In 1983, Aizenman, Chayes, Chayes, Fröhlich, and Russo [1] proved that 2-dimensional Bernoulli plaquette percolation in \(\mathbb {Z}^3\) exhibits a sharp phase transition for the event that a large rectangular loop is “bounded by a surface of plaquettes.” We extend this result both to \((d-1)\) -dimensional plaquette percolation in \(\mathbb {Z}^d,\) and to a dependent model of plaquette percolation called the plaquette random-cluster model. As a consequence, we obtain a sharp phase transition for Wilson loop expectations in \((d-2)\) -dimensional q-state Potts hyperlattice gauge theory on \(\mathbb {Z}^d\) dual to that of the Potts model. Our proof is unconditional for Ising lattice gauge theory, but relies on a regularity conjecture for the random-cluster model in slabs when \(q>2.\) We also further develop the general theory of the i-plaquette random cluster model and its relationship with \((i-1)\) -dimensional Potts lattice gauge theory.
