\section{The six-vertex model as a gradient model}\label{sec:gradient}

This section places the six-vertex model in Sheffield's framework~\cite{Sheffield}. In a shifted height
variable the ice rule becomes a nearest-neighbour constraint and the weights become an interaction
along the diagonals of each vertex (Lemma~\ref{lem:weights}, Figure~\ref{fig:vertices}). This makes the
model a perturbed Lipschitz simply attractive potential, so Sheffield's variational principle applies
(Proposition~\ref{prop:LSAP}, Lemma~\ref{lem:SFE}); but for $c\ne1$ no nearest-neighbour potential has the
same specification (Proposition~\ref{prop:notNN}), so his uniqueness theorems do not apply. The last
subsection introduces the pairs with residual energies on which the swaps of Section~\ref{sec:swaps} act.

\subsection{Heights, blocks and weights}

The faces of $\Z^2$ form the set $F=\Z^2$, a face being indexed by its lower left corner. For $x\in F$
the \emph{block} $B(x)=\{x,x+e_1,x+e_2,x+e_1+e_2\}$ is the set of four faces around one vertex of
$\Z^2$. Its two \emph{diagonals} are $d^+_x=\{x,x+e_1+e_2\}$ and $d^-_x=\{x+e_1,x+e_2\}$; $\Diag$ is the
set of all diagonals and $d^\perp$ denotes the other diagonal of the block of $d$. Faces are
\emph{NN-adjacent} if $|x-y|_1=1$, and \emph{$*$-adjacent} if they are NN-adjacent or form a diagonal.

Fix the parity convention $h(x)\equiv x_1+x_2\pmod2$ and put $\psi(x):=(h(x)+x_1+x_2)/2$, the
representation of \cite[Section~4.4]{LammersTassy}. Let $\Omega$ be the set of $\psi:F\to\Z$ with
$\psi(x+e_i)-\psi(x)\in\{0,1\}$ for all $x$ and $i=1,2$. Height functions of arrow configurations and
elements of $\Omega$ correspond bijectively up to additive constants, and arrow configurations
correspond to gradients of elements of $\Omega$, equivariantly under translations. For
$\psi\in\Omega$ and a diagonal $d$ put
\[
\mathsf a(\psi;d^+_x):=\ind{\psi(x+e_1+e_2)-\psi(x)=1},\qquad \mathsf a(\psi;d^-_x):=\ind{\psi(x+e_2)=\psi(x+e_1)} ;
\]
in both cases $\mathsf a(\psi;d)=1$ exactly when $h$ takes the same value on the two faces of $d$.

\begin{lemma}[Vertex weights]\label{lem:weights}
In every block $\mathsf a(\psi;d^+_x)+\mathsf a(\psi;d^-_x)\in\{1,2\}$, and the weight of the vertex surrounded by
$B(x)$ is
\[
w_x(\psi)=c^{\,\mathsf a(\psi;d^+_x)+\mathsf a(\psi;d^-_x)-1}.
\]
\end{lemma}

\begin{proof}
Going around the vertex, the height makes four steps $\pm1$ with sum zero. If the steps alternate in
sign, the heights are $m,m+1,m,m+1$, both diagonal pairs are equal, and these are the two
configurations in which the horizontal arrows both point in or both point out. Otherwise the heights
are $m,m+1,m+2,m+1$ up to rotation, one diagonal pair is equal and the other differs by $2$, and the
weight is $1$.
\end{proof}

This is the form of the weight used in \cite[(248)]{DKLM}, where the weight of a configuration is $c$ to the
number of diagonals across which the height is constant.

\input{fig-vertices}

For a finite set $\Lambda\subset F$ and $\psi\in\Omega$, let $\gamma_\Lambda(\cdot\mid\psi)$ be the
probability measure on the elements of $\Omega$ that agree with $\psi$ off $\Lambda$, with weights
proportional to $\prod_{x:\,B(x)\cap\Lambda\ne\emptyset}w_x$. The kernels $\gamma_\Lambda$ form the
\emph{six-vertex specification}; they commute with adding a constant to $\psi$. A \emph{Gibbs measure on
heights} is a probability measure $\nu$ on $\Omega$ with $\nu\gamma_\Lambda=\nu$ for every finite
$\Lambda$. A \emph{gradient Gibbs measure} is a probability measure $\mu$ on gradient configurations
$\nabla\psi$ (equivalently, on arrow configurations) with $\mu\gamma_\Lambda=\mu$ for every finite
$\Lambda$, where $\gamma_\Lambda$ acts on $\psi$ modulo constants. A gradient Gibbs measure is
\emph{smooth} if it is the gradient law of a Gibbs measure on heights, and \emph{rough} otherwise
\cite[p.~10]{Sheffield}.

