$[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$ but not
$\\{z,v_{1},v_{2},\cdots,v_{r}\\}$, and is of distance at least $\eta/2K$ from
$z$. Here, we have used the fact (which follows from (6.73)) that
$\big{|}\partial_{w}\mathcal{A}^{(\bm{v},\bm{s})}(w)/\mathcal{A}^{(\bm{v},\bm{s})}(w)\big{|}$
tends to $0$ as $w$ tends to $\infty$. Along the contour $\omega$,
$\big{|}{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{|}\lesssim 1/n\eta$, so we
conclude that

(6.77)
$\displaystyle\frac{\partial_{z}{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)}=\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\frac{\log{\mathcal{B}}(w){\rm
d}w}{(w-z)^{2}}+\operatorname{\mathcal{O}}\left(\frac{1}{n\eta^{2}}\right).$

This finishes the proof of (6.34).

By first integrating (6.76) from $z_{2}$ to $z_{1}$, and then applying (6.33),
we get

(6.78)
$\displaystyle\begin{split}\phantom{{}={}}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(z_{1})-\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(z_{2})&=\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\Big{(}\log{\mathcal{B}}(w)+\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}\Big{)}\left(\frac{1}{w-z_{1}}-\frac{1}{w-z_{2}}\right){\rm
d}w\\\
&=b(z_{1})-b(z_{2})+\operatorname{\mathcal{O}}\left(\frac{1}{n\eta}\right).\end{split}$

We can also take $z_{2}=\infty$ in (6.78) to obtain

(6.79)
$\displaystyle\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)-\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(\infty)=b(z)+\operatorname{\mathcal{O}}\left(\frac{1}{n\eta}\right).$

Thanks to the relation (6.79), (6.36) implies (6.35). Thus we only need to
prove (6.36). The relation (6.79) is almost the same as (6.45), but with an
error depending on $\eta$. The same induction argument after (6.45)
establishes (6.36), and 6.9.

∎

##  7\. Improved Edge Estimates

In this section, we strengthen the weak estimate 6.7 through bootstrapping,
and conclude the proof of Propositions 5.5 and 6.4. We collect some notations,
and miscellaneous derivative bounds in Section 7.1. We prove the first
statement (6.14) of 6.4 in Section 7.2, and the second statement (6.15) in
Section 7.3. Finally, in Section 7.4, we prove 5.5 using 6.4 as input.

###  7.1. Notations and Preliminary Estimates

In this section, we collect some notations, and miscellaneous derivative
bounds, which will be used to prove 6.4 in Sections 7.2 and 7.3.

To prove (6.14), we take $r=1$, $s_{1}=-1$, and $v_{1}=v$ in (6.27),

(7.1)
$\displaystyle\begin{split}\partial_{z}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)-b(z)\big{)}&=\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}\frac{{\rm
d}w}{(w-z)^{2}},\end{split}$

where the contour $\omega\subseteq\mathscr{D}$ encloses
$[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$ but not $z$,
$\overline{z}$, or $v$. We recall from (6.24) that (6.14) is equivalent to

(7.2)
$\displaystyle\bigg{|}\partial_{z}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)-b(z)\big{)}\Big{|}_{z=v}\bigg{|}\lesssim\frac{\big{|}\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]\big{|}}{n\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)}.$

By the definition (6.27), $b(z)=\operatorname{\mathcal{O}}(1/z)$ as
$z\rightarrow\infty$. Thus, by integrating (7.1) from $z$ to $\infty$, we get

(7.3)
$\displaystyle\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)-\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(\infty)-b(z)$
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}\frac{{\rm
d}w}{w-z}.$

By taking a contour integral on both sides of (7.3), we get for any
$k_{1}+k_{2}\geqslant 1$ that

(7.4)
$\displaystyle\begin{split}\frac{1}{2\pi\mathrm{i}}\oint_{\omega^{\prime}}\frac{\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}&=\frac{1}{(2\pi\mathrm{i})^{2}}\oint_{w\in\omega}\oint_{w^{\prime}\in\omega^{\prime}}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w{\rm
d}w^{\prime}}{(w-w^{\prime})(w^{\prime}-z)^{k_{1}}(w^{\prime}-\overline{z})^{k_{2}}}\\\
&=-\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}},\end{split}$

where the contour $\omega^{\prime}\subseteq\mathscr{D}$ encloses $\omega$ but
not $z,\overline{z}$, or $v,\overline{v}$. In the following, we analyze (7.4),
which is more general than (7.1). In particular, by taking $k_{1}=2$ and
$k_{2}=0$ in (7.4), we recover (7.1), where we are using the facts that
$\log\mathcal{A}^{(\bm{v},\bm{s})}(\infty)$ is bounded (in a way dependent on
the $v_{j}$, by (6.32)); that $b(\infty)=0$; and that
$\log\mathcal{A}^{(\bm{v},\bm{s})}(z)-b(z)$ has no poles outside of
$\omega^{\prime}$.

We recall the spectral region
$\mathscr{D}({\mathfrak{r}})=\big{\\{}z\in\mathscr{D}:\operatorname{dist}(z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}])\geqslant{\mathfrak{r}}\big{\\}}$.
Let $v\in\mathscr{D}\setminus\mathscr{D}({\mathfrak{r}})$. We will later take
either $z\in\\{v,\overline{v}\\}$, or
$z\in\mathscr{D}\cap\mathscr{D}(\mathfrak{r})$ in (7.1). We divide them into
three cases. If $z\in\\{v,\overline{v}\\}$, then we deform the contour
$\omega$ in (7.4) to see that its right side equals

(7.5)
$\displaystyle\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{v,\overline{v}}}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}-\frac{1}{2\pi\mathrm{i}}\oint_{\omega+}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}.$

If $z\in\mathscr{D}({\mathfrak{r}})\setminus\mathscr{D}(2{\mathfrak{r}})$ we
deform the contour $\omega$ in (7.4) to

(7.6)
$\displaystyle\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{v,\overline{v}}}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}+\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{z,\overline{z}}}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}-\frac{1}{2\pi\mathrm{i}}\oint_{\omega+}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}.$

If $z\in\mathscr{D}(2\mathfrak{r})$, we deform the contour $\omega$ in (7.4)
to

(7.7)
$\displaystyle\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{v,\overline{v}}}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}-\frac{1}{2\pi\mathrm{i}}\oint_{\widehat{\omega}+}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}.$

Here, in all three cases, the contour $\omega_{v,\overline{v}}$ encloses $v$
and $\overline{v}$; the contour $\omega_{z,\overline{z}}$ encloses $z$ and
$\overline{z}$; the contour
$\omega+\subseteq\mathscr{D}\cap\mathscr{D}(2{\mathfrak{r}})$ encloses
$[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$ and
$v,\overline{v},z,\overline{z}$; and the contour
$\widehat{\omega}+\subseteq\mathscr{D}({\mathfrak{r}})\setminus\mathscr{D}(2{\mathfrak{r}})$
encloses $[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$ and
$v,\overline{v}$, but not $z,\overline{z}$. See Figure 13.

$v$$\bar{v}$$\omega_{v,\bar{v}}$$\omega_{v,\bar{v}}$$\widehat{\omega}+$$v$$\bar{v}$$\omega_{v,\bar{v}}$$\omega_{v,\bar{v}}$$\omega+$$z$$\bar{z}$$\omega_{z,\bar{z}}$$\omega_{z,\bar{z}}$
Figure 13.  Shown to the left is the case $z\in\mathscr{D}(2{\mathfrak{r}})$,
where the contour $\omega_{v,\overline{v}}$ consists of two small circles
enclosing $v,\overline{v}$ respectively, and the contour
$\widehat{\omega}+\subseteq\mathscr{D}({\mathfrak{r}})\setminus\mathscr{D}(2{\mathfrak{r}})$
encloses $[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$ and
$v,\overline{v}$ but not $z$. Shown to the right is the case
$z\in\mathscr{D}({\mathfrak{r}})\setminus\mathscr{D}(2{\mathfrak{r}})$ or
$z=v,\bar{v}$, where the contour $\omega_{v,\overline{v}}$ encloses
$v,\overline{v}$, the contour $\omega_{z,\overline{z}}$ encloses
$z,\overline{z}$, and the contour
$\omega+\subseteq\mathscr{D}(2{\mathfrak{r}})$ encloses
$[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$ and
$v,\overline{v},z,\overline{z}$.

We recall ${\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(z)$ from (6.26),

$\displaystyle\begin{split}&{\mathcal{E}}^{({\bm{v}},\bm{s})}(z)={\mathcal{E}}^{({\bm{v}},\bm{s})}_{0}(z)+\frac{\psi^{({\bm{v}},\bm{s})}(z)-1}{f(z)+1},\\\
&{\mathcal{E}}^{({\bm{v}},\bm{s})}_{0}(z)=\left(\frac{f(z+1/n)}{f(z+1/n)+1}\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(z+1/n){\mathcal{B}}(z+1/n)}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(z){\mathcal{B}}(z)}\frac{\tilde{g}(z)}{\tilde{g}(z+1/n)}e^{\frac{1}{n}\psi(z)}-\frac{f(z)}{f(z)+1}\right),\end{split}$

and $b(z)$ from (6.10),

$\displaystyle
b(z)=\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\frac{\log{\mathcal{B}}(w){\rm
d}w}{w-z}=-\log{\mathcal{B}}(z)+\frac{1}{2\pi\mathrm{i}}\oint_{\omega+}\frac{\log{\mathcal{B}}(w){\rm
d}w}{w-z},$

where the new contour
$\omega+\subseteq\mathscr{D}\cap\mathscr{D}({\mathfrak{r}})$ encloses both
$[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$ and $z$. We introduce

$\displaystyle\hat{g}(z)=\tilde{g}(z)\exp\Bigg{(}-{\frac{1}{2\pi\mathrm{i}}\oint_{\omega+}\frac{\log{\mathcal{B}}(w){\rm
d}w}{w-z}}\Bigg{)},$

where the contour $\omega+\subseteq\mathscr{D}\cap\mathscr{D}({\mathfrak{r}})$
encloses both $[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$ and $z$.
Similar to $\tilde{g}(z)$ (by 6.1), the function $\hat{g}(z)$ is analytic in
$\Lambda$ and does not have zeros or poles in $\Lambda$. Moreover, we have

(7.8)
$\displaystyle\big{|}\partial_{z}\log\hat{g}(z)\big{|}\leqslant\big{|}\partial_{z}\log\tilde{g}(z)\big{|}+\frac{1}{2\pi}\left|\oint_{\omega+}\frac{\log{\mathcal{B}}(w)}{(w-z)^{2}{\rm
d}w}\right|\lesssim
1,\quad\left|\frac{\hat{g}(z)}{\hat{g}(z+1/n)}-1\right|\lesssim\frac{1}{n},$

by taking the contour $\omega+$ to be bounded away from $z$, together with the
fact that $\mathcal{B}(w)=\big{(}f(w)+1\big{)}(w-\mathfrak{a})$ is bounded
above and below for $w$ bounded away from
$[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$.

With $\hat{g}(z)$, we can rewrite ${\mathcal{E}}^{({\bm{v}},\bm{s})}_{0}(z)$
as

(7.9)
$\displaystyle\begin{split}\frac{f(z+1/n)}{f(z+1/n)+1}\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(z+1/n)e^{-b(z+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)e^{-b(z)}}\frac{\hat{g}(z)}{\hat{g}(z+1/n)}e^{\frac{1}{n}\psi(z)}-1\right)+\frac{f(z+1/n)}{f(z+1/n)+1}-\frac{f(z)}{f(z)+1}.\end{split}$

To estimate (7.5), (7.6) and (7.7), for any function $h$, which is analytic in
small neighborhoods of $z,\overline{z}$, and $k\geqslant 0$, we let
${\mathcal{D}}_{z}^{(k)}h$ denote the set of values of the form

(7.10)
$\displaystyle\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{z,\overline{z}}}\frac{h(w)}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}\mathrm{d}w,\quad
k_{1}+k_{2}=k+1,$

where the contour $\omega_{z,\overline{z}}$ encloses $z,\overline{z}$. Observe
that the expression (7.10) contains the first $k$ derivatives of $h$ (up to
some constant) at $z,\overline{z}$, by taking $k_{1}=k+1,k_{2}=0$ or
$k_{1}=0,k_{2}=k+1$. Although ${\mathcal{D}}_{z}^{(k)}h$ is a set of numbers,
with a slight abuse of notation, later we will simply treat
${\mathcal{D}}_{z}^{(k)}h$ as a number, which can be any number in
${\mathcal{D}}_{z}^{(k)}h$. We consider the product of two functions,

$\displaystyle\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{z,\overline{z}}}\frac{h_{1}(w)h_{2}(w)}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}\mathrm{d}w.$

If $k_{2}=0$ (and, similarly, if $k_{1}=0$), by the chain rule we have that

$\displaystyle\phantom{{}={}}\frac{(k_{1}-1)!}{2\pi\mathrm{i}}\oint_{\omega_{z,\overline{z}}}\frac{h_{1}(w)h_{2}(w)}{(w-z)^{k_{1}}}\mathrm{d}w=\partial_{z}^{k_{1}-1}\big{(}h_{1}(z)h_{2}(z)\big{)},$

is a linear combination of

$\displaystyle\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{z,\overline{z}}}\frac{h_{1}(w)}{(w-z)^{k_{1}-a}}\mathrm{d}w\cdot\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{z,\overline{z}}}\frac{h_{2}(w)}{(w-z)^{a+1}}\mathrm{d}w,\quad
0\leqslant a<k_{1}.$

More generally, for $k_{1},k_{2}\geqslant 1$ we have

$\displaystyle\begin{split}(k_{1}-1)!(k_{2}-1)!&\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{z,\overline{z}}}\frac{h_{1}(w)h_{2}(w)}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}\mathrm{d}w\\\
&=\partial_{z}^{k_{1}-1}\partial_{\overline{z}}^{k_{2}-1}\left(\frac{h_{1}(z)h_{2}(z)-h_{1}(\overline{z})h_{2}(\overline{z})}{z-\overline{z}}\right)\\\
&=\partial_{z}^{k_{1}-1}\partial_{\overline{z}}^{k_{2}-1}\left(\frac{h_{1}(z)-h_{1}(\overline{z})}{z-\overline{z}}h_{2}(z)+\frac{h_{2}(z)-h_{2}(\overline{z})}{z-\overline{z}}h_{1}(\overline{z})\right),\end{split}$

which can be written as a linear combination of terms in the form

$\displaystyle\frac{1}{2\pi\mathrm{i}}\oint\frac{h_{1}(w)\mathrm{d}w}{(w-z)^{k_{1}-a}(w-\overline{z})^{k_{2}}}\cdot\frac{1}{2\pi\mathrm{i}}\oint\frac{h_{2}(w)\mathrm{d}w}{(w-z)^{a+1}},\quad
0\leqslant a<k_{1},$

and

$\displaystyle\frac{1}{2\pi\mathrm{i}}\oint\frac{h_{2}(w)\mathrm{d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}-a}}\cdot\frac{1}{2\pi\mathrm{i}}\oint\frac{h_{1}(w)\mathrm{d}w}{(w-\overline{z})^{a+1}},\quad
0\leqslant a<k_{2}.$

Therefore, ${\mathcal{D}}_{z}^{(k)}(h_{1}h_{2})$ is a linear combination of
${\mathcal{D}}_{z}^{(k_{1})}(h_{1}){\mathcal{D}}_{z}^{(k_{2})}(h_{2})$, with
$k_{1}+k_{2}=k$ (and uniformly bounded coefficients). We write this by

(7.11)
$\displaystyle{\mathcal{D}}_{z}^{(k)}(h_{1}h_{2})\in\text{LinComb}_{k_{1}+k_{2}=k}{\mathcal{D}}_{z}^{(k_{1})}(h_{1}){\mathcal{D}}_{z}^{(k_{2})}(h_{2}).$

If $h(w)$ is uniformly bounded (that is, if
$\big{|}h(w)\big{|}=\mathcal{O}(1)$) for $|w-z|,|w-\bar{z}|\lesssim\eta$, we
can take the contour $\omega_{z,\overline{z}}$ in (7.10) as two small circles
centered around $z,\bar{z}$ with radius of order $\eta$. Then we have the
trivial bound for any element of ${\mathcal{D}}_{z}^{(k)}h$, given by

(7.12)
$\displaystyle|{\mathcal{D}}_{z}^{(k)}h|=\left|\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{z,\overline{z}}}\frac{h(w)\mathrm{d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}\right|\lesssim\oint_{\omega_{z,\overline{z}}}\frac{\mathrm{d}w}{\eta^{k_{1}+k_{2}}}\lesssim\frac{1}{\eta^{k}},\qquad\text{for
$k_{1}+k_{2}=k+1$},$

which will be used repeatedly in the remaining of this section.

For the complex slope $f(z)$, we recall from (6.6) that
$\mathop{\mathrm{Im}}\big{[}f(x+0\mathrm{i})/(f(x+\mathrm{i})+1)\big{]}$
defines a negative measure on ${\mathbb{R}}$ with support $I$. Thanks to
(6.5), as in the fourth statement of 5.3 we have for any $z\in\mathscr{D}$ and
$k\geqslant 1$ that

(7.13) $\displaystyle\left|\frac{f(z)}{f(z)+1}\right|\lesssim
1,\quad\left|{\mathcal{D}}_{z}^{(k)}\frac{f}{f+1}\right|\lesssim\frac{|\mathop{\mathrm{Im}}[f(z)/f(z)+1]|}{\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{k-1}},\quad\frac{|\mathop{\mathrm{Im}}[f(z)/f(z)+1]|}{\operatorname{Im}z}\lesssim\frac{1}{\operatorname{dist}(z,I)}.$

We remark that the second estimate in (7.13) is slightly more general than
(5.51), but it can be proven essentially the same way as in (5.54), by

$\displaystyle\begin{split}\left|{\mathcal{D}}^{(k)}_{z}\frac{f}{f+1}\right|&=\left|\displaystyle\frac{1}{\pi}\int_{I}\frac{k!\mathop{\mathrm{Im}}\big{[}f(x+\mathrm{i}0)/(f(x+\mathrm{i}0)+1)\big{]}{\rm
d}x}{(x-z)^{k_{1}}(x-\bar{z})^{k_{2}}}\right|+\operatorname{\mathcal{O}}(1)\\\
&\lesssim\Bigg{|}\displaystyle\frac{1}{\operatorname{dist}(z,I)^{k_{1}+k_{2}-2}}\displaystyle\int_{I}\displaystyle\frac{k!\mathop{\mathrm{Im}}\big{[}f(x+\mathrm{i}0)/(f(x+\mathrm{i}0)+1)\big{]}\mathrm{d}x}{|x-z|^{2}}\Bigg{|}+\mathcal{O}(1)\\\
&\lesssim\frac{\big{|}\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]\big{|}}{\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{k-1}},\end{split}$

where $k_{1}+k_{2}=k+1$.

The last two relations in (7.13) together with (7.11) implies for any
$k\geqslant 1$ and $j\geqslant 0$,

(7.14)
$\displaystyle\begin{split}{\mathcal{D}}_{z}^{(k)}\left(\frac{f}{f+1}\right)^{j}&\lesssim\sum\left|\prod_{u=1}^{j}{\mathcal{D}}_{z}^{(\tilde{k}_{u})}\left(\frac{f}{f+1}\right)\right|\\\
&\lesssim\sum\left(\frac{|\mathop{\mathrm{Im}}[f(z)/f(z)+1]|}{\mathop{\mathrm{Im}}[z]}\right)^{j}\frac{1}{\operatorname{dist}(z,I)^{\sum_{u=1}^{j}(k_{u}-1)}}\\\
&\lesssim\sum\frac{|\mathop{\mathrm{Im}}[f(z)/f(z)+1]|}{\mathop{\mathrm{Im}}[z]}\frac{1}{\operatorname{dist}(z,I)^{j-1}}\frac{1}{\operatorname{dist}(z,I)^{k-j}}\lesssim\frac{|\mathop{\mathrm{Im}}[f(z)/f(z)+1]|}{\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{k-1}},\end{split}$

where the sum is over index sets
$(\tilde{k}_{1},\tilde{k}_{2},\ldots,\tilde{k}_{j})$ such that
$\sum_{u=1}^{j}\tilde{k}_{u}=k$; and in the second line we used the last
estimate in (7.13).

###  7.2. Proof of Statement (5.71)

We recall from our 6.1 that
$\mathscr{D}\subseteq\big{\\{}z\in\Lambda:\operatorname{dist}(z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}])\geqslant
n^{{\mathfrak{c}}-1}\big{\\}}$. We fix a large constant

(7.15) $\displaystyle q=4\lceil{R/{\mathfrak{c}}}\rceil.$

For any $z\in\mathscr{D}$, we define the control parameters

(7.16) $\displaystyle\begin{aligned}
\Phi_{k}(z)=\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{k}}+\frac{1}{\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}^{q}}\left(\frac{\log
n}{n}\right)^{q-k},\qquad&\text{for $0\leqslant k\leqslant q$};\\\
\Phi_{k}(v)=\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}^{-k},\qquad&\text{for
$k>q$}.\end{aligned}$

Since $q\geqslant 4/{\mathfrak{c}}$, for $k=1$, we have

(7.17)
$\displaystyle\Phi_{1}(z)=\frac{\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)}+\operatorname{\mathcal{O}}\left(n\left(\frac{\log
n}{n^{\mathfrak{c}}}\right)^{q}\right)\asymp\frac{\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)},$

where we used the lower bound
$|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|/\mathop{\mathrm{Im}}[z]\gtrsim 1$ from
(6.4).

