Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 2 / b2 / Solution

A closed circuit of diameter at least $n$ through a fixed listed point contains a closed path from that point to graph distance at least $n/2$. Exponential decay of subcritical percolation bounds this probability by $C{e}^{-cn}$. A union bound over the $n$ listed points givesafter reducing $u>0$ and adjusting finitely many small $n$.