%
\section{From the Wilson action to a diluted Villain model}\label{sec:2}

In this section we bound the Wilson loop by an average of Villain loops over a
Bernoulli set of frozen plaquettes, and we show that frozen plaquettes are
insulating walls.

\paragraph{Complexes and fields.} Let $X$ be a finite $2$-dimensional cell
complex with oriented edges $E(X)$, oriented plaquettes $P(X)$ and incidence
numbers $[p:e]$. For $\theta\in\mathbb{T}^{E(X)}$, $\mathbb{T}=\RR/2\pi\ZZ$, we set
$\theta_p:=\sum_e[p:e]\theta_e$, and for $n\in\ZZ^{P(X)}$, $(\partial
n)_e:=\sum_pn_p[p:e]$. An \emph{integer $1$-cycle} is a $C\in\ZZ^{E(X)}$ with zero
boundary, $\theta_C:=\sum_eC_e\theta_e$, and $Z_2(X):=\ker\partial$. The Villain
weight of Definition~\ref{def:1.1}, normalized by $\int\vt_t\,d\theta/2\pi=1$, is
$\vt_t(\theta)=\sum_{n\in\ZZ}e^{-n^2/(2t)}e^{in\theta}=(2\pi t)^{1/2}\sum_{m\in\ZZ}e^{-t(\theta-2\pi m)^2/2}>0$
by Poisson summation, and $\vt_t\to1$ uniformly as $t\to0$. Let
$\mu^{V}_{X,t}\propto\prod_p\vt_{t_p}(\theta_p)\prod_ed\theta_e$ for
$t\in(0,\infty)^{P(X)}$, and
$\mu^{W}_{X,\beta}\propto\prod_pe^{\beta\cos\theta_p}\prod_ed\theta_e$. For $\tau>0$,
$D\subseteq P(X)$ and $s>0$ let $t^{\tau,D,s}_p$ be $\tau$ off $D$ and $s$ on
$D$. Then the \emph{frozen expectation}
$g_{X,\tau}(C;D):=\lim_{s\to\infty}\ex{\cos\theta_C}^{V}_{X,t^{\tau,D,s}}$
exists by Lemma~\ref{lem:2.3}, and the plaquettes of $D$ are \emph{frozen}.

\paragraph{Ginibre's inequality.} Let $\mu\propto\exp(\sum_AJ_A\cos(A\cdot\phi))\,d\phi$
on a torus $\mathbb{T}^m$, where $A$ runs over a finite family of integer
vectors and $J_A\ge0$. Then for all integer vectors $B,B'$
\begin{equation}\label{eq:2.ginibre}
\ex{\cos(B\cdot\phi)}_\mu\ge0,\qquad
\ex{\cos(B\cdot\phi)\cos(B'\cdot\phi)}_\mu-\ex{\cos(B\cdot\phi)}_\mu\ex{\cos(B'\cdot\phi)}_\mu\ge0,
\end{equation}
by \cite[Propositions~2 and~3]{Gi70} applied to the torus of \cite[Example~4]{Gi70},
with $S$ the set of functions $\cos(m\cdot\phi)$. In particular,
$\partial\ex{\cos(B\cdot\phi)}_\mu/\partial J_A\ge0$ for all integer vectors $B$.

\subsection{Monotonicity}\label{ssec:2.1}

\begin{theorem}[Monotonicity for the Wilson action]\label{thm:2.1}
Let $C$ be an integer $1$-cycle supported in a box $\Lam\subset\Zthree$, in the
U(1) Wilson model with free boundary conditions. (a)
$\ex{e^{i\theta_C}}^{W}_{\Lam,\beta}=\ex{\cos\theta_C}^{W}_{\Lam,\beta}\in[0,1]$.
(b) $\ex{\cos\theta_C}^{W}_{\Lam,\beta}$ is nondecreasing in $\beta$, and
nondecreasing in $\Lam$ under inclusion. (c) Along every increasing sequence
of boxes $\Lam_k$ exhausting $\Zthree$ the limit
$\ex{\cos\theta_C}^{W}_{\beta}:=\lim_k\ex{\cos\theta_C}^{W}_{\Lam_k,\beta}$
exists, equals the supremum of $\ex{\cos\theta_C}^{W}_{\Lam,\beta}$ over all
boxes $\Lam$ containing the support of $C$, and is nondecreasing in $\beta$.
\end{theorem}

\begin{proof}
(a) and the monotonicity in $\beta$ are \eqref{eq:2.ginibre} with $J_p=\beta$,
together with the substitution $\theta\mapsto-\theta$. For the volume monotonicity, let
$\Lam\subseteq\Lam'$ be boxes. The expectation on
$\Lam'$ with coupling $s\in[0,\beta]$ on the plaquettes not in $\Lam$ is
nondecreasing in $s$ and equals the one on $\Lam$ at $s=0$, where the edges
outside $\Lam$ interact with nothing. (c) follows, since each box lies in
$\Lam_k$ for large $k$.
\qed
\end{proof}

