\section{The limit in domains}\label{sec:domains}

This section proves Corollary~\ref{cor:domains}. Subsection~\ref{subsec:same} checks that \cite{OAI225},
\cite{DKLM} and \cite{ADHLM} define the same full-plane measure and observables. Subsection~\ref{subsec:member}
shows that the full-plane statement of \cite{OAI225} gives membership in $\ICI$, whatever convention is used
for points on grid lines. Subsection~\ref{subsec:harm} writes out a harmonicity statement at positive mesh
used in the proof of \cite[Theorem~1.3]{ADHLM}.

Throughout this section $c\in[1,2]$ and $\Delta=1-c^2/2$. For a mesh $\delta>0$ and a point $u\in\R^2$
off the grid lines of $\delta\Z^2$, $h_\delta(u)$ is the height of the face of $\delta\Z^2$ containing $u$, a
face being identified with its lower left corner. A \emph{grid-line rule} assigns to each $\delta$ and each
point $p$ on a grid line a face of $\delta\Z^2$ whose closure contains $p$, without reference to the
configuration, and $h_\delta(p)$ is the height of that face. The \emph{floor rule} takes the face with lower
left corner $(\delta\lfloor p_1/\delta\rfloor,\delta\lfloor p_2/\delta\rfloor)$; it is the convention of
\cite[(7)]{DKLM} and \cite[Section~1.1]{OAI225}.

Let $G(x,y):=-\frac1{2\pi}\log|x-y|$, let $\Gamma$ be the full-plane Gaussian free field with covariance
kernel $G$, and for points $u_1,u_1',u_2,u_2'$ put
\begin{equation}\label{eq:K}
\mathcal K(u_1,u_1';u_2,u_2'):=G(u_1',u_2')-G(u_1',u_2)-G(u_1,u_2')+G(u_1,u_2),
\end{equation}
the covariance of the increments $\Gamma(u_1')-\Gamma(u_1)$ and $\Gamma(u_2')-\Gamma(u_2)$.

\begin{definition}[The set $\ICI$]\label{def:ICI}
Fix a grid-line rule. Following \cite[(11)]{ADHLM}, $\ICI$ is the set of $\Delta\in[-1,\frac12]$ such that,
for every $k\ge1$ and all distinct points $u_1,u_1',\dots,u_k,u_k'\in\R^2$,
\begin{equation}\label{eq:ICI}
\E_{\Pp_c}\Bigl[\prod_{i=1}^k\bigl(h_\delta(u_i)-h_\delta(u_i')\bigr)\Bigr]\ \xrightarrow[\delta\to0]{}\
v^{k/2}\,\E\Bigl[\prod_{i=1}^k\bigl(\Gamma(u_i)-\Gamma(u_i')\bigr)\Bigr],\qquad v=\frac{2}{\arccos\Delta}.
\end{equation}
\end{definition}

In \cite[footnote~2]{ADHLM} the ambiguity on grid lines is left open, with the observation that any
deterministic rule fixed in advance resolves it and that it has no effect on their results. The results below hold for every grid-line rule.

\begin{definition}[The full-plane statement at $c$]\label{def:O1}
Following \cite[Theorem~1.1(3)]{OAI225}, the \emph{full-plane statement holds at $c$} if, for every
$k\ge1$ and with $h_\delta$ defined by the floor rule,
\begin{equation}\label{eq:O1}
\E_{\Pp_c}\Bigl[\prod_{i=1}^k\bigl(h_\delta(u_i')-h_\delta(u_i)\bigr)\Bigr]\ \xrightarrow[\delta\to0]{}\
v(c)^{k/2}\sum_{\mathfrak p}\prod_{\{i,j\}\in\mathfrak p}\mathcal K(u_i,u_i';u_j,u_j'),\qquad v(c)=\frac1{\arcsin(c/2)},
\end{equation}
uniformly on compact sets of tuples $(u_i,u_i')_{i\le k}$ with $\{u_i,u_i'\}\cap\{u_j,u_j'\}=\emptyset$ for
$i\ne j$, where the sum is over the pairings $\mathfrak p$ of $\{1,\dots,k\}$ and is zero for odd $k$.
\end{definition}

Since $\arccos(1-c^2/2)=2\arcsin(c/2)$, the two variances agree, $v(c)=2/\arccos\Delta$, and we write $v$
for both. The differences in \eqref{eq:ICI} and \eqref{eq:O1} are taken in opposite orders, each as in
its source.

