that, observe that the Cauchy-Schwarz inequality and Lemma 4.5 give

$\displaystyle\Big{(}\sum_{y\in
Y}\phi[z_{0}\leftrightarrow\Lambda_{s}(y)|E]\Big{)}^{2}$
$\displaystyle\leq\sum_{y,z\in
Y}\phi[z_{0}\leftrightarrow\Lambda_{s}(y),z_{0}\leftrightarrow\Lambda_{s}(z)|E]$
$\displaystyle\leq C\sum_{y\in Y}\phi[z_{0}\leftrightarrow\Lambda_{s}(y)|E]$
$\displaystyle\qquad+C\sum_{y\neq z\in
Y}(\tfrac{s}{|z-y|})^{2}\Big{[}\phi[z_{0}\leftrightarrow\Lambda_{s}(y)|E]+\phi[z_{0}\leftrightarrow\Lambda_{s}(z)|E]\Big{]}$
$\displaystyle\leq C^{\prime}\sum_{y\in
Y}\phi[z_{0}\leftrightarrow\Lambda_{s}(y)|E],$

which implies that the sum is at most $C^{\prime}$, and therefore concludes
the proof. ∎

###  4.3 Proof of Theorem 2.4

We prove the first identity (the second follows from the same argument). It
suffices to show that

$\frac{{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{1}[\mathrm{lmax}(\infty)=0^{+}]}{{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{1}[\mathrm{lmax}(\infty)=0]}=\frac{\sin\alpha}{\sin\beta}.$

First of all, as seen in Theorem 3.8, for every $\varepsilon>0$, there exists
$R>0$ such that for every $\alpha,\beta\in(\varepsilon,\pi-\varepsilon)$ and
$q\in[1,4]$, for every event $A$,

$|{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}_{1}^{2}[A]-\phi_{\mathbb{L}_{1}}^{0}[A|\\{0\leftrightarrow\partial\Lambda_{R},\mathrm{lmax}(0)=0\\}\cup\\{0^{+}\leftrightarrow\partial\Lambda_{R},\mathrm{lmax}(0^{+})=0^{+}\\}]|\leq\varepsilon.$

The convergence in Theorem 3.8 is uniform in $q\in[1,4]$ and
$\alpha,\beta\in(\varepsilon,\pi-\varepsilon)$ (as shown by Proposition 3.9).
Also, the eigenvalues $\lambda_{N}^{(k)}(\theta)$ are continuous in $\theta$
and $q$. As a consequence, we only need to prove the claim for
$\alpha,\beta\neq\pi/2$ and $1\leq q<4$.

We now focus on this case. Applying Propositions 2.5 and 2.6 gives that

$\displaystyle\Big{|}\frac{{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{1}[\mathrm{lmax}(\infty)=0^{+}]}{{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{1}[\mathrm{lmax}(\infty)=0]}-\frac{\lambda^{(k)}(\beta)}{\lambda^{(k+3)}(\beta)}\times\frac{1-\lambda^{(k+3)}_{N}(\alpha)/\lambda^{(k)}_{N}(\alpha)}{1-\lambda^{(k+3)}_{N}(\beta)/\lambda^{(k)}_{N}(\beta)}\Big{|}\leq
C\Big{(}\frac{\lambda_{N}^{(k)}(\beta)}{\lambda_{N}^{(k+3)}(\beta)}-1\Big{)}^{\eta}.$
(44)

Below, $o(1)$ denotes a quantity tending to 0 as $N$ tends to infinity.
Theorem 2.7 implies that for $k\in[N^{1/2},2N^{1/2}]$,

$\frac{1-\lambda^{(k+3)}_{N}(\alpha)/\lambda^{(k)}_{N}(\alpha)}{1-\lambda^{(k+3)}_{N}(\beta)/\lambda^{(k)}_{N}(\beta)}=\frac{\log\lambda^{(k)}_{N}(\alpha)-\log\lambda^{(k+3)}_{N}(\alpha)}{\log\lambda^{(k)}_{N}(\beta)-\log\lambda^{(k+3)}_{N}(\beta)}+o(1)$

and

$\frac{\lambda^{(k)}_{N}(\beta)}{\lambda_{N}^{(k+3)}(\beta)}=1+o(1)$

so (44) implies that

$\frac{{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{1}[\mathrm{lmax}(\infty)=0^{+}]}{{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{1}[\mathrm{lmax}(\infty)=0]}=\frac{\log\lambda^{(k)}_{N}(\alpha)-\log\lambda^{(k+3)}_{N}(\alpha)}{\log\lambda^{(k)}_{N}(\beta)-\log\lambda^{(k+3)}_{N}(\beta)}+o(1).$
(45)

At this stage, we use Theorem 2.7 one more time to notice that

$\frac{\log\lambda^{(N^{1/2})}_{N}(\alpha)-\log\lambda^{(2N^{1/2})}_{N}(\alpha)}{\log\lambda^{(N^{1/2})}_{N}(\beta)-\log\lambda^{(2N^{1/2})}_{N}(\beta)}=\frac{\sin\alpha}{\sin\beta}+o(1).$

We deduce that there exists $k_{-}$ between $N^{1/2}$ and $2N^{1/2}$ such that

$\frac{\log\lambda^{(k_{-})}_{N}(\alpha)-\log\lambda^{(k_{-}+3)}_{N}(\alpha)}{\log\lambda^{(k_{-})}_{N}(\beta)-\log\lambda^{(k_{-}+3)}_{N}(\beta)}\geq\frac{\sin\alpha}{\sin\beta}-o(1)$

and similarly $k_{+}$ such that

$\frac{\log\lambda^{(k_{+})}_{N}(\alpha)-\log\lambda^{(k_{+}+3)}_{N}(\alpha)}{\log\lambda^{(k_{+})}_{N}(\beta)-\log\lambda^{(k_{+}+3)}_{N}(\beta)}\leq\frac{\sin\alpha}{\sin\beta}+o(1).$

Applying (45) to $k_{+}$ and $k_{-}$ and letting $N$ tend to infinity
concludes the proof.

##  5 Homotopy distance: proof of Theorem 2.2

###  5.1 Encoding of homotopy classes

In the introduction, we were not precise in the way we compute homotopy
classes. We now remedy this approximation. Recall that
$\mathbb{B}_{\eta}:=\eta\mathbb{Z}^{2}\cap[-1/\eta,1/\eta]^{2}$. Consider the
set of oriented edges $\vec{E}$:

$\vec{E}:=\\{(x,y):x,y\in\mathbb{B}_{\eta}:\|x-y\|_{2}=\eta\\}.$

Below, we see an oriented edge $(x,y)$ as a segment joining the endpoints $x$
and $y$ with an orientation from $x$ to $y$.

Let $\mathcal{W}$ be the set of finite words on the alphabet $\vec{E}$ and
denote the empty word by $\emptyset$. Define the “cyclic” equivalence relation
$\sim$ on $\mathcal{W}$ by $(u_{i})_{1\leq i\leq p}\sim(v_{j})_{1\leq j\leq
q}$ if and only if $p=q$ and there exists $k\in[1,p]$ such that $u_{1}\ldots
u_{p}=v_{k}\ldots v_{p}v_{1}\ldots v_{k-1}$. Define the set of cyclic words as
the quotient ${\mathcal{C}\mathcal{W}}:=\mathcal{W}/\sim$.

We also wish to work with a reduced representation of cyclic words. Let
$\preceq$ be the (smallest) order relation on $\mathcal{C}\mathcal{W}$ such
that for any word $u=u_{1}\cdots u_{p}$, any integer $1\leq k<p$ (resp. $1\leq
k\leq p$) such that $u_{k+1}=(x,y)$ and $u_{k}=(y,x)$, and $v=u_{1}\cdots
u_{k-1}u_{k+2}\cdots u_{p}$, we have $v\preceq u$. It is straightforward to
check that there exists a smallest $\underline{u}\preceq u$ for every $u$. We
call this the reduced word of $u$.

######  Definition 5.1 (Homotopy classes).

For a non-self-intersecting smooth loop
$\gamma\subset\mathbb{R}^{2}\setminus\mathbb{B}_{\eta}$, let
$u=\mathbf{u}(\gamma)$ be the word associated to $\gamma$ defined as follows:
orient the loop counterclockwise, fix $a\in\gamma$ not on an oriented
edge444Changing $a$ will correspond to a rerooting of the loop and will lead
to the same cyclic word., and let $\mathbf{u}(\gamma)=u_{1}\cdots u_{k}$,
where $u_{i}$ is the $i$-th (when going counter-clockwise along the curve
starting from $a$) oriented edge $(x,y)$ crossed by $\gamma$ in such a way
that $x$ is on the left of the crossing and $y$ on the right. Then, the
homotopy class $[\gamma]_{\eta}$ of $\gamma$ is the reduced word
$\underline{u}(\gamma)$.

######  Remark 5.2.

The previous definitions are sufficient for the proof of Theorem 2.2. In
preparation for the proof of Theorem 2.3 in the next section, we explain how
homotopy classes in other spaces considered in this paper are encoded.
Consider a collection of non-self-intersecting loops
$(\ell_{x}:x\in\mathbb{B}_{\eta})$ satisfying that the right-most point
$z(\ell_{x})$ in $\ell_{x}$ (if there is more than one such point, consider
the lowest one) belongs to $B(x,\tfrac{1}{4}\eta)$ for every
$x\in\mathbb{B}_{\eta}$. We wish to compute homotopy classes in the space
$\mathbb{R}^{2}\setminus\cup_{x\in\mathbb{B}_{\eta}}\ell_{x}$ in a way that is
consistent with the definition of $[\cdot]$ above. Define the segment $(x,y)$
for $x$ and $y$ neighbors to be the oriented segment from $z(\ell_{x})$ to
$z(\ell_{y})$. Encode the homotopy classes of loops in
$\mathbb{R}^{2}\setminus\cup_{x\in\mathbb{B}_{\eta}}\ell_{x}$ using reduced
words in the same way as above with segments between $z(\ell_{x})$ and
$z(\ell_{y})$ playing the role of the segment between $x$ and $y$ in the case
of $\mathbb{R}^{2}\setminus\mathbb{B}_{\eta}$.

Figure 19: In dashed black, the segments corresponding to the oriented edges
in $\mathbb{B}_{\eta}$. Also, the crosses correspond to the points
$z(\ell_{x})$ for the loops. Finally, the associated segments are depicted in
dashed red.

###  5.2 Proof of Theorem 2.2

###### Proof of Theorem 2.2.

We show the result for the Camia-Newman distance. The version of the result
for the Schramm-Smirnov distance follows readily from known implications
between the former and the latter (see e.g. [11, Theorem 7]). In this proof,
fix $\kappa,\eta,\delta>0$ such that

$\kappa>12\sqrt{2}\eta\geq 1000\delta.$

Below, a word $v=v_{1}\cdots v_{\ell}\in\mathcal{W}$ is a subword of
$u\in\mathcal{C}\mathcal{W}$ if there exists $k$ such that $v_{i}=u_{k+i}$ for
all $1\leq i\leq\ell$. We extend the order relation to non-cyclic words and
can therefore talk of $\underline{v}$. Also, we call ${\rm diam}(v)$ the
maximal Euclidean distance between two centers of edges in
$\\{v_{1},\dots,v_{k}\\}$ and say that $v$ intersects $B(0,1/\kappa)$ if it
contains a letter which is incident to a point of $\mathbb{B}_{\eta}\cap
B(0,1/\kappa)$. With these definitions, introduce the events

$\displaystyle\mathrm{Normal}(\kappa,\eta)$
$\displaystyle:=\Big{\\{}\begin{array}[]{c}\nexists\text{ a loop $\gamma$ such
that\leavevmode\nobreak\ ${\bf u}(\gamma)$ contains a subword $v$ }\\\
\text{intersecting $B(0,1/\kappa)$ with $\underline{v}=\emptyset$ and ${\rm
diam}(v)\geq\tfrac{1}{3}\kappa$}\end{array}\Big{\\}},$
$\displaystyle\mathrm{Dense}(\eta)$ $\displaystyle:=\\{\text{every face of
$\eta\mathbb{Z}^{2}\cap B(0,1/\eta)$ contains a loop in $\mathcal{F}_{0}$ and
one in $\mathcal{F}_{1}$}\\}.$

Now, consider $\omega_{\delta}\sim\phi_{\delta\mathbb{L}(\alpha)}$ and
$\overline{\omega}^{\prime}_{\delta}$ of
$\omega_{\delta}\sim\phi_{\delta\mathbb{L}(\pi/2)}$ and let
$\mathcal{F}_{i}(\omega_{\delta})$ and
$\mathcal{F}_{i}(\omega^{\prime}_{\delta})$ be the collections of loops
(primal or dual depending on $i$) in $\overline{\omega}_{\delta}$ and
$\overline{\omega}^{\prime}_{\delta}$ respectively.

#### Claim 1.

For every $\kappa>12\sqrt{2}\eta$, one has

$\\{d_{\mathbf{H}}(\omega_{\delta},\omega^{\prime}_{\delta})\leq\eta\\}\cap\\{\omega_{\delta},\omega^{\prime}_{\delta}\in\mathrm{Normal}(\kappa,\eta)\cap\mathrm{Dense}(\eta,\delta)\\}\subset\\{d_{\mathbf{CN}}(\omega_{\delta},\omega^{\prime}_{\delta})\leq\kappa\\}.$
(46)

###### Proof.

Consider a loop $\gamma\in\mathcal{F}_{i}(\omega_{\delta})$ that is included
in $B(0,1/\kappa)$, we need to prove that there exists
$\gamma^{\prime}\in\mathcal{F}_{i}(\omega_{\delta}^{\prime})$ such that
$d(\gamma^{\prime},\gamma)\leq\kappa.$ Since the same can be done for
$\mathcal{F}_{i}(\omega_{\delta}^{\prime})$, this will conclude the proof of
(46).

Write
$\boldsymbol{u}(\gamma)=\underline{u}_{1}v^{1}\underline{u}_{2}v^{2}\cdots\underline{u}_{k}v^{k}$,
where $\underline{u}_{i}$ are the letters of $\underline{u}=[\gamma]_{\eta}$
and $v^{1},\dots,v^{k}$ are words satisfying
$\underline{v}^{1}=\dots=\underline{v}^{k}=\emptyset$ (such a decomposition
exists but may not be unique). We justify in the next paragraph that
$\omega_{\delta}\in{\rm Normal}(\kappa,\eta)$ implies that
$d(\gamma,\boldsymbol{\gamma})\leq\kappa/2$ for any non-self-crossing smooth
curve $\boldsymbol{\gamma}$ satisfying
$\mathbf{u}(\boldsymbol{\gamma})=[\gamma]_{\eta}=\underline{u}$.

To prove that $d(\gamma,\boldsymbol{\gamma})\leq\kappa/2$, consider a
parametrization of $\gamma$ on $[0,1]$ and let $t_{i}$ be the first time
$t\geq t_{i-1}$ such that $\gamma(t_{i})\in\underline{u}_{i}$ (where we
consider $t_{0}=0$) and parametrize $\boldsymbol{\gamma}$ on $[0,1]$ in such a
way that $\boldsymbol{\gamma}(t_{i})\in\underline{u}_{i}$. Then, we claim that
for every $t\in[0,1]$, $|\gamma(t)-\boldsymbol{\gamma}(t)|\leq\kappa/2$.
Indeed, we know that for $t_{i}\leq t<t_{i+1}$, $\boldsymbol{\gamma}(t)$
belongs to the face of $\eta\mathbb{Z}^{2}$ that contains $\underline{u}_{i}$
and $\underline{u}_{i+1}$ and $\gamma(t)$ belongs to one of the faces
bordering $\underline{u}_{i}$, $\underline{u}_{i+1}$, or one of the letters in
$v^{i}$. Now, the diameter of $v^{i}$ is smaller than $\kappa/3$, and we
therefore deduce that $\gamma(t)$ is within distance
$\kappa/3+2\sqrt{2}\eta\leq\kappa/2$ of $\boldsymbol{\gamma}(t)$, hence
$d(\gamma,\boldsymbol{\gamma})\leq\kappa/2$.

We are now ready to conclude. Assume first that $\gamma$ surrounds at most one
point $x\in\mathbb{B}_{\eta}$. Then, $[\gamma]_{\eta}$ is either the empty
word, or a word made of four letters corresponding to edges incident to $x$.
As a consequence, we may choose $\boldsymbol{\gamma}$ with a diameter which is
smaller than $2\sqrt{2}\eta$ and such that
$u(\boldsymbol{\gamma})=[\gamma]_{\eta}$. Also, since
$\omega^{\prime}_{\delta}\in{\rm Dense}(\eta)$, there exists a loop
$\gamma^{\prime}\in\mathcal{F}_{i}(\omega_{\delta}^{\prime})$ included in one
of the faces intersected by $\boldsymbol{\gamma}$. We obtain immediately that
$d(\boldsymbol{\gamma},\gamma^{\prime})\leq 2\sqrt{2}\eta$ and therefore
$d(\gamma,\gamma^{\prime})\leq\kappa/2+2\sqrt{2}\eta\leq\kappa$.

Let us now assume that $\gamma$ surrounds at least two points in
$\mathbb{B}_{\eta}$. Being included in $B(0,1/\kappa)$, $\gamma$ cannot
surround all the points in $\mathbb{B}_{\eta}$. The fact that $d_{\bf
H}(\omega_{\delta},\omega^{\prime}_{\delta})\leq\eta$ thus implies the
existence of $\gamma^{\prime}\in\mathcal{F}_{i}^{\prime}$ such that
$[\gamma]_{\eta}=[\gamma^{\prime}]_{\eta}$. Since $\gamma^{\prime}$ must
intersect $B(0,1/\kappa)$ as well, and $\omega_{\delta}$ and
$\omega^{\prime}_{\delta}$ are in ${\rm Normal}(\kappa,\eta)$, we obtain that
$d(\gamma,\boldsymbol{\gamma})\leq\kappa/2$ and
$d(\gamma^{\prime},\boldsymbol{\gamma})\leq\kappa/2$ for every
$\boldsymbol{\gamma}$ with
$u(\boldsymbol{\gamma})=[\gamma]_{\eta}=[\gamma^{\prime}]_{\eta}$. The
triangular inequality gives that $d(\gamma^{\prime},\gamma)\leq\kappa$. This
concludes the proof.∎

We now turn to another claim.

#### Claim 2.

There exist $c,C\in(0,\infty)$ such that for every $\beta$,

$\phi_{\delta\mathbb{L}(\beta)}[\mathrm{Normal}(\kappa,\eta)]\geq
1-\tfrac{C}{\eta^{2}\kappa^{2}}\exp[-c\kappa/\eta].$ (47)

###### Proof.

Let $A(\kappa,\eta)$ be the event that there exists a crossing of the
rectangle $R:=[0,\kappa/3]\times[0,\kappa/12]$ whose cluster in the strip
$\mathbb{R}\times[0,\kappa/12]$ surrounds no vertex in $\eta\mathbb{Z}^{2}$.
We claim the existence of $c>0$ such that

$\phi_{\delta\mathbb{L}(\beta)}[A(\kappa,\eta)]\leq C\exp[-c\kappa/\eta].$
(48)

To prove (48), let $\mathbf{N}$ be the number of clusters that contain a
vertical crossing of $R$ and for $i\leq\mathbf{N}$, call $\Gamma_{i}$ the
right-boundary of the $i$-th cluster $\mathcal{C}_{i}$ in $R$ crossing $R$
when starting counting clusters from the right. Let $\Omega_{i}$ be the set of
vertices in $\mathbb{R}\times[0,\kappa/12]$ on the left of (and including)
$\Gamma_{i}$ (see Figure 20 for a picture). Note that $\Gamma_{i}$ is
measurable in terms of the edges on $\Gamma_{i}$ or on its right, and that it
induces wired boundary conditions on $\Gamma_{i}$ for the measure in
$\Omega_{i}$. Let $X_{i}$ be a maximal set of vertices in
$\Omega_{i}\cap\eta\mathbb{Z}^{2}$ that are at a distance at least $4\eta$ of
each other555This is a technical statement enabling to use the comparison
between boundary conditions “independently” in each of the boxes
$\Lambda_{2\eta}(x)$., but at a distance at most $\eta$ of $\Gamma_{i}$. Note
that one may easily construct such a set of cardinality at least
$\lfloor\kappa/(48\eta)\rfloor-1$. Then, for every $x\in X_{i}$, by (RSW)
there exists an open path disconnecting $\Lambda_{\eta}(x)$ from
$\partial\Lambda_{2\eta}(x)$ in $\Omega_{i}$ with probability larger than
$c_{0}>0$, even when we enforce free boundary conditions on
$\partial\Lambda_{2\eta}(x)$. The comparison between boundary conditions thus
implies that

$\phi_{\delta\mathbb{L}(\beta)}[\nexists x\in X_{i}\text{ surrounded by
}\mathcal{C}_{i}|\Gamma_{i}]\leq(1-c_{0})^{|X_{i}|}\leq(1-c_{0})^{\lfloor\kappa/(48\eta)\rfloor-1}.$

It remains to sum over $i$ and use that
$\phi_{\delta\mathbb{L}(\beta)}[\mathbf{N}]\leq C_{0}$ (Proposition 3.7) to
get (48):

$\displaystyle\phi_{\delta\mathbb{L}(\beta)}[A(\kappa,\eta)]$
$\displaystyle\leq(1-c_{0})^{\lfloor\kappa/(48\eta)\rfloor-1}\phi_{\delta\mathbb{L}(\beta)}[\mathbf{N}]\leq
C_{0}(1-c_{0})^{\lfloor\kappa/(48\eta)\rfloor-1}.$

We are now in a position to conclude. Consider the square box
$B_{1}:=[-\kappa/24,\kappa/24]^{2}$ as well as the rectangle
$R_{1}^{R}:=[\kappa/12,\kappa/6]\times[-\kappa/6,\kappa/6]$ and its rotations
$R_{1}^{T}$, $R_{1}^{L}$, and $R_{1}^{B}$ by angles $\pi/2$, $\pi$, and
$3\pi/2$, respectively. Also, consider a collection of translates
$(B_{i},R_{i}^{R},R_{i}^{T},R_{i}^{L},R_{i}^{B})$ of
$(B_{1},R_{1}^{R},R_{1}^{T},R_{1}^{L},R_{1}^{B})$ such that the boxes $B_{i}$
cover $B(0,1/\eta-\kappa/3)$. Note that the probability that a
translate/rotation/dual of $A(\kappa,\eta)$ occurs for some $R_{i}^{\\#}$ is
bounded by the right-hand side of (48). Write $A_{\rm global}$ for the event
that the rotation/translation of $A$ or its dual occurs for some
$R_{i}^{\\#}$. The union bound implies that

$\displaystyle\phi_{\delta\mathbb{L}(\beta)}[A_{\rm global}(\kappa,\eta)]$
$\displaystyle\leq\frac{C_{1}}{\eta^{2}\kappa^{2}}(1-c_{0})^{\lfloor\kappa/(48\eta)\rfloor-1}.$

We next prove that ${\rm Normal}(\kappa,\eta)$ occurs as soon as $A_{\rm
global}(\kappa,\eta)$ does.