\subsection{The Hartman\textendash Watson mixture and the annealed coupling field}\label{ssec:2.2}

\begin{lemma}[Hartman\textendash Watson mixture]\label{lem:2.2}
For every $\beta>0$ there is a unique probability measure $\omb$ on $(0,\infty)$ such
that, for all $\nu\ge0$ and $\theta\in\mathbb{T}$,
\begin{equation}\label{eq:2.hw}
\int_{(0,\infty)}e^{-\nu^2u/2}\,\omb(du)=\frac{I_\nu(\beta)}{I_0(\beta)},\qquad
\frac{e^{\beta\cos\theta}}{I_0(\beta)}=\int_{(0,\infty)}\vt_{1/u}(\theta)\,\omb(du),
\end{equation}
and $\omb((0,\eps))\le e^{\nu^2\eps/2}I_\nu(\beta)/I_0(\beta)$ for all $\eps>0$
and $\nu\ge0$.
\end{lemma}

\begin{proof}
By \cite[Theorem~1.3]{HW74}, $I_\nu(\beta)/I_0(\beta)=\int_0^\infty
e^{-\nu^2t}\,W(dt,\beta)$ ($0\le\nu<\infty$) with a distribution function
$W(\cdot,\beta)$ on $[0,\infty)$. We let $\omb$ be its image under $u=2t$, which has no
atom at $0$ because $I_\nu(\beta)\le(\beta/2)^\nu e^{\beta^2/4}/\Gamma(\nu+1)\to0$
as $\nu\to\infty$. The second identity follows from the first by Fourier
inversion and Fubini, as $\sum_nI_{\abs n}(\beta)/I_0(\beta)=e^\beta/I_0(\beta)<\infty$
(for the XY model it is \cite[Lemma~9.6]{AHPS}). The tail bound is the
Chernoff bound $\one_{u<\eps}\le e^{\nu^2(\eps-u)/2}$. It is unique, since a finite measure on $[0,\infty)$
is determined by its Laplace transform on $[0,\infty)$, which the first identity of \eqref{eq:2.hw} gives
at $\nu^2/2$ for every real $\nu\ge0$.
\qed
\end{proof}

\begin{lemma}\label{lem:2.3}
Let $X$ be a finite $2$-complex and $C$ an integer $1$-cycle. (a) For
$t\in(0,\infty)^{P(X)}$, $\ex{\sin\theta_C}^{V}_{X,t}=0$ and
$\ex{\cos\theta_C}^{V}_{X,t}\ge0$. (b) $\ex{\cos\theta_C}^{V}_{X,t}$ is
nondecreasing in each $t_p$. (c) (a) and (b) hold when some plaquettes are
removed (weight $1$, the limit $t_p\to0$); a removed plaquette gives a value no
larger than any $t_p>0$. (d) For fixed couplings off $D\subseteq P(X)$, the
limit as the couplings on $D$ tend to $\infty$ (jointly, or all equal to $s\to\infty$; by monotonicity the two
limits agree) exists, lies in $[0,1]$, and is
nondecreasing in each remaining coupling and in $D$; in particular $g_{X,\tau}(C;D)\in[0,1]$
exists and is nondecreasing in $D$ (for inclusion) and in $\tau$.
\end{lemma}