For $z\in\mathscr{D}$,
$\Phi_{k}(z)\lesssim\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}^{-k}$.
Thus for $k+k^{\prime}>q$, we have
$\Phi_{k}(z)\Phi_{k^{\prime}}(z)\lesssim\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}^{-k-k^{\prime}}=\Phi_{k+k^{\prime}}(z)$.
For $k+k^{\prime}\leqslant q$,

$\displaystyle\begin{split}\Phi_{k}(z)\Phi_{k^{\prime}}(z)&\lesssim\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{k}}\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{k^{\prime}}}+\Phi_{k}(z)\frac{1}{\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}^{q}}\left(\frac{\log
n}{n}\right)^{q-{k^{\prime}}}\\\
&\qquad+\Phi_{k^{\prime}}(z)\frac{1}{\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}^{q}}\left(\frac{\log
n}{n}\right)^{q-{k}}\\\
&\lesssim\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{k+k^{\prime}}}+\frac{1}{\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}^{q}}\left(\frac{\log
n}{n}\right)^{q-k-k^{\prime}}=\Phi_{k+k^{\prime}}(z),\end{split}$

where for the last line, we used
$\Phi_{k}(z)\lesssim\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}^{-k}\ll(n/\log
n)^{k}$. Thus $\Phi_{k}(z)$ satisfies

(7.18)
$\displaystyle\Phi_{k}(z)\Phi_{k^{\prime}}(z)\lesssim\Phi_{k+k^{\prime}}(z).$

The statement (7.2) will eventually be a consequence of the following
proposition, which states that if we have some (weak) a priori estimate on
${\mathcal{A}}^{({\bm{v}},\bm{s})}(z)$, then the dynamical loop equation can
be used to obtain an improved estimate on
${\mathcal{A}}^{({\bm{v}},\bm{s})}(z)$.

######  Proposition 7.1.

Adopt 6.1. Suppose that $r=1$, $s_{1}=-1$, and
$v_{1}=v\in\mathscr{D}\setminus\mathscr{D}({\mathfrak{r}})$; let $z$ be a
complex number with either $z\in\\{v,\overline{v}\\}$ or
$z\in\mathscr{D}\cap\mathscr{D}(\mathfrak{r})$. Further assume for any
$1\leqslant k\leqslant 2q$ (as defined in (7.15)) we have

(7.19)
$\displaystyle\Big{|}{\mathcal{D}}_{z}^{(k)}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{|}\leqslant\log
n\cdot\Phi_{k}(v).$

Then, we have the improved estimate

(7.20)
$\displaystyle\Big{|}{\mathcal{D}}_{z}^{(k)}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{|}\leqslant{\mathfrak{C}}\Phi_{k}(v),\quad
1\leqslant k\leqslant 2q,$

for some constant ${\mathfrak{C}}=\mathfrak{C}(R)>1$.

###### Proof.

In the following we prove (7.20) for $z=v$; the proofs in the other cases when
$z=\overline{v}$ or $z\in\mathscr{D}\cap\mathscr{D}({\mathfrak{r}})$ follow
from essentially the same argument. We denote
$\eta=\operatorname{dist}\big{(}v,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}$.
Thanks to (6.79), for any $z\in\mathscr{D}$ with
$\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}\gtrsim\eta$,
we have that

(7.21)
$\displaystyle\left|\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)-\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(\infty)-b(z)\right|\lesssim\frac{1}{n\eta}\lesssim
1.$

For $q\leqslant k\leqslant 2q$, (7.20) follows from (7.21) and the trivial
bound (7.12),

$\displaystyle\Big{|}{\mathcal{D}}_{v}^{(k)}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{|}$
$\displaystyle\lesssim\frac{1}{\eta^{k}}=\Phi_{k}(v).$

For $k\leqslant q$, we recall that from (7.5),
${\mathcal{D}}_{v}^{(k)}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)+b(w)\big{)}$
is a collection of values in the form,

(7.22)
$\displaystyle\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{v,\overline{v}}}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-v)^{k_{1}}(w-\overline{v})^{k_{2}}}-\frac{1}{2\pi\mathrm{i}}\oint_{\omega+}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-v)^{k_{1}}(w-\overline{v})^{k_{2}}},\quad k_{1}+k_{2}=k+1,$

where the contour $\omega_{v,\overline{v}}$ encloses $v,\overline{v}$, and the
contour $\omega+\subseteq\mathscr{D}({\mathfrak{r}})$ encloses
$[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$ and $v,\overline{v}$.

From our construction, ${\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)$ is analytic
for $w\in\mathscr{D}$, since all the denominators in (7.9) are bounded away
from $0$. However, $\psi^{({\bm{v}},\bm{s})}(w)-1=1/n(w-v)$ has a pole at $v$.
Along $\big{\\{}w\in\mathbb{C}:|w-v|=\eta/3\big{\\}}$ and
$\big{\\{}w\in\mathbb{C}:|w-\overline{v}|=\eta/3\big{\\}}$, (6.28) gives
$\big{|}{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{|}\lesssim 1/n\eta$. So, we
can expand $\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}$ as

(7.23)
$\displaystyle\begin{split}\sum_{j=1}^{q}&\frac{(-1)^{j-1}\big{(}{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}^{j}}{j}+\operatorname{\mathcal{O}}\left(\frac{1}{(n\eta)^{q+1}}\right)\\\
&=\sum_{j_{1}+j_{2}=1}^{q}\frac{(-1)^{j_{1}+j_{2}-1}}{j_{1}+j_{2}}{j_{1}+j_{2}\choose
j_{1}}\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}^{j_{1}}\left(\frac{\psi^{({\bm{v}},\bm{s})}-1}{f(w)+1}\right)^{j_{2}}+\operatorname{\mathcal{O}}\left(\frac{1}{(n\eta)^{q+1}}\right)\\\
&=\sum_{j_{1}+j_{2}=1}^{q}\frac{(-1)^{j_{1}+j_{2}-1}}{j_{1}+j_{2}}{j_{1}+j_{2}\choose
j_{1}}\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}^{j_{1}}\left(\frac{1}{f(w)+1}\right)^{j_{2}}\frac{1}{n^{j_{2}}(w-v)^{j_{2}}}+\operatorname{\mathcal{O}}\left(\frac{1}{(n\eta)^{q+1}}\right).\end{split}$

Thanks to the trivial bound (7.12), the contribution from the error term
$\operatorname{\mathcal{O}}((n\eta)^{-(q+1)})$ is bounded by
$\operatorname{\mathcal{O}}((n\eta)^{-(q+1)}\eta^{-k})$, which is smaller than
$\Phi_{k}(v)$. By plugging (7.23) into the first term in (7.22), we get
expressions involving linear combinations (with bounded coefficients) of
quantities of the form

(7.24)
$\displaystyle\begin{split}\phantom{{}={}}\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{v,\overline{v}}}&\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}^{j_{1}}\left(\frac{1}{f(w)+1}\right)^{j_{2}}\frac{\mathrm{d}w}{n^{j_{2}}(w-v)^{k_{1}+j_{2}}(w-\overline{v})^{k_{2}}}\\\
&\qquad\qquad\in\frac{1}{n^{j_{2}}}{\mathcal{D}}_{v}^{(k_{1}+k_{2}+j_{2}-1)}\left(\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}^{j_{1}}\left(\frac{1}{f(w)+1}\right)^{j_{2}}\right),\end{split}$

where $1\leqslant j_{1}+j_{2}\leqslant q$. Thanks to (7.11), (7.24) decomposes
to sum of products involving

(7.25)
$\displaystyle\frac{1}{n^{j_{2}}}{\mathcal{D}}_{v}^{(\hat{k}_{1})}\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}^{j_{1}}{\mathcal{D}}_{v}^{(\hat{k}_{2})}\left(\frac{1}{f(w)+1}\right)^{j_{2}}\quad\hat{k}_{1}+\hat{k}_{2}=k_{1}+k_{2}+j_{2}-1=k+j_{2}.$

In the following we show that under our hypothesis (7.19), for
$0\leqslant\hat{k}\leqslant 2q$,

(7.26)
$\displaystyle\big{|}{\mathcal{D}}_{v}^{(\hat{k})}({\mathcal{E}}_{0}^{({\bm{v}},\bm{s})})\big{|}$
$\displaystyle\lesssim\Phi_{\hat{k}}(v),$

which together with (7.18) and (7.11), would imply

(7.27)
$\displaystyle{\mathcal{D}}_{v}^{(\hat{k}_{1})}\left({\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}\right)^{j_{1}}\lesssim\sum\left|\prod_{u=1}^{j_{1}}{\mathcal{D}}_{v}^{(\tilde{k}_{u})}\left({\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}\right)\right|\lesssim\prod_{u=1}^{j_{1}}\Phi_{\tilde{k}_{u}}(v)\lesssim\Phi_{\hat{k}_{1}}(v),\quad
j_{1}\geqslant 1,$

where the sum is over index sets
$(\tilde{k}_{1},\tilde{k}_{2},\ldots,\tilde{k}_{j_{1}})$ such that
$\sum_{u=1}^{j_{1}}\tilde{k}_{u}=\hat{k}_{1}$.

For $q\leqslant{\hat{k}}\leqslant 2q$, (7.26) follows from the trivial bound
(7.12),
${\mathcal{D}}_{v}^{({\hat{k}})}({\mathcal{E}}_{0}^{({\bm{v}},\bm{s})})\lesssim\eta^{-\hat{k}}=\Phi_{\hat{k}}(v)$.
For ${\hat{k}}\leqslant q$, we recall the expression of
${\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}$ from (7.9). For the last term in (7.9)
we have

(7.28)
$\displaystyle\left|{\mathcal{D}}_{v}^{({\hat{k}})}\left(\frac{f(w+1/n)}{f(w+1/n)+1}-\frac{f(w)}{f(w)+1}\right)\right|\lesssim\frac{|\mathop{\mathrm{Im}}[f(v)/f(v)+1]|}{n\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{\hat{k}}}\leqslant\Phi_{\hat{k}}(v),\quad{\hat{k}}\geqslant
0,$

where we used (7.13). Next we need to estimate the first term in (7.9)

$\displaystyle{\mathcal{D}}_{v}^{({\hat{k}})}\Bigg{(}\frac{f(w+1/n)}{f(w+1/n)+1}\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\frac{\hat{g}(w)}{\hat{g}(w+1/n)}e^{\frac{1}{n}\psi(w)}-1\right)\Bigg{)},$

which, thanks to (7.11), is a linear combination of

(7.29)
$\displaystyle{\mathcal{D}}_{v}^{(\hat{k}-\hat{k}^{\prime})}\left(\frac{f(w+1/n)}{f(w+1/n)+1}\right){\mathcal{D}}_{v}^{(\hat{k}^{\prime})}\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\frac{\hat{g}(w)}{\hat{g}(w+1/n)}e^{\frac{1}{n}\psi(w)}-1\right).$

If $\hat{k}^{\prime}=0$, we can use (7.8) to bound the second term of (7.29)
as

(7.30)
$\displaystyle\begin{split}\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(v+1/n)e^{-b(v+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(v)e^{-b(v)}}\frac{\hat{g}(v)}{\hat{g}(v+1/n)}e^{\frac{1}{n}\psi(v)}-1=\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(v+1/n)e^{-b(v+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(v)e^{-b(v)}}-1+\operatorname{\mathcal{O}}\left(\frac{1}{n}\right).\end{split}$

For the first term on the right side of (7.30), by a Taylor expansion, using
(7.19), we have

(7.31)
$\displaystyle\begin{split}\Bigg{|}\log\bigg{(}\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(v+1/n)e^{-b(v+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(v)e^{-b(v)}}\bigg{)}\Bigg{|}&\lesssim\left|\sum_{j=1}^{q}\frac{\partial_{v}^{j}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(v)-b(v)\big{)}}{n^{j}}+\operatorname{\mathcal{O}}\left(\frac{1}{(n\eta)^{q+1}}\right)\right|\\\
&\lesssim\sum_{j=1}^{q}n^{-j}\log n\cdot\Phi_{j}(v)+\frac{1}{(n\eta)^{q+1}}\\\
&\lesssim\frac{\log
n|\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]|}{n^{2}\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)}+\operatorname{\mathcal{O}}\left(\frac{1}{n}\right)\lesssim\frac{\big{|}\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]\big{|}}{n\mathop{\mathrm{Im}}v}\ll
1,\end{split}$

where we used $\Phi_{j}(v)/n^{j}$ is decreasing in $j$, $q\geqslant
1/{\mathfrak{c}}$ thus $1/(n\eta)^{q+1}\leqslant 1/n$, and the bound
$n\operatorname{dist}(v,I)\geqslant n^{{\mathfrak{c}}}\gg\log n$. By plugging
(7.31) into (7.30), we get

(7.32)
$\displaystyle\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(v+1/n)e^{-b(v+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(v)e^{-b(v)}}\frac{\hat{g}(v)}{\hat{g}(v+1/n)}e^{\frac{1}{n}\psi(v)}-1=\operatorname{\mathcal{O}}\left(\frac{|\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]|}{n\mathop{\mathrm{Im}}[v]}\right).$

Then for $\hat{k}^{\prime}=0$, by plugging (7.32) into (7.29), and using
(7.13), we have if $\hat{k}\geqslant 1$,

(7.33)
$\displaystyle|\eqref{e:linear}|\lesssim\frac{|\mathop{\mathrm{Im}}[f(v)/f(v)+1]|}{\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{\hat{k}-1}}\left(\frac{|\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]|}{n\mathop{\mathrm{Im}}[v]}\right)\leqslant\Phi_{\hat{k}}(v),$

or $\hat{k}=0$,

(7.34)
$\displaystyle|\eqref{e:linear}|\lesssim\left(\frac{|\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]|}{n\mathop{\mathrm{Im}}[v]}\right)\leqslant\Phi_{0}(v).$

For $\hat{k}^{\prime}\geqslant 1$, using (7.11) again, the second term on the
right side of (7.29) is a linear combination of quantities of the form

(7.35)
$\displaystyle\begin{split}&\phantom{{}={}}\left|{\mathcal{D}}_{v}^{(\hat{k}^{\prime}-\hat{k}^{\prime\prime})}\left(\frac{\hat{g}(w)}{\hat{g}(w+1/n)}e^{\frac{1}{n}\psi(w)}\right){\mathcal{D}}_{v}^{(\hat{k}^{\prime\prime})}\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\right)\right|\\\
&\qquad\qquad\qquad\qquad\qquad\lesssim\left(\frac{1}{n}\right)^{\bm{1}(\hat{k}^{\prime}-\hat{k}^{\prime\prime}\geqslant
1)}\left|{\mathcal{D}}_{v}^{(\hat{k}^{\prime\prime})}\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\right)\right|,\end{split}$

where we used (7.8), as well as the analyticity of $\hat{g}$ and $\psi$ on
$\Lambda$ (enabling the use of (7.12) with $\eta$ of order $1$), to bound the
first term.

If $\hat{k}^{\prime\prime}=0$, we can bound the last term on the right side of
(LABEL:e:llc) by $\operatorname{\mathcal{O}}(1)$ using (7.31). By plugging
(LABEL:e:llc) into (7.29) and using (7.13), the sum of such terms with
$\hat{k}^{\prime\prime}=0$ is bounded

(7.36)
$\displaystyle\frac{\big{|}\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]\big{|}}{\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{\hat{k}-\hat{k}^{\prime}-1}}\left(\frac{1}{n}\right)^{\bm{1}(\hat{k}^{\prime}-\hat{k}^{\prime\prime}\geqslant
1)}\lesssim\Phi_{\hat{k}}(v).$

To estimate the second term on the right side of (LABEL:e:llc) with
$1\leqslant\hat{k}^{\prime\prime}\leqslant q$, we first show that

(7.37)
$\displaystyle\left|{\mathcal{D}}_{v}^{(\hat{k}^{\prime\prime})}\log\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\right)\right|\lesssim\Phi_{\hat{k}^{\prime\prime}}(v).$

By a Taylor expansion, we have

(7.38)
$\displaystyle\begin{split}\phantom{{}={}}\Bigg{|}{\mathcal{D}}_{v}^{(\hat{k}^{\prime\prime})}\log\bigg{(}&\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\bigg{)}\Bigg{|}\\\
&\lesssim\left|{\mathcal{D}}_{v}^{({\hat{k}^{\prime\prime}})}\left(\sum_{j=1}^{q-{\hat{k}^{\prime\prime}}}\frac{\partial_{w}^{j}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}}{n^{j}}+\operatorname{\mathcal{O}}\left(\frac{1}{(n\eta)^{q-{\hat{k}^{\prime\prime}}+1}}\right)\right)\right|\\\
&\lesssim\sum_{j=1}^{q-{\hat{k}^{\prime\prime}}}n^{-j}\Big{|}{\mathcal{D}}_{v}^{({\hat{k}^{\prime\prime}}+j)}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{|}+\left|{\mathcal{D}}_{v}^{({\hat{k}^{\prime\prime}})}\operatorname{\mathcal{O}}\left(\frac{1}{(n\eta)^{q-{\hat{k}^{\prime\prime}}+1}}\right)\right|\\\
&\lesssim\sum_{j=1}^{q-{\hat{k}^{\prime\prime}}}n^{-j}\log
n\cdot\Phi_{{\hat{k}^{\prime\prime}}+j}(v)+\frac{1}{n\eta}\frac{1}{\eta^{q}}\frac{1}{n^{q-{\hat{k}^{\prime\prime}}}}\lesssim\Phi_{\hat{k}^{\prime\prime}}(v),\end{split}$

where in the third inequality we used our assumption (7.19) to bound the sum,
and the trivial bound (7.12) for the last term. In the last line, we used that
$n\operatorname{dist}(v,I)\geqslant n^{{\mathfrak{c}}}\gg\log n$, so
$n^{-j}\log
n\cdot\Phi_{{\hat{k}^{\prime\prime}}+j}(v)\lesssim\Phi_{\hat{k}^{\prime\prime}}(v)$
for $j\geqslant 1$. Thanks to (7.11) and the relation (7.18), (7.38) implies
that

(7.39)
$\displaystyle\Bigg{|}{\mathcal{D}}_{v}^{({\hat{k}^{\prime\prime}})}\bigg{(}\log\Big{(}\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\Big{)}\bigg{)}^{j}\Bigg{|}\lesssim\Phi_{\hat{k}^{\prime\prime}}(v),\quad
j\geqslant 1.$

Using (7.39) as input, we get the following bound for the second term on the
right side of (LABEL:e:llc),

(7.40)
$\displaystyle\begin{split}&\phantom{{}={}}\left|{\mathcal{D}}_{v}^{(\hat{k}^{\prime\prime})}\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\right)\right|\\\
&\qquad=\left|{\mathcal{D}}_{v}^{(\hat{k}^{\prime\prime})}\Bigg{(}1+\frac{1}{j!}\sum_{j=1}^{q-\hat{k}^{\prime\prime}}\bigg{(}\log\Big{(}\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\Big{)}\bigg{)}^{j}+\operatorname{\mathcal{O}}\bigg{(}\frac{1}{(n\eta)^{q-\hat{k}^{\prime\prime}+1}}\bigg{)}\Bigg{)}\right|\lesssim\Phi_{\hat{k}^{\prime\prime}}(v),\end{split}$

for $1\leqslant\hat{k}^{\prime\prime}\leqslant q$. By plugging (LABEL:e:llc2)
and (LABEL:e:llc) into (7.29), the sum of such terms with
$\hat{k}^{\prime\prime}\geqslant 1$ is bounded by

(7.41)
$\displaystyle\begin{split}&\sum_{\hat{k}^{\prime\prime}=1}^{\hat{k}^{\prime}}\frac{\big{|}\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]|}{\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{\hat{k}-\hat{k}^{\prime}-1}}\left(\frac{1}{n}\right)^{\bm{1}(\hat{k}^{\prime}-\hat{k}^{\prime\prime}\geqslant
1)}\Phi_{\hat{k}^{\prime\prime}}(v)\lesssim\frac{\big{|}\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]\big{|}}{\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{\hat{k}-\hat{k}^{\prime}-1}}\Phi_{\hat{k}^{\prime}}(v)\\\
&\lesssim\frac{1}{\operatorname{dist}(v,I)^{\hat{k}-\hat{k}^{\prime}}}\left(\frac{\big{|}\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]\big{|}}{n\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{\hat{k}^{\prime}}}+\frac{1}{\eta^{q}}\left(\frac{\log
n}{n}\right)^{q-\hat{k}^{\prime}}\right)\leqslant\Phi_{\hat{k}}(v),\end{split}$

where we used that $\Phi_{\hat{k}^{\prime\prime}}(v)$ is nondecreasing in
$\hat{k}^{\prime\prime}$ in the first line; in the second line we used the
last estimate in (7.13).

