\section{Introduction}\label{sec:intro}

\subsection{Where things stand}\label{subsec:roadmap}

The six-vertex model on $\Z^2$ assigns an arrow to every edge so that each vertex has two incoming
and two outgoing arrows. We take the weights $a=b=1$ and $c>0$: a vertex has weight $c$ if its two
horizontal arrows both point into it or both point out of it, and weight $1$ otherwise. We write
$\Delta:=1-c^2/2$, so that $c\in[1,2]$ corresponds to $\Delta\in[-1,\frac12]$. An arrow configuration
has a \emph{height function} $h$ on the faces of $\Z^2$, defined up to an additive constant: adjacent
faces differ by $1$, and the face to the left of an arrow is the higher one.

For these weights, three recent works together come close to proving that the height function
converges to a Gaussian free field throughout $-1\le\Delta\le\frac12$:
\begin{itemize}
\item Duminil-Copin, Kozlowski, Lammers and Manolescu~\cite{DKLM} construct the \emph{plane measure}
$\Pp_c$, the limit of the model on a torus conditioned to have no net horizontal flux
(Definition~\ref{def:plane}), work with a spin representation of it that goes back to Lis \cite{Lis,Lis22}, and prove the Gaussian free field limit of its height function for
$-1\le\Delta\le-\frac12$.
\item Averous, Duminil-Copin, He, Lammers and Manolescu~\cite{ADHLM} prove that the height function in
discrete approximations of simply connected domains converges to a Gaussian free field at every $\Delta$
for which the full-plane correlations of $\Pp_c$ converge to those of a Gaussian free field.
\item A manuscript of OpenAI~\cite{OAI225}, which is not refereed, states that full-plane input for every
$c\in(0,2]$, computing correlations by Bethe ansatz.
\end{itemize}
Figure~\ref{fig:roadmap} shows how these pieces fit. Checking them against each other, we found one step
that is not proved in the literature. For $\sqrt2\le c<2$ the Bethe ansatz of \cite{OAI225} is carried out
in sectors that are only nearly balanced, and the state $\rho_c$ it produces is identified with $\Pp_c$
by citing a characterisation of $\Pp_c$ as the unique minimiser of the specific free energy. That
characterisation, and uniqueness of the zero-slope phase more generally, is stated in \cite{DKLM} as a
remark without proof, and by Ryan~\cite{Ryan} with a sketch that applies Sheffield's uniqueness theory
for simply attractive potentials~\cite{Sheffield}. Sheffield's theory is proved for potentials that couple
nearest neighbours, and the six-vertex model is not of that form in height coordinates when $c\ne1$
(Proposition~\ref{prop:notNN} and Figure~\ref{fig:vertices}): its weight is an interaction along the
diagonals of every vertex.

The main result of this paper proves the uniqueness for $1<c\le2$ (Theorem~\ref{thm:main}), and with it the
characterisation (Corollary~\ref{cor:minimiser}). This supplies the missing identification, given two
properties of the near-balanced state stated in \cite{OAI225} (Corollary~\ref{cor:ID}). The remaining
checks, that the three works concern the same measure and that one step of \cite{ADHLM} at positive mesh
follows from their hypotheses, are in Section~\ref{sec:domains}.

\input{fig-roadmap}

\subsection{Main results}

A probability measure on arrow configurations is a \emph{gradient Gibbs measure} of the six-vertex
model if, for every finite set $\Lambda$ of faces, the conditional law of the heights on $\Lambda$ given the
heights off $\Lambda$ is the six-vertex weight restricted to the admissible fillings
(Section~\ref{sec:gradient}; this is Sheffield's notion~\cite{Sheffield} for the six-vertex potential).
Such a measure is \emph{smooth} if it is the gradient of a Gibbs measure on heights and \emph{rough}
otherwise \cite[p.~10]{Sheffield}. For a measure invariant under the translations of $\Z^2$, the
\emph{slope} is the vector of mean height increments $(\E[h(e_1)-h(0)],\E[h(e_2)-h(0)])$. We call a
gradient Gibbs measure that is invariant and ergodic under the translations of $\Z^2$ a \emph{phase}.

