Skip to main content
SpringerLink
Log in

Convergence Rates for Subcritical Threshold-One Contact Processes on Lattices

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In this paper we are concerned with threshold-one contact processes on lattices. We show that the probability that the origin is infected converges to 0 at an exponential rate I in the subcritical case. Furthermore, we give a limit theorem for I as the degree of the lattice grows to infinity. Our results also hold for classic contact processes on lattices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andjel, E.D., Liggett, T.M., Mountford, T.: Clustering in one-dimensional threshold voter models. Stoch. Process. Appl. 42, 73–90 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Springer, New York (1997)

    Google Scholar 

  3. Cox, J.T., Durrett, R.: Nonlinear voter models. Random Walks, Brownian Motion and Interacting Particle Systems. A Festschrift in Honor of Frank Spiter, pp. 189–201. Birkhäuser, Boston (1991)

    Chapter  Google Scholar 

  4. Durrett, R.: Probability: Theory and Examples, 4th edn. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  5. Fontes, L.R., Schonmann, R.H.: Threshold \(\theta \ge 2\) contact processes on homogeneous trees. Probab. Theory Relat. Fields 141, 513–541 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Griffeath, D.: The basic contact process. Stoch. Process. Appl. 11, 151–186 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  7. Griffeath, D.: The binary contact path process. Ann. Probab. 11, 692–705 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  8. Handjani, S.: The complete convergence theorem for coexistent threshold voter models. Ann. Probab. 27, 226–245 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Liggett, T.M.: Interacting Particle Systems. Springer, New York (1985)

    Book  MATH  Google Scholar 

  10. Liggett, T.M.: Coexistence in threshold voter models. Ann. Probab. 22, 764–802 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, New York (1999)

    Book  MATH  Google Scholar 

  12. Mountford, T., Schonmann, R.H.: The survival of large dimensional threshold contact processes. Ann. Probab. 37, 1483–1501 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Xue, X.F.: Critical density points for threshold voter models on homogeneous trees. J. Stat. Phys. 146, 423–433 (2012)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Xue, X.F.: Asymptotic behavior of critical infection rates for threshold-one contact processes on lattices and regular trees. J. Theor. Probab. 28, 1447–1467 (2014)

    Article  Google Scholar 

  15. Xue, X.F.: Fluid limit of threshold voter models on tori. J. Stat. Phys. 159, 274–293 (2015)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Acknowledgments

The author is grateful to the reviewers for their useful comments and to the financial support from the National Natural Science Foundation of China with Grant Number 11171342 and China Postdoctoral Science Foundation (No. 2015M571095).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Xiaofeng Xue

Authors
  1. Xiaofeng Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaofeng Xue.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xue, X. Convergence Rates for Subcritical Threshold-One Contact Processes on Lattices. J Stat Phys 162, 371–386 (2016). https://doi.org/10.1007/s10955-015-1419-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-015-1419-2

Keywords

Search

Navigation