I have very few ideas to how solve this problem.
Let $ D = \{ z^2 : 0 \leq \Re(z) , \Im(z) \leq 1 \}$. Discretize $D$ with $D_{\delta}$ by a honeycomb lattice of mesh size $\delta$ and color the hexagons of $D_{\delta}$ in black or white with probability 1/2 independently. Let $P_{\delta} $ denote the probability that the points $1/4$ and $-1/4$ are separated by a black path from the points $3/4 + i $ and $-3/4 + i $ in $D_{\delta}$. Show that $$ \lim_{\delta \to 0 } P_{\delta} = 1/2 $$
Problem 1: I don't get very well what shape has the domain $D$.
Idea: Carleson's formulation of Cardy's formula says what follows: If we have a Jordan domain $( \Omega, a_1,a_2,a_3,a_4)$, with $a_1,a_2,a_3,a_4 \in \partial \Omega $ in counterclockwise order, then it is known that there exist a unique conformal from $ \Omega$ to the equilateral triangle $ \Delta$ with vertices $1, \pm \frac{\sqrt{3}}{3} i $. Then if we discretize $\Omega$ by $\Omega_{\delta}$ $$ \lim_{\delta \to 0} \mathbb{P} \{ [a_1a_2] \leftrightarrow_w [a_3a_4] \} = \Re( \varphi(a_4) )$$ where $\leftrightarrow_w$ means that there exists a white path connecting $[a_1a_2]$ to $[a_3a_4]$.
But I truly believe improbable that I have to found explicitly a conformal map between $D$ and $\Delta$, unless it is easier than what I expect, so maybe there is some percolation argumentation that make this problem more easy than one would expect. Some ideas?