\begin{proof}
On a cube of $\Zthree$ with couplings in $(0,\infty)$, (b) is proved in
\cite[Corollary~1.6]{CG25}, and also in \cite[Lemma~C.1]{Gu80} for plaquette-dependent
couplings. We need the lemma for boxes and adapt these results to them as follows. Removal, $t_p\to0$,
is dominated convergence in \eqref{eq:2.dual}, whose terms decrease as
$t_p\downarrow0$ to those of the complex without $p$, and (c) follows. A box
is a cube with the outside plaquettes removed, and an integer $1$-cycle $C$ has
$\theta_C=\theta_\ell$ for one closed path $\ell$. Freezing, $t_p\to\infty$, is a
monotone limit in $[0,1]$. This yields (d), as $t^{\tau,D,s}\le t^{\tau,D',s}$
for $D\subseteq D'$, $s\ge\tau$. For (a) we use $\theta\mapsto-\theta$ and the fact that the value
with all plaquettes removed is $\int\cos\theta_C\,d\theta/(2\pi)^{\abs E}\ge0$.
This proves the lemma on boxes. In Section~\ref{S:proofsA} we prove the lemma on all finite $2$-complexes, by verifying Ginibre's
hypothesis for the Villain weights directly.
\qed
\end{proof}

\paragraph{The annealed coupling field.} Fix $X$ and $\beta>0$. Let
$Z(t):=\int\prod_p\vt_{t_p}(\theta_p)\,d\theta/(2\pi)^{\abs{E(X)}}>0$ for
$t\in(0,\infty)^{P(X)}$, and let $\mu_{\mathrm{ann}}$ be the probability measure on
$(0,\infty)^{P(X)}$ with density proportional to $Z(1/u)$, $1/u:=(1/u_p)_p$,
with respect to $\omb^{\otimes P(X)}$. The normalization is finite and positive
by \eqref{eq:2.hw} and Fubini. We also put $\thJ(u):=\sum_{n\in\ZZ}e^{-n^2u/2}$ and
$\delta_\beta(\eps):=\int_{(0,\eps)}\thJ(u)\,\omb(du)$.

\begin{lemma}[Annealed coupling field]\label{lem:2.4}
Let $X$ be a finite $2$-complex, $\beta>0$ and $\eps>0$.
(i) For every bounded measurable $F$ on $\mathbb{T}^{E(X)}$,
$\ex{F}^{W}_{X,\beta}=\int\ex{F}^{V}_{X,1/u}\,\mu_{\mathrm{ann}}(du)$.
(ii) (Tilt bound.) For every $p\in P(X)$ and every
$u_{-p}:=(u_q)_{q\ne p}$,
$\mu_{\mathrm{ann}}(u_p<\eps\mid u_{-p})\le\delta_\beta(\eps)$.
(iii) Under $\mu_{\mathrm{ann}}$ the random set $D_\eps(u):=\{p:u_p<\eps\}$ is
stochastically dominated by $\Ber(\delta')^{\otimes P(X)}$ for every
$\delta'\in[\min(\delta_\beta(\eps),1),1]$: $\int F(D_\eps(u))\,\mu_{\mathrm{ann}}(du)\le\EE_{D\sim\Ber(\delta')^{\otimes P(X)}}F(D)$
for every nondecreasing function $F$ on the subsets of $P(X)$.
(iv) $\delta_\beta(\eps)\le\inf_{\nu\ge0}e^{\nu^2\eps/2}\sum_{n\in\ZZ}I_{\sqrt{n^2+\nu^2}}(\beta)/I_0(\beta)$.
\end{lemma}

\begin{proof}
(i) We multiply the second identity of \eqref{eq:2.hw} over $p$, integrate
against $F$ and against $1$, and use Fubini.

(ii) The tilt is bounded because, as a function of one coupling, $Z$ is a combination of the functions
$e^{-n^2u_p/2}$ with nonnegative coefficients, each at most the constant
coefficient. We fix $p$ and $u_{-p}$. As
$\mu_{\mathrm{ann}}$ has density proportional to $Z(1/u)$ with respect to a
product measure, the conditional law of $u_p$ given $u_{-p}$ has density
proportional to $u_p\mapsto Z(1/u)$ with respect to $\omb$. Expanding
$\vt_{1/u_p}(\theta_p)=\sum_ne^{-n^2u_p/2}e^{in\theta_p}$, an absolutely
convergent series, we obtain $Z(1/u)=\sum_{n\in\ZZ}e^{-n^2u_p/2}A_n$,
$A_n:=\int e^{in\theta_p}\prod_{q\ne p}\vt_{1/u_q}(\theta_q)\,d\theta/(2\pi)^{\abs{E}}$,
with $A_0>0$ independent of $u_p$. Here $A_n/A_0=\ex{e^{in\theta_p}}$ in the system
with $p$ removed. Since $n\theta_p=\theta_{C'}$ for the integer $1$-cycle
$C'=n\,\partial p$, Lemma~\ref{lem:2.3}(a),(c) shows that $A_n$ is real and
$0\le A_n\le A_0$. Hence $1\le Z(1/u)/A_0\le\thJ(u_p)$, where the lower
bound comes from the term $n=0$, and $\mu_{\mathrm{ann}}(u_p<\eps\mid u_{-p})
=\int_{(0,\eps)}(Z/A_0)\,d\omb\big/\int_{(0,\infty)}(Z/A_0)\,d\omb
\le\int_{(0,\eps)}\thJ\,d\omb=\delta_\beta(\eps)$.

