Advertisement

Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 685–708 | Cite as

The box-crossing property for critical two-dimensional oriented percolation

  • H. Duminil-Copin
  • V. Tassion
  • A. Teixeira
Article
  • 172 Downloads

Abstract

We consider critical oriented Bernoulli percolation on the square lattice \(\mathbb {Z}^2\). We prove a Russo–Seymour–Welsh type result which allows us to derive several new results concerning the critical behavior:
  • We establish that the probability that the origin is connected to distance n decays polynomially fast in n.

  • We prove that the critical cluster of 0 conditioned to survive to distance n has a typical width \(w_n\) satisfying \(\varepsilon n^{2/5}\le w_n\le n^{1-\varepsilon }\) for some \(\varepsilon >0\).

The sub-linear polynomial fluctuations contrast with the supercritical regime where \(w_n\) is known to behave linearly in n. It is also different from the critical picture obtained for non-oriented Bernoulli percolation, in which the scaling limit is non-degenerate in both directions. All our results extend to the graphical representation of the one-dimensional contact process.

Keywords

Percolation Oriented percolation Critical behaviour Contact process Renormalization 

Mathematics Subject Classification

60K35 82B43 82C43 

Notes

Acknowledgements

We are grateful to Daniel Valesin for the careful reading of the first version of this article and for their very helpful comments. The work of the two first authors was supported by a grant from the Swiss NSF and the NCCR SwissMap also funded by the Swiss NSF. The project was initiated during a stay of the third author to the Université de Genève, and the authors are grateful to the institution for making such a stay possible. AT was supported by CNPq grants 306348/2012-8 and 478577/2012-5 and by FAPERJ grant 202.231/2015.

References

  1. 1.
    Balister, P., Bollobás, B., Stacey, A.: Improved upper bounds for the critical probability of oriented percolation in two dimensions. Random Struct. Algorithms 5(4), 573–589 (1994). (English)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bezuidenhout, C., Grimmett, G.: The critical contact process dies out. Ann. Probab. 18(4), 1462–1482 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Broadbent, S.R., Hammersley, J.M.: Percolation processes. I. Crystals and mazes. Proc. Camb. Philos. Soc. 53, 629–641 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Belitsky, V., Ritchie, T.L.: Improved lower bounds for the critical probability of oriented bond percolation in two dimensions. J. Stat. Phys. 122(2), 279–302 (2006). (English)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duminil-Copin, H., Tassion, V.: A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Comm. Math. Phys. 343(2), 725–745 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Duminil-Copin, H., Tassion, V.: Rsw and box-crossing property for planar percolation. In: Proceedings of the International Congress of Mathematical Physics (2016)Google Scholar
  7. 7.
    Durrett, R., Griffeath, D.: Supercritical contact processes on \({ Z}\). Ann. Probab. 11(1), 1–15 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Durrett, R., Schonmann, R.H., Tanaka, N.I.: The contact process on a finite set. III. The critical case. Ann. Probab. 17(4), 1303–1321 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Durrett, R., Schonmann, R.H., Tanaka, N.I.: Correlation lengths for oriented percolation. J. Stat. Phys. 55(5–6), 965–979 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Durrett, R., Tanaka, N.I.: Scaling inequalities for oriented percolation. J. Stat. Phys. 55(5–6), 981–995 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Durrett, R.: Oriented percolation in two dimensions. Ann. Probab. 12(4), 999–1040 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Galves, A., Presutti, E.: Edge fluctuations for the one-dimensional supercritical contact process. Ann. Probab. 15(3), 1131–1145 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Griffeath, D.: The basic contact processes. Stoch. Process. Appl. 11(2), 151–185 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grimmett, G.: Percolation, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321. Springer, Berlin (1999)Google Scholar
  15. 15.
    Harris, T.E.: Contact interactions on a lattice. Ann. Probab. 2, 969–988 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Harris, T.E.: Additive set-valued Markov processes and graphical methods. Ann. Probab. 6(3), 355–378 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kuczek, T.: The central limit theorem for the right edge of supercritical oriented percolation. Ann. Probab. 17(4), 1322–1332 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  2. 2.Département de mathématiquesUniversité de GenéveGeneva 4Switzerland
  3. 3.Instituto Nacional de Matemática Pura e Aplicada - IMPARio de JaneiroBrazil

Personalised recommendations