For integers $N\ge1$ and $L$ even, let $\Pp^{\mathrm{bal}}_{N,L}$ be the six-vertex measure on the torus
with $N$ columns and $L$ rows conditioned on every vertical column of horizontal edges having equally
many left and right arrows.

\begin{definition}[The plane measure]\label{def:plane}
For $c\in[1,2]$,
\[
\Pp_c:=\lim_{L\to\infty,\ L\text{ even}}\ \lim_{N\to\infty}\ \Pp^{\mathrm{bal}}_{N,L}
\]
in the local weak topology. The limit exists and is invariant under the automorphisms of $\Z^2$ by
\cite[Theorem~2.2]{DKLM}, where it is denoted $\Pp_{\Z^2}$.
\end{definition}

\begin{theorem}[Uniqueness of the zero-slope phase]\label{thm:main}
Let $a=b=1$ and $1<c\le2$. 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 equal to $\Pp_c$.
\end{theorem}

The \emph{specific free energy} of a translation-invariant measure $\mu$ on arrow configurations is
$\SFE(\mu)=-\mathsf h(\mu)-\E_\mu[\log w_0]+\text{const}$, with $\mathsf h$ the entropy per face and $w_0$
the weight of the vertex at the origin (Section~\ref{subsec:FE}).

\begin{corollary}[Variational characterisation]\label{cor:minimiser}
Let $a=b=1$ and $1<c\le2$. The plane measure $\Pp_c$ is the only probability measure on arrow
configurations that is invariant under the translations of $\Z^2$ and minimises the specific free
energy.
\end{corollary}

Corollary~\ref{cor:minimiser} follows from Theorem~\ref{thm:main} in three steps (Section~\ref{sec:proofs}):
almost every ergodic component of a minimiser is a minimiser; it has slope $0$ because the surface
tension is strictly convex \cite[Theorem~13.16]{LammersTassy}; and it is a Gibbs measure by Sheffield's
variational principle \cite[Theorem~2.5.2]{Sheffield}. A translation-invariant measure that is
\emph{flip-symmetric}, that is, invariant under $h\mapsto-h$, has slope $0$, so Theorem~\ref{thm:main} also
gives, in the gradient formulation, the form of the remark in \cite{DKLM} that concerns flip-symmetric
ergodic Gibbs measures. At $c=1$ the conclusion of
Theorem~\ref{thm:main} follows from Sheffield's theorem for nearest-neighbour potentials
(Remark~\ref{rem:c1}).

Theorem~\ref{thm:main} has two ingredients.

\begin{theorem}[Two distinct zero-slope phases are smooth]\label{thm:smooth}
Let $a=b=1$ and $c>1$. Let $\mu_1\ne\mu_2$ be gradient Gibbs measures of the six-vertex model that are
invariant and ergodic under the translations of $\Z^2$ and have slope $0$. Then $\mu_1$ and $\mu_2$ are
both smooth.
\end{theorem}

\begin{theorem}[The plane measure is ergodic and rough]\label{thm:plane}
Let $a=b=1$ and $1\le c\le2$. The plane measure $\Pp_c$ is a gradient Gibbs measure of the six-vertex
model, it is trivial on the tail $\sigma$-field and hence ergodic under the translations of $\Z^2$, it has
slope $0$, and it is rough: no Gibbs measure on heights, translation-invariant or not, has gradient law
$\Pp_c$.
\end{theorem}

Theorem~\ref{thm:main} follows at once: if $\mu\ne\Pp_c$, both would be smooth by
Theorem~\ref{thm:smooth}, but $\Pp_c$ is rough. Theorem~\ref{thm:smooth} is the six-vertex form of the
part of Sheffield's Chapter~8 that makes two distinct ergodic phases of the same slope smooth
\cite[Sections~8.4--8.6]{Sheffield}. It holds for every $c>1$; the hypothesis $c\le2$ enters only through
Theorem~\ref{thm:plane}, which rests on the spin representation of \cite[Section~14]{DKLM}.

