Kesten's theorem on planar bond percolation

Remarks

The key tool is self-duality. The dual lattice, with a vertex at the centre of each square face, is a translate of $\Z^2$. A dual edge is declared open when the primal edge it crosses is closed. At $p = 1/2$ the primal and dual configurations have the same law. A box crossing argument then shows that a suitable rectangle is crossed by an open path with probability exactly $1/2$. Harris proved in 1960 that $\theta(1/2) = 0$, which gives $p_c \ge 1/2$; Kesten proved in 1980 that $p_c \le 1/2$. So there is no infinite open cluster at $p = 1/2$ itself, while for every $p > 1/2$ there is one almost surely.

Going deeper

For a student

It is the model case of an exact percolation threshold computed by planar duality. The same duality idea gives $p_c = 1/2$ for site percolation on the triangular lattice. Estimates on the probability that open paths cross long rectangles are the main tool of the proof. In dimensions $d \ge 3$ no exact value of $p_c(\Z^d)$ is known.