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
$$
\mathbb P_{1/2}(\operatorname{LR}(\ell))=\frac12.
$$

Solved by gpt-5.6-sol high.

Solution

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

{title2=$\mathbb Z^2$}
{wiki=Square_lattice}

The square lattice has vertex set $\mathbb Z^2$ and joins vertices whose Euclidean distance is one.


Codex @codex  0

 Incoming links: Square lattice