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  \pdfinfo{/Title (Area law for three-dimensional U(1) lattice gauge theory with the Wilson action at every coupling) /Author (Tejasvi Singh Tomar)}
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\title{Area law for three-dimensional U(1) lattice gauge theory with the
Wilson action at every coupling}
\titlerunning{Area law for 3D U(1) Wilson lattice gauge theory}
\author{Tejasvi Singh Tomar}
\authorrunning{Tejasvi Singh Tomar}
\institute{ORCID: \url{https://orcid.org/0000-0003-4668-3639}}
\date{\normalsize Preprint. DOI: 10.5281/zenodo.23445737\\[0.5ex] October 2026}

\maketitle

\begin{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 $\beta>0$
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 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
$\beta$ is at most an average, over a sparse Bernoulli set of frozen
plaquettes, of the Villain loop at coupling $K\beta$, 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 $(1-\eta)RT$ outside an event of probability at most $e^{-RT+O(R+T)}$, with
$\eta\to0$ as $K\to\infty$. 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($n$) 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\textendash Glashow model at infinite Higgs coupling.
\keywords{lattice gauge theory \and confinement \and Wilson loop \and area law
\and Coulomb gas \and Debye screening \and Villain action}
\subclass{81T25 \and 82B20 \and 60K35}
\end{abstract}

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\appendix
\inputsec{appA}
\inputsec{appB}
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\inputsec{appF}
\inputsec{appG}

\section*{Declarations}
\noindent\textbf{Use of AI tools.} Generative AI tools were used to find and
prove the results, to write the programs and the Lean formalization, to write out the proofs of Part~S4 and to
referee them, and to draft the text.

\noindent\textbf{Formalization.} Theorem~\ref{thm:1.2}, Corollaries~\ref{cor:1.3}
and~\ref{cor:1.4}, Proposition~\ref{prop:9.1} and all results of Appendix~\ref{app:G} are formalized in
Lean~4 with Mathlib and checked by the Lean kernel, using only the three standard axioms of Lean and
Mathlib, as described in the companion paper \cite{TomLean}.

\noindent\textbf{Conflict of interest.} The author declares no conflict of
interest.

\noindent\textbf{Data availability.} No data were generated or analysed. The
scripts behind the numerical checks of Section~\ref{app:E} of the
supplementary material and behind Appendix~\ref{app:G} are in the file
\texttt{scripts.zip} of the \mbox{Zenodo} record of this paper,
\url{https://doi.org/10.5281/zenodo.23445737}. The exact computation in the
proof of Theorem~\ref{thm:G.8}, two sign conditions in exact rational interval
arithmetic, is carried out by the program \texttt{cube\_amonotone.py} of this
file, and the same inequalities are decided by the Lean kernel in the
formalization \cite{TomLean}. No other proof depends on the scripts.

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