Both $\{\#C_N\geq\theta\#H_N/2\}$ and $\{0\in C_N\}$ are increasing events. The first has probability at least $\theta/2$ by part (a2), while the second has probability at least $\theta$ because an infinite origin cluster reaches $h_N$. The <Harris-FKG inequality> gives
$$
\mathbb P(A_N)\geq\frac{\theta}{2}\theta=\frac{\theta^2}{2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 2 / a3 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-209/2/a3/solution