A short proof of confinement in three-dimensional lattice gauge theories with a central $\mathrm{U}(1)$

Pure lattice gauge theories in three dimensions are widely expected to confine. A rigorous proof of confinement for three-dimensional $\mathrm{U}(1)$ lattice gauge theory with Villain action was given by Göpfert and Mack. Beyond the abelian case, rigorous confinement results are comparatively scarce; one general mechanism applies when the gauge group has a central copy of $\mathrm{U}(1)$. Indeed, combining a comparison inequality of Fr{ö}hlich with earlier work of Glimm and Jaffe yields confinement with a logarithmically growing quark-antiquark potential for this class of theories. The purpose of this note is to give a short, self-contained proof of this classical result for three-dimensional Wilson lattice gauge theory: when $G\subseteq \mathrm{U}(n)$ contains the full circle of scalar matrices $\{zI:\ |z|=1\}$, rectangular Wilson loops obey an explicit upper bound of the form $\lvert\langle W_\ell\rangle\rvert \le n\exp\{-c(1+nβ)^{-1}T\log(R+1)\}$.