% % \providecommand{\ZZ}{\mathbb{Z}} \providecommand{\RR}{\mathbb{R}} \providecommand{\one}{\mathbf{1}} \providecommand{\dist}{\operatorname{dist}} \providecommand{\ex}[1]{\langle #1\rangle} \providecommand{\abs}[1]{\lvert #1\rvert} \providecommand{\vt}{\tilde v} \providecommand{\omb}{\omega_{\beta}} \providecommand{\Kst}{K_{*}} \section{Comparing Wilson and Villain loops by correlation inequalities} \label{app:G} Numbers \ref{app:G}.$n$ refer to the statements and subsections of this appendix, while equation numbers (E.$n$), and statement numbers cited with \cite{GM82}, refer to Appendix~E of \cite{GM82}. Results for the Wilson action are often obtained by comparison with the Villain action, or by redoing the Villain analysis with additional estimates \cite[Section~6 and Appendix~B]{FS81KT}, \cite{FS82}. For upper bounds on Wilson loops, which is the direction needed for confinement, Theorem~\ref{thm:2.5} bounds the Wilson loop by an average of Villain loops in which a sparse random set of plaquettes is frozen. Freezing turns the problem into a Villain model on a domain with insulating walls, and this is where most of the length of this paper lies. It is natural to ask whether a better comparison avoids frozen plaquettes. Two facts are known. First, the character coefficients $I_k(\beta)/I_0(\beta)$ of the Wilson weight, with $I_k(\beta)\sim(\beta/2)^k/k!$ as $k\to\infty$, decay more slowly than those of any Villain weight, $e^{-k^2/(2t)}$, so no Villain plaquette weight dominates the Wilson weight coefficient by coefficient (Section~\ref{ssec:1.3}). Second, Ginibre's inequality \cite{Gi70} transfers to Villain weights through complexes that realize them as limits: cable graphs for spin systems, credited to Bellissard in \cite[Section~3]{FS81} and developed in \cite[Appendix~A]{AHPS} and \cite{DG25,DvEG}, and carpet graphs for lattice gauge theory \cite{CG25}. Such arguments give inequalities for all characters at once. To our knowledge, the following box version of the first fact is new. On every finite box, an inequality between Wilson and Villain loops that holds for all charges, even with a perimeter factor, forces the Villain side to contain arbitrarily strong couplings (Corollary~\ref{cor:G.4} and Proposition~\ref{prop:G.6}). The proof pairs a lower bound from Ginibre's inequality with the spin-wave upper bound for Villain loops, which we extend to Villain mixtures (Theorem~\ref{thm:G.2} and Remark~\ref{rem:G.5}). We complement it with three related results. Wilson loops are not monotone in the character coefficients of the plaquette weights (Theorem~\ref{thm:G.8}, proved by an exact computation). Every representation of a weight as a mixture of Villain weights carries an explicit minimal mass of strong couplings (Proposition~\ref{prop:G.10}). Freezing plaquettes has increasing returns: the effect of freezing two plaquettes can exceed the product of the effects of freezing each of them (Proposition~\ref{prop:G.12}). Section numbers of \cite{FS81KT} and \cite{FS81} refer to their preprints IHES/P/81/25 and IHES/P/80/10. \subsection{Setting}\label{ssec:G.1} \paragraph{Complexes and loops.