Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 209 / 2 / 1 / Solution

Suppose for contradiction that the critical probability for site percolation on the triangular lattice satisfied $p_c>1/2$. Then $p=1/2$ would be subcritical, so exponential decay of subcritical percolation would give constants $C,c>0$ such that the probability that a fixed site has an open path to distance $N$ is at most $C{e}^{-cN}$.

Every left-to-right open crossing of an $N$ by $N$ rhombus contains a site on its left side joined to distance at least $N$. There are $O(N)$ possible starting sites, so the union bound would implyPlanar self-duality and symmetry instead make this crossing probability exactly $1/2$ at every $N$. This contradiction proves $p_c\le 1/2$.