We conclude from (7.33), (7.34), (7.36) and (LABEL:e:8112) that for
$0\leqslant\hat{k}\leqslant q$,

(7.42)
$\displaystyle\begin{split}&\phantom{{}={}}\Bigg{|}{\mathcal{D}}_{v}^{(\hat{k})}\bigg{(}\frac{f(z+1/n)}{f(z+1/n)+1}\Big{(}\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(z+1/n)e^{-b(z+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)e^{-b(z)}}\frac{\hat{g}(z)}{\hat{g}(z+1/n)}e^{\frac{1}{n}\psi(z)}-1\Big{)}\bigg{)}\Bigg{|}\lesssim\Phi_{\hat{k}}(v).\end{split}$

The claim (7.26) follows from combining (7.28) and (LABEL:e:secondt).

Now we can use (7.14) and (7.27) to upper bound (7.25). There are several
cases, If $\hat{k}_{1}=0$, then $\hat{k}_{2}=k+j_{2}$. We may assume that
$j_{2}\geqslant 1$, for otherwise the second factor in (7.25) vanishes. In
this case, (7.25) becomes

$\displaystyle|\eqref{e:tt1copy}|\lesssim\frac{1}{n^{j_{2}}}{\mathcal{D}}_{v}^{(k+j_{2})}\left(\left(\frac{1}{f(w)+1}\right)^{j_{2}}\right)\lesssim\frac{\mathop{\mathrm{Im}}[f(v)/f(v)+1]}{n^{j_{2}}\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{k+j_{2}-1}}\lesssim\Phi_{k}(v).$

If $\hat{k}_{2}=0$, then $\hat{k}_{1}=k+j_{2}$. We may assume that
$j_{1}\geqslant 1$ (for otherwise the first factor in (7.25) vanishes). In
this case, (7.25) becomes

$\displaystyle|\eqref{e:tt1copy}|\lesssim\frac{1}{n^{j_{2}}}{\mathcal{D}}_{v}^{k+j_{2}}\left(({\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w))^{j_{1}}\right)\lesssim\frac{\Phi_{k+j_{2}}(v)}{n^{j_{2}}}\lesssim\Phi_{k}(v).$

If $\hat{k}_{1},\hat{k}_{2}\geqslant 1$, then we may assume that
$j_{1},j_{2}\geqslant 1$, and (7.25) becomes

$\displaystyle|\eqref{e:tt1copy}|\lesssim\Phi_{\hat{k}_{1}}(v)\frac{\mathop{\mathrm{Im}}[f(v)/f(v)+1]}{n^{j_{2}}\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{\hat{k}_{2}-1}}\lesssim
n^{-(j_{2}-1)}\Phi_{\hat{k}_{1}}(v)\Phi_{\hat{k}_{2}-1}(v)\lesssim
n^{-(j_{2}-1)}\Phi_{k+j_{2}-1}(v)\lesssim\Phi_{k}(v).$

It follows that the first term in (7.22) is bounded by

(7.43)
$\displaystyle\left|\frac{1}{2\pi\mathrm{i}}\oint_{\omega_{v,\overline{v}}}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-v)^{k_{1}}(w-\overline{v})^{k_{2}}}\right|\lesssim\Phi_{k}(v).$

For the second term in (7.22), we may deform $\omega+$ so that it is bounded
away from $v$. Then, any $w\in\omega+$ is bounded away from $v$, so (6.26)
gives

$\displaystyle\left|\frac{\psi^{({\bm{v}},\bm{s})}(w)-1}{f(w)+1}\right|\lesssim\frac{1}{n},\qquad\text{meaning}\qquad\big{|}{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{|}=\big{|}{\mathcal{E}}^{({\bm{v}},\bm{s})}_{0}(w)\big{|}+\operatorname{\mathcal{O}}\left(\frac{1}{n}\right).$

Thus it follows

(7.44)
$\displaystyle\left|\frac{1}{2\pi\mathrm{i}}\oint_{\omega+}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-v)^{k_{1}}(w-\overline{v})^{k_{2}}}\right|\lesssim\max_{w\in\omega+}\big{|}{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{|}\lesssim\frac{1}{n}+\max_{w\in\omega+}\big{|}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{|}.$

For the last term on the right side, recall the expression of
${\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)$ from (7.9)

(7.45)
$\displaystyle\begin{split}|{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)|&\lesssim\left|\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\frac{\hat{g}(w)}{\hat{g}(w+1/n)}e^{\frac{1}{n}\psi(w)}-1\right|+\left|\frac{f(w+1/n)}{f(w+1/n)+1}-\frac{f(w)}{f(w)+1}\right|\\\
&\lesssim\frac{|\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]|}{n\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)}+\frac{1}{n}\lesssim\Phi_{1}(v)\lesssim\Phi_{k}(v),\end{split}$

where in the first line, the first term on the right side is bounded as in
(7.32), the second term is bounded by using (7.13) and the fact that $w$ is
bounded away from $[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$; for the
second line, we used (6.4). We conclude from plugging (7.43), (7.44) and
(7.45) into (7.5) that

$\displaystyle\Big{|}{\mathcal{D}}_{v}^{(k)}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{|}=|\eqref{e:toest1}|\lesssim\Phi_{k}(v).$

This finishes the proof of 7.1, by taking ${\mathfrak{C}}$ larger than the
above implicit constant. ∎

Now we can establish the first statement of 6.4, given by equation (6.14).

###### Proof of (6.14).

We recall from (7.2) and (7.17) that (6.14) is equivalent to

$\displaystyle\Big{|}{\mathcal{D}}_{v}^{(1)}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{|}\lesssim\Phi_{1}(v),$

where $\Phi_{1}(v)$ is explicitly given by (7.17). Thanks to 7.1, a weak
estimate of the form,

$\displaystyle\Big{|}{\mathcal{D}}_{v}^{(k)}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{|}\leqslant\log
n\cdot\Phi_{k}(v),\qquad\text{for $v\in\mathscr{D}$},$

for $k\in[\\![{1,2q}]\\!]$, implies the improved estimate

$\displaystyle\Big{|}{\mathcal{D}}_{v}^{(k)}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w))\big{)}\Big{|}\leqslant{\mathfrak{C}}\Phi_{k}(v),\quad
1\leqslant k\leqslant 2q.$

For $v\in\mathscr{D}\cap\mathscr{D}({\mathfrak{r}})$ then
$\operatorname{dist}(v,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}])\geqslant{\mathfrak{r}}$,
6.7 gives that

$\displaystyle\left|\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\frac{\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}\right|=\left|\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\frac{\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-z)^{k_{1}}(w-\overline{z})^{k_{2}}}\right|\lesssim\frac{1}{n},$

so that
$\big{|}{\mathcal{D}}_{z}^{(k)}(\log{\mathcal{A}}^{({\bm{v}},\bm{s})}-b)\big{|}\lesssim\Phi_{k}(v)$
for $z=v,\overline{v}$ or $z\in\mathscr{D}({\mathfrak{r}})$. In particular, if
we take $\eta_{0}={\mathfrak{r}}$, then (7.20) holds for $v\in\mathscr{D}$
with
$\operatorname{dist}\big{(}v,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}\geqslant\eta_{0}$.
We define the sequence

$\displaystyle\eta_{i+1}=\eta_{i}-n^{-3q},\quad i\geqslant 0.$

We show by induction on $i$ that the claim (6.14) holds for
$v\in\mathscr{D}\cap\\{w:\operatorname{dist}(w,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}])\geqslant\eta_{i}\\}$,
whenever $\eta_{i}\gg 1/n$ (meaning $i=\mathcal{O}(n^{3q})$). Thanks to (7.21)
and the trivial bound (7.12), we have for any $z,v\in\mathscr{D}$, and
$1\leqslant k\leqslant 2q$,

$\displaystyle\big{|}\partial_{v}{\mathcal{D}}_{z}^{(k)}(\log{\mathcal{A}}^{({\bm{v}},\bm{s})}-b)\big{|}\lesssim
n^{k+1}\leqslant n^{2q+1}.$

In particular
${\mathcal{D}}_{z}^{(k)}(\log{\mathcal{A}}^{({\bm{v}},\bm{s})}-b)$ is
Lipschitz in $v$ with Lipschitz constant at most $n^{2q+1}$. Therefore if
(7.20) holds for $v\in\mathscr{D}$ with
$\operatorname{dist}\big{(}v,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}\geqslant\eta_{i}$,
then (7.19) holds for $v\in\mathscr{D}$ with
$\operatorname{dist}\big{(}v,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}\geqslant\eta_{i+1}$.
And 7.1 implies that (7.20) also holds for $v\in\mathscr{D}$ with
$\operatorname{dist}\big{(}v,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}\geqslant\eta_{i+1}$.
In this way, by repeatedly using 7.1, we conclude that
$\big{|}{\mathcal{D}}_{z}^{(k)}(\log{\mathcal{A}}^{({\bm{v}},\bm{s})}-b)\big{|}\lesssim\Phi_{k}(v)$
holds for any $v\in\mathscr{D}$ and $z\in\\{v,\overline{v}\\}$ or
$z\in\mathscr{D}({\mathfrak{r}})$. ∎

###  7.3. Proof of Statement (6.15)

We next proceed to prove the second statement of 6.4, given by (6.15). To that
end, we take $R=4p$, $s_{1}=s_{2}=\cdots=s_{2p-1}=1$,
$s_{2p}=s_{2p+1}=\cdots=s_{4p-2}=-1$, and $s_{4p-1}=-1,v_{4p-1}=-v$. For any
$0\leqslant r\leqslant 2p-1$, the derivative
$\partial_{z}\partial_{v_{1}}\partial_{v_{2}}\cdots\partial_{v_{r}}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}$
after specializing at $z=v,v_{j}=v_{j+2p-1}\in\\{v,\overline{v}\\}$, gives the
$(r+1)$-th joint cumulants of

(7.46)
$\displaystyle\sum_{i=1}^{m}\frac{1}{v-x_{i}-e_{i}/n}-\sum_{i=1}^{m}\frac{1}{v-x_{i}},\quad\sum_{i=1}^{m}\frac{1}{\overline{v}-x_{i}-e_{i}/n}-\sum_{i=1}^{m}\frac{1}{\overline{v}-x_{i}}.$

For any subset $J\subseteq[\\![{1,2p-1}]\\!]$, we denote
$\partial_{V_{J}}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}=(\prod_{j\in
J}\partial_{v_{j}})\log{\mathcal{A}}^{({\bm{v}},\bm{s})}$. In order to
establish (6.15), it suffices to show that for any subset222Here, $J$ does not
contain repeated elements (for otherwise the associated cumulant below is
equal to $0$). $J\subseteq[\\![{1,2p-1}]\\!]$ with $0\leqslant|J|=r\leqslant
2p-1$ we have

(7.47)
$\displaystyle\left|\partial_{z}\partial_{V_{J}}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)|_{z=v,v_{j}=v_{j+2p-1}\in\\{v,\overline{v}\\}}\right|\lesssim\frac{|\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]|}{n^{r}\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{2r}}.$

Indeed, assuming (7.47), we deduce that the $(r+1)$-th joint cumulants of
(7.46) are bounded by

$\displaystyle\displaystyle\frac{\big{|}\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]\big{|}}{n^{r}\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{2r}}.$

Then, the left side of (6.15) is a combination of the first $(2p)$-th
cumulants of (7.46)

(7.48)
$\displaystyle\begin{split}\phantom{{}={}}\mathbb{E}_{\mathbb{Q}}&\left|\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}-\mathbb{E}_{\mathbb{Q}}\left[\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}\right]\right|^{2p}\\\
&\lesssim\sum_{\sum(r_{u}+1)=2p}\prod_{u}\frac{\big{|}\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]\big{|}}{n^{r_{u}}\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{2r_{u}}}\\\
&\lesssim\left(\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{2}}\right)^{p}+\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n^{2p-1}\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{4p-2}},\end{split}$

where the sum is over positive integer sequences $(r_{1},r_{2},\ldots,r_{k})$
such that $\sum_{u=1}^{k}(r_{u}+1)=2p$. To get the last line, we used the fact
that the largest term in the sum is either obtained when $k=p$,
$r_{1}=r_{2}=\cdots=r_{p}=1$ or when $k=1$, $r_{1}=2p-1$.

Set $R=4p$, and then define the parameter $q>1$ as in (7.15). Further denote
the control parameters by setting

(7.49) $\displaystyle\begin{aligned}
\Phi_{k,0}(z)=\Phi_{k}(z),\qquad&\text{for $k\geqslant 0$};\\\
\Phi_{k,\ell}(z)=\frac{\big{|}\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]\big{|}}{n^{\ell}\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{k+2\ell-1}}+\frac{1}{\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}^{q}}\left(\frac{\log
n}{n}\right)^{q-k-\ell},\qquad&\text{for $\ell\geqslant 1$ and $q\leqslant
k+\ell$};\\\
\Phi_{k,\ell}(z)=\operatorname{dist}\big{(}z,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}^{-k-\ell},\qquad&\text{for
$\ell\geqslant 1$ and $k+\ell>q$}.\end{aligned}$

Since $q\geqslant 4R/{\mathfrak{c}}\geqslant 16p/{\mathfrak{c}}$, for $k=1$
and $\ell\leqslant 2p-1$, we have

(7.50)
$\displaystyle\Phi_{1,\ell}(z)=\frac{\big{|}\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]\big{|}}{n^{\ell}\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{2\ell}}+\operatorname{\mathcal{O}}\left(n^{2p}\left(\frac{\log
n}{n^{\mathfrak{c}}}\right)^{q}\right)\asymp\frac{\big{|}\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]\big{|}}{n^{\ell}\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{2\ell}},$

where we used the lower bound
$\big{|}\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]\big{|}/\mathop{\mathrm{Im}}[z]\gtrsim
1$ from (6.4). Moreover, similarly to (7.18), one can check that
$\Phi_{k,\ell}(z)$ satisfies

(7.51)
$\displaystyle\Phi_{k,\ell}(z)\Phi_{k^{\prime},\ell^{\prime}}(z)\lesssim\Phi_{k+k^{\prime},\ell+\ell^{\prime}}(z).$

The statement (7.47) will be deduced as a consequence of the following two
propositions, which state that if we have some (weak) a priori estimates on
$\partial_{V_{J}}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)$, then the dynamical
loop equation can be used to obtain improved estimates on
$\partial_{V_{J}}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(z)$.

######  Proposition 7.2.

Take $s_{1}=s_{2}=\cdots=s_{2p-1}=1$, $s_{2p}=s_{2p+1}=\cdots=s_{4p-2}=-1$,
and $s_{4p-1}=-1,v_{4p-1}=-v$, for some $v\in\mathbb{C}$. The following two
statements hold.

  1. (1)

Suppose $v\in\mathscr{D}({\mathfrak{r}})$. Assume for any
$z\in\mathscr{D}({\mathfrak{r}})$ and $J\subseteq[\\![{1,2p-1}]\\!]$ with
$1\leqslant k+|J|\leqslant 2q$ that

(7.52)
$\displaystyle\bigg{|}{\mathcal{D}}_{z}^{(k)}\Big{(}\partial_{V_{J}}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{)}\Big{|}_{v_{j}=v_{j+2p-1}\in\\{v,\overline{v}\\}}\bigg{|}\lesssim\frac{\log
n}{n^{1\vee|J|}}.$

Then we have the improved estimate

(7.53)
$\displaystyle\bigg{|}{\mathcal{D}}_{z}^{(k)}\Big{(}\partial_{V_{J}}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{)}\Big{|}_{v_{j}=v_{j+2p-1}\in\\{v,\overline{v}\\}}\bigg{|}\leqslant\frac{{\mathfrak{C}}}{n^{1\vee|J|}},$

for some constant $\mathfrak{C}=\mathfrak{C}(R)>1$.

  2. (2)

Suppose $v\in\mathscr{D}$. Assume for any $z=v,\overline{v}$ or
$z\in\mathscr{D}({\mathfrak{r}})$ $J\subseteq[\\![{1,2p-1}]\\!]$ with
$1\leqslant k+|J|\leqslant 2q$ that

(7.54)
$\displaystyle\bigg{|}{\mathcal{D}}_{z}^{(k)}\Big{(}\partial_{V_{J}}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{)}\Big{|}_{v_{j}=v_{j+2p-1}\in\\{v,\overline{v}\\}}\bigg{|}\leqslant\log
n\cdot\Phi_{k,|J|}(v).$

Then we have the improved estimate

(7.55)
$\displaystyle\bigg{|}{\mathcal{D}}_{z}^{(k)}\Big{(}\partial_{V_{J}}(\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\Big{)}\Big{|}_{v_{j}=v_{j+2p-1}\in\\{v,\overline{v}\\}}\bigg{|}\leqslant{\mathfrak{C}}\Phi_{k,|J|}(v),$

for some constant ${\mathfrak{C}}=\mathfrak{C}(R)>1$.

Let us mention that the first statement of 7.2 is nearly a special case of the
second statement, except that in the former we allow
$\operatorname{dist}(v,I)$ to grow with $n$. As such, these two parts of the
proposition are very closely related; the only reason why we separated their
statements is that the definition of $\Phi_{k,|J|}(v)$ involves the function
$f(v)$, whose domain is $\Lambda$ and is thus not defined for general
$v\in\mathscr{D}(\mathfrak{r})$.

###### Proof of 7.2.

Here, we only establish the second statement of 7.2. The proof of the first
statement is entirely analogous, by replacing every appearance of
$\Phi_{k,|J|}(v)$ below with the quantity $1/n^{1\vee|J|}$.

In the following we prove (7.55) for $z=v$; the proofs in the other cases that
$z=\overline{v}$ or $z\in\mathscr{D}({\mathfrak{r}})$ follow from essentially
the same argument. To simplify notation, we take $|J|=r\leqslant 2q-1$ and
$J=\\{1,2,\cdots,r\\}$; we also denote
$\eta=\operatorname{dist}\big{(}v,[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}$.
For $q\leqslant k+r\leqslant 2q$, using (7.21), the trivial bound (7.12)
implies

(7.56)
$\displaystyle\left|{\mathcal{D}}^{(k)}_{z}\partial_{v_{1}}\partial_{v_{2}}\cdots\partial_{v_{r}}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\big{|}_{z=v,v_{i}=v_{j+2p-1}\in\\{v,\overline{v}\\}}\right|\lesssim\frac{1}{\eta^{k+r}}=\Phi_{k,r}(v).$

It therefore suffices to consider the case $1\leqslant k+r\leqslant q$, in
which case the proof will be similar to that of 7.1. Specifically, by taking
derivative with respect to $v_{1},v_{2},\cdots,v_{r}$ on both sides of (7.4),
then specializing $z=v$ and $v_{i}\in\\{v,\overline{v}\\}$, we get

(7.57)
$\displaystyle\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\frac{\partial_{v_{1}}\partial_{v_{2}}\cdots\partial_{v_{r}}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}}{(w-v)^{k_{1}}(w-\overline{v})^{k_{2}}}=-\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\frac{\partial_{v_{1}}\partial_{v_{2}}\cdots\partial_{v_{r}}\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}{\rm
d}w}{(w-v)^{k_{1}}(w-\overline{v})^{k_{2}}}.$

By expanding $\log\big{(}1+{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{)}$ as in
(7.23), we must estimate

(7.58)
$\displaystyle\begin{split}\frac{1}{2\pi\mathrm{i}}\oint_{\omega}&\partial_{v_{1}}\partial_{v_{2}}\cdots\partial_{v_{r}}\left(\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}^{j_{1}}\left(\frac{\psi^{({\bm{v}},\bm{s})}(w)-1}{f(w)+1}\right)^{j_{2}}\frac{1}{(w-v)^{k_{1}}(w-\overline{v})^{k_{2}}}\right){\rm
d}w\\\
&=\frac{1}{2\pi\mathrm{i}}\oint_{\omega}\sum\prod_{u=1}^{j_{1}}\partial_{V_{J_{u}}}\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}\prod_{u=1}^{j_{2}}\partial_{V_{J^{\prime}_{u}}}\big{(}\psi^{({\bm{v}},\bm{s})}(w)-1\big{)}\frac{1}{\big{(}f(w)+1\big{)}^{j_{2}}}\frac{{\rm
d}w}{(w-v)^{k_{1}}(w-\overline{v})^{k_{2}}},\end{split}$

where the sum is over all index sets
$J_{1},J_{2},\ldots,J_{j_{1}},J_{1}^{\prime},J_{2}^{\prime},\ldots,J_{j_{2}}^{\prime}$
such that
$\bigcup_{u=1}^{j_{1}}J_{u}\cup\bigcup_{u=1}^{j_{2}}J^{\prime}_{u}=[\\![{1,r}]\\!]$,
and $j_{1}+j_{2}\leqslant q$. Thanks to the trivial bound (7.12), the
contribution from the error term in (7.23) is at most
$\big{|}{\mathcal{E}}^{({\bm{v}},\bm{s})}(w)\big{|}^{q+1}\eta^{-k-r}=\operatorname{\mathcal{O}}\big{(}(n\eta)^{-q-1}\eta^{-k-r}\big{)}$,
which is smaller than $\Phi_{k,r}(v)$ for $\eta\geqslant n^{\mathfrak{c}-1}$
and $q\geqslant 4R/\mathfrak{c}$. We recall the expression of
$\psi^{({\bm{v}},\bm{s})}(w)$ from (6.20),

$\displaystyle\psi^{({\bm{v}},\bm{s})}(w)-1=\displaystyle\prod_{j=1}^{4p-1}\bigg{(}1-\displaystyle\frac{1}{n(v_{j}-w)}\bigg{)}^{-s_{j}}-1.$

If $|w-v_{j}|\gtrsim\eta$, then $\big{|}1/n(v_{j}-w)\big{|}\lesssim 1/n\eta\ll
1$, so $\psi^{({\bm{v}},\bm{s})}(w)-1$ can be Taylor expanded as a sum of
terms in the form

(7.59) $\displaystyle\prod_{j\in L}\frac{1}{n(w-v_{j})},$

where $L$ is a multi-set of $[\\![{1,4p-1}]\\!]$, with $1\leqslant|L|\leqslant
q$, plus an error $\operatorname{\mathcal{O}}((n\eta)^{-(q+1)})$. Again,
thanks to the trivial bound (7.12), the contribution from such error term in
(7.58) is $\operatorname{\mathcal{O}}((n\eta)^{-(q+1)}\eta^{-(k+r)})$, which
is smaller than $\Phi_{k,r}(v)$. The derivative of (7.59) with respect to
$V_{J^{\prime}_{u}}$ is nonzero only if $J^{\prime}_{u}\subseteq L$. If this
is the case, then

(7.60) $\displaystyle\partial_{V_{J^{\prime}_{u}}}\prod_{j\in
L}\frac{1}{n(w-v_{j})}=\prod_{j\in J^{\prime}_{u}}\frac{\text{multiplicity of
$j$ in $L$}}{n(w-v_{j})^{2}}\prod_{j\in L\setminus
J^{\prime}_{u}}\frac{1}{n(w-v_{j})},$

where $L\setminus J^{\prime}_{u}$ is the multi-set obtained from $L$ by
removing a copy of each element in $J_{u}^{\prime}$.