} We use the notation of Section~\ref{sec:2} for a finite $2$-dimensional cell complex $X$: $E(X)$, $P(X)$, $[p:e]$, $\theta_p$, $\partial$, $Z_2(X)$ and $\theta_C$. For $n\in\ZZ^{P(X)}$ we have $\theta_{\partial n}=\sum_pn_p\theta_p$, and we set $\abs{\partial n}=\sum_e\abs{(\partial n)_e}$. We let $L(X)=\ker(\partial\otimes\RR)\subseteq\RR^{P(X)}$, with the standard inner product of $\RR^{P(X)}$. Since $\partial$ is an integer matrix, $Z_2(X)$ is a lattice of full rank in $L(X)$. We write $\delta_p\in\ZZ^{P(X)}$ for the indicator of $p$. In a box of $\Zthree$ every integer $1$-cycle is a boundary $\partial n$. \paragraph{Weights.} A \emph{plaquette weight} is an even continuous function $w\colon\mathbb{T}\to[0,\infty)$ whose Fourier coefficients $c_k(w)=\int_{\mathbb{T}}w(\varphi)e^{-ik\varphi}\,d\varphi/2\pi$ are nonnegative and summable, with $c_0(w)=1$. The Wilson weight is $w^{W}_{\beta}(\varphi)=e^{\beta\cos\varphi}/I_0(\beta)$, with $c_k(w^{W}_{\beta})=I_k(\beta)/I_0(\beta)$. The Villain weight $\vt_t$ of Section~\ref{sec:2} has $c_k(\vt_t)=e^{-k^2/(2t)}$ for $t>0$, and $\vt_0=1$ is a removed plaquette. A weight is of \emph{Ginibre type} if it is proportional to $e^{f}$, where $f(\varphi)=\sum_{j=1}^J\lambda_j\cos j\varphi$ with all $\lambda_j\ge0$. The weight $w^{W}_{\beta}$ is of Ginibre type. A \emph{Villain mixture with couplings at most $\tau$} is $w=\int_{[0,\tau]}\vt_t\,\nu(dt)$ for a probability measure $\nu$ on $[0,\tau]$. Its coefficients satisfy $c_k(w)\le e^{-k^2/(2\tau)}$ for $k\neq0$. \paragraph{Gibbs measures.} Given weights $w=(w_p)_{p\in P(X)}$, let $\mu_{X,w}$ be the probability measure on $\mathbb{T}^{E(X)}$ proportional to $\prod_pw_p(\theta_p)\prod_ed\theta_e/2\pi$, with expectation $\ex{\cdot}_{X,w}$. If $w_p=w^{W}_{\beta}$ for all $p$, this is the Wilson expectation $\ex{\cdot}^{W}_{X,\beta}$ of Section~\ref{sec:2}. For $n\in\ZZ^{P(X)}$ the Wilson loop of $\partial n$ is $\cos\theta_{\partial n}$. We use one monotonicity property: if every $w_p$ is of Ginibre type, with coefficients $\lambda_{p,j}$, then $\ex{\cos\theta_{\partial n}}_{X,w}$ is nondecreasing in every $\lambda_{p,j}$, by Ginibre's inequality \eqref{eq:2.ginibre}. \emph{Villain monotonicity} states that $\ex{\cos\theta_{\partial n}}_{X,w}$ is nondecreasing in every $t_p$ if $w_p=\vt_{t_p}$. It is proved on the lattice in \cite[Lemma~C.1]{Gu80}, on the cubes $\{x\in\ZZ^d:\abs{x}_\infty\le L\}$ in \cite[Corollary~1.6]{CG25}, and on finite two-dimensional complexes in Lemma~\ref{lem:2.3}(b). We do not use it in this appendix. \begin{lemma}\label{lem:G.1} For weights $w$ and $n\in\ZZ^{P(X)}$, \[ \ex{\cos\theta_{\partial n}}_{X,w}=\frac{\hat Z(n)}{\hat Z(0)},\qquad \hat Z(n)=\sum_{m\in Z_2(X)}\prod_{p\in