Cardy's formula and conformal invariance for critical bond percolation on the square lattice

Theorem 1.2: for critical bond percolation on $\delta\mathbb Z^2$ in a Jordan domain with four uniformly approximated marks, all boundary edges sampled and no external connections, the probability of an open crossing between the arcs $[a,b]$ and $[c,d]$ converges to Cardy's hypergeometric function of the cross-ratio. The $q=1$ case of Theorem 1.1 gives chordal $\mathrm{SLE}_6$ convergence of the exploration interface and CLE$_6$ convergence of all nested loops. The crossing result is proved directly, not from the curve limit. It does not treat other lattices or general boundary conditions beyond those stated.

Theorem 1.2: for critical bond percolation on $\delta\mathbb Z^2$ in a Jordan domain with four uniformly approximated marks, all boundary edges sampled and no external connections, the probability of an open crossing between the arcs $[a,b]$ and $[c,d]$ converges to Cardy's hypergeometric function of the cross-ratio. The $q=1$ case of Theorem 1.1 gives chordal $\mathrm{SLE}_6$ convergence of the exploration interface and CLE$_6$ convergence of all nested loops. The crossing result is proved directly, not from the curve limit. It does not treat other lattices or general boundary conditions beyond those stated.