\section{The plane measure: proof of Theorem~\ref*{thm:plane}}\label{sec:plane}

Throughout this section $1\le c\le2$. The plane measure is a gradient Gibbs measure of slope $0$ by
\cite{DKLM} (Subsection~\ref{subsec:planeGibbs}). Its spin representation provides nested open circuits
around every face, with a Markov property across each circuit (Subsection~\ref{subsec:circuits}); these give
tail triviality (Subsection~\ref{subsec:tail}) and roughness (Subsection~\ref{subsec:rough},
Figure~\ref{fig:circuits}).

\emph{Notation.} We use the notation of \cite[Section~14]{DKLM}. The faces split
into even faces $F^\circ$, forming a square lattice with the diagonal adjacency $E^\circ$, and odd faces
$F^\bullet$. For a height function $h$, the spins are $\sigma^\circ(u)=+$ if and only if $h(u)\in4\Z$ for
$u$ even, and $\sigma^\bullet(v)=+$ if and only if $h(v)\in1+4\Z$ for $v$ odd; for NN-adjacent faces $u$
even and $v$ odd, $h(v)-h(u)=\sigma^\circ(u)\sigma^\bullet(v)$ \cite[(247)]{DKLM}. An \emph{even domain} $\mathcal V$
is bounded by a self-avoiding circuit $\partial\mathcal V$ of $E^\circ$. For an even domain $\mathcal V$ and
$\#\in\{+,-\}$, $\mu^\#_{\mathcal V}$ is the measure of \cite[Definition~14.5]{DKLM} on triples
$(\sigma^\circ,\omega,\sigma^\bullet)$, with $\omega\subset E^\circ(\mathcal V)$ a set of open edges. Under $\mu^+_{\mathcal V}$ the
height function with $h=0$ on $\partial\mathcal V$ has law proportional to $\ind{h|_{\partial\mathcal V}\equiv0}\,c^{\#A(h)}$
\cite[(248)]{DKLM}, where $A(h)$ is the set of diagonals of $\mathcal V$ on which $h$ is constant; this is the
six-vertex measure on $\mathcal V$ with that boundary condition, up to the blocks that straddle $\partial\mathcal V$,
which contribute a constant. Let $\mu_{\Z^2}$ be the full-plane measure of \cite[Theorem~17.1]{DKLM}: the
common limit of $\mu^+_{\mathcal V}$ and $\mu^-_{\mathcal V}$ as $\mathcal V\uparrow\Z^2$ \cite[(306)]{DKLM}.

\subsection{The plane measure is a gradient Gibbs measure of slope zero}\label{subsec:planeGibbs}

\begin{lemma}[Gibbs property, invariance, slope]\label{lem:planeGibbs}\leavevmode
\begin{enumerate}
\item[(a)] The gradient law of $h$ under $\mu_{\Z^2}$ is $\Pp_c$, and it is a gradient Gibbs measure of the
six-vertex model.
\item[(b)] $\Pp_c$ is invariant under the translations of $\Z^2$ and has slope $0$.
\item[(c)] Under $\mu^+_{\mathcal V}$, conditionally on the spins, the edges $xy\in E^\circ(\mathcal V)\setminus\partial\mathcal V$ are
independent, with $\omega(xy)=0$ if $\sigma^\circ(x)\ne\sigma^\circ(y)$, $\omega(xy)=1$ if
$\sigma^\circ(x)=\sigma^\circ(y)$ and $\sigma^\bullet$ disagrees across $xy$, and $\omega(xy)$ Bernoulli$(1/c)$ if
both agree. This rule depends on the spins only through $\nabla h$, and it holds under $\mu_{\Z^2}$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) By \cite[(248)]{DKLM}, for a finite set $\Lambda$ of faces and an even domain $\mathcal V$ such that every block
meeting $\Lambda$ consists of faces of $\mathcal V$ not on $\partial\mathcal V$, the conditional law under $\mu^+_{\mathcal V}$ of $h$ on
$\Lambda$ given $h$ off $\Lambda$ is $\gamma_\Lambda$. Both sides of the DLR equation for a local test function
are local, so the equation passes to the limit $\mathcal V\uparrow\Z^2$. By \cite[Lemma~18.7(iii)]{DKLM} the gradient
law of $h$ under $\mu_{\Z^2}$ is $\Pp_{\Z^2}=\Pp_c$.
(b) Invariance is part of \cite[Theorem~2.2]{DKLM}. The measure $\mu^+_{\mathcal V}$ is invariant under $h\mapsto-h$
by (248), so the limit is, and $\mu_{\Z^2}[h(x+e_i)-h(x)]=0$.
(c) This is \cite[(238)--(240)]{DKLM}. The agreement pattern is $A(h)$, a function of $\nabla h$; the rule
is local and passes to the limit. At $c=1$ the last case is deterministic.
\end{proof}

