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 β > 0 there are σ(β) > 0 and R1(β) < ∞ such that rectangular Wilson loops of sides R, T ≥ R1(β) satisfy ⟨cos θC⟩ ≤ exp(−σ(β)RT + c(R + T)), uniformly in the finite box, with an absolute constant c and an explicit lower bound on σ(β). For the Villain action this is the theorem of Göpfert and Mack. For the Wilson action it is cited as known but has remained open. The proof rests on a comparison. The Wilson plaquette weight is a mixture of Villain weights whose coupling has a very light upper tail. Hence, by Ginibre's inequality, the Wilson loop at coupling β is at most an average, over a sparse Bernoulli set of frozen plaquettes, of the Villain loop at coupling Kβ, and frozen plaquettes are insulating walls for the monopole gas. A variational regularization (built on the smallest minimizer of a cut functional) freezes more plaquettes, which only increases the loop, and produces slit domains on which the multiscale analysis of Göpfert and Mack runs with uniform constants, with wall fields graded by their periods around the handles. On these domains the loop decays like the exponential of the minimal spanning area with frozen plaquettes free. A Peierls estimate shows that this area is at least (1 − η)RT outside an event of probability at most e−RT+O(R+T), with η → 0 as K → ∞. We carry over the steps cited from Brydges–Federbush and Göpfert–Mack with explicit changes, and we prove in full a tree-graph counting step of Brydges whose printed deduction is incomplete. As corollaries, we obtain the area law at every coupling for all closed subgroups of U(n) containing the scalars, also with a Higgs field blind to the centre (as for U(2) with an adjoint Higgs field), and for the Georgi–Glashow model at infinite Higgs coupling. The PDF contains the paper and its supplementary material (Sections S1 to S3). Files: u1-wilson-area-law.pdf (the paper with the supplementary material), source.zip (the LaTeX source) and scripts.zip (the scripts behind the numerical checks of Section S3.1 and the threshold values of Section S1.1.2, with their outputs; no proof depends on them). Mathematics Subject Classification (2020): 81T25, 82B20, 60K35.