Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 2 / b4

By planar duality on the triangular lattice, a finite open cluster containing the origin is surrounded by a closed circuit. If its diameter exceeds $n$, an outermost surrounding circuit reaches distance of order $n$ and crosses the positive coordinate ray at some $(m,0)$ with $m\ge cn$. Part (b3), with constants rescaled, therefore gives