F(z)\big{)}\partial_{ttx}F(z)+\mathfrak{b}_{xx}\big{(}\nabla
F(z)\big{)}\partial_{txx}F(z)$
$\displaystyle\qquad+\Big{(}\partial_{tt}F(z)\partial_{u}\mathfrak{b}_{tt}\big{(}\nabla
F(z)\big{)}+\partial_{tx}F(z)\partial_{v}\mathfrak{b}_{tt}\big{(}\nabla
F(z)\big{)}\Big{)}\partial_{tt}F(z)$
$\displaystyle\qquad+2\Big{(}\partial_{tt}F(z)\partial_{u}\mathfrak{b}_{tx}\big{(}\nabla
F(z)\big{)}+\partial_{tx}F(z)\partial_{v}\mathfrak{b}_{tx}\big{(}\nabla
F(z)\big{)}\Big{)}\partial_{tx}F(z)$
$\displaystyle\qquad+\Big{(}\partial_{tt}F(z)\partial_{u}\mathfrak{b}_{xx}\big{(}\nabla
F(z)\big{)}+\partial_{tx}F(z)\partial_{v}\mathfrak{b}_{xx}\big{(}\nabla
F(z)\big{)}\Big{)}\partial_{xx}F(z),$

we have, for any sequence
$\bm{Q}^{(3)}=\big{(}\bm{q}^{(k)}\big{)}_{k\in\llbracket 1,3\rrbracket}$ with
$\bm{q}^{(k)}=\big{(}q_{0}^{(k)},q_{1}^{(k)},\ldots,q_{k}^{(k)}\big{)}$ for
each $k\in\llbracket 1,3\rrbracket$,

$\displaystyle\mathfrak{v}_{1,0}\big{(}\bm{Q}^{(3)}\big{)}$
$\displaystyle=\mathfrak{b}_{tt}\big{(}\bm{q}^{(1)}\big{)}q_{3}^{(3)}+2\mathfrak{b}_{tx}\big{(}\bm{q}^{(1)}\big{)}q_{2}^{(3)}+\mathfrak{b}_{xx}\big{(}\bm{q}^{(1)}\big{)}q_{1}^{(3)}+\Big{(}q_{2}^{(2)}\partial_{u}\mathfrak{b}_{tt}\big{(}\bm{q}^{(1)}\big{)}+q_{1}^{(2)}\partial_{v}\mathfrak{b}_{tt}\big{(}\bm{q}^{(1)}\big{)}\Big{)}q_{2}^{(2)}$
$\displaystyle\quad+2\Big{(}q_{2}^{(2)}\partial_{u}\mathfrak{b}_{tx}\big{(}\bm{q}^{(1)}\big{)}+q_{1}^{(2)}\partial_{v}\mathfrak{b}_{tx}\big{(}\bm{q}^{(1)}\big{)}\Big{)}q_{1}^{(2)}+\Big{(}q_{2}^{(2)}\partial_{u}\mathfrak{b}_{xx}\big{(}\bm{q}^{(1)}\big{)}+q_{1}^{(2)}\partial_{v}\mathfrak{b}_{xx}\big{(}\bm{q}^{(1)}\big{)}\Big{)}q_{0}^{(2)}.$

######  Remark 5.3.

For any function
$F\in\mathcal{C}^{2}(\overline{\mathfrak{R}})\cap\operatorname{Adm}(\mathfrak{R})$
satisfying (3.25), the sequence $\bm{Q}_{F}^{(m)}(z)$ is consistent, for each
integer $m\geq 1$ and point $z\in\mathfrak{R}$; this follows from applying
$\partial_{t}^{i}\partial_{x}^{j}$ to (3.25) (for each
$(i,j)\in\mathbb{Z}_{\geq 0}^{2}$ with $i+j\leq m+2)$.

Observe that consistency imposes $\binom{m}{2}$ relations (one for each
$(i,j)\in\mathbb{Z}_{\geq 0}$ with $i+j\leq m-2$) between the
$\binom{m+2}{2}-1$ coordinates of $\bm{Q}$; this suggests that there are
$\binom{m+2}{2}-1-\binom{m}{2}=2m$ “free parameters” describing a consistent
$\bm{Q}$. The following lemma indicates this to indeed be the case, taking
these $2m$ parameters to be the first two coordinates
$\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}_{k\in\llbracket 1,m\rrbracket}$ of each
vector in $\bm{Q}$.

######  Lemma 5.4.

Fix an integer $m\geq 1$ and $m$ pairs of real numbers
$\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}\in(\mathbb{R}^{2})^{m}$ with
$q_{0}^{(1)}<0$.

  1. (1)

There is a unique consistent sequence of $m$ vectors
$\bm{Q}=\big{(}\bm{q}^{(k)}\big{)}\in\mathbb{R}^{2}\times\mathbb{R}^{3}\times\cdots\times\mathbb{R}^{m+1}$
so that, for each $k\in\llbracket 1,m\rrbracket$, the first two coordinates of
$\bm{q}^{(k)}$ are $\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}$.

