In this paper we consider independent site percolation in a triangulation of $\mathbb{R}^2$ given by adding $\sqrt{2}$-long diagonals to the usual graph $\mathbb{Z}^2$. We conjecture that $p_c=\frac{1}{2}$ for any such graph, and prove it for almost every such graph.
A Lemma for Color Switching on the Square Lattice
$SLE_6$ and 2-d critical bond percolation on the square lattice
Conformal invariance of the exploration path in 2-d critical bond percolation in the square lattice
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