\subsection{Sheffield's framework}

\begin{proposition}[A perturbed Lipschitz simply attractive potential]\label{prop:LSAP}
For every $c>0$ the six-vertex specification is, in the variable $\psi$ with values in $\Z$, the
gradient specification of a perturbed Lipschitz simply attractive potential in the sense of
\cite[Section~1.1.4]{Sheffield}.
\end{proposition}

\begin{proof}
Fix $K>2|\log c|$. Let $\Phi$ be the nearest-neighbour potential whose pair function on each pair
$(x,x+e_i)$ is $V(t)=K$ for $t\in[0,1]$ and $V(t)=+\infty$ otherwise. It is positive, convex and
invariant under $\Z^2$, and it confines $\psi(x+e_i)-\psi(x)$ to $[0,1]$, hence to $\{0,1\}$: a
Lipschitz simply attractive potential. Let $\Psi$ assign to every diagonal $d$ the energy
$-\log c\cdot \mathsf a(\psi;d)$. By Lemma~\ref{lem:weights}, the exponential of minus the sum of $\Psi$ over
the two diagonals of a block is $c$ times the vertex weight, so $\Phi+\Psi$ has the six-vertex
specification. The potential $\Psi$ has finite range and is invariant under $\Z^2$, and $\Phi$
strictly dominates it: for a finite nonempty $\Lambda$ and a configuration of finite $\Phi$-energy,
the $\Phi$-energy of the pairs meeting $\Lambda$ is at least $2K|\Lambda|$, since every face of
$\Lambda$ is the lower endpoint of two nearest-neighbour pairs, while the $\Psi$-energy of the pairs
meeting $\Lambda$ is at most $4|\log c|\,|\Lambda|$ in absolute value, since every face lies in four
diagonals. The ratio is at most $2|\log c|/K<1$.
\end{proof}

\begin{proposition}[Not a nearest-neighbour model in heights]\label{prop:notNN}
For every $c\ne1$, no nearest-neighbour gradient potential on the heights of faces has the
six-vertex specification. For $c=1$ the six-vertex model is a Lipschitz simply attractive potential.
\end{proposition}

\begin{proof}
Let $x$ be a face whose four neighbours have height $m$; then $h(x)\in\{m-1,m+1\}$. Each of the four
vertices of $x$ is surrounded by $x$, two of these neighbours, which are diagonal to each other and
have equal heights, and one face $y$ diagonal to $x$. By Lemma~\ref{lem:weights} its weight is $c$ if
$h(x)=h(y)$ and $1$ otherwise. If $k_+$ and $k_-$ of the four diagonal faces have heights $m+1$ and
$m-1$, then, given the configuration outside $x$, the ratio of the probabilities of $h(x)=m+1$ and
$h(x)=m-1$ is $c^{k_+-k_-}$. Both $k_+=4$ and $k_-=4$ occur: take all faces of the parity class of $x$
other than $x$ at height $m+1$, respectively $m-1$, and all faces of the other class at height $m$.
For a nearest-neighbour potential the conditional law of $h(x)$ depends only on the heights of its
four neighbours, so the ratio would be the same in the two cases, forcing $c^4=c^{-4}$. For $c=1$ we
have $\Psi=0$ in the proof of Proposition~\ref{prop:LSAP}.
\end{proof}

\subsection{Free energy}\label{subsec:FE}

Let $\mathbb X$ be the subshift of finite type on the alphabet $\{0,1\}^2$ (the two increments of
$\psi$ at a face) whose configurations are those in which the signed cyclic sum of the four increments
around every block vanishes, that is, the gradients of elements of $\Omega$; its constraint involves
the faces $x$, $x+e_1$ and $x+e_2$. For a
translation-invariant probability measure $\mu$ on $\mathbb X$ put
\[
P(\mu):=\mathsf h(\mu)+\mu(\log w_0),\qquad P^*:=\sup_\mu P(\mu),
\]
with $\mathsf h(\mu)$ the Kolmogorov--Sinai entropy of the $\Z^2$-action. We call $\mu$ an
\emph{equilibrium state} if $P(\mu)=P^*$. Sheffield's \emph{specific free energy} $\SFE$ and
\emph{surface tension} $\sigma_c(u)$, the infimum of $\SFE$ over translation-invariant gradient
measures of slope $u$, are defined in \cite[Section~2.3 and Chapter~4]{Sheffield}.

\begin{lemma}[Free energy and equilibrium states]\label{lem:SFE}\leavevmode
\begin{enumerate}
\item[(i)] For translation-invariant measures supported on admissible configurations,
$\SFE(\mu)=-P(\mu)+\mathrm{const}$ with a constant that does not depend on $\mu$.
\item[(ii)] $\sigma_c$ is convex and even, and $\sigma_c(0)=\min_u\sigma_c(u)=-P^*+\mathrm{const}$. Hence
$\SFE(\mu)=\sigma_c(0)$ if and only if $\mu$ is an equilibrium state.
\item[(iii)] Every gradient Gibbs measure of the six-vertex model that is invariant and ergodic under
the translations of $\Z^2$ and has slope $0$ is an equilibrium state.
\end{enumerate}
\end{lemma}

