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

TL;DR

This paper proves a sharp phase transition in Potts lattice gauge theories and related models, extending classical percolation results to higher dimensions and dependent models, with implications for Wilson loop expectations.

Contribution

It extends sharp phase transition results to higher-dimensional plaquette percolation and Potts gauge theories, including dependent models like the plaquette random-cluster model.

Findings

Established sharp phase transition for $(d-2)$-dimensional Wilson loops.

Extended Aizenman et al.'s result to higher dimensions and dependent models.

Unconditional proof for Ising gauge theory, conditional for $q>2$.

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