(iii) Let $Y_p:=\one[u_p\ge\eps]$ and fix an order $s_1,s_2,\dots$ of $P(X)$.
Since $Y_{s_1},\dots,Y_{s_j}$ are functions of $u_{-s_{j+1}}$, (ii) and the tower
property give $\PP(Y_{s_{j+1}}=1\mid Y_{s_1},\dots,Y_{s_j})\ge1-\min(\delta_\beta(\eps),1)$.
By \cite[Lemma~1.1]{LSS97} this yields the domination for $\delta'=\min(\delta_\beta(\eps),1)$,
and a monotone coupling shows that $\EE F(D)$ is nondecreasing in $\delta'$.

(iv) is the
Chernoff bound $\one_{u<\eps}\le e^{\nu^2(\eps-u)/2}$ integrated against
$\thJ(u)\,\omb(du)$, with \eqref{eq:2.hw}.
\qed
\end{proof}

\begin{theorem}[Wilson below diluted Villain]\label{thm:2.5}
For every finite $2$-complex $X$, integer $1$-cycle $C$, $\beta>0$,
$\eps>0$ and $\delta'\in[\min(\delta_\beta(\eps),1),1]$,
\begin{equation}\label{eq:2.W}
0\le\ex{\cos\theta_C}^{W}_{X,\beta}\le\EE_{D\sim\Ber(\delta')^{\otimes P(X)}}\,g_{X,1/\eps}(C;D).
\end{equation}
\end{theorem}

\begin{proof}
Lemma~\ref{lem:2.4}(i) represents the Wilson model as a Villain model with random
couplings $1/u_p$, and by monotonicity we round all couplings up, the weak
ones to $1/\eps$ and the strong ones to $\infty$. Positivity is a case of
\eqref{eq:2.ginibre}. For fixed $u$ and $D=D_\eps(u)$ we have $1/u_p\le1/\eps$ off
$D$, and hence Lemma~\ref{lem:2.3}(b),(d) give
$\ex{\cos\theta_C}^{V}_{X,1/u}\le g_{X,1/\eps}(C;D_\eps(u))$. We integrate this against
$\mu_{\mathrm{ann}}(du)$, use Lemma~\ref{lem:2.4}(i) once more, and apply
Lemma~\ref{lem:2.4}(iii) to the function $g_{X,1/\eps}(C;\cdot)$, which is nondecreasing
by Lemma~\ref{lem:2.3}(d).
\qed
\end{proof}

\subsection{The density of frozen plaquettes}\label{ssec:2.3}

From now on we take $\eps=1/(K\beta)$, so that the Villain coupling is $\tau=K\beta$,
and set $\delta:=\delta_\beta(1/(K\beta))$ and $\cdel(K):=-(\log\delta)/\beta$, a
function of $K$ and $\beta$, although the notation shows only $K$. Moreover $\delta>0$: $\thJ\ge1$ gives
$\delta\ge\omb((0,1/(K\beta)))$, and $\omb((0,\eps))=0$ would give $I_\nu(\beta)/I_0(\beta)\le e^{-\nu^2\eps/2}$
for all $\nu\ge0$ by \eqref{eq:2.hw}, against $I_\nu(\beta)\ge(\beta/2)^\nu/\Gamma(\nu+1)$. For
$K>e/2$ let $\astar=\astar(K)>e/2$ be the solution of $\astar=K\log(2\astar)$
(unique on $(e/2,\infty)$, where $a\mapsto a/\log 2a$ increases), and put
$\cdelsp(K):=\astar\bigl(\tfrac12\log2\astar-1\bigr)+1$.