  2. (2)

For any real numbers $\varepsilon\in(0,1)$ and $B>1$, there exists a constant
$C=C(\varepsilon,B,m)>1$ so that, if

(5.3) $\displaystyle
q_{0}^{(1)}\leq-\varepsilon,\quad\text{and}\quad\max_{1\leq k\leq
m}\Big{(}\big{|}q_{0}^{(k)}\big{|}+\big{|}q_{1}^{(k)}\big{|}\Big{)}\leq B,$

then the consistent sequence $\bm{Q}$ sastisfies
$\big{|}q_{j}^{(k)}\big{|}\leq C$ for each $k\in\llbracket 1,m\rrbracket$ and
$j\in\llbracket 0,k\rrbracket$.

###### Proof.

Observe for any $(i,j)\in\mathbb{Z}_{>0}^{2}$ that

$\displaystyle\partial_{t}^{i}\partial_{x}^{j}\Bigg{(}\displaystyle\sum_{i^{\prime},j^{\prime}\in\\{t,x\\}}\mathfrak{b}_{i^{\prime}j^{\prime}}\big{(}\nabla
F(z)\big{)}\partial_{i^{\prime}}\partial_{j^{\prime}}F(z)\Bigg{)}=\mathfrak{b}_{tt}\big{(}\nabla
F(z)\big{)}\partial_{t}^{i+2}\partial_{x}^{j}F(z)+\mathfrak{u}_{i,j}\Big{(}\big{(}\partial_{t}^{i^{\prime}}\partial_{x}^{j^{\prime}}F(z)\big{)}_{i^{\prime},j^{\prime}}\Big{)},$

for some function $\mathfrak{u}_{i,j}$ that depends polynomially (as the
$\mathfrak{b}_{ij}(u,v)$ from (3.26) are polynomials in $(u,v)$) only on those
derivatives $\partial_{t}^{i^{\prime}}\partial_{x}^{j^{\prime}}F(z)$ such that
either $i^{\prime}\leq i+1$ or $i^{\prime}+j^{\prime}<i+j+2$. As such, for any
sequence of $m$ vectors
$\bm{Q}=\big{(}\bm{q}^{(1)},\bm{q}^{(2)},\ldots,\bm{q}^{(m)}\big{)}\in\mathbb{R}^{2}\times\mathbb{R}^{3}\times\cdots\mathbb{R}^{m+1}$,
we have

(5.4)
$\displaystyle\mathfrak{v}_{i,j}\big{(}\bm{q}^{(1)},\bm{q}^{(2)},\ldots,\bm{q}^{(i+j+2)}\big{)}=q_{i+2}^{(i+j+2)}\mathfrak{b}_{tt}\big{(}\bm{q}^{(1)}\big{)}+\mathfrak{u}_{i,j}\Big{(}\big{(}q_{i^{\prime}}^{(k^{\prime})}\big{)}_{i^{\prime},k}\Big{)},$

where again $\mathfrak{u}_{i,j}$ is a polynomial in those
$q_{i^{\prime}}^{(k^{\prime})}$ with either $i^{\prime}\leq i+1$ or
$k^{\prime}\leq i+j+1$. Thus,

(5.5)
$\displaystyle\mathfrak{v}_{i,j}\big{(}\bm{q}^{(1)},\bm{q}^{(2)},\ldots,\bm{q}^{(i+j+2)}\big{)}=0,\quad\text{if
and only if}\quad
q_{i+2}^{(i+j+2)}=-\displaystyle\frac{\mathfrak{u}_{i,j}\Big{(}\big{(}q_{i^{\prime}}^{(k^{\prime})}\big{)}_{i^{\prime},k^{\prime}}\Big{)}}{\mathfrak{b}_{tt}\big{(}\bm{q}^{(1)}\big{)}}.$

Hence since $q_{0}^{(1)}<0$, meaning
$\mathfrak{b}_{tt}\big{(}\bm{q}^{(1)}\big{)}>0$, the consistency of $\bm{Q}$
provides a unique solution for $q_{i+2}^{(i+j+2)}$, in terms of all
$\big{(}q_{i^{\prime}}^{(k^{\prime})}\big{)}$ such that either $i^{\prime}\leq
i+1$ or $k^{\prime}\leq i+j+1$. In particular, given the pairs
$\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}$, this by induction on the
lexicographic pair $(i+j+2,i+2)$ determines a unique consistent solution for
the remaining entries of $\bm{Q}$, establishing the first statement of the
lemma.