By (c), the pair $(\nabla h,\omega)$ can be realised from any random gradient field with law $\Pp_c$ by
sampling $\omega$ with this rule, and then has the law it has under $\mu_{\Z^2}$. The measure $\mu_{\Z^2}$ is
invariant under the global flip of both spins, since $\mu^+_{\mathcal V}$ and $\mu^-_{\mathcal V}$ are exchanged by it and have
the same limit; statements below about $(\nabla h,\omega)$ do not depend on the global sign.

\subsection{Circuits and the Markov property}\label{subsec:circuits}

An \emph{$\omega$-circuit} is a self-avoiding circuit of $E^\circ$ all of whose edges are open. The height
is constant on its faces, since $\sigma^\circ$ is constant along open edges and diagonally adjacent even
faces with equal $\sigma^\circ$ have equal heights. Fix the odd face $o:=(1,0)$. An $\omega$-circuit $\mathfrak C$
\emph{surrounds} $o$ if $o$ is a face of the even domain $\mathcal V_{\mathfrak C}$ that it bounds.

\begin{lemma}[Nested circuits]\label{lem:circuits}
$\mu_{\Z^2}$-almost surely, $o$ is surrounded by infinitely many $\omega$-circuits on which
$\sigma^\circ=+$ and infinitely many on which $\sigma^\circ=-$. Hence there is almost surely an infinite
nested family of $\omega$-circuits surrounding $o$ in which consecutive circuits have different heights.
The same holds with $o$ replaced by any finite box.
\end{lemma}

\begin{proof}
By \cite[Corollary~16.6]{DKLM}, for $r\ge4$, a point $x$, and an even domain $\mathcal V$ whose boundary surrounds
$\mathrm{Ball}_{2r}(x)$, the $\mu^+_{\mathcal V}$-probability that the annulus $\mathrm{Ball}_{2r}(x)\setminus\mathrm{Ball}_r(x)$
contains an $\omega$-circuit with $\sigma^\circ=-$ around $x$ is at least a constant $c_{\mathrm{circ}}>0$,
conditionally on any event measurable with respect to the even spins and the edges not meeting
$\mathrm{Ball}_{2r}(x)$; the same holds for $\sigma^\circ=+$. The balls are $\ell^\infty$-boxes. Applying this to
the annuli at scales $2^ir$ from the outside in, and passing to the limit with
\cite[Theorem~17.1]{DKLM}, gives infinitely many circuits of each sign almost surely. Two
$\omega$-circuits surrounding $o$ either share a face or are nested, because $E^\circ$ is planar; circuits
with different $\sigma^\circ$ share no face and carry different heights. For a box, the same annulus
argument applies around its centre.
\end{proof}

\emph{Outermost circuits.} Consider a property of $\omega$-circuits that is inherited by the circuit bounding
the outer face of $\mathcal V_{\mathfrak C}\cup \mathcal V_{\mathfrak C'}$; we use ``surrounds $o$ and lies in a given domain'', possibly
together with ``has height different from a given value'', which is inherited because circuits that
share a face have the same height. If two circuits with such a property surround $o$, the outer
boundary of $\mathcal V_{\mathfrak C}\cup \mathcal V_{\mathfrak C'}$ is again a self-avoiding $\omega$-circuit with the property: both domains
contain $o$, so the union has no pinch point. Hence among finitely many candidates there is a unique
outermost one, as in \cite[proofs of Theorem~16.4 and Lemma~17.3]{DKLM}.