\subsection{The two works concern the same measure}\label{subsec:same}

\begin{proposition}\label{prop:same}
The plane law of \cite[Section~1.1]{OAI225} is the measure $\Pp_c$ of Definition~\ref{def:plane}, and the
height functions and their rescalings in \cite{OAI225}, \cite{DKLM} and \cite{ADHLM} agree off the grid
lines. In particular \eqref{eq:O1} and \eqref{eq:ICI} concern the same expectations, up to the
grid-line rule.
\end{proposition}

\begin{proof}
\emph{Weights.} In \cite{DKLM} the vertex types $1$--$2$, $3$--$4$ and $5$--$6$ have weights $a$, $b$ and
$c$, and types $5$ and $6$ are the two vertices whose horizontal arrows both point in or both point
out. In \cite{OAI225} those two vertices have weight $c$ and the other four weight $1$. With $a=b=1$ the
finite-volume measures on a torus coincide.
\emph{Conditioning and limits.} Both condition on the balanced sector with $L$ even, and both let
$N\to\infty$ first and then $L\to\infty$ in the local weak topology; the limit exists by
\cite[Theorem~2.2]{DKLM}.
\emph{Heights.} In all three works the face to the left of an arrow is one unit higher, a point off the
grid lines takes the value of the face containing it, and $h_\delta(u)=h(u/\delta)$. The kernel $G$ is the
same.
\emph{The measure in \cite[(11)]{ADHLM}.} The condition there is formulated for ``the unique
flip-symmetric Gibbs measure'', with a reference to \cite{DKLM} for its definition. The measure defined in
\cite{DKLM} is the limit of Definition~\ref{def:plane}, and the proofs of \cite{ADHLM} use that limit:
their full-plane measure is the one of \cite[Sections~14--17]{DKLM}, whose gradient marginal is $\Pp_c$ by
\cite[Lemma~18.7]{DKLM}, and it is reached from balanced cylinders in \cite[Theorem~6.3]{ADHLM}. For
Read as in the remark after \cite[(6)]{DKLM}, as the unique flip-symmetric gradient Gibbs measure that is
ergodic under translations, the phrase designates $\Pp_c$ for $1\le c\le2$ (Theorem~\ref{thm:main} and
Remark~\ref{rem:c1}); without ergodicity it designates no single measure, since a phase of nonzero slope
averaged with its flip is flip-symmetric; and no Gibbs measure on heights has gradient law $\Pp_c$
(Theorem~\ref{thm:plane}).
\end{proof}

\begin{remark}[The construction of \cite{OAI225} for $1\le c<\sqrt2$]\label{rem:lower}
For $c<\sqrt2$ the construction of \cite{OAI225} uses the balanced sector, so the state it studies is the
limit of Definition~\ref{def:plane}. Sections~4 and~8 of \cite{OAI225} describe the change of regime at $c=1$
and its Section~3 at $c=\sqrt2$; for $1\le c<\sqrt2$ the direct argument given there for $c<1$ applies, with
one exception. At $c=1$ a step that compares the states obtained with the roles of the two axes exchanged
uses a matrix that is singular there. At $c=1$ the comparison follows instead from the invariance of
$\Pp_c$ under the automorphisms of $\Z^2$ \cite[Theorem~2.2]{DKLM}, on the assumption, which we have not
verified, that the state obtained in \cite{OAI225} with the axes exchanged is the limit of
Definition~\ref{def:plane} with the axes exchanged.
\end{remark}

\subsection{Membership}\label{subsec:member}

\begin{proposition}[Independence of the grid-line rule]\label{prop:rule}
If the full-plane statement holds at $c$, then \eqref{eq:ICI} holds for all distinct points and every
grid-line rule; in particular $\Delta\in\ICI$.
\end{proposition}

\begin{proof}
Let $u_1,u_1',\dots,u_k,u_k'$ be distinct; they satisfy the disjointness condition of
Definition~\ref{def:O1}. By Wick's formula,
$\E[\prod_{i=1}^kv^{1/2}(\Gamma(u_i')-\Gamma(u_i))]=v^{k/2}\sum_{\mathfrak p}\prod_{\{i,j\}\in\mathfrak p}\mathcal K(u_i,u_i';u_j,u_j')$,
the right side of \eqref{eq:O1}. The products in \eqref{eq:ICI} and \eqref{eq:O1} differ by the factor
$(-1)^k$, which is $1$ for even $k$, and for odd $k$ both limits vanish. Hence \eqref{eq:ICI} holds
under the floor rule.