To see this, assume ${\rm Normal}(\kappa,\eta)$ fails and consider a loop
$\gamma$ and a subword $v$ of $u(\gamma)$ with $\underline{v}=\emptyset$ of
maximal diameter among the subwords intersecting $B(0,1/\kappa)$. The first
and last letters of $v$ must necessarily border the same face $f$. Consider
two times $s$ and $t$ such that $\gamma(s),\gamma(t)\in f$ and $\gamma([s,t])$
has $v$ as an encoding word, and close $\gamma([s,t])$ into a non-self-
crossing loop $\ell(\gamma)$ by going from $\gamma(s)$ to $\gamma(t)$ inside
the face $f$. Note that $f$ must intersect a box $B_{i}$, and that
$\ell(\gamma)$ must cross one of the four rectangles $R_{i}^{\\#}$ around it.
Now, outside of $f$, $\ell(\gamma)$ is identical to $\gamma$ so it has either
primal edges of $\omega_{\delta}$ bordering it on its interior or dual edges.
In the former case, the non occurrence of $A(\kappa,\eta)$ for the rectangle
mentioned above implies that $\ell(\gamma)$ must necessarily surround a point
in $\mathbb{B}_{\eta}$, which contradicts the fact that
$\underline{v}=\emptyset$. In the latter case, the non occurrence of the dual
of $A(\kappa,\eta)$ implies the same claim. ∎

We are now in a position to conclude the proof of the theorem. Claim 1 implies
that

$\displaystyle\mathbf{P}[d_{\mathbf{H}}(\omega_{\delta},\omega^{\prime}_{\delta})\leq\eta,d_{\mathbf{CN}}(\omega_{\delta},\omega^{\prime}_{\delta})>\kappa]$
$\displaystyle\leq\phi_{\delta\mathbb{L}(\alpha)}[{\rm
Normal}(\kappa,\eta)^{c}]+\phi_{\delta\mathbb{L}(\pi/2)}[{\rm
Normal}(\kappa,\eta)^{c}]$
$\displaystyle\qquad+\phi_{\delta\mathbb{L}(\alpha)}[{\rm
Dense}(\eta)^{c}]+\phi_{\delta\mathbb{L}(\pi/2)}[{\rm Dense}(\eta)^{c}].$

Now, Claim 2 applied to $\beta$ equal to $\alpha$ or $\tfrac{\pi}{2}$ gives

$\phi_{\delta\mathbb{L}(\alpha)}[{\rm
Normal}(\kappa,\eta)^{c}]+\phi_{\delta\mathbb{L}(\pi/2)}[{\rm
Normal}(\kappa,\eta)^{c}]\leq\tfrac{1}{2}\kappa,$

provided $\eta=\eta(\kappa)>0$ is chosen small enough.

Finally, since any vertex with four closed edges incident to it gives rise to
a small loop in $\mathcal{F}_{1}$, and similarly for $\mathcal{F}_{0}$ when
considering the dual graph, the finite-energy property immediately implies
that

$\phi_{\delta\mathbb{L}(\alpha)}[{\rm
Dense}(\eta)^{c}]+\phi_{\delta\mathbb{L}(\pi/2)}[{\rm
Dense}(\eta)^{c}]\leq\frac{2C}{\eta^{2}}\exp[-c(\eta/\delta)^{2}]\leq\tfrac{1}{2}\kappa,$

provided $\delta=\delta(\eta)$ small enough. The last three displayed
inequalities conclude the proof of the theorem.∎

Figure 20: An example with two clusters crossing. Note that the set $X_{i}$
does not have to be included in the rectangle $R$. A way to construct a large
set of points with the properties of $X_{i}$ is to choose for each $1\leq
j<\lfloor\kappa/48\eta\rfloor-1$, on each line
$\\{(x,y)\in\mathbb{R}^{2}:y=(2j+1)\eta\\}$, a vertex of $\Omega_{i}$ which is
at a distance smaller than $\eta$ of $\Gamma_{i}$.

##  6 Universality in isoradial rectangular graphs: proof of Theorem 2.3

The section is divided in six subsections. In the first one, we recall the
setting of the proof and introduce some convenient notation. In the second
one, we define the notion of nails, give a precise definition of
$\mathcal{H}$, and introduce the formal definition of our coupling. The third
one explains how one can couple the increments of the maximal coordinates of
nails with independent increments that have the law of increments in a track-
exchange on the IIC. In the fourth subsection, we will explore the combination
of several increments into so-called compounded steps corresponding to
bringing down one track from its starting to its ending position. The fifth
subsection shows that the speed that can be associated to the evolution of a
compounded step is approximately zero. Finally, the last subsection contains
the proof of the theorem.

###  6.1 Setting of the proof

Below, fix $\alpha\in(0,\pi/2)$ (the case $\alpha>\pi/2$ can be obtained by a
global reflection with respect to the $y$-axis). We further assume, except
when otherwise stated, that

$\cos\alpha\notin\mathbb{Q}.$ (49)

This assumption plays an implicit role in the definition of the coupling, and
is essential in the proof of Proposition 6.14, see Remark 6.15.

Figure 21: The quantities $h(t)$, $a(t)$, $b(t)$ and $d(t)$. One may deduce
from the picture that $\lfloor t/(2N)\rfloor=12$ as there are 12 tracks of
angle $\alpha$ stacked above level $-N$.

Also, let $0<\varepsilon\ll\eta\ll 1$ be fixed along the whole section (they
will be chosen appropriately in the proof of the theorem at the end of the
section). Set

$\mathbb{B}_{\eta}(N):=\eta N\mathbb{Z}^{2}\cap[-N,N]^{2}$

(note that it is not quite the blow up by a factor $N$ of $\mathbb{B}_{\eta}$)
and

$T:=2N\times\lceil 2N/\sin\alpha\rceil.$

As in Section 2, $\mathbb{L}^{(0)}$ is the isoradial lattice with angles

$\alpha_{j}=\alpha_{j}(\alpha,N):=\begin{cases}\ \alpha&\text{ if }j\geq N,\\\
\ \tfrac{\pi}{2}&\text{ if }j<N.\end{cases}$

Recall the successive transformations $\mathbf{T}_{j(t)}$ of the lattice
described in Section 2: at time $0\leq t<T$, the track to be descended is
$t_{j(t)}$ with

$j(t):=N+(2N+1)\lfloor t/(2N)\rfloor-t.$

and the graph $\mathbb{L}^{(t)}$ obtained from the graph $\mathbb{L}^{(0)}$ by
applying successively the maps $\mathbf{T}_{j(s)}$ for $0\leq s<t$.

We use the following four convenient quantities (see Figure 21):

$\displaystyle h(t)$ $\displaystyle:=\text{the second coordinate of the
horizontal line $t_{j(t)}^{-}$ in $\mathbb{L}^{(t)}$},$ $\displaystyle a(t)$
$\displaystyle:=h(2N\lfloor t/(2N)\rfloor)+\sin\alpha,$ $\displaystyle b(t)$
$\displaystyle:=h(2N\lfloor t/(2N)\rfloor-1)+\sin\alpha-1\text{ if }t>2N\text{
and }:=-N\text{ if }t\leq 2N,$ $\displaystyle d(t)$
$\displaystyle:=\min\\{h(t)-b(t),a(t)-h(t)\\}.$

Note that $b(t)$ is the top of the track of angle $\alpha$ below $t_{j(t)}$
(except when $t<2N$ in which case we set it to be $-N$ by convention), and
$a(t)$ is the bottom of the track of angle $\alpha$ above $t_{j(t)}$.

###  6.2 Definition of nails, marked nails, and the coupling $\mathbf{P}$

Recall the definition, for a (primal) cluster $\mathcal{C}$ in a configuration
$\omega$, of ${\rm T}(\mathcal{C})$, ${\rm B}(\mathcal{C})$ and ${\rm
R}(\mathcal{C})$, which are respectively the maximal second, minimal second
and maximal first coordinates of a vertex in $\mathcal{C}$. Define ${\rm
Vspan}(\mathcal{C}):={\rm T}(\mathcal{C})-{\rm B}(\mathcal{C})$.

######  Definition 6.1 (Nail).

For $x=(x_{1},x_{2})\in\mathbb{B}_{\eta}(N)$ and a configuration $\omega$,
call a (primal) cluster $\mathcal{C}$ of $\omega$ on some $\mathbb{L}^{(t)}$ a
nail (near $x$) if

${\rm Vspan}(\mathcal{C})\geq\varepsilon N\qquad\text{and}\qquad\max\\{|{\rm
T}(\mathcal{C})-x_{2}|,|{\rm B}(\mathcal{C})-x_{2}|,|{\rm
R}(\mathcal{C})-x_{1}|\\}\leq\sqrt{\eta\varepsilon}N.$

We now define the coupling of the measures $\phi_{\mathbb{L}^{(t)}}$ for
$0\leq t\leq T$.

Figure 22: In black, the nails (i.e. the elements indexed by $I(\omega)$) and
in bold black the marked nails. The red segments depict the information
provided by $({\rm T}^{(t)},{\rm B}^{(t)},{\rm R}^{(t)})$. The blue loops are
the loops surrounding at least two marked nails, i.e. the loops contributing
to $[\cdot]_{\bullet,0}$ and $[\cdot]_{\bullet,1}$. The grey area are the
boxes $\Lambda_{\sqrt{\varepsilon\eta}N}(x)$, which one should think of
potentially much bigger than the minimal size $O(\varepsilon N)$ of nails, but
much smaller than the minimal distance $\eta N$ between vertices of
$\mathbb{B}_{\eta}(N)$.

Step 0 of the coupling $\mathbf{P}$. Sample
$\omega^{(0)}\sim\phi_{\mathbb{L}^{(0)}}$.

Index the nails near all points in $\mathbb{B}_{\eta}$ in $\omega^{(0)}$ by
integers $1,\dots,M=M(\omega^{(0)})$, and let $\mathcal{C}(\omega^{(0)},i)$ be
the nail indexed by $i$. Define $I^{(0)}:=\\{1,\dots,M\\}$. Write ${\rm
T}^{(0)}$, ${\rm B}^{(0)}$ and ${\rm R}^{(0)}$ for the functions from
$I^{(0)}$ into $\mathbb{R}$ giving, for every $i\in I^{(0)}$ and $A\in\\{{\rm
T},{\rm B},{\rm R}\\}$, $A^{(0)}(i):=A(\mathcal{C}(\omega^{(0)},i))$. Also set
${\rm Vspan}(i):={\rm Vspan}(\mathcal{C}(\omega^{(0)},i))$.

For each $x=(x_{1},x_{2})\in\mathbb{B}_{\eta}(N)$, choose, if it exists,
$i_{x}\in I^{(0)}$ such that

${\rm Vspan}(i_{x})\geq 2\varepsilon N\qquad\text{and}\qquad\max\\{|{\rm
T}(i_{x})-x_{2}|,|{\rm B}(i_{x})-x_{2}|,|{\rm
R}(i_{x})-x_{1}|\\}\leq(\sqrt{\eta\varepsilon}-\varepsilon)N$ (50)

(if there is more than one, pick the smallest such integer). Call
$\mathcal{C}(\omega^{(0)},i_{x})$ the marked nail near $x$. Let
$I_{\bullet}\subset I^{(0)}$ be the indexes corresponding to the marked nails
near each $x\in\mathbb{B}_{\eta}(N)$, with the understanding that there may be
some $x$ for which there is no such marked nail (we will see later in this
section that, with a very large probability, there is a marked nail near every
$x\in\mathbb{B}_{\eta}(N)$).

Finally, if there exists a marked nail near every $x\in\mathbb{B}_{\eta}$,
introduce the two multisets666Formally, these are functions from the set of
homotopy classes, or in our case of reduced words, into non-negative integers.
$[\cdot]^{(0)}_{\bullet,0}$ and $[\cdot]^{(0)}_{\bullet,1}$ gathering the
homotopy classes (in the sense of Remark 5.2) in the full plane minus the
marked nails $\mathbb{R}^{2}\setminus\\{\mathcal{C}(\omega,i):i\in
I_{\bullet}\\}$ of the loops in $\mathcal{F}_{0}(\omega^{(0)})$ and
$\mathcal{F}_{1}(\omega^{(0)})$ that surround at least two but not all marked
nails. At this stage, we insist on the fact that $[\cdot]^{(0)}_{\bullet,0}$
and $[\cdot]^{(0)}_{\bullet,1}$ are multisets as there may be more than one
loop in $\omega^{(0)}$ of a given homotopy class. If there exists
$x\in\mathbb{B}_{\eta}$ that does not have a marked nail near it, simply set
$[\cdot]^{(0)}_{\bullet,0}=[\cdot]^{(0)}_{\bullet,1}=\emptyset$.

To lighten the notation, we write

$\mathcal{H}^{(0)}:=(I^{(0)},{\rm T}^{(0)},{\rm B}^{(0)},{\rm
R}^{(0)},[\cdot]^{(0)}_{\bullet,0},[\cdot]^{(0)}_{\bullet,1}).$

Fix now $0\leq t<T$ and assume that $\omega^{(t)}$ and
$\mathcal{H}^{(t)}=(I^{(t)},{\rm T}^{(t)},{\rm B}^{(t)},{\rm
R}^{(t)},[\cdot]^{(t)}_{\bullet,0},[\cdot]^{(t)}_{\bullet,1})$ have been
constructed (for the latter, the way it is given in terms of $\omega^{(t)}$ is
explained for $t+1$ below), where

  * •

$I^{(t)}$ is a subset of $\mathbb{Z}_{>0}$,

  * •

${\rm T}^{(t)},{\rm B}^{(t)},{\rm R}^{(t)}$ are functions from $I^{(t)}$ to
$\mathbb{R}$,

  * •

$[\cdot]^{(t)}_{\bullet,0}$ and $[\cdot]^{(t)}_{\bullet,1}$ are multisets with
elements in homotopy classes in
$\mathbb{R}^{2}\setminus\\{\mathcal{C}(\omega^{(t)},i):i\in I_{\bullet}\\}$ if
$I_{\bullet}\subset I^{(t)}$, and equal to $\emptyset$ otherwise.

Step $t$ to $t+\tfrac{1}{2}$ of the coupling $\mathbf{P}$. Sample
$\omega^{(t+1/2)}\sim\phi_{\mathbb{L}^{(t)}}[\,\cdot\,|\mathcal{H}=\mathcal{H}^{(t)}],$

where $\omega\in\\{\mathcal{H}=H\\}$ (with $H=(I,{\rm T},{\rm B},{\rm
R},[\cdot]_{\bullet,0},[\cdot]_{\bullet,1})$ a possible realization of
$\mathcal{H}^{(t)}$) denotes the event that

  * (i)

there exists an indexation of the nails in $\omega$ by $I$ (call
$\mathcal{C}(\omega,i)$ the nail indexed by $i\in I$);

  * (ii)

$A(\mathcal{C}(\omega,i))=A(i)$ for every $i\in I$ and $A\in\\{{\rm T},{\rm
B},{\rm R}\\}$;

  * (iii)

if $I_{\bullet}\subset I$, the further requirement that $[\cdot]_{\bullet,0}$
and $[\cdot]_{\bullet,1}$ are giving the homotopy classes of the loops of
$\omega$ that surround at least two but not all marked nails.

Step $t+\tfrac{1}{2}$ to $t+1$ of the coupling $\mathbf{P}$. Set
$\omega^{(t+1)}:=\mathbf{T}_{j(t)}(\omega^{(t+1/2)})$ (remember that
$\mathbf{T}_{j(t)}$ is a map sending a configuration to a random
configuration).

Due to the previous step, $\omega^{(t+1/2)}$ necessarily satisfies the event
$\\{\mathcal{H}=\mathcal{H}^{(t)}\\}$. Consider the indexation of the nails of
$\omega^{(t+1/2)}$ by $I^{(t)}$ given by (i) and write
$\mathcal{C}(\omega^{(t+1/2)},i)$ for the nail indexed by $i$.

Since the track-exchange map $\mathbf{T}_{j(t)}$ is obtained as a sequence of
star-triangle transformations, one can check that a nail
$\mathcal{C}(\omega^{(t+1/2)},i)$ is transformed into a cluster $\mathcal{C}$
in $\omega^{(t+1)}$. If $\mathcal{C}$ is still a nail in $\omega^{(t+1)}$,
include $i$ in $I^{(t+1)}$ and define $A^{(t+1)}(i)=A(\mathcal{C})$ for
$A\in\\{{\rm T},{\rm B},{\rm R}\\}$. If this is not the case, then do not
include $i$ in $I^{(t+1)}$. Finally, for each “new” nail
$\mathcal{C}^{\prime}$ in $\omega^{(t+1)}$, i.e. a nail that was not in
$\omega^{(t+1/2)}$, pick an integer $i$ that was not used in any of the
$I^{(s)}$ for $s\leq t$ and include it in $I^{(t+1)}$. As before, let
$A^{(t+1)}(i)=A(\mathcal{C}^{\prime})$ for $A\in\\{{\rm T},{\rm B},{\rm
R}\\}$. Note that nails may appear or disappear when applying
$\mathbf{T}_{j(t)}$ since extrema of clusters may move during the star-
triangle transformations, which may alter the validity of the conditions of
being a nail.

If $I_{\bullet}\subset I^{(t+1)}$, define $[\cdot]^{(t+1)}_{\bullet,0}$ and
$[\cdot]^{(t+1)}_{\bullet,1}$ to be the multisets giving the homotopy classes
in $\mathbb{R}^{2}\setminus\\{\mathcal{C}(\omega^{(t+1)},i):i\in
I_{\bullet}\cap I^{(t+1)}\\}$ of the loops surrounding at least two but not
all marked nails. Otherwise, set
$[\cdot]^{(t+1)}_{\bullet,0}=[\cdot]^{(t+1)}_{\bullet,1}=\emptyset$.

Set

$\mathcal{H}^{(t+1)}:=(I^{(t+1)},{\rm T}^{(t+1)},{\rm B}^{(t+1)},{\rm
R}^{(t+1)},[\cdot]^{(t+1)}_{\bullet,0},[\cdot]^{(t+1)}_{\bullet,1}).$

From now on, call $\mathbf{P}$ the coupling thus obtained.

######  Remark 6.2.

Let us mention that a marked nail can disappear at some time $t$, meaning that
some $i\in I_{\bullet}$ can be in $I^{(t)}$ but not in $I^{(t+1)}$, but no new
marked nail may appear. The disappearance of a marked nail affects
significantly the notion of homotopy and we stop keeping track of it (hence
the convention of denoting
$[\cdot]^{(t)}_{\bullet,0}=[\cdot]^{(t)}_{\bullet,1}=\emptyset$ and to
consider it as an empty condition in (iii) of the definition of
$\mathcal{H}=\mathcal{H}^{(t)}$). We will see that the condition (50) on our
marked nails guarantees a posteriori that the marked nails do not disappear
during the whole process with high probability. We will also see that
$[\cdot]^{(t)}_{\bullet,0}$ and $[\cdot]^{(t)}_{\bullet,1}$ are preserved, and
therefore equal to their values at time 0.

###  6.3 Controlling one single time step using IIC increments

In this section, we wish to connect our configurations $(\omega^{(t)}:0\leq
t<T)$ with IIC measures in order to be able to control the displacements of
the nails during the process. We therefore “decorate” our coupling by
enhancing it with additional configurations sampled according to IIC measures.

Recall the lattice $\mathbb{L}^{(i)}$ with horizontal tracks of angle $\beta$
except for $t_{i}$ which has angle $\alpha$. In this section, $\beta$ is fixed
to be $\pi/2$. Recall also the IIC measures
${\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}_{i}$, $\Psi_{i}$ and  $\Psi$
defined on $\mathbb{L}^{(i)}$ and $\mathbb{L}^{(0)}$ respectively.

