the event $\\{\Lambda_{r}\equiv 0\\}$ that every edge with one endpoint in
$\Lambda_{r-1}$ is closed. The previous study of $\Delta_{p_{c}}(r,R)$ thus
implies that for any $r\leq R/2$,

$\displaystyle\phi_{\Lambda_{R}}^{0}[A(x,y)]-\phi_{\Lambda_{R}}^{0}[A(x,y)\,|\,\Lambda_{r}\equiv
0]\asymp\Delta_{p_{c}}(r,R)\big{[}\phi_{\Lambda_{R}}^{0}[A(x,y)]-\phi_{\Lambda_{R}}^{0}[A(x,y)\,|\,\Lambda_{R/2}\equiv
0]\big{]}.$

We deduce that

$\sum_{y\in\partial\Lambda_{R}}(4-d_{y})\big{[}\phi_{\Lambda_{R}}^{0}[A(x,y)]-\phi_{{\rm
Ann}(r,R)}^{0}[A(x,y)]\big{]}\asymp\Delta_{p_{c}}(r,R)\Sigma_{R},$ (6.11)

where

$\Sigma_{R}:=\sum_{y\in\partial\Lambda_{R}}(4-d_{y})\big{[}\phi_{\Lambda_{R}}^{0}[A(x,y)]-\phi_{\Lambda_{R}}^{0}[A(x,y)\,|\,\Lambda_{R/2}\equiv
0]\big{]}$ (6.12)

is a constant that depends on $R$ only. Inserting (6.10) and (6.11) into (6.9)
gives

$\displaystyle\Delta_{p_{c}}(r,R)\Sigma_{R}\asymp\pi_{1}^{+}(R)\pi_{2}(r,R)\sum_{k\leq
r/2}\phi_{\mathbb{U}}^{0}[(k,0)\longleftrightarrow\partial\Lambda_{r}],$

for any $r\leq R/2$. Divide the relation above for an arbitrary $r\leq R/2$
with the same relation for $r=R/2$ (we use that
$\Delta_{p_{c}}(r/2,R)\asymp\Delta_{p_{c}}(r,R)$) to find

$\Delta_{p_{c}}(r,R)\asymp\pi_{2}(r,R)\,\frac{\sum_{k\leq
r/2}\phi_{\mathbb{U}}^{0}[(k,0)\longleftrightarrow\partial\Lambda_{r}]}{\sum_{k\leq
R/4}\phi_{\mathbb{U}}^{0}[(k,0)\longleftrightarrow\partial\Lambda_{R}]}\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{2}(r,R)(r/R).$ (6.13)

The inequality above is obtained by summing over $j=0,\dots,R/r$ and
$k=1,\dots,r/2$ the inequality below :

$\phi_{\mathbb{U}}^{0}[(k+rj/2,0)\longleftrightarrow\partial\Lambda_{R}]\leq\phi_{\mathbb{U}}^{0}[(k,0)\longleftrightarrow\partial\Lambda_{r}],$

which is a direct consequence of the comparison between boundary conditions
(CBC).

###### Remark 6.8

Applying (6.13) with $r=1$, the comparison between boundary conditions (CBC)
and (2.10) imply that

$\pi_{1}(R)\Delta_{p_{c}}(R)\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\frac{\pi_{1}(R)\pi_{2}(R)}{\sum_{k\leq
R/4}\phi_{\mathbb{U}}^{0}[(k,0)\longleftrightarrow\partial\Lambda_{R}]}\geq\frac{\pi_{1}(R)\pi_{2}(R)}{\tfrac{1}{4}R\pi_{1}(3R/4)}\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{2}(R)/R\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!R^{c-2}.$

####  6.2.3 Stability of $\Delta_{p}(r,R)$: proof of Proposition 6.4

The proof is based on the following quantity. Call a subgraph $\Omega$ of
$\mathbb{Z}^{2}$ an topological $R$-annulus if it is of the form
$\Lambda_{R}\setminus H$ with $H$ a simply connected domain such that
$\Lambda_{R/8}\subset H\subset\Lambda_{R/4}$; see Figure 13. The boundary of
$\Omega$ is split into $\partial\Lambda_{R}$ and $\partial_{\rm
in}\Omega:=\partial\Omega\setminus\partial\Lambda_{R}$. Set

$\mathbf{N}_{\Omega}(p):=\sum_{y\in\partial_{\rm
in}\Omega}\phi_{\Omega,p}^{0}[y\longleftrightarrow\partial\Lambda_{R/2}].$

The first observation is the following lemma stating that for $q=4$ the
quantity $\mathbf{N}_{\Omega}(p)$ does not depend on the choice of the
$R$-annulus $\Omega$, or on the choice of $p$ such that $R\leq L(p)$.

###### Lemma 6.9

For every $p\geq p_{c}$ and $R\leq L(p)$,

$\displaystyle\sup_{\Omega}\mathbf{N}_{\Omega}(p^{*})\asymp\sup_{\Omega}\mathbf{N}_{\Omega}(p_{c})\asymp\inf_{\Omega}\mathbf{N}_{\Omega}(p_{c})\asymp\inf_{\Omega}\mathbf{N}_{\Omega}(p),$
(6.14)

where the infimum and supremum are taken over topological $R$-annuli.

Figure 13: In grey the domain $\mathscr{H}$ (which is an topological
$R$-annulus) carved by the event $F$ (in blue). In red, the exploration path
$\gamma=\gamma(\omega)$ going around the boundary vertex $x$.

##### Proof

We start by proving that

$\displaystyle\sup_{\Omega}\mathbf{N}_{\Omega}(p_{c})\asymp\Sigma_{R}/\pi_{1}^{+}(R)\asymp\inf_{\Omega}\mathbf{N}_{\Omega}(p_{c}),$
(6.15)

where $\Sigma_{R}$ was introduced in (6.12) in the previous section.

Fix $\Omega$ an topological $R$-annulus and set $x:=(0,R)$. Then, due to
(2.1), for every $y\in\partial_{\rm in}\Omega$,

$\displaystyle\phi_{\Omega,p_{c}}^{0}[y\longleftrightarrow x\text{ and
}\partial_{\rm
in}\Omega\stackrel{{\scriptstyle*}}{{\longleftrightarrow}}\partial\Lambda_{R}]\asymp\pi_{1}^{+}(R)\phi_{\Omega,p_{c}}^{0}[y\longleftrightarrow\partial\Lambda_{R/2}].$

In particular, we have that

$\displaystyle\mathbf{N}_{\Omega}(p_{c})\asymp\frac{1}{\pi_{1}^{+}(R)}\sum_{y\in\partial_{\rm
in}\Omega}\phi_{\Omega,p_{c}}^{0}[A(x,y)].$ (6.16)

Subtracting (6.8) for the domains $\Lambda_{R}$ and $\Omega$, and following
the proof of Proposition 6.3, we find that

$\sum_{y\in\partial_{\rm
in}\Omega}\phi_{\Omega,p_{c}}^{0}[A(x,y)]\asymp\Sigma_{R},$

which concludes the proof of (6.15).

We now prove that for every $p$ and $R\leq L(p)$,

$\inf_{\Omega}\mathbf{N}_{\Omega}(p)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!R^{2}\pi_{4}(p,R)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\sup_{\Omega}\mathbf{N}_{\Omega}(p),$
(6.17)

where the infimum and supremum are taken over topological $R$-annuli. Note
that this inequality implies the result. Indeed, since
$\mathbf{N}_{\Omega}(p)$ is increasing in $p$ and
$\pi_{4}(p^{*},R)=\pi_{4}(p,R)$ (by duality), the previous inequality implies
that for $p\geq p_{c}$ and $R\leq L(p)=L(p^{*})$,

$\displaystyle\sup_{\Omega}\mathbf{N}_{\Omega}(p_{c})\geq\sup_{\Omega}\mathbf{N}_{\Omega}(p^{*})\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!R^{2}\pi_{4}(p^{*},R)=R^{2}\pi_{4}(p,R)\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\inf_{\Omega}\mathbf{N}_{\Omega}(p)\geq\inf_{\Omega}\mathbf{N}_{\Omega}(p_{c}),$

which combines with (6.15) to give the result. We now focus on the proof of
(6.17). We proceed similarly to [DMT20, Prop 6.8], but from inside out.