To establish the second, observe from (3.26) that
$\mathfrak{b}_{tt}\big{(}\bm{q}^{(1)}\big{)}\geq\varepsilon^{2}$, as
$q_{0}^{(1)}\leq-\varepsilon$. Thus, since the $\mathfrak{u}_{i,j}$ are
polynomials with coefficients and degrees only dependent on $i,j\in\llbracket
1,m\rrbracket$, it follows by (5.5) (5.3), and induction on the lexicographic
pair $(i+j+2,i+2)\in\llbracket 2,m\rrbracket^{2}$ that for each
$(i,j)\in\llbracket 0,m\rrbracket^{2}$ there exists a constant
$C_{i,j}=C_{i,j}(\varepsilon,B)>1$ such that
$\big{|}q_{i+2}^{(i+j+2)}\big{|}\leq C_{i,j}$. This (together with the fact
from (5.3) that $\big{|}q_{0}^{(k)}\big{|}+\big{|}q_{1}^{(k)}\big{|}\leq B$
for each $k\in\llbracket 1,m\rrbracket$) yields the second statement of the
lemma, with $C=\max_{i+j\leq m-2}C_{i,j}+B$. ∎

######  Definition 5.5.

Adopting the notation of 5.4, we call $\bm{Q}=\big{(}\bm{q}^{(k)}\big{)}$ the
_consistent extension_ of the sequence of $m$ pairs
$\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}\in(\mathbb{R}^{2})^{m}$.

We conclude this section with the following two lemmas. The first is a
continuity bound, saying that if two sequences of real pairs are close, then
their consistent extensions are as well. The second implies that, if
$\widetilde{\bm{Q}}$ is a sequence that “almost satisfies” the consistency
relations (5.2), then there is a consistent sequence $\bm{Q}$ close to it.

######  Lemma 5.6.

For any integer $m\geq 1$ and real numbers $\varepsilon\in(0,1)$ and $B>1$,
there exists a constant $C=C(\varepsilon,B,m)>1$ such that the following
holds. Let $\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}\in(\mathbb{R}^{2})^{m}$ and
$\big{(}\widetilde{q}_{0}^{(k)},\widetilde{q}_{1}^{(k)}\big{)}\in(\mathbb{R}^{2})^{m}$
be two sequences of $m$ real number pairs such that

$\displaystyle\displaystyle\max\big{\\{}q_{0}^{(1)},\widetilde{q}_{0}^{(1)}\\}\leq-\varepsilon;\qquad\displaystyle\max_{1\leq
k\leq
m}\Big{(}\big{|}q_{0}^{(k)}\big{|}+\big{|}q_{1}^{(k)}\big{|}+\big{|}\widetilde{q}_{0}^{(k)}\big{|}+\big{|}\widetilde{q}_{1}^{(k)}\big{|}\Big{)}\leq
B.$

Letting $\bm{Q}=\big{(}\bm{q}^{(k)}\big{)}$ and
$\widetilde{\bm{Q}}=\big{(}\widetilde{\bm{q}}^{(k)}\big{)}$ denote the
consistent extensions of $\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}$ and
$\big{(}\widetilde{q}_{0}^{(k)},\widetilde{q}_{1}^{(k)}\big{)}$, respectively,
we have

$\displaystyle\big{\|}\bm{q}^{(k)}-\widetilde{\bm{q}}^{(k)}\big{\|}\leq
C\displaystyle\sum_{k^{\prime}=1}^{k}\Big{(}\big{|}q_{0}^{(k^{\prime})}-\widetilde{q}_{0}^{(k^{\prime})}\big{|}+\big{|}q_{1}^{(k^{\prime})}-\widetilde{q}_{1}^{(k^{\prime})}\big{|}\Big{)},\qquad\text{for
each $k\in\llbracket 1,m\rrbracket$}.$

###### Proof.

Throughout, we recall the notation from the proof of 5.4. First observe from
the second part of 5.4 that there exists a constant
$C_{1}=C_{1}(\varepsilon,B,m)>1$ such that

(5.6)
$\displaystyle\displaystyle\sum_{k=1}^{m}\Big{(}\big{|}\bm{q}^{(k)}\big{|}+\big{|}\widetilde{\bm{q}}^{(k)}\big{|}\Big{)}\leq
C_{1}.$

Next, due to (5.6); the definition (3.26) of $\mathfrak{b}_{tt}$; and the fact
that
$\max\big{\\{}q_{0}^{(1)},\widetilde{q}_{0}^{(1)}\big{\\}}\leq-\varepsilon$,
there exists a constant $C_{2}=C_{2}(\varepsilon,m)>1$ such that

(5.7)
$\displaystyle\bigg{|}\displaystyle\frac{1}{\mathfrak{b}_{tt}\big{(}\bm{q}^{(1)}\big{)}}-\displaystyle\frac{1}{\mathfrak{b}_{tt}\big{(}\widetilde{\bm{q}}^{(1)}\big{)}}\bigg{|}\leq
C_{2}\big{|}\bm{q}^{(1)}-\widetilde{\bm{q}}^{(1)}\big{|};\qquad\bigg{|}\displaystyle\frac{1}{\mathfrak{b}_{tt}\big{(}\bm{q}^{(1)}\big{)}}\bigg{|}+\bigg{|}\displaystyle\frac{1}{\mathfrak{b}_{tt}\big{(}\widetilde{\bm{q}}^{(1)}\big{)}}\bigg{|}\leq
C_{2}.$

Moreover, by (5.6); the fact that the $\mathfrak{u}_{i,j}$ (satisfying (5.5))
are polynomials that only depend on the $q_{i^{\prime}}^{(k^{\prime})}$ with
either $i^{\prime}\leq i+1$ or $k^{\prime}\leq i+j+1$; and the fact that the
coefficients and degrees of $\mathfrak{u}_{i,j}$ only depend on
$i,j\in\llbracket 1,m\rrbracket$, there exists a constant
$C_{3}=C_{3}(\varepsilon,B,m)>1$ such that

