For $\omega\in\{0,1\}^{E(G)}$, write $o(\omega)$ and $c(\omega)$ for its numbers of open and closed edges and $k(\omega)$ for the number of connected components of the open spanning subgraph. The free <random-cluster model> is
$$
\mathbb P_{\mathrm{FK}}^{p,q}(\omega)
=\frac1{Z_G(p,q)}
p^{o(\omega)}(1-p)^{c(\omega)}q^{k(\omega)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 204 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-204/3/c/solution