Let $F$ be the event that in ${\rm Ann}(R/8,R/4)$ there exists an open circuit
surrounding $\Lambda_{R/8}$ which is connected to $\Lambda_{R/8}$, as well as
a dual-open circuit, which is necessarily outside of the open one; see Figure
13. If $F$ occurs, let $\mathscr{H}=\mathscr{H}(\omega)$ be the graph formed
of the union of all open clusters that intersect $\Lambda_{R/8}$, along with
all finite components of $\mathbb{Z}^{2}$ minus the said union. Then, due to
the definition of $F$, $\Lambda_{R/8}\subset\mathscr{H}\subset\Lambda_{R/4}$.
Write $\Omega(\omega)=\Lambda_{R}\setminus\mathscr{H}$ for the random
topological $R$-annulus formed by removing $\mathscr{H}$.

When the measure $\phi^{0}_{\Lambda_{R},p}$ is conditioned on
$F\cap\\{\Omega(\omega)=\Omega\\}$, its restriction to $\Omega$ is
$\phi^{0}_{\Omega,p}$. Notice that if $y\in\partial_{\rm in}\Omega$ is
connected to $\partial\Lambda_{R/2}$ by an open path, then a four arm event to
distance $R/8$ occurs around $y$. Thus, using the quasi-multiplicativity (Mix)
and Theorem 2.1 (to bound the probability of $F$ from below), we find that

$\displaystyle R^{2}\pi_{4}(p,R)\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\phi_{\Lambda_{R},p}[\mathbf{N}_{\Omega(\omega)}(p)\,\mathbbm{1}_{F}]\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\inf_{\Omega}\mathbf{N}_{\Omega}(p).$

Conversely, the separation of arms for the four-arm event and Theorem 2.1 show
that for each $y\in{\rm Ann}(5R/32,7R/32)$,

$\displaystyle\phi_{\Lambda_{R},p}[F\cap\\{y\in\partial_{\rm
in}\Omega(\omega)\\}\cap\\{y\longleftrightarrow\partial\Lambda_{R/2}\\}]\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{4}(p,R).$

Summing over all $y$ we find

$\displaystyle R^{2}\pi_{4}(p,R)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\phi_{\Lambda_{R},p}[\mathbf{N}_{\Omega(\omega)}(p)\mathbbm{1}_{F}]\leq\sup_{\Omega}\mathbf{N}_{\Omega}(p).$

$\square$

###### Remark 6.10

The previous proof shows as a byproduct that $\pi_{4}(p,R)\asymp\pi_{4}(R)$
for every $R\leq L(p)$. It is somehow surprising that the stability of
$\pi_{4}$ below the correlation length can be directly extracted from the
parafermionic observable. Recall, however, that this is only valid for $q=4$.

###### Remark 6.11

The previous proof also implies that $R^{2}\pi_{4}(R)$ is of the order of
$\mathbf{N}_{\Omega}(p_{c})$ for every topological $R$-annulus $\Omega$. In
particular, taking $\Omega=\Lambda_{R/8}$ gives that

$R^{2}\pi_{4}(R)\asymp R\pi_{1}^{+}(R).$ (6.18)

The conjectural scaling limit of the model suggests that $\pi_{1}^{+}(R)\asymp
R^{-1}$, which would imply that $\pi_{4}(R)\asymp R^{-2}$. We see that in this
case the fact that $\Delta_{p_{c}}(R)\gg\pi_{4}(R)$ is crucial for the bound
(6.2).

Figure 14: By sliding the boxes along a given path one by one from the
exterior towards $y$, one finds the first time that the twice larger box
$\Lambda_{2\cdot 8^{k}}(z_{k})$ disconnects $y$ from the boundary. Note that
by construction the first time it stops cannot be the quite the same for
different $k$. In particular, the box $\Lambda_{8^{k}/2}(z_{k})$ being at a
distance $8^{k}/2$ of the boundary of $\Lambda_{8^{k}}(z_{k})$, it is also at
such a distance of $\partial_{\rm in}\Omega$. Since on contrary
$\Lambda_{8^{\ell}/2}(z_{\ell})$ is within a distance $8^{\ell}\times 3/2$ of
it, we immediately deduce that the two boxes do not intersect as soon as
$\ell<k$.

##### Proof of Proposition 6.4

We treat the case $p>p_{c}$; the case $p<p_{c}$ can be done similarly. Fix
$R\leq L(p)$. The inequality (5.27) implies that

$\displaystyle\Big{|}\log\frac{\Delta_{p}(r,R)}{\Delta_{p_{c}}(r,R)}\Big{|}$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\underbrace{\int_{p_{c}}^{p}\Big{(}\sum_{\ell=1}^{R}\ell\Delta_{u}(\ell)\Big{)}du}_{(A)}+\underbrace{\int_{p_{c}}^{p}\Big{(}\sum_{\ell=R}^{L(u)}\ell\Delta_{u}(\ell)\Delta_{u}(R,\ell)\Big{)}du}_{(B)}$

and we need to prove that $(A)$ and $(B)$ are bounded by constants that are
independent of $R$ and $p>p_{c}$. The bound on $(B)$ is easy to obtain since
(5.1) shows that $(B)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\phi_{p}[\mathscr{C}(\Lambda_{R})]-\phi_{p_{c}}[\mathscr{C}(\Lambda_{R})]\leq
1$. We therefore focus on $(A)$.

Choose the topological $R$-annulus $\Omega$ minimizing
$\mathbf{N}_{\Omega}(p)$ and observe that by Lemma 6.9,
$\mathbf{N}_{\Omega}(p)\asymp\mathbf{N}_{\Omega}(p_{c})$. We claim that

$\displaystyle\tfrac{d}{du}\log\mathbf{N}_{\Omega}(u)\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\sum_{\ell=1}^{R}\ell\Delta_{u}(\ell).$
(6.21)

Observe that (6.21) implies that $(A)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\log\mathbf{N}_{\Omega}(p)-\log\mathbf{N}_{\Omega}(p_{c})\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!1$ which concludes the proof of the
proposition. Thus, we only need to prove (6.21), which we do next.

Start by observing that

$\displaystyle\tfrac{d}{du}\log\mathbf{N}_{\Omega}(u)=\sum_{y\in\partial_{\rm
in}\Omega}\tfrac{d}{du}\phi_{\Omega,u}^{0}[y\longleftrightarrow\partial\Lambda_{R/2}]=\tfrac{1}{u(1-u)}\sum_{y\in\partial_{\rm
in}\Omega}\,\sum_{e\in\Omega}\mathrm{Cov}[y\longleftrightarrow\partial\Lambda_{R/2},\omega_{e}].$
(6.22)

Fix some $y\in\partial_{\rm in}\Omega$ and set $N:=\lfloor\log_{8}R\rfloor$.
Then, there exist points $z_{1},\dots,z_{N-2}$ in $\Omega$ such that for each
$k$, $\Lambda_{8^{k}}(z_{k})\subset\Omega$ but $y$ is not connected to
$\partial\Lambda_{R}$ in the subgraph $\Omega\setminus\Lambda_{2\cdot
8^{k}}(z_{k})$; the latter condition includes the situation where
$y\in\Lambda_{2\cdot 8^{k}}(z_{k})$. See Figure 14 and its caption for an
explanation of this elementary fact. For $k<\ell$, since $\Lambda_{2\cdot
8^{k}}(z_{k})$ intersects the boundary of $\Omega$, but
$\Lambda_{8^{\ell}}(z_{\ell})\subset\Omega$, we conclude that
$\Lambda_{8^{k}/2}(z_{k})$ and $\Lambda_{8^{\ell}/2}(z_{\ell})$ do not
intersect. Thus, the boxes $(\Lambda_{8^{k}/2}(z_{k}))_{k\leq N-2}$ are
pairwise disjoint.