P(X)}c_{\abs{n_p+m_p}}(w_p). \] \end{lemma} \begin{proof} Each series $w_p(\theta_p)=\sum_kc_k(w_p)e^{ik\theta_p}$ converges absolutely and uniformly. We expand the product of the weights times $e^{i\theta_{\partial n}}$ and integrate term by term. The term indexed by $l\in\ZZ^{P(X)}$ contributes $\prod_pc_{\abs{l_p}}(w_p)$ if $\partial(l+n)=0$, and nothing otherwise. Writing $l=m-n$ with $m\in Z_2(X)$ and replacing $m$ by $-m$ gives $\hat Z(n)$. The substitution $\theta\mapsto-\theta$ shows that $\ex{e^{i\theta_{\partial n}}}_{X,w}=\ex{\cos\theta_{\partial n}}_{X,w}$. The term $m=0$ gives $\hat Z(0)\ge1$. \qed \end{proof} For Villain weights on a box, Lemma~\ref{lem:G.1} is Lemma~\ref{lem:2.7}. \subsection{All-charge comparisons}\label{ssec:G.2} \begin{theorem}\label{thm:G.2} Let $n_0\in\ZZ^{P(X)}$ with $\partial n_0\neq0$, and put $\kappa=\dist(n_0,L(X))^2$. Then $\kappa>0$, and the following hold. \begin{itemize} \item[(a)] If every $w_p$ is a Villain mixture with couplings at most $\tau$, then $\ex{\cos k\theta_{\partial n_0}}_{X,w}\le e^{-\kappa k^2/(2\tau)}$ for every $k\in\ZZ$. \item[(b)] If every $w_p$ is of Ginibre type and $\partial\delta_{p_0}\neq0$, then $\ex{\cos k\theta_{p_0}}_{X,w}\ge c_k(w_{p_0})$ for every $k\ge0$. In particular $\ex{\cos k\theta_{p_0}}_{X,w}\ge I_k(\beta)/I_0(\beta)$ if $w_{p_0}=w^{W}_{\beta}$. \end{itemize} \end{theorem} The proof of (a) uses the following elementary fact. \begin{lemma}\label{lem:G.3} Let $\Gamma$ be a lattice of full rank in a finite-dimensional real inner product space $L$ with norm $\|\cdot\|$, and let $x\in L$. Then $\sum_{m\in\Gamma}e^{-\|m+x\|^2/2}\le\sum_{m\in\Gamma}e^{-\|m\|^2/2}$. \end{lemma} \begin{proof} By Poisson summation the left side equals $c\sum_{\xi\in\Gamma^*}e^{-2\pi^2\|\xi\|^2}e^{2\pi i\langle\xi,x\rangle}$, where $\Gamma^*$ is the dual lattice and $c>0$ depends only on $\Gamma$. All coefficients are positive, so the sum is largest at $x=0$. \qed \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:G.2}] Since $\partial n_0\neq0$, the vector $n_0$ does not lie in $L(X)$, so $\kappa>0$. (a) Write $w_p=\int\vt_t\,\nu_p(dt)$ and $\nu=\bigotimes_p\nu_p$. We have $c_j(\vt_t)=e^{-j^2/(2t)}$, read as $\one_{j=0}$ when $t=0$, and $c_j(w_p)=\int c_j(\vt_t)\,\nu_p(dt)$ by Fubini's theorem. By Lemma~\ref{lem:G.1} and Tonelli's theorem, $\hat Z(n)=\int\hat Z_t(n)\,\nu(dt)$ for $n\in\ZZ^{P(X)}$, where \[ \hat Z_t(n)=\sum_{m\in Z_2(X)}\prod_{p\in P(X)}e^{-(n_p+m_p)^2/(2t_p)}. \] It suffices to show $\hat Z_t(kn_0)\le e^{-\kappa k^2/(2\tau)}\hat Z_t(0)$ for every $t\in[0,\tau]^{P(X)}$. Let first $t\in(0,\tau]^{P(X)}$. We equip $\RR^{P(X)}$ with the inner product $\langle x,y\rangle_t=\sum_px_py_p/t_p$ and its norm $\|\cdot\|_t$, so that $\hat Z_t(n)=\sum_{m\in Z_2(X)}e^{-\|n+m\|_t^2/2}$. Let $\Pi_t$ be the orthogonal projection onto $L(X)$ for this inner product, and $\kappa_t=\|n_0-\Pi_tn_0\|_t^2$. For $m\in Z_2(X)$ we have $\|kn_0+m\|_t^2=k^2\kappa_t+\|k\Pi_tn_0+m\|_t^2$. Lemma~\ref{lem:G.3} with $\Gamma=Z_2(X)$ and $x=k\Pi_tn_0$ gives $\hat Z_t(kn_0)\le e^{-k^2\kappa_t/2}\hat Z_t(0)$. Since $t_p\le\tau$, we have $\|x\|_t^2\ge\abs{x}^2/\tau$ for all $x$, and hence $\kappa_t\ge\kappa/\tau$. If some coordinates of $t$ vanish, we replace them by $s\in(0,\tau]$. As $s\downarrow0$, every term of the resulting sums decreases to the corresponding term for $t$, and the inequality passes to the limit by dominated convergence. (b) By Ginibre's inequality, $\ex{\cos k\theta_{p_0}}_{X,w}$ decreases when all coefficients $\lambda_{p,j}$ with $p\neq p_0$ are lowered to $0$. Then $\mu_{X,w}$ is proportional to $w_{p_0}(\theta_{p_0})\prod_ed\theta_e/2\pi$. Since $\partial\delta_{p_0}$ is a nonzero integer vector, $\theta\mapsto\theta_{p_0}$ is a continuous homomorphism of $\mathbb{T}^{E(X)}$ onto $\mathbb{T}$, so it pushes Haar measure to Haar measure. Hence the expectation equals $\int\cos(k\varphi)w_{p_0}(\varphi)\,d\varphi/2\pi=c_k(w_{p_0})$. \qed \end{proof} \begin{corollary}\label{cor:G.4} Let $p_0\in P(X)$ with $\partial\delta_{p_0}\neq0$, and let $\beta>0$. Let $v=(v_p)_{p\in P(X)}$ be Villain mixtures with couplings at most $\tau$. For all constants $a,b\ge0$ there is $k\ge1$ with \[ \ex{\cos k\theta_{p_0}}^{W}_{X,\beta}>ae^{bk\abs{\partial\delta_{p_0}}}\,\ex{\cos k\theta_{p_0}}_{X,v}. \] In particular no inequality $\ex{\cos\theta_{\partial n}}^{W}_{X,\beta}\le ae^{b\abs{\partial n}}\ex{\cos\theta_{\partial n}}_{X,v}$ holds for all $n\in\ZZ^{P(X)}$. \end{corollary} \begin{proof} Let $\kappa_{p_0}=\dist(\delta_{p_0},L(X))^2>0$. By Theorem~\ref{thm:G.2} and $I_k(\beta)\ge(\beta/2)^k/k!\ge(\beta/2k)^k$, the left side is at least $(\beta/2k)^k/I_0(\beta)$ and the right side is at most $ae^{bk\abs{\partial\delta_{p_0}}-\kappa_{p_0}k^2/(2\tau)}$. The ratio of the two bounds tends to infinity as $k\to\infty$. \qed \end{proof} \begin{remark}\label{rem:G.5} (i) For $w_p=\vt_\tau$ on every plaquette, part (a) is the classical spin-wave upper bound: the Villain loop is at most the value $e^{-\kappa k^2/(2\tau)}$ of the same observable for the Gaussian spin wave. It follows from the decoupling of the Villain model into a spin wave and a Coulomb gas \cite[Proposition~5.2]{GS23}, and in three dimensions it gives the bound of Glimm and Jaffe \cite{GJ77}, see \cite[Corollary~7.3]{GS23}. In the proof above, Lemma~\ref{lem:G.3} bounds the Coulomb-gas factor by $1$. Part (a) adds uniformity over Villain mixtures with couplings at most $\tau$. (ii) For a box $\Lam$ of $\Zthree$, $L(\Lam)$ is spanned by the boundaries of the cubes of $\Lam$. These are the stars of the dual graph of Section~\ref{ssec:2.4}, whose sites are the cubes and the exterior ground site, as a network with unit conductances. The matrix of the orthogonal projection onto their span is the transfer-current matrix $Y$ \cite[Section~2.4]{LP16}. Hence $\dist(\delta_p,L(\Lam))^2=1-Y(p,p)$, and $Y(p,p)$ is the effective resistance across $p$ \cite[Section~4.2]{LP16}. For a single cube, $\kappa=5/6$. \end{remark} Comparisons are often made through auxiliary complexes. Let $G$ be a finite two-dimensional complex with plaquette weights $w$, and let $\gamma\in\ZZ^{P(G)}$ with $\partial\gamma\neq0$. The \emph{weight realized by $(G,\gamma,w)$} is the density $w_G$ of the law of $\theta_{\partial\gamma}$ under $\mu_{G,w}$ with respect to $d\varphi/2\pi$. Its coefficients are $c_k(w_G)=\ex{\cos k\theta_{\partial\gamma}}_{G,w}$. If a copy of $G$ is glued into a plaquette $p$ of $X$ along $\partial\gamma=\partial\delta_p$ in place of $w_p$, then integrating out the new edges replaces $w_p$ by $w_G$. Cable graphs and carpet graphs are of this kind. \begin{proposition}\label{prop:G.6} In this setting the following hold. \begin{itemize} \item[(a)] If every plaquette $p$ of $G$ carries a weight of Ginibre type, with coefficients $\lambda_{p,j}$, then each $c_k(w_G)$ is nondecreasing in every $\lambda_{p,j}$. \item[(b)] If every plaquette of $G$ carries a Villain mixture with couplings at most $\tau$, then $c_k(w_G)\le e^{-\kappa_Gk^2/(2\tau)}$ with $\kappa_G=\dist(\gamma,L(G))^2>0$. \end{itemize} In particular, let $\beta>0$. If $(G,\gamma,w)$ realizes $w^{W}_{\beta}$ and every weight of $G$ is of Ginibre type, then raising any of the coefficients $\lambda_{p,j}$ produces a weight $w'$ with $c_k(w')\ge I_k(\beta)/I_0(\beta)$ for all $k$, and no complex whose plaquettes carry Villain mixtures with bounded couplings realizes such a weight. \end{proposition} \begin{proof} Part (a) is Ginibre's inequality applied to $\cos k\theta_{\partial\gamma}$ on $G$, and part (b) is Theorem~\ref{thm:G.2}(a) on $G$ with $n_0=\gamma$. The last assertion follows, since $(\beta/2)^k/(k!\,I_0(\beta))$ is not bounded by $e^{-\kappa_Gk^2/(2\tau)}$ for large $k$. \qed \end{proof} \begin{remark}\label{rem:G.7} Ginibre's inequality, Villain monotonicity, complexes as in Proposition~\ref{prop:G.6}, and the annealed Hartman\textendash Watson representation with monotone rounding of the couplings (Theorem~\ref{thm:2.5}) all give inequalities for all characters at once. By Corollary~\ref{cor:G.4} none of them bounds Wilson loops by Villain loops with bounded couplings, even on one box. If the Villain side of such a comparison is a Villain mixture, its mixing measure gives positive mass to arbitrarily large couplings, and we bound the mass of strong couplings from below in Section~\ref{ssec:G.4}. This is why the comparison of Theorem~\ref{thm:2.5} keeps a random set of frozen plaquettes. \end{remark} \subsection{Monotonicity in character coefficients}\label{ssec:G.3} Ginibre's inequality gives monotonicity in the Fourier coefficients of the logarithm of a weight, and Villain monotonicity raises all coefficients $c_k(\vt_t)$ at once. Wilson loops are not monotone in each coefficient $c_k(w_p)$ of the weights themselves. Let $X$ be the boundary of the unit cube, with its $8$ vertices, $12$ edges