After specializing $v_{j}\in\\{v,\overline{v}\\}$, the right side of (7.60) is
in the form

$\displaystyle\frac{1}{n^{\ell_{1u}+\ell_{2u}-|J_{u}^{\prime}|}(w-v)^{\ell_{1u}}(w-\overline{v})^{\ell_{2u}}},\quad
2|J^{\prime}_{u}|\leqslant\ell_{1u}+\ell_{2u}\leqslant q+|J^{\prime}_{u}|.$

Therefore after specializing $v_{j}\in\\{v,\overline{v}\\}$, the leading order
term of
$\prod_{u=1}^{j_{2}}\partial_{V_{J^{\prime}_{u}}}\big{(}\psi^{({\bm{v}},\bm{s})}(w)-1\big{)}$
is a linear combination (with bounded coefficients) of terms of the form

(7.61)
$\displaystyle\frac{1}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J^{\prime}_{u}|}(w-v)^{\ell_{1}}(w-\overline{v})^{\ell_{2}}},\quad
2\sum_{u}|J^{\prime}_{u}|\leqslant\ell_{1}+\ell_{2}\leqslant
j_{2}q+\sum_{u=1}^{j_{2}}|J_{u}^{\prime}|,$

where $\ell_{1}=\sum_{u}\ell_{1u}$ and $\ell_{2}=\sum_{u}\ell_{2u}$. By
plugging (7.61) into (7.58), we get

(7.62)
$\displaystyle\begin{split}\frac{1}{2\pi\mathrm{i}}&\displaystyle\frac{1}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J^{\prime}_{u}|}}\oint_{\omega}\prod_{u=1}^{j_{1}}\partial_{V_{J_{u}}}\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}\frac{1}{\big{(}f(w)+1\big{)}^{j_{2}}}\frac{{\rm
d}w}{(w-v)^{k_{1}+\ell_{1}}(w-\overline{v})^{k_{2}+\ell_{2}}}\\\
&=-\frac{1}{2\pi\mathrm{i}}\displaystyle\frac{1}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J^{\prime}_{u}|}}\oint_{\omega_{v,\overline{v}}}\prod_{u=1}^{j_{1}}\partial_{V_{J_{u}}}\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}\frac{1}{\big{(}f(w)+1\big{)}^{j_{2}}}\frac{{\rm
d}w}{(w-v)^{k_{1}+\ell_{1}}(w-\overline{v})^{k_{2}+\ell_{2}}}\\\
&\qquad+\frac{1}{2\pi\mathrm{i}}\displaystyle\frac{1}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J^{\prime}_{u}|}}\oint_{\omega+}\prod_{u=1}^{j_{1}}\partial_{V_{J_{u}}}\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}\frac{1}{\big{(}f(w)+1\big{)}^{j_{2}}}\frac{{\rm
d}w}{(w-v)^{k_{1}+\ell_{1}}(w-\overline{v})^{k_{2}+\ell_{2}}},\end{split}$

where the contour $\omega_{v,\overline{v}}$ encloses $v,\overline{v}$, and the
contour $\omega+\subseteq\mathscr{D}(2{\mathfrak{r}})$ encloses
$[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]$ and $v,\overline{v}$. See
Figure 13. Thanks to (7.11), the first term on the right side of (7.62)
decomposes to sum of terms of the form

(7.63)
$\displaystyle\frac{1}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J^{\prime}_{u}|}}{\mathcal{D}}_{v}^{(\hat{k}_{1})}\left(\prod_{u=1}^{j_{1}}\partial_{V_{J_{u}}}({\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w))\right){\mathcal{D}}_{v}^{(\hat{k}_{2})}\left(\frac{1}{f(w)+1}\right)^{j_{2}},$

where
$\hat{k}_{1}+\hat{k}_{2}=k_{1}+k_{2}+\ell_{1}+\ell_{2}-1=k+\ell_{1}+\ell_{2}$.

In the following we show that under our hypothesis (7.54), for
$0\leqslant\hat{k}+|\hat{J}|\leqslant 2q$,

(7.64)
$\displaystyle\left|{\mathcal{D}}_{v}^{(\hat{k})}(\partial_{V_{\hat{J}}}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})})\right|$
$\displaystyle\lesssim\Phi_{\hat{k},|\hat{J}|}(v),\quad\hat{J}\subseteq[\\![{1,r}]\\!].$

Together with (7.11) and (7.51), (7.64) would imply for $j_{1}\geqslant 1$
that

(7.65)
$\displaystyle{\mathcal{D}}_{v}^{(\hat{k}_{1})}\left(\prod_{u=1}^{j_{1}}\partial_{V_{J_{u}}}\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}\right)\lesssim\sum\prod_{u=1}^{j_{1}}\left|{\mathcal{D}}_{v}^{(\tilde{k}_{u})}\left(\partial_{V_{J_{u}}}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}\right)\right|\lesssim\prod_{u=1}^{j_{1}}\Phi_{\tilde{k}_{u},|J_{u}|}(v)\lesssim\Phi_{\hat{k},\sum_{u}|J_{u}|}(v),$

where the sum is over index sets
$(\tilde{k}_{1},\tilde{k}_{2},\ldots,\tilde{k}_{j_{1}})$ such that
$\sum_{u=1}^{j_{1}}\tilde{k}_{u}=\hat{k}$.

If $\hat{J}=\emptyset$, (7.64) is (7.26). In the following we assume that
$|\hat{J}|\geqslant 1$. For $q\leqslant\hat{k}+|\hat{J}|\leqslant 2q$, (7.64)
follows from the trivial bound
$\left|{\mathcal{D}}_{v}^{(\hat{k})}(\partial_{V_{\hat{J}}}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})})\right|\lesssim
1/\eta^{\hat{k}+|\hat{J}|}=\Phi_{\hat{k},|\hat{J}|}$. For
$\hat{k}+|\hat{J}|\leqslant q$, we recall the expression of
${\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}$ from (7.9), and we need to estimate

$\displaystyle{\mathcal{D}}_{v}^{(\hat{k})}\left(\frac{f(w+1/n)}{f(w+1/n)+1}\frac{\hat{g}(w)}{\hat{g}(w+1/n)}e^{\frac{1}{n}\psi(w)}\partial_{V_{\hat{J}}}\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\right)\right),$

which, thanks to (7.11), is a linear combination of

(7.66)
$\displaystyle{\mathcal{D}}_{v}^{(\hat{k}-\hat{k}^{\prime})}\left(\frac{f(w+1/n)}{f(w+1/n)+1}\frac{\hat{g}(w)}{\hat{g}(w+1/n)}e^{\frac{1}{n}\psi(w)}\right){\mathcal{D}}_{v}^{(\hat{k}^{\prime})}\partial_{V_{\hat{J}}}\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\right).$

By essentially the same argument as for (7.39), we have

(7.67)
$\displaystyle{\mathcal{D}}_{v}^{(\hat{k}^{\prime})}\partial_{V_{\hat{J}}}\left(\log\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\right)\right)^{j}\lesssim\Phi_{\hat{k}^{\prime},|\hat{J}|}(v),\quad\hat{k}^{\prime}\geqslant
0.$

Using (7.67) as input, we get the following bound for the second term on the
right side of (7.66),

(7.68)
$\displaystyle\begin{split}\phantom{{}={}}&\left|{\mathcal{D}}_{v}^{(\hat{k}^{\prime})}\partial_{V_{\hat{J}}}\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\right)\right|\\\
&\qquad=\left|{\mathcal{D}}_{v}^{(\hat{k}^{\prime})}\left(\frac{1}{j!}\sum_{j=1}^{q-\hat{k}^{\prime}}\partial_{V_{\hat{J}}}\left(\log\left(\frac{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w+1/n)e^{-b(w+1/n)}}{{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)e^{-b(w)}}\right)\right)^{j}+\operatorname{\mathcal{O}}\left(\frac{1}{(n\eta)^{q-\hat{k}^{\prime}+1}\eta^{|\hat{J}|}}\right)\right)\right|\lesssim\Phi_{\hat{k}^{\prime},|\hat{J}|}(v),\end{split}$

for any $\hat{k}^{\prime}\geqslant 0$.

By plugging (LABEL:e:linearl2) into (7.66), if $\hat{k}^{\prime}=\hat{k}$, we
have $|\eqref{e:linearl}|\lesssim\Phi_{\hat{k},|\hat{J}|}(v)$. For
$k^{\prime}\leqslant\hat{k}-1$, using (7.8) and (7.13), we can bound the first
term in (7.29) as

(7.69)
$\displaystyle\left|{\mathcal{D}}_{v}^{(\hat{k}-\hat{k}^{\prime})}\left(\frac{f(w+1/n)}{f(w+1/n)+1}\frac{\hat{g}(w)}{\hat{g}(w+1/n)}e^{\frac{1}{n}\psi(w)}\right)\right|\lesssim\frac{\big{|}\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]\big{|}}{\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{\hat{k}-\hat{k}^{\prime}-1}}.$

By plugging (LABEL:e:linearl2) and (7.69) into (7.66), the sum of terms with
$\hat{k}^{\prime}\leqslant\hat{k}-1$ is bounded by

$\displaystyle\sum_{\hat{k}^{\prime}=0}^{\hat{k}-1}\frac{\big{|}\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]\big{|}}{\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{\hat{k}-\hat{k}^{\prime}-1}}\Phi_{\hat{k}^{\prime},|\hat{J}|}(v)\lesssim\Phi_{\hat{k},|\hat{J}|}(v),$

where we have used the last inequality in (7.13). This finishes the proof of
(7.64).

Now we can use (7.14) and (7.65) to upper bound (7.63). There are two cases,
namely, either $\hat{k}_{2}=0$ or $\hat{k}_{2}\geqslant 1$:

(7.70)
$\displaystyle\begin{split}|\eqref{e:tt1copy2}|&\lesssim\frac{\Phi_{\hat{k}_{1},\sum_{u}|J_{u}|}(v)}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J_{u}^{\prime}|}}\left(\bm{1}(\hat{k}_{2}=0)+\frac{\big{|}\mathop{\mathrm{Im}}[f(v)/(f(v)+1)]\big{|}\bm{1}(\hat{k}_{2}\geqslant
1)}{\mathop{\mathrm{Im}}[v]\operatorname{dist}(v,I)^{\hat{k}_{2}-1}}\right)\\\
&\lesssim\frac{\Phi_{\hat{k}_{1}+\hat{k}_{2},\sum_{u}|J_{u}|}(v)}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J_{u}^{\prime}|}}=\frac{\Phi_{k+\ell_{1}+\ell_{2},\sum_{u}|J_{u}|}(v)}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J_{u}^{\prime}|}}\leqslant\frac{\Phi_{k+2\sum_{u}|J^{\prime}_{u}|,\sum_{u}|J_{u}|}(v)}{n^{\sum_{u}|J_{u}^{\prime}|}}\lesssim\Phi_{k,r}(v),\end{split}$

where for the second to last inequality we used that
$\ell_{1}+\ell_{2}\geqslant 2\sum_{u}|J^{\prime}_{u}|$ from (7.61), and
$\Phi_{k,\ell}(v)/n^{k}$ is deceasing in $k$: for the last inequality, we used
$\sum_{u}|J_{u}^{\prime}|+\sum_{u}|J_{u}|=r$. For the second term in (7.62),
since $w\in\omega+$ is bounded away from $v$, we have

(7.71)
$\displaystyle\begin{split}&\left|\frac{1}{2\pi\mathrm{i}}\displaystyle\frac{1}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J^{\prime}_{u}|}}\oint_{\omega+}\prod_{u=1}^{j_{1}}\partial_{V_{J_{u}}}\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}\frac{1}{\big{(}f(w)+1\big{)}^{j_{2}}}\frac{{\rm
d}w}{(w-v)^{k_{1}+\ell_{1}}(w-\overline{v})^{k_{2}+\ell_{2}}}\right|\\\
&\qquad\lesssim\max_{w\in\omega+}\bigg{|}\frac{\prod_{u=1}^{j_{1}}\partial_{V_{J_{u}}}\big{(}{\mathcal{E}}_{0}^{({\bm{v}},\bm{s})}(w)\big{)}}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J^{\prime}_{u}|}}\bigg{|}\lesssim\frac{\Phi_{0,\sum_{u}|J_{u}|}(v)}{n^{\ell_{1}+\ell_{2}-\sum_{u}|J^{\prime}_{u}|}}\lesssim\frac{\Phi_{0,\sum_{u}|J_{u}|}(v)}{n^{\sum_{u}|J^{\prime}_{u}|}}\lesssim\Phi_{k,r}(v),\end{split}$

where we used (7.64) and $\ell_{1}+\ell_{2}\geqslant
2\sum_{u}|J^{\prime}_{u}|$ from (7.61) in the second line. We conclude from
plugging (7.70) and (LABEL:e:outomega) into (7.62)

$\displaystyle\Big{|}{\mathcal{D}}_{z}^{(k)}\partial_{v_{1}}\partial_{v_{2}}\cdots\partial_{v_{r}}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}-b(w)\big{)}\big{|}_{z=v,v_{j}=v_{j+2p-1}\in\\{v,\bar{v}\\}}\Big{|}=|\eqref{e:toest3}|\lesssim\Phi_{k,r}(v).$

This finishes the proof of 7.1. ∎

Now we can establish the second statement of 6.4, given by (6.15).

###### Proof of (6.15).

We recall from (7.47) and (7.48) that (6.15) follows from the estimates of the
derivatives of $\log{\mathcal{A}}^{({\bm{v}},\bm{s})}-b$:
$\big{|}{\mathcal{D}}_{z}^{(1)}\partial_{V_{J}}\log({\mathcal{A}}^{({\bm{v}},\bm{s})}-b(w))|_{z=v,v_{j}=v_{j+2p-1}\in\\{v,\bar{v}\\}}\big{|}\lesssim\Phi_{1,{|J|}}(v)$,
for any $J\subseteq[\\![{1,2p-1}]\\!]$, where $\Phi_{1,|J|}(v)$ is explicitly
given in (7.50). Thanks to Propositions 7.2, if we have some weak estimates
for
$\big{|}{\mathcal{D}}_{z}^{(k)}\partial_{V_{J}}(\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w))|_{v_{j}=v_{j+2p-1}\in\\{v,\overline{v}\\}}\big{|}$,
then it implies better estimates. Next we show that (7.52) holds for
$\operatorname{dist}\big{(}\\{z,v\\},[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}\asymp
n$. In fact, for $J=\emptyset$, (7.52) follows from 6.7. For $|J|\geqslant 1$,
thanks to (7.21), and the trivial bound (7.12) (with $\eta\asymp n$), we have

$\displaystyle\Big{|}{\mathcal{D}}_{z}^{(k)}\partial_{V_{J}}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\big{|}_{v_{j}=v_{j+2p-1}\in\\{v,\bar{v}\\}}\Big{|}\lesssim\eta^{-k-|J|}\lesssim\displaystyle\frac{1}{n^{1\vee|J|}}.$

Thus, (7.52) holds for
$\operatorname{dist}\big{(}\\{z,v\\},[{\mathfrak{a}}^{\prime},{\mathfrak{b}}^{\prime}]\big{)}\asymp
n$.

Then a continuity argument quickly implies that

$\displaystyle\Big{|}{\mathcal{D}}_{z}^{(k)}\partial_{V_{J}}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\big{|}_{v_{j}=v_{j+2p-1}\in\\{v,\overline{v}\\}}\Big{|}\lesssim\displaystyle\frac{1}{n^{1\vee|J|}},\quad\text{holds
for any $z,v\in\mathscr{D}({\mathfrak{r}})$};$

we omit further details, since they are very similar to the one implemented in
the proof of (6.14) (by repeatedly using the first statement of 7.2). In
particular, for any $v\in\mathscr{D}({\mathfrak{r}})$, the assumption (7.54)
in 7.2 holds. Then, by a continuity argument that is again very similar to the
one used in the proof of (6.14) (by repeated use of the second statement of
7.2), we conclude that

$\displaystyle\Big{|}{\mathcal{D}}_{z}^{(k)}\partial_{V_{J}}\big{(}\log{\mathcal{A}}^{({\bm{v}},\bm{s})}(w)-b(w)\big{)}\big{|}_{v_{j}=v_{j+2p-1}\in\\{v,\overline{v}\\}}\Big{|}\lesssim\Phi_{k,|J|}(v),\quad\text{holds
for any $v\in\mathscr{D}$ and
$z\in\\{v,\bar{v}\\}\cup\mathscr{D}({\mathfrak{r}})$}.$

This finishes the proof of (6.15). ∎

###  7.4. Proof of 5.5

Finally, in Section 7.4, we prove 5.5 using 6.4 as input.

###### Proof of 5.5.

We recall the relation between $\mathbb{P}$ and $\mathbb{Q}$ from (6.12), and
$b(z)$ from (6.10). Then

$\displaystyle\partial_{z}b(z)=\displaystyle\frac{1}{2\pi\mathrm{i}}\displaystyle\oint_{\omega}\displaystyle\frac{\log\mathcal{B}(w)dw}{(w-z)^{2}},$

where the contour $\omega\subseteq\Lambda$ contains
$[\mathfrak{a},\mathfrak{b}]$, but not $z$. The transition probability (4.9)
is a special case of $\mathbb{P}$, and so 5.5 follows if we show

(7.72)
$\displaystyle\begin{split}&\phantom{{}={}}\mathbb{E}_{\mathbb{P}}\left[\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}\right]=\partial_{z}b(z)+\operatorname{\mathcal{O}}\left(\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)}\right).\end{split}$

and

(7.73)
$\displaystyle\begin{split}&\phantom{{}={}}\mathbb{E}_{\mathbb{P}}\left|\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}-\mathbb{E}_{\mathbb{P}}\left[\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}\right]\right|^{2p}\\\
&\qquad\qquad\qquad\qquad\qquad\lesssim\left(\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{2}}\right)^{p}+\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n^{2p-1}\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{4p-2}}.\end{split}$

We notice the trivial bound for the exponent involved in $\mathbb{P}$, given
by

(7.74) $\displaystyle\left|\sum_{1\leqslant i,j\leqslant
m}\frac{e_{i}e_{j}}{n^{2}}\kappa(x_{i},x_{j})+\operatorname{\mathcal{O}}(1/n)\right|\lesssim
1.$

Then, by (6.14) and (6.12), (LABEL:e:improve1P) is equivalent to

(7.75)
$\displaystyle\begin{split}\phantom{{}={}}\mathbb{E}_{\mathbb{Q}}&\Bigg{[}\bigg{(}\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}-\partial_{z}b(z)\bigg{)}\\\
&\quad\times\bigg{(}e^{\sum_{i,j}\frac{e_{i}e_{j}}{n^{2}}\kappa(x_{i},x_{j})+\operatorname{\mathcal{O}}(1/n)}-\mathbb{E}_{\mathbb{Q}}\left[e^{\sum_{i,j}\frac{e_{i}e_{j}}{n^{2}}\kappa(x_{i},x_{j})+\mathcal{O}(1/n)}\right]\bigg{)}\Bigg{]}=\operatorname{\mathcal{O}}\left(\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)}\right).\end{split}$

We will prove (7.75) by the Cauchy–Schwarz inequality. In the following, we
estimate the variances of the first and second term in (7.75).

