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

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 $p_c\le 1/2$.

The Russo-Seymour-Welsh theorem and self-duality give a scale-independent $c>0$ such that every annuluscontains a closed circuit surrounding the origin with probability at least $c$. Choose every second annulus so that the corresponding site sets are disjoint. Their circuit events are then independent, and the Borel-Cantelli lemmas say that infinitely many of them occur almost surely.

Every such closed circuit separates the origin from infinity, so the origin cannot belong to an infinite open cluster. By translation invariance, the same holds for every site. Since the triangular lattice is countable, a countable union shows that almost surely no infinite open cluster exists at $p=1/2$.