\subsection{The identification step and the limit in domains}\label{subsec:gap}

The manuscript~\cite{OAI225} computes correlations by Bethe ansatz, which works in one sector of the
transfer matrix at a time. For $1\le c<\sqrt2$ its construction uses the balanced sector, so it concerns
$\Pp_c$ directly (see Remark~\ref{rem:lower} for the endpoint $c=1$). For $\sqrt2\le c<2$ it uses the sectors with $n_L:=L/2-\lfloor L^\varepsilon\rfloor$ right
arrows per column, for a fixed small $\varepsilon>0$ \cite[Section~3]{OAI225}, and studies
\begin{equation}\label{eq:rho}
\rho_c:=\lim_{L\to\infty,\ L\text{ even}}\ \lim_{N\to\infty}\ \Pp^{(n_L)}_{N,L},
\end{equation}
where $\Pp^{(n)}_{N,L}$ is the torus measure conditioned on $n$ right arrows per column. The theorem about
$\Pp_c$ then requires the identification
\begin{equation}\label{eq:ID}\tag{ID}
\rho_c=\Pp_c .
\end{equation}
In \cite[Section~4, ``Identification when $c\ge1$'']{OAI225} this is derived from the characterisation of
$\Pp_c$ as the unique minimiser of the free energy, with a reference to the remark of \cite{DKLM}.
Corollary~\ref{cor:minimiser} proves that characterisation. We use two statements of \cite{OAI225}, which
we do not reprove:
\begin{itemize}
\item[(O1)] the limit \eqref{eq:rho} exists \cite[Sections~3 and~4]{OAI225};
\item[(O2)] $\rho_c$ minimises the specific free energy among translation-invariant measures
\cite[Section~4, the rectangle bound and the tiling argument]{OAI225}.
\end{itemize}

\begin{corollary}[Identification]\label{cor:ID}
Let $a=b=1$ and $\sqrt2\le c<2$. If \textup{(O1)} and \textup{(O2)} hold, then $\rho_c=\Pp_c$.
\end{corollary}

The argument is the one of \cite[Section~4]{OAI225}, with Corollary~\ref{cor:minimiser} in place of the
remark it cites. For $\sqrt3\le c\le2$, that is $-1\le\Delta\le-\frac12$, the full-plane statement is the
main theorem of \cite{DKLM} and does not depend on \cite{OAI225}. The range in which the manuscript's
full-plane theorem depended on the identification is therefore $\sqrt2\le c<\sqrt3$, that is
$-\frac12<\Delta\le0$. There it no longer rests on an unproved characterisation; it remains only as strong
as the rest of \cite{OAI225}, which we do not assess.

Averous, Duminil-Copin, He, Lammers and Manolescu~\cite{ADHLM} denote by $\ICI$ the set of $\Delta$ at which
the full-plane correlations of the height function under $\Pp_c$ converge to those of a Gaussian free
field \cite[(11)]{ADHLM}. Section~\ref{sec:domains} checks that \cite{OAI225} and \cite{ADHLM} concern the same
full-plane measure and observables, which gives the following.

\begin{corollary}[Domains]\label{cor:domains}
Let $c\in[1,2]$ and $\Delta=1-c^2/2$. If the full-plane statement of \cite[Theorem~1.1(3)]{OAI225} holds at
$c$ (Definition~\ref{def:O1}), then $\Delta\in\ICI$, and \cite[Theorems~1.3 and~1.4]{ADHLM} hold at $\Delta$
with the constant $\delta_0$ of \cite[Theorem~1.3]{ADHLM} allowed to depend on $\Delta$.
\end{corollary}

Corollary~\ref{cor:domains} is an implication between the results of three preprints and is only as strong
as they are. For $0<\Delta\le\frac12$ the construction of \cite{OAI225} uses the balanced sector; for
$-\frac12<\Delta\le0$ it uses the identification of Corollary~\ref{cor:ID}; for $-1\le\Delta\le-\frac12$ the
full-plane statement is the main theorem of \cite{DKLM} (which \cite{ADHLM} record for $-1<\Delta\le-\frac12$).

