Decomposition of global 2-SLE for $\kappa \in (4,8)$ and an application for critical FK-Ising model

We consider global 2-SLE${}_{\kappa }$ $({\eta }_1,{\eta }_2)$ in a topological rectangle with $\kappa \in (4,8)$. We derive the law of a random hitting point of the curves and show that, conditional on this random hitting point, the pair of two curves has the same law as Gaussian free field flow lines with proper boundary data. Using a similar idea, we derive the asymptotic of the probability for ${\eta }_1\cap {\eta }_2=\mathrm{\varnothing }$. As an application, we derive the asymptotic of the probability for the existence of two disjoint open paths in critical FK-Ising model.