011).Contact processes on the integers. Doctoral Thesis, Heriot-Watt University.Google Scholar
[18] Tzioufas, A. (2016).Highly supercritical oriented percolation in two dimensions revisited. Preprint. Available at http://arxiv.org/abs/arXiv:1311.2952.Google Scholar