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 $\mathbb{P}(nu_1 \leftrightarrow nu_2 \leftrightarrow nw)^{2}\,/\,\mathbb{P}(nu_1 \leftrightarrow nu_2)\cdot\mathbb{P}(nu_1 \leftrightarrow nw)\cdot\mathbb{P}(nu_2 \leftrightarrow nw)$ converges to $K_F$ as $n \to \infty$, where $x\leftrightarrow y$ denotes the event that $x$ 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 $s>0$.

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