The stochastic bounds from part (b) hold uniformly in every box. Passing to the increasing limit for the local increasing event $A$ gives
$$
\mathbb P_{p/(2-p)}^{\mathrm{bond}}(A)
\leq\lim_{N\to\infty}P_N(A)
\leq\mathbb P_p^{\mathrm{bond}}(A).
$$
Thus the infinite-volume free random-cluster measure lies stochastically between Bernoulli bond percolation at parameters $p/(2-p)$ and $p$.


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 3 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-209/3/e/solution