Now, for any $e\in\Lambda_{8^{k}/2}(z_{k})$ for one such $k$, a direct
application of Theorem 4.8 implies that

$\displaystyle\mathrm{Cov}[y\longleftrightarrow\partial\Lambda_{R/2},\omega_{e}]\asymp\Delta_{u}(8^{k})\phi_{\Omega,u}^{0}[y\longleftrightarrow\partial\Lambda_{R/2}].$

Summing over $e$ and keeping in mind that all covariances in (6.22) are non-
negative gives

$\displaystyle\tfrac{d}{du}\phi_{\Omega,u}^{0}[y\longleftrightarrow\partial\Lambda_{R/2}]$
$\displaystyle\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\sum_{k=1}^{N-2}\sum_{e\in\Lambda_{8^{k}/2}(z_{k})}\mathrm{Cov}[y\longleftrightarrow\partial\Lambda_{R/2},\omega_{e}]$
$\displaystyle\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\sum_{k=1}^{N-2}8^{2k}\Delta_{u}(8^{k})\phi_{\Omega,u}^{0}[y\longleftrightarrow\partial\Lambda_{R/2}]$
$\displaystyle\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\Big{(}\sum_{\ell=1}^{R}\ell\Delta_{u}(\ell)\Big{)}\phi_{\Omega,u}^{0}[y\longleftrightarrow\partial\Lambda_{R/2}].$

For the last inequality, we used the fact that
$\Delta_{u}(\ell)\asymp\Delta_{u}(8^{k})$ for any $8^{k}\leq\ell\leq 8^{k+1}$.
Finally, summing over $y$, we find

$\displaystyle\tfrac{d}{du}\mathbf{N}_{\Omega}(u)\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\mathbf{N}_{\Omega}(u)\sum_{\ell=1}^{R}\ell\Delta_{u}(\ell),$

which concludes the proof of (6.21) and of the whole proposition. $\square$

##  7 Proofs of the stability theorems

In this section, we prove stability results for crossing probabilities, arm
events and $\Delta_{p}$. First, observe that due to the previous section,
Propositions 5.1 and 5.6 immediately lead to the following corollary.

###### Corollary 7.1

(i) Fix $\eta>0$. For every $p\in(0,1)$ and every $\eta$-regular quad
$(\mathscr{D},a,b,c,d)$ at scale $R\leq L(p)$,

$\tfrac{{\rm d}}{{\rm d}p}\phi_{p}[\mathscr{C}(\mathscr{D})]\asymp
R^{2}\Delta_{p}(R)+\sum_{\ell=R}^{L(p)}\ell\Delta_{p}(\ell)\Delta_{p}(R,\ell),$
(7.1)

where the constants in $\asymp$ depend on $\eta$.

(ii) For every $p\in(0,1)$, $\sigma\in\\{0,1\\}^{k}$ and $r\leq R\leq L(p)$,

$\tfrac{{\rm d}}{{\rm d}p}\log\phi_{p}[A_{\sigma}(r,R)]\asymp
R^{2}\Delta_{p}(R)+\sum_{\ell=R}^{L(p)}\ell\Delta_{p}(\ell)\Delta_{p}(R,\ell),$
(7.2)

where the constants in $\asymp$ depend on $\sigma$.

(iii) For every $p\in(0,1)$ and two edges $e$ and $f$ at a distance $R\leq
L(p)$ of each other,

$\big{|}\tfrac{{\rm d}}{{\rm
d}p}\log\mathrm{Cov}_{p}(\omega_{e},\omega_{f})\big{|}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!R^{2}\Delta_{p}(R)+\sum_{\ell=R}^{L(p)}\ell\Delta_{p}(\ell)\Delta_{p}(R,\ell).$
(7.3)

##### Proof

By (1.22) and (1.17), for any $p\in(0,1)$ and $R\leq L(p)$,
$\sum_{\ell=1}^{R}\ell\Delta_{p}(\ell)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!R^{2}\Delta_{p}(R)$. Insert this into
Propositions 5.1 and 5.6 to obtain the desired results. $\square$

We next prove the stability of crossing probabilities and arm events
probabilities stated in Theorem 1.4.

##### Proof of Theorem 1.4

Fix $p\neq p_{c}$. We prove (1.11); the stability for arm events can be
deduced similarly. Fix some $\eta$-regular discrete quad
$(\mathscr{D},a,b,c,d)$ at some scale $R\geq 1$. The inequality (1.11) is
trivial when $R>L(p)$, and we focus on the case where $R\leq L(p)$. Applying
(5.1) to $u$ between $p_{c}$ and $p$, we find,

$\displaystyle\tfrac{{\rm d}}{{\rm d}u}\phi_{u}[\mathscr{C}(\mathscr{D})]$
$\displaystyle\asymp
R^{2}\Delta_{u}(R)+\sum_{\ell=R}^{L(p)}\ell\Delta_{u}(\ell)\Delta_{u}(R,\ell)+\sum_{\ell=L(p)}^{L(u)}\ell\Delta_{u}(\ell)\Delta_{u}(R,\ell)$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!(\tfrac{R}{L(p)})^{\delta}\,\Big{[}L(p)^{2}\Delta_{u}(L(p))+\sum_{\ell=L(p)}^{L(u)}\ell\Delta_{u}(\ell)\Delta_{u}(L(p),\ell)\Big{]},$
(7.6)

where the inequality is due to a simple computation based on the quasi-
multiplicativity of $\Delta_{p}$ and the two inequalities from (1.22) (the
constant $\delta$ is actually given by this displayed equation). The terms for
$\ell\geq L(p)$ are dominated by the corresponding sum in the second line.

Corollary 7.1 applied to $\mathscr{C}(\Lambda_{L(p)})$ together with (7.6)
imply that

$\displaystyle\tfrac{{\rm d}}{{\rm
d}u}\phi_{u}[\mathscr{C}(\mathscr{D})]\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!(\tfrac{R}{L(p)})^{\delta}\,\tfrac{{\rm
d}}{{\rm d}u}\phi_{u}[\mathscr{C}(\Lambda_{L(p)})].$

Integrate the above between $p_{c}$ and $p$ to find

$\displaystyle|\phi_{p}[\mathscr{C}(\mathscr{D})]-\phi_{p_{c}}[\mathscr{C}(\mathscr{D})]|\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!(\tfrac{R}{L(p)})^{\delta}|\phi_{p}[\mathscr{C}(\Lambda_{L(p)})]-\phi_{p_{c}}[\mathscr{C}(\Lambda_{L(p)})]|\leq(\tfrac{R}{L(p)})^{\delta}.$

$\square$

###### Remark 7.2

One may also obtain the following improvement of (1.12) (similar to (1.11)):

$\displaystyle\exp[-C(R/L(p))^{\delta}]\leq\frac{\pi_{1}(p,R)}{\pi_{1}(p_{c},R)}\leq\exp[C(R/L(p))^{\delta}]\qquad\text{
for all\leavevmode\nobreak\ $R\leq L(p)$},$ (7.7)

where $\delta,C>0$ are universal constants depending only on $q$.

Moreover, since Corollary 7.1 applies to logarithmic derivatives of any arm
event probabilities, (7.7) also applies to $\pi_{\sigma}(p,R)$ and
$\pi_{\sigma}^{+}(p,R)$, with $C$ depending on the color sequence $\sigma$.
Notice, however, that we do not claim to prove stability for probabilities of
arm events in the half-plane with boundary conditions strictly on the half-
plane.

We now turn to the proof of the stability of $\Delta_{p}$. Let us note that
for $q=4$, the stability of $\Delta_{p}$ was proved also in Section 6.2, as a
step towards (1.22).