\begin{lemma}\label{lem:2.6}
Let $K>e/2$ and $\beta\ge\pi^{-2}$. Then
\begin{equation}\label{eq:2.cdelta}
\log\delta\le-\beta\Bigl(\cdelsp(K)-\frac{1}{4\astar}\Bigr)+\log\bigl((2\astar\beta+5)\pi\sqrt\beta\bigr)+\frac12 ,
\end{equation}
so that $\cdel(K)\ge\cdelsp(K)-1/(4\astar)-O_K(\log\beta/\beta)$. The function
$K\mapsto\cdelsp(K)/(2K)$ is increasing on $(e/2,\infty)$; it equals
$14.9662$ at $K=377.3635$, and it is at least $14.9663$ for $K\ge\Kst=377.38$.
Consequently, for every $K>\Kst$ there are $\epsilon_K>0$ and $\beta_1(K)$
such that
\begin{equation}\label{eq:2.threshold}
\frac{\cdel(K)}{2K}\ge14.9662+\epsilon_K\qquad(\beta\ge\beta_1(K)).
\end{equation}
\end{lemma}

The number $14.9662$ is $6\pi^2\vCb(0)=14.96613052\ldots$ rounded up. Here
$\vCb(0)=(-\Delta_{\Zthree})^{-1}(0,0)=0.2527310\ldots$ is half of Watson's integral
\cite{Wat39} in the normalization of \cite{Zu11}, whose closed form is \cite[(3.6)]{Zu11} (Section~\ref{ssec:D.1}),
and the Debye exponent $\pi^2\vCb(0)=2.4943550871\ldots$ gives
$\log(1/\mt)=\pi^2\vCb(0)\,\tau+O(\log\tau)$ (Definition~\ref{def:3.1}). Thus, for $K>\Kst$,
\eqref{eq:2.threshold} says $\delta^{1/2}\le\mt^{\,6}e^{-\epsilon_K\tau+O(\log\tau)}$.
In the Peierls count of Lemma~\ref{lem:5.16} this means that one mark, of
probability at most $\delta^{1/2}$ (Definition~\ref{def:3.6}), outweighs the entropy $(\mt^{-1})^{6+o(1)}$ of
a root and a part.
For the first assertion of Theorem~\ref{thm:1.2} the value of $\Kst$ is not needed: with
$t:=\log2\astar(K)$ we have $K=e^t/(2t)$ and $\cdelsp(K)/(2K)=\frac14t(t-2)+te^{-t}$, which tends
to $\infty$ with $K$ (as $t>\log2K$), so every sufficiently large $K$ satisfies \eqref{eq:2.threshold}.

\begin{proof}
Roughly speaking, the heavy dual tail of the Wilson weight comes from $\omb$ near $u=0$.
Applying Lemma~\ref{lem:2.4}(iv) at its saddle point $\nu=\astar\beta$, with the series
of $I_\mu$, log-convexity of $\Gamma$, Stirling's inequality and
$I_0(\beta)\ge e^{\beta-1/2}/(\pi\sqrt\beta)$ for $\beta\ge\pi^{-2}$, we obtain
\eqref{eq:2.cdelta} (Section~\ref{ssec:D.3} gives the computation, the monotonicity and the two
values). For $K>\Kst$,
$\epsilon_K:=\frac12\bigl(\cdelsp(K)/(2K)-14.9662\bigr)\ge9\cdot10^{-5}$, since
$\cdelsp(K)/(2K)$ is increasing and, by the computation in Section~\ref{ssec:D.3},
equals $14.96639$ at $\Kst$. For these $K$ the value $14.9663$ of the statement gives
$\epsilon_K\ge5\cdot10^{-5}$, and as $1/(8K\astar)<10^{-7}$ for $K\ge\Kst$, either lower bound on $\epsilon_K$
shows that \eqref{eq:2.cdelta} implies \eqref{eq:2.threshold} once the
$O_K(\log\beta/\beta)$ term is below $K\epsilon_K$.
\qed
\end{proof}

\subsection{The dual representation and the frozen limit}\label{ssec:2.4}

Let $\Lam'$ be a box, regarded as a closed cubical complex with cubes
$\Lam'_3$, plaquettes $\Lam'_2$ and boundary plaquettes $\bdtwo\Lam'$, and let
$\partial:\ZZ^{\Lam'_3}\to\ZZ^{\Lam'_2}$ be the boundary map, cubes oriented
outward. Since the complex is contractible, $\partial$ is injective,
$Z_2(\Lam')=\partial\ZZ^{\Lam'_3}$, and each integer $1$-cycle $C$ in $\Lam'$
bounds an integer $2$-chain $n_0$.