For another rule and a point $p$, let $\zeta_\delta(p)$ be the centre of the face whose height defines
$h_\delta(p)$. Then $h_\delta(p)$ is the floor-rule value at $\zeta_\delta(p)$, which lies off the grid lines,
and $|\zeta_\delta(p)-p|\le\delta/\sqrt2$. Let $d_0>0$ be the smallest distance between two of the points
and $\mathcal Q$ the set of tuples within distance $d_0/4$ of $(u_i,u_i')_{i\le k}$ in each coordinate. It
is compact and consists of tuples of distinct points, so the convergence in \eqref{eq:O1} is uniform on
$\mathcal Q$ and the limit is continuous there. For $\delta<d_0/4$ the tuple of centres lies in $\mathcal Q$ and
tends to $(u_i,u_i')_{i\le k}$, so the expectation in \eqref{eq:ICI}, the floor-rule expectation at the
tuple of centres, converges to the value of the limit at $(u_i,u_i')_{i\le k}$.
\end{proof}

A rule whose choice of face depends on the configuration, such as taking the largest adjacent value, is
not covered: it changes $h_\delta(p)$ by a bounded local random variable whose correlation with a distant
increment is not controlled by \eqref{eq:O1}.

\subsection{Discrete harmonicity}\label{subsec:harm}

The proof of \cite[Theorem~1.3]{ADHLM} uses, besides membership in $\ICI$ and inputs available for every
$c\in[1,2]$, one statement about the full-plane measure at positive mesh: the correlation between a
circle-average defect of the height function and that of its reflection in a line vanishes as
$\delta\to0$ \cite[(113)]{ADHLM}. The reference given there is a proposition of \cite{DKLM} on the
harmonicity of the limiting correlation functions \cite[Proposition~7.2]{DKLM}. The analogous step for $-1\le\Delta\le-\frac12$ is stated in one sentence in
\cite[after (143)]{DKLM}, by uniform convergence and harmonicity of the limiting covariance. We write the
argument out at positive mesh, uniformly over the lines involved, from membership in $\ICI$ alone.

\begin{lemma}[Discrete harmonicity]\label{lem:harm}
Let $\Delta\in\ICI$ for some grid-line rule, and let $h_\delta$ be defined with any grid-line rule. Fix
$r>0$, $\eta>0$, $C_0\ge0$ and a compact set $\mathcal L$ of pairs $(x,\ell)$ with $x\in\R^2$ and $\ell$ a line
at distance at least $r+\eta$ from $x$; write $p\mapsto p^\dagger$ for the reflection in $\ell$. For each
$\delta$ and each $(x,\ell)$ let $f\mapsto f^\dagger$ be a map on the faces of $\delta\Z^2$ such that the centre
of $f^\dagger$ is within $C_0\delta$ of the reflection of the centre of $f$, and put $h^\dagger(f):=h(f^\dagger)$.
With $\mathcal D_rF(x):=F(x)-\avint_{\partial B_r(x)}F(s)\,\dd s$,
\begin{equation}\label{eq:harm}
\sup_{(x,\ell)\in\mathcal L}\Bigl|\E_{\Pp_c}\bigl[\mathcal D_rh_\delta(x)\cdot\mathcal D_rh^\dagger_\delta(x)\bigr]\Bigr|
\ \xrightarrow[\delta\to0]{}\ 0,
\end{equation}
the supremum being also over all such face maps.
\end{lemma}

\begin{proof}
Let $\mathcal F^{(\delta)}_2(u_1,u_1';u_2,u_2'):=\E_{\Pp_c}[(h_\delta(u_1')-h_\delta(u_1))(h_\delta(u_2')-h_\delta(u_2))]$ with
$h_\delta$ defined by the floor rule, and let $D_2$ be the set of tuples with
$\{u_1,u_1'\}\cap\{u_2,u_2'\}=\emptyset$.

\emph{Step 1: uniform convergence.} We claim that $\mathcal F^{(\delta)}_2\to v\,\mathcal K$ uniformly on compact
subsets of $D_2$. If the convergence failed on some compact set along $\delta_n\to0$, choose a countable
dense set of tuples of distinct points with no coordinate in $\bigcup_n\delta_n\Z$. At these tuples no
grid-line rule is involved, so \eqref{eq:ICI} with $k=2$ gives pointwise convergence along $\delta_n$, and
\cite[Corollary~4.6]{DKLM}, valid for $c\in[1,2]$, upgrades pointwise convergence on a countable dense
set to uniform convergence on compact sets, a contradiction.

\emph{Step 2: face centres.} For a point $p$ let $\zeta(p)$ be the centre of the face that defines
$h_\delta(p)$ under the rule of the lemma, and $\zeta^\dagger(p)$ the centre of the image of that face under
$f\mapsto f^\dagger$. Then $h_\delta(p)$ and $h^\dagger_\delta(p)$ are the floor-rule values at $\zeta(p)$ and
$\zeta^\dagger(p)$, with $|\zeta(p)-p|\le\delta/\sqrt2$ and $|\zeta^\dagger(p)-p^\dagger|\le(C_0+1)\delta$, the second
because the reflection is an isometry.