We notice that
$\big{|}\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]/(n\operatorname{Im}z)\big{|}\ll 1$
for $z\in\mathscr{D}$. We can replace the inner expectation in (6.15) by
$\partial_{z}b(z)$, and the error is negligible:

(7.76)
$\displaystyle\begin{split}&\phantom{{}={}}\mathbb{E}_{\mathbb{Q}}\left|\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}-\partial_{z}b(z)\right|^{2p}\\\
&\qquad\qquad\lesssim\left|\mathbb{E}_{\mathbb{Q}}\left[\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}\right]-\partial_{z}b(z)\right|^{2p}\\\
&\qquad\qquad\qquad\qquad+\mathbb{E}_{\mathbb{Q}}\left|\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}-\mathbb{E}_{\mathbb{Q}}\left[\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}\right]\right|^{2p}\\\
&\qquad\qquad\lesssim\left(\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{2}}\right)^{p}+\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n^{2p-1}\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{4p-2}}.\end{split}$

In particular, by taking $p=1$, we have

(7.77)
$\displaystyle\begin{split}\mathbb{E}_{\mathbb{Q}}\left|\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}-\partial_{z}b(z)\right|^{2}\lesssim\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{2}}.\end{split}$

For the second term in (7.75), we recall that $\kappa(z,w)$ is analytic in a
neighborhood of $[{\mathfrak{a}},{\mathfrak{b}}]$. The Laplace transform of
$\sum_{ij}e_{i}e_{j}\kappa(x_{i},x_{j})/n^{2}$ can be computed as an integral
of its expectation under deformation of $\mathbb{Q}$ as

$\displaystyle\mathbb{E}_{\mathbb{Q}}\left[\exp\Bigg{(}t\sum_{1\leq i,j\leq
m}\frac{e_{i}e_{j}}{n^{2}}\kappa(x_{i},x_{j})\Bigg{)}\right]=1+\int_{0}^{t}\mathbb{E}_{\mathbb{Q}}\left[\sum_{1\leqslant
i,j\leqslant
n}\frac{e_{i}e_{j}}{n^{2}}\kappa(x_{i},x_{j})\exp\Bigg{(}\theta\sum_{1\leq
i,j\leq m}\frac{e_{i}e_{j}}{n^{2}}\kappa(x_{i},x_{j})\Bigg{)}\right]{\rm
d}\theta.$

This can be analyzed the same way as in [Hua21, Section 8.2], using the loop
equation of the deformed measure, and it gives

(7.78) $\displaystyle\mathbb{E}_{\mathbb{Q}}\left|\exp\bigg{(}\sum_{1\leq
i,j\leq
m}\frac{e_{i}e_{j}}{n^{2}}\kappa(x_{i},x_{j})\bigg{)}-\mathbb{E}_{\mathbb{Q}}\Bigg{[}\exp\bigg{(}\sum_{1\leq
i,j\leq
m}\frac{e_{i}e_{j}}{n^{2}}\kappa(x_{i},x_{j})\bigg{)}\Bigg{]}\right|^{2}\lesssim\frac{1}{n}.$

Then using (7.74), we conclude from (7.78) that

(7.79) $\displaystyle\mathbb{E}_{\mathbb{Q}}\left|\exp\bigg{(}\sum_{1\leq
i,j\leq
m}\frac{e_{i}e_{j}}{n^{2}}\kappa(x_{i},x_{j})+\operatorname{\mathcal{O}}(1/n)\bigg{)}-\mathbb{E}_{\mathbb{Q}}\Bigg{[}\exp\bigg{(}\sum_{ij}\frac{e_{i}e_{j}}{n^{2}}\kappa(x_{i},x_{j})\bigg{)}\Bigg{]}\right|^{2}\lesssim\frac{1}{n}.$

Using (7.77) and (7.79), (7.75) follows from the Cauchy-Schwarz inequality,

$\displaystyle|\eqref{e:smallg}|\lesssim\sqrt{\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{2}}}\frac{1}{\sqrt{n}}=\operatorname{\mathcal{O}}\left(\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)}\right).$

This finishes the proof of (LABEL:e:improve1P).

To prove (LABEL:e:improve2P), similarly to in (LABEL:e:improve2w), using
(LABEL:e:improve1P) as input, we can replace the inner expectation in
(LABEL:e:improve2P) by $\partial_{z}b(z)$. In this way, (LABEL:e:improve2P) is
equivalent to

(7.80)
$\displaystyle\mathbb{E}_{\mathbb{P}}\left|\sum_{i=1}^{m}\frac{1}{z-x_{i}-e_{i}/n}-\frac{1}{z-x_{i}}-\partial_{z}b(z)\right|^{2p}\lesssim\left(\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{2}}\right)^{p}+\frac{|\mathop{\mathrm{Im}}[f(z)/(f(z)+1)]|}{n^{2p-1}\mathop{\mathrm{Im}}[z]\operatorname{dist}(z,I)^{4p-2}}.$

Thanks to the relation (6.12) between $\mathbb{P}$ and $\mathbb{Q}$, and the
trivial bound (7.74), we conclude (7.80) from (LABEL:e:improve2w) and (7.79).
∎

##  Appendix A Thin Slice of Polygonal Domains

In this section, we check that thin slices of polygonal domains with its
(tilted) limiting continuum height function satisfy Assumption 2.3. Throughout
this section, we recall the notation from Section 2.2. We will focus on
polygonal domains $\mathfrak{P}\subset\mathbb{R}^{2}$, given as follows.

######  Definition A.1.

A subset $\mathfrak{P}\subset\mathbb{R}^{2}$ is _polygonal_ if it is simply-
connected and its boundary $\partial\mathfrak{P}$ consists of a finite union
of line segments, each of which is parallel to an axis of $\mathbb{T}$. The
domain333We assume throughout that all vertices of $n\mathfrak{P}$ are in
$\mathbb{Z}^{2}$. $\mathsf{P}=\mathsf{P}_{n}=n\mathfrak{P}\subset\mathbb{T}$,
is then tileable and is therefore associated with a (unique, up to global
shift) boundary height function $\mathsf{h}=\mathsf{h}_{n}$. By translating
$\mathfrak{P}$ if necessary, we will assume that
$\mathfrak{P}\subset\mathbb{R}\times\mathbb{R}_{\geqslant 0}$ and that
$(0,0)\in\partial\mathfrak{P}$. Then by shifting $\mathsf{h}$ if necessary, we
will further suppose that $\mathsf{h}(0,0)=0$. Under this notation, we set
$h:\partial\mathfrak{P}\rightarrow\mathbb{R}$ by $h(u)=n^{-1}\mathsf{h}(nu)$
for each $u\in\partial\mathfrak{P}$. Moreover, we abbreviate
$\mathrm{Adm}(\mathfrak{P})=\mathrm{Adm}(\mathfrak{P};h)$;
$\mathfrak{L}(\mathfrak{P})=\mathfrak{L}(\mathfrak{P};h)$; and
$\mathfrak{A}(\mathfrak{P})=\mathfrak{A}(\mathfrak{P};h)$, they do not depend
on the above choice of global shift fixing $h$.

The following result from [KO07, ADPZ20] describes properties of the limit
shape $H^{*}$ and complex slope $f_{t}(x)$ for polygonal domains
${\mathfrak{P}}$ as in Definition A.1.

######  Proposition A.2 ([KO07, ADPZ20]).

Adopt the notation of Definition A.1, and assume that the domain
$\mathfrak{R}=\mathfrak{P}$ polygonal with at least $6$ sides. Then following
statements hold.

  1. (1)

For any $(x,t)\in\mathfrak{P}\setminus\mathfrak{L}(\mathfrak{P})$, we have
$\nabla H^{*}(x,t)\in\big{\\{}(0,0),(1,0),(1,-1)\big{\\}}$.

  2. (2)

The arctic boundary $\mathfrak{A}=\mathfrak{A}(\mathfrak{P})$ is an algebraic
curve, and its singularities are all either ordinary cusps or tacnodes.

  3. (3)

Fix $(x_{0},t_{0})\in\overline{\mathfrak{L}}$. There exists a neighborhood
$\mathfrak{U}\subset\mathbb{C}^{2}$ of $(x_{0},t_{0})$ and a real analytic
function $Q_{0}:\mathfrak{U}\rightarrow\mathbb{C}$ such that, for any
$(x,t)\in\mathfrak{U}\cap\overline{\mathfrak{L}}$, we have

(A.1) $\displaystyle
Q_{0}\big{(}f_{t}(x)\big{)}=x\big{(}f_{t}(x)+1\big{)}-tf_{t}(x).$

There exists a nonzero rational function $Q$ such that, for any
$(x,t)\in\overline{\mathfrak{L}}$, we have

(A.2) $\displaystyle
Q\bigg{(}f_{t}(x),x-\displaystyle\frac{tf_{t}(x)}{f_{t}(x)+1}\bigg{)}=0.$

  4. (4)

For any $(x,t)\in\overline{\mathfrak{L}}$,
$(x,t)\in\mathfrak{A}({\mathfrak{P}})$ if and only if
$\big{(}f_{t}(x),x-tf_{t}(x)/(f_{t}(x)+1)\big{)}$ is a double root of $Q_{0}$
from (A.1).

###### Proof.

A.2 follows essentially from results in [KO07, ADPZ20]. Its first statement is
[ADPZ20, Theorem 1.10], and the second statement follows from [ADPZ20, Theorem
1.2, Theorem 1.10]. It follows from [ADPZ20, Theorem 1.5] that the complex
slope $f_{t}(x)$ extends continuously to a real number on the arctic boundary.

For any connected component
${\mathfrak{L}}^{\prime}\subset{\mathfrak{L}}({\mathfrak{P}})$ of the liquid
region, with arctic boundary ${\mathfrak{A}}^{\prime}$, we consider the
following map from the closure of the liquid region
$\overline{\mathfrak{L}}^{\prime}={\mathfrak{L}}^{\prime}\cup{\mathfrak{A}}^{\prime}$
to $\mathbb{CP}^{2}$

(A.3)
$\displaystyle(x,t)\in\overline{\mathfrak{L}}\mapsto(f(x,t),z(x,t)):=\left(f_{t}(x),x-t\frac{f_{t}(x)}{f_{t}(x)+1}\right)\in\mathbb{CP}^{2}.$

The map (A.3) restricted to the liquid region ${\mathfrak{L}}^{\prime}$ is an
injection to its image, since we can recover $(x,t)$ from its image as

(A.4) $\displaystyle t=-\frac{{\rm Im}[z(x,t)]}{{\rm
Im}\left[f(x,t)/(f(x,t)+1)\right]},\quad x=z(x,t)+t\frac{f(x,t)}{f(x,t)+1}.$

Moreover, for $(x,t)$ in the liquid region ${\mathfrak{L}}^{\prime}$,
$\mathop{\mathrm{Im}}[f(x,t)]<0$ and the complex Burgers equation (2.2)
implies
$\partial_{t}f(x,t)/\partial_{x}f(x,t)=-f(x,t)/(f(x,t)+1)\not\in{\mathbb{R}}$.
Therefore, the first coordinate of the map (A.3), $(x,t)\mapsto f(x,t)$ is a
local homeomorphism and there exists a function $Q_{0}$ such that
$z(x,t)=Q_{0}(f(x,t))/(f(x,t)+1)$ in a small neighborhood of
${\mathfrak{U}}\subseteq{\mathfrak{L}}^{\prime}$ of $(x_{0},t_{0})$. The
complex Burgers equation (2.2) implies that $Q_{0}$ satisfies the
Cauchy–Riemann equations. It follows that $Q_{0}$ is analytic, and the image
of the map (A.3) restricted to the liquid region ${\mathfrak{L}}^{\prime}$
gives a Riemann surface:
$\\{(f,z)=(f(x,t),z(x,t))\in\mathbb{C}^{2}:(x,t)\in{\mathfrak{L}}^{\prime}\\}$
with local charts given by $Q_{0}$.

Assuming ${\mathfrak{L}}^{\prime}$ is simply connected (we will show it in
A.3), [ADPZ20, Theorem 5.1] gives a decomposition of the map $(x,t)\mapsto
f(x,t)$. It implies for any $(x_{0},t_{0})\in{\mathfrak{A}}^{\prime}$, there
exists a small neighborhood ${\mathfrak{U}}$ of it, such that the map
$(x,t)\mapsto f(x,t)$ is injective for
$(x,t)\in{\mathfrak{U}}\cap\overline{\mathfrak{L}}^{\prime}$. Thus the same
argument as above, there exists a continuous map $Q_{0}$ such that
$z(x,t)=Q_{0}(f(x,t))/(f(x,t)+1)$ for
$(x,t)\in{\mathfrak{U}}\cap\overline{{\mathfrak{L}}^{\prime}}$, where
${\mathfrak{U}}$ is a small neighborhood of
$(x_{0},t_{0})\in{\mathfrak{A}}^{\prime}$, and $Q_{0}$ is analytic on
$f({\mathfrak{U}}\cap{\mathfrak{L}}^{\prime})$. By Schwartz reflection
principle, we can extends $Q_{0}$ to a real analytic function in a
neighborhood of $f(x_{0},t_{0})$. This and the discussion above gives (A.1).

Since the image of the arctic boundary ${\mathfrak{A}}^{\prime}$ under the map
(A.3) is real, we can glue the image of the map (A.3) restricted to the liquid
region ${\mathfrak{L}}^{\prime}$, and its complex conjugate along the image of
the arctic boundary to get an immersed Riemann surface in $\mathbb{CP}^{2}$:
$\\{(f,z)=(f(x,t),z(x,t))\in\mathbb{CP}^{2}:(x,t)\in\overline{\mathfrak{L}}^{\prime}\\}\cup\\{(f,z)=(\overline{f(x,t)},\overline{z(x,t)})\in\mathbb{CP}^{2}:(x,t)\in\overline{\mathfrak{L}}^{\prime}\\}$.
Since $Q_{0}$ is real analytic, the local charts around any point
$(f(x_{0},t_{0}),z(x_{0},t_{0}))$ for
$(x_{0},t_{0})\in{\mathfrak{A}}^{\prime}$ on the arctic boundary are given by
$z=Q_{0}(f)/(f+1)$. $f(x,t)$ with its complex conjugate and $z(x,t)$ with its
complex conjugate are two meromorphic function on this Riemann surface. By
[Jos13, Theorem 5.8.1], there exists a rational function $Q^{\prime}$ such
that $Q^{\prime}(f(x,t),z(x,t))=0$ for any
$(x,t)\in\overline{{\mathfrak{L}}^{\prime}}$, which gives (A.2) by taking
product of $Q^{\prime}$ corresponding to each connected component of
${\mathfrak{L}}({\mathfrak{P}})$.

The fourth statement follows from the facts that
$(x,t)\in\mathfrak{A}(\mathfrak{P})$ if and only if $f_{t}(x)\in\mathbb{R}$,
by (2.6), and that any root of $Q_{0}$ is real if and only if it is a double
root, as $Q_{0}$ is real anaytic (see also the discussion at the end of [KO07,
Section 1.6]).

∎

######  Lemma A.3.

Adopting the notation of A.2, any connected component
$\mathfrak{L}^{\prime}\subseteq\mathfrak{L}(\mathfrak{P})$ is simply
connected.

###### Proof.

Assume to the contrary that ${\mathfrak{L}}^{\prime}$ is not simply connected.
Then there exists a closed curve $\gamma\subseteq{\mathfrak{L}}^{\prime}$,
which is not homotopy equivalent to a point in ${\mathfrak{L}}^{\prime}$. It
follows that the curve $\gamma$ encloses a frozen region
${\mathfrak{F}}\subseteq{\mathfrak{P}}\setminus\overline{{\mathfrak{L}}({\mathfrak{P}})}$.
From Definition A.1, the polygonal domain ${\mathfrak{P}}$ is simply
connected, so the curve $\gamma$ separates the frozen region ${\mathfrak{F}}$
from the boundary $\partial{\mathfrak{P}}\subset\overline{{\mathfrak{P}}}$;
see Figure 14. We denote the boundary of ${\mathfrak{F}}$ by
${\mathfrak{A}}^{\prime}=\partial\mathfrak{F}\subseteq{\mathfrak{A}}({\mathfrak{P}})$.

$\partial{\mathfrak{P}}$${\mathfrak{F}}$$\gamma$${\mathfrak{F}}$${\mathfrak{A}}^{\prime}$${\mathfrak{L}}({\mathfrak{P}})$$\xi$$u$$v$
Figure 14.  Shown to the left is the frozen region ${\mathfrak{F}}$, which is
separated from the boundary $\partial{\mathfrak{P}}$ by a curve
$\gamma\in{\mathfrak{L}}^{\prime}$. Shown to the right are two points $u,v$ on
the boundary ${\mathfrak{A}}^{\prime}$ of ${\mathfrak{F}}$ such that
$H^{*}(v)>H^{*}(u)$.

By the first part of A.2, for almost any $(x,t)\in\mathfrak{F}$ we have
$\nabla H^{*}\in\big{\\{}(0,0),(1,0),(1,-1)\big{\\}}$. We consider two cases
for the behavior of $\nabla H^{*}$ inside ${\mathfrak{F}}$; either $\nabla
H^{*}|_{\mathfrak{F}}$ is continuous or discontinuous. If $\nabla H^{*}$ is
discontinuous at some point $w\in{\mathfrak{F}}$, then [DSS10, Theorem 1.3]
implies that there exists a segment connecting $w$ and
$\partial{\mathfrak{P}}$ staying outside the liquid region, which contradicts
the fact that $\gamma\in{\mathfrak{L}}^{\prime}$ separates $\mathfrak{F}$ from
$\partial{\mathfrak{P}}$. In the second case, $\nabla H^{*}$ is continuous
inside ${\mathfrak{F}}$; thus, it takes a constant value in
$\big{\\{}(0,0),(1,0),(1,-1)\big{\\}}$. Without loss of generality, we may
assume that $\nabla H^{*}\equiv(0,0)$ on ${\mathfrak{F}}$, so
$H^{*}|_{\mathfrak{F}}$ is constant. By continuity, $H^{*}$ is the same
constant on the boundary ${\mathfrak{A}}^{\prime}$ of ${\mathfrak{F}}$.

By the second part, the arctic boundary ${\mathfrak{A}}^{\prime}$ is an
algebraic curve with finitely many singularities, which are all either first
order cusps or tacnodes; by [ADPZ20, Theorem 1.3(c)], at every nonsingular
point $\xi\in{\mathfrak{A}}^{\prime}$, the boundary ${\mathfrak{A}}^{\prime}$
is locally strictly convex; and, by [ADPZ20, Theorem 1.5, Theorem 1.10], the
unit tangent vector $\tau(\xi)$ to ${\mathfrak{A}}^{\prime}$ at any
$\xi\in\mathfrak{A}^{\prime}$ depends continously on $\xi$. Therefore, when
$\xi$ moves along ${\mathfrak{A}}^{\prime}$, the tangent vector angle winds
around the unit circle $\mathbb{S}^{1}$ at least once. Hence, there exists a
nonsingular point $\xi\in{\mathfrak{A}}^{\prime}$ such that the unit tangent
vector $\tau(\xi)$ has positive $x$-coordinate and negative $y$-coordinate.

Shift the tangent line at $\xi$ in the normal direction to
$\mathfrak{A}^{\prime}$, slightly into the liquid region, such that it
intersects ${\mathfrak{A}}^{\prime}$ at two points, $u$ and $v$. By the local
strict convexity of $\mathfrak{A}^{\prime}$ at $\xi$, the segment from $u$ to
$v$ (which is parallel to $\tau(\xi)$) remains strictly inside the liquid
region $\mathfrak{L}(\mathfrak{P})$; see Figure 14. In the liquid region,
$\nabla H^{*}$ stays inside strictly the triangle $\mathcal{T}$ from (2.1).
Since $v-u$ is parallel to $\tau(\xi)$, which has positive $x$-coordinate and
negative $y$-coordinate, it follows that $\nabla H^{*}(z)\cdot\tau(\xi)>0$ for
any $z\in\mathfrak{L}(\mathfrak{P})$. Thus,

(A.5) $\displaystyle H^{*}(v)-H^{*}(u)=\int_{0}^{1}\nabla H^{*}\big{(}\theta
v+(1-\theta)u\big{)}\cdot(v-u)\rm d\theta>0,$

which contradicts the fact that $H^{*}$ is constant on
${\mathfrak{A}}^{\prime}$. Hence, ${\mathfrak{L}}^{\prime}$ is simply
connected. ∎

######  Assumption A.4.

Under the notation of Definition A.1, assume the following four properties
hold.

  1. (1)

The arctic boundary $\mathfrak{A}=\mathfrak{A}(\mathfrak{P})$ has no tacnode
singularities.

  2. (2)

No cusp singularity of $\mathfrak{A}$ is also a tangency location of
$\mathfrak{A}(\mathfrak{P})$.

  3. (3)

There exists an axis $\ell$ of $\mathbb{T}$ such that any line connecting two
distinct cusp singularities of $\mathfrak{A}$ is not parallel to $\ell$.

  4. (4)

Any intersection point between $\mathfrak{A}$ and $\partial\mathfrak{P}$ must
be a tangency location of $\mathfrak{A}$.

It has been proven in [AH21, Proposition 4.18], under A.4, for any
$u\in\overline{{\mathfrak{L}}({\mathfrak{P}})}$, there exists a trapezoid
${\mathfrak{D}}(u)\subseteq{\mathfrak{P}}\cap({\mathbb{R}}\times[{\mathfrak{t}}_{1},{\mathfrak{t}}_{2}])$
in the form (2.8), containing $u$ (flip ${\mathfrak{P}}$ if necessary), with
liquid region ${\mathfrak{L}}$, and complex slope
$f:{\mathfrak{L}}\mapsto\mathbb{H}^{-}$ satisfying 2.7. For all
${\mathfrak{t}}_{1}\leqslant t\leqslant{\mathfrak{t}}_{2}$, we define slices
of the liquid region (along the horizontal line $y=t$) by
$I_{t}=\big{\\{}x:(x,t)\in\overline{\mathfrak{L}}\big{\\}}$. Then the
following constraints hold:

  1. (1)

The boundary height function $h$ is constant along both
$\partial_{\operatorname{ea}}(\mathfrak{D})$ and
$\partial_{\operatorname{we}}(\mathfrak{D})$.