Let us give ourselves i.i.d. families of random variables $\omega^{(t)}_{A,j}$
indexed by $0\leq t<T$, $j\in\mathbb{Z}$, and $A\in\\{{\rm T},{\rm B},{\rm
R}\\}$, with laws ${\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}_{j}$,
$\Psi_{j}$, and  $\Psi$ if $A={\rm T},{\rm B},{\rm R}$ respectively (when
$A={\rm R}$ there is no need for a subscript $j$ but we will use this
convenient “unified” notation).

We also give ourselves independent $\\{0,1\\}$-valued random variables
$X^{(t)}_{H}(i)$, indexed by all possible values $H=(I,{\rm T},{\rm B},{\rm
R},[\cdot]_{\bullet,0},[\cdot]_{\bullet,1})$ of $\mathcal{H}^{(t)}$ and $i\in
I$, satisfying

$\mathbf{P}[X^{(t)}_{H}(i)=1]=\phi_{\mathbb{L}^{(t)}}[({\rm
R}(i),h(t))\in\mathcal{C}(\omega,i)\,|\,\mathcal{H}=H]$

(it is possible that $({\rm R}(i),h(t))$ is not a vertex of
$\mathbb{L}^{(t)}$, in which case $X_{H}^{(t)}(i)$ is 0 almost surely). To get
an intuition on these variables, in the coupling below, the fact that
$X_{\mathcal{H}^{(t)}}^{(t)}(i)$ is equal to 1 will detect whether the cluster
$\mathcal{C}(\omega,i)$ has a right extremum on $t_{j(t)}^{-}$.

We are now ready to define the coupling of the process $(\omega^{(t)}:0\leq
t<T)$ with the random variables introduced above. Below, the steps $0$ and
$t+1/2$ to $t+1$ are done exactly as in the previous section. We therefore
focus on the steps $t$ to $t+1/2$. We will sample $\omega^{(t+1/2)}$ from
$\omega^{(t)}$ in a few steps, coupled to the variables introduced in the two
last paragraphs. Nevertheless, notice that the law of $\omega^{(t+1/2)}$ given
$\omega^{(t)}$ is the same as in the previous section. For this reason, we
keep denoting this bigger coupling $\mathbf{P}$.

For $0\leq t<T$ and a nail $\mathcal{C}$, call a vertex $x\in\mathbb{L}^{(t)}$
a

  * •

Top $t$-extremum (of $\mathcal{C}$) if $x\in t_{j(t)}^{-}\cup t_{j(t)-1}^{-}$
and its second coordinate equals ${\rm T}(\mathcal{C})$,

  * •

Bottom $t$-extremum (of $\mathcal{C}$) if $x\in t_{j(t)}^{-}\cup
t_{j(t)+1}^{-}$ and its second coordinate equals ${\rm B}(\mathcal{C})$,

  * •

Right $t$-extremum (of $\mathcal{C}$) if $x\in t_{j(t)}^{-}$ and its first
coordinate equals ${\rm R}(\mathcal{C})$,

  * •

Fake right $t$-extremum (of $\mathcal{C}$) if $x\in t_{j(t)}^{-}$, its first
coordinate is strictly larger than ${\rm R}(\mathcal{C})-\cos\alpha$, and the
vertex of $\mathcal{C}$ with maximal first coordinate is below $b(t)$.

We also use vertical $t$-extremum to denote a top or bottom $t$-extremum.

######  Remark 6.3.

Note that for $A(\mathcal{C})$ to be possibly modified by the track-exchange,
there must exist a $A$ $t$-extremum or fake right $t$-extremum in the case
$A={\rm R}$.

Complete description of the coupling $\mathbf{P}$ from time $t$ to $t+1/2$ Fix
$0\leq t<T$ and assume that $I_{\bullet}$, $\omega^{(t)}$, and
$\mathcal{H}^{(t)}$ have been defined. We divide the construction of
$\omega^{(t+1/2)}$ in four cases; which case applies is determined by
$\mathcal{H}^{(t)}$.

#### Case 0: Two distinct nails contain vertical $t$-extrema.

In such case, sample $\omega^{(t+1/2)}$ as in previous section independently
of the variables $\omega^{(t)}_{A,j}$ and $X_{H}^{(t)}$.

#### Case 1: A unique nail contains a vertical $t$-extremum and it is a top
one.

Let $i$ be the index of this nail. Proceed as follows: Step 1. Sample the
random variable $\mathbf{x}$ in such a way that for every
$x\in\mathbb{L}^{(t)}$,
$\mathbf{P}[\mathbf{x}=x]=\phi_{\mathbb{L}^{(t)}}[\mathrm{lmax}(\mathcal{C}(\omega,i))=x|\mathcal{H}=\mathcal{H}^{(t)}].$
Step 2. Sample $\omega$ according to
$\phi_{\mathbb{L}^{(t)}}[\,\cdot\,|\mathrm{lmax}(\mathcal{C}(\omega,i))=\mathbf{x},\mathcal{H}=\mathcal{H}^{(t)}]$
and sample $\omega^{(t+1/2)}=\omega$ on the set $\Omega(\mathbf{x},\omega)$ of
edges outside of $\Lambda_{d(t)^{1/3}}(\mathbf{x})$ that are connected in
$\omega$ to the complement of $\Lambda_{d(t)^{1/2}}(\mathbf{x})$ (see Figure
26). Step 3. Sample $(\omega^{(t+1/2)},\omega_{h(t)-T^{(t)}(i)}^{\rm T})$ (for
the first one we only need to sample the remaining edges) using the coupling
between
$\phi_{\mathbb{L}^{(t)}}[\,\cdot\,|\mathrm{lmax}(\mathcal{C}(\omega^{(t+1/2)},i))=\mathbf{x},\mathcal{H}=\mathcal{H}^{(t)},\omega^{(t+1/2)}_{|\Omega(\mathbf{x},\omega^{(t+1/2)})}]\quad\text{and}\quad{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}_{j(t)-T^{(t)}(i)}$
which is maximizing the probability that $\omega^{(t+1/2)}$ and the translate
of $\omega_{{\rm T},j(t)-T^{(t)}(i)}^{(t)}$ by $\mathbf{x}$ coincide on
$\Lambda_{d(t)^{1/4}}(\mathbf{x})$.

#### Case 2: A unique nail contains a vertical $t$-extremum and it is a bottom
one.

Proceed exactly as in the previous step with ${\rm B}$ instead of ${\rm T}$
and $\Psi$ instead of  $\Psi$ .

#### Case 3: No nail contains a vertical $t$-extremum.

Proceed as follows, Step -1. Couple777Note that $X^{(t)}$ and
$\widetilde{X}^{(t)}$ have the same marginal laws, but that in the former the
random variables $X^{(t)}_{\mathcal{H}^{(t)}}(i)$ are independent, while in
the latter they are not. in the best possible way the random variables
$\displaystyle X^{(t)}$ $\displaystyle:=(X^{(t)}_{\mathcal{H}^{(t)}}(i):i\in
I^{(t)})\qquad\text{ and
}\qquad\widetilde{X}^{(t)}=(\widetilde{X}^{(t)}(i):i\in I^{(t)}),$ where
$\widetilde{X}^{(t)}$ has the law of the random variable $(\mathbf{1}[({\rm
R}(i),h(t))\in\mathcal{C}(\omega,i)]:i\in I^{(t)})$ with
$\omega\sim\phi_{\mathbb{L}^{(t)}}[\,\cdot\,|\mathcal{H}=\mathcal{H}^{(t)}]$.
Step 0. If $X^{(t)}\neq\widetilde{X}^{(t)}$ or $\sum_{i\in
I^{(t)}}\widetilde{X}^{(t)}(i)\neq 1$, sample independently the random
variables $\omega^{(t)}_{A,j}$ and the random variable
$\omega^{(t+1/2)}\sim\phi_{\mathbb{L}^{(t)}}[\,\cdot\,|\mathcal{H}=\mathcal{H}^{(t)},X=\widetilde{X}^{(t)}],$
where the event $X=\widetilde{X}^{(t)}$ means that $({\rm R}(i),h(t))$ belongs
to the nail indexed by $i$ if and only if $\widetilde{X}^{(t)}(i)=1$. Step 1.
Otherwise, set $\mathbf{x}:=({\rm R}^{(t)}(i),h(t))$ with $i$ the integer such
that $\widetilde{X}^{(t)}(i)=1$. Step 2–3. Proceed as in Case 1 with ${\rm R}$
instead of ${\rm T}$ and  $\Psi$ instead of  $\Psi$ , except that we further
condition at each step on $X=\widetilde{X}^{(t)}$.

######  Remark 6.4.

We will see in the next section that Case 0 occurs very rarely, hence we do
not bother coupling efficiently the true configuration with an IIC
configuration in this case. In Step 3 of Case 1, it could be that the best
coupling is terrible due to the fact that $\omega^{(t+1/2)}$ does something
strange on $\Omega(\mathbf{x},\omega^{(t+1/2)})$. Yet, this will be shown to
occur with only small probability and the best coupling guarantees equality of
the two configurations with very large probability. Finally, in Step 0 of Case
3, the best coupling (which depends on $\mathcal{H}^{(t)}$) is typically
making the two random variables equal. We will see that in this case there is
typically a single $i$ for which $X^{(t)}(i)=1$.

We now turn to an important proposition describing the increments
$A^{(t+1)}(i)-A^{(t)}(i)$ for the nails $i\in I^{(t)}$ in terms of increments
of IIC variables, called the IIC displacement random variables.

######  Definition 6.5 (IIC displacements).

Sample a configuration according to
${\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}_{j}$, apply $\mathbf{T}_{j}$,
and call $\delta^{\rm IIC}_{j}{\rm T}$ the maximal $y$-coordinate of a vertex
of the incipient infinite cluster after the transformation. Similarly, sample
a configuration according to $\Psi_{j}$, apply $\mathbf{T}_{j}$, and call
$\delta^{\rm IIC}_{j}{\rm B}$ the maximal $y$-coordinate of a vertex of the
incipient infinite cluster after the transformation. Finally, sample a
configuration according to  $\Psi$ , apply $\mathbf{T}_{0}$, and call
$\delta^{\rm IIC}{\rm R}$ the maximal $x$-coordinate of a vertex of the
incipient infinite cluster after the transformation.

######  Remark 6.6.

The effect of a track exchange on the top and right of a cluster is described
in Figures 23 and 24. Notice that $\delta^{\rm IIC}_{1}{\rm
T}\in\\{0,\sin\alpha\\}$, $\delta^{\rm IIC}_{0}{\rm
T}\in\\{-1,\sin\alpha-1\\}$, and $\delta^{\rm IIC}_{j}{\rm T}=0$ for other
values of $j$. Similarly, $\delta_{0}^{\rm IIC}{\rm
B}\in\\{\sin\alpha,\sin\alpha-1\\}$, $\delta_{-1}^{\rm IIC}{\rm
B}\in\\{0,-1\\}$, and $\delta_{j}^{\rm IIC}{\rm B}=0$ for other values of $j$.
For the right, note that $\delta^{\rm IIC}{\rm
R}\in\\{0,-1,\cos\alpha-1,\cos\alpha\\}.$

######  Remark 6.7.

The effect of a track exchange implies that ${\rm T}^{(t+1)}(i)-{\rm
T}^{(t)}(i)$ and ${\rm B}^{(t+1)}(i)-{\rm B}^{(t)}(i)$ belong to
$\\{-1,\sin\alpha-1,0,\sin\alpha\\}$. For ${\rm R}^{(t+1)}(i)-{\rm
R}^{(t)}(i)$, the situation is more complicated since there may be a fake
right $t$-extremum, whose coordinate is therefore not equal to ${\rm
R}(\mathcal{C}(\omega^{(t+1/2)},i))$, that jumps right and after the
transformation has a first coordinate equal to ${\rm
R}(\mathcal{C}(\omega^{(t+1)},i))$. Nevertheless, one always has $|{\rm
R}^{(t+1)}(i)-{\rm R}^{(t)}(i)|\leq 1$.

Let $\delta_{j}^{\rm IIC}A^{(t)}$ be the displacement constructed out of the
IIC configuration $\omega_{A,j}^{(t)}$ by applying $\mathbf{T}_{j(t)}$. Note
that the $\delta_{j}^{\rm IIC}A^{(t)}$ form families of i.i.d. random
variables with law $\delta_{j}^{\rm IIC}A$. For $i\in I^{(t)}$, define the
random variables

$\displaystyle\delta^{{\rm err}}{\rm T}^{(t)}(i)$ $\displaystyle:={\rm
T}^{(t+1)}(i)-{\rm T}^{(t)}(i)-\delta_{h(t)-T^{(t)}(i)}^{\rm IIC}{\rm
T}^{(t)}(i),$ $\displaystyle\delta^{{\rm err}}{\rm B}^{(t)}(i)$
$\displaystyle:={\rm B}^{(t+1)}(i)-{\rm
B}^{(t)}(i)-\delta_{h(t)-B^{(t)}(i)}^{\rm IIC}{\rm B}^{(t)}(i),$
$\displaystyle\delta^{{\rm err}}{\rm R}^{(t)}(i)$ $\displaystyle:={\rm
R}^{(t+1)}(i)-{\rm R}^{(t)}(i)-X^{(t)}_{\mathcal{H}^{(t)}}(i)\,\delta^{{\rm
IIC}}{\rm R}^{(t)}(i).$

Figure 23:  Different environments around $0$ picked according to the
incipient infinite cluster measures with a track of angle $\alpha$ at height
$1$ (left) and $0$ (right). The bold edges and points are part of the
incipient infinite clusters. The different outcomes of the transformations
give different top-most points for the infinite cluster.
Left two diagrams: Two possible outcomes of the track-exchange
$\mathbf{T}_{1}$ corresponding to $\delta^{\rm IIC}_{1}{\rm T}=0$ and
$\delta^{\rm IIC}_{1}{\rm T}=\sin\alpha$. The first outcome occurs certainly
when the gray rhombus contains a primal edge, and with positive probability
when it contains a dual one; in the latter case, the second outcome is also
possible.
Right three diagrams: Three possible outcomes of the track-exchange
$\mathbf{T}_{0}$ corresponding to $\delta^{\rm IIC}_{0}{\rm T}=-1$ and
$\delta^{\rm IIC}_{0}{\rm T}=\sin\alpha-1$, respectively. The first outcome
occurs only when the edge below and to the left of $0$ is the unique open edge
adjacent to $0$.  Figure 24: When performing a track exchange between $t_{0}$
and $t_{-1}$, the vertex $0$ is modified locally so that the coordinate of the
right-most point in $t_{0}^{-}\cup t_{-1}^{-}$ moves by either $-1$ (first
line) or $\cos\alpha$ (second line). The first outcome occurs certainly when
both gray rhombi contain primal edges, and with positive probability
otherwise; the second outcome may only occur when at least one of the two gray
rhombi contains a dual edge. When the second outcome occurs, $\delta^{\rm
IIC}{\rm R}=\cos\alpha$. For the first outcome, $\delta^{\rm IIC}{\rm R}$ may
take values $0$, $\cos\alpha-1$, or $-1$. The first two values appear if the
incipient infinite cluster contains a vertex below $0$ with first coordinate
$0$, or one above $0$, with first coordinate $\cos\alpha-1$, respectively. The
same outcomes occur for any environment in $t_{0}$ and $t_{-1}$ to the left of
$0$.

We are now ready to present the main statement of this section.

######  Proposition 6.8 (Properties of the coupling).

For $\varepsilon,\eta>0$, the coupling $\mathbf{P}$ satisfies the following
properties:

  * (o)

for every $0\leq t\leq T$, $\omega^{(t)}\sim\phi_{\mathbb{L}^{(t)}}$;

  * (i)

$([\cdot]^{(t)}_{\bullet,0},[\cdot]^{(t)}_{\bullet,1})=([\cdot]^{(0)}_{\bullet,0},[\cdot]^{(0)}_{\bullet,1})$
for every $t<\tau$, where $\tau:=\inf\\{t>0:I_{\bullet}\not\subset
I^{(t)}\\}$;

  * (ii)

for every $t\geq 0$, the variables $(\delta_{j}^{\rm
err}A^{(s)}(i),\delta^{\rm IIC}_{j}A^{(s)}(i),X^{(s)}_{H}(i):A,i,j,H,s<t)$ are
independent of the $(\delta^{\rm IIC}_{j}A^{(t)}(i),X^{(t)}_{H}(i):A,i,j,H)$;

  * (iii)

there exist $C,c\in(0,\infty)$ such that for every $t\geq 0$,

$\mathbf{E}[{\rm Err}^{(t)}]\leq\frac{C}{d(t)N^{c}},$

where $\displaystyle{\rm Err}^{(t)}:=\sum_{i\in I^{(t)}}M^{(t)}(i)$ with
$M^{(t)}(i):=|\delta^{{\rm err}}{\rm T}^{(t)}(i)|+|\delta^{{\rm err}}{\rm
B}^{(t)}(i)|+|\delta^{{\rm err}}{\rm R}^{(t)}(i)|$.

######  Remark 6.9.

Let us mention that we expect the bound of (iii) to be valid with $N^{1+c}$
instead of $d(t)N^{c}$ in the denominator. The reason for the appearance of
$d(t)$ is due to the fact that we do not, at this stage, know how to prove
(15) for generic sequences of angles $\boldsymbol{\alpha}$. In retrospect, our
rotation invariance result shows that (15) does hold for arbitrary sequences,
but only post factum.

Figure 25: We depicted the example of a nail having two top $t$-extrema.

The rest of the subsection is dedicated solely to proving Proposition 6.8.
This proof is tedious, but does not involve particularly innovative ideas (the
heavy lifting was done when properly defining the coupling). In the first
reading, one may skip the proof and focus on the next sections first.

Before diving into the proof of this proposition, let us start with a lemma.
Define

$\displaystyle{\rm BAD}_{1}(t)$ $\displaystyle:=\\{\text{a nail contains a
vertical and a right $t$-extremum}\\},$ $\displaystyle{\rm BAD}_{2}(t)$
$\displaystyle:=\\{\text{a nail contains two vertical $t$-extrema $x,y$
satisfying }|x-y|\geq d(t)^{1/5}\\},$ $\displaystyle{\rm BAD}_{3}(t)$
$\displaystyle:=\\{\text{two nails contain a $t$-extremum}\\},$
$\displaystyle{\rm BAD}_{4}(t)$ $\displaystyle:=\\{\text{a nail contains a
fake right $t$-extremum}\\},$ $\displaystyle{\rm BAD}_{5}(t)$
$\displaystyle:=\Big{\\{}\begin{array}[]{c}\text{a nail $i$ contains a right
$t$-extremum and a vertex $(x_{1},x_{2})$}\\\ \text{with ${\rm
R}^{(t)}(i)-1<x_{1}<{\rm R}^{(t)}$ and $|x_{2}-h(t)|\geq
d(t)^{1/5}$}\end{array}\Big{\\}}.$

######  Lemma 6.10.

There exist $C,c\in(0,\infty)$ such that for every $0\leq t<T$ and $1\leq
i\leq 4$,

$\phi_{\mathbb{L}^{(t)}}[{\rm BAD}_{i}(t)]\leq\frac{C}{d(t)N^{c}}.$

###### Proof.

We divide into the different events ${\rm BAD}_{i}(t)$.

#### Bound on the probability of ${\rm BAD}_{1}(t)$.

Assume that the vertical $t$-extremum is a top $t$-extremum (the bottom case
is treated similarly) and let $x$ be the right $t$-extremum of the nail. In
this case, there must be a three-arm event in the bottom-left quarterplane
translated by $x$, and going from $\Lambda_{1}(x)$ to
$\partial\Lambda_{\varepsilon N}(x)$. We therefore deduce from (13) that the
probability of this is bounded by $C/(\varepsilon N)^{2+c}$. Summing over
$O(N)$ possible values of $x$ – recall that since it is a right $t$-extremum,
$x$ is within a distance $\sqrt{\varepsilon\eta}N$ of one of the points in
$\mathbb{B}_{\eta}(N)$ – gives the required bound.

#### Bound on the probability of ${\rm BAD}_{2}(t)$.

In this case, the two $t$-extrema are either both in the top direction, or
both in the bottom one (since ${\rm Vspan}\geq\varepsilon N>1+\sin\alpha$ for
$N$ large enough). Let us assume it is the former that happens and let $x$ and
$y$ be the two $t$-extrema. We assume that $x$ is on the left of $y$. Also,
note that they have to be exactly at the same height as they belong to the
same nail. The following must therefore occur (see Figure 25 for a picture):

  * •

a $3$-arm event in the half-plane below $x$ from $x$ to
$\partial\Lambda_{|x-y|/2}(x)$;

  * •

a $3$-arm event in the half-plane below $y$ from $y$ to
$\partial\Lambda_{|x-y|/2}(y)$;

  * •

a $3$-arm event in the half-plane below $x$ from $\Lambda_{2|x-y|}(x)$ to
$\partial\Lambda_{\varepsilon N}(x)$ (this may be an empty condition if
$|x-y|\geq\tfrac{1}{2}\varepsilon N$);

  * •

a 1-arm event from $\Lambda_{\varepsilon N}(x)$ to $\Lambda_{N}$ in
$\mathbb{Z}\times[-N,N]$ (this last condition is only relevant in case
$x\notin\Lambda_{N}$, which may occur since nails may be very long in the left
direction, and have maxima far on the left of $\Lambda_{N}$ – obviously, this
is atypical, but should be taken care of nonetheless).