(5.8) $\displaystyle\begin{aligned}
&\bigg{|}\mathfrak{u}_{i,j}\Big{(}\big{(}q_{i^{\prime}}^{(k^{\prime})}\big{)}_{i^{\prime},k^{\prime}}\Big{)}-\mathfrak{u}_{i,j}\Big{(}\big{(}\widetilde{q}_{i^{\prime}}^{(k^{\prime})}\big{)}_{i^{\prime},k^{\prime}}\Big{)}\bigg{|}\leq
C_{3}\Bigg{(}\displaystyle\sum_{i^{\prime}=0}^{i+1}\displaystyle\sum_{k^{\prime}=1}^{i+j+2}\big{|}q_{i^{\prime}}^{(k^{\prime})}-\widetilde{q}_{i^{\prime}}^{(k^{\prime})}\big{|}+\displaystyle\sum_{i^{\prime}=0}^{i+2}\displaystyle\sum_{k^{\prime}=1}^{i+j+1}\big{|}q_{i^{\prime}}^{(k^{\prime})}-\widetilde{q}_{i^{\prime}}^{(k^{\prime})}\big{|}\Bigg{)};\\\
&\bigg{|}\mathfrak{u}_{i,j}\Big{(}\big{(}q_{i^{\prime}}^{(k^{\prime})}\big{)}_{i^{\prime},k^{\prime}}\Big{)}\bigg{|}+\bigg{|}\mathfrak{u}_{i,j}\Big{(}\big{(}\widetilde{q}_{i^{\prime}}^{(k^{\prime})}\big{)}_{i^{\prime},k^{\prime}}\Big{)}\bigg{|}\leq
C_{3}.\end{aligned}$

Thus, (5.5), (5.7), and (5.8) together yield a constant
$C_{4}=C_{4}(\varepsilon,B,m)>1$ so that

$\displaystyle\big{|}q_{i+2}^{(i+j+2)}-\widetilde{q}_{i+2}^{(i+j+2)}\big{|}\leq
C_{4}\Bigg{(}\displaystyle\sum_{i^{\prime}=0}^{i+1}\displaystyle\sum_{k^{\prime}=1}^{i+j+2}\big{|}q_{i^{\prime}}^{(k^{\prime})}-\widetilde{q}_{i^{\prime}}^{(k^{\prime})}\big{|}+\displaystyle\sum_{i^{\prime}=0}^{i+2}\displaystyle\sum_{k^{\prime}=1}^{i+j+1}\big{|}q_{i^{\prime}}^{(k^{\prime})}-\widetilde{q}_{i^{\prime}}^{(k^{\prime})}\big{|}\Bigg{)}.$

Induction on the lexicographic pair $(i+j+2,i+2)\in\llbracket
2,m\rrbracket^{2}$ then gives a constant $C_{i,j}=C_{i,j}(\varepsilon,B,m)>1$
for any $(i,j)\in\llbracket 0,m\rrbracket^{2}$ with $i+j+2\leq m$ such that

$\displaystyle\big{|}q_{i+2}^{(i+j+2)}-\widetilde{q}_{i+2}^{(i+j+2)}\big{|}\leq
C_{i,j}\displaystyle\sum_{k^{\prime}=1}^{i+j+2}\Big{(}\big{|}q_{0}^{(k^{\prime})}-\widetilde{q}_{0}^{(k^{\prime})}\big{|}+\big{|}q_{1}^{(k^{\prime})}-\widetilde{q}_{1}^{(k^{\prime})}\big{|}\Big{)}.$

Summing over all such $(i,j)$ with $i+j=k-2$, for any $k\in\llbracket
1,m\rrbracket$, we deduce the lemma. ∎

######  Lemma 5.7.

For any integer $m\geq 1$ and real numbers $\varepsilon\in(0,1)$ and $B>1$,
there exists a constant $C=C(\varepsilon,B,m)>1$ such that the following
holds. Let $\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}\in(\mathbb{R}^{2})^{m}$
denote a sequence of $m$ pairs of real numbers satisfying (5.3), and let
$\bm{Q}=\big{(}\bm{q}^{(k)}\big{)}$ denote its consistent extension. Let
$\widetilde{\bm{Q}}=\big{(}\widetilde{\bm{q}}^{(k)}\big{)}\in\mathbb{R}^{2}\times\mathbb{R}^{3}\times\cdots\times\mathbb{R}^{m+1}$
be such that $\widetilde{q}_{0}^{(k)}=q_{0}^{(k)}$ and
$\widetilde{q}_{1}^{(k)}=q_{1}^{(k)}$ for each $k\in\llbracket 1,m\rrbracket$.
If $\big{|}\widetilde{q}_{j}^{(k)}\big{|}\leq B$ for each $k\in\llbracket
1,m\rrbracket$ and $j\in\llbracket 0,k\rrbracket$, then