\begin{lemma}[Dual representation]\label{lem:2.7}
Let $t\in(0,\infty)^{\Lam'_2}$, let $C$ be an integer $1$-cycle in $\Lam'$, and
let $n_0$ be any integer $2$-chain of $\Lam'$ with $\partial n_0=C$. Then
\begin{equation}\label{eq:2.dual}
\ex{\cos\theta_C}^{V}_{\Lam',t}=\frac{\sum_{k\in\ZZ^{\Lam'_3}}\prod_{p}e^{-(n_0+\partial k)_p^2/(2t_p)}}{\sum_{k\in\ZZ^{\Lam'_3}}\prod_pe^{-(\partial k)_p^2/(2t_p)}} .
\end{equation}
\end{lemma}

\begin{proof}
By the expansion
$\prod_p\vt_{t_p}(\theta_p)=\sum_{n\in\ZZ^{\Lam'_2}}\prod_pe^{-n_p^2/(2t_p)}e^{i(\partial n)\cdot\theta}$,
which converges absolutely and has even coefficients, $\ex{e^{i\theta_C}}=\ex{\cos\theta_C}$
is the ratio of the sums over
$\partial n=C$ and over $\partial n=0$. Since $Z_2(\Lam')=\partial\ZZ^{\Lam'_3}$ and $\partial$ is injective,
the chains $n=n_0+\partial k$,
$k\in\ZZ^{\Lam'_3}$, parametrize the first set bijectively for every choice of $n_0$ (a point that matters when
frozen plaquettes surround an edge of $C$).
\qed
\end{proof}

Dually, cubes are sites, an interior
plaquette is a bond between its two cubes, and a boundary plaquette is a bond
to an exterior ground site, where $k=0$. We now fix $\tau>0$ and a set $F\subseteq\Lam'_2$ of plaquettes to be frozen, and
let $t^{s}_p$ be $\tau$ for $p\notin F$ and $s$ for $p\in F$. The \emph{unfrozen graph} has
the cubes and the ground as vertices and the plaquettes not in $F$ as bonds, and $\mathcal K_F$ denotes
the set of its components not containing the ground. A cube
with all faces in $F$ is such a component. Fix $c_\KK\in\KK$ and put
$W:=\{\sum_{\KK\in\mathcal K_F}a_\KK\one_\KK:a\in\ZZ^{\mathcal K_F}\}$ and
$W':=\{k\in\ZZ^{\Lam'_3}:k(c_\KK)=0\ \text{for all}\ \KK\in\mathcal K_F\}$. Then
$\ZZ^{\Lam'_3}=W\oplus W'$, and $(\partial w)_p=0$ for $w\in W$, $p\notin F$.

\begin{lemma}\label{lem:2.8}
In this setting, for every integer $1$-cycle $C$ in $\Lam'$ and every integer
$2$-chain $n_0$ of $\Lam'$ with $\partial n_0=C$,
\begin{equation}\label{eq:2.frozen}
\lim_{s\to\infty}\ex{\cos\theta_C}^{V}_{\Lam',t^s}
=\frac{\sum_{k\in W'}\exp\bigl(-\frac{1}{2\tau}\sum_{p\notin F}(n_0+\partial k)_p^2\bigr)}{\sum_{k\in W'}\exp\bigl(-\frac{1}{2\tau}\sum_{p\notin F}(\partial k)_p^2\bigr)}.
\end{equation}
The quadratic form $k\mapsto\sum_{p\notin F}(\partial k)_p^2$ is positive
definite on $W'\otimes\RR$, so both sums converge; the summands are invariant
under $W$, so the sums run over $\ZZ^{\Lam'_3}/W$ ($k$ modulo one constant on
each component in $\mathcal K_F$) and do not depend on the choice of $W'$. In
particular every cube all of whose faces are frozen drops out.
\end{lemma}

\begin{proof}
Freezing deletes dual bonds. The only issue is the soft directions $W$, one
zero mode for each component in $\mathcal K_F$, isolated cubes included, such
as the cubes outside $\Lam'$ in Proposition~\ref{prop:2.9}. Along these directions only the frozen weights act, at strength $1/s$. Poisson summation over $w\in W$,
where $k=w'+w$ and the quadratic form of this summation is positive definite because $\partial$ is injective,
gives a prefactor that
cancels in the ratio times a factor in $[0,2]$ tending to $1$. A $w'$ with
$\one_{F^c}\partial w'=0$ is constant on components and vanishes on that of the
ground, which proves the positive definiteness stated in the lemma. Dominated convergence then yields
\eqref{eq:2.frozen} (see Section~\ref{S:proofsA} for the details of this proof).
\qed
\end{proof}

