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

2D percolation path exponents x^P_l 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 x^P_l = \(l²-1\)/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, D_EP = 2-x^P₃ = 4/3, and also explains the absence of narrow gate fjords, which was originally noted by Grossman and Aharony.