Factorization Formulas for $2D$ Critical Percolation, Revisited

We consider critical site percolation on the triangular lattice in the upper half-plane. Let $u_1,u_2$ be two sites on the boundary and $w$ a site in the interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a paper from 2007 that the ratio $$\mathbb{P}$(nu_1 \leftrightarrow nu_2 \leftrightarrow nw)^{2},/,$\mathbb{P}$(nu_1 \leftrightarrow nu_2)\cdot$\mathbb{P}$(nu_1 \leftrightarrow nw)\cdot$\mathbb{P}$(nu_2 \leftrightarrow nw)$convergesto$K_F$as$n \to \infty$,where$x\leftrightarrow y$denotestheeventthat$x$and$y$ are in the same open cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in 2012 an analog of this factorization in the scaling limit. We prove, using their result and a generalized coupling argument, the earlier mentioned prediction. Furthermore we prove a factorization formula for the probability $$\mathbb{P}$(nu_2 \leftrightarrow [nu_1,nu_1+s];, nw \leftrightarrow [nu_1,nu_1+s])$,where$s>0$.