Uniform connected-subgraph limit of the random-cluster model
= Uniform connected-subgraph limit of the random-cluster model
On a finite connected graph, fixing $p=1/2$ and sending $q\downarrow0$ makes the random-cluster model converge to the uniform law on connected spanning subgraphs: the factor $q^{k(\omega)}$ selects the minimum possible component count $k=1$, while all surviving configurations have equal remaining weight.