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

$2\mathrm{D}$ percolation path exponents $x_{\ell }^P$ describe probabilities for traversals of annuli by $\ell$ nonoverlapping paths on either occupied or vacant clusters, with at least one of each type. We relate the probabilities rigorously to amplitudes of $O(N=1\mathrm{})$ models whose exponents, believed to be exact, yield $x_{\ell }^P=({\ell }^2-1\mathrm{})/12$. This extends to half-integers the Saleur–Duplantier exponents for $k=\ell /2$ clusters, yields the exact fractal dimension of the external cluster perimeter, $D_{\mathrm{EP}}{=2-x}_3^P=4\mathrm{}/3$, and also explains the absence of narrow gate fjords, which was originally noted by Grossman and Aharony.