The Plaquette Random Cluster Model and Potts Lattice Gauge Theory

The classical q-state Potts model of interacting spins in the integer lattice Zd – of which the Ising model is the special case q=2 – is one of the most important models in statistical physics and probability. It is often studied via a coupling with the Fortuin-Kasteleyn random cluster model of dependent bond percolation. In our talk, we describe how to generalize these models to higher-dimensional cubical complexes by defining q-state Potts lattice gauge theory and the plaquette random cluster model. When q is a prime integer, we show that the expectation of a Wilson loop variable in Potts lattice gauge theory (an analogue of its namesake in quantum field theory) equals the probability that the loop is null-homologous in the corresponding plaquette random cluster model. We also prove that the i-dimensional plaquette random cluster model in the 2i-dimensional torus exhibits a sharp phase transition in the sense of homological percolation: that is, the emergence of giant cycles which are non-trivial homology classes in the ambient torus.