  2. (2)

Either $\partial_{\operatorname{no}}(\mathfrak{D})$ is packed with respect to
$h$ or there exists
$\mathfrak{t}^{\prime}>\mathfrak{t}_{2}-{\mathfrak{t}}_{1}$ such that $H^{*}$
admits an extension to time ${\mathfrak{t}}_{1}+\mathfrak{t}^{\prime}$.

  3. (3)

There exists $\widetilde{\mathfrak{t}}\in[\mathfrak{t}_{1},\mathfrak{t}_{2}]$
such that the following holds. For
$t\in[\mathfrak{t}_{1},\widetilde{\mathfrak{t}}]$, the set $I_{t}$ consists of
one nonempty interval, and for
$t\in(\widetilde{\mathfrak{t}},\mathfrak{t}_{2}]$, the set $I_{t}$ consists of
two nonempty disjoint intervals.444Observe if
$\widetilde{\mathfrak{t}}=\mathfrak{t}_{2}$, then $I_{t}$ always only consists
of one nonempty interval.

  4. (4)

Any tangency location along $\partial\overline{\mathfrak{L}}$ is of the form
$\min I_{t}$ or $\max I_{t}$, for some
$t\in(\mathfrak{t}_{1},\mathfrak{t}_{2})$. At most one tangency location is of
the form $\min I_{t}$, and at most one is of the form $\max I_{t}$.

  5. (5)

There exists an algebraic curve $Q$ such that, for any
$(x,t)\in\overline{\mathfrak{L}}$, we have

(A.6) $\displaystyle
Q\bigg{(}f_{t}(x),x-\displaystyle\frac{tf_{t}(x)}{f_{t}(x)+1}\bigg{)}=0.$

Furthermore, the curve $Q$ “approximately comes from a polygonal domain” in
the following sense. There exists a polygonal domain $\mathfrak{P}$ satisfying
A.4 with liquid region $\mathfrak{L}(\mathfrak{P})$; a connected component
$\mathfrak{L}_{0}(\mathfrak{P})\subseteq\mathfrak{L}(\mathfrak{P})$; and a
real number $\alpha\in\mathbb{R}$ with $|\alpha-1|<n^{-\delta}$ such that, if
$Q_{\mathfrak{L}_{0}}$ is the algebraic curve associated with
$\mathfrak{L}_{0}$ from A.2, then

(A.7) $\displaystyle
Q(u,v)=\displaystyle\frac{u+1}{\alpha^{-1}u+1}Q_{\mathfrak{L}_{0}}(\alpha^{-1}u,v).$

We remark that in (A.7), if $\alpha=1$, then the height function $H^{*}$ on
${\mathfrak{D}}$ is the restriction of the limiting continuum height function
on ${\mathfrak{P}}$. For $\alpha\neq 1$ and $|\alpha-1|<n^{-\delta}$, $H^{*}$
is a slightly tilted version of that.

With a slight abuse of notation, we denote the shifted version of
${\mathfrak{D}},{\mathfrak{P}},H^{*},f$ by time ${\mathfrak{t}}_{1}$, still by
${\mathfrak{D}},{\mathfrak{P}},H^{*},f$, and
${\mathfrak{t}}={\mathfrak{t}}_{2}-{\mathfrak{t}}_{1}$. Then the trapezoid
domain ${\mathfrak{D}}$ and limiting continuum height function $H^{*}$
satisfies the first four statements in 2.3. In particular, thanks to
Assumption A.4, the cusp location on ${\mathfrak{A}}({\mathfrak{D}})$ (if it
exists) is not a tangency location. We denote the extended trapezoid domain by
${\mathfrak{D}}^{\prime}\subseteq{\mathbb{R}}\times[0,{\mathfrak{t}}^{\prime}]$.
Let
${\mathfrak{D}}^{t}={\mathfrak{D}}^{\prime}\cap({\mathbb{R}}\times[t,{\mathfrak{t}}^{\prime}])$.
And we still denote the extended complex slope constructed using (A.6) and
(A.7) by $f$, and the extended liquid region
$\widetilde{\mathfrak{L}}={\mathfrak{L}}({\mathfrak{D}}^{\prime})$. Then it
follows from A.2, (A.6) and (A.7), Item (a) and (b) in the fifth statement of
2.3 holds. Next we show that (2.14) is a bijection to its image provided
${\mathfrak{t}}^{\prime}$ is small enough.

######  Proposition A.5.

There exists a constant $c=c({\mathfrak{P}})>0$. If $\mathfrak{t}^{\prime}<c$,
then for any $t\in[0,{\mathfrak{t}}^{\prime}]$ the map

(A.8)
$\displaystyle\pi:(x,r)\in{\mathfrak{L}}({\mathfrak{D}})\cap{\mathfrak{D}}^{t}\mapsto
x+(t-r)\frac{f_{r}(x)}{f_{r}(x)+1}\in\mathbb{H}^{+}$

is a bijection.

###  A.1. Proof of Propositions A.5

To show (A.8) is a bijection, we will use the following criterion for the
global homemorphism from [MO63, Theorem 1].

######  Theorem A.6.

Fix an integer $d\geqslant 2$. Let $X\subset\mathbb{R}^{d}$ denote a compact
subset, whose boundary $\partial X$ is an irreducible separating set of
${\mathbb{R}}^{d}$. Let $f:X\rightarrow\mathbb{R}^{d}$ be a continuous mapping
that is locally one-to-one on $X\setminus Z$, where $Z\subset X$ is such that
$Z\cap X^{\circ}$ ($X^{\circ}$ is the interior of $X$) is a discrete set and
$\partial X\setminus Z$ is not empty. If $f|_{\partial X}$ is one-to-one, then
$f$ is a homeomorphism of $X$ onto $f(X)$.

The map $\pi$ in (A.8) satisfies the assumptions of A.6 with $Z$ the critical
point of $\pi$. It is a bijection, if it is one-to-one on the boundary. The
boundary of the liquid region
${\mathfrak{L}}({\mathfrak{D}}^{\prime})\cap{\mathfrak{D}}^{t}$ consists of
three parts: the north boundary, the south boundary, and the arctic curve
boundary

(A.9)
$\displaystyle\big{\\{}(x,{\mathfrak{t}}^{\prime})\in\overline{{\mathfrak{L}}({\mathfrak{D}}^{\prime})}\big{\\}}\cup\big{\\{}(x,t)\in\overline{{\mathfrak{L}}({\mathfrak{D}}^{\prime})}\big{\\}}\cup\big{\\{}(x,r)\in\partial\overline{{\mathfrak{L}}({\mathfrak{D}}^{\prime})}:r\in(t,{\mathfrak{t}}^{\prime})\big{\\}}.$

######  Lemma A.7.

The map (A.8) maps the south boundary and the arctic curve bijectively to an
interval in ${\mathbb{R}}$.

###### Proof.

Any $(x,t)$ on the south boundary
$\big{\\{}(x,t)\in\overline{{\mathfrak{L}}({{\mathfrak{D}}^{\prime}})}\big{\\}}$
is mapped to $z=x$ by (3.22). At any point $(x,r)$ on the arctic boundary of
${\mathfrak{L}}({\mathfrak{D}}^{\prime})\cap{\mathfrak{D}}^{t}$ such that
$(x,r)\in\mathfrak{A}({\mathfrak{D}}^{\prime})$ with
$r\in(t,{\mathfrak{t}}^{\prime})$, the tangent vector to
$(x,r)\in\mathfrak{A}({\mathfrak{D}}^{\prime})$ has slope

(A.10) $\displaystyle\frac{f_{r}(x)+1}{f_{r}(x)},$

by (3.6). $(x,r)$ is mapped to $(f,z)$ by (A.8) as given by

(A.11) $\displaystyle
f=f_{r}(x),\quad\frac{r-t}{x-z}=\frac{f_{r}(x)+1}{f_{r}(x)}.$

Comparing with (A.10), the right side of the second expression in (A.11) is
the slope of the tangent vector of arctic curve at $(x,r)$. Thus $(z,t)$ is
the intersection of the tangent line to the arctic curve at $(x,r)$ with the
bottom boundary of ${\mathfrak{D}}^{t}$; See Figure 15. When we move along the
south boundary and the arctic curve counterclockwise, its image
$z\in{\mathbb{R}}$ also moves in negative direction. The projection map (A.8)
maps the south boundary and the arctic curve bijectively to an interval in
${\mathbb{R}}$.

∎

$y=t$$y={\mathfrak{t}}^{\prime}$${\mathfrak{a}}(t)$${\mathfrak{b}}(t)$$(x,r)$$(z,t)$$\mathfrak{L}({\mathfrak{D}}^{\prime})\cap{\mathfrak{D}}^{t}$
Figure 15.  Any point $(x,r)$ on the arctic curve is mapped to $(f,z)$ by
(3.22), where $(z,t)$ is the intersection of the tangent to the arctic curve
at $(x,r)$, and the bottom boundary of ${\mathfrak{D}}^{t}$ as shown in the
left figure.

###### Proof of A.5.

By our assumption, the north boundary
$\big{\\{}(x,{\mathfrak{t}}^{\prime})\in\overline{{\mathfrak{L}}({{\mathfrak{D}}^{\prime}})}\big{\\}}$
is a single interval. It is mapped by (A.8) to a curve in the upper complex
plane (namely, we have $\mathop{\mathrm{Im}}z>0$), except for at its two end
points (which are mapped to the real line). Thanks to Theorem A.6 and Lemma
A.7, A.5 follows if we show that there exists a universal constant
$c({\mathfrak{P}})>0$ such that, for any $0\leqslant\delta\leqslant
c({\mathfrak{P}})$,

(A.12)
$\displaystyle(x,{\mathfrak{t}}^{\prime})\in{\mathfrak{L}}({\mathfrak{D}}^{\prime})\mapsto
x-\delta\frac{f_{{\mathfrak{t}}^{\prime}}(x)}{f_{{\mathfrak{t}}^{\prime}}(x)+1}\in\mathbb{H}^{+},$

is a bijection onto its image. Thanks to the third statement of Proposition
A.2, locally around $\big{(}f_{t}(x),x\big{)}$, there exists a real analytic
function $Q_{0}$ in one variable such that,

(A.13) $\displaystyle
Q_{0}\big{(}f_{t}(x)\big{)}=x\big{(}f_{t}(x)+1\big{)}-tf_{t}(x).$

In the following, only consider the case $\alpha=1$, for the general case when
$|\alpha-1|=\operatorname{o}(1)$ is entirely analogous. There are two cases:
the north boundary $\partial_{\rm
no}{\mathfrak{L}}({\mathfrak{D}}^{\prime})=\\{(x,{\mathfrak{t}}^{\prime})\in{\mathfrak{L}}({\mathfrak{D}}^{\prime})\\}$
is close to a horizontal tangency location; or $\partial_{\rm
no}{\mathfrak{L}}({\mathfrak{D}}^{\prime})$ is bounded away from horizontal
tangency locations. We discuss these cases separately. Fix a small constant
${\mathfrak{c}}_{0}>0$.

$t={\mathfrak{s}}$$t={\mathfrak{t}}^{\prime}$$t={\mathfrak{t}}^{\prime}-\delta$$\mathscr{D}$$t={\mathfrak{t}}^{\prime}$$t={\mathfrak{t}}^{\prime}-\delta$$(E_{c},t_{c})$$(E_{c},t_{c})$$t={\mathfrak{t}}^{\prime}$$t={\mathfrak{t}}^{\prime}-\delta$
Figure 16.  Shown to the left is the case that the north boundary
$\partial_{\rm
no}{\mathfrak{L}}({\mathfrak{D}}^{\prime})=\\{(x,{\mathfrak{t}}^{\prime})\in{\mathfrak{L}}({\mathfrak{D}}^{\prime})\\}$
is close to a horizontal tangency location. Shown to the middle is the case
that $\partial_{\rm no}{\mathfrak{L}}({\mathfrak{D}}^{\prime})$ is close to a
cusp location $(E_{c},t_{c})$ and ${\mathfrak{t}}^{\prime}>t_{c}$. Shown to
the right is the case that $\partial_{\rm
no}{\mathfrak{L}}({\mathfrak{D}}^{\prime})$ is close to a cusp location
$(E_{c},t_{c})$ and ${\mathfrak{t}}^{\prime}<t_{c}$.

First suppose that $\partial_{\rm no}{\mathfrak{L}}({\mathfrak{D}}^{\prime})$
is close to a horizontal tangency location at time ${\mathfrak{s}}$ with
distance less than ${\mathfrak{c}}_{0}$; see Figure 16 (flip
${\mathfrak{D}}^{\prime}$ if necessary). We can consider the liquid region
$\mathscr{D}$ between times ${\mathfrak{t}}^{\prime}-\delta$ and
${\mathfrak{s}}$, and the projection map

(A.14) $\displaystyle(x,t)\in\mathscr{D}\mapsto
x+({\mathfrak{t}}^{\prime}-\delta-t)\frac{f_{t}(x)}{f_{t}(x)+1}\in\mathbb{H}^{+}.$

Similarly to Lemma A.7, (A.14) maps the boundary $\partial\mathscr{D}$
bijectively to ${\mathbb{R}}\cup\\{\infty\\}$. To use A.6 we further compose
(A.14) with a map $\mathbb{H}^{+}\cup\\{\infty\\}$ to the unit disk
${\mathbb{D}}$. Then by taking $X={\mathbb{D}}$ in A.6 we conclude that (A.14)
is a bijection. By taking $t={\mathfrak{t}}^{\prime}$, we get that (A.12) is a
bijection.

We next address the remaining two cases, namely, when $\partial_{\rm
no}{\mathfrak{L}}({\mathfrak{D}}^{\prime})$ is distance at least
${\mathfrak{c}}_{0}$ away from horizontal tangency locations. Then, there
exists a universal constant ${\mathfrak{C}}_{0}>0$ depending on
${\mathfrak{c}}_{0}$ such that,for $(x,t)$ distance ${\mathfrak{c}}_{0}$ away
from any horizontal tangency location, we have

(A.15)
$\displaystyle\big{|}f_{t}(x)+1\big{|}\geqslant{\mathfrak{C}}_{0}^{-1};\qquad\left|\frac{f_{t}(x)}{f_{t}(x)+1}\right|\leqslant{\mathfrak{C}}_{0}.$

In what follows, we fix small constants
$0<{\mathfrak{c}}<{\mathfrak{c}}^{\prime}$, which will be chosen later (and
will depend only on $\mathfrak{P}$).

For $(x,t)$ at least distance ${\mathfrak{c}}$ away from the arctic curve,
there is a universal constant ${\mathfrak{C}}$ (depending on ${\mathfrak{c}}$)
such that

(A.16)
$\displaystyle\left|\partial_{x}\frac{f_{t}(x)}{f_{t}(x)+1}\right|\leqslant{\mathfrak{C}}.$

Assume that there exist $(X,{\mathfrak{t}}^{\prime})$ and
$(X^{\prime},{\mathfrak{t}}^{\prime})$ with $X<X^{\prime}$ such that

(A.17) $\displaystyle
X-\delta\frac{f_{{\mathfrak{t}}^{\prime}}(X)}{f_{{\mathfrak{t}}^{\prime}}(X)+1}=X^{\prime}-\delta\frac{f_{{\mathfrak{t}}^{\prime}}(X^{\prime})}{f_{{\mathfrak{t}}^{\prime}}(X^{\prime})+1},$

then (A.15) gives that

(A.18)
$\displaystyle|X-X^{\prime}|=\delta\left|\frac{f_{{\mathfrak{t}}^{\prime}}(X)}{f_{{\mathfrak{t}}^{\prime}}(X)+1}-\frac{f_{{\mathfrak{t}}^{\prime}}(X^{\prime})}{f_{{\mathfrak{t}}^{\prime}}(X^{\prime})+1}\right|\leqslant
2\delta{\mathfrak{C}}_{0}\leqslant\displaystyle\frac{{\mathfrak{c}}}{2},$

provided we take $\delta\leqslant{\mathfrak{c}}/8{\mathfrak{C}}_{0}$. If the
interval $\big{\\{}(y,{\mathfrak{t}}^{\prime}):X\leqslant y\leqslant
X^{\prime}\big{\\}}$ is distance ${\mathfrak{c}}$ away from the arctic curve,
then

$\displaystyle|X-X^{\prime}|=\delta\left|\frac{f_{{\mathfrak{t}}^{\prime}}(X)}{f_{{\mathfrak{t}}^{\prime}}(X)+1}-\frac{f_{{\mathfrak{t}}^{\prime}}(X^{\prime})}{f_{{\mathfrak{t}}^{\prime}}(X^{\prime})+1}\right|\leqslant\delta|X-X^{\prime}|\sup_{X\leqslant
y\leqslant
X^{\prime}}\left|\partial_{y}\frac{f_{{\mathfrak{t}}^{\prime}}(y)}{f_{{\mathfrak{t}}^{\prime}}(y)+1}\right|\leqslant\delta{\mathfrak{C}}|X-X^{\prime}|<|X-X^{\prime}|,$

which leads to a contradiction, provided we take $\delta\leqslant
1/2{\mathfrak{C}}$.

It remains to consider the case when the interval
$\big{\\{}(y,{\mathfrak{t}}^{\prime}):X\leqslant y\leqslant
X^{\prime}\big{\\}}$ is of distance at most $\mathfrak{c}$ from the arctic
curve. Let us first consider the case that
$\big{\\{}(y,{\mathfrak{t}}^{\prime}):X\leqslant y\leqslant
X^{\prime}\big{\\}}$ is of distance at most ${\mathfrak{c}}$ from a cusp
location $(E_{c},t_{c})$ of the arctic boundary for $\mathfrak{P}$; see Figure
16 (by our assumption, the cusp point upward). There are two cases, either
${\mathfrak{t}}^{\prime}\geqslant t_{c}$ or ${\mathfrak{t}}^{\prime}\leqslant
t_{c}$. The proofs are essentially the same, so we will only discuss the
former case. We construct a curve $\mathscr{C}$ connecting two points on
$t={\mathfrak{t}}^{\prime}-\delta$, such that $(X,{\mathfrak{t}}^{\prime})$
and $(X^{\prime},{\mathfrak{t}}^{\prime})$ are inside the liquid region
${\mathfrak{L}}({\mathfrak{P}})$ between $\mathscr{C}$ and
$t={\mathfrak{t}}^{\prime}-\delta$. We denote the liquid region between
$\mathscr{C}$ and $t={\mathfrak{t}}^{\prime}-\delta$ by $\mathscr{D}$; see
Figure 17. We will show that the map

(A.19)
$\displaystyle(x,t)\in\mathscr{D}\subseteq{\mathfrak{L}}({\mathfrak{P}})\mapsto
x+({\mathfrak{t}}^{\prime}-\delta-t)\frac{f_{t}(x)}{f_{t}(x)+1}\in\mathbb{H}^{+},$

is a bijection. By taking $t={\mathfrak{t}}^{\prime}$ and $x=X,X^{\prime}$,
this will rule out (A.17). By Lemma A.7, (A.19) maps the bottom boundary of
$\mathscr{D}$ bijectively to an interval in ${\mathbb{R}}$, and $\mathscr{C}$
to a curve in the upper half plane. Thanks to A.6, the claim that (A.19) is a
bijection would follow if we show that the map (A.19) restricted to
$\mathscr{C}$ is one-to-one.

$t={\mathfrak{t}}^{\prime}$$t={\mathfrak{t}}^{\prime}-\delta$$(E_{c},t_{c})$$\mathscr{D}$$\mathscr{C}$$X$$X^{\prime}$$t={\mathfrak{t}}^{\prime}$$t={\mathfrak{t}}^{\prime}-\delta$$(E_{c},t_{c})$$\mathscr{D}$$\mathscr{C}$$X$$X^{\prime}$
Figure 17.  Shown to the left is the curve $\mathscr{C}$ around a cusp
location, connecting two points on $t={\mathfrak{t}}^{\prime}-\delta$, and
$(X,{\mathfrak{t}}^{\prime}),(X^{\prime},{\mathfrak{t}}^{\prime})$ are inside
the liquid region between $\mathscr{C}$ and
$t={\mathfrak{t}}^{\prime}-\delta$. Shown to the right is the curve
$\mathscr{C}$ around a edge point $(E({\mathfrak{s}}),{\mathfrak{s}})$,
connecting one point on $t={\mathfrak{t}}^{\prime}-\delta$ and one point on
the arctic curve, and
$(X,{\mathfrak{t}}^{\prime}),(X^{\prime},{\mathfrak{t}}^{\prime})$ are inside
the liquid region between $\mathscr{C}$ and
$t={\mathfrak{t}}^{\prime}-\delta$. .