##### Proof of Theorem 1.6(iii).

Fix $p\neq p_{c}$, $R\leq L(p)$ and two edges $e$ and $f$ at a distance $R$ of
each other. For $u$ between $p$ and $p_{c}$, Corollary 7.1 gives

$\displaystyle\Big{|}\tfrac{{\rm d}}{{\rm
d}u}\log\mathrm{Cov}_{u}(\omega_{e},\omega_{f})\big{|}$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!R^{2}\Delta_{u}(R)+\sum_{\ell=R}^{L(u)}\ell\Delta_{u}(\ell)\Delta_{u}(R,\ell)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\tfrac{{\rm d}}{{\rm
d}u}\phi_{u}[\mathscr{C}(\Lambda_{R})].$

By integrating the above between $p_{c}$ and $p$, we find

$\frac{\mathrm{Cov}_{p}(\omega_{e},\omega_{f})}{\mathrm{Cov}_{p_{c}}(\omega_{e},\omega_{f})}\asymp
1.$ (7.8)

Apply now (5.7) to deduce that

$\Delta_{p}(R)\asymp\sqrt{\mathrm{Cov}_{p}(\omega_{e},\omega_{f})}\asymp\sqrt{\mathrm{Cov}_{p_{c}}(\omega_{e},\omega_{f})}\asymp\Delta_{p_{c}}(R).$

$\square$

###### Remark 7.3

It will come to no surprise to the reader that the same type of improved
stability as in (7.7) may be shown for the covariance. Getting the same result
for $\Delta_{p}$ itself seems more difficult as we crucially rely on an up-to-
constant relation between $\Delta_{p}$ and the covariance, and that the
derivative of $\Delta_{p}$ itself is less obvious.

##  8 Derivation of the scaling relations

This section is dedicated to proving the scaling relations (Theorems 1.9, 1.10
and 1.11). The proof of the near-critical scaling relations (Theorem 1.11) is
based on the stability below the characteristic length (Theorem 1.4 and
Theorem 1.6(iii)). With the latter results at our disposal, the proofs of the
critical and near-critical scaling relations (1.24)–(1.29) are very close to
those for Bernoulli percolation and contain no significant innovation. For
this reason, we are voluntarily quick on these proofs, trying to merely recall
the crucial ingredients.

The main novelties in this section are the independent proof of the scaling
relation involving the magnetic field (Section 8.3), and the derivation of the
scaling relation involving $\alpha$.

###  8.1 Scaling relations at criticality: proof of Theorem 1.9

In this section we work with $p=p_{c}$ and we drop it from the notation. We
will prove a stronger result, which implies (1.24) when taking
$x\in\partial\Lambda_{R}$.

###### Lemma 8.1

Fix $1\leq q\leq 4$. For every $R\geq 1$ and every $x\in\Lambda_{R}$,

$\pi_{1}(R)^{2}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\phi_{\mathbb{Z}^{2}}[0\xleftrightarrow{\Lambda_{2R}}x,0\longleftrightarrow\partial\Lambda_{R}]\leq\phi_{\mathbb{Z}^{2}}[0\longleftrightarrow
x]\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(|x|)^{2}.$ (8.1)

##### Proof

The middle inequality is obvious. For the right one, observe that if $0$ and
$x$ are connected, then $0\longleftrightarrow\partial\Lambda_{r}$ and
$x\longleftrightarrow\partial\Lambda_{r}(x)$, where $r:=\lfloor|x|/3\rfloor$.
The invariance under translations of $\phi_{\mathbb{Z}^{2}}$, the mixing
property (Mix) and (2.11) give that

$\phi_{\mathbb{Z}^{2}}[0\longleftrightarrow
x]\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(r)^{2}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(|x|)^{2}.$

We now prove the left inequality. The FKG inequality (FKG) implies that

$\displaystyle\phi_{\mathbb{Z}^{2}}[0\stackrel{{\scriptstyle\Lambda_{2R}}}{{\longleftrightarrow}}x,0\longleftrightarrow\partial\Lambda_{R}]$
$\displaystyle\geq\phi_{\mathbb{Z}^{2}}[A_{R},0\longleftrightarrow\partial\Lambda_{2R},x\longleftrightarrow\partial\Lambda_{3R}(x)]$
$\displaystyle\geq\phi_{\mathbb{Z}^{2}}[A_{R}]\pi_{1}(2R)\pi_{1}(3R)$
$\displaystyle\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(R)^{2},$

where we used (RSW’) and (2.11). $\square$

Recall that ${\sf C}$ is the cluster of $0$. Let ${\rm rad}({\sf
C}):=\max\\{r:{\sf C}\text{ intersects }\partial\Lambda_{r}\\}$ be the radius
of ${\sf C}$.

###### Lemma 8.2

For every $R\geq 1$,

$\displaystyle R^{2}\pi_{1}(R)$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\phi_{\mathbb{Z}^{2}}[|{\sf C}|\,|R\leq{\rm
rad}({\sf C})<4R]\leq\sqrt{\phi_{\mathbb{Z}^{2}}[|{\sf C}|^{2}\,|R\leq{\rm
rad}({\sf C})<4R]}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!R^{2}\pi_{1}(R).$ (8.6)

##### Proof

Fix $n\geq 1$. The inequality in the middle is the Cauchy-Schwarz inequality.
For the first one, observe that (8.1) and (RSW’) imply that

$\displaystyle\phi_{\mathbb{Z}^{2}}[|{\sf C}|\mathbbm{1}_{R\leq{\rm rad}({\sf
C})<4R}]\geq\sum_{x\in\Lambda_{R}}\phi_{\mathbb{Z}^{2}}[0\xleftrightarrow{\Lambda_{2R}}x,\Lambda_{2R}\not\longleftrightarrow\partial\Lambda_{4R}]\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!|\Lambda_{R}|\pi_{1}(R)^{2}.$

Divide the above by $\phi_{\mathbb{Z}^{2}}[R\leq{\rm rad}({\sf
C})<4R]\leq\pi_{1}(R)$ to obtain the first inequality in (8.6).

We turn to the last inequality of (8.6). We have

$\displaystyle\phi_{\mathbb{Z}^{2}}[|{\sf C}|^{2}\mathbbm{1}_{R\leq{\rm
rad}({\sf C})<4R}]$
$\displaystyle\leq\sum_{x,y\in\Lambda_{4R}}\phi_{\mathbb{Z}^{2}}[0\longleftrightarrow
x,0\longleftrightarrow y,0\longleftrightarrow\partial\Lambda_{R}].$

Fix $x,y\in\Lambda_{4R}$ and assume first that $|x|\leq|y|\leq|x-y|$. Set
$\ell:=|x|$ and $k:=|y|$. Observe that the event on the right induces the
following events which are listed along with the order – up to constants – of
their probabilities (which are obtained thanks to (2.11)):

  * •

$0\longleftrightarrow\partial\Lambda_{\ell/4}$ – of probability of order
$\pi_{1}(\ell)$;

  * •

$x\longleftrightarrow\partial\Lambda_{\ell/4}(x)$ – of probability of order
$\pi_{1}(\ell)$;

  * •

$y\longleftrightarrow\partial\Lambda_{k/4}(y)$ – of probability of order
$\pi_{1}(k)$;

  * •

$\partial\Lambda_{\min\\{2\ell,k/4\\}}\longleftrightarrow\partial\Lambda_{k}$
– of probability of order $\pi_{1}(k)/\pi_{1}(\ell)$;

  * •

$\partial\Lambda_{\min\\{2k,R\\}}\longleftrightarrow\partial\Lambda_{R}$ – of
probability of order $\pi_{1}(R)/\pi_{1}(k)$.

