  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 $$p^{\text{# of plaquettes}}(1-p)^{\text{# of
complementary plaquettes}}q^{\mathbf{ b}_{i-1}},$$ where $q\geq 1$ is a real
parameter and $\mathbf{b}_{i-1}$ 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, Fr\"olich, 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
$p_{\mathrm{sd}} \mathrel{\vcenter{:}}= \frac{\sqrt{q}}{1+\sqrt{q}}$ in the
sense of homological percolation. This implies a qualitative change in the
generalized Swendsen--Wang dynamics from local to non-local behavior.
