A Sharp Deconfinement Transition for Potts Lattice Gauge Theory in Codimension Two

In 1983, Aizenman, Chayes, Chayes, Fröhlich, and Russo [1] proved that 2-dimensional Bernoulli plaquette percolation in $Z^3$ 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 $Z^d,$ 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 $Z^d$ 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.