Several iterations of the mixing property (Mix) imply that

$\displaystyle\phi_{\mathbb{Z}^{2}}[0\longleftrightarrow
x,0\longleftrightarrow y,0\longleftrightarrow\partial\Lambda_{R}]$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(\ell)\pi_{1}(k)\pi_{1}(R).$ (8.9)

Observe now that for $1\leq\ell\leq k\leq 4R$, there are $8\ell$ vertices
$x\in\mathbb{Z}^{2}$ with $|x|=\ell$ and $8k$ vertices $y$ with $|y|=k$. Thus,

$\displaystyle\sum_{\begin{subarray}{c}x,y\in\Lambda_{4R}\\\
|x|\leq|y|\leq|x-y|\end{subarray}}\phi_{\mathbb{Z}^{2}}[0\longleftrightarrow
x,0\longleftrightarrow y,0\longleftrightarrow\partial\Lambda_{R}]$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\sum_{1\leq\ell\leq k\leq
4R}\ell\pi_{1}(\ell)k\pi_{1}(k)\pi_{1}(R)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!R^{4}\pi_{1}(R)^{3},$

where in the last line we used that

$\sum_{k=1}^{R}k\pi_{1}(k)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!R^{2}\pi_{1}(R)$ (8.10)

which is a consequence of (2.10) and (2.11). The same upper bound may be
obtained for any of the other five possible orderings of $|x|,|y|,|x-y|$.
Overall, we conclude that

$\displaystyle\phi_{\mathbb{Z}^{2}}[|{\sf C}|^{2}\mathbbm{1}_{R\leq{\rm
rad}({\sf C})<4R}]\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!R^{4}\pi_{1}(R)^{3},$

which gives the result by dividing by

$\displaystyle\phi_{\mathbb{Z}^{2}}[R\leq{\rm rad}({\sf C})<4R]$
$\displaystyle\geq\phi_{\mathbb{Z}^{2}}[0\longleftrightarrow\partial\Lambda_{R},\Lambda_{R}\not\longleftrightarrow\partial\Lambda_{2R}]\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(R),$ (8.13)

where in the last inequality we used the mixing property (Mix) and (RSW’).
$\square$

##### Proof of Theorem 1.9

Lemma 8.1 applied with $R=2|x|$ and (2.11) directly imply (1.24). We turn to
the proof of (1.25). Fix $R\geq 1$ and $r:=\varphi(R)$. Let us start with the
lower bound on $\phi_{\mathbb{Z}^{2}}[|{\sf C}|\geq R]$. Let $c\in(0,1)$ be
the constant appearing in the first bound
$\\!{\begin{array}[]{c}<\\\\[-10.0pt] \scriptstyle{\frown}\end{array}}\\!$ of
(8.6). Then, using the definition of $\varphi$, (8.6), the Paley–Zygmund
inequality and (8.13), we find that

$\displaystyle\phi_{\mathbb{Z}^{2}}[|{\sf C}|\geq\tfrac{c}{2}R]$
$\displaystyle\geq\phi_{\mathbb{Z}^{2}}[|{\sf
C}|\geq\tfrac{c}{2}r^{2}\pi_{1}(r)]$
$\displaystyle\geq\phi_{\mathbb{Z}^{2}}[|{\sf
C}|\geq\tfrac{1}{2}\phi_{\mathbb{Z}^{2}}(|{\sf C}|\,|\,r\leq{\rm rad}({\sf
C})<4r)]$ $\displaystyle\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\frac{\phi_{\mathbb{Z}^{2}}[|{\sf
C}|\,|\,r\leq{\rm rad}({\sf C})<4r]^{2}}{\phi_{\mathbb{Z}^{2}}[|{\sf
C}|^{2}\,|\,r\leq{\rm rad}({\sf C})<4r]}\,\phi_{\mathbb{Z}^{2}}[r\leq{\rm
rad}({\sf C})<4r]$ $\displaystyle\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\phi_{\mathbb{Z}^{2}}[r\leq{\rm rad}({\sf
C})<4r]$ $\displaystyle\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(r).$

This concludes the proof of the lower bound since (2.11) implies that
$\phi(\tfrac{c}{2}R)\asymp\phi(R)=r$.

We turn to the complementary upper bound. Using the Markov inequality in the
third line and the definition of $\varphi$ and (8.1) in the fourth, we obtain
that

$\displaystyle\phi_{\mathbb{Z}^{2}}[|{\sf C}|\geq R]$
$\displaystyle=\phi_{\mathbb{Z}^{2}}[|{\sf C}|\geq R,{\rm rad}({\sf
C})>r]+\phi_{\mathbb{Z}^{2}}[|{\sf C}|\geq R,{\rm rad}({\sf C})\leq r]$
$\displaystyle\leq\pi_{1}(r)+\phi_{\mathbb{Z}^{2}}[|{\sf
C}\cap\Lambda_{r}|\geq R]$
$\displaystyle\leq\pi_{1}(r)+\frac{1}{R}\sum_{u\in\Lambda_{r}}\phi_{\mathbb{Z}^{2}}[0\longleftrightarrow
u]$ $\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(r)+\frac{1}{r^{2}\pi_{1}(r)}\sum_{u\in\Lambda_{r}}\pi_{1}(|u|)^{2}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(r),$

where the last inequality follows from (2.10) via the following computation

$\frac{1}{k}\sum_{k=1}^{r}\frac{k\pi_{1}(k)^{2}}{r\pi_{1}(r)^{2}}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\frac{1}{r}\sum_{k=1}^{r}(k/r)^{1-2c}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!1.$

$\square$

###  8.2 Scaling relations in the near-critical regime: proof of Theorem 1.11

By Theorem 1.3 and (2.11), we have that $L(p)\asymp\xi(p)$ and
$\pi_{1}(L(p))\asymp\pi_{1}(\xi(p))$, so we only need to show (1.27)–(1.30)
with $L(p)$ instead of $\xi(p)$.

####  8.2.1 Proof of (1.27) (scaling relation between $\beta$, $\nu$, and
$\xi_{1}$)

On the one hand, the stability below the characteristic length (Theorem 1.4)
gives

$\displaystyle\theta(p)\leq\pi_{1}(p,L(p))\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(L(p)).$

On the other hand, the FKG inequality (FKG), (RSW’), and Corollary 2.14 imply
that

$\displaystyle\theta(p)\geq\phi_{p}[0\longleftrightarrow\partial\Lambda_{2L(p)},A_{L(p)},\Lambda_{L(p)}\longleftrightarrow\infty]\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(p,2L(p))\geq\pi_{1}(2L(p))\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(L(p)).$

####  8.2.2 Proof of (1.28) (scaling relation between $\gamma$, $\nu$ and
$\xi_{1}$)

We start with the case $p<p_{c}$. First, Proposition 2.12 can be improved to

$c\pi_{1}(R)^{2}\exp[-C|x|/L(p)]\leq\phi_{p}[0\longleftrightarrow x]\leq
C\pi_{1}(R)^{2}\exp[-c|x|/L(p)],$ (8.14)

where $R:=\min\\{|x|,L(p)\\}$ and $c,C>0$ are uniform constants. Indeed, one
simply needs to combine the proofs of Proposition 2.12 and Lemma 8.1 in a
standard fashion.

Now, (1.28) follows by summing (8.14) over every $x\in\mathbb{Z}^{2}$ and
using that, by (2.10),

$\pi_{1}(R)\leq C(L(p)/R)^{1/2-c}\pi_{1}(L(p)).$ (8.15)

We now turn to the case $p>p_{c}$.

The only additional difficulty comes from the fact that we need to force 0 and
$x$ not to be connected to infinity. For the lower bound, sum over every
$x\in\Lambda_{L(p)/2}$ the following inequality

$\phi_{p}[0\longleftrightarrow
x,0\not\longleftrightarrow\infty]\geq\phi_{p}[\Lambda_{L(p)}\not\longleftrightarrow\infty]\phi_{\Lambda_{L(p)},p}^{0}[0\longleftrightarrow
x]\geq c\pi_{1}(R)^{2},$

where the first inequality is due to the spatial Markov property (SMP), and
the second inequality follows (RSW), ($p$-MON) and an argument similar to
Lemma 8.1.

For the upper bound, let $A_{x}^{*}$ be the event that there exists a circuit
in $\omega^{*}$ surrounding 0 and $x$. The FKG inequality in the first
inequality, Corollary 2.14 and (CBC) in the second, and a reasoning similar to
Lemma 8.1 in the third (based on (Mix) and Theorem 1.4) imply that

$\displaystyle\phi_{p}[0\longleftrightarrow x,0\not\longleftrightarrow\infty]$
$\displaystyle\leq\phi_{p}[0\longleftrightarrow x]\phi_{p}[A_{x}^{*}]$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\phi_{\Lambda_{R},p}^{1}[0\longleftrightarrow\partial\Lambda_{R/2}]^{2}\exp(-c|x|/L)$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(R)^{2}\exp(-c|x|/L).$

Summing this inequality over $x\in\mathbb{Z}^{2}$ gives the upper bound.

