Kesten's theorem p_c = 1/2 on the square lattice
/ Kesten: KESS-ten /
For bond percolation on the square lattice Z^2 the critical probability is exactly 1/2 — a clean, rational, fully proved value, in contrast to the merely-numerical p_c of almost every other lattice. Kesten's 1980 theorem is the landmark result of two-dimensional percolation, and its proof wove together duality, the correlation inequalities, and the now-central idea of crossing probabilities (the Russo-Seymour-Welsh theory). It is the rigorous confirmation of a value physicists had long believed on symmetry grounds.
The reason the answer is 1/2 rather than anything else is self-duality. The planar dual of Z^2 is again a copy of Z^2, and there is a beautiful coupling: place a dual edge crossing each primal edge, declaring the dual edge open exactly when the primal edge is closed. Then a left-right open crossing of a rectangle in the primal lattice is blocked precisely by a top-bottom open crossing in the dual lattice. If the primal is at parameter p, the dual is at parameter 1-p, so the natural self-dual point is p = 1-p, i.e. p = 1/2. Harris (1960) had used duality plus FKG to show theta(1/2) = 0, hence p_c >= 1/2 (no percolation at 1/2). The hard direction, p_c <= 1/2, is Kesten's: he proved that for p > 1/2 percolation occurs, using a sharp-threshold / RSW box-crossing argument to upgrade the equal-crossing fact at 1/2 into an infinite cluster just above it. Together they pin p_c exactly to 1/2.
Two cautions keep this honest. First, the clean value 1/2 is special to BOND percolation on Z^2; site percolation on Z^2 has p_c approximately 0.5927 with no closed form, and the self-dual triangular SITE lattice is the one with p_c = 1/2 for sites. Second, '1/2' refers to the location of the transition, not to the density there: Kesten and Harris together also give theta(1/2) = 0, so there is no infinite cluster AT criticality on Z^2 — the transition is continuous. The whole edifice rests on planar duality, which is why these exact values are a two-dimensional luxury with no analogue in d >= 3.
Consider an (n+1)-by-n rectangle in Z^2 bond percolation at p = 1/2. By the duality coupling, the probability of a left-right open primal crossing plus the probability of a top-bottom open dual crossing equals 1, and by the lattice symmetry these two are equal, so each is exactly 1/2. This exact crossing identity at p = 1/2 is the seed from which RSW grows uniform crossing bounds for all aspect ratios.
Self-duality at p = 1/2: primal and dual crossings are complementary and, by symmetry, equally likely.
p_c = 1/2 is exact only for BOND percolation on Z^2; site percolation on Z^2 has no closed form. And 1/2 is the transition location, not the density there — theta(1/2) = 0, so there is no infinite cluster at criticality.