$\displaystyle\big{|}\bm{q}^{(k)}-\widetilde{\bm{q}}^{(k)}\big{|}\leq
C\displaystyle\sum_{h=2}^{k}\displaystyle\sum_{i=0}^{h-2}\Big{|}\mathfrak{v}_{i,h-i-2}\big{(}\widetilde{\bm{q}}^{(1)},\widetilde{\bm{q}}^{(2)},\ldots,\widetilde{\bm{q}}^{(h)}\big{)}\Big{|},\qquad\text{for
each $k\in\llbracket 1,m\rrbracket$}.$

###### Proof.

Throughout, we recall the notation from the proof of 5.4. Following the
beginning of the proof of 5.6, we deduce the existence of a constant
$C_{1}=C_{1}(\varepsilon,B,m)>1$ such that (5.6) holds (by the assumption of
this lemma, as well as the second part of 5.4); a constant
$C_{2}=C_{2}(\varepsilon,m)>1$ such that (5.7) holds (due to (5.6); the
definition (3.26) of $\mathfrak{b}_{tt}$; and the fact that
$q_{0}^{(1)}=\widetilde{q}_{0}^{(1)}\leq-\varepsilon$); and a constant
$C_{3}=C_{3}(\varepsilon,B,m)>1$ such that (5.8) holds (by (5.6); the fact
that the $\mathfrak{u}_{i,j}$ are polynomials that only depend on the
$q_{i^{\prime}}^{(k^{\prime})}$ with either $i^{\prime}\leq i+1$ or
$k^{\prime}\leq i+j+1$; and the fact the coefficients and degrees of
$\mathfrak{u}_{i,j}$ only depend on $i,j\in\llbracket 1,m\rrbracket$).

Next, by (5.4), we have

$\displaystyle
q_{i+2}^{(i+j+2)}=-\displaystyle\frac{\mathfrak{u}_{i,j}\Big{(}\big{(}q_{i^{\prime}}^{(k^{\prime})}\big{)}_{i^{\prime},k^{\prime}}\Big{)}}{\mathfrak{b}_{tt}\big{(}\bm{q}^{(1)}\big{)}};\qquad\widetilde{q}_{i+2}^{(i+j+2)}=\displaystyle\frac{\mathfrak{v}_{i,j}\big{(}\widetilde{\bm{q}}^{(1)},\widetilde{\bm{q}}^{(2)},\ldots,\widetilde{\bm{q}}^{(i+j+2)}\big{)}}{\mathfrak{b}_{tt}\big{(}\widetilde{\bm{q}}^{(1)}\big{)}}-\displaystyle\frac{\mathfrak{u}_{i,j}\Big{(}\big{(}\widetilde{q}_{i^{\prime}}^{(k^{\prime})}\big{)}_{i^{\prime},k^{\prime}}\Big{)}}{\mathfrak{b}_{tt}\big{(}\widetilde{\bm{q}}^{(1)}\big{)}}.$

This, together with (5.7) and (5.8), yields a constant
$C_{4}=C_{4}(\varepsilon,m)>1$ so that

$\displaystyle\big{|}q_{i+2}^{(i+j+2)}-\widetilde{q}_{i+2}^{(i+j+2)}\big{|}\leq
C_{4}\Bigg{(}\displaystyle\sum_{i^{\prime}=0}^{i+1}\displaystyle\sum_{k^{\prime}=1}^{i+j+2}\big{|}q_{i^{\prime}}^{(k^{\prime})}-\widetilde{q}_{i^{\prime}}^{(k^{\prime})}\big{|}$
$\displaystyle+\displaystyle\sum_{i^{\prime}=0}^{i+2}\displaystyle\sum_{k^{\prime}=1}^{i+j+1}\big{|}q_{i^{\prime}}^{(k^{\prime})}-\widetilde{q}_{i^{\prime}}^{(k^{\prime})}\big{|}$
$\displaystyle+\Big{|}\mathfrak{v}_{i,j}\big{(}\widetilde{\bm{q}}^{(1)},\widetilde{\bm{q}}^{(2)},\ldots,\widetilde{\bm{q}}^{(i+j+2)}\big{)}\Big{|}\Bigg{)}.$

Induction on the lexicographic pair $(i+j+2,i+2)\in\llbracket
2,m\rrbracket^{2}$ (together with the fact that
$\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}=\big{(}\widetilde{q}_{0}^{(k)},\widetilde{q}_{1}^{(k)}\big{)}$,
for each $k\in\llbracket 1,m\rrbracket$) then gives a constant
$C_{i,j}=C_{i,j}(\varepsilon,B,m)>1$ for any $(i,j)\in\llbracket
0,m\rrbracket^{2}$ with $i+j+2\leq m$ such that

$\displaystyle\big{|}q_{i+2}^{(i+j+2)}-\widetilde{q}_{i+2}^{(i+j+2)}\big{|}\leq
C_{i,j}\displaystyle\sum_{h=2}^{i+j+2}\displaystyle\sum_{i^{\prime}=0}^{i+2}\Big{|}\mathfrak{v}_{i^{\prime},h-i^{\prime}-2}\big{(}\widetilde{\bm{q}}^{(1)},\widetilde{\bm{q}}^{(2)},\ldots,\widetilde{\bm{q}}^{(h)}\big{)}\Big{|}.$

Summing over all such $(i,j)$ with $i+j=k-2$, for any $k\in\llbracket
2,m\rrbracket$ (recall $\bm{q}^{(1)}=\widetilde{\bm{q}}^{(1)}$, addressing the
case $k=1$), we deduce the lemma. ∎

###  5.2. Concentrating Profiles

In this section we discuss families of (deterministic) height profiles that
are limit shapes for systems of non-intersecting Brownian bridges that admit a
(nearly) optimal concentration bound. They are given by the below definition.

