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On the Growth of the One-Dimensional Reverse Immunization Contact Processes

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

Published online by Cambridge University Press:  14 July 2016

A. Tzioufas
A. Tzioufas*
Affiliation:
Heriot-Watt University
*
∗ Postal address: Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: [email protected]
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Abstract

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We are concerned with the variation of the supercritical nearest-neighbours contact process such that first infection occurs at a lower rate; it is known that the process survives with positive probability. Regarding the rightmost infected of the process started from one site infected and conditioned to survive, we specify a sequence of space-time points at which its behaviour regenerates and, thus, obtain the corresponding strong law and central limit theorem. We also extend complete convergence in this case.

Keywords

Contact processKuczek-type argument

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82C22: Interacting particle systems
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 611 - 623
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Bezuidenhout, C. E. and Grimmett, G. (1990). The critical contact process dies out. Ann. Prob. 18, 1462–1482.Google Scholar
[2] Durrett, R. (1980). On the growth of one-dimensional contact processes. Ann. Prob. 8, 890–907.Google Scholar
[3] Durrett, R. (1984). Oriented percolation in two dimensions. Ann. Prob. 12, 999–1040.Google Scholar
[4] Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks/Cole, Pacific Grove, CA.Google Scholar
[5] Durrett, R. (1995). Ten Lectures on Particle Systems (Lecture Notes Math. 1608). Springer, New York.Google Scholar
[6] Durrett, R. and Griffeath, D. (1983). Supercritical contact processes on {Z}. Ann. Prob. 11, 1–15.Google Scholar
[7] Durrett, R. Schinazi, R. B. (2000). Boundary modified contact processes. J. Theoret. Prob. 13, 575–594.Google Scholar
[8] Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems (Lecture Notes Math. 724). Springer, Berlin.Google Scholar
[9] Harris, T. E. (1972). Nearest-neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9, 66–89.Google Scholar
[10] Kuczek, T. (1989). The central limit theorem for the right edge of supercritical oriented percolation. Ann. Prob. 17, 1322–1332.Google Scholar
[11] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
[12] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.Google Scholar
[13] Stacey, A. (2003). Partial immunization processes. Ann. Appl. Prob. 13, 669–690.Google Scholar