Fortuin–Kasteleyn random cluster model

Remarks

For $q = 1$ the factor $q^{k(\omega)}$ is $1$, and $\phi_{p, 1}$ is Bernoulli bond percolation: each edge is open independently with probability $p$. For integer $q \ge 2$ and $p = 1 - e^{-\beta}$, the Edwards–Sokal coupling relates $\phi_{p, q}$ to the $q$-state Potts model with energy $-\sum_{\set{x, y} \in E} \delta_{\sigma_x, \sigma_y}$ at inverse temperature $\beta$. Under this coupling, $\phi_{p, q}$ is the law of the open edges, and the Potts spins are obtained by giving each open cluster an independent uniform colour. Two Potts spins at $x$ and $y$ are then correlated through the event that $x$ and $y$ lie in the same open cluster. Non-example: the measure proportional to $p^{\abs{\omega}} (1 - p)^{\abs{E \setminus \omega}} q^{\abs{\omega}}$ is not of this form for $q \ne 1$, since its extra weight counts edges, not components; it is just Bernoulli percolation with another parameter. The partition function $Z$ is, up to a change of variables, the Tutte polynomial of $G$.