Consider critical site percolation on the triangular lattice with mesh-size $\delta$. Let $D$ be the domain $H\backslash[0;i]$ where $H$ is the complex upper half plane. Compute the probability $$ \lim_{\delta\rightarrow 0} P_\delta ([a; 0]\leftrightarrow [c;\infty) \text{in} D) $$ that there is an open path from the segment $[a; 0]$ to the segment $[c; \infty)$, $a < 0 < c$.
From Cardy-Smirnov's formula for $\text{SLE}_6$ I know that $$ \lim_{\delta\rightarrow 0} P_\delta ([a; 0]\leftrightarrow [c;\infty) \text{in} H) \sim \int_0^C \frac{dx}{x^{2/3}(1-x)^{2/3}}, C = -\frac{a}{c-a} $$ and the conformal mapping from $D$ to $H$ is $g(z)=\sqrt{z^2+1}$. Thus, $a'=-\sqrt{a^2+1}$ and $c'=\sqrt{c^2+1}$ and $$ \lim_{\delta\rightarrow 0} P_\delta ([a'; -1]\leftrightarrow [c';\infty) \text{in} H') \sim \int_{C_1}^{C_2} \frac{dx}{x^{2/3}(1-x)^{2/3}}, C_2 = -\frac{a'}{c'-a'}, C_1 = \frac{1}{c'+1} $$ Is this expression correct? Do i need to apply inverse transform to get $P_\delta$ in $D$ from this expression? Thanks.