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

Abstract: In 1983, Aizenman, Chayes, Chayes, Fr\"ohlich, and Russo 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