For each $x\in H_N$, the event that $x$ belongs to an infinite open cluster implies that $x$ has an open path to $h_N$. Translation invariance gives probability $\theta$ for the former event. Therefore
$$
\mathbb E\#C_N=\sum_{x\in H_N}\mathbb P(x\in C_N)
\geq\theta\#H_N.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 2 / a1 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-209/2/a1/solution