######  Definition 5.8.

Fix real numbers $a<b$ and two functions $f,g:[a,b]\rightarrow\mathbb{R}$ with
$f<g$. Denote the strip domain
$\mathfrak{S}=\mathfrak{P}_{f;g}\subset\mathbb{R}^{2}$, and let
$\mathfrak{H}:\overline{\mathfrak{S}}\rightarrow\mathbb{R}$ be a function.
Given real numbers $C>1$ and $c,\delta\in\big{(}0,\frac{1}{2}\big{)}$, we say
that $\mathfrak{H}$ is _$(C;c;\delta)$ -concentrating_ if it is satisfies the
equation (3.25) for each $z\in\mathfrak{S}$, and the following holds for any
integer $n\geq 1$. There exists an integer $n^{\prime}\in[cn,n]$; starting
data $\bm{u}\in\overline{\mathbb{W}}_{n^{\prime}}$; and ending data
$\bm{v}\in\overline{\mathbb{W}}_{n^{\prime}}$ on $\mathfrak{S}$ so that,
sampling $n^{\prime}$ non-intersecting Brownian bridges
$\bm{y}=(y_{1},y_{2},\ldots,y_{n^{\prime}})\in\llbracket
1,n^{\prime}\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$ from the measure
$\mathfrak{Q}_{f;g}^{\bm{u};\bm{v}}(n^{-1})$ and denoting the associated
height function by $\mathsf{H}^{\bm{y}}$ (recall (3.6)), we have

$\displaystyle\mathbb{P}\Bigg{[}\displaystyle\sup_{z\in\mathfrak{S}}\big{|}n^{-1}\mathsf{H}^{\bm{y}}(z)-\mathfrak{H}(z)\big{|}\geq
n^{\delta-1}\Bigg{]}\leq Ce^{-c(\log n)^{2}}.$

We will be interested in height functions that are
$(C;c;\delta)$-concentrating with small $\delta>0$. While it is plausible that
essentially any solution of (3.25) on $\mathfrak{R}$ has this property, it is
not transparent to us how to show this directly. We instead show the following
result stating that, if
$\bm{Q}\in\mathbb{R}^{2}\times\mathbb{R}^{3}\times\cdots\times\mathbb{R}^{m+1}$
is an admissible sequence of $m$ vectors in the sense of Definition 5.1, then
(for small $\delta$) there exists a $(C;c;\delta)$-concentrating height
function $\mathfrak{H}$ whose full-$m$ local profile at $z=(0,0)$ is given by
$\bm{Q}$; see Figure 5 for a depiction. It will be established in Section 7.2
below.

Figure 5. Shown above is a depiction of 5.9 and 5.11.

######  Proposition 5.9.

For any integer $m\geq 1$ and real numbers $\varepsilon\in(0,1)$ and $B>1$,
there exist constants $\nu=\nu(\varepsilon,B,m)\in(0,1)$,
$c=c(\varepsilon,B,m)>0$, and $C=C(\varepsilon,B,m)>1$ such that the following
holds. Fix a sequence of $m$ pairs
$\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}\in(\mathbb{R}^{2})^{m}$ satisfying
(5.3); denote its consistent extension by $\bm{Q}$. There is a strip domain
$\mathfrak{S}\subset\mathbb{R}^{2}$ of width $\nu$, such that
$[0,\nu]\times[-\nu,\nu]\subseteq\mathfrak{S}\subseteq[0,\nu]\times[-C\nu,C\nu]$,
and a function $\mathfrak{H}:\overline{\mathfrak{S}}\rightarrow\mathbb{R}$,
such that $\|\mathfrak{H}\|_{\mathcal{C}^{m+1}(\overline{\mathfrak{S}})}\leq
C$ and $\sup_{z\in\mathfrak{S}}\partial_{x}\mathfrak{H}(z)\leq-C^{-1}$, which
satisfies the following two properties.

  1. (1)

The full-$m$ local profile of $\mathfrak{H}$ satisfies
$\lim_{z\rightarrow(0,0)}\bm{Q}_{\mathfrak{H}}^{(m)}(z)=\bm{Q}$.

  2. (2)

For any real number $\delta\in(0,1)$, there exists a constant
$C_{1}=C_{1}(\varepsilon,\delta,B,m)>1$ such that $\mathfrak{H}$ is
$(C_{1};c;\delta)$-concentrating.