We deduce from (10) that

$\phi_{\mathbb{L}^{(t)}}[x,y\text{ top $t$-extrema of the same
nail}]\leq\frac{C}{|x-y|^{4}}\Big{(}\frac{|x-y|}{\varepsilon
N}\Big{)}^{2}\exp\big{(}-\frac{c|x|}{N}\big{)}.$ (51)

Summing over $x$ and $y$ at a distance $d(t)^{1/4}$ of each other (with $y$ on
the right of $x$ and left of the right-side of $\Lambda_{N}$) gives that

$\phi_{\mathbb{L}^{(t)}}[{\rm
BAD}_{2}(t)]\leq\frac{C^{\prime}}{\varepsilon^{2}Nd(t)^{1/4}}.$ (52)

#### Bound on the probability of ${\rm BAD}_{3}(t)$.

Assume that $x$ and $y$ are the right $t$-extrema of two different nails. The
cases of the top or bottom $t$-extrema are actually simpler to handle (and
they give better bounds).

First, assume that $|x-y|\leq 2d(t)$ and that $y$ is on the right of $x$. In
this case, for $x$ and $y$ to be right $t$-extrema of their respective nails,
there must be

  * •

a $3$-arm event in the half-plane on the left of $x$ from $x$ to
$\partial\Lambda_{|x-y|/2}(x)$;

  * •

a $3$-arm event in the half-plane on the left of $y$ from $y$ to
$\partial\Lambda_{|x-y|/2}(y)$;

  * •

a $5$-arm event in the half-plane on the left of $y$ from
$\Lambda_{2|x-y|}(y)$ to $\partial\Lambda_{d(t)}(y)$ (this may be an empty
condition if $|x-y|\geq d(t)/2$);

  * •

a $3$-arm event888In fact even a $5$-arm event occurs up to
$\partial\Lambda_{\varepsilon N/2-|x-y|}(y)$. Also, once this is observed, one
may wonder why we distinguish between the third and fourth bullets since in
both cases a $5$-arm event occurs. The reason comes from the fact that the
estimate used in both cases is not quite the same, since one occurs in an area
with all but one track having a transverse angle equal to $\tfrac{\pi}{2}$,
while the second occurs in a “mixed” lattice. in the half-plane on the left of
$y$ from $\Lambda_{d(t)}(y)$ to $\partial\Lambda_{\varepsilon N/2}(y)$.

Using (15) (twice), (16), and (12), we deduce that

$\phi_{\mathbb{L}^{(t)}}[x,y\text{ right $t$-extrema of distinct
nails}]\leq\frac{C}{|x-y|^{4}}\Big{(}\frac{|x-y|}{d(t)}\Big{)}^{2+c_{0}}\Big{(}\frac{2d(t)}{\varepsilon
N}\Big{)}^{1+c_{1}}.$ (53)

Now, assume $|x-y|>2d(t)$ and assume that $y$ is right of $x$. In this case,
for $x$ and $y$ to be at the right-most ends of their respective nails, there
must be

  * •

a $3$-arm event in the half-plane on the left of $x$ from $x$ to
$\partial\Lambda_{d(t)}(x)$;

  * •

a $3$-arm event in the half-plane on the left of $y$ from $y$ to
$\partial\Lambda_{d(t)}(y)$;

  * •

a $3$-arm event in the half-plane on the left of $x$ from $\Lambda_{d(t)}(x)$
to $\partial\Lambda_{|x-y|/2}(x)$;

  * •

a $3$-arm event in the half-plane on the left of $y$ from $\Lambda_{d(t)}(y)$
to $\partial\Lambda_{|x-y|/2}(y)$;

  * •

a $3$-arm event999Again, there is even a $5$-arm event up to
$\partial\Lambda_{\varepsilon N/2-|x-y|}(y)$. in the half-plane on the left of
$y$ from $\Lambda_{2|x-y|}(y)$ to $\partial\Lambda_{\varepsilon N/2}(y)$ (this
condition is empty if $|x-y|\geq\varepsilon N/4$).

Using (15) (twice) and (12) (three times), we deduce that

$\phi_{\mathbb{L}^{(t)}}[x,y\text{ right $t$-extrema of distinct
nails}]\leq\frac{C}{d(t)^{4}}\Big{(}\frac{d(t)}{|x-y|}\Big{)}^{2+2c_{1}}\Big{(}\frac{2|x-y|}{\varepsilon
N}\Big{)}^{1+c_{1}}.$ (54)

Let us now sum on $y$ and then on $x$ to obtain

$\displaystyle\phi_{\mathbb{L}^{(t)}}[$ $\displaystyle\text{two distinct nails
in }I^{(t)}\text{ contain a right $t$-extremum}]$ $\displaystyle\leq
2CN\Big{[}\sum_{k=1}^{2d(t)}\frac{1}{k^{4}}\Big{(}\frac{k}{d(t)}\Big{)}^{2+c_{0}}\Big{(}\frac{d(t)}{\varepsilon
N}\Big{)}^{1+c_{1}}+\sum_{k=2d(t)}^{2N}\frac{1}{d(t)^{4}}\Big{(}\frac{d(t)}{k}\Big{)}^{2+2c_{1}}\Big{(}\frac{k}{\varepsilon
N}\Big{)}^{1+c_{1}}\Big{]}$
$\displaystyle\leq\frac{C_{1}}{d(t)^{1+c_{0}}}\Big{(}\frac{d(t)}{\varepsilon
N}\Big{)}^{c_{1}}+\frac{C_{2}}{d(t)^{2}}\Big{(}\frac{d(t)}{\varepsilon
N}\Big{)}^{c_{1}}\leq\frac{C_{3}}{d(t)(\varepsilon N)^{c_{3}}},$

where we choose $c_{3}:=\min\\{c_{0},c_{1}\\}$.

Doing the same with other types of $t$-extrema implies the bound for ${\rm
BAD}_{3}(t)$.

#### Bound on the probability of ${\rm BAD}_{4}(t)$.

For $x$ to be a fake right $t$-extremum of a nail $\mathcal{C}$, there must
exist $y=(y_{1},y_{2})\in\mathcal{C}$ with $y_{2}\leq b(t)$ and $y_{1}={\rm
R}(\mathcal{C})$. As a consequence, there must be

  * •

a $3$-arm event in the half-plane on the left of $x$ from $x$ to
$\partial\Lambda_{d(t)/2}(x)$;

  * •

a $3$-arm event in the half-plane on the left of $x$ from
$\Lambda_{d(t)/2}(x)$ to $\partial\Lambda_{|x-y|/2}(x)$;

  * •

a $3$-arm event in the half-plane on the left of $y$ from $y$ to
$\partial\Lambda_{|x-y|/2}(y)$;

  * •

a $3$-arm event in the half-plane on the left of $y$ from
$\Lambda_{2|x-y|}(y)$ to $\partial\Lambda_{\varepsilon N/2}(x)$.

If we denote $E(x,y)$ the previous event, we deduce that

$\phi_{\mathbb{L}^{(t)}}[E(x,y)]\leq\frac{C}{d(t)^{2}}\Big{(}\frac{d(t)}{|x-y|}\Big{)}^{1+c_{1}}\Big{(}\frac{2}{|x-y|}\Big{)}^{1+c_{1}}\Big{(}\frac{4|x-y|}{\varepsilon
N}\Big{)}^{1+c_{1}}.$ (55)

Summing over $y$ and then over $x$ gives,

$\phi_{\mathbb{L}^{(t)}}[{\rm
BAD}_{4}(t)]\leq\sum_{x,y}\phi_{\mathbb{L}^{(t)}}[E(x,y)]\leq\sum_{x}\frac{C}{d(t)}\frac{C_{2}}{N^{1+c_{1}}}\leq\frac{C_{3}}{d(t)N^{c_{1}}}.$
(56)

#### Bound on the probability of ${\rm BAD}_{5}(t)$.

The bound follows from a combination of the arguments for ${\rm BAD}_{2}(t)$
and ${\rm BAD}_{4}(t)$. We leave it to the reader. ∎

Figure 26: A picture of the set $\Omega(\omega^{(t+1/2)},x)$ (in grey). The
dark grey cluster is the unique cluster of the annulus crossing from inside to
outside. Everything outside $\Lambda_{d(t)^{1/2}}(x)$ is included in
$\Omega(\omega^{(t+1/2)},x)$. The red part is the place where we try to couple
as best as possible $\omega^{(t+1/2)}$ and $\mathbf{x}+\omega^{(t)}_{{\rm
T},j(t)-T^{(t)}(i)}$. In the proof of Proposition 6.8, note that when applied
to $\omega^{(t+1/2)}\in E(x)$ (which means that there exists a unique cluster
crossing the annulus), all the conditions on the $t$-extrema of other nails
and homotopy classes of loops are not impacted by what happens inside
$\Omega(\omega^{(t+1/2)},\mathbf{x})^{c}$. As a consequence, there is a
“screening” property and the conditioning inside is simply the existence of
the red arms, meaning that $\mathbf{x}$ is equal to the left-most top-most
vertex in the cluster that is crossing the annulus.

###### Proof of Proposition 6.8.

By construction of the coupling, (o) and (ii) are trivial.

Property (i) follows from the locality of the star-triangle operations and the
common way of measuring
$([\cdot]^{(t)}_{\bullet,0},[\cdot]^{(t)}_{\bullet,1})$. Indeed, since
$[\cdot]^{(t-1)}_{\bullet,0}=[\cdot]^{(t-1/2)}_{\bullet,0}$ by definition of
$\omega^{(t-1/2)}$, it suffices to check that
$[\cdot]^{(t-1/2)}_{\bullet,0}=[\cdot]^{(t)}_{\bullet,0}$. Now,
$I_{\bullet}\subset I^{(s)}$ for every $s\leq t$ so there are nails near every
$x\in\mathbb{B}_{\eta}(N)$. Recall the definition of oriented edges $(x,y)$
from Section 5.1. Since the star-triangle transformations modify the lowest
right-most point of marked nails of $\omega^{(t-1/2)}$, but do so only
locally, while preserving the connections outside of $t_{j(t)}^{-}$, we
immediately get that the reduced words are not modified by the track-exchange,
and therefore $[\cdot]^{(t-1/2)}_{\bullet,0}=[\cdot]^{(t)}_{\bullet,0}$.

We therefore focus on proving (iii). We divide the analysis of ${\rm
Err}^{(t)}$ depending on the case used for the coupling. Below, constants
$c_{i},C_{i}$ are universal and independent of everything else except
$\varepsilon$ and $\eta$.

#### Error in Case 0

If Case 0 holds, then $\omega^{(t)}\in{\rm BAD}_{3}(t)$. Due to the coupling
generated by the track-exchange, the displacement of any $t$-extremum is at
most 1 so all variables $\delta^{\rm err}A^{(t)}(i)$ are deterministically
bounded by 2. Thus, Markov’s inequality implies that for every $\lambda\geq
0$,

$\displaystyle\mathbf{E}[{\rm Err}^{(t)}\mathbf{1}_{\mathrm{Case}\ 0}]$
$\displaystyle\leq 6\mathbf{E}[|I^{(t)}|\mathbf{1}_{\omega^{(t)}\in{\rm
BAD}_{3}(t)}]\leq 6\lambda\mathbf{P}[\omega^{(t)}\in{\rm
BAD}_{3}(t)]+6\mathbf{P}[|I^{(t)}|>\lambda].$

Lemma 6.10 and Proposition 3.7 imply that by choosing $\lambda$ to be a large
multiple of $\log N$, we obtain

$\displaystyle\mathbf{E}[{\rm Err}^{(t)}\mathbf{1}_{\mathrm{Case}\ 0}]$
$\displaystyle\leq\frac{C_{2}}{d(t)N^{c_{2}}}.$

#### Error in Cases 1 and 2

We deal with Case 1 as Case 2 can be treated in the same way. Since
$\mathbf{x}$ can take values $x\in S^{(t)}$ only, and that by (10),

$\phi_{\mathbb{L}^{(t)}}[\mathrm{lmax}(\mathcal{C}(\omega,i))=x,\mathbf{x}=x,\mathcal{H}=\mathcal{H}^{(t)}]\leq\frac{C_{3}}{N^{2}}\exp[-c_{3}\tfrac{|x|}{N}],$
(57)

(as in the bound of the probability of ${\rm BAD}_{2}(t)$, we need to account
for the possibility that $x$ is far on the left of $\Lambda_{N}$), it suffices
to show that

$\mathbf{P}[{\rm Err}^{(t)}\neq
0|\omega^{(t+1/2)}\in\\{\mathrm{lmax}(\mathcal{C}(\omega,i))=x,\mathbf{x}=x,\mathcal{H}=\mathcal{H}^{(t)}\\}]$

is small and to plug it in (57) above. Then, summing over $x$ and applying a
manipulation similar to Case 0 will conclude the proof.

For $(A,i)$ in $\\{({\rm B},i),i\\}\cup\\{({\rm T},i),i\neq\mathbf{i}\\}$,
with $\mathbf{i}$ the unique integer such that
$\mathcal{C}(\omega,\mathbf{i})$ contains a top $t$-extremum, we immediately
find that

$\delta^{\rm err}A^{(t)}(i)=A^{(t+1)}(i)-A^{(t)}(i)=\delta^{\rm
IIC}A^{(t)}(i)=0.$

Therefore, the errors can only come from the evolution of ${\rm
T}^{(t)}(\mathbf{i})$ and the ${\rm R}^{(t)}(i)$ for $i\in I^{(t)}$. We treat
the case $A={\rm T}$ and $A={\rm R}$ separately.

Below, we fix $x$ and set $\omega^{{\rm T}}:=\omega^{({\rm T})}_{h(t)-{\rm
T}^{(t)}(i)}.$

#### Error from the top $t$-extremum

We start with $|\delta^{\rm err}{\rm T}^{(t)}(\mathbf{i})|$, which can come
from a number of facts (see Figure 27):

  * (i)

$\omega^{(t+1/2)}$ and $x+\omega^{\rm T}$ are not equal on
$\Lambda_{d(t)^{1/4}}(x)$;

  * (ii)

there is another top $t$-extremum in
$\mathcal{C}(\omega^{(t+1/2)},\mathbf{i})$ at a distance at least $d(t)^{1/5}$
of $x$;

  * (iii)

there are two top $t$-extrema in $\omega^{\rm T}$ at a distance at least
$d(t)^{1/5}$ of each other.

  * (iv)

$\omega^{(t+1/2)}$ and $x+\omega^{\rm T}$ are equal on
$\Lambda_{d(t)^{1/4}}(x)$ but the track-exchange operators outputs on
$\Lambda_{d(t)^{1/5}}(x)$ are different in $\omega^{(t+1/2)}$ and $\omega^{\rm
T}$;

Indeed, if none of (i)–(iv) occurs, then (i) gives that $\omega^{(t+1/2)}$ and
$x+\omega^{\rm T}$ coincide on $\Lambda_{d(t)^{1/4}}(x)$, (iv) guarantees that
the result of the track-exchange output is the same in
$\Lambda_{d(t)^{1/5}}(x)$. Finally, the absence of other top $t$-extrema in
either $\omega^{(t+1/2)}$ and $\omega^{T}$ guarantees that the change of
height is measured by what happens within $\Lambda_{d(t)^{1/5}}(x)$.

#### Subcase (i).

Let $E(x)$ be the event that there is a unique cluster in $\omega^{(t+1/2)}$
crossing the annulus $\Lambda_{d(t)^{1/2}}(x)\setminus\Lambda_{d(t)^{1/3}}(x)$
from outside to inside. Note that this event is measurable in terms of
$\omega^{(t+1/2)}$ restricted to $\Omega(x,\omega^{(t+1/2)})$. Furthermore,
observe that this event has a “screening effect” (see Figure 26) implying

$\phi_{\mathbb{L}^{(t)}}[\,\cdot\,|\mathrm{lmax}(\mathcal{C}(\omega,\mathbf{i}))=x,\mathbf{x}=x,\mathcal{H}=\mathcal{H}^{(t)},\omega^{(t+1/2)}_{|\Omega(x,\omega^{(t+1/2)})}]=\phi_{\Omega(x,\omega^{(t+1/2)})^{c}}^{\xi}[\,\cdot\,|\mathrm{lmax}(\mathcal{C})=x],$

where $\xi$ are the boundary conditions induced by $\omega^{(t+1/2)}$ on the
graph $\mathbb{L}^{(t)}\setminus\Omega(x,\omega^{(t+1/2)})$, and $\mathcal{C}$
is the unique cluster crossing the annulus
$\Lambda_{d(t)^{1/2}}(x)\setminus\Lambda_{d(t)^{1/3}}(x)$.

Therefore, the mixing property of the IIC given by Proposition 3.9 implies
that on $\omega^{(t+1/2)}\in E(x)$, the coupling does not give equality with
probability at most $C_{3}d(t)^{-c_{3}}$. Combined with (57) (and using
$|\delta^{\rm err}{\rm T}^{(t)}|\leq 2$), we deduce that

$\displaystyle\mathbf{E}[|\delta^{\rm err}{\rm
T}^{(t)}(\mathbf{i})|\mathbf{1}_{(i),\mathrm{lmax}(\mathcal{C}(\omega^{(t+1/2)},\mathbf{i}))=x}]$
$\displaystyle\leq\frac{2C_{3}}{d(t)^{c_{3}}}\mathbf{P}[\mathrm{lmax}(\mathcal{C}(\omega^{(t+1/2)},\mathbf{i}))=x]+2\mathbf{P}[\mathrm{lmax}(\mathcal{C}(\omega^{(t+1/2)},\mathbf{i}))=x,\omega^{(t+1/2)}\notin
E(x)]$ $\displaystyle\leq\frac{C_{4}}{N^{2}d(t)^{c_{4}}}\exp[-c|x|/N],$

where in the last inequality we used (10) and (14).

#### Subcase (ii).

In this case, $\omega^{(t+1/2)}$ belongs to ${\rm BAD}_{2}(t)$ so Lemma 6.10
gives

$\mathbf{E}[|\delta^{\rm err}{\rm T}^{(t)}(\mathbf{i})|\mathbf{1}_{(ii)}]\leq
2\phi_{\mathbb{L}^{(t)}}[{\rm BAD}_{2}(t)]\leq\frac{C_{5}}{d(t)N^{c_{5}}}$

(we directly provided the estimate summed over $x$ in this case as it follows
from the statement of Lemma 6.10).

#### Subcase (iii).

First, since by construction $\omega^{\rm T}$ is independent of the event
$\omega^{(t+1/2)}\in\\{\mathrm{lmax}(\mathcal{C}(\omega,i))=x,\mathbf{x}=x,\mathcal{H}=\mathcal{H}^{(t)}\\}$,
it suffices to prove that

$\mathbf{P}[\omega^{\rm T}\text{ contains a top $t$-extremum outside
$\Lambda_{d(t)^{1/5}}$}]\leq\frac{C_{6}}{d(t)^{1/4}}.$

To see this, simply sum over $y\in\mathbb{Z}\setminus\Lambda_{d(t)^{1/4}}$ the
probability of $y$ being a top $t$-extremum, which can easily be proved to be
of order $C_{7}/|y|^{2}$. We conclude that

$\mathbf{P}[|\delta^{\rm err}{\rm
T}^{(t)}(\mathbf{i})|\mathbf{1}_{(iii),\mathrm{lmax}(\mathcal{C}(\omega^{(t+1/2)},\mathbf{i}))=x}]\leq\frac{C_{8}}{N^{2}d(t)^{1/4}}\exp[-c|x|/N].$

#### Subcase (iv).

To be in this case, it must be that in the intersection of the annulus
$\Lambda_{d(t)^{1/4}}(x)\setminus\Lambda_{d(t)^{1/5}}(x)$ with
$t_{j(t)}^{-}\cup t_{j(t)-1}^{-}$ on the left and the right of $x$, there is
no pair of closed edges on top of each other since the existence of such edges
decouple the star-triangle transformations on their left and right as seen in
the paragraph above Definition 3.11. We deduce that

$\mathbf{E}[|\delta^{\rm err}{\rm
T}^{(t)}(\mathbf{i})|\mathbf{1}_{(iv),\mathrm{lmax}(\mathbf{i})=x}]\leq\frac{C_{8}\exp[-c_{8}d(t)^{1/4}]}{N^{2}}\exp[-c|x|/N].$

Figure 27: The different cases zoomed at a distance $d(t)^{1/2}$ around
$\mathbf{x}$. The configuration $\omega^{(t+1/2)}$ is depicted in black, and
$\mathbf{x}+\omega_{{\rm T},j}^{(t)}$ in blue.

#### Error from the right $t$-extrema

On the one hand, there can exist $i\in I^{(t)}$ such that ${\rm
R}^{(t+1)}(i)\neq{\rm R}^{(t)}(i)$. Yet, this can occur only when some
$\mathcal{C}(\omega^{(t+1/2)},i)$ contains a right $t$-extremum, i.e. when
$\omega^{(t+1/2)}\in{\rm BAD}_{1}(t)\cup{\rm BAD}_{3}(t)$. On the other hand,
there can exist $i$ such that $X^{(t)}_{\mathcal{H}^{(t)}}(i)=1$ and
$\delta^{\rm IIC}{\rm R}^{(t)}(i)\neq 0$. Yet, the probability that
$X^{(t)}_{\mathcal{H}^{(t)}}(i)=1$ is such that

$\mathbf{P}[X^{(t)}_{\mathcal{H}^{(t)}}(i)=1|\mathcal{H}^{(t)}]\leq\mathbf{P}[\omega^{(t+1/2)}\in{\rm
BAD}_{1}(t)\cup{\rm BAD}_{3}(t)|\mathcal{H}^{(t)}].$

By proceeding in the same way as in Case 0, and using Lemma 6.10, we deduce
that

$\mathbf{E}\Big{[}\Big{(}\sum_{i\in I^{(t)}}|\delta^{\rm err}{\rm
R}^{(t)}(i)|\Big{)}\mathbf{1}_{\text{Case 1}}]\leq
2\mathbf{E}[|I^{(t)}|\mathbf{1}_{\omega^{(t+1/2)}\in{\rm BAD}_{1}(t)\cup{\rm
BAD}_{3}(t)}]\leq\frac{C_{9}}{d(t)N^{c_{9}}}$

(again here we directly give the summed error as it is provided by Lemma
6.10).

