Path Crossing Exponents and the External Perimeter in 2D Percolation

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

1999-01-04

arxiv.org/abs/cond-mat/9901018v2

Phys. Rev. Lett. v. 83, 1359 (1999).

Physics

Condensed Matter

Statistical Mechanics

4 pages, 2 figures (EPSF). Revised presentation

Scientific paper

10.1103/PhysRevLett.83.1359

2D Percolation path exponents $x^{\cal P}_{\ell}$ describe probabilities for traversals of annuli by $\ell$ non-overlapping paths, each 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 $x^{\cal P}_{\ell}=({\ell}^2-1)/12$. This extends to half-integers the Saleur--Duplantier exponents for $k=\ell/2$ clusters, yields the exact fractal dimension of the external cluster perimeter, $D_{EP}=2-x^{\cal P}_3=4/3$, and also explains the absence of narrow gate fjords, as originally found by Grossman and Aharony.

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