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 $Ce^{-cn}$. A union bound over the $n$ listed points gives
$$
\mathbb P(B_n)\leq Cn e^{-cn}\leq n e^{-u n}
$$
after reducing $u>0$ and adjusting finitely many small $n$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-209/2/b2/solution