#### Error in Case 3

In Case 3, no error is made for ${\rm T}$ and ${\rm B}$, and we only need to
control the error due to movements of ${\rm R}^{(t)}(i)$. Also, the error
strictly after Step 0 can be treated in exactly the same way as in Case 1.
Indeed, any such error either implies the occurrence of ${\rm BAD}_{5}(t)$ or
is generated by the configuration in the box of size $d(t)$ around
$\mathbf{x}$, in which case we use (15) instead of (10) as for Case 1.

The only new type of errors we need to control are those in Case 0, and they
are of three types:

  * (i)

$X^{(t)}$ and $\widetilde{X}^{(t)}$ do not couple,

  * (ii)

$X^{(t)}=\widetilde{X}^{(t)}$ but there is no $i$ with
$\widetilde{X}^{(t)}(i)=1$,

  * (iii)

$X^{(t)}=\widetilde{X}^{(t)}$ but there are two $i$ with
$\widetilde{X}^{(t)}(i)=1$.

We divide our analysis between the different cases.

#### Subcase (i).

For every $i\in I^{(t)}$, $X^{(t)}(i)$ and $\widetilde{X}^{(t)}(i)$ have the
same law. Using the inclusion-exclusion principle, we see that for the best
coupling between the two random variables, we have that

$\mathbf{P}[X^{(t)}(i)\neq\widetilde{X}^{(t)}(i)|\mathcal{H}^{(t)}]\leq
C_{13}\mathbf{P}[|\widetilde{X}^{(t)}|\geq 2|\mathcal{H}^{(t)}].$

Yet, using an argument similar to Case 2, we find that

$\displaystyle\mathbf{E}\Big{[}{\rm Err}^{(t)}\mathbf{1}_{\text{Case
2},X^{(t)}\neq\widetilde{X}^{(t)}}\Big{]}$ $\displaystyle\leq
2\mathbf{E}[|I^{(t)}|\mathbf{1}(X^{(t)}\neq\widetilde{X}^{(t)})]$
$\displaystyle\leq
2C_{13}\mathbf{E}[|I^{(t)}|\mathbf{1}(|\widetilde{X}^{(t)}|\geq 2)]$
$\displaystyle=2C_{13}\mathbf{E}[|I^{(t)}|\mathbf{1}(\omega^{(t+1/2)}\in{\rm
BAD}_{3}(t))]$

(the last equality is due to the fact that
$\omega^{(t+1/2)}\in\\{X=\widetilde{X}^{(t)}\\}$). Then, we conclude using
Lemma 6.10 as before.

#### Subcase (ii).

In this case, $\delta^{\rm IIC}{\rm R}^{(t)}(i)=0$ for every $i\in I^{(t)}$.
Yet, for ${\rm R}^{(t+1)}(i)$ to be different from ${\rm R}^{(t)}(i)$, it must
be that $\mathcal{C}(\omega^{(t+1/2)},i)$ contains a fake right $t$-extremum
(as it does not contain a right $t$-extremum on the event
$X=\widetilde{X}^{(t)}$). Therefore, $\omega^{(t+1/2)}$ must contain a fake
right $t$-extremum, i.e. that $\omega^{(t+1/2)}\in{\rm BAD}_{4}(t)$. We deduce
the result from Lemma 6.10.

#### Subcase (iii).

In this case, $\omega^{(t+1/2)}$ must contain two right $t$-extrema.
Therefore, $\omega^{(t+1/2)}\in{\rm BAD}_{3}(t)$ again and the proof follows
from Lemma 6.10 and an argument similar to Case 0. ∎

###  6.4 Compounded time steps

We now group steps into so-called compounded time steps corresponding to the
action of a single track going down from its initial position to its final
one. More precisely, for $0\leq k<\lceil 2N/\sin\alpha\rceil$ we study the
steps $t\in[\tau_{k},\tau_{k+1})$, where $\tau_{k}:=2kN$ (it will be important
that the time steps correspond to the action of the same track of angle
$\alpha$, here the $(k+1)$-st one) to be pushed down.

First, introduce the speeds of the IIC in each direction, a notion which will
be useful in the next sections. Note that the definition below does not
immediately seem to be connected to the speed of a process. We will see later
that it will in fact correspond to the speed (or “drift”) of extrema of nails
when bringing tracks down.

######  Definition 6.11 (Speed in each direction).

Define

$\displaystyle v_{{\rm T}}$
$\displaystyle:=\sin\alpha-\frac{\mathbf{P}[\delta^{\rm IIC}_{1}{\rm
T}=0]}{\mathbf{P}[\delta^{\rm IIC}_{0}{\rm
T}=\sin\alpha-1]}\qquad\qquad\qquad\qquad v_{{\rm
B}}:=\sin\alpha-\frac{\mathbf{P}[\delta^{\rm IIC}_{0}{\rm
B}=\sin\alpha-1]}{\mathbb{P}[\delta^{\rm IIC}_{-1}{\rm B}=0]},$ $\displaystyle
v_{{\rm R}}$ $\displaystyle:=\frac{{\bf E}[\delta^{\rm IIC}{\rm R}]}{{\bf
P}[\delta^{\rm IIC}{\rm R}\in\\{0,-1\\}]}.$

For $i\in I^{(\tau_{k})}$, introduce the random time at which $i$ ceases to be
the index of a nail:

$\tau_{\rm end}(i):=\min\\{s:i\notin I^{(s)}\\}$

(it is equal to $T$ if such an $s$ does not exist). Let $\mathcal{F}_{k}$ be
the $\sigma$-algebra containing all the variables

$(\omega^{(s)}:s\leq\tau_{k})\ ,\ (\omega^{(s+1/2)}:s<\tau_{k})\ ,\
(X^{(s)}_{H}(i):s<\tau_{k})\ ,\ (\delta^{\rm IIC}_{j}A^{(s)}(i):s<\tau_{k}).$

Recall the definition of $M^{(s)}(i)$ from the previous section.

######  Proposition 6.12 (Compounded time step for $A={\rm T}$ or ${\rm B}$).

There exist $c,C\in(0,\infty)$ such that for $A\in\\{{\rm T},{\rm B}\\}$,
$i\in\mathbb{Z}_{>0}$, $0\leq k<\lceil 2N/\sin\alpha\rceil$, there exist
random variables $\Delta^{\rm IIC}A^{(k)}(i)$ and $\Delta^{\rm err}A^{(k)}(i)$
such that a.s. for every $i\in I^{(\tau_{k})}$,

  * (i)

$A^{(\tau_{\rm end}\wedge\tau_{k+1})}(i)-A^{(\tau_{k})}(i)=\Delta^{\rm
IIC}A^{(k)}(i)+\Delta^{\rm err}A^{(k)}(i)$;

  * (ii)

$\mathbf{E}[\exp(c|\Delta^{\rm IIC}A^{(k)}(i)|)|\mathcal{F}_{k}]\leq C$;

  * (iii)

$\displaystyle\mathbf{E}[|\Delta^{\rm err}A^{(k)}(i)|\,|\mathcal{F}_{k}]\leq
C\,\mathbf{E}\Big{[}\sum_{s=\tau_{k}}^{\tau_{\rm
end}\wedge\tau_{k+1}}M^{(s)}(i)\big{|}\mathcal{F}_{k}\Big{]}$;

  * (iv)

$\displaystyle\mathbf{E}[\Delta^{\rm
IIC}A^{(k)}(i)|\mathcal{F}_{k}]=\begin{cases}0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\
\ \text{ if }A^{(\tau_{k})}(i)\leq b(\tau_{k}),\\\
v_{A}+O(e^{-c|A^{(\tau_{k})}(i)-b(\tau_{k})|}+\mathbf{P}[\tau_{\rm
end}<\tau_{k+1}|\mathcal{F}_{k}])\ \ \ \ \text{ otherwise}.\end{cases}$

######  Remark 6.13.

The $O(\cdot)$ quantity in $(iv)$ comes from the fact that there are two types
of errors: the first term comes from cases where $A^{(\tau_{k})}(i)$ is close
to the bottom $b(\tau_{k})$ in which case the process described in the next
proof could be stopped because the track reaches its final position, and the
second from the fact that the nail $i$ can cease to be indexed during the
interval $[\tau_{k},\tau_{k+1})$.

###### Proof.

We treat the case of $A={\rm T}$. The case of ${\rm B}$ is similar. In the
whole proof, fix $k$ and $i\in I^{(\tau_{k})}$. To lighten the notation we
omit $i$ in the notation.

#### Case 1 ${\rm T}^{(\tau_{k})}\leq b(\tau_{k})$.

In this case, the coupling $\mathbf{P}$ is such that ${\rm T}^{(\tau_{\rm
end}\wedge\tau_{k+1})}-{\rm T}^{(\tau_{k})}=0$ so we may define

$\Delta^{\rm IIC}{\rm T}^{(k)}=\Delta^{\rm err}{\rm T}^{(k)}:=0.$

#### Case 2 ${\rm T}^{(\tau_{k})}>b(\tau_{k})$.

We first describe how a track-exchange at time $\tau_{k}\leq s<\tau_{k+1}$
modifies the value of ${\rm T}^{(s)}$. There are three possibilities:

  * •

${\rm T}^{(s)}\notin\\{h(s)-1,h(s)\\}$, in which case ${\rm T}^{(s)}$ is not
altered;

  * •

${\rm T}^{(s)}=h(s)-1$ which happens exactly once. In this case, ${\rm T}$ may
either stay put or increase by $\sin\alpha$. In the former case, ${\rm T}$ is
not altered by subsequent steps and in the latter ${\rm T}^{(s+1)}=h(s+1)$;

  * •

${\rm T}^{(s)}=h(s)$, which implies that either $T^{(s-1)}=h(s-1)$ and the
track-exchange “dragged down” the top of the nail, or ${\rm
T}^{(s-1)}=h(s-1)-1$ but the track-exchange failed to increase the top. In
this case, ${\rm T}^{(s)}$ may either increase by $\sin\alpha-1$ in which case
it will not move at subsequent steps, or decrease by $-1$, in which case ${\rm
T}^{(s+1)}=h(s+1)$.

From the previous discussion, we find

${\rm T}^{(\tau_{\rm end}\wedge\tau_{k+1})}-{\rm
T}^{(\tau_{k})}=\sin\alpha-(\sigma-\tau),$ (58)

where $\tau$ is the first (and unique) time for which ${\rm
T}^{(\tau)}=h(\tau)-1$ and $\sigma$ is the time defined by

$\sigma:=\begin{cases}\tau&\text{ if }{\rm T}^{(\tau+1)}={\rm
T}^{(\tau)}+\sin\alpha,\\\ \inf\\{s\in(\tau,\tau_{\rm
end}\wedge\tau_{k+1}):{\rm T}^{(s+1)}-{\rm T}^{(s)}\neq-1\\}&\text{ if }{\rm
T}^{(\tau+1)}={\rm T}^{(\tau)}\text{ and }s\text{ exists},\\\ \tau_{\rm
end}\wedge\tau_{k+1}&\text{ otherwise}.\end{cases}$

To define $\Delta^{\rm IIC}{\rm T}$, we use a similar formula except that we
consider the random variables $\delta_{j}^{\rm IIC}{\rm T}^{(s)}$ instead of
the true increments:

$\displaystyle\Delta^{\rm IIC}{\rm T}^{(k)}$
$\displaystyle:=\sin\alpha-(\sigma^{\rm IIC}-\tau),$ (59)

where

$\sigma^{\rm IIC}:=\begin{cases}\tau&\text{ if }\delta^{\rm IIC}_{1}{\rm
T}^{(\tau)}=\sin\alpha,\\\ \inf\\{s\in(\tau,\tau_{\rm
end}\wedge\tau_{k+1}):\delta_{0}^{\rm IIC}{\rm T}^{(s)}\neq-1\\}&\text{ if
}\delta^{\rm IIC}_{1}{\rm T}^{(\tau)}=0\text{ and such an }s\text{ exists},\\\
\tau_{\rm end}\wedge\tau_{k+1}&\text{ otherwise}\end{cases}$

(note that $\tau_{\rm end}$ is still a function of the true increments).

Finally, we set

$\displaystyle\Delta^{\rm err}{\rm T}^{(k)}$ $\displaystyle:=\sigma^{\rm
IIC}-\sigma.$ (60)

We are now in a position to derive our proposition. First, (i) is satisfied by
construction and (58)–(60). The definition of $\sigma^{\rm IIC}$ from
independent “trial” events immediately leads to (ii). For (iv), we have that

$\displaystyle\mathbf{E}[\Delta^{\rm IIC}{\rm T}^{(k)}|\mathcal{F}_{k}]$
$\displaystyle=\sin\alpha-\mathbf{E}[\sigma^{\rm IIC}-\tau|\mathcal{F}_{k}]$
$\displaystyle=\sin\alpha-\frac{\mathbf{P}[\delta^{\rm IIC}_{1}{\rm
T}=0]}{1-\mathbf{P}[\delta^{\rm IIC}_{0}{\rm T}=-1]}+O(e^{-c|{\rm
T}^{(t)}-b(t)|}+\mathbf{P}[\tau_{\rm end}<\tau_{k+1}|\mathcal{F}_{k}]),$

where the error term comes from the fact that $\sigma^{\rm IIC}$ can be equal
to $\tau_{\rm end}\wedge\tau_{k+1}$. More precisely, when $\tau_{\rm
end}\geq\tau_{k+1}$, we obtain the first error since $\tau_{k+1}-\tau\geq{\rm
T}^{(t)}-b(t)$ and the difference is a geometric random variable, and when
$\tau_{\rm end}<\tau_{k+1}$, we obtain the second term in the $O(\cdot)$.

It only remains to prove (iii), i.e. to bound $\mathbf{E}[|\sigma^{\rm
IIC}-\sigma|\,|\mathcal{F}_{k}]$. In order to do it, introduce further random
times defined recursively by $\sigma^{\rm IIC}_{0}=\sigma^{\rm IIC}$ and

$\sigma^{\rm IIC}_{\ell+1}:=\inf\\{s\in(\sigma^{\rm IIC}_{\ell},\tau_{\rm
end}):\delta_{0}^{\rm IIC}{\rm T}^{(s)}\neq-1\\}$

when $s$ exists and $\sigma^{\rm IIC}_{\ell+1}=\tau_{\rm end}\wedge\tau_{k+1}$
otherwise (note that for $\ell$ large enough, the sequence becomes stationary
at $\tau_{\rm end}\wedge\tau_{k+1}$, which is compatible with the formula
below). We have

$\sigma^{\rm IIC}-\sigma=\sum_{\tau\leq s\leq\sigma^{\rm
IIC}}\mathbf{1}_{s>\sigma}-\sum_{\ell\geq 0}\sum_{s=\sigma^{\rm
IIC}_{\ell}+1}^{\sigma^{\rm IIC}_{\ell+1}}\mathbf{1}_{s\leq\sigma}.$

Now, let $\mathbf{X}$ denote the sum of the $|\delta^{\rm err}{\rm T}^{(s)}|$
for $s\in[\tau,\tau_{\rm end}\wedge\tau_{k+1}]$. On the one hand, for
$\tau\leq s\leq\sigma^{\rm IIC}$ to satisfy $s>\sigma$, it must be that
$\delta^{\rm err}{\rm T}^{(r)}\neq 0$ for some $r\in[\tau,s)$ and that
$\delta^{\rm IIC}_{0}{\rm T}^{(r^{\prime})}=-1$ for every
$r^{\prime}\in(r,s)$. Independence provided by Proposition 6.8(ii) implies
that

$\displaystyle\mathbf{E}\Big{[}$ $\displaystyle\sum_{\tau\leq s\leq\sigma^{\rm
IIC}}\mathbf{1}_{s>\sigma}\big{|}\mathcal{F}_{k}\Big{]}$
$\displaystyle\leq\sum_{s\geq r\geq\tau}\mathbf{P}[\forall
r^{\prime}\in(\tau,r),\delta^{\rm err}{\rm T}^{(r^{\prime})}=0;\delta^{\rm
err}{\rm T}^{(r)}\neq 0;\forall s^{\prime}\in(r,s),\delta^{\rm IIC}_{0}{\rm
T}^{(s^{\prime})}=-1|\mathcal{F}_{k}]$
$\displaystyle\leq\frac{\mathbf{P}[\mathbf{X}\geq\sin\alpha|\mathcal{F}_{k}]}{\mathbf{P}[\delta^{\rm
IIC}_{0}{\rm T}=\sin\alpha-1]}.$

On the other hand, for $\sigma^{\rm IIC}_{\ell}<s\leq\sigma^{\rm
IIC}_{\ell+1}$ to be smaller than $\sigma$, it must be that $\delta^{\rm
err}{\rm T}^{(\sigma^{\rm IIC}_{l})}=-\sin\alpha$ for every $0\leq l\leq\ell$,
and that $\delta^{\rm IIC}_{0}{\rm T}^{(r)}=-1$ for every $r\in[\sigma^{\rm
IIC}_{\ell},s]$ so that by independence of the variables $\delta^{\rm
IIC}_{0}{\rm T}^{(r)}$ for $r>\sigma^{\rm IIC}_{\ell}$ and $\delta^{\rm
err}{\rm T}^{(\sigma_{l}^{\rm IIC})}$ for $l\leq\ell$, we get in a fairly
similar fashion to the previous displayed equation that

$\displaystyle\mathbf{P}\Big{[}\sum_{s=\sigma^{\rm IIC}_{\ell}+1}^{\sigma^{\rm
IIC}_{\ell+1}}\mathbf{1}_{s\leq\sigma}\big{|}\mathcal{F}_{k}\Big{]}$
$\displaystyle\leq\sum_{j\geq 0}\mathbf{P}[\delta^{\rm IIC}_{0}{\rm
T}=-1]^{j}\mathbf{P}[\mathbf{X}\geq\ell\sin\alpha|\mathcal{F}_{k}]$
$\displaystyle\leq
C\,\mathbf{P}[\mathbf{X}\geq\ell\sin\alpha|\mathcal{F}_{k}].$

The claim follows by summing over $\ell$. ∎

We now treat the impact of compounded steps on ${\rm R}$.

######  Proposition 6.14 (Compounded time step for $A={\rm R}$).

There exist $c,C\in(0,\infty)$ such that for $i\in\mathbb{Z}_{>0}$ and $0\leq
k<\lceil 2N/\sin\alpha\rceil$, there exist random variables $\Delta^{\rm
IIC}{\rm R}^{(k)}(i)$ and $\Delta^{\rm err}{\rm R}^{(k)}(i)$ such that a.s.
for every $i\in I^{(\tau_{k})}$,

  * (i)

${\rm R}^{(\tau_{\rm end}\wedge\tau_{k+1})}(i)-{\rm
R}^{(\tau_{k})}(i)=\Delta^{\rm IIC}{\rm R}^{(k)}(i)+\Delta^{\rm err}{\rm
R}^{(k)}(i)$;

  * (ii)

$\mathbf{E}[\exp(c|\Delta^{\rm IIC}{\rm R}^{(k)}(i)|)|\mathcal{F}_{k}]\leq C$;

  * (iii)

$\displaystyle\mathbf{E}[|\Delta^{\rm err}{\rm
R}^{(k)}(i)|\,|\mathcal{F}_{k}]\leq C\mathbf{E}\Big{[}\sum_{s=t}^{\tau_{\rm
end}\wedge\tau_{k+1}}M^{(s)}(i)\big{|}\mathcal{F}_{k}\Big{]}$;

  * (iv)

$\mathbf{E}[\Delta^{\rm IIC}{\rm R}^{(k)}(i)|\mathcal{F}_{k}]$ is equal to

$\begin{cases}0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad\
\text{ if }{\rm R}^{(\tau_{k})}(i)\notin k\cos\alpha+\mathbb{Z},\\\
\displaystyle v_{\rm R}+O\Big{(}\mathbf{E}\Big{[}\sum_{s=\tau_{k}}^{\tau_{\rm
end}\wedge\tau_{k+1}}M^{(s)}(i)\big{|}\mathcal{F}_{k}\Big{]}+\mathbf{P}[\tau_{\rm
end}<\tau_{k+1}|\mathcal{F}_{k}]\Big{)}\quad\text{otherwise}.\end{cases}$

######  Remark 6.15.

When $\cos\alpha\notin\mathbb{Q}$, the condition ${\rm R}^{(\tau_{k})}\notin
k\cos\alpha+\mathbb{Z}$ implies that no right-most vertex can belong to the
area below the $(k+1)$-st track. The information on ${\rm R}$ therefore gives
more than simply the first-coordinate of the right-most point, it also
provides information on its vertical position. This is not necessary true for
rational values of $\cos\alpha$. In this case, one should therefore record
this information more explicitly. We chose to restrict ourselves to $\alpha$
with $\cos\alpha$ irrational as we will see it is sufficient to get our
result.

