Abstract
We establish convergence to SLE6 of the Exploration Processes for the correlated bond-triangular type models studied by two of us in an earlier work (Chayes and Lei in Rev. Math. Phys. 19:511–565, 2007). This (rigorously) establishes that these models are in the same universality class as the standard site percolation model on the triangular lattice. In the context of these models, the result is proven for all domains with boundary Minkowski dimension less than two. Moreover, the proof of convergence applies in the context of general critical 2D percolation models and for general domains, under the stipulation that Cardy’s Formula can be established for domains in this generality.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aizenman, M.: The geometry of critical percolation and conformal invariance. In: Bai-lin, H. (ed.) Proceedings STATPHYS19, Xiamen, 1995. World Scientific, Singapore (1995)
Aizenman, M., Burchard, A.: Hölder regularity and dimension bounds for random curve. Duke Math. J. 99(3), 419–453 (1999)
Binder, I., Chayes, L., Lei, H.K.: On convergence to SLE6 II: discrete approximations and extraction of Cardy’s formula for general domains. J. Stat. Phys. (2010). doi:10.1007/s10955-010-0053-2
Bollobás, B., Riordan, O.: Percolation. Cambridge University Press, Cambridge (2006)
Borgs, C., Chayes, J.T., Randall, D.: The van den Berg–Kesten–Reimer inequality: a review. In: Bramson, M., Durrett, R. (eds.) Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten, pp. 159–173. Birkhauser, Basel (1999). Also available at http://www.math.gatech.edu/randall/reprints.html
Camia, F., Newman, C.M.: Two-dimensional critical percolation: the full scaling limit. Commun. Math. Phys. 268(1), 1–38 (2006)
Camia, F., Newman, C.M.: Critical percolation exploration path and SLE6: a proof of convergence. Available at arXiv:math.PR/0604487 (2006)
Chayes, J.T., Chayes, L.: Percolation and random media. In: Osterwalder, K., Stora, R. (eds.) Les Houches Session XLIII: Critical Phenomena, Random Systems and Gauge Theories, pp. 1001–1042. Elsevier, Amsterdam (1986)
Chayes, L., Lei, H.K.: Cardy’s formula for certain models of the bond-triangular type. Rev. Math. Phys. 19, 511–565 (2007)
Grimmett, G.: Percolation. Springer, Berlin (1999)
Kesten, H.: Analyticity properties and power law estimates of functions in percolation theory. J. Stat. Phys. 25(4), 717–756 (1981)
Kesten, H.: Percolation Theory for Mathematicians. Birkhäuser, Basel (1982)
Kesten, H.: Scaling relations for 2D-percolation. Commun. Math. Phys. 109, 109–156 (1987)
Kesten, H., Sidoravicius, V., Zhang, Y.: Almost all words are seen in critical site percolation on the triangular lattice. Electron. J. Probab. 3(10), 1–75 (1998)
Lawler, G.F.: Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence (2005)
Lawler, G.F., Schramm, O., Werner, W.: One-arm exponent for critical 2D percolation. Electron. J. Probab. 7, 13 (2002). (electronic)
Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004)
Reimer, D.: Proof of the van den Berg–Kesten conjecture. Comb. Probab. Comput. 9(1), 27–32 (2000)
Smirnov, S.: Critical percolation in the plane: conformal invariance. Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sr. I Math. 333, 239–244 (2001). Also available at http://www.math.kth.se/~stas/papers/percras.ps
Smirnov, S.: Towards conformal invariance of 2d lattice models. In: Proceedings of the International Congress of Mathematicians, Madrid, Spain (2006)
van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab. 22, 556–569 (1985)
Werner, W.: Lectures on two-dimensional critical percolation. arXiv:0710.0856
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Binder, I., Chayes, L. & Lei, H.K. On Convergence to SLE6 I: Conformal Invariance for Certain Models of the Bond-Triangular Type. J Stat Phys 141, 359–390 (2010). https://doi.org/10.1007/s10955-010-0052-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-0052-3