\begin{lemma}[Strong Markov property]\label{lem:Markov}
Let $\mathcal C$ be a random self-avoiding circuit of $E^\circ$ such that, for every deterministic circuit
$\mathfrak C_0$, the event $\{\mathcal C=\mathfrak C_0\}$ is measurable with respect to the spins and edges on $\mathfrak C_0$ and
outside $\mathcal V_{\mathfrak C_0}$ and is contained in $\{\mathfrak C_0\subset\omega\}$. Then under $\mu_{\Z^2}$, conditionally on
$\{\mathcal C=\mathfrak C_0\}$ and on the configuration on $\mathfrak C_0$ and outside $\mathcal V_{\mathfrak C_0}$, the configuration inside $\mathcal V_{\mathfrak C_0}$
has law $\mu^\flat_{\mathcal V_{\mathfrak C_0}}$, where $\flat$ is the value of $\sigma^\circ$ on $\mathfrak C_0$. In terms of heights, the
field $h-h(\mathfrak C_0)$ inside $\mathcal V_{\mathfrak C_0}$ has the law of $h$ under $\mu^+_{\mathcal V_{\mathfrak C_0}}$, and the joint law of
$(h-h(\mathfrak C_0),\omega)$ inside is invariant under $h-h(\mathfrak C_0)\mapsto-(h-h(\mathfrak C_0))$ with $\omega$ unchanged.
\end{lemma}

\begin{proof}
For even domains $\mathcal V\subset B$, local $f$ of the configuration in $\mathcal V$ and local $g$ of the configuration
outside $\mathcal V$, \cite[Lemma~14.12]{DKLM} gives
$\mu^\#_B[f\cdot g\cdot\ind{\partial\mathcal V\subset\omega^\flat}]=\mu^\flat_{\mathcal V}[f]\cdot\mu^\#_B[g\cdot\ind{\partial\mathcal V\subset\omega^\flat}]$,
where $\omega^\flat$ denotes open edges with $\sigma^\circ=\flat$. Both sides are limits of local quantities, so
letting $B\uparrow\Z^2$ with \cite[(306)]{DKLM} gives the same identity for $\mu_{\Z^2}$, and a monotone class
argument extends it to bounded $g$ measurable with respect to the configuration outside $\mathcal V$. Summing
over the deterministic circuits $\mathfrak C_0$ gives the statement for $\mathcal C$, as in
\cite[(299) and (310)]{DKLM}. Since $\mu^+_{\mathcal V}$ and $\mu^-_{\mathcal V}$ are exchanged by the global flip and have the same
gradient law, the field $h-h(\mathfrak C_0)$ inside has the law of $h$ under $\mu^+_{\mathcal V_{\mathfrak C_0}}$, and (248) and
Lemma~\ref{lem:planeGibbs}(c) give the symmetry, with $\omega$ unchanged because the rule for $\omega$
depends on $h$ only through $A(h)$, which is invariant under $h\mapsto-h$.
\end{proof}

The circuits used below satisfy the hypothesis of Lemma~\ref{lem:Markov}: the outermost $\omega$-circuit
surrounding $o$ in a box is determined by the edges outside it, and the outermost $\omega$-circuit
surrounding $o$ strictly inside a given circuit with height different from that circuit is determined
by the spins and edges outside it together with the sign on it.

\input{fig-circuits}

\subsection{Tail triviality}\label{subsec:tail}

\begin{lemma}[Tail triviality]\label{lem:tail}
$\mu_{\Z^2}$ is trivial on the tail $\sigma$-field of $(\sigma^\circ,\omega,\sigma^\bullet)$. Hence $\Pp_c$ is
tail-trivial and ergodic under the translations of $\Z^2$.
\end{lemma}

