The i-dimensional plaquette random-cluster model on a finite cubical complex is the random complex of i-plaquettes with each configuration having probability proportional to \(p^{\#\text { of plaquettes}}\left( 1-p \right) ^{\#\text { of complementary plaquettes}}q^{\textbf{b}_{i-1}},\) where \(q\ge 1\) is a real parameter and \(\textbf{b}_{i-1}\) denotes the rank of the \((i-1)\) -homology group with coefficients in a specified coefficient field. When q is prime and the coefficient field is \(\mathbb {F}_q\) , this model is coupled with the \((i-1)\) -dimensional q-state Potts lattice gauge theory. We prove that the probability that an \((i-1)\) -cycle in \(\mathbb {Z}^d\) is null-homologous in the plaquette random-cluster model equals the expectation of the corresponding generalized Wilson loop variable. This provides the first rigorous justification for a claim of Aizenman, Chayes, Chayes, Frölich, and Russo that there is an exact relationship between Wilson loop variables and the event that a loop is bounded by a surface in an interacting system of plaquettes. We also prove that the i-dimensional plaquette random-cluster model on the 2i-dimensional torus exhibits a sharp phase transition at the self-dual point \(p_{\textrm{sd}} :=\frac{\sqrt{q}}{1+\sqrt{q}}\) in the sense of homological percolation. This implies a qualitative change in the generalized Swendsen–Wang dynamics from local to non-local behavior.