and $6$ faces oriented by the outward normal, and fix a face $p_0$. For a weight $w$ and $x\ge0$ let $w_x=w+2(x-c_1(w))\cos\varphi$, so that $c_1(w_x)=x$ and $c_k(w_x)=c_k(w)$ for $k\neq\pm1$. Put $w_x$ on $p_0$ and $w$ on the other five faces, and let $\mathcal{W}(x)=\hat Z(\delta_{p_0})/\hat Z(0)$ as in Lemma~\ref{lem:G.1}. When $w_x\ge0$, this is $\ex{\cos\theta_{p_0}}$. \begin{theorem}\label{thm:G.8} Let $w$ be either the Wilson weight $w^{W}_{8}$ or the Villain weight $\vt_t$ with $e^{-1/(2t)}=15/16$. Then $\mathcal{W}$ is strictly decreasing on $[0,\infty)$. In particular, for $00$ for $w=\vt_t$. Moreover, if every face carries $w+2s\cos\varphi$, then $\frac{d}{ds}\ex{\cos\theta_{p_0}}$ is negative at $s=0$. \end{theorem} \begin{proof} The complex $X$ is a sphere, so $Z_2(X)=\ZZ\one_{P(X)}$, where $\one_{P(X)}=\sum_p\delta_p$. Write $c_k=c_k(w)$. Lemma~\ref{lem:G.1} with $n=\delta_{p_0}$ and $m=-h\one_{P(X)}$ gives $\hat Z(\delta_{p_0})=\sum_{h\in\ZZ}c'_{\abs{h-1}}c_{\abs{h}}^5$ and $\hat Z(0)=\sum_hc'_{\abs{h}}c_{\abs{h}}^5$, where $c'$ denotes the coefficients of the weight on $p_0$. Collecting the terms that contain $c'_1=x$ we get \[ \mathcal{W}(x)=\frac{A_C+xB_C}{A_0+xB_0},\qquad \begin{aligned} A_0&=1+2\textstyle\sum_{h\ge2}c_h^6, & B_0&=2c_1^5,\\ A_C&=c_1^5+\textstyle\sum_{h\ge3}c_{h-1}c_h^5+\sum_{h\ge1}c_{h+1}c_h^5, & B_C&=1+c_2^5. \end{aligned} \] Hence $\mathcal{W}'(x)=S/(A_0+xB_0)^2$ with $S=A_0B_C-A_CB_0$, a number independent of $x$. It remains to show $S<0$. If every face carries $w+2s\cos\varphi$, the derivative at $s=0$ has the sign of $U=Z_C'Z_0-Z_CZ_0'$, where $Z_C=A_C+c_1B_C$, $Z_0=A_0+c_1B_0$, $Z_C'=B_C+5c_1^4(1+c_2)$ and $Z_0'=12c_1^5$. We evaluate $S$ and $U$ in exact rational arithmetic with intervals. Both are homogeneous of degree $11$ in $(c_k)$, so for $w^{W}_{8}$ we may replace $c_k$ by $a_k=I_k(8)=\sum_{m\ge0}4^{2m+k}/(m!\,(m+k)!)$. The partial sums with $m<80$ are rational lower bounds. Consecutive terms have ratio at most $16/(81(81+k))$ beyond $m=80$, so we bound the remainder by a geometric series. Comparing the series term by term gives $I_{k+1}(8)\le 4I_k(8)/(k+1)$, and with it we bound the parts of the sums over $h$ with $\abs{h}>40$ by geometric series. For $\vt_t$ the coefficients $c_k=(15/16)^{k^2}$ are rational, and we bound the parts with $\abs{h}>40$ again by geometric series. All operations are exact, with endpoints rounded outward to $400$ significant bits. Running the program \texttt{cube\_amonotone.py} of the script archive released with the paper (see the Data availability statement), we obtain $S<0$ and $U<0$ in both cases. Finally, $\min w^{W}_{8}=e^{-8}/I_0(8)$ is attained at $\varphi=\pi$, and $\vt_t>0$ is continuous. For $00$. Heuristically, for $w^{W}_{\beta}$ or $\vt_t$ the dual variable $h$ of the proof is close to a centered Gaussian of variance about $\beta/6$ or $t/6$. On $p_0$ the loop replaces $h$ by $h-1$, whose law is close to a Gaussian