Let $f_{c}=f_{t_{c}}(E_{c})$, and denote the map

(A.20)
$\displaystyle(x,t)\in{\mathfrak{L}}({\mathfrak{P}})\mapsto(f,w)=\left(f_{t}(x),x+(t_{c}-t)\frac{f_{t}(x)}{f_{t}(x)+1}\right).$

Here, $w=w(x,t)$ is a function of $x,t$; if the context is clear we will
simply write it as $w$. We can interpret $f=f_{t_{c}}(w)$ as a function of
$w$. We will soon see that after restricted to a fan-shaped neighborhood of
$E_{c}$, $f_{t_{c}}(w)$ is a single valued function. With the above notations,
locally around $(f_{c},E_{c})$ we can rewrite (A.13) as

(A.21) $\displaystyle Q_{1}(f):=\frac{Q_{0}(f)}{f+1}+\frac{t_{c}f}{f+1}=w.$

Since $(E_{c},t_{c})$ is a cusp of the arctic boundary for $\mathfrak{P}$, it
is characterized by the equations

$\displaystyle Q_{0}(f_{c})=E_{c}(f_{c}+1)-t_{c}f_{c};\qquad
Q_{0}^{\prime}(f_{c})=E_{c}-t_{c};\qquad Q_{0}^{\prime\prime}(f_{c})=0.$

We also have that $Q_{0}^{\prime\prime\prime}(f_{c})\asymp 1$, since $Q_{0}$
does not admit any quadruple roots by A.2. It follows from (A.21) (and the
fact that no cusp is simultaneously a tangency location) that

$\displaystyle Q_{1}(f_{c})=E_{c};\qquad Q_{1}^{\prime}(f_{c})=0;\qquad
Q_{1}^{\prime\prime}(f_{c})=0;\qquad|Q_{0}^{\prime\prime\prime}(f_{c})|\asymp
1.$

This, together with (A.21), implies that, locally around $E_{c}$,
$f=f_{t_{c}}(w)$ has cube root behavior, namely,

(A.22) $\displaystyle f_{t_{c}}(w)=f_{c}+\sqrt[3]{C(w)(w-E_{c}}),$

for some real analytic function $C(w)$. Let us discuss the choice for the cube
root in (A.22); recall that in (A.20), we must have
$\mathop{\mathrm{Im}}f=\mathop{\mathrm{Im}}[f_{t}(x)]<0$ and
$\mathop{\mathrm{Im}}w=(t_{c}-t)\mathop{\mathrm{Im}}\big{[}f_{t}(x)/(f_{t}(x)+1)\big{]}>0$.
Therefore, for $t>t_{c}$ in (A.20) (here we used that the cusp points upward),
its image corresponds to the branch of $f_{t_{c}}(w)$ in the lower half plane.
Also $\mathop{\mathrm{Im}}[f_{t_{c}}(x)]=-\pi\partial_{x}H_{t_{c}}(x)<0$ for
$x\in(E_{c}-\varepsilon,E_{c})\cup(E_{c},E_{c}+\varepsilon)$ for some small
$\varepsilon>0$. Thus the branch of $f_{t_{c}}(w)$ has to be the one in the
lower half plane corresponding to the fan-shaped region with argument
$(-2\pi/3,-\pi/3)$; See Figure 18. Then $f_{t_{c}}(w)$ is a single valued
function restricted to a neighborhood of $E_{c}$ after removing the branch
cut.

Next we consider the preimage of (A.20) for $\big{(}f_{t_{c}}(w),w\big{)}$
with $w$ on the circle $w=E_{c}+{\mathfrak{c}}^{\prime}e^{\mathrm{i}\theta}$,
where $\mathfrak{c}^{\prime}$ is chosen so that $f(w)$ is single valued on
this circle (and ${\mathfrak{c}}$ chosen to be smaller than
${\mathfrak{c}}^{\prime}$). Since
$\big{|}f_{t}(x)/(f_{t}(x)+1)\big{|}\leqslant{\mathfrak{C}}_{0}$ by (A.15), we
have

$\displaystyle{\mathfrak{c}}^{\prime}=|w-E_{c}|=\left|x-E_{c}+(t_{c}-t)\frac{f_{t}(x)}{f_{t}(x)+1}\right|\leqslant{\mathfrak{C}}_{0}\big{(}|x-E_{c}|+|t_{c}-t|\big{)}.$

Thus $(x,t)$ is bounded away from $(E_{c},t_{c})$, with
$\operatorname{dist}\big{(}(x,t),(E_{c},t_{c})\big{)}\geqslant{\mathfrak{c}}^{\prime}/2{\mathfrak{C}}_{0}$.
Therefore, if we take
${\mathfrak{c}}\leqslant{\mathfrak{c}}^{\prime}/4{\mathfrak{C}}_{0}$, then by
(A.18) $(X,{\mathfrak{t}}^{\prime})$ and $(X,{\mathfrak{t}}^{\prime})$ will be
inside the preimage curve. For
$\mathop{\mathrm{Im}}[w]\leqslant-{\mathfrak{c}}^{\prime}/2$, the
corresponding $(x,t)$ satisfies

$\displaystyle
t_{c}-t=\frac{\mathop{\mathrm{Im}}[w]}{\mathop{\mathrm{Im}}[f/(f+1)]}=\frac{\mathop{\mathrm{Im}}[w]}{\mathop{\mathrm{Im}}[f]}|f+1|^{2}\geqslant\frac{{\mathfrak{c}}^{\prime}}{2{\mathfrak{C}}_{0}^{2}\mathop{\mathrm{Im}}[\sqrt[3]{C(w){\mathfrak{c}}^{\prime}e^{\mathrm{i}\theta}}]}\geqslant\frac{{\mathfrak{c}}^{\prime}}{C{\mathfrak{C}}_{0}^{2}},$

for some large constant $C$ depending on $C(w)$. If we take $\delta\leqslant
c({\mathfrak{P}})\leqslant{\mathfrak{c}}^{\prime}/C{\mathfrak{C}}_{0}^{2}$,
then the preimage will intersect $t={\mathfrak{t}}^{\prime}-\delta$. This
gives the desired curve $\mathscr{C}$; see Figure 18.

Next we consider the image of $\mathscr{C}$ under the map (A.19)

(A.23) $\displaystyle(x,t)\in\mathscr{C}\mapsto
x+({\mathfrak{t}}^{\prime}-\delta-t)\frac{f_{t}(x)}{f_{t}(x)+1}=w-(t_{c}-{\mathfrak{t}}^{\prime}+\delta)\frac{f_{t_{c}}(w)}{f_{t_{c}}(w)+1},$

where $w=w(x,t)$ is from (A.20). For $(x,t)\in\mathscr{C}$, $w$ is on the
circle $w=E_{c}+{\mathfrak{c}}^{\prime}e^{\mathrm{i}\theta}$. Since it is
bounded away from $E_{c}$, namely $|w-E_{c}|={\mathfrak{c}}^{\prime}$, we have

$\displaystyle\left|\partial_{w}\frac{f_{t_{c}}(w)}{f_{t_{c}}(w)+1}\right|\leqslant\frac{\partial_{w}f_{t_{c}}(w)}{|f_{t_{c}}(w)+1|^{2}}\leqslant{\mathfrak{C}}_{0}^{2}\big{|}\partial_{w}\sqrt[3]{C(w)(w-E_{c})}\big{|}\leqslant
C{\mathfrak{C}}_{0}^{2}({\mathfrak{c}}^{\prime})^{-2/3},$

where the constant $C$ depends on $C(w)$. Therefore, if we take
$|t_{c}-{\mathfrak{t}}^{\prime}+\delta|\leqslant{\mathfrak{c}}+c({\mathfrak{P}})$
much smaller than $({\mathfrak{c}}^{\prime})^{2/3}/C{\mathfrak{C}}_{0}^{2}$
then, by the implicit function theorem, the map

$\displaystyle w\mapsto
w-(t_{c}-{\mathfrak{t}}^{\prime}+\delta)\frac{f_{t_{c}}(w)}{f_{t_{c}}(w)+1},$

is a bijection from the curve $\big{\\{}w(x,t):(x,t)\in\mathscr{C}\big{\\}}$
onto its image. We conclude that the map (A.19) is one-to-one after restricted
to $\mathscr{C}$. Thus, the map (A.19) is a bijection.

$t={\mathfrak{t}}^{\prime}$$t={\mathfrak{t}}^{\prime}-\delta$$t=t_{c}$$\mathscr{C}$$(x,t)$$\mapsto$$w=w(x,t)=E_{c}+{\mathfrak{c}}^{\prime}e^{\mathrm{i}\theta}$$E_{c}$$\mapsto$$f_{t_{c}}(w)=f_{c}+\sqrt[3]{C(w)(w-E_{c})}$
Figure 18.  For $t>t_{c}$ in (A.20) (dark shaded area), its image satisfies
$\mathop{\mathrm{Im}}[w]>0$ and $\mathop{\mathrm{Im}}[f_{t_{c}}(w)]<0$. It
must be the branch of $f_{t_{c}}(w)$ in the lower half plane. After restricted
to this branch, the preimage of the circle
$w=E_{c}+{\mathfrak{c}}^{\prime}e^{\mathrm{i}\theta}$, is bounded away from
$(E_{c},t_{c})$ and intersects $t={\mathfrak{t}}^{\prime}-\delta$. The
preimage between two intersections with $t={\mathfrak{t}}^{\prime}-\delta$
gives the desired curve $\mathscr{C}$.

Next we consider the case when $(X,{\mathfrak{t}}^{\prime})$ and
$(X^{\prime},{\mathfrak{t}}^{\prime})$ are close to a point
$\big{(}E({\mathfrak{s}}),{\mathfrak{s}}\big{)}$ (which is bounded away from a
cusp location) on the arctic curve, namely,
$\operatorname{dist}\big{(}\big{\\{}(y,{\mathfrak{t}}^{\prime}):X\leqslant
y\leqslant
X^{\prime}\big{\\}},(E({\mathfrak{s}}),{\mathfrak{s}})\big{)}\leqslant
2{\mathfrak{c}}$. Then we can construct a curve $\mathscr{C}$ connecting a
point on the horizontal level $t={\mathfrak{t}}^{\prime}-\delta$ to a point on
the arctic curve, such that $(X,{\mathfrak{t}}^{\prime})$ and
$(X^{\prime},{\mathfrak{t}}^{\prime})$ are inside the liquid region
${\mathfrak{L}}({\mathfrak{P}})$ between $\mathscr{C}$ and
$t={\mathfrak{t}}^{\prime}-\delta$. We denote this region by $\mathscr{D}$;
see Figure 17. We can consider the same map (A.20), by replacing the cusp
location $(E_{c},t_{c})$ by $\big{(}E({\mathfrak{s}}),{\mathfrak{s}}\big{)}$.
Instead of cube root behavior (A.22), $f_{{\mathfrak{s}}}(w)$ has square root
behavior
$f_{\mathfrak{s}}(w)=f_{\mathfrak{s}}(w)+\sqrt{C(w)\big{(}w-E(\mathfrak{s})\big{)}}$
in a neighborhood of $E({\mathfrak{s}})$. Then, by following essentially the
same argument as used in the cusp case, we will have that the map (A.19) is
also a bijection on $\mathscr{D}$.

∎

##  Appendix B Initial Estimates

In this section we study the image of the map (3.11), and prove Proposition
3.2 for the special case $\beta=H_{t}^{*}$. This will serve as the initial
estimates for the general case by a deformation argument.

By Item (c) in the fifth statement of our 2.3, the projection map (3.11) is a
bijection. Together with its complex conjugate, it maps two copies of the
liquid region ${\mathcal{L}}({\mathfrak{D}}^{\prime})\cap{\mathfrak{D}}^{t}$
gluing along the south boundary and arctic curve to a domain
$\mathscr{D}_{0}$. We can interpret $f$ in (3.10) as a function of
$z\in\mathscr{D}_{0}$ as $f=f_{t}^{*}(z)$. The north boundary
$I_{{\mathfrak{t}}^{\prime}}^{*}=[E_{1}({\mathfrak{t}}^{\prime}),E_{2}({\mathfrak{t}}^{\prime})]$
of ${\mathcal{L}}({\mathfrak{D}}^{\prime})\cap{\mathfrak{D}}^{t}$ is a single
interval, it is mapped to a curve

(B.1)
$\displaystyle\mathscr{C}^{*}(x):=x-({\mathfrak{t}}^{\prime}-t)\frac{f_{{\mathfrak{t}}^{\prime}}^{*}(x)}{f_{{\mathfrak{t}}^{\prime}}^{*}(x)+1},\quad
x\in I_{{\mathfrak{t}}^{\prime}}^{*}.$

We denote the contour
$\mathscr{C}_{0}:=\big{\\{}\mathscr{C}(x)\cup\overline{\mathscr{C}(x)}:x\in
I_{{\mathfrak{t}}^{\prime}}^{*}\big{\\}}$, which is also the boundary curve of
$\mathscr{D}_{0}$.

######  Proposition B.1.

The contour $\mathscr{C}_{0}$ defined as in (B.1) is an analytic Jordan curve.
There exists a closed set $\mathscr{A}_{0}$ containing an annulus neighborhood
of $\mathscr{C}_{0}$ (shrinking ${\mathfrak{t}}^{\prime}$ if necessary) such
that the following holds for $z\in\mathscr{A}_{0}$

(B.2) $\displaystyle
1/{\mathfrak{C}}\leqslant|f_{t}^{*}(z)|\leqslant{\mathfrak{C}},\quad|f_{t}^{*}(z)+1|\geqslant
1/{\mathfrak{C}},\quad|\partial_{z}f_{t}^{*}(z)|\leqslant{\mathfrak{C}}$

and
$1+({\mathfrak{t}}^{\prime}-t)\partial_{z}(f_{t}^{*}(z)/(f_{t}^{*}(z)+1))=0$
has two zeros (counting multiplicity) at
$z=\mathscr{C}^{*}(E_{1}({\mathfrak{t}}^{\prime})),\mathscr{C}^{*}(E_{1}({\mathfrak{t}}^{\prime}))$.

###### Proof.

By shrinking ${\mathfrak{t}}^{\prime}$ if necessary, we can extend
$f^{*}_{t}(z)$ to a neighborhood of the contour $\mathscr{C}_{0}$. Moreover,
we can also choose the time slice $t={\mathfrak{t}}^{\prime}$ bounded away
from tangency locations and branch points. Thus by taking $\mathscr{A}_{0}$
the closure of a sufficiently small annulus neighborhood of $\mathscr{C}_{0}$,
(B.2) holds.

Locally around $(f_{r}^{*}(x),x)$ for
$(x,r)\in{\mathfrak{L}}({\mathfrak{D}}^{\prime})\cap{\mathfrak{D}}^{t}$ there
exists an analytic function $Q_{0}$ such that

(B.3) $\displaystyle Q_{0}(f_{r}^{*}(x))=x(f_{r}^{*}(x)+1)-rf_{r}^{*}(x).$

Using (3.10), we can rewrite (B.3) in terms of $(f_{t}^{*}(z),z)$,

(B.4) $\displaystyle
Q_{0}(f_{t}^{*}(z))=x(f_{r}^{*}(x)+1)-rf_{r}^{*}(x)=z(f_{t}^{*}(z)+1)-tf_{t}^{*}(z).$

The two end points
$E_{1}({\mathfrak{t}}^{\prime}),E_{2}({\mathfrak{t}}^{\prime})$ of the
$I_{{\mathfrak{t}}^{\prime}}^{*}$ are characterized by
$\partial_{f}Q_{0}(f_{{\mathfrak{t}}^{\prime}}^{*}(E_{i}({\mathfrak{t}}^{\prime})))=E_{i}({\mathfrak{t}}^{\prime})-{\mathfrak{t}}^{\prime}$,
$i\in\\{1,2\\}$. Since they are not cusp locations, we have
$\partial^{2}_{f}Q_{0}(f_{{\mathfrak{t}}^{\prime}}^{*}(E_{i}({\mathfrak{t}}^{\prime})))\neq
0$. Using the relation (B.4), we can write the derivatives of $Q_{0}$ in terms
of the derivatives of $f_{t}^{*}(z)$. By taking derivatives with respect to
$z$ on both sides of (B.4), and rearranging, we get

(B.5)
$\displaystyle\partial_{f}Q_{0}(f_{t}^{*}(z))=z-t+\frac{f_{t}^{*}(z)+1}{\partial_{z}f_{t}^{*}(z)},\quad\partial^{2}_{f}Q_{0}(f_{t}^{*}(z))=-\frac{(f_{t}^{*}(z)+1)^{3}}{(\partial_{z}f_{t}^{*}(z))^{3}}\partial_{z}^{2}\left(\frac{f_{t}^{*}(z)}{f_{t}^{*}(z)+1}\right).$

For
$z=\mathscr{C}^{*}(x)=x+(t-{\mathfrak{t}}^{\prime})f_{t}^{*}(z)/(f_{t}^{*}(z)+1)$
for some $x\in I_{\mathfrak{t}}^{\prime*}$, we can rewrite the first relation
in (B.5) as

(B.6)
$\displaystyle\partial_{f}Q_{0}(f_{t}^{*}(z))=\partial_{f}Q_{0}(f_{{\mathfrak{t}}^{\prime}}^{*}(x))=(x-{\mathfrak{t}}^{\prime})+\frac{1}{f_{t}^{*}(z)+1}\left({\mathfrak{t}}^{\prime}-t+\frac{1}{\partial_{z}(f_{t}^{*}(z)/(f_{t}^{*}(z)+1))}\right).$

The second term on the righthand side of (B.6) vanishes only if $x$ is one of
the end points
$E_{1}({\mathfrak{t}}^{\prime}),E_{2}({\mathfrak{t}}^{\prime})$. Thus we
conclude that along the contour $\mathscr{C}_{0}$,
$1+({\mathfrak{t}}^{\prime}-t)\partial_{z}(f_{t}^{*}(z)/(f_{t}^{*}(z)+1))=0$
only at
$z=\mathscr{C}^{*}(E_{1}({\mathfrak{t}}^{\prime})),\mathscr{C}^{*}(E_{1}({\mathfrak{t}}^{\prime}))$.
Since $(E_{i}({\mathfrak{t}}^{\prime}),{\mathfrak{t}}^{\prime})$ are not cusp
locations, the second relation in (B.5) implies that
$\partial_{z}^{2}(f_{t}^{*}(z)/(f_{t}^{*}(z)+1))\neq 0$ at these two points.
By taking $\mathscr{A}_{0}$ the closure of a sufficiently small annulus
neighborhood of $\mathscr{C}_{0}$,
$1+({\mathfrak{t}}^{\prime}-t)\partial_{z}(f_{t}^{*}(z)/(f_{t}^{*}(z)+1))=0$
has two zeros (counting multiplicity) at
$z=\mathscr{C}^{*}(E_{1}({\mathfrak{t}}^{\prime})),\mathscr{C}^{*}(E_{1}({\mathfrak{t}}^{\prime}))$.

Next we show the contour $\mathscr{C}_{0}$ is real analytic. In fact, from the
defining relation (B.1), the contour $\mathscr{C}_{0}$ can also be constructed
from $f_{t}^{*}(z)$ as
$\\{w\in{\mathbb{C}}:\mathop{\mathrm{Im}}[w]+({\mathfrak{t}}^{\prime}-t)\mathop{\mathrm{Im}}[f_{t}^{*}(w)/(f_{t}^{*}(w)+1)]=0\\}$.
For $z=\mathscr{C}^{*}(x)$ and $x\neq
E_{1}({\mathfrak{t}}^{\prime}),E_{2}({\mathfrak{t}}^{\prime})$, we have
$\partial_{z}(z+({\mathfrak{t}}^{\prime}-t)(f_{t}^{*}(z)/(f_{t}^{*}(z)+1)))\neq
0$. Analytic implicit function theorem implies that
$\\{w\in{\mathbb{C}}:\mathop{\mathrm{Im}}[w]+({\mathfrak{t}}^{\prime}-t)\mathop{\mathrm{Im}}[f_{t}^{*}(w)/(f_{t}^{*}(w)+1)]=0\\}$
is analytic around $w=z$. Next we discuss the case
$z=\mathscr{C}^{*}(E_{i}({\mathfrak{t}}^{\prime}))$, with $i\in\\{1,2\\}$. By
Taylor expansion around $z$,
$f_{t}^{*}(w)/(f_{t}^{*}(w)+1)=f_{t}^{*}(z)/(f_{t}^{*}(z)+1)-(w-z)/({\mathfrak{t}}^{\prime}-t)+a_{2}(w-z)^{2}+a_{3}(w-z)^{3}\cdots$,
where $a_{2}\neq 0$. Let $w=z+u+\mathrm{i}v$. In a neighborhood of $z$, the
contour $\mathscr{C}_{0}$ is characterized by

$\displaystyle\frac{\mathop{\mathrm{Im}}[w]+({\mathfrak{t}}^{\prime}-t)\mathop{\mathrm{Im}}[f_{t}^{*}(w)/(f_{t}^{*}(w)+1)]}{({\mathfrak{t}}^{\prime}-t)\mathop{\mathrm{Im}}[w]}=2a_{2}u+a_{3}\frac{\mathop{\mathrm{Im}}[(u+\mathrm{i}v)^{2}]}{v}+\cdots=0,$

where by diving $\mathop{\mathrm{Im}}[w]$, we remove the trivial solution that
$w\in{\mathbb{R}}$. Since $a_{2}\neq 0$, the derivative with respect to $u$ is
nonzero. The analytic implicit function theorem gives that $\mathscr{C}_{0}$
is real analytic around $x=E_{i}({\mathfrak{t}}^{\prime})$. ∎

######  Proposition B.2.

Proposition 3.2 holds for $\beta=H_{t}^{*}$.

