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 $m\geq c n$. Part (b3), with constants rescaled, therefore gives
$$
\mathbb P(0\text{ lies in a finite open cluster of diameter}>n)
\leq Ce^{-cn}.
$$


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-209/2/b4/solution