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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Asymmetry of near-critical percolation interfaces
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by Pierre Nolin and Wendelin Werner PDF
J. Amer. Math. Soc. 22 (2009), 797-819 Request permission

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
  • 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.
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 22 (2009), 797-819
  • MSC (2000): Primary 60K35, 82B27, 82B43
  • DOI: https://doi.org/10.1090/S0894-0347-08-00619-X
  • MathSciNet review: 2505301