####  8.2.3 Proof of (1.29) (scaling relation between $\iota$ and $\nu$)

Assume $p>p_{c}$, the case $p<p_{c}$ is identical. Write $L=L(p)$. Use
Corollary 7.1 and integrate the derivative of
$g(p):=\phi_{p}[\mathscr{C}(\Lambda_{L})]$ between $p_{c}$ and $p$ to get

$\displaystyle\int_{p_{c}}^{p}L^{2}\Delta_{u}(L)du\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!g(p)-g(p_{c})\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!1.$ (8.20)

In the other direction, let $p_{0}\in[p_{c},p]$ be such that $L(p_{0})=RL(p)$
for some $R>1$. Theorem 1.4 and the definition of $L(p)$ implies that

$g(p)-g(p_{0})\geq
g(p)-g(p_{c})-CR^{-\varepsilon}\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!1$ (8.21)

provided that $R$ is large enough. For $u\in[p_{0},p]$, Corollary 7.1,
together with the quasi-multiplicativity property Theorem 1.6(ii), implies

$\displaystyle g^{\prime}(u)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!L(u)^{2}\Delta_{u}(L(u))\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!L^{2}\Delta_{u}(L).$

(Note that the constant in $\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!$ depends on $R$.) Integrating the previous
displayed equation between $p_{0}$ and $p$ and then using (8.21) gives

$\displaystyle 1\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!g(p)-g(p_{0})\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\int_{p_{0}}^{p}L^{2}\Delta_{u}(L)du\leq\int_{p_{c}}^{p}L^{2}\Delta_{u}(L)du.$

Together with (8.20), the previous displayed equation and the stability of
$\Delta_{u}(L)$ given by Theorem 1.6(iii) show that

$\displaystyle
1\asymp\int_{p_{c}}^{p}L^{2}\Delta_{u}(L)du\asymp(p-p_{c})L^{2}\Delta_{p_{c}}(L),$

which concludes the proof.

####  8.2.4 Proof of (1.30) (scaling relation between $\iota$ and $\alpha$)

A straightforward computation involving (1.13) shows that

$\displaystyle f^{\prime\prime}(p)=2\frac{{\rm d}}{{\rm
d}p}\phi_{p}[\omega_{e}]=2\sum_{f}\mathrm{Cov}_{p}(\omega_{e},\omega_{f}),$

where $e$ is a fixed edge of $\mathbb{Z}^{2}$ and the sum is over all edges
$f$. By Lemma 5.3 applied to $\mathscr{D}$ formed of the single edge $e$, we
find

$\mathrm{Cov}_{p}(\omega_{e};\omega_{f})\asymp\Delta_{p}(\ell\wedge
L(p))^{2}e^{-c\ell/L(p)}\asymp\Delta_{p_{c}}(\ell\wedge
L(p))^{2}e^{-c\ell/L(p)},$

where $\ell$ is the distance between $e$ and $f$. For the second equivalence,
we use Theorem 1.6(iii). Summing the displayed equation above over all edges
$f$, we conclude that

$\displaystyle
f^{\prime\prime}(p)\asymp\sum_{\ell=1}^{L(p)}\ell\Delta_{p_{c}}(\ell)^{2},$

as required.

###### Remark 8.3

The above shows that if the phase transition is of second order (meaning that
$f^{\prime\prime}(p)$ diverges as $p$ tends to $p_{c}$), then
$\sum_{\ell}\ell\Delta_{p}(\ell)^{2}$ diverges, which, using the
interpretation of crossing probabilities in terms of $\Delta_{p}(\ell)$,
implies that the crossing probabilities of quads for the infinite-volume
measure are not differentiable at $p_{c}$.

###### Remark 8.4

When $\sum_{\ell\geq 1}\ell\Delta(\ell)^{2}$ converges, the computation above
simply proves that $f^{\prime\prime}(p)$ remains bounded uniformly in $p$.
Nevertheless, it is easy to deduce by differentiating one more time that for
$p\neq p_{c}$ (we drop $p$ from the notation),

$\displaystyle f^{\prime\prime\prime}(p)$
$\displaystyle\leq\sum_{f,g}\phi[\omega_{e}\omega_{f}\omega_{g}]-\phi[\omega_{e}\omega_{f}]\phi[\omega_{g}]-\phi[\omega_{f}\omega_{g}]\phi[\omega_{e}]-\phi[\omega_{e}\omega_{g}]\phi[\omega_{f}]+2\phi[\omega_{e}]\phi[\omega_{f}]\phi[\omega_{g}]$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\sum_{\ell\leq
L(p)}\ell\Delta(\ell)\Delta(\ell^{\prime},\ell)\sum_{\ell^{\prime}\leq\ell}\ell^{\prime}\Delta(\ell^{\prime})^{2}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!L(p)^{4}\Delta(L(p))^{3}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\frac{1}{|p-p_{c}|^{3}L(p)^{2}}.$

If one defines $\alpha$ in this framework by the formula
$f^{\prime\prime\prime}(p)=|p-p_{c}|^{-\alpha-1-o(1)}$, we deduce that
$\alpha\leq 2-2\nu$. The converse bound does not follow by the same
computation since the summand on the first line is not of definite sign; at
the time of writing the matching lower bound on $f^{\prime\prime\prime}(p)$
remains unproved.

###  8.3 Scaling relation with magnetic field: proof of Theorem 1.10

We insist one last time on the fact that this section is independent of the
rest of the paper. Below, we work with the graph $\mathbb{Z}^{2}$ with the
addition of the ghost. We drop $p_{c}$ and $q$ but keep $h$ in the notation
except when it is equal to $0$, in which case we omit it as well. We start by
a lemma relating certain quantities at $h=0$ with the corresponding quantities
at $h\geq 0$.

Set
$\tilde{\pi}_{1}(h,R):=\phi_{\Lambda_{R},h}^{1}[0\leftrightarrow\partial\Lambda_{R}]$,
where connections need to occur in $\mathbb{Z}^{2}$. Additionally, let
$A_{R}^{*}$ be the event that there exists a dual circuit in ${\rm Ann}(R,2R)$
surrounding $\Lambda_{R}$ and

$\displaystyle\beta(h,R):=\phi_{{\rm Ann}(R,2R),h}^{1}[A_{R}^{*}].$

Finally, for $C>1$, define

$\displaystyle h_{c}(R)=h_{c}(R,C):=\inf\big{\\{}h>0:\,\exists\ r\leq R\text{
such that }\tilde{\pi}_{1}(h,r)>C\tilde{\pi}_{1}(r)\text{ or
}\beta(h,r)<C^{-1}\beta(r)\big{\\}}.$

###### Lemma 8.5

For any $C>1$, there exists $\varepsilon>0$ such that for every $R$,

$\displaystyle h_{c}(R)R^{2}\pi_{1}(R)\geq\varepsilon.$ (8.22)

Due to the definition of $h_{c}$, crossing estimates as in Theorem 2.1 apply
in the regime of $(r,h)$ with $r\leq R$ and $h\leq h_{c}(R)$. Indeed, the
crossing probabilities in the primal model increase with $h$, which ensures
the lower bounds. For the upper bounds, observe that $\beta(h,r)$ involves the
boundary conditions that render dual crossings least likely. Theorem 2.1
applied at $p_{c}$ combined with the definition of $h_{c}(R)$ implies the
uniform positivity of $\beta(r,h)$ for $r\leq R$ and $h\leq h_{c}(R)$. The FKG
inequality and the monotonicity of boundary conditions imply lower bounds for
crossing probabilities in the dual model, as claimed.

As a consequence, a similar proof to that of Lemma 8.1 applies for all $r\leq
R$ and $0\leq h\leq h_{c}(R)$, and yields

$\displaystyle\phi_{{\rm
Ann}(r,2r),h}^{1}[y\longleftrightarrow\partial\Lambda_{2r}]$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}({\rm dist}(y,\partial{\rm
Ann}(r,2r)))$ for $y\in{\rm Ann}(r,2r)$ and (8.25)
$\displaystyle\phi_{\Lambda_{r},h}^{1}[0\longleftrightarrow
y,0\longleftrightarrow\partial\Lambda_{r}]$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(r)\pi_{1}(\|y\|\wedge{\rm
dist}(y,\partial\Lambda_{n}))$ $\displaystyle\text{ for \leavevmode\nobreak\
$y\in\Lambda_{r}$},$ (8.28)

where the constants in $\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!$ depend on $C$.

