Path Crossing Exponents and the External Perimeter in 2D Percolation

Percolation path crossing exponents describe probabilities for $\ell$ non-overlapping traversing paths, each of either occupied sites or vacancies. We show, for collections including at least one of each, that in 2D the exponents are those of an $O(N=1)$ loop model. This extends the earlier identification by Saleur and Duplantier of $k$ spanning cluster exponents, for which $\ell=2k$. The results yield $D_{EP}=4/3$ for the fractal dimension of the accessible external cluster perimeter, and explain the absence of narrow gate fjords, in agreement with the original findings of Grossman and Aharony.