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Phys. Rev. Lett. 83, 1359–1362 (1999)

Path-Crossing Exponents and the External Perimeter in 2D Percolation

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Michael Aizenman1, Bertrand Duplantier2, and Amnon Aharony3
1Department of Physics, Jadwin Hall, Princeton University, Princeton, New Jersey 08544
and Department of Mathematics, Fine Hall, , Princeton, New Jersey 08544
2Service de Physique Théorique de Saclay, 91191 Gif-sur-Yvette Cedex, France
and Institut Henri Poincaré 11, rue Pierre et Marie Curie, 75231 Paris Cedex 05, France
3School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received 6 January 1999; published in the issue dated 16 August 1999

2D percolation path exponents xℓP describe probabilities for traversals of annuli by ℓ nonoverlapping paths on either occupied or vacant clusters, with at least one of each type. We relate the probabilities rigorously to amplitudes of O(N = 1) models whose exponents, believed to be exact, yield xℓP = (ℓ2-1)/12. This extends to half-integers the Saleur–Duplantier exponents for k = ℓ/2 clusters, yields the exact fractal dimension of the external cluster perimeter, DEP = 2-x3P = 4/3, and also explains the absence of narrow gate fjords, which was originally noted by Grossman and Aharony.

© 1999 The American Physical Society

URL:
http://link.aps.org/doi/10.1103/PhysRevLett.83.1359
DOI:
10.1103/PhysRevLett.83.1359
PACS:
64.60.Ak, 05.45.Df, 05.50.+q, 64.60.Fr
 
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