##### Proof of Lemma 8.5

Fix $C>1$ and $r\leq R$. All constants in the signs
$\\!{\begin{array}[]{c}<\\\\[-10.0pt] \scriptstyle{\frown}\end{array}}\\!$
below are allowed to depend on $C$, but not on $r$ or $R$. The differential
formula [Gri06, Thm. 3.12] reads

$\displaystyle\tfrac{d}{dh}\tilde{\pi}_{1}(h,r)=\tfrac{1}{1-e^{-h}}\sum_{y\in\Lambda_{r}}\phi_{\Lambda_{r},h}^{1}[0\leftrightarrow\partial\Lambda_{r},\omega_{y\mathfrak{g}}=1]-\phi_{\Lambda_{r},h}^{1}[0\leftrightarrow\partial\Lambda_{r}]\phi_{\Lambda_{r},h}^{1}[\omega_{y\mathfrak{g}}=1].$
(8.29)

Let us analyse the right-hand side of the above. Recall that ${\sf C}$ is the
cluster of the origin for the connectivity in $\mathbb{Z}^{2}$. First we show
that the vertices $y\notin{\sf C}$ have a negative contribution to (8.29). Fix
some $y\in\Lambda_{r}$. For a set $\mathscr{C}\subset\Lambda_{r}$ containing
$0$ and $y\notin\mathscr{C}$,

$\displaystyle\phi_{\Lambda_{r},h}^{1}[0\leftrightarrow\partial\Lambda_{r},\omega_{y\mathfrak{g}}=1,{\sf
C}=\mathscr{C}]$
$\displaystyle=\phi_{\Lambda_{r},h}^{1}[\phi_{\Lambda_{r},h}^{1}(\omega_{y\mathfrak{g}}=1\,|\,{\sf
C}=\mathscr{C})\,\mathbbm{1}_{0\leftrightarrow\partial\Lambda_{r}}\mathbbm{1}_{{\sf
C}=\mathscr{C}}]$
$\displaystyle\leq\phi_{\Lambda_{r},h}^{1}[\omega_{y\mathfrak{g}}=1]\phi_{\Lambda_{r},h}^{1}[0\leftrightarrow\partial\Lambda_{r},{\sf
C}=\mathscr{C}].$

The first inequality is due (SMP) and (CBC) since
$\phi_{\Lambda_{r},h}^{1}[\cdot\,|\,{\sf C}=\mathscr{C}]$ is a random-cluster
measure with free boundary conditions on
$(\Lambda_{r}\setminus\mathscr{C})\cup\\{\mathfrak{g}\\}$, and is
stochastically dominated by the restriction of $\phi_{\Lambda_{r},h}^{1}$ to
$(\Lambda_{r}\setminus\mathscr{C})\cup\\{\mathfrak{g}\\}$.

Summing the above over every $\mathscr{C}\subset\Lambda_{r}$, we find that

$\displaystyle\phi_{\Lambda_{r},h}^{1}[0\leftrightarrow\partial\Lambda_{r},\omega_{y\mathfrak{g}}=1]-\phi_{\Lambda_{r},h}^{1}[\omega_{y\mathfrak{g}}=1]\phi_{\Lambda_{r},h}^{1}[0\leftrightarrow\partial\Lambda_{r}]\leq\phi_{\Lambda_{r},h}^{1}[0\leftrightarrow
y,0\leftrightarrow\partial\Lambda_{r},\omega_{y\mathfrak{g}}=1].$

Plugging this inequality in (8.29) gives

$\displaystyle\tfrac{d}{dh}\tilde{\pi}_{1}(h,r)\leq\tfrac{1}{1-e^{-h}}\sum_{y\in\Lambda_{r}}\phi_{\Lambda_{r},h}^{1}[0\leftrightarrow
y,0\leftrightarrow\partial\Lambda_{r},\omega_{y\mathfrak{g}}=1]\leq\sum_{y\in\Lambda_{r}}\phi_{\Lambda_{r},h}^{1}[0\leftrightarrow
y,0\leftrightarrow\partial\Lambda_{r}],$

where the second inequality comes from (SMP), which implies that

$\phi_{\Lambda_{r},h}^{1}[\omega_{v\mathfrak{g}}=1\,|0\leftrightarrow
y,0\leftrightarrow\partial\Lambda_{r}]\leq 1-e^{-h}.$

Now, assuming that $h\leq h_{c}(R)$, (8.28) applies, and we conclude that

$\displaystyle\tfrac{d}{dh}\tilde{\pi}_{1}(h,r)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\sum_{y\in\Lambda_{r}}\pi_{1}(r)\pi_{1}(\|y\|\wedge{\rm
dist}(y,\partial\Lambda_{r}))\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!r\pi_{1}(r)\sum_{k=1}^{r/2}\pi_{1}(k)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!r^{2}\pi_{1}(r)^{2}.$

The second inequality uses the fact that there are at most order $r$ vertices
at distance $k$ from $0$ or $\partial\Lambda_{r}$; the third one is a standard
consequence of (2.10) and (2.11). Keeping in mind that
$\pi_{1}(r)\leq\tilde{\pi}_{1}(h,r)$, the above implies

$\displaystyle\tfrac{d}{dh}\log\tilde{\pi}_{1}(h,r)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!r^{2}\pi_{1}(r).$ (8.32)

A similar computation, where ${\sf C}$ is replaced by the cluster of
$\partial\Lambda_{2r}$ implies that

$\displaystyle-\tfrac{d}{dh}\log\beta(h,r)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\sum_{y\in{\rm Ann}(r,2r)}\phi_{{\rm
Ann}(r,2r),h}^{1}[y\leftrightarrow\partial\Lambda_{2r}]\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!r^{2}\pi_{1}(r).$ (8.37)

We are now in a position to conclude. Let $c_{0}$ be a constant larger than
the constants involved in the inequalities
$\\!{\begin{array}[]{c}<\\\\[-10.0pt] \scriptstyle{\frown}\end{array}}\\!$ in
(8.32) and (8.37). Then, for $\varepsilon>0$, integrate these two inequalities
for $h$ between $0$ and
$h^{\prime}=\min\\{h_{c}(R),\varepsilon/(R^{2}\pi_{1}(R))\\}$. We find

$\displaystyle\left.\begin{array}[]{r}\log\tilde{\pi}_{1}(h,r)-\log\tilde{\pi}_{1}(r)\\\
\log\beta(r)-\log\beta(h,r)\end{array}\right\\}\leq
c_{0}h^{\prime}r^{2}\pi_{1}(r)\leq\varepsilon c_{0}.$

Now, for $\varepsilon<(\log C)/c_{0}$, the above shows that
$h^{\prime}<h_{c}(R)$, which implies (8.22). $\square$

##### Proof of Theorem 1.10

Fix $h>0$. Again, $\mathsf{C}$ is the cluster of the origin when considering
connections in $\mathbb{Z}^{2}$ only. We start with the lower bound. We have

$\displaystyle\phi_{h}[0\longleftrightarrow\mathfrak{g}]$
$\displaystyle\geq\phi_{h}[0\longleftrightarrow\mathfrak{g},|{\sf C}|\geq
1/h]\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\phi_{h}[|{\sf C}|\geq
1/h]\geq\phi_{0}[|{\sf C}|\geq 1/h]\geq\pi_{1}(\varphi(1/h)),$

where the fourth inequality is due to (1.25), the third to monotonicity in $h$
($h$-MON), and the second to the fact that conditioned on $|{\sf C}|\geq 1/h$,
there is a positive probability for $0$ to be connected to $\mathfrak{g}$.
This last property can be easily deduced from the finite energy property,
which states that every edge connecting $\mathbb{Z}$ to $\mathfrak{g}$ has a
probability larger than $(1-e^{-h})/(1+(q-1)e^{-h})$ and smaller than $h$ of
being open, regardless of the states of other edges.