\emph{Step 3: an integral formula.} $\mathcal D_rh_\delta(x)$ is the average over $s\in\partial B_r(x)$ of
$h_\delta(x)-h_\delta(s)$, and similarly for $h^\dagger_\delta$. All integrands are bounded for fixed $\delta$,
so by Fubini's theorem
\[
\E_{\Pp_c}\bigl[\mathcal D_rh_\delta(x)\,\mathcal D_rh^\dagger_\delta(x)\bigr]
=\avint_{\partial B_r(x)}\avint_{\partial B_r(x)}\mathcal F^{(\delta)}_2\bigl(\zeta(s),\zeta(x);\zeta^\dagger(t),\zeta^\dagger(x)\bigr)\,\dd s\,\dd t ,
\]
in which every argument of $\mathcal F^{(\delta)}_2$ is a face centre.

\emph{Step 4: passage to the limit.} Let $\mathcal Q$ be the set of tuples $(s,x;t^\dagger,x^\dagger)$ with
$(x,\ell)\in\mathcal L$ and $s,t\in\partial B_r(x)$; it is compact. The discs $\bar B_r(x)$ and $\bar B_r(x^\dagger)$
lie on opposite sides of $\ell$ at distance at least $\eta$ from it, so every point of the first pair is at
distance at least $2\eta$ from every point of the second. Let $\mathcal Q'$ be the closed
$\eta/2$-neighbourhood of $\mathcal Q$, a compact subset of $D_2$ on which $\mathcal K$ is uniformly continuous. For
$(C_0+1)\delta\le\eta/2$ the argument of $\mathcal F^{(\delta)}_2$ above lies in $\mathcal Q'$ within $(C_0+1)\delta$ of
$(s,x;t^\dagger,x^\dagger)$, so Step~1 gives
\[
\sup_{(x,\ell)\in\mathcal L}\Bigl|\E_{\Pp_c}\bigl[\mathcal D_rh_\delta(x)\,\mathcal D_rh^\dagger_\delta(x)\bigr]
-v\avint_{\partial B_r(x)}\avint_{\partial B_r(x)}\mathcal K(s,x;t^\dagger,x^\dagger)\,\dd s\,\dd t\Bigr|\longrightarrow0 .
\]

\emph{Step 5: the limit vanishes.}
$\mathcal K(s,x;t^\dagger,x^\dagger)=G(x,x^\dagger)-G(x,t^\dagger)-G(s,x^\dagger)+G(s,t^\dagger)$. For
$y\in\{x^\dagger,t^\dagger\}$ the function $G(\cdot,y)$ is harmonic near $\bar B_r(x)$, so its average over
$\partial B_r(x)$ is $G(x,y)$, and the average of $\mathcal K$ over $s$ vanishes for every $t$.
\end{proof}

\begin{remark}[Use in \cite{ADHLM}]\label{rem:use}
In the proof of \cite[Proposition~7.1]{ADHLM} the statement is needed at the origin, for reflections in
horizontal lines that separate the support of the circle average from a family of rectangles, at a
distance from the origin bounded below by $r$ plus a positive margin and bounded above in terms of the
data of the proposition. The reflection of faces in \cite{ADHLM} is defined through face centres; it maps
faces to faces, so that $C_0=0$, when the line is a lattice translate of the lines of
\cite[(107)]{ADHLM}, and in general any choice of a face whose closure contains the reflected centre has
$C_0\le1$. Lemma~\ref{lem:harm} gives the required bound uniformly over such lines. It is a statement at
fixed $\Delta$, which is why $\delta_0$ may depend on $\Delta$ in Corollary~\ref{cor:domains}.
\end{remark}

\begin{proof}[Proof of Corollary~\ref{cor:domains}]
Proposition~\ref{prop:rule} gives $\Delta\in\ICI$ for every grid-line rule, and the theorems of
\cite{ADHLM} are stated for every $\Delta\in\ICI$; Remark~\ref{rem:use} explains the role of
Lemma~\ref{lem:harm} in their proof. Apart from this step, the inputs of \cite{ADHLM} for $\Delta\in\ICI$
that we have traced are stated in their sources for all $c\in[1,2]$ or all $c\ge1$; we have not traced
the proofs of \cite[Proposition~3.5 and Corollary~3.6]{ADHLM} and of the diagonal case of
\cite[Theorem~6.3]{ADHLM}.
\end{proof}