The Potts model offers a fundamental example of phase transition in statistical mechanics, and has been fruitfully studied using a coupling with the random-cluster model of percolation. We explore an analogous relationship in higher dimensions between Potts lattice gauge theory and a coupled random-cluster model of plaquette percolation. The associated Wilson loop variables can then be understood in terms of the topology of the cubical complex formed by the plaquettes. We also show that percolation with i-dimensional plaquettes in a 2i-dimensional torus exhibits a phase transition in its global topological properties.
No background in topology or percolation theory will be assumed.
This is based on joint work with Benjamin Schweinhart.