###### Proof.

Again, we fix $k$ and $i$ and drop $i$ from the notation. We first describe
how a track-exchange for $\tau_{k}\leq s<\tau_{k+1}$ modifies the value of
${\rm R}^{(s)}$. There are three possibilities:

  * •

${\rm R}^{(s)}\in k\cos\alpha+\mathbb{Z}$ and $({\rm R}^{(s)},h(s))$ does not
belong to the nail $\mathcal{C}(\omega^{(s)},i)$, in such case ${\rm
R}^{(s+1)}={\rm R}^{(s)}$,

  * •

${\rm R}^{(s)}\in k\cos\alpha+\mathbb{Z}$ and $({\rm R}^{(s)},h(s))$ belongs
to the cluster. In such case, the track-exchange creates a change of
$\cos\alpha$, $\cos\alpha-1$, or $-1$ (see Figure 24). In the former case,
${\rm R}^{(s+1)}={\rm R}^{(s)}+\cos\alpha$ and the next track-exchanges will
not impact the maximum. In the latter, ${\rm R}^{(s)}$ can change by values in
$[-1,0]\cap(\mathbb{Z}+\\{0,\dots,k-2,k-1,k+1\\}\cos\alpha)$ due to the
possible existence of other vertices that are not affected by the track-
exchange but had almost-maximal first coordinate101010In fact, essentially the
only other two values that are possible are $0$ if there is another extremum
of the cluster in the square region below $t_{j(s)}^{-}$, or $\cos\alpha-1$ if
there is a “near” extremum in the square region above height $h(s)$ that
becomes the right-most point after the transformation. For the increment to be
different from $0$, $\cos\alpha-1$ or $-1$, it must be that
$\mathcal{C}(\omega^{(s)},i)$ contains a vertex below $b(t)$ with first
coordinate in $({\rm R}^{(s)}-1,{\rm R}^{(s)})$, which has small probability
as shown in Lemma 6.10..

  * •

${\rm R}^{(s)}\notin k\cos\alpha+\mathbb{Z}$. In such a case, there is only
one possibility for ${\rm R}^{(s+1)}$ not to be equal to ${\rm R}^{(s)}$,
which is that there exists a fake right $s$-extremum and that the track-
exchange implies an increase of $\cos\alpha$ locally, which leads to ${\rm
R}^{(s+1)}=x_{1}+\cos\alpha$ and no further change can occur.

Now, recall the definition of $X^{(s)}_{H}=X^{(s)}_{H}(i)$ from the previous
section, and introduce

$\displaystyle\Delta^{\rm IIC}{\rm R}^{(k)}$
$\displaystyle:=\sum_{\tau_{k}\leq s\leq\sigma^{\rm
IIC}}X^{(s)}_{\mathcal{H}^{(s)}}\delta^{\rm IIC}{\rm R}^{(s)},$

where

$\sigma^{\rm IIC}:=\begin{cases}\min\\{s\in[\tau_{k},\tau_{\rm
end}\wedge\tau_{k+1}):\delta^{\rm IIC}{\rm
R}^{(s)}\in\\{\cos\alpha,\cos\alpha-1\\}\text{ and
}X^{(s)}_{\mathcal{H}^{(s)}}=1\\}&\text{if }s\text{ exists},\\\ \tau_{\rm
end}\wedge\tau_{k+1}&\text{otherwise},\end{cases}$

and

$\Delta^{\rm err}{\rm R}^{(k)}:={\rm R}^{(\tau_{\rm
end}\wedge\tau_{k+1})}-{\rm R}^{(\tau_{k})}-\Delta^{\rm IIC}{\rm R}^{(k)}\,.$

By definition, (i) and (ii) are satisfied. The proof of (iii) follows the same
steps as the proof of (iii) in Proposition 6.12 (we leave the details to the
reader), except in the case corresponding to the third bullet above, i.e. that
${\rm R}^{(s)}\notin k\cos\alpha+\mathbb{Z}$ but ${\rm R}^{(s+1)}\neq{\rm
R}^{(s)}$. Yet, in this case $M^{(s)}\neq 0$ and no further error is made at
later times.

For (iv), note that if $R^{(\tau_{k})}\notin k\cos\alpha+\mathbb{Z}$, then
$X^{(s)}_{\mathcal{H}^{(s)}}$ is always equal to 0 and $\Delta^{\rm IIC}{\rm
R}=0$. If, on the contrary, $R^{(\tau_{k})}\in k\cos\alpha+\mathbb{Z}$, define

$\widetilde{\sigma}^{\rm IIC}:=\min\\{s\geq\tau_{k}:\delta^{\rm IIC}{\rm
R}^{(s)}\in\\{\cos\alpha,\cos\alpha-1\\}\text{ and }Y^{(s)}=1\\},$

where $Y^{(s)}=X_{\mathcal{H}^{(s)}}^{(s)}$ for $s\leq\tau_{\rm
end}\wedge\tau_{k+1}$ and 1 for $s>\tau_{\rm end}\wedge\tau_{k+1}$. Also
define $\widetilde{\Delta}^{\rm IIC}{\rm R}^{(k)}$ using the same formulas as
for $\Delta^{\rm IIC}{\rm R}^{(k)}$ but with $\widetilde{\sigma}^{\rm IIC}$
instead of $\sigma^{\rm IIC}$. Then, a direct computation gives

$\mathbf{E}[\widetilde{\Delta}^{\rm IIC}{\rm R}^{(k)}|\mathcal{F}_{k}]=v_{\rm
R}.$

Moreover,

$\displaystyle|\mathbf{E}[\Delta^{\rm IIC}{\rm
R}^{(k)}|\mathcal{F}_{k}]-v_{\rm R}|$
$\displaystyle\leq\mathbf{E}[\widetilde{\sigma}^{\rm IIC}-\sigma^{\rm
IIC}|\mathcal{F}_{k}]\leq C_{0}\mathbf{P}[\widetilde{\sigma}^{\rm
IIC}\neq\sigma^{\rm IIC}|\mathcal{F}_{k}]$ $\displaystyle\leq
C_{0}\mathbf{P}[\forall s\leq\tau_{\rm
end}\wedge\tau_{k+1}:X^{(s)}_{\mathcal{H}^{(s)}}=1,\delta^{\rm
IIC}R^{(s)}=-1|\mathcal{F}_{k}].$

To estimate the probability on the right, observe that if $\tau_{\rm
end}\geq\tau_{k+1}$ and $\delta^{\rm IIC}R^{(s)}=-1$ for every $s$ such that
$X^{(s)}_{\mathcal{H}^{(s)}}=1$, it must be that $M^{(s)}\neq 0$ for at least
one $s$ since otherwise ${\rm R}^{(\tau_{k+1})}\in{\rm
R}^{(\tau_{k})}+\mathbb{Z}$, which is impossible. We therefore obtain that

$\displaystyle\mathbf{P}[\forall s\leq\tau_{\rm
end}\wedge\tau_{k+1}:X^{(s)}_{\mathcal{H}^{(s)}}=1$ $\displaystyle,\delta^{\rm
IIC}R^{(s)}=-1|\mathcal{F}_{k}]$ $\displaystyle\leq\mathbf{P}[\tau_{\rm
end}<\tau_{k+1}|\mathcal{F}_{k}]+\mathbf{E}\Big{[}\sum_{s=\tau_{k}}^{\tau_{\rm
end}\wedge\tau_{k+1}}M^{(s)}\big{|}\mathcal{F}_{k}\Big{]}.$

This concludes the proof of the proposition. ∎

###  6.5 Speed of the drift

In this section, we compute $v_{A}$ for $A={\rm T},{\rm B},{\rm R}$. We start
with the first two.

######  Proposition 6.16.

We have $v_{{\rm T}}=v_{{\rm B}}=0$.

###### Proof.

We treat the case of $v_{\rm T}$ (the case of $v_{\rm B}$ is the same).
Introduce $\mathbf{l}_{N}:=\mathrm{lmax}((0,-N))$ (i.e. the left-most highest
vertex in the cluster of $(0,-N)$) and let $\mathbf{E}$ be the coupling
between $\omega_{1}\sim\phi_{\mathbb{L}_{1}}$ and
$\omega_{0}\sim\phi_{\mathbb{L}_{0}}$ obtained by setting
$\omega_{0}=\mathbf{T}_{1}(\omega_{1})$. Also, let $\Delta{\rm T}$ be the
difference between the top height of the cluster of $(0,-N)$ in $\omega_{1}$
and in $\omega_{0}$.

We find that

$\displaystyle\phi_{\mathbb{L}_{0}}[\mathbf{l}_{N}\in t_{1}^{-}]$
$\displaystyle=\mathbf{P}[\mathbf{l}_{N}\in t_{1}^{-}\text{ in }\omega_{0}]$
(61) $\displaystyle=\mathbf{P}[\mathbf{l}_{N}\in t_{1}^{-}\text{ in
}\omega_{1}\text{ and }\Delta{\rm
T}=\sin\alpha-1]+\mathbf{P}[\mathbf{l}_{N}\in t_{0}^{-}\text{ in
}\omega_{1}\text{ and }\Delta{\rm T}=\sin\alpha]$
$\displaystyle=\phi_{\mathbb{L}_{1}}[\mathbf{l}_{N}\in
t_{1}^{-}]\mathbf{P}[\delta_{0}^{\rm
IIC}=\sin\alpha-1]+\phi_{\mathbb{L}_{1}}[\mathbf{l}_{N}\in
t_{0}^{-}]\mathbf{P}[\delta_{1}^{\rm IIC}=\sin\alpha]+o_{N}(1),$

where in the second step we used that we may couple the increment $\Delta{\rm
T}$ with an IIC increment exactly as we did in the previous section (to
estimate the error, one needs to perform a reasoning similar to the error in
the top extremum in Case 1 of the coupling).

Using the same coupling, we also see that $\mathbf{l}_{N}\in t_{0}^{-}\cup
t_{1}^{-}$ in $\omega_{0}$ if and only if it does in $\omega_{1}$, so we get
that

$\phi_{\mathbb{L}_{0}}[\mathbf{l}_{N}\in t_{0}^{-}\cup
t_{1}^{-}]=\phi_{\mathbb{L}_{1}}[\mathbf{l}_{N}\in t_{0}^{-}\cup t_{1}^{-}].$
(62)

Decomposing on the possible values of $\mathbf{l}_{N}$ (like in the proof of
Proposition 2.6) and using the mixing of the IIC (Proposition 3.9), we also
find that for $i=0,1$,

$\phi_{\mathbb{L}_{i}}[\mathbf{l}_{N}\in t_{1}^{-}|\mathbf{l}_{N}\in
t_{0}^{-}\cup
t_{1}^{-}]={\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{i}[\mathrm{lmax}(\infty)=0^{+}]+o_{N}(1).$
(63)

Dividing (61) by (62) and plugging (63) into it, we find that

$\displaystyle{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{0}[\mathrm{lmax}$
$\displaystyle(\infty)=0^{+}]$ (64)
$\displaystyle={\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{1}[\mathrm{lmax}(\infty)=0^{+}]\mathbf{P}[\delta_{0}^{\rm
IIC}=\sin\alpha-1]+{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{1}[\mathrm{lmax}(\infty)=0]\mathbf{P}[\delta_{1}^{\rm
IIC}=\sin\alpha].$

Theorem 2.4 applied to $\beta=\tfrac{\pi}{2}$ gives

${\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{0}[\mathrm{lmax}(\infty)=0^{+}]={\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{1}[\mathrm{lmax}(\infty)=0]=1-{\text{\rotatebox[origin={c}]{180.0}{$\Psi$}}}^{2}_{1}[\mathrm{lmax}(\infty)=0^{+}]=\frac{1}{1+\sin\alpha},$

which, when inserted in (64) and multiplied by $1+\sin\alpha$, gives $v_{\rm
T}=0$. ∎

Next, we turn to the lateral speed $v_{\rm R}$, whose value is deduced from
the one of $v_{{\rm T}}$.

######  Proposition 6.17.

We have $v_{{\rm R}}=0$.

The idea of the proof is to obtain the right displacement of the cluster as
the top displacement in a rotated version of the process. Below, we mention
not only horizontal tracks but also vertical tracks and their track-exchanges.
We believe that at this point the reader may easily make sense of these
transformations so we omit the details of the definitions. Also, we refer to
[18] for more information.

Figure 28: The initial and final graphs $\mathbb{B}^{(0)}$ and
$\mathbb{B}^{(2M+1)}$ of the two processes are the same, but the intermediate
graphs $(\mathbb{B}^{(t)})_{0<t\leq 2M}$ (top) and
$(\widetilde{\mathbb{B}}^{(t)})_{0<t<M}$ (bottom) are different (the figure
depicts the diamond graphs). In the top process, horizontal tracks are
exchanged successively, by pushing the gray rhombus from right to left; in the
bottom process, the rhombus is pushed downwards, effectively exchanging
vertical tracks. Throughout the two processes, we record the right-most
coordinate of the union of all the clusters intersecting the base.

###### Proof.

We refer to Figure 28 for an illustration. Consider $M$ and $N$ two integers
satisfying $M=N^{2}$. Consider the graph $\mathbb{B}^{(0)}$ formed of $2M+2$
“horizontal” tracks $t_{-M},\dots,t_{M},t_{\alpha}$ of transverse angles
$\pi/2$ for the first $2M+1$ and transverse angle $\alpha$ for $t_{\alpha}$
and $M$ “vertical” tracks $s_{0},\dots,s_{M-1}$ of transverse angle $0$. In
addition to the intersection between the vertical and horizontal tracks,
$\mathbb{B}^{(0)}$ contains also the intersection of tracks
$t_{-M},\dots,t_{M}$ with $t_{\alpha}$; these occur at the right side of the
graph (note that $t_{\alpha}$ is not straight and does a sharp turn at the
top-right corner of the rectangle). Translate $\mathbb{B}^{(0)}$ so that $0$
is the vertex left of $s_{0}$ and below $t_{0}$.

We perform track-exchanges via star-triangle transformations applied to the
graph $\mathbb{B}^{(0)}$. Contrary to the other parts of the paper, where
exchanged tracks change name, here the tracks will conserve their indexing
during track-exchanges. Define recursively $\mathbb{B}^{(t+1)}$ for $0\leq
t\leq 2M$ as obtained from $\mathbb{B}^{(t)}$ by performing the track-exchange
$\mathbf{T}_{M-t}$ between $t_{M-t}$ and $t_{\alpha}$ which is a composition
of $M$ star-triangle transformations.

We follow the extrema of the set $\mathcal{C}_{\text{base}}$ obtained as the
union of the primal clusters intersecting the base $\\{0\\}\times[-N,N]$. Let
$(\omega^{(t)},{\rm R}^{(t)})_{0\leq t\leq 2M+1}$ be obtained as follows:

  * •

the initial step is defined by sampling $\omega^{(0)}$ according to
$\phi_{\mathbb{B}^{(0)}}^{0}$ and setting ${\rm R}^{(0)}$ to be the maximal
first coordinate of vertices in $\mathcal{C}_{\rm base}$;

  * •

at time $0\leq t\leq 2M$, sample a configuration

$\omega^{(t+1/2)}\sim\phi_{\mathbb{B}^{(t)}}^{0}[\,\cdot\,|{\rm
R}(\mathcal{C}_{\text{base}})={\rm R}^{(t)}],$

let $\omega^{(t+1)}:=\mathbf{T}_{M-t}(\omega^{(t)})$, and set ${\rm
R}^{(t+1)}$ to be the maximal first coordinate of a vertex in
$\mathcal{C}_{\rm base}$.

Following the proofs of the previous sections (with some additional
simplifications in this context, for instance $\tau_{\rm end}$ does not need
to be introduced) we find that

$\displaystyle{\rm R}^{(2M+1)}-{\rm R}^{(0)}=\Delta^{\rm IIC}{\rm
R}+\Delta^{\rm err}{\rm R},$ (65)

with the equivalent of (iii) and (iv) of Proposition 6.14 being that a.s.,

$\displaystyle\mathbf{E}[|\Delta^{\rm err}{\rm R}|\,|\mathcal{F}_{0}]$
$\displaystyle\leq C\,\mathbf{E}\Big{[}\sum_{t=0}^{2M}{\rm
Err}^{(t)}\big{|}\mathcal{F}_{0}\Big{]},$ $\displaystyle\mathbf{E}[\Delta^{\rm
IIC}{\rm R}|\mathcal{F}_{0}]$ $\displaystyle=v_{\rm
R}+O\Big{(}\mathbf{E}\Big{[}\sum_{t=0}^{2M}{\rm
Err}^{(t)}\big{|}\mathcal{F}_{0}\Big{]}\Big{)},$

where ${\rm Err}^{(t)}$ is defined in a similar fashion to Proposition 6.14,
but with $\mathcal{C}_{\rm base}$ playing the role of the union of the nails
now.

Let $h(t)$ be the height of the bottom of $t_{\alpha}$ at time $t$,
$d(t):=\min\\{M-h(t),h(t)+M\\}$. Following an argument similar to the proof of
Proposition 6.8, the mistake contributing to ${\rm Err}^{(t)}$ can be of three
types:

  * $(i)$

$({\rm R}^{(t)},h(t))\in\mathcal{C}_{\rm base}$ and ${\rm
R}^{(t)}\leq\sqrt{N}$,

  * $(ii)$

$({\rm R}^{(t)},h(t))\in\mathcal{C}_{\rm base}$ and ${\rm R}^{(t)}\geq
M-\sqrt{N}$,

  * $(iii)$

$({\rm R}^{(t)},h(t))\in\mathcal{C}_{\rm base}$ and $\sqrt{N}\leq{\rm
R}^{(t)}\leq M-\sqrt{N}$, but the true configuration and the IIC
configurations are not coupled in the box of radius $\min\\{d(t),N\\}^{1/4}$
around $({\rm R}^{(t)},h(t))$,

  * $(iv)$

$({\rm R}^{(t)},h(t))\in\mathcal{C}_{\rm base}$ and there is a vertex
$x=(x_{1},x_{2})\in\mathcal{C}_{\rm base}$ with $x_{1}\in({\rm R}^{(t)}-1,{\rm
R}^{(t)}]$ and $|x_{2}-h(t)|\geq d(t)^{1/5}$.

Recalling that the error is deterministically bounded by 2, we therefore have
that

$\mathbf{E}[{\rm Err}^{(t)}]\leq
2(\mathbf{P}[(i)]+\mathbf{P}[(ii)]+\mathbf{P}[(iii)]+\mathbf{P}[(iv)]).$ (66)

We now bound the probabilities of the events $(i)$, $(ii)$, and $(iii)$
separately. For $(i)$, (RSW) immediately implies the existence of $c>0$ such
that for every $0\leq t\leq 2M$,

$\mathbf{P}[(i)]=\phi_{\mathbb{B}^{(t)}}^{0}[(i)]\leq\phi_{\mathbb{B}^{(t)}}^{0}[{\rm
R}^{(t)}\leq\sqrt{N}]\leq\exp[-c\sqrt{N}].$

To estimate $(ii)$ and $(iii)$, let us first estimate, for $x\in t_{M-t}^{+}$,
the probability of the event $E(x)$ that $({\rm R}^{(t)},h(t))=x$ and ${\rm
base}\longleftrightarrow x$. Let $s(x)$ be the distance between $x$ and
$\partial\mathbb{B}^{(t)}$. We have that

$\phi_{\mathbb{B}^{(t)}}^{0}[E(x)]\leq\phi_{\mathbb{B}^{(t)}}\Big{[}\\{{\rm
base}\longleftrightarrow\partial\Lambda_{d(t)}(x)\\}\cap A_{010,x}^{\rm
R}(s(x),\tfrac{d(t)}{2})\cap A_{010,x}^{\rm R}(0,\tfrac{s(x)}{2})\Big{]},$
(67)

where $A_{010,x}^{\rm R}(r,R)$ is the translate of $A_{010}^{\rm R}(r,R)$ by
$x$. Using the mixing property, (RSW) for the first event on the right-hand
side, an argument similar to (12) for the third, and (15) for the fourth, we
obtain that

$\phi_{\mathbb{B}^{(t)}}^{0}[E(x)]\leq
C\big{(}\frac{N}{\max\\{N,|x|\\}}\big{)}^{c}\times\big{(}\frac{d(t)}{|x|}\big{)}^{c}\times\big{(}\frac{s(x)}{d(t)}\big{)}^{1+c}\times\big{(}\frac{1}{s(x)}\big{)}^{2}.$

Summing over the $x\in t_{M-t}^{+}$ that are at a distance at most $\sqrt{N}$
from the right-hand side of $\mathbb{B}^{(t)}$, we obtain that

$\displaystyle\mathbf{P}[(ii)]=\phi_{\mathbb{B}^{(t)}}^{0}[(ii)]$
$\displaystyle\leq C\big{(}\frac{N}{M}\big{)}^{2c}\times\frac{1}{d(t)}.$ (68)

For $(iii)$, using the same argument as in Proposition 6.8 in the first step
and summing over $x$ in the second

$\displaystyle\mathbf{P}[(iii)]$
$\displaystyle\leq\frac{C}{\min\\{d(t),N\\}^{c}}\sum_{x\in
t_{M-t}^{+}}\phi_{\mathbb{B}^{(t)}}^{0}[E(x)]\leq\frac{C}{\min\\{d(t),N\\}^{c}}\times\frac{1}{M^{c}d(t)^{1-c}}.$
(69)

The bound on $(iv)$ can be obtained as in Lemma 6.10:

$\displaystyle\mathbf{P}[(iv)]$ $\displaystyle\leq\frac{C}{d(t)M^{c}}.$ (70)

Plugging (67)–(70) into (66) gives

$\mathbf{E}[{\rm
Err}^{(t)}]=O\big{(}\exp(-c\sqrt{N})+\frac{1}{d(t)}\big{(}\frac{N}{M}\big{)}^{2c}+\frac{1}{\min\\{d(t),N\\}^{c}d(t)^{1-c}M^{c}}+\frac{1}{d(t)M^{c}}\big{)}.$
(71)

When summing over $t$ and using that $M=N^{2}$, we deduce that

$\mathbf{E}\Big{[}\sum_{t}{\rm Err}^{(t)}\Big{]}=O(N^{-c}).$

Overall, we find that

$\displaystyle\mathbf{E}[{\rm R}^{(2M+1)}]-\mathbf{E}[{\rm R}^{(0)}]$
$\displaystyle=\mathbf{E}[\Delta^{\rm err}{\rm R}]+\mathbf{E}[\Delta^{\rm
IIC}{\rm R}]=v_{\rm R}+O(N^{-c}).$

Now, define a similar sequence of graphs $\widetilde{\mathbb{B}}^{(t)}$ for
$0\leq t\leq M$ by setting $\widetilde{\mathbb{B}}^{(0)}=\mathbb{B}^{(0)}$ and
obtaining $\widetilde{\mathbb{B}}^{(t+1)}$ from $\widetilde{\mathbb{B}}^{(t)}$
by performing the track-exchange between $s_{M-1-t}$ and $t_{\alpha}$. Also,
define a Markov chain $(\widetilde{\rm R}^{{(t)}})_{0\leq t\leq M}$ as before.
Following again the same reasoning as in the previous sections, and observing
that the behaviour of $\widetilde{\rm R}^{{(t)}}$ under the track-exchange of
vertical tracks is the same as the behaviour of the top of a cluster when
exchanging horizontal tracks, we obtain using a reasoning similar to
Propositions 6.12 and 6.8 (with the same adaptation as above) that

$\displaystyle\mathbf{E}[\widetilde{\rm R}^{(M)}]-\mathbf{E}[\widetilde{\rm
R}^{(0)}]=v_{\rm T}+O(N^{-c})+O(\phi_{\mathbb{B}^{(0)}}^{0}[e^{-c|{\rm
R}^{(0)}|}])=v_{\rm T}+O(N^{-c})$

(in the second equality we used (67)). Here $v_{\rm T}$ refers to passing a
track with transverse angle $\alpha+\pi/2$, or equivalently $\pi/2-\alpha$ by
symmetry.

