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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Asymmetry of near-critical percolation interfaces


Authors: Pierre Nolin and Wendelin Werner
Journal: J. Amer. Math. Soc. 22 (2009), 797-819
MSC (2000): Primary 60K35, 82B27, 82B43
Published electronically: September 16, 2008
MathSciNet review: 2505301
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Abstract: We study the possible scaling limits of percolation interfaces in two dimensions on the triangular lattice. When one lets the percolation parameter $ p(N)$ vary with the size $ N$ of the box that one is considering, three possibilities arise in the large-scale limit. It is known that when $ p(N)$ does not converge to $ 1/2$ fast enough, then the scaling limits are degenerate, whereas if $ p(N) - 1 / 2$ goes to zero quickly, the scaling limits are SLE(6) as when $ p=1/2$. We study some properties of the (non-void) intermediate regime where the large scale behavior is neither SLE(6) nor degenerate. We prove that in this case, the law of any scaling limit is singular with respect to that of SLE(6), even if it is still supported on the set of curves with Hausdorff dimension equal to $ 7/4$.


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Additional Information

Pierre Nolin
Affiliation: DMA, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France, and Laboratoire de Mathématiques, Bât. 425, Université Paris-Sud 11, 91405 Orsay Cedex, France
Email: pierre.nolin@ens.fr

Wendelin Werner
Affiliation: DMA, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France, and Laboratoire de Mathématiques, Bât. 425, Université Paris-Sud 11, 91405 Orsay Cedex, France
Email: wendelin.werner@math.u-psud.fr

DOI: http://dx.doi.org/10.1090/S0894-0347-08-00619-X
PII: S 0894-0347(08)00619-X
Keywords: Near-critical percolation, off-critical regime, massive scaling limit, random fractal interfaces, SLE
Received by editor(s): December 4, 2007
Published electronically: September 16, 2008
Additional Notes: This research was supported by the Agence Nationale pour la Recherche under the grant ANR-06-BLAN-0058.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.