\begin{proof}
Let $A$ be an event depending on the configuration in $\mathrm{Ball}_r$, $T$ a tail event, and $\varepsilon>0$.
By \cite[Theorem~17.1]{DKLM} there is $r'>r$ with $|\mu^\pm_{\mathcal V}(A)-\mu_{\Z^2}(A)|\le\varepsilon$ for every even domain
$\mathcal V$ whose boundary surrounds $\mathrm{Ball}_{r'}$. For $R>r'$ let $\mathcal G_R$ be the event that some
$\omega$-circuit in $\mathrm{Ball}_R$ surrounds $\mathrm{Ball}_{r'}$, and let $\mathcal C$ be the outermost one. The
event $T$ is measurable with respect to the configuration outside $\mathcal V_{\mathcal C}$, so by
Lemma~\ref{lem:Markov},
$|\mu_{\Z^2}(A\cap T\cap\mathcal G_R)-\mu_{\Z^2}(A)\,\mu_{\Z^2}(T\cap\mathcal G_R)|\le\varepsilon$. Letting $R\to\infty$
($\mu_{\Z^2}(\mathcal G_R)\to1$ by Lemma~\ref{lem:circuits}) and $\varepsilon\to0$ gives
$\mu_{\Z^2}(A\cap T)=\mu_{\Z^2}(A)\mu_{\Z^2}(T)$, so $\mu_{\Z^2}(T)\in\{0,1\}$. The gradient field is a function of
the spins. For ergodicity, let $I$ be a translation-invariant event and $A$ a local event with
$\Pp_c(I\,\triangle\,A)<\varepsilon$. By tail triviality and reverse martingale convergence,
$\sup_{B}|\Pp_c(A\cap B)-\Pp_c(A)\Pp_c(B)|\to0$ over events $B$ measurable outside $\Lambda_N$ as $N\to\infty$, so
$\Pp_c(A\cap\theta_xA)\to\Pp_c(A)^2$ as $|x|\to\infty$, while $|\Pp_c(A\cap\theta_xA)-\Pp_c(I)|\le2\varepsilon$. Hence
$\Pp_c(I)=\Pp_c(I)^2$.
\end{proof}

Ergodicity and extremality of the full-plane spin measure are also proved in
\cite[Proposition~4.11]{GlazmanLammers}, for $0<a,b\le c\le a+b$; for $c\in[\sqrt3,2]$, Lis proves that the
torus measures converge to an ergodic, polynomially mixing limit \cite[Theorem~1.1]{Lis}. We
included the argument above because it uses only the Markov property of Lemma~\ref{lem:Markov}, which
is needed again below.

\subsection{Roughness}\label{subsec:rough}

\begin{proposition}[Roughness]\label{prop:rough}
There is no Gibbs measure $\nu$ on heights, translation-invariant or not, whose gradient law is $\Pp_c$.
\end{proposition}

\begin{proof}
Suppose that $\nu$ exists, and let $h$ denote the honest height under $\nu$. Sample $\omega$ given $\nabla h$
as in Lemma~\ref{lem:planeGibbs}(c), and let $Q$ be the joint law of $(h,\omega)$; then $(\nabla h,\omega)$ has
its law under $\mu_{\Z^2}$.

\emph{Step 1: the honest exterior does not bias the interior.} Let $\mathcal C$ be a circuit as in
Lemma~\ref{lem:Markov}, and $\mathcal H_{\mathcal C}$ the $\sigma$-field generated by $\mathcal C$, by $\omega$ on $\mathcal C$ and
outside $\mathcal V_{\mathcal C}$, and by the honest heights of the faces not strictly inside $\mathcal C$. We claim that,
conditionally on $\mathcal H_{\mathcal C}$, the pair $(h-h(\mathcal C),\omega)$ inside $\mathcal V_{\mathcal C}$ has the law given
by Lemma~\ref{lem:Markov}. Let $\Lambda$ be a box containing the region where $\mathcal C$ lies (for the circuits
used below, a fixed box) and $b$ a face outside $\Lambda$. By the Gibbs property of $\nu$ and the
covariance of $\gamma_\Lambda$ under constant shifts, conditionally on the honest heights outside $\Lambda$ the
gradient field on $\Lambda$ and across its boundary has a law that depends only on the gradient field
outside $\Lambda$. Since the complement of $\Lambda$ is connected, $h(b)$ is therefore conditionally
independent of $(\nabla h,\omega)$ given the gradient field outside $\Lambda$; this uses neither translation
invariance nor any other property of $\nu$. The $\sigma$-field $\mathcal H_{\mathcal C}$ is generated by gradient
information on $\mathcal C$ and outside it together with $h(b)$, so the conditional law of the interior given
$\mathcal H_{\mathcal C}$ equals its conditional law given the gradient information alone, which is the one in
Lemma~\ref{lem:Markov}.