with the same variance and mean $-5/6$. When the variance exceeds $1$, shifting a Gaussian lowers the probability of $\{\abs{h}=1\}$ to second order. \end{remark} \subsection{Villain-mixture representations}\label{ssec:G.4} A \emph{Villain-mixture representation} of a plaquette weight $w$ is a probability measure $\omega$ on $(0,\infty)$ with $c_k(w)=\int e^{-k^2u/2}\,\omega(du)$ for all $k\in\ZZ$. The variable $u$ is the inverse coupling, so that $w=\int\vt_{1/u}\,\omega(du)$. For $w^{W}_{\beta}$ the Hartman\textendash Watson law $\omb$ of Lemma~\ref{lem:2.2} is such a representation \cite[Theorem~1.3]{HW74}. Yor \cite{Yo80} described its density, and Patie \cite[Remark~2.7]{Pat09} relates it to the infinitely divisible laws of Hartman \cite{Har76}. It is used in comparisons in \cite[(9.10)]{AHPS} and in Section~\ref{sec:2}. By Lemma~\ref{lem:2.2} it is the only probability measure that reproduces $I_\nu(\beta)/I_0(\beta)$ for every real $\nu\ge0$. \begin{proposition}\label{prop:G.10} Let $\omega$ be a Villain-mixture representation of a weight $w$ with coefficients $c_k$, let $\eps>0$, and let $1\le k_1e^{-k_1^2\eps/2}$. Then \[ \omega((0,\eps))\ \ge\ \bigl(c_{k_1}-e^{-k_1^2\eps/2}\bigr)^{k_2^2/(k_2^2-k_1^2)}\,c_{k_2}^{-k_1^2/(k_2^2-k_1^2)}. \] Moreover $\omega((0,\eps))\le e^{k^2\eps/2}c_k$ for every $k$. \end{proposition} In terms of couplings, $\omega((0,\eps))$ is the mass of couplings above $1/\eps$. For $\omega=\omb$ the last bound is the tail bound of Lemma~\ref{lem:2.2}. \begin{proof} Let $F(s)=\int_{(0,\eps)}e^{-su}\,\omega(du)$ for $s\ge0$. By H\"older's inequality $\log F$ is convex. Put $s_i=k_i^2/2$. Then $F(s_2)\le c_{k_2}$, and $F(s_1)\ge c_{k_1}-\int_{[\eps,\infty)}e^{-s_1u}\omega(du)\ge c_{k_1}-e^{-s_1\eps}>0$. Convexity on $[0,s_2]$ gives $\log F(s_1)\le(1-s_1/s_2)\log F(0)+(s_1/s_2)\log F(s_2)$, that is \[ \log F(0)\ge\frac{s_2\log F(s_1)-s_1\log F(s_2)}{s_2-s_1}, \] and the bounds on $F(s_1)$ and $F(s_2)$ give the claim. The upper bound is the Chernoff bound $\one_{u<\eps}\le e^{k^2(\eps-u)/2}$. \qed \end{proof} \begin{remark}\label{rem:G.11} (i) Only integer frequencies constrain $\omega$, and the representation is then not unique in general. Suppose some representation has a density bounded below by a positive constant on an interval $[u_1,u_2]\subset(0,\infty)$. In the variable $x=e^{-u/2}$ the constraints read $\int x^{k^2}\,\tilde\omega(dx)=c_k$. Let $[a,b]$ be a closed interval inside $(e^{-u_2/2},e^{-u_1/2})$. Since $\sum_{k\ge1}k^{-2}<\infty$, the functions $x^{k^2}$, $k\ge0$, do not span a dense subspace of $C[a,b]$ \cite{CE43}, \cite[Theorem~29]{Alm07}. Hence some signed measure $\zeta\neq0$ on $[a,b]$ annihilates all of them. The image of $\zeta$ under a dilation $x\mapsto sx$ annihilates them too, since $\int(sx)^{k^2}\zeta(dx)=s^{k^2}\int x^{k^2}\zeta(dx)=0$. We average these images against a continuous probability density in $s$ supported near $1$. The average is $h(x)\,dx$ with $h$ continuous and supported in $[e^{-u_2/2},e^{-u_1/2}]$, and it annihilates all $x^{k^2}$. As the support of the density shrinks to $\{1\}$, the averages converge weakly to $\zeta$, so $h\neq0$ for a small enough support. Adding a small multiple of $h\,dx$ gives another representation. Proposition~\ref{prop:G.10} applies to all of them. (ii) Numerical. For $w^{W}_{\beta}$ with $\eps=1/(K\beta)$ and $\beta\to\infty$, we choose $k_1$ just above the crossing of $I_k(\beta)/I_0(\beta)$ and $e^{-k^2\eps/2}$ and $k_2=k_1+1$. At leading order Proposition~\ref{prop:G.10} then gives mass at least $e^{-63.02\tau}$ above the coupling $\tau=K\beta$ when $K=\Kst$, while the upper bound gives $e^{-29.93\tau}$, the order of the density $\delta$ of frozen plaquettes in Lemma~\ref{lem:2.6}. Hence, at leading order, the analogue of the condition \eqref{eq:2.threshold} on the density of frozen plaquettes fails for every representation when $K\le45.3$. \end{remark} \subsection{Freezing has increasing returns}\label{ssec:G.5} Let $X$ and $p_0$ be as in Theorem~\ref{thm:G.8}, let $\tau>0$, and let $D$ be a set of faces with $p_0\notin D$. Put $\vt_\tau$ on the faces outside $D$ and $\vt_s$ on those in $D$, and let $g(D)=\lim_{s\to\infty}\ex{\cos\theta_{p_0}}$. By monotone convergence in Lemma~\ref{lem:G.1} the limit exists and is given by the dual formula with $c_k=1$ on $D$, which is finite since $p_0\notin D$. In the notation of Section~\ref{sec:2}, $g(D)$ is the frozen expectation $g_{X,\tau}(\partial\delta_{p_0};D)$. The effect of a frozen set is not bounded by the product of the effects of its plaquettes. \begin{proposition}\label{prop:G.12} For $\tau\ge3$ and distinct faces $p,p'\neq p_0$, \[ g(\{p,p'\})\,g(\emptyset)>g(\{p\})\,g(\{p'\}). \] Hence $D\mapsto\log g(D)$ is not submodular, and $g(D)/g(\emptyset)\le\prod_{q\in D}g(\{q\})/g(\emptyset)$ fails for $D=\{p,p'\}$. \end{proposition} \begin{proof} By Lemma~\ref{lem:G.1} the value $g(D)$ depends only on $d=\abs{D}$. Write $g_d$ for it. The dual formula gives $g_d=\sum_he^{-((h-1)^2+(5-d)h^2)/(2\tau)}\big/\sum_he^{-(6-d)h^2/(2\tau)}$. Completing the square, \[ \log g_d=-\frac{5-d}{2\tau(6-d)}+\log R_d,\qquad R_d=\frac{\Theta_d(1/(6-d))}{\Theta_d(0)},\quad \Theta_d(y)=\sum_{h\in\ZZ}e^{-(6-d)(h-y)^2/(2\tau)}. \] By Poisson summation $\Theta_d(y)$ is a positive multiple of $1+2\sum_{m\ge1}e^{-2\pi^2\tau m^2/(6-d)}\cos(2\pi my)$. This gives $(1-2\varrho)/(1+2\varrho)\le R_d\le1$ with $\varrho=e^{-\pi^2\tau/3}/(1-e^{-\pi^2\tau/3})$. The claim is $\log g_2-2\log g_1+\log g_0>0$. The leading terms contribute $\frac1\tau(-\frac38+\frac45-\frac5{12})=\frac1{120\tau}$, and the terms $\log R_d$ contribute at least $-2\lambda$ with $\lambda=\log\frac{1+2\varrho}{1-2\varrho}$. For $\tau\ge3$ we have $\varrho<5.2\cdot10^{-5}$ and $\lambda<2.1\cdot10^{-4}$, so $240\tau\lambda<1$ at $\tau=3$. Since $\tau\lambda$ decreases in $\tau$ for $\tau\ge3$, the claim follows. \qed \end{proof}