Factorization formulas for 2D critical percolation, revisited

We consider critical site percolation on the triangular lattice in the upper half-plane. Let u1,u2 be two sites on the boundary and w a site in the interior. It was predicted by Simmons et al. (2007) that the ratio P(nu1↔nu2↔nw)2/P(nu1↔nu2)⋅P(nu1↔nw)⋅P(nu2↔nw) converges to KF as n→∞, where x↔y denotes that x and y are in the same cluster, and KF is a constant. Beliaev and Izyurov (2012) proved an analog of this in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for P(nu2↔[nu1,nu1+s];nw↔[nu1,nu1+s]), where s>0.