Let us now derive the upper bound. Let $\varepsilon$ be the quantity given by
Lemma 8.5 for some fixed $C>1$. Choose $R$ to be the largest integer such that
$hR^{2}\pi_{1}(R)\leq\varepsilon$. Notice that this implies that $h\leq
h_{c}(R)$. We deduce that

$\displaystyle\phi_{h}[0\longleftrightarrow\mathfrak{g}]$
$\displaystyle\leq\pi_{1}(h,R)+\phi_{h}[0\longleftrightarrow\mathfrak{g},\,0\not\longleftrightarrow\partial\Lambda_{R}]$
$\displaystyle\leq\pi_{1}(h,R)+\sum_{y\in\Lambda_{R}}\phi_{h}[0\longleftrightarrow
y,\,\omega_{y\mathfrak{g}}=1]$
$\displaystyle\leq\pi_{1}(h,R)+h\sum_{y\in\Lambda_{R}}\phi_{h}[0\longleftrightarrow
y],$

where the last inequality uses the finite energy property. Now, use that
$h\leq h_{c}(R)$ to deduce that
$\pi_{1}(h,R)\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(R)$ and that, similarly to Lemma
8.1, $\phi_{h}[0\longleftrightarrow y]\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(|y|)^{2}$. We deduce from the above
that

$\displaystyle\phi_{h}[0\longleftrightarrow\mathfrak{g}]$
$\displaystyle\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(R)+h\sum_{y\in\Lambda_{R}}\pi_{1}(|y|)^{2}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(R)+hR^{2}\pi_{1}(R)^{2}\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(R),$

where in the second inequality we used a computation similar to (1.28).
Observe that, with the notation of Theorem 1.9,
$R=\varphi(\varepsilon/h)\\!{\begin{array}[]{c}>\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\varphi(1/h)$, so

$\phi_{h}[0\longleftrightarrow\mathfrak{g}]\\!{\begin{array}[]{c}<\\\\[-10.0pt]
\scriptstyle{\frown}\end{array}}\\!\pi_{1}(\varphi(1/h)),\leavevmode\nobreak\
$

which concludes the proof. $\square$

##### Acknowledgments

The first author is supported by the ERC CriBLaM, the NCCR SwissMAP, the Swiss
NSF and an IDEX Chair from Paris-Saclay. The second author is supported by the
NCCR SwissMAP and the Swiss NSF. We thank Matan Harel for useful discussions
at an early stage of the project, as well as Jeremy Dubout and Robin Pierre
Kaufmann who worked on parts of Sections 2.4 and 3 during their respective
Master thesis with the two authors. Finally, we are extremely thankful to
Vincent Tassion, whose discussions on this project have been very inspiring
and absolutely crucial.

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# Reply to Comments on the article “M. O. Katanaev, Complete separation of
variables in the geodesic Hamilton–Jacobi equation in four dimensions, Physica
Scripta (2023), 98, 104001” by V. V. Obukhov, K. E. Osetrin, and A. V.
Shapovalov

M. O. Katanaev

Steklov Mathematical Institute,
119991, Moscow, ul. Gubkina, 8
and
Lobachevsky Institute of Mathematics and Mechanics,
Kazan Federal University, Kazan, Russia E-mail<EMAIL_ADDRESS>
It is good that the authors of the Comments did not find a mistake in my
papers [1, 2]. The main criticism of the Comments is that my papers “do not
contain any new results” as compared to the paper [3]. I knew this paper when
I wrote mine and even cited it in [1].

Let me comment the article [3]. V. Shapovalov formulated the main result as:

“Theorem 1. In a Stäckel space, and only in such, there exists a (priviledged)
coordinate system $u$ wherein the metric tensor has the form

$\begin{split}&\overset{\circ}{g}{}^{ir}=f^{-1}_{\sigma\nu}(u)h^{ir}_{\nu}\left(\overset{\nu}{u}\right),\\\
h^{ir}_{\nu}\equiv\delta^{i}_{i_{\nu}}\delta^{r}_{r_{\nu}}h^{i_{\nu}r_{\nu}}&+\big{(}\delta^{i}_{i_{0}}\delta^{r}_{r_{\nu}}+\delta^{i}_{r_{\nu}}\delta^{r}_{i_{0}}\big{)}h^{i_{0}r_{\nu}}+\delta^{i}_{i_{0}}\delta^{r}_{r_{0}}h^{i_{0}r_{0}}_{\nu}.\qquad\text{''}\end{split}$

The matrix $f$ in this “theorem” is described but there is no definition of
$h^{i_{\nu}r_{\nu}}$, $h^{i_{0}r_{\nu}}$, and $h^{i_{0}r_{0}}_{\nu}$ in the
entire paper. Therefore it is not a theorem, because there is no statement.
Moreover, on page 793, V. Shapovalov wrote: ”Since Theorem 1 has no bearing on
this question and since its proof is very cumbersome, we omit it”. As far as I
know, the “proof” was never published.

Therefore I cannot criticize the “result” of V. Shapovalov, because there is
neither theorem nor proof. In addition, the paper does not contain any
example. It is clear that “Theorem 1” by V. Shapovalov is not related to
theorems proved in [1]. For example, the Stäckel matrix, which is crucial for
the whole construction, denoted by $f$ in [3], depends only on coordinates.
The analog of this matrix denoted by $b$ in Eq.(94) [1] depends also
explicitly and nontrivially on parameters entering the complete integrals of
the Hamilton–Jacobi equation. The difference is evident.

In papers [1, 2], I listed ten classes of separable metrics in four
dimensions. As far as I know, this is a new result. This list is not mentioned
in any of the papers cited in the Comments. In particular, the authors of the
Comments wrote: “These theorems have been proved half a century ago by
Vladimir N. Shapovalov. This also applies to the complete list [3, 4] of
metrics, integrals of motion and complete integrals of the free
Hamilton–Jacobi equations in the privileged (separable) coordinate systems for
an arbitrary 4-dimensional Stäckel space $V_{4}$ given in the article [2]”. It
is easily checked that there are no lists of metrics, integrals of motion and
complete integrals in [3, 4].

Moreover, V. V. Obukhov and K. E. Osetrin published two books on the subject
in Russian [5, 6]. On page 22 of [6], it is written: “It can be easily shown
that there are only 7 types of Stäckel spaces of Lorentz signature $(+,-,-,-)$
– three isotropic: (3.1), (2.1), and (1.1) and four nonisotropic: (3.0),
(2.0), (1,0), and (0.0)” (the translation is mine). Afterwards, the separable
metrics are listed. We see that several separable metrics are missed. This
statement shows clearly that the list of separable metrics in four dimensions
was not known to the authors of the Comments at least in 2007.

Acknowledgments

This paper has been supported by the Kazan Federal University Strategic
Academic Leadership Program.

Many thanks to V. V. Obukhov, K. E. Osetrin, and A. V. Shapovalov for the
interest in my papers.

## References

  * [1] M. O. Katanaev.  Complete separation of variables in the geodesic Hamilton-Jacobi equation.  arXiv:2305.02222 [gr-qc], 2023.
  * [2] M. O. Katanaev.  Complete separation of variables in the geodesic Hamilton-Jacobi equation in four dimensions.  Phys. Scr., 98(10):104001, 2023.  DOI 10.1088/1402-4896/acf251.
  * [3] V. N. Shapovalov.  Stäckel spaces.  Siberian Math. J., 20(5):790–800, 1980.
  * [4] V. N. Shapovalov.  Symmetry and separation of variables in Hamilton–Jacobi equations. I.  Sov. Phys. J., 21:1124–1129, 1978.
  * [5] V. V. Obukhov.  Stäckel spaces in gravity theory.  Tomsk State Pedagogical U., Tomsk, 2006.  [In Russian].
  * [6] V. V. Obukhov and K. E. Osetrin.  Classification problems in gravity theory.  Tomsk State Pedagogical U., Tomsk, 2007.  [In Russian].