\subsection{Strategy of the proofs}\label{subsec:strategy}

\emph{Theorem~\ref{thm:smooth}.} We follow Sheffield's cluster-swapping argument
\cite[Chapter~8]{Sheffield}, which compares two Gibbs measures through independent samples
$\psi_1,\psi_2$ of each and their difference $D=\psi_2-\psi_1$. For a level $\ell$, exchanging the two
samples on a region where $D\ne\ell$, bounded by faces where $D=\ell$, preserves the nearest-neighbour
constraint. It preserves the joint weight only up to the diagonal interactions that straddle the
boundary of the region. Call a diagonal across which $D-\ell$ has the same sign, while $D=\ell$ on the
other diagonal, a \emph{saddle}: separating its two faces costs a factor $q=c^{-2}$. We pay for it with an
independent field of nonnegative integer \emph{residual energies} on the diagonals, in the spirit of
the coupling of Fortuin--Kasteleyn and Edwards--Sokal \cite{FK,EdwardsSokal}: separating a saddle lowers
its residual energy by one, so a saddle whose residual energy is $0$ cannot be separated and is an
\emph{open bridge}. The clusters of $\{D\ne\ell\}$ joined by open bridges are exactly the sets that can be
exchanged without changing the joint weight (Figure~\ref{fig:swap}). This is where the argument departs
from Sheffield's: his residual energies live on the nearest-neighbour edges, ours on the diagonals, and
clusters must be joined across saddles.

The swap identities sort the possible patterns of infinite clusters (Section~\ref{sec:swaps}):
\begin{itemize}
\item no infinite cluster at any level: then the two copies are exchangeable and $\mu_1=\mu_2$;
\item infinite clusters of one sign only, at every level: impossible;
\item a highest level with an infinite plus cluster, or a lowest level with an infinite minus cluster:
that level defines a canonical height, and both measures are smooth;
\item infinite clusters of both signs at every level: the remaining case.
\end{itemize}
In the remaining case we fold the pair (Section~\ref{sec:fold}): on alternate bands of levels the copies
are exchanged, so that the difference is folded into a strip (Figure~\ref{fig:fold}). The folded pair has
the same free energy as the original, so it is again an equilibrium state; since the six-vertex model
has hard constraints, we deduce that it is a Gibbs measure from the Lanford--Ruelle theorem for subshifts
of finite type~\cite{BGMT}. Each infinite cluster of the folded pair lies in a \emph{double band}, a pair
of adjacent bands, and a Burton--Keane argument~\cite{BurtonKeane} shows that there are finitely many.
If some double band carries an infinite cluster, the lowest such one defines a canonical height and both
measures are smooth; if none does, a symmetry of the folded pair gives a contradiction.

\emph{Theorem~\ref{thm:plane}.} In the spin representation in the form of \cite[Section~14]{DKLM}, the plane measure has infinitely
many nested open circuits around every face, with a Markov property across each circuit and a reflection
symmetry inside it. If $\Pp_c$ were the gradient of a Gibbs measure $\nu$ on heights, the height at a face
would be the height of an outer circuit plus a sum of increments whose signs are independent fair coin
flips given everything else, and anti-concentration~\cite{Erdos} would make its law spread out without
bound (Figure~\ref{fig:circuits}).

