Area law for three-dimensional U(1) lattice gauge theory with the Wilson action at every coupling
We prove that three-dimensional U(1) lattice gauge theory with the Wilson action confines static charges at every coupling. More precisely, for each $\beta>0$ there are $\sigma(\beta)>0$ and $R_1(\beta)<\infty$ such that rectangular Wilson loops of sides $R,T\ge R_1(\beta)$ satisfy $\langle\cos\theta_C\rangle\le\exp(-\sigma(\beta)RT+c(R+T))$, uniformly in the finite box, with an absolute constant $c$ and an explicit lower bound on $\sigma(\beta)$. For the Villain action this is the theorem of G\"opfert and Mack. For the Wilson action it is cited as known but has remained open. The proof…
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