Observe that 5.9 fixes the local profile of $\mathfrak{H}$ at the point
$(0,0)$ on the boundary of $\partial\mathfrak{R}$ . However, it will be useful
to (at least approximately) fix the local profile of $\mathfrak{H}$ at some
point in the interior of $\mathfrak{R}$, to which end we have the following
lemma and corollary; see Figure 5 for a depiction of the latter.

######  Lemma 5.10.

For any integer $m\geq 1$ and real numbers $\varepsilon\in(0,1)$,
$\nu\in(0,1)$, and $B>1$, there exist constants
$c=c(\varepsilon,B,m)\in\big{(}0,\frac{\nu}{2}\big{)}$ and
$C=C(\varepsilon,B,m)>1$ such that the following holds. Fix a consistent
sequence
$\bm{Q}=\big{(}\bm{q}^{(k)}\big{)}\in\mathbb{R}^{2}\times\mathbb{R}^{3}\times\cdots\times\mathbb{R}^{m+1}$;
denoting
$\bm{q}^{(k)}=\big{(}q_{0}^{(k)},q_{1}^{(k)},\ldots,q_{k}^{(k)}\big{)}\in\mathbb{R}^{k+1}$
for each $k\in\llbracket 1,m\rrbracket$, assume that (5.3) holds. Fix a real
number $\rho\in(0,c)$ and set

(5.9) $\displaystyle
r_{0}^{(k)}=\displaystyle\sum_{i=0}^{m-k}\displaystyle\frac{(-\rho)^{i}}{i!}q_{i}^{(k+i)},\quad\text{and}\quad
r_{1}^{(k)}=\displaystyle\sum_{i=0}^{m-k}\displaystyle\frac{(-\rho)^{i}}{i!}q_{i+1}^{(k+i)},\qquad\text{for
each $k\in\llbracket 1,m\rrbracket$}.$

Let
$\bm{R}\in\mathbb{R}^{2}\times\mathbb{R}^{3}\times\cdots\times\mathbb{R}^{m+1}$
denote the consistent extension of the sequence of $m$ pairs
$\big{(}r_{0}^{(k)},r_{1}^{(k)}\big{)}\in(\mathbb{R}^{2})^{m}$. If there
exists a function $\mathfrak{H}:[0,\nu]\times[-\nu,\nu]\rightarrow\mathbb{R}$
satisfying (3.25) on $[0,\nu]\times[-\nu,\nu]$, with

(5.10) $\displaystyle\|\mathfrak{H}\|_{\mathcal{C}^{m+1}}\leq
B,\quad\text{and}\quad\displaystyle\lim_{z\rightarrow(0,0)}\bm{Q}_{\mathfrak{H}}^{(m)}(z)=\bm{R},$

then $\big{|}\bm{q}_{\mathfrak{H}}^{(k)}(\rho,0)-\bm{q}^{(k)}\big{|}\leq
C\rho^{m-k+1}$ for each $k\in\llbracket 1,m\rrbracket$, where
$\bm{q}_{\mathfrak{H}}^{(k)}$ is as defined in (5.1).

###### Proof.

Throughout this proof, we set $q_{0}^{(0)}=0$. First observe that, since (5.3)
holds, 5.4 yields a constant $C_{1}=C_{1}(\varepsilon,B,m)>1$ such that

(5.11) $\displaystyle\displaystyle\max_{k\in\llbracket
0,m\rrbracket}\displaystyle\max_{j\in\llbracket
0,k\rrbracket}\big{|}q_{j}^{(k)}\big{|}\leq C_{1}.$

This, together with the fact that $q_{0}^{(1)}\leq-\varepsilon$ (by (5.3))
implies (by the definition (5.9) of $r_{0}^{(k)}$ and $r_{1}^{(k)}$) that
there exist constants $c_{1}=c_{1}(\varepsilon,B,m)>0$ and
$C_{2}=C_{2}(\varepsilon,B,m)>1$ such that for $\rho\in(0,c_{1})$ we have
$r_{0}^{(1)}\leq-\frac{\varepsilon}{2}$ and $\big{|}r_{i}^{(k)}\big{|}\leq
C_{2}$, for each $i\in\\{0,1\\}$ and $k\in\llbracket 1,m\rrbracket$. Hence,
the second part of 5.4 yields a constant $C_{3}=C_{3}(\varepsilon,B,m)>1$ such
that

(5.12) $\displaystyle
r_{0}^{(1)}\leq-\frac{\varepsilon}{2};\qquad\displaystyle\max_{k\in\llbracket
0,m\rrbracket}\displaystyle\max_{j\in\llbracket
0,k\rrbracket}\big{|}r_{j}^{(k)}\big{|}\leq C_{3}.$

Next, define the (not necessarily consistent) sequence of $m+1$ vectors
$\widetilde{\bm{R}}=\big{(}\widetilde{\bm{r}}^{(k)}\big{)}_{k\in\llbracket
0,m\rrbracket}\in\mathbb{R}\times\mathbb{R}^{2}\times\cdots\times\mathbb{R}^{m+1}$,
where
$\widetilde{\bm{r}}^{(k)}=\big{(}\widetilde{r}_{0}^{(k)},\widetilde{r}_{1}^{(k)},\ldots,\widetilde{r}_{k}^{(k)}\big{)}\in\mathbb{R}^{k+1}$
for each $k\in\llbracket 0,m\rrbracket$, by