\subsection{One box with insulating walls}\label{ssec:2.5}

For a box $\Lam'$, $D\subseteq\Lam'_2\setminus\bdtwo\Lam'$ and $\tau>0$,
$\gfix_{\Lam',\tau}(C;D):=g_{\Lam',\tau}(C;D\cup\bdtwo\Lam')$ is the Villain
expectation on $\Lam'$ with the plaquettes of $D$ and the boundary plaquettes
frozen. We write $C\subset\Lam'\setminus\partial\Lam'$ if all edges
of $C$ lie in $\Lam'$ and none lies in the boundary of $\Lam'$.

\begin{proposition}\label{prop:2.9}\label{eq:2.onebox}
Let $C$ be an integer $1$-cycle, $\Lam'$ a box with
$C\subset\Lam'\setminus\partial\Lam'$, $K,\beta>0$, $\tau=K\beta$, $\delta$ as
in Section~\ref{ssec:2.3} and $\delta':=\min(\delta,1)$. For every finite box
$\Lam\supseteq C$,
$\ex{\cos\theta_C}^{W}_{\Lam,\beta}\le\EE_{D\sim\Ber(\delta')^{\otimes(\Lam'_2\setminus\bdtwo\Lam')}}\,\gfix_{\Lam',\tau}(C;D)$.
\end{proposition}

\begin{proof}
We use volume monotonicity and then freeze everything outside $\Lam'$. Let
$\Lam''\supseteq\Lam\cup\Lam'$ be a box and
$F_{\mathrm{out}}:=\Lam''_2\setminus(\Lam'_2\setminus\bdtwo\Lam')$. Then
Theorem~\ref{thm:2.1}(b), Theorem~\ref{thm:2.5} on $X=\Lam''$ with
$\eps=1/(K\beta)$, and Lemma~\ref{lem:2.3}(d) give
$\ex{\cos\theta_C}^{W}_{\Lam,\beta}\le\EE_{D''}\,g_{\Lam'',\tau}(C;D\cup F_{\mathrm{out}})$,
where $D''\sim\Ber(\delta')^{\otimes\Lam''_2}$ and $D:=D''\cap(\Lam'_2\setminus\bdtwo\Lam')$.
By Lemma~\ref{lem:2.8} on $\Lam''$ and on $\Lam'$ ($n_0$ in $\Lam'$), the right
side is $\gfix_{\Lam',\tau}(C;D)$, as the unfrozen plaquettes are the same, the
cubes outside $\Lam'$ are isolated, and the ground is not reached.
\qed
\end{proof}

\paragraph{Slit graphs.} Let $\Lam'$ be a box, $\FF\subseteq\Lam'_2$ with
$\bdtwo\Lam'\subseteq\FF$, and $\XX\subseteq\Lam'_3$ a set of cubes all of whose
faces lie in $\FF$. The \emph{slit graph} $\Gp=(V,E)$ has $V:=\Lam'_3\setminus\XX$ and
$E:=E(V)\setminus S$, with $E(V)$ the nearest-neighbour pairs in $V$ and $S$
those whose common plaquette lies in $\FF$. Conversely, each triple
$(\Lam',\XX,S)$, $S$ a set of bonds with both ends in $\Lam'_3\setminus\XX$,
arises in this way, with $\FF:=F(\XX)\cup F_S\cup\bdtwo\Lam'$, where $F(\XX)$ is the set of faces
of the cubes of $\XX$ and $F_S$ is the set of plaquettes dual to $S$. The phrase ``every slit graph''
means every such triple. As $E$ omits the bonds of $S$, a slit graph is in general not an induced subgraph
of $\Zthree$. Its components are denoted $\KK$ (an isolated vertex is a
component), its vertices have degree at most $6$, and
$d_{\Gp}(x,y)\ge\norm{x-y}_1$, with $d_{\Gp}=\infty$ across components. We orient
a bond $b=(x,y)\in E$ and its plaquette $p_b$ compatibly, with $p_b$ oriented as in $\partial
x$. For an integer $2$-chain $\Xi$ of $\Lam'$ with $\partial\Xi=C$ let
$\xi_b\in\ZZ$ be the coefficient of $p_b$ in $\Xi$, which for the flat rectangle
spanned by $C$ is the signed crossing number, in $\{0,\pm1\}$, and for
$k\in\ZZ$ put
\begin{equation}\label{eq:2.ZG}
Z_{\Gp}(k,\Xi):=\sum_{[m]}\exp\Bigl(-\frac{1}{2\tau}\sum_{b=(x,y)\in E}\bigl(m(y)-m(x)-k\xi_b\bigr)^2\Bigr),
\qquad \WG(C):=\frac{Z_{\Gp}(1,\Xi)}{Z_{\Gp}(0,\Xi)},
\end{equation}
where $[m]$ runs over the classes of $m\in\ZZ^V$ modulo functions constant on
each component.

