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Area law for three-dimensional U(1) lattice gauge theory with the Wilson action at every coupling

A mix of human-written and AI-generated textHuman understanding: all partsmath-ph — Mathematical Physicsmath.PR — Probability

Contributed by Tejasvi Singh Tomar ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

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\beta>0 there are σ(β)>0\sigma(\beta)>0 and R1(β)<∞R_1(\beta)<\infty such that rectangular Wilson loops of sides R,T≥R1(β)R,T\ge R_1(\beta) satisfy ⟨cos⁡θC⟩≤exp⁡(−σ(β)RT+c(R+T))\langle\cos\theta_C\rangle\le\exp(-\sigma(\beta)RT+c(R+T)), uniformly in the finite box, with an absolute constant cc and an explicit lower bound on σ(β)\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 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\beta is at most an average, over a sparse Bernoulli set of frozen plaquettes, of the Villain loop at coupling KβK\beta, 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. It produces slit domains on which the multiscale analysis of G\"opfert and Mack, with wall fields graded by their periods around the handles, runs with uniform constants. 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(1-\eta)RT outside an event of probability at most e−RT+O(R+T)e^{-RT+O(R+T)}, with η→0\eta\to0 as K→∞K\to\infty. We carry over the steps cited from Brydges\textendash Federbush and G\"opfert\textendash Mack with explicit changes and prove them in full in Part S4 of the supplementary material, including 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(nn) 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\textendash Glashow model at infinite Higgs coupling.

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Version history

  1. v1Submitted by Tejasvi Singh TomarInitial depositCurrentOct 26, 2026