By definition, ${\rm R}^{(0)}$ and $\widetilde{\rm R}^{(0)}$ have the same
law. Observe that $\mathbb{B}^{(2M+1)}=\widetilde{\mathbb{B}}^{(M)}$ and since
our transformations ensure that the random-cluster law is preserved,
$\widetilde{\rm R}^{(M)}$ has the same law as ${\rm R}^{(2M+1)}$. Thus,

$\displaystyle v_{\rm R}=v_{\rm T}+O(N^{-c})=O(N^{-c}),$

where in the last equality we used Proposition 6.16. Letting $N$ go to
infinity concludes the proof. ∎

###  6.6 Proof of Theorem 2.3

We start with a lemma gathering the estimates obtained on the increments of
the extrema.

######  Lemma 6.18 (Nails do not move).

There exist $c_{0},C_{0}\in(0,\infty)$ such that for every $N$,

$\displaystyle\mathbf{P}[\exists i\in I^{(0)},\exists t\leq\tau_{\rm
end}(i),\exists A\in\\{{\rm T},{\rm B},{\rm R}\\},|A^{(t)}(i)-A^{(0)}(i)|\geq
N^{1-c_{0}}]\leq\frac{C_{0}}{N^{c_{0}}}.$

###### Proof.

First of all, observe that it is sufficient to control the increments at
compounded steps $\tau_{k}$ since $A^{(t)}(i)$ is between $A^{(\tau_{k})}(i)$
and $A^{(\tau_{k+1}\wedge\tau_{\rm end}(i))}(i)$ for every
$t\in[\tau_{k},\tau_{k+1}]$. For this reason, we only focus on compounded
steps and introduce the time $\tau^{\prime}_{\rm end}(i)$ denoting the integer
$k$ such that $\tau_{k}\leq\tau_{\rm end}(i)<\tau_{k+1}$. For each $i\in
I^{(0)}$, introduce the processes indexed by integer times $0\leq K<\lceil
2N/\sin\alpha\rceil$,

$\displaystyle\boldsymbol{\Sigma}_{A,i}(K)$
$\displaystyle:=\sum_{k=0}^{K\wedge\tau_{\rm
end}^{\prime}(i)}\mathbf{E}[\Delta^{\rm IIC}A^{(k)}(i)|\mathcal{F}_{k}],$
$\displaystyle\mathbf{M}_{A,i}(K)$
$\displaystyle:=\sum_{k=0}^{K\wedge\tau_{\rm end}^{\prime}(i)}\Delta^{\rm
IIC}A^{(k)}(i)-\boldsymbol{\Sigma}_{A,i}(K),$
$\displaystyle\boldsymbol{\Delta}^{\rm err}A(i,K)$
$\displaystyle:=\sum_{k=0}^{K\wedge\tau_{\rm end}^{\prime}(i)}|\Delta^{\rm
err}A^{(k)}(i)|.$

We now bound the probability that each one of these processes is large, which
by (i) of Propositions 6.12–6.14 will bound the probability that
$|A^{(\tau_{K+1})}-A^{(0)}|$ is large.

Below, the constants $c,C\in(0,\infty)$ are introduced to satisfy Propositions
3.4, 6.8, 6.12, and 6.14. They are fixed all along the proof. The other
constants $c_{i},C_{i}$ are independent of everything and should be thought of
as being respectively much smaller than $c$ and much larger than $C$.

We start with the easiest process, which is the last one. Note that the
process is increasing and non-negative. Markov’s inequality and Propositions
6.12–6.14(iii) imply that for every $i\in I^{(0)}$,

$\displaystyle\mathbf{P}[\boldsymbol{\Delta}^{\rm err}A(i,\lceil
2N/\sin\alpha\rceil-1)\geq N^{1-c_{0}}|\mathcal{F}_{0}]$
$\displaystyle\leq\frac{1}{N^{1-c_{0}}}\mathbf{E}[\boldsymbol{\Delta}^{\rm
err}A(i,\lceil 2N/\sin\alpha\rceil-1)|\mathcal{F}_{0}]$
$\displaystyle\leq\frac{C}{N^{1-c_{0}}}\mathbf{E}\big{[}\sum_{0\leq
t<T}M^{(t)}(i)\big{|}\mathcal{F}_{0}\big{]}.$

Summing over on $i\in I^{(0)}$, averaging on $\mathcal{F}_{0}$ gives that

$\displaystyle\mathbf{P}\Big{[}\exists A,\exists i\in I^{(0)},\exists
K:\boldsymbol{\Delta}^{\rm err}A(i,K)\geq N^{1-c_{0}}\Big{]}$
$\displaystyle\leq\frac{C}{N^{1-c_{0}}}\mathbf{E}\Big{[}\sum_{0\leq t<T}{\rm
Err}^{(t)}\Big{]}\leq\frac{C_{1}\log N}{N^{c-c_{0}}},$ (72)

where in the second inequality we used Proposition 6.8(iii).

Let us now turn to the second process, which is a martingale with increments
that have uniform exponential moments because of Propositions 6.12–6.14(ii).
We deduce from a trivial modification of the Azuma-Hoeffding inequality (to
accommodate the unbounded increments, simply truncate the martingale
increments at $N^{c_{1}}$ and bound the error by the probability that there
exists a single increment larger than $N^{c_{1}}$) that for every $i\in
I^{(0)}$,

$\mathbf{P}\big{[}\exists
A:\max_{K}|\mathbf{M}_{A,i}(K)|>N^{3/4}\,\big{|}\,\mathcal{F}_{0}\big{]}\leq\exp[-c_{2}N^{c_{2}}].$

By averaging on $\mathcal{F}_{0}$ and using that $|I^{(0)}|$ has uniformly
bounded expectation (by Proposition 3.7), we deduce that

$\mathbf{P}\big{[}\exists A,\exists i\in
I^{(0)}:\max_{K}|\mathbf{M}_{A,i}(K)|>N^{3/4}\big{]}\leq\exp[-c_{3}N^{c_{3}}].$

It only remains to prove the following inequality:

$\mathbf{P}[\exists A,\exists i\in
I^{(0)},\max_{K}|\boldsymbol{\Sigma}_{A,i}(K)|\geq\tfrac{1}{2}N^{1-c_{0}}]\leq
N^{-c_{4}}.$ (73)

In order to prove this, let $\mathbf{N}_{1}$ be the number $k$ such that there
exists $i\in I^{(\tau_{k})}$ and $A\in\\{{\rm T},{\rm B}\\}$ such that
$|A^{(\tau_{k})}(i)-b(\tau_{k})|\leq N^{c_{5}}$, and $\mathbf{N}_{2}$ the
number of $k$ such that $I^{(\tau_{k})}\not\subset I^{(\tau_{k}+1)}$.

Propositions 6.12–6.14(iv) give that for every $A$, $i$, and $K$,

$|\boldsymbol{\Sigma}_{A,i}(K)|\leq
N\exp(-cN^{c_{5}})+C_{1}\mathbf{N}_{1}+C_{1}\sum_{0\leq k<\lceil
2N/\sin\alpha\rceil}\mathbf{P}[\tau_{\rm end}^{\prime}=k|\mathcal{F}_{k}].$

Since the last term on the right has an expectation which is bounded by the
expectation of $\mathbf{N}_{2}$, the Markov property implies that for
$c_{0}<c$ and $N$ large enough,

$\displaystyle\mathbf{P}[\exists A,\exists i\in
I^{(0)},\max_{K}|\boldsymbol{\Sigma}_{A,i}(K)|\geq\tfrac{1}{2}N^{1-c_{0}}]\leq\frac{2C_{1}\mathbf{E}[\mathbf{N}_{1}+\mathbf{N}_{2}]}{N^{1-c_{0}}}+2N^{c_{0}}\exp(-cN^{c_{5}}).$
(74)

Yet, for each time $t$ it is a direct consequence of Propositions 3.4 that for
$c_{5}$ sufficiently small,

$\mathbf{P}[\exists i\in I^{(t)},\exists A\in\\{{\rm T},{\rm
B}\\}:|A^{(t)}(i)-b(t)|\leq N^{c_{5}}]\leq C_{2}N^{c_{5}-c}$

so $\mathbf{E}[\mathbf{N}_{1}]\leq C_{3}N^{1+c_{5}-c}$.

Now, pick $c_{6}<c/2$. To have $I^{(\tau_{k})}\not\subset I^{(\tau_{k+1})}$,
it must be that one of the following three things occurs:

  * •

there exists $i\in I^{(\tau_{k})}$ with ${\rm
Vspan}^{(\tau_{k})}(i)\leq\varepsilon N+2N^{1-c_{6}}$ or $\max\\{|{\rm
T}^{(\tau_{k})}(i)-x_{2}|,|{\rm B}^{(\tau_{k})}-x_{2}|,|{\rm
R}^{(\tau_{k})}(i)-x_{1}|\\}\geq\sqrt{\eta\varepsilon}N-2N^{1-c_{6}}$,

  * •

$\Delta^{\rm err}A^{(k)}(i)\geq N^{1-c_{6}}$ for some $A$ and $i\in
I^{(\tau_{k})}$,

  * •

$\Delta^{\rm IIC}A^{(k)}(i)\geq N^{1-c_{6}}$ for some $A$ and $i\in
I^{(\tau_{k})}$.

Using Propositions 3.4 again, the first item occurs with probability
$O(N^{c_{6}-c})$. The second item occurs with probability $O(N^{c_{6}-c})$ by
the same computation as (72). The last item occurs with probability
$O(\exp(-cN^{1-c_{6}}))$ by (ii) of Propositions 6.12 and 6.14. The bound
$c_{6}<c/2$ gives

$\mathbf{E}[\mathbf{N}_{2}]\leq C_{4}N^{1-c_{6}}.$

Moreover, by picking $c_{0}\ll c_{i}$ small enough and plugging the two
expectation estimates into (74) implies (73). This concludes the proof. ∎

We now turn to a second lemma stating that with large probability, marked
nails exist near every $x\in\mathbb{B}_{\eta}(N)$ at time 0, or in other words
when defining $I_{\bullet}$ at the first step of the coupling, we get
$|I_{\bullet}|=|\mathbb{B}_{\eta}(N)|$.

######  Lemma 6.19 (Nails exist).

There exist $c,C\in(0,\infty)$ such that for every $0<\varepsilon\ll\eta$,

$\mathbf{P}[|I_{\bullet}|=|\mathbb{B}_{\eta}(N)|]\geq
1-\frac{C}{\eta^{2}}\big{(}\frac{\varepsilon}{\eta}\big{)}^{c}.$

###### Proof.

Set $\kappa:=(\eta\varepsilon^{3})^{1/4}$. There are $O(1/\eta^{2})$ elements
in $\mathbb{B}_{\eta}(N)$. Furthermore, for fixed $x\in\mathbb{B}_{\eta}(N)$,
the non-existence of a “markable” nail near $x$ requires the existence of a
dual path from $\Lambda_{2\varepsilon N}(x)$ to $\Lambda_{\kappa N}(x)$ or a
primal path from $\Lambda_{\kappa N}(x)$ to
$\Lambda_{((\eta\varepsilon)^{1/2}-\varepsilon)N}(x)$ (otherwise there exists
a primal circuit in the annulus $\Lambda_{\kappa
N}(x)\setminus\Lambda_{2\varepsilon N}(x)$ that is not connected to
$\partial\Lambda_{((\eta\varepsilon)^{1/2}-\varepsilon)N}(x)$ and therefore
constitutes a nail at $x$ that we may mark. Using (9) and the assumption that
$\varepsilon\ll\eta$ concludes the proof. ∎

###### Proof of Theorem 2.3.

We start by assuming that $\cos\alpha\notin\mathbb{Q}$. Consider
$1\gg\eta\gg\varepsilon>0$ and assume in particular that
$C\varepsilon^{c}/\eta^{2+c}\leq\eta/2$, where $c$ and $C$ are the constants
of Lemma 6.19. Also, we assume $N$ is large enough that
$C_{0}/N^{c_{0}}\leq\eta/2$, where $c_{0}$ and $C_{0}$ are the constants of
Lemma 6.18.

The two previous lemmata imply immediately that provided that $\varepsilon$ is
sufficiently small with probability $1-\eta$, marked nails exist near all
points in $\mathbb{B}_{\eta}(N)$ and belong to $I^{(t)}$ for every $0\leq
t<T$. By Proposition 6.8(i), we deduce that
$([\cdot]^{(T)}_{\bullet,0},[\cdot]^{(T)}_{\bullet,1})=([\cdot]^{(0)}_{\bullet,0},[\cdot]^{(0)}_{\bullet,1})$.

Now, the homotopy classes with respect to $\mathbb{B}_{\eta}(N)$ and with
respect to marked nails are equal for any loop that remains at a distance
$\sqrt{\eta\varepsilon}N$ of $\mathbb{B}_{\eta}(N)$. Since all loops (in
$\overline{\omega}^{(0)}$ and $\overline{\omega}^{(T)}$) surrounding at least
two but not all points in $\mathbb{B}_{\eta}(N)$ have a diameter which is
larger than $\eta N$, (RSW) immediately implies that they satisfy the previous
property with probability larger than $1-\eta$ provided
$\varepsilon=\varepsilon(\eta)>0$ is chosen small enough.

In particular, when setting $\eta_{0}=\sqrt{\eta}$ and assuming that
$2\eta\leq\eta_{0}$, we obtain that the rescaled configurations
$\omega_{\delta}^{(0)}$ and $\omega_{\delta}^{(T)}$ satisfy

$\mathbf{P}[d_{\mathbf{H}}(\omega_{\delta}^{(0)},\omega_{\delta}^{(T)})\leq\eta_{0}]\leq
2\eta\leq\eta_{0}.$

It remains to observe that thanks to properties of the track-exchange
operators (see Remark 3.12), the law of the homotopy classes around
$\mathbb{B}_{\eta}(N)$ is the same under $\phi_{\mathbb{L}(\pi/2)}$ and
$\phi_{\mathbb{L}^{(0)}}$ (and similarly under $\phi_{\mathbb{L}(\alpha)}$ and
$\phi_{\mathbb{L}^{(T)}}$). As a consequence, we may construct a coupling
between $\widetilde{\omega}_{\delta}\sim\phi_{\delta\mathbb{L}(\pi/2)}$ and
$\widetilde{\omega}_{\delta}^{\prime}\sim\phi_{\delta\mathbb{L}(\pi/2)}$ by
first using Remark 3.12 to couple $\omega_{\delta}$ and
$\omega_{\delta}^{(0)}$ in such a way that the homotopy classes of loops
surrounding one but not all points in $\mathbb{B}_{\eta}$ are the same, then
use the coupling constructed above, and finally couple $\omega_{\delta}^{(T)}$
with $\omega^{\prime}_{\delta}$ using Remark 3.12 again. Overall, we exactly
proved that the rescaled version of $\mathbf{P}$ satisfies the properties of
the statement of our theorem for $\eta_{0}$, so the proof is finished.

To get the result for $\cos\alpha$ rational, simply take the coupling obtained
as the weak limit of couplings with $\alpha_{n}$ satisfying
$\cos\alpha_{n}\notin\mathbb{Q}$ and tending to $\alpha$. One easily checks
that the limit makes sense and satisfies all the requested properties as the
bounds are continuous in $\alpha$ (note that one may also directly define the
coupling in this setting, being careful with the vertical position of right-
most points, see Remark 6.15 again). We insist that this limit should be taken
at $N$ (or equivalently $\delta>0$) fixed.

∎

##  7 Proofs of the main theorems

###  7.1 Proofs of the results for the random-cluster model

###### Proof of Theorem 1.2.

We prove the result for the Schramm-Smirnov topology but a similar proof works
for the Camia-Newman one. By Theorem 2.2, it suffices to construct a coupling
of $(\omega_{\delta},\omega^{\prime}_{\delta})$ with
$\omega,\omega^{\prime}\sim\phi_{\delta\mathbb{L}(\pi/2)}$ for which the
distance $d_{\bf H}(\omega_{\delta},e^{i\alpha}\omega^{\prime}_{\delta})$ is
typically small.

#### Case of $\Omega=\mathbb{R}^{2}$

We start with a coupling on the full space $\delta\mathbb{L}(\pi/2)$. Let
$\sigma_{u}$ be the reflection with respect to the line $e^{iu}\mathbb{R}$.

Fix $\varepsilon>0$ and choose $\eta\leq\varepsilon/4$ so that Theorem 2.2
implies that for every coupling of
$\omega_{\delta}\sim\phi_{\delta\mathbb{L}(\pi/2)}$ and
$\omega^{\prime}_{\delta}\sim\phi_{\delta\mathbb{L}(\pi/2)}$, we have

$\mathbf{P}[d_{\bf
H}(\omega_{\delta},e^{i\alpha}\omega^{\prime}_{\delta})\leq\eta,d_{\bf
SS}(\omega_{\delta},e^{i\alpha}\omega^{\prime}_{\delta})>\tfrac{\varepsilon}{2}]\leq\tfrac{\varepsilon}{4}.$
(75)

Now, construct an explicit coupling $\mathbb{P}$ between
$\omega_{\delta}\sim\phi_{\delta\mathbb{L}(\pi/2)}$ and
$\omega^{\prime}_{\delta}\sim\phi_{\delta\mathbb{L}(\pi/2)}$ as follows:
sample $\omega^{\prime}_{\delta}\sim\phi_{\delta\mathbb{L}(\pi/2)}$ and couple
$\sigma_{0}\omega^{\prime}_{\delta}$ with
$\omega_{\delta}^{\alpha}\sim\phi_{\delta\mathbb{L}(\alpha)}$ using Theorems
2.2 and 2.3 (this is doable since
$\sigma_{0}\omega_{\delta}^{\prime}\sim\phi_{\delta\mathbb{L}(\pi/2)}$) and
(75) in such a way that

$\mathbb{P}[d_{\bf
SS}(\sigma_{0}\omega_{\delta}^{\prime},\omega_{\delta}^{\alpha})\geq\tfrac{\varepsilon}{2}]\leq\tfrac{\varepsilon}{2},$
(76)

then, couple $\sigma_{\alpha/2}\omega_{\delta}^{\alpha}$ with
$\omega_{\delta}\sim\phi_{\delta\mathbb{L}(\pi/2)}$ by Theorems 2.2 and 2.3
(this is doable since
$\sigma_{\alpha/2}\omega_{\delta}^{\alpha}\sim\phi_{\delta\mathbb{L}(\alpha)}$)
in such a way that

$\mathbb{P}[d_{\bf
SS}(\omega_{\delta},\sigma_{\alpha/2}\omega_{\delta}^{\alpha})\geq\tfrac{\varepsilon}{2}]\leq\tfrac{\varepsilon}{2}.$
(77)

Since

$\displaystyle d_{\bf
SS}(\omega_{\delta},e^{i\alpha}\omega^{\prime}_{\delta})$
$\displaystyle=d_{\bf
SS}(\omega_{\delta},\sigma_{\alpha/2}\sigma_{0}\omega^{\prime}_{\delta})=d_{\bf
SS}(\sigma_{\alpha/2}\omega_{\delta},\sigma_{0}\omega^{\prime}_{\delta})$ (78)
$\displaystyle\leq d_{\bf
SS}(\sigma_{\alpha/2}\omega_{\delta},\omega_{\delta}^{\alpha})+d_{\bf
SS}(\omega_{\delta}^{\alpha},\sigma_{0}\omega^{\prime}_{\delta})=d_{\bf
SS}(\omega_{\delta},\sigma_{\alpha/2}\omega_{\delta}^{\alpha})+d_{\bf
SS}(\omega_{\delta}^{\alpha},\sigma_{0}\omega^{\prime}_{\delta}).$

The result then follows by combining (76)–(78).

