Electronic Journal of Probability

Multitype Contact Process on $\mathbb{Z}$: Extinction and Interface

Daniel Valesin

Full-text: Open access

Abstract

We consider a two-type contact process on the integers. Both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only if, in the initial configuration, it is confined to a finite interval and surrounded by infinitely many individuals of the other type. Additionally, we show that if both types are present in finite number in the initial configuration, then there is a positive probability that they are both present for all times. Finally, it is shown that, starting from the configuration in which all sites to the left of the origin are occupied by type 1 particles and all sites to the right of the origin are occupied by type 2 particles, the process defined by the size of the interface area between the two types is stochastically tight.

Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 73, 2220-2260.

Dates
Accepted: 18 December 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819858

Digital Object Identifier
doi:10.1214/EJP.v15-836

Mathematical Reviews number (MathSciNet)
MR2748404

Zentralblatt MATH identifier
1226.60137

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

Keywords
Interacting Particle Systems

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Valesin, Daniel. Multitype Contact Process on $\mathbb{Z}$: Extinction and Interface. Electron. J. Probab. 15 (2010), paper no. 73, 2220--2260. doi:10.1214/EJP.v15-836. https://projecteuclid.org/euclid.ejp/1464819858


Export citation

References

  • Andjel, Enrique D.; Miller, Judith R.; Pardoux, Etienne Survival of a single mutant in one dimension. Electron. J. Probab. 15 (2010), no. 14, 386-408.
  • Andjel, Enrique D.; Mountford, Thomas; Pimentel, Leandro P. R.; Valesin, Daniel Tightness for the interface of the one-dimensional contact process. Bernoulli 16, Number 4 (2010), 909-925.
  • Belhaouari, S.; Mountford, T.; Sun, Rongfeng; Valle, G. Convergence results and sharp estimates for the voter model interfaces. Electron. J. Probab. 11 (2006), no. 30, 768-801.
  • Belhaouari, S.; Mountford, T.; Valle, G. Tightness for the interfaces of one-dimensional voter models. Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 421-442.
  • Cox, J. T.; Durrett, R. Hybrid zones and voter model interfaces. Bernoulli 1 (1995), no. 4, 343-370.
  • Durrett, Richard Lecture notes on particle systems and percolation. The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1988. viii+335 pp. ISBN: 0-534-09462-7.
  • Durrett, Rick Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8, 60-01.
  • Durrett, Rick; Swindle, Glen Are there bushes in a forest- Stochastic Process. Appl. 37 (1991), no. 1, 19-31.
  • Kuczek, Thomas The central limit theorem for the right edge of supercritical oriented percolation. Ann. Probab. 17 (1989), no. 4, 1322-1332.
  • Lawler, Gregory F.; Limic, Vlada Random walk: a modern introduction. Cambridge Studies in Advanced Mathematics, 123. Cambridge University Press, Cambridge, 2010. xii+364 pp. ISBN: 978-0-521-51918-2.
  • Levin, David A.; Peres, Yuval; Wilmer, Elizabeth L. Markov chains and mixing times. With a chapter by James G. Propp and David B. Wilson. American Mathematical Society, Providence, RI, 2009. xviii+371 pp. ISBN: 978-0-8218-4739-8.
  • Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4.
  • Liggett, Thomas M. Stochastic interacting systems: contact, voter and exclusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 324. Springer-Verlag, Berlin, 1999. xii+332 pp. ISBN: 3-540-65995-1.
  • Mountford, Thomas S.; Sweet, Ted D. An extension of Kuczek's argument to nonnearest neighbor contact processes. J. Theoret. Probab. 13 (2000), no. 4, 1061-1081.
  • Neuhauser, Claudia Ergodic theorems for the multitype contact process. Probab. Theory Related Fields 91 (1992), no. 3-4, 467-506.
  • Spitzer, Frank Principles of random walks. Second edition. Graduate Texts in Mathematics, Vol. 34. Springer-Verlag, New York-Heidelberg, 1976. xiii+408.
  • Sturm, Anja; Swart, Jan M. Tightness of voter model interfaces. Electron. Commun. Probab. 13 (2008), 165-174.