###### Proof.

For $\beta=H_{t}^{*}$, the first statement in 3.2 follows from the fifth
statement of 2.3.
$\tilde{\mathscr{U}}_{t}:=\tilde{\mathscr{U}}_{t}^{H_{t}^{*}}=\mathscr{D}_{0}$
from the discussion above (B.1), and
$\mathscr{U}_{t}:=\mathscr{U}_{t}^{H_{t}^{*}}$ is the regime from
$\mathscr{D}_{0}$ by removing the interval
$I_{t}^{*}=\\{x:(x,t)\in{\mathcal{L}}({\mathfrak{D}}^{\prime})\\}$. Moreover,
$f_{t}^{*}(z)=f(z;H_{t}^{*},t)$ is analytic on $\mathscr{U}_{t}$.

Next, we show that for $x\in\tilde{\mathscr{U}}_{t}\cap{\mathbb{R}}$,

(B.7)
$\displaystyle\arg^{*}f^{*}_{t}(x+0\mathrm{i})=-\pi(\partial_{x}H^{*}_{t}(x)+\bm{1}_{(-\infty,{\mathfrak{a}}(t)]}(x)+\bm{1}_{({\mathfrak{b}}(t),\infty]}(x)).$

For south boundary of
${\mathfrak{L}}({\mathfrak{D}}^{\prime})\cap{\mathfrak{D}}^{t}$, $x\in
I_{t}^{*}=\\{x:(x,t)\in{\mathcal{L}}({\mathfrak{D}}^{\prime})\\}$, the
argument of the complex slope $f^{*}_{t}(x)$ is given explicitly by the
defining relation (3.1),

(B.8)
$\displaystyle\arg^{*}f_{t}^{*}(x+0\mathrm{i})=\arg^{*}f^{*}_{t}(x)=-\pi\partial_{x}H_{t}^{*}(x).$

For $z\in(\tilde{\mathscr{U}}_{t}\cap{\mathbb{R}})\setminus I_{t}^{*}$, it
corresponds to a point on the arctic curve
$(x,r)\in\partial\overline{{\mathfrak{L}}({{\mathfrak{D}}^{\prime}})}$ with
$r\in(t,{\mathfrak{t}}^{\prime})$, given by (A.11); See Figure 15,

(B.9) $\displaystyle
f^{*}_{t}(z)=f^{*}_{r}(x),\quad\frac{r-t}{x-z}=\frac{f^{*}_{r}(x)+1}{f^{*}_{r}(x)}.$

Since the segment from $(z,t)$ to $(x,r)$ stays in the frozen region (we
interpret the region on the left of $\partial_{\rm we}({\mathfrak{D}}^{t})$ a
frozen region consisting of lozenges of type $2$, and the region on the right
of $\partial_{\rm es}({\mathfrak{D}}^{t})$ a frozen region consisting of
lozenges of type $3$), there are several cases. If the frozen region consists
of type 1 lozenges, then
$\partial_{x}H^{*}_{t}(z)+\bm{1}_{(-\infty,{\mathfrak{a}}(t)]}(z)+\bm{1}_{({\mathfrak{b}}(t),\infty]}(z)=0$
and $f^{*}_{t}(z)=f^{*}_{r}(x)\in[0,+\infty]$; If the frozen region consists
of type 2 lozenges, then
$\partial_{x}H^{*}_{t}(z)+\bm{1}_{(-\infty,{\mathfrak{a}}(t)]}(z)+\bm{1}_{({\mathfrak{b}}(t),\infty]}(z)=1$
and $f^{*}_{t}(z)=f^{*}_{r}(x)\in[-\infty,-1]$; If the frozen region consists
of type 3 lozenges, then
$\partial_{x}H^{*}_{t}(z)+\bm{1}_{(-\infty,{\mathfrak{a}}(t)]}(z)+\bm{1}_{({\mathfrak{b}}(t),\infty]}(z)=1$
and $f^{*}_{t}(z)=f^{*}_{r}(x)\in[-1,0]$. In all cases we have

(B.10)
$\displaystyle\arg^{*}f^{*}_{t}(z)=\arg^{*}f^{*}_{r}(x)=-\pi(\partial_{x}H^{*}_{t}(z)+\bm{1}_{(-\infty,{\mathfrak{a}}(t)]}(z)+\bm{1}_{({\mathfrak{b}}(t),\infty]}(z)).$

This finishes the proof of (B.7).

The above discussion gives that for $x\in\mathscr{U}_{t}\cap{\mathbb{R}}$ with
$x<{\mathfrak{a}}(t)$, $f^{*}_{t}(x)\in(-\infty,-1)$ and for
$x\in\mathscr{U}_{t}\cap{\mathbb{R}}$ with $x>{\mathfrak{b}}(t)$, we have
$f^{*}_{t}(x)\in(0,-1)$. Thanks to (3.10), we have

$\displaystyle\frac{|\mathop{\mathrm{Im}}[f^{*}_{t}(x)/(f^{*}_{t}(x)+1)]|}{|\mathop{\mathrm{Im}}x|}=\frac{1}{r-t}\geqslant\frac{1}{{\mathfrak{t}}^{\prime}-t}\gtrsim
1.$

This gives the second statement of 3.2.

We denote the Stieltjes transform of $\partial_{x}H_{t}^{*}$ as
$m_{t}^{*}(z)=m_{t}(z;H_{t}^{*},t)$, and its modification
$\tilde{m}^{*}_{t}(z)$, which is the Stieltjes transform of the measure
$\partial_{x}H^{*}_{t}(x)+\bm{1}_{(-\infty,{\mathfrak{a}}(t)]}(x)+\bm{1}_{({\mathfrak{b}}(t),\infty]}(x)$.
Then for any $x\in\tilde{\mathscr{U}}_{t}\cap{\mathbb{R}}$, (B.10) implies

(B.11) $\displaystyle\begin{split}\lim_{\eta\rightarrow
0+}\mathop{\mathrm{Im}}[\tilde{m}^{*}_{t}(x\pm\mathrm{i}\eta)]=\mp\pi(\partial_{x}H^{*}_{t}(x)+\bm{1}_{(-\infty,{\mathfrak{a}}(t)]}(x)+\bm{1}_{({\mathfrak{b}}(t),\infty]}(x))=\lim_{\eta\rightarrow
0+}\arg^{*}f^{*}_{t}(x).\end{split}$

We take the ratio of $e^{\tilde{m}^{*}_{t}(z)}$ and $f_{t}(z;\beta,s)$,

(B.12) $\displaystyle
f^{*}_{t}(z)=e^{\tilde{m}^{*}_{t}(z)}\tilde{g}^{*}_{t}(z)=e^{m^{*}_{t}(z)}g^{*}_{t}(z),$

then $\tilde{g}^{*}_{t}(z)$ is well defined in the domain
$\tilde{\mathscr{U}}_{t}$. It is analytic on
$\tilde{\mathscr{U}}_{t}\setminus[{\mathfrak{a}}(t),{\mathfrak{b}}(t)]$, and
(B.11) implies that it is continuous on
$[{\mathfrak{a}}(t),{\mathfrak{b}}(t)]$. Morera’s theorem implies that
$\tilde{g}^{*}_{t}(z)$ extends to an analytic function on
$\tilde{\mathscr{U}}_{t}$. From the construction, $\tilde{g}^{*}_{t}(z)$ is
real analytic, i.e.
$\overline{\tilde{g}^{*}_{t}(z)}=\tilde{g}^{*}_{t}(\bar{z})$, and is positive
on $\tilde{\mathscr{U}}_{t}\cap{\mathbb{R}}$. Moreover, in
${\mathscr{U}}_{t}$, $f^{*}_{t}(z)$ and $e^{\tilde{m}^{*}_{t}(z)}$ does not
have zeros or poles. Thus $\tilde{g}^{*}_{t}(z)$ does not have zeros or poles
in ${\mathscr{U}}_{t}$. Using the defining relation (3.27), we conclude that
$g^{*}_{t}(z)$ is a meromorphic function in the domain
$\tilde{\mathscr{U}}_{t}$. It is positive on the interval
$({\mathfrak{a}}(t),{\mathfrak{b}}(t))$, has a simple pole at
${\mathfrak{a}}(t)$, has a simple zero at ${\mathfrak{b}}(t)$, and does not
have other poles or zeros. This gives the third statement of 3.2.

For $\beta=H_{t}^{*}$, the fourth statement of 3.2 is an empty statement.

∎

##  Appendix C Proof of Propositions 3.2

In this Section we prove Proposition 3.2. To study general boundary profile
$\beta$, we interpolate $\beta$ with $H_{t}^{*}$. We collect facts about
Shiffer kernels in Section C.1. In Section C.2, we construct a deformation
$f(z;\theta)$ parameterized by $0\leqslant\theta\leqslant 1$, starting from
$f(z;0)=f_{t}^{*}(z)$ and show if the solution of the deformation satisfies
certain properties, then $f(z;1)=f(z;\beta,t)$ as in Proposition 3.2, and it
satisfies all the properties in Proposition 3.2. Finally in Section C.3, we
analyze these deformation formulas, and conclude the proof of Proposition 3.2.

###  C.1. Schiffer kernel

Let $\mathscr{D}$ be a simply connected domain in the complex plane
${\mathbb{C}}$ with boundary given by an analytic curve $\mathscr{C}$. We
further assume $\mathscr{D}$ is symmetric with respect to the real axis. Let
$\phi(z)$ be the Riemann map from $\mathscr{D}\cap\mathbb{H}^{+}$ to the upper
half plane $\mathbb{H}^{+}$. By the Schwarz reflection principle, it extends
to $\mathscr{D}$, satisfying $\phi(\bar{z})=\overline{\phi(z)}$. The Green’s
function of $\mathscr{D}$ is given by

$\displaystyle G(z,z^{\prime})=\log\big{|}\phi(z)-\phi(z^{\prime})\big{|}.$

The Schiffer kernel $B(z,z^{\prime}){\rm d}z{\rm d}z^{\prime}$ from [Eyn18,
Sch46] on $\mathscr{D}$ is given in terms of the Riemann map $\phi(z)$ or the
derivatives of the Green’s function by

(C.1) $\displaystyle
B(z,z^{\prime})=\partial_{z}\partial_{z^{\prime}}G(z,z^{\prime})=\frac{\phi^{\prime}(z)\phi^{\prime}(z^{\prime})}{\big{(}\phi(z)-\phi(z^{\prime})\big{)}^{2}}.$

It is a symmetric meromorphic bilinear differential (namely, a meromorphic
$1$-form in $z$ tensored by one in $z^{\prime}$, satisfying
$B(z,z^{\prime})=B(z^{\prime},z)$); it has double poles on the diagonal,

(C.2) $\displaystyle
B(z,z^{\prime})=\frac{1}{(z-z^{\prime})^{2}}+K(z,z^{\prime}),$

where $K(z,z^{\prime})$ is holomorphic.

Since the curve $\mathscr{C}$ is real analytic, we show that $\phi(z)$ and the
Schiffer kernel $B(z,z^{\prime})$ can be analytically extended to the contour
$\mathscr{C}$. For $z\in\mathscr{C}\cap\mathbb{H}^{+}$, namely
$\mathop{\mathrm{Im}}[z]>0$, this follows from that Riemann mapping function
can be extended analytically to the boundary which contains an analytic arc.
If $z$ is one of points where $\mathscr{C}\cap{\mathbb{R}}$, then in a small
neighborhood of $z$, we can parametrize $\mathscr{C}$ as $\gamma(t)$ for
$t\in(-\varepsilon,\varepsilon)$, with $\gamma(0)=z$ and
$\gamma(-t)=\overline{\gamma(t)}$. $\gamma(t)$ is an analytic function on a
small disk $\mathbb{D}$ centered around $0$. Then we can consider
$\phi(\gamma(z))$, and extend it to $\mathbb{D}$ by Schwarz reflection
principle; See Figure 19. Then locally around $z$,

(C.3) $\displaystyle\phi(z+\Delta z)=\phi(z)+C(\Delta z)\Delta z^{2},$

for some analytic function $C(\Delta z)$. Using the relation (C.1), we can
also extend the Schiffer kernel to the contour $\mathscr{C}$, and
$B(z,z^{\prime})=0$ if $z$ is one of points where
$\mathscr{C}\cap{\mathbb{R}}$.

$\mathbb{D}$$\gamma(t)$$E_{1}({\mathfrak{t}}^{\prime})$$\phi(z)$ Figure 19.
We can extend $\phi(z)$ analytically to a neighborhood of $z$ where
$\mathscr{C}$ intersect ${\mathbb{R}}$, by the Schwarz reflection principle,
such that $\phi(z+\Delta z)=\phi(z)+C(\Delta z)\Delta z^{2},$.

###  C.2. Deformation Formulas

To analyze $f(z;\beta,t)$ for boundary profile $\beta$ close to $H_{t}^{*}$,
we interpolate $\beta$ with $H_{t}^{*}$,

$\displaystyle\beta^{\theta}=(1-\theta)\beta^{0}+\theta\beta^{1},\quad
0\leqslant\theta\leqslant 1,\quad\beta^{0}=H_{t}^{*},\quad\beta^{1}=\beta.$

We show that for boundary profile $\beta$ sufficiently close to $H_{t}^{*}$,
$f(z;\beta,t)$ can be obtained by solving the following system of equations:
Set $f(z;0)=f_{t}^{*}(z)$ and $\mathscr{C}(x;0)=\mathscr{C}^{*}(x)$ from
(B.1). We construct their deformations as $f(z;\theta),\mathscr{C}(x;\theta)$
satisfying the following differential equations:

(C.4)
$\displaystyle\begin{split}&\partial_{\theta}\mathscr{C}(x;\theta)=F(z;\theta)\partial_{\theta}\log
f_{s}(z;\theta)|_{z=\mathscr{C}(x;\theta)}\\\
&F(z;\theta):=\frac{-({\mathfrak{t}}^{\prime}-t)f(z;\theta)/(f(z;\theta)+1)^{2})}{1+({\mathfrak{t}}^{\prime}-t)\partial_{z}(f(z;\theta)/(f(z;\theta)+1))}.\end{split}$

and

(C.5) $\displaystyle\partial_{\theta}\log
f(z;\theta)=-\int_{{\mathfrak{a}}(t)}^{{\mathfrak{b}}(t)}B(z,y;\theta)(\beta^{1}(y)-\beta^{0}(y)){\rm
d}y,\quad
z\in\mathscr{D}_{\theta}\setminus[{\mathfrak{a}}(t),{\mathfrak{b}}(t)].$

where
$\mathscr{C}_{\theta}=\\{\mathscr{C}(x;\theta)\cup\overline{\mathscr{C}(x;\theta)}:x\in
I_{{\mathfrak{t}}^{\prime}}^{*}\\}$, $\mathscr{D}_{\theta}$ is the region
enclosed by $\mathscr{C}_{\theta}$, and $B(z,z^{\prime};\theta)$ (recall from
(C.1)) is the Schiffer kernel of the domain $\mathscr{D}_{\theta}$. We remark
that if $\mathscr{C}_{\theta}$ is an analytic Jordan curve, and the righthand
side of (LABEL:e:dthetaw), $F(z;\theta)\partial_{\theta}\log f_{s}(z;\theta)$,
is analytic in a neighborhood of $\mathscr{C}_{\theta}$, then the differential
equations (LABEL:e:dthetaw) and (C.5) are well defined.

To understand the righthand side of (LABEL:e:dthetaw), we define a function

(C.6) $\displaystyle
G(z,z^{\prime};\theta):=\frac{B(z,z^{\prime};\theta)}{1+({\mathfrak{t}}^{\prime}-t)\partial_{z}(f(z;\theta)/(f(z;\theta)+1))},\quad
z,z^{\prime}\in\mathscr{D}_{\theta}\setminus[{\mathfrak{a}}(t),{\mathfrak{b}}(t)].$

Then using (C.5) and (C.6), for $z\in\mathscr{C}_{\theta}$, we have

(C.7) $\displaystyle F(z;\theta)\partial_{\theta}\log
f(z;\theta)=\frac{({\mathfrak{t}}^{\prime}-t)f(z;\theta)}{(f(z;\theta)+1)^{2}}\int_{{\mathfrak{a}}(t)}^{{\mathfrak{b}}(t)}G(z,y;\theta)(\beta^{1}(y)-\beta^{0}(y)){\rm
d}y.$

######  Proposition C.1.

If $\mathscr{C}_{\theta}$ is an analytic Jordan curve, and the righthand side
of (LABEL:e:dthetaw), $F(z;\theta)\partial_{\theta}\log f_{s}(z;\theta)$, is
analytic on a closed set $\mathscr{A}_{\theta}$ containing an annulus
neighborhood of $\mathscr{C}_{\theta}$, the derivative of the Schiffer kernel
$B(z,z^{\prime};\theta)$ with respect to $\theta$ is given by

(C.8)
$\displaystyle\begin{split}\partial_{\theta}B(z,z^{\prime};\theta)&=-\frac{1}{2\pi\mathrm{i}}\int_{\omega}B(z,w;\theta)F(w;\theta)\partial_{\theta}\log
f(w;\theta)B(w,z^{\prime};\theta){\rm d}w\\\ &-\bm{1}(\text{$z$ outside
$\omega$})\partial_{z}(F(z;\theta)\partial_{\theta}\log
f(z;\theta)B(z,z^{\prime};\theta))\\\ &-\bm{1}(\text{$z^{\prime}$ outside
$\omega$})\partial_{z^{\prime}}(F(z^{\prime};\theta)\partial_{\theta}\log
f_{s}(z^{\prime};\theta)B(z,z^{\prime};\theta)),\quad
z,z^{\prime}\in\mathscr{D}_{\theta},\end{split}$

where the contour $\omega\subseteq\mathscr{A}_{\theta}$ encloses
$[{\mathfrak{a}}(t),{\mathfrak{b}}(t)]$. The derivative of
$G(z,z^{\prime};\theta)$ with respect to $\theta$ is given by

(C.9)
$\displaystyle\begin{split}&\phantom{{}={}}\partial_{\theta}G(z,z^{\prime};\theta)=-\frac{1}{2\pi\mathrm{i}}\int_{\omega}G(z,w;\theta)F(w;\theta)\partial_{\theta}\log
f_{s}(w;\theta)B(w,z^{\prime};\theta){\rm d}w\\\
&-\frac{({\mathfrak{t}}^{\prime}-t)f(z;\theta)}{(f(z;\theta)+1)^{2}}\partial_{z}G(z,z^{\prime};\theta)\int_{{\mathfrak{a}}(t)}^{{\mathfrak{b}}(t)}G(z,y;\theta)(\beta^{1}(y)-\beta^{0}(y)){\rm
d}y\\\ &-\bm{1}(\text{$z^{\prime}$ outside
$\omega$})\partial_{z^{\prime}}(F(z^{\prime};\theta)\partial_{\theta}\log
f_{s}(z^{\prime};\theta)G(z,z^{\prime};\theta))\\\ \end{split}$

where the contour $\omega\subseteq\mathscr{A}_{\theta}$ encloses
$[{\mathfrak{a}}(t),{\mathfrak{b}}(t)]$ but not $z$.

###### Proof.

We recall the defining relation (C.1) of the Schiffer kernel, given as the
derivatives of the Green’s function. The variational formula of Green’s
function has been studied in [Sch46] under more general conditions. By taking
derivatives of the variational formula of Green’s function, see [Hua20a,
Theorem 3.7], we get for $z,z^{\prime}\in\mathscr{D}_{\theta}$,

$\displaystyle\begin{split}&\phantom{{}={}}\partial_{\theta}B(z,z^{\prime};\theta)=\frac{1}{2\pi\mathrm{i}}\int_{x\in
I^{*}_{{\mathfrak{t}}^{\prime}}}B(z,\mathscr{C}(x;\theta);\theta)\partial_{\theta}\mathscr{C}(x;\theta)B(\mathscr{C}(x;\theta),z^{\prime};\theta){\rm
d}\mathscr{C}(x;\theta)\\\ &-\frac{1}{2\pi\mathrm{i}}\int_{x\in
I^{*}_{{\mathfrak{t}}^{\prime}}}B(z,\overline{\mathscr{C}(x;\theta)};\theta)\partial_{\theta}\overline{\mathscr{C}(x;\theta)}B(\overline{\mathscr{C}(x;\theta)},z^{\prime};\theta){\rm
d}\overline{\mathscr{C}(x;\theta)}\\\
&=-\frac{1}{2\pi\mathrm{i}}\int_{w\in\mathscr{C}_{\theta}}B(z,w;\theta)F(w;\theta)\partial_{\theta}\log
f(w;\theta)B(w,z^{\prime};\theta){\rm d}w,\end{split}$

where we used (LABEL:e:dthetaw) for the last line. By our assumption
$F(w;\theta)\partial_{\theta}\log f(w;\theta)$ is analytic for
$w\in\mathscr{A}_{\theta}$. We can get the case when $z$ or $z^{\prime}$ is
outside the contour $\omega$, by deforming the contour $\mathscr{C}_{\theta}$.
Since the Schiffer kernel $B(z,w;\theta)$ has a double pole at $w=z$, see
(C.2), we have