The box-crossing property for critical two-dimensional oriented percolation
- 184 Downloads
Abstract
-
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 \(\varepsilon >0\).
Keywords
Percolation Oriented percolation Critical behaviour Contact process RenormalizationMathematics Subject Classification
60K35 82B43 82C43Notes
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.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.Bezuidenhout, C., Grimmett, G.: The critical contact process dies out. Ann. Probab. 18(4), 1462–1482 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Broadbent, S.R., Hammersley, J.M.: Percolation processes. I. Crystals and mazes. Proc. Camb. Philos. Soc. 53, 629–641 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.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.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.Durrett, R., Griffeath, D.: Supercritical contact processes on \({ Z}\). Ann. Probab. 11(1), 1–15 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
- 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.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.Durrett, R., Tanaka, N.I.: Scaling inequalities for oriented percolation. J. Stat. Phys. 55(5–6), 981–995 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Durrett, R.: Oriented percolation in two dimensions. Ann. Probab. 12(4), 999–1040 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Galves, A., Presutti, E.: Edge fluctuations for the one-dimensional supercritical contact process. Ann. Probab. 15(3), 1131–1145 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Griffeath, D.: The basic contact processes. Stoch. Process. Appl. 11(2), 151–185 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Grimmett, G.: Percolation, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321. Springer, Berlin (1999)Google Scholar
- 15.Harris, T.E.: Contact interactions on a lattice. Ann. Probab. 2, 969–988 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Harris, T.E.: Additive set-valued Markov processes and graphical methods. Ann. Probab. 6(3), 355–378 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Kuczek, T.: The central limit theorem for the right edge of supercritical oriented percolation. Ann. Probab. 17(4), 1322–1332 (1989)MathSciNetCrossRefzbMATHGoogle Scholar