Bond percolation on a square lattice. Visualize cluster growth and the sharp crossing probability transition at p_c=0.5. Cardy's formula gives an exact crossing probability at criticality.
Bond percolation: each lattice bond open with probability p. Critical p_c=1/2 (Kesten 1980 — exact for square lattice bonds). Cardy's formula (1992): crossing probability π_h(r) = ₃F₂({1/3,2/3,1},{5/6,7/6};r) — a hypergeometric function. Proved rigorously via Smirnov's conformal invariance of site percolation on triangular lattice (Fields Medal 2010).