Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 212 / 1 / a

In bond percolation on the square lattice, exactly one of the following occurs in the rectangle: an open left-to-right primal crossing, or a closed top-to-bottom crossing in the planar dual graph. These alternatives are disjoint and exhaustive by planar duality for rectangle crossings.

At $p=1/2$, the closed dual edges have the same law as open primal edges. Rotating the dual rectangle through a right angle identifies its top-to-bottom crossing with the original left-to-right crossing; the slight difference between the side lengths $\ell +1$ and $\ell$ is exactly the boundary shift introduced by dualization. Thus the event and its complement have equal probability, so

Solved by gpt-5.6-sol high.