\emph{Step 2: a random walk of circuit heights.} This is the mechanism of \cite[Section~4]{GlazmanLammers}.
Fix $R$. Let $\mathcal C_0$ be the outermost $\omega$-circuit
surrounding $o$ in $\mathrm{Ball}_R$, and $a_0:=h(\mathcal C_0)$. Recursively, as long as one exists, let
$\mathcal C_i$ be the outermost $\omega$-circuit surrounding $o$ strictly inside $\mathcal C_{i-1}$ with
$h(\mathcal C_i)\ne h(\mathcal C_{i-1})$. Let $T=T_R$ be the number of steps, with $T_R:=0$ if $\mathcal C_0$ does not
exist; $a_i:=h(\mathcal C_i)$, $m_i:=|a_i-a_{i-1}|\ge2$, $\varepsilon_i:=\operatorname{sign}(a_i-a_{i-1})$, and
$Y:=h(o)-a_T$. For $1\le i\le T$ let $\Xi_i$ be the map that reflects $h$ inside $\mathcal C_{i-1}$ about
$a_{i-1}$ and then reflects the new $h$ inside $\mathcal C_i$ about its new height, with $\omega$ unchanged.
Each of the two reflections preserves $Q$, by Step~1 and the symmetry in Lemma~\ref{lem:Markov}. Both
preserve $T$ and the circuits $\mathcal C_0,\dots,\mathcal C_T$ as sets: a circuit that crosses
$\mathcal C_{i-1}$ shares a face with it and so has height $a_{i-1}$, which the first reflection fixes, and
the defining properties of the circuits are invariant under reflection of the interior. The composition
changes $\varepsilon_i$ to $-\varepsilon_i$ and leaves $a_0$, all $m_j$, all $\varepsilon_j$ with $j\ne i$, and $Y$
unchanged; inside $\mathcal C_i$ it is the shift $h\mapsto h-2\varepsilon_im_i$. Therefore, conditionally on
$\mathcal G:=\sigma(a_0,T,(\mathcal C_i),(m_i),Y)$, the signs $\varepsilon_1,\dots,\varepsilon_T$ are independent and
uniform. Since $h(o)=a_0+\sum_{i\le T}\varepsilon_im_i+Y$, Lemma~\ref{lem:LO} gives, for every integer $j$,
\[
\nu(h(o)=j)\le \E_Q[\beta(T_R)],\qquad \beta(T):=\binom{T}{\lfloor T/2\rfloor}2^{-T}.
\]

\emph{Step 3.} By Lemma~\ref{lem:circuits} there is almost surely an infinite nested family of
$\omega$-circuits around $o$ in which consecutive circuits have different heights. If $\mathrm{Ball}_R$
contains $2m$ such circuits, then by induction $\mathcal C_i$ exists for $i\le m$ and its domain contains
that of the $2i$-th of them, so $T_R\ge m$. Hence $T_R\to\infty$ almost surely as $R\to\infty$,
$\E_Q[\beta(T_R)]\to0$, and $\nu(h(o)=j)=0$ for every $j$, which is absurd.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:plane}]
Lemma~\ref{lem:planeGibbs}, Lemma~\ref{lem:tail} and Proposition~\ref{prop:rough}.
\end{proof}

The hypotheses $1\le c\le2$ enter through the results of \cite{DKLM} used here: Theorem~2.2,
Corollary~16.6, Theorem~17.1 and Lemma~18.7, all of which rest on \cite[Theorem~16.4]{DKLM}; and through
the coupling of Lemma~\ref{lem:planeGibbs}(c), which needs $c\ge1$.