\subsection{Relation to earlier work}

\emph{Uniqueness.} At $c=1$ the six-vertex model is the uniform measure on graph homomorphisms, a
nearest-neighbour simply attractive model, and uniqueness of the zero-slope phase follows from
\cite[Theorem~9.1.1]{Sheffield}; Chandgotia, Peled, Sheffield and Tassy prove delocalisation and the
absence of translation-invariant Gibbs measures on heights there \cite{CPST}. At $c=2$ Glazman and Peled
prove uniqueness of the flat Gibbs state \cite[Corollary~2.1]{GlazmanPeled}, and for $c>2$ they prove that
the height function is localised: its extremal Gibbs states are Gibbs measures on heights, invariant
under parity-preserving translations \cite[Theorem~2]{GlazmanPeled}. A proof of uniqueness through
roughness must therefore use $c\le2$. For $1<c<2$, uniqueness of the zero-slope phase is asserted in
\cite{DKLM} (remark after (6)) and in \cite[Section~2.1]{Ryan}, in both cases by reference to Sheffield's
theory; Ryan's sketch regards the six-vertex potential as simply attractive and applies
\cite[Lemma~8.7.1]{Sheffield}. A remark of Lammers and Tassy after \cite[Theorem~13.16]{LammersTassy} states
that the six-vertex potential can be written as a simply attractive potential when
$c_+c_-\ge\max\{a_+a_-,b_+b_-\}$ in their notation, which reads $c\ge1$ when $a=b=1$; we do not know which
representation is intended, and Proposition~\ref{prop:notNN} shows that for $a=b=1$ it cannot use
nearest-neighbour interactions between face heights when $c\ne1$. We use the strict convexity proved in
\cite[Theorem~13.16]{LammersTassy} itself, not the remark. The planar arguments of
\cite[Chapter~9]{Sheffield} do not extend beyond nearest-neighbour interactions
\cite[Section~10.4.3]{Sheffield}; our proof extends the non-planar ones of Chapter~8. Cluster swapping has
recently been extended to spin models by van Engelenburg, Heeney and Lis \cite{vEHL}.

\emph{The plane measure.} Ergodicity of the full-plane spin measure is known \cite{GlazmanLammers}, and for
$c\in[\sqrt3,2]$ Lis proves that the torus measures have an ergodic limit \cite{Lis}
(Section~\ref{subsec:tail}). Delocalisation, in the form of variance growth, is proved by Duminil-Copin,
Karrila, Manolescu and Oulamara for $1\le c\le2$ \cite{DKMO} and by Glazman and Lammers
\cite{GlazmanLammers}, whose random walk of circuit heights \cite[Section~4]{GlazmanLammers} is also the
mechanism of our proof of roughness. Roughness in Sheffield's sense, that no Gibbs measure on heights has
gradient law $\Pp_c$, is what the proof of Theorem~\ref{thm:main} needs. Ryan states, citing Lammers and
Ott, that for $a,b\le c$ roughness in this sense is equivalent to delocalisation \cite[Section~2.1]{Ryan};
we give a direct proof.

\emph{Variational principles.} Lammers and Tassy prove that minimisers of the specific free energy are
Gibbs measures for a class of Lipschitz height models that includes the six-vertex model
\cite[Theorem~4.4]{LammersTassy}. We need the same conclusion for the folded pair of
Section~\ref{sec:fold}, which carries residual energies and a hard constraint between the two copies, and
obtain it from the Lanford--Ruelle theorem of \cite{BGMT}.

\subsection{Organisation}

Sections~\ref{sec:gradient}--\ref{sec:fold} prove Theorem~\ref{thm:smooth}, Section~\ref{sec:plane} proves
Theorem~\ref{thm:plane}, Section~\ref{sec:proofs} proves Theorem~\ref{thm:main} and Corollaries~\ref{cor:minimiser}
and~\ref{cor:ID}, and Section~\ref{sec:domains} treats the limit in domains.