\begin{proof}
(i) Sheffield defines $\SFE(\mu)$ as the limit of $|\Lambda_n|^{-1}$ times the relative entropy of the
restriction of $\mu$ to $\Lambda_n$ with respect to $e^{-H^\circ_{\Lambda_n}}$ times counting measure
\cite[Section~2.3]{Sheffield}. For a measure on admissible configurations this is the Shannon entropy
of the gradient pattern on $\Lambda_n$ with a minus sign, plus $\mu(H^\circ_{\Lambda_n})$, and
$H^\circ_{\Lambda_n}=K\cdot\#\{\text{NN pairs in }\Lambda_n\}-\log c\cdot\#\{d\subset\Lambda_n:\mathsf a(\psi;d)=1\}$
in the notation of Proposition~\ref{prop:LSAP}. Dividing by $|\Lambda_n|$ gives
$\SFE(\mu)=-P(\mu)+2K-\log c$.
(ii) $\SFE$ is affine, so $\sigma_c$ is convex \cite[Chapter~4]{Sheffield}. The map
$\psi\mapsto-\psi+x_1+x_2$, that is $h\mapsto-h$, preserves $\Omega$ and every $w_x$ and reverses the
slope, so $\sigma_c$ is even and $\sigma_c(0)\le\sigma_c(u)$ for all $u$. The infimum of $\SFE$ over
all translation-invariant measures is $\inf_u\sigma_c(u)=\sigma_c(0)$, which is $-P^*$ plus the
constant of~(i).
(iii) By Proposition~\ref{prop:LSAP}, \cite[Theorem~6.3.1]{Sheffield} applies: an ergodic gradient
Gibbs measure of finite slope $u$ has $\SFE=\sigma_c(u)$. (Slope $0$ is $(\frac12,\frac12)$ in the coordinates
$\psi$, an interior point of the set $U_\Phi=(0,1)^2$ of slopes of finite surface tension.) With $u=0$ and~(ii) the measure is an
equilibrium state.
\end{proof}

\subsection{Triples: pairs with residual energies}\label{subsec:triples}

From now on until the end of Section~\ref{sec:fold}, $c>1$ and $q:=c^{-2}\in(0,1)$.

Sheffield compares two Gibbs measures through triplets made of two independent samples and an
independent field of residual energies on the edges \cite[Sections~8.1.3 and~8.3]{Sheffield}. For the
six-vertex model the interaction lives on the diagonals, so the residual energies do too, and since the
weight on a diagonal takes two values, an integer-valued residual energy suffices. Its role is the one of
the edge variable in the Edwards--Sokal coupling: given the two samples, it decides which diagonals may
be cut by a swap without changing the joint weight (Lemma~\ref{lem:FKswap}).

A \emph{triple} is $\om=(\psi_1,\psi_2,n)$ with $\psi_1,\psi_2\in\Omega$ and $n:\Diag\to\Z_{\ge0}$, taken
modulo the common shifts $(\psi_1,\psi_2)\mapsto(\psi_1+t,\psi_2+t)$, so that the \emph{difference}
$D:=\psi_2-\psi_1$ is an \emph{honest} integer-valued function, not only a gradient. A \emph{lift} of a
gradient triple $(\nabla\psi_1,\nabla\psi_2,n)$ is a triple with these gradients; its lifts differ by the
shifts $(\psi_1,\psi_2)\mapsto(\psi_1,\psi_2+t)$, $t\in\Z$. Let $\Om$ be the space of triples. The number
$n_d$ is the \emph{residual energy} of the diagonal $d$, and we put
\[
\tau_d(\om):=n_d-\tfrac12\bigl(\mathsf a(\psi_1;d)+\mathsf a(\psi_2;d)\bigr).
\]
Let $\pi$ be the product over $\Diag$ of the geometric law $\pi_0(j)=(1-q)q^j$, $j\ge0$. For finite
$\Lambda\subset F$ let $E(\Lambda)$ be the set of diagonals with a face in $\Lambda$, and let
$\gb_\Lambda(\cdot\mid\om)$ be the kernel that resamples $\psi_1$ and $\psi_2$ on $\Lambda$ from
$\gamma_\Lambda(\cdot\mid\psi_1)\otimes\gamma_\Lambda(\cdot\mid\psi_2)$ and $n$ on $E(\Lambda)$ from
$\pi$. Since $c^{\mathsf a}q^n=q^{n-\mathsf a/2}$, the density of $\gb_\Lambda(\cdot\mid\om)$ on the countable set of
triples that agree with $\om$ off $(\Lambda,E(\Lambda))$ is proportional to
\begin{equation}\label{eq:density}
\prod_{d\in E(\Lambda)}q^{\tau_d}.
\end{equation}
If $\mu_1,\mu_2$ are gradient Gibbs measures, the law $\mub:=\mu_1\otimes\mu_2\otimes\pi$ of the
\emph{gradient triple} $(\nabla\psi_1,\nabla\psi_2,n)$ is invariant under every $\gb_\Lambda$.