#### Case of a bounded simply connected domain $\Omega$ with $C^{1}$-smooth
boundary

To obtain the result in a finite domain, we use the domain Markov property and
the fact that one may approximate $\phi_{\Omega_{\delta}}^{0}$ by asking that
there exists a loop $\Gamma$ within distance $\eta$ of $\partial\Omega$ in the
infinite-volume measure. More precisely, let $A(\Omega,\eta)$ be the event
that there exists a loop
$\boldsymbol{\Gamma}\in\mathcal{F}_{0}(\omega_{\delta})$ which is included in
$\Omega$ and such that $d(\boldsymbol{\Gamma},\partial\Omega)\leq\eta$ ($d$ is
the distance between loops defined in the introduction). Note that whether
$A(\Omega,\eta)$ occurs or not can be measured in the Schramm-Smirnov topology
(we leave this as an exercise).

Now, fix $\varepsilon_{0}>0$. We use the characterization of the Schramm-
Smirnov distance provided in [32, Proposition 3.9]. There exists a family of
non-degenerate quads $Q_{1},\dots,Q_{n}$ in $\Omega$ such that if the sets of
quads in $Q_{1},\dots,Q_{n}$ that are crossed are the same in
$\omega_{\delta}$ and $\omega^{\prime}_{\delta}$, then $d_{\bf
SS}(\omega_{\delta},\omega^{\prime}_{\delta})\leq\varepsilon_{0}$. In
particular, we deduce that if $H_{\vec{Q}}(I)$ denotes the event that $Q_{i}$
is crossed if and only if $i\in I$, then there exists a coupling $\mathbf{P}$
between $\omega_{\delta}\sim\phi_{\Omega_{\delta}}^{0}$ and
$\omega^{\prime}_{\delta}\sim\phi_{e^{i\alpha}\Omega_{\delta}}^{0}$ such that

$\mathbf{P}[d_{\bf
SS}(\omega_{\delta},\omega^{\prime}_{\delta})\geq\varepsilon_{0}]\leq\varepsilon_{0}$

if and only if for every $I\subset\\{1,\dots,n\\}$,

$|\phi_{\Omega_{\delta}}^{0}[H_{\vec{Q}}(I)]-\phi_{e^{i\alpha}\Omega_{\delta}}^{0}[H_{e^{i\alpha}\vec{Q}}(I)]|\leq\varepsilon_{0}/2^{n}=:\varepsilon.$
(79)

Now, the infinite volume result above implies that for every
$\delta<\delta_{0}(\Omega,\eta,\varepsilon)$,

$|\phi_{\delta\mathbb{Z}^{2}}[H_{\vec{Q}}(I)|A(\Omega,\eta)]-\phi_{\delta\mathbb{Z}^{2}}[H_{e^{i\alpha}\vec{Q}}(I)|A(e^{i\alpha}\Omega,\eta)]|\leq\tfrac{1}{2}\varepsilon.$
(80)

We therefore wish to prove that

$|\phi_{\Omega_{\delta}}^{0}[H_{\vec{Q}}(I)]-\phi_{\delta\mathbb{Z}^{2}}[H_{\vec{Q}}(I)|A(\Omega,\eta)]|\leq\tfrac{1}{2}\varepsilon.$
(81)

The same can be done for the rotated version, so that the previous displayed
equations imply (79) and conclude the proof.

To get (81), let $\boldsymbol{\Omega}_{\delta}$ be the interior of the outer-
most loop in $\mathcal{F}_{0}(\omega)$ satisfying the conditions of
$A(\Omega,\eta)$. Using the spatial Markov property, it suffices to show that

$|\phi_{\Omega_{\delta}}^{0}[H_{\vec{Q}}(I)]-\phi_{\boldsymbol{\Omega}_{\delta}}^{0}[H_{\vec{Q}}(I))]|\leq\tfrac{1}{2}\varepsilon.$
(82)

Note that there is a clear increasing coupling between
$\boldsymbol{\omega}_{\delta}\sim\phi_{\boldsymbol{\Omega}_{\delta}}^{0}$ and
$\omega_{\delta}\sim\phi_{\Omega_{\delta}}^{0}$
($\boldsymbol{\omega}_{\delta}\leq\omega_{\delta}$ because of
$\boldsymbol{\Omega}_{\delta}\subset\Omega_{\delta}$), so that for
$\omega_{\delta}$ to belong to $H_{\vec{Q}}(I)$ but not
$\boldsymbol{\omega}_{\delta}$ or vice versa, it must be that one of the quads
$Q_{i}$ must be crossed in one but not in the other. We deduce that it
suffices to show that for every possible realization of
$\boldsymbol{\Omega}_{\delta}$,

$\phi_{\Omega_{\delta}}^{0}[\mathcal{C}(Q_{i})]-\phi_{\boldsymbol{\Omega}_{\delta}}^{0}[\mathcal{C}(Q_{i})]\leq\tfrac{1}{2n}\varepsilon.$
(83)

Therefore, the result boils down to the following.

#### Claim

For every $\epsilon>0$, every bounded simply connected domain $\Omega$ with
$C^{1}$-smooth boundary, and every quad $Q$ inside $\Omega$, there exists
$\eta=\eta(\Omega,Q,\epsilon)>0$ such that for every
$\Omega^{\prime}\subset\Omega$ with
$d(\partial\Omega^{\prime},\partial\Omega)\leq\eta$,

$\phi_{\Omega_{\delta}}^{0}[\mathcal{C}(Q)]\leq\phi_{\Omega^{\prime}_{\delta}}^{0}[\mathcal{C}(Q)]+\epsilon$

for $\delta$ small enough.

###### Proof.

We only sketch the proof. Consider first the “epigraph“ domains indexed by
continuous functions $f$ from $[-2,2]$ to $\mathbb{R}$ given by

$\Omega(f):=\\{x=(x_{1},x_{2})\in\mathbb{R}^{2}:x_{1}\in(-2,2),f(x_{1})<x_{2}<2\\}$

(see Figure 29). Define $\Lambda:=[-1,1]^{2}$. For $\alpha>0$, a
straightforward yet quite lengthy application of the techniques
developed111111The whole fo Section 4 of [24] should be adapted to finite
domains and considering the covariance of crossing events with edges on the
boundary of the domain. in [24, Lemma 5.3] implies that for every $f\leq-2$
and $1\leq k\leq\tfrac{1}{2}\lfloor 1/\alpha\rfloor=:K$,

$\phi^{0}_{\Omega(f)_{\delta}}[\mathcal{C}(\Lambda)]-\phi^{0}_{\Omega(f+\alpha)_{\delta}}[\mathcal{C}(\Lambda)]\leq
C\big{(}\phi^{0}_{\Omega(f+k\alpha)_{\delta}}[\mathcal{C}(\Lambda)]-\phi^{0}_{\Omega(f+(k+1)\alpha)_{\delta}}[\mathcal{C}(\Lambda)]\big{)}.$

Summing over $1\leq k\leq K$, we deduce that

$\phi^{0}_{\Omega(f)_{\delta}}[\mathcal{C}(\Lambda)]-\phi^{0}_{\Omega(f+\alpha)_{\delta}}[\mathcal{C}(\Lambda)]\leq\frac{C}{K}\big{(}\phi^{0}_{\Omega(f)_{\delta}}[\mathcal{C}(\Lambda)]-\phi^{0}_{\Omega(f+K\alpha)_{\delta}}[\mathcal{C}(\Lambda)]\big{)}\leq\frac{C}{K}\leq
4C\alpha.$ (84)

Note that a similar argument works for any rotation, translate, or rescaling
of the domains above.

We now use our assumption that $\partial\Omega$ is $C^{1}$-smooth. Since
$\partial\Omega$ is given by a curve $\gamma$ which is $C^{1}$ and has non-
vanishing differential, one may find (see Figure 29) constants
$\kappa=\kappa(\Omega)>0$ and $C=C(\Omega)>0$, functions
$f_{s}:[-2,2]\rightarrow(-\infty,-2]$ and
$T_{s}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ for $1\leq s\leq S$, where $S$
depends on $\Omega$ (through the modulus of continuity of the derivative for
the function parametrizing $\partial\Omega$) but not on $\eta$, satisfying the
following properties:

  * •

$T_{s}$ is the composition of a rotation, a translation, and the
multiplication by $\kappa$;

  * •

$T_{s}(\Omega(f_{s}))$ is included in $\Omega$ for every $s$;

  * •

for all $\eta$ small enough, $\\{x\in\Omega:d(x,\Omega^{c})\leq\eta\\}$ is
included in the union of the sets

$A_{s}:=T_{s}(\\{x=(x_{1},x_{2}):x_{1}\in[-1,1],f(x_{1})<x_{2}<f(x_{1})+C\eta\\}).$

Introducing the domains $\Omega_{s}:=\Omega\setminus\bigcup_{t=1}^{s}A_{t}$,
and using again [24] for the first and second inequalities, one can prove the
existence of $C_{i}=C_{i}(\Omega,Q,\kappa)>0$ such that

$\displaystyle\phi_{\Omega_{s-1}}^{0}[\mathcal{C}(Q)]-\phi_{\Omega_{s}}^{0}[\mathcal{C}(Q)]$
$\displaystyle\leq
C_{1}\big{(}\phi_{\Omega_{s-1}}^{0}[\mathcal{C}(T_{s}(\Lambda))]-\phi_{\Omega_{s}}^{0}[\mathcal{C}(T_{s}(\Lambda))]\big{)}$
$\displaystyle\leq
C_{2}\big{(}\phi_{\Omega(f_{s})}^{0}[\mathcal{C}(\Lambda)]-\phi_{\Omega(f_{s}+C\eta)}^{0}[\mathcal{C}(\Lambda)]\big{)}$
(85) $\displaystyle\leq C_{3}\eta,$

where the last line is due to (84) applied to $\alpha=C\eta$.

Choose $\eta=\eta(\Omega,\epsilon,S)>0$ small enough. Summing (85) over $s$
gives

$\phi_{\Omega_{\delta}}^{0}[\mathcal{C}(Q)]-\phi^{0}_{\Omega_{\delta}^{\prime}}[\mathcal{C}(Q)]\leq\sum_{s=0}^{S-1}\phi_{\Omega_{s-1}}^{0}[\mathcal{C}(Q)]-\phi_{\Omega_{s}}^{0}[\mathcal{C}(Q)]\leq\epsilon.$

∎

Figure 29: On the left, an example of a domain $\Omega(f)$. Note that the sets
$\Omega(f+k\alpha)$ have a nested structure (the red part denotes
$\Omega(f)\setminus\Omega(f+\alpha)$). On the right, the impact of changing
the boundary is compared with the impact of changing the boundary in a family
of subdomains which are images by simple transformations of domains
$\Omega(f)$ (with potentially different functions $f$). The existence of such
a decomposition is made possible by the fact that the boundary of $\Omega$ is
$C^{1}$.

#### Case of a (possibly unbounded) simply connected domain $\Omega$ with
$C^{1}$-smooth boundary

For every $\varepsilon>0$, to determine the Schramm-Smirnov distance up to a
precision of $\varepsilon>0$, only quads in $B(0,1/\varepsilon)$ need to be
considered. Consider a bounded domain $\Omega^{(\varepsilon)}$ that coincides
with $\Omega$ on $B(0,1/\varepsilon^{C})$. By the mixing property, one has
that for every $\delta>0$ and every event $E$ depending on edges in
$\delta\mathbb{Z}^{2}\cap B(0,1/\varepsilon)$ only,

$|\phi_{\Omega^{(\varepsilon)}_{\delta}}[E]-\phi_{\Omega_{\delta}}[E]|\leq
C_{\rm mix}\varepsilon^{\rm c_{\rm mix}(C-1)}\phi_{\Omega_{\delta}}[E].$

Now, take the domain $\Omega^{(\varepsilon)}$ very large but finite, equal to
$\Omega$ up to large distance. Using the invariance by rotation in
$\Omega^{(\varepsilon)}_{\delta}$ and taking $\delta$ to 0 then $\varepsilon$
to 0 concludes the proof.∎

###### Proof of Corollary 1.3.

When one considers a quad $Q$ that remains at a distance at least
$\varepsilon$ of the boundary of $\Omega$, the result follows directly from
Theorem 1.2 and the measurability of $\mathcal{C}(Q)$ in the Schramm-Smirnov
topology (note that the event gets rewritten as $Q\in\omega$ when $\omega$ is
seen as an element of $\mathcal{H}$).

Now, when $1\leq q<4$, to get the result without any assumption on the
distance to the boundary, note that for a quad $Q$, there exists a quad
$Q^{\prime}$ that is such that its distance to $\partial\Omega$ is at least
$\varepsilon$, and which is in Hausdorff distance at a distance at most
$2\varepsilon$ from $Q$. Using the strong version of crossing estimates from
[25], we obtain easily (this type of reasoning is now classical, see for
instance [24, Lemma 3.12] for an example) that

$|\phi_{\Omega_{\delta}}[\mathcal{C}(Q)]-\phi_{\Omega_{\delta}}[\mathcal{C}(Q^{\prime})]|\leq
C\varepsilon^{c}$

for two constants $C>0$ and $c>0$. The result follows readily by first
choosing $\varepsilon$ small enough and then letting $\delta$ tend to zero and
use the rotational invariance result for $Q^{\prime}$. ∎

###### Proof of Corollary 1.4.

We use a conditional mixing argument due to Garban, Pete, and Schramm [34,
Section 3] in the case of Bernoulli percolation and that can be extended to
the random-cluster model using crossing estimates. Consider the Euclidean ball
$B_{n}$ of radius $n$, and its boundary $\partial B_{n}$. Introduce the
quantities

$\epsilon(n,N):=\phi_{\mathbb{Z}^{2}}^{0}[0\longleftrightarrow
B_{N}^{c}|B_{n}\longleftrightarrow
B_{N}^{c}]\quad\text{and}\quad\epsilon(n):=\lim_{N\rightarrow\infty}\epsilon(n,N).$

The statement of conditional mixing from [34] implies the following claim (in
[34] it is stated for the four-arm event, but a similar – in fact simpler –
argument can be performed for the one-arm event, see e.g. Proposition 5.3 of
the same paper). For every $\beta,\varepsilon>0$, there exists
$\eta=\eta(\beta,\varepsilon)>0$ such that for every $\Omega$ and every
$x_{1},\dots,x_{n}$ at a distance $\varepsilon$ of each other and of the
boundary, and every partition $P$ of $(x_{1},\dots,x_{n})$,

$\Big{|}\phi_{\Omega_{\delta}}^{0}[\mathcal{E}(P,x_{1},\dots,x_{n})]-\epsilon({\tfrac{\eta}{\delta}})^{n}\phi_{\Omega_{\delta}}^{0}[\mathcal{E}(P,B_{\eta/\delta}(x_{1}),\dots,B_{\eta/\delta}(x_{n}))]\Big{|}\leq\beta\phi_{\Omega_{\delta}}^{0}[\mathcal{E}(P,x_{1},\dots,x_{n})],$

where $\mathcal{E}(P,B_{\eta/\delta}(x_{1}),\dots,B_{\eta/\delta}(x_{n}))$ is
the event that the balls $B_{\eta/\delta}(x_{i})$ are connected to each other
if and only if they belong to the same element of the partition $P$. The same
formula applies in the rotated measure.

We conclude, by observing that $e^{i\alpha}B_{\eta/\delta}(x_{i})$ and
$B_{\eta/\delta}(e^{i\alpha}x_{i})$ are equal, and that the event
$\mathcal{E}(P,B_{\eta/\delta}(x_{1}),\dots,B_{\eta/\delta}(x_{n}))$ is
measurable in the Schramm-Smirnov topology, so that its probability or the
probability of its rotation by an angle of $\alpha$ are close to each other by
Theorem 1.2. ∎

###  7.2 Proofs of the theorems for the other models

###### Proof of Corollary 1.6.

Fix $\tau_{1},\dots,\tau_{n}\in\mathbb{T}_{q}$. Let
$I_{i}\subset\\{x_{1},\dots,x_{n}\\}$ be the sets of $x_{j}$ such that
$\tau_{j}=i$ and call a partition $P=(P_{1},\dots,P_{k})$ of
$\\{x_{1},\dots,x_{n}\\}$ compatible with $\tau$ if each $P_{j}$ is included
in one of the $I_{i}$. Also, let $|P|=k$ be the number of elements in the
partition. The Edwards-Sokal coupling implies that

$\mu_{\Omega_{\delta}}[\sigma_{x_{i}}=\tau_{i},1\leq i\leq
n]=\sum_{\text{compatible
}P}q^{-|P|}\phi^{0}_{\Omega_{\delta}}[\mathcal{E}(P,x_{1},\dots,x_{n})].$

We deduce the corollary by using Corollary 1.4. ∎

###### Proof of Corollary 1.8.

Fix $x_{1},\dots,x_{2n}\in(\delta\mathbb{Z}^{2})^{*}$ and let
$\Gamma_{1},\dots,\Gamma_{2n}$ be the exterior-most loops in
$\overline{\omega}_{\delta}$ that surround one $x_{i}$ but not the others.
Also, let $\Gamma$ be the smallest loop surrounding all the $x_{i}$. Finally,
let
$\overline{\mathcal{F}}_{\delta}=\overline{\mathcal{F}}_{\delta}(x_{1},\dots,x_{2n})$
be the set of loops in $\overline{\omega}_{\delta}$ that surround at least one
of the $\Gamma_{i}$ and are surrounded by $\Gamma$ (including $\Gamma$ and all
$\Gamma_{i}$).

Also, let $\mathbf{N}$ be the number of loops surrounding the origin and
introduce, for a curve $\gamma$ and $\delta>0$,

$c_{\delta}(\gamma):=\phi_{\overline{\gamma}_{\delta}}^{0}[\mathbf{N}]-\phi_{B_{\delta}}^{0}[\mathbf{N}],$

where $\overline{\gamma}$ is the domain surrounded by $\gamma$.

It is shown in [22] that there exits a function $F$ taking possible
realizations of $\overline{\mathcal{F}}_{\delta}$ as arguments and outputting
a complex number such that

  * •

$\mathbb{E}^{\rm
6V}_{\delta\mathbb{Z}^{2}}\big{[}\prod_{i=1}^{n}(h_{x_{2i}}-h_{x_{2i-1}})\big{]}=\phi_{\delta\mathbb{Z}^{2}}[F(\overline{\mathcal{F}}_{\delta})]$,

  * •

$F(\overline{\mathcal{F}}_{\delta})=G(\overline{\mathcal{F}}_{\delta},c_{\delta}(\Gamma_{1}),\dots,c_{\delta}(\Gamma_{2n}))$
where $G(\overline{\mathcal{F}}_{\delta},c_{1},\dots,c_{2n})$ is a function
that depends on $\overline{\mathcal{F}}_{\delta}$ only through the inclusions
between the different loops.

  * •

$|F(\overline{\mathcal{F}}_{\delta})|\leq
C(\log|\overline{\mathcal{F}}_{\delta}|)^{n}$ where
$|\overline{\mathcal{F}}_{\delta}|$ is the number of loops in
$\overline{\mathcal{F}}_{\delta}$.

Now, the conditional mixing from [34] easily implies that
$c_{\delta}(\Gamma)-c_{\delta}(e^{i\alpha}\Gamma)$ tends to 0 as $\delta$
tends to 0. So we deduce the result from Theorem 1.2 for the Camia-Newman
distance using dominated convergence. One may be slightly worried about the
fact that $c_{\delta}(\Gamma)$ is not bounded from below and can be very
negative, but this is treated by the third point: for
$|F(\overline{\mathcal{F}}_{\delta})|$ to exceed $\log k$, there must be at
least $k$ loops in $\overline{\mathcal{F}}_{\delta}$, an event which occurs
with probability smaller than $C_{n}k^{-c}$ for some constant $c>0$
independent of everything.∎

######  Remark 7.1.

The previous connection between the six-vertex model on the medial lattice and
the random-cluster model on the primal one extends to generic weights $a$ and
$b$. The outcome is a random-cluster model with isoradial weights. When
choosing the corresponding isoradial embedding for the primal lattice, and
defining the six-vertex model on the associated medial lattice, one obtains a
rotational invariance for every $a,b,c$ with
$\Delta:=(a^{2}+b^{2}-c^{2})/2ab\in[-1,-1/2]$.

#### Acknowledgments

The first author was funded by the ERC CriBLaM. The third and fourth authors
were funded by the Swiss FNS. The first, third, fourth and fifth authors were
partially funded by the NCCR SwissMap and the Swiss FNS.

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