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 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 -dimensional plaquette percolation in 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 -dimensional -state Potts hyperlattice gauge theory on 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 We also further develop the general theory of the -plaquette random cluster model and its relationship with -dimensional Potts lattice gauge theory.