Graphical representations are a powerful tool in the study of lattice spin models. We will present an extension of this idea to higher dimensions wherein Potts lattice gauge theory --- which assigns spins to edges rather than vertices --- is coupled with a dependent plaquette percolation called the plaquette random cluster model. We show that Wilson loop expectations in Potts lattice gauge theory equal the probability of a topological event in the plaquette random cluster model. Applications include a proofs that Wilson loop expectations in three dimensions undergo a sharp phase transition from area law to perimeter law behavior, and that the self-dual point in four dimensions is the threshold for the existence of giant plaquette surfaces in the sense of homological percolation. Based on joint work with Paul Duncan. If time permits, I will discuss recent work with Summer Eldridge and Malin Forsström on an extension of this representation to the Potts lattice Higgs model.