\paragraph{Coarse and fine.} In Sections~\ref{sec:3} to~\ref{sec:mincut} the
hole set $\XX$ is a set of coarse sites and the frozen set $S$ is a set of coarse
bonds, as in Definition~\ref{def:3.7}, with the coarse slit graph
$(\Lamo\setminus\XX,E(\Lamo\setminus\XX)\setminus S)$. In view of Definition~\ref{def:3.4}, a coarse face carries
between $r_0^2$ and $\bar r_0^2$ fine plaquettes. On
the fine lattice the hole cubes are the fine cubes in sites of $\XX$, $F(\XX)$
is the set of their faces (Definition~\ref{def:3.6}), $F_S$ is the set of fine
plaquettes in the coarse faces dual to the bonds of $S$, and
\begin{equation}\label{eq:2.FF}
\FF:=F(\XX)\cup F_S\cup\bdtwo\Lamo .
\end{equation}
For the output $(\XX,S)=(\XX(D),S(D))$ of Theorem~\ref{thm:3.20} the fine slit
graph of this $\FF$ is the \emph{output} $\Gp=\Gp(D)$. Its vertices are the fine
cubes of $\Lamo$ outside the holes, and a bond at a vertex is missing from $E$
exactly when it leads into a hole, is frozen (dual to $F_S$), or leaves $\Lamo$.

\begin{proposition}[Frozen plaquettes are insulating walls]\label{prop:2.10}\label{eq:2.insul}
Let $\Lam'$, $\FF$, $\XX$ and $\Gp$ be as above, $\tau>0$, and $C$ an integer
$1$-cycle in $\Lam'$. For every $D\subseteq\FF\setminus\bdtwo\Lam'$,
$\gfix_{\Lam',\tau}(C;D)\le\gfix_{\Lam',\tau}(C;\FF\setminus\bdtwo\Lam')=\WG(C)$,
and $\WG(C)$ depends neither on the choice of $\Xi$ nor on the choice of $\XX$.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:2.3}(d), freezing raises the loop, and the inequality follows. Since freezing
also deletes a dual bond, holes, sheets, cuts and box walls are admissible. For
the identity we apply Lemma~\ref{lem:2.8} with $F=\FF$, $n_0=\Xi$. Here no
component reaches the ground, cubes with all faces in $\FF$ (those of $\XX$
among them) are isolated and drop out, and
$(\Xi+\partial k)_{p_b}=\xi_b+k(x)-k(y)$ gives $Z_{\Gp}(1,\Xi)/Z_{\Gp}(0,\Xi)$ with
$m=k|_V$. A change $\Xi'=\Xi+\partial U$ is absorbed by $m\mapsto m-U|_V$. On
outputs, $D\subseteq\FF(D)$ for all $D$ by Theorem~\ref{thm:3.20}, and hence
the inequality applies to every $D$.
\qed
\end{proof}

\begin{definition}[Min-cut area]\label{def:2.11}
For $\FF$ as in \eqref{eq:2.FF}, the \emph{min-cut area} of $C$ in $\Gp$ is
$\AG(C):=\min\{\#(\supp\Sigma\setminus\FF):\ \Sigma$ an integer $2$-chain of
$\Lamo$ with $\partial\Sigma=C\}$.
\end{definition}