(5.13)
$\displaystyle\widetilde{r}_{i}^{(k)}=\displaystyle\sum_{i^{\prime}=0}^{m-k}\displaystyle\frac{(-\rho)^{i^{\prime}}}{i^{\prime}!}q_{i+i^{\prime}}^{(k+i^{\prime})},$

so in particular

(5.14)
$\displaystyle\widetilde{r}_{0}^{(k)}=r_{0}^{(k)};\qquad\widetilde{r}_{1}^{(k)}=r_{1}^{(k)}.$

Let us express the $q_{i}^{(k)}$ through the $\widetilde{r}_{i}^{(k)}$. To
that end, define the polynomials
$A_{0},A_{1}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by

(5.15) $\displaystyle
A_{0}(t,x)=\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{j=0}^{k}\displaystyle\frac{q_{j}^{(k)}}{j!(k-j)!}t^{j}x^{k-j};\qquad
A_{1}(t,x)=\displaystyle\sum_{k=0}^{m}\displaystyle\sum_{j=0}^{k}\displaystyle\frac{\widetilde{r}_{j}^{(k)}}{j!(k-j)!}t^{j}x^{k-j}.$

so that

(5.16) $\displaystyle\begin{aligned}
A_{0}(t-\rho,x)&=\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{j=0}^{k}\displaystyle\sum_{i=0}^{j}\displaystyle\frac{(-\rho)^{j-i}}{(k-j)!i!(j-i)!}q_{j}^{(k)}t^{i}x^{k-j}\\\
&=\displaystyle\sum_{i=0}^{m}\displaystyle\sum_{j^{\prime}=0}^{m-i}\displaystyle\frac{t^{i}x^{j^{\prime}}}{i!j^{\prime}!}\displaystyle\sum_{k^{\prime}=0}^{m-i-j^{\prime}}\displaystyle\frac{(-\rho)^{k^{\prime}}}{k^{\prime}!}q_{i+k^{\prime}}^{(i+j^{\prime}+k^{\prime})}=\displaystyle\sum_{i=0}^{m}\displaystyle\sum_{j^{\prime}=0}^{m-i}\displaystyle\frac{\widetilde{r}_{i}^{(i+j^{\prime})}}{i!j^{\prime}!}t^{i}x^{j^{\prime}}=A_{1}(t,x),\end{aligned}$

where in the second equality we changed variables
$(j^{\prime},k^{\prime})=(k-j,j-i)$ (and used the fact that $q_{0}^{(0)}=0$),
in the third we used the definition (5.13) of $\widetilde{r}_{i}^{(i+j)}$, and
in the fourth we again changed variables $(i,j^{\prime})=(j,k-j)$. It follows
(again since $q_{0}^{(0)}=0$) that

$\displaystyle\displaystyle\sum_{k=0}^{m}\displaystyle\sum_{j=0}^{k}\displaystyle\frac{t^{j}x^{k-j}}{j!(k-j)!}q_{j}^{(k)}=A_{0}(t,x)=A_{1}(t+\rho,x)$
$\displaystyle=\displaystyle\sum_{k=0}^{m}\displaystyle\sum_{j=0}^{k}\displaystyle\sum_{i=0}^{j}\displaystyle\frac{\rho^{j-i}}{(k-j)!i!(j-i)!}\widetilde{r}_{j}^{(k)}t^{i}x^{k-j}$
$\displaystyle=\displaystyle\sum_{i=0}^{m}\displaystyle\sum_{j^{\prime}=0}^{m-i}\displaystyle\frac{t^{i}x^{j^{\prime}}}{i!j^{\prime}!}\displaystyle\sum_{k^{\prime}=0}^{m-i-j^{\prime}}\displaystyle\frac{\rho^{k^{\prime}}}{k^{\prime}!}\widetilde{r}_{i+k^{\prime}}^{(i+j^{\prime}+k^{\prime})}$
$\displaystyle=\displaystyle\sum_{k=0}^{m}\displaystyle\sum_{j^{\prime}=0}^{k}\displaystyle\frac{t^{j}x^{k-j}}{j!(k-j)!}\displaystyle\sum_{i^{\prime}=0}^{m-k}\displaystyle\frac{\rho^{i^{\prime}}}{i^{\prime}!}\widetilde{r}_{j+i^{\prime}}^{(k+i^{\prime})},$

where in the fourth equality we again changed variables
$(j^{\prime},k^{\prime})=(k-j,j-i)$, and in the fifth we changed variables
$(i,j^{\prime},k^{\prime})=(j,k-j,i^{\prime})$. Comparing coefficients of
$t^{j}x^{k-j}$, we deduce

(5.17) $\displaystyle
q_{i}^{(k)}=\displaystyle\sum_{i^{\prime}=0}^{m-k}\displaystyle\frac{\rho^{i^{\prime}}}{i^{\prime}!}\widetilde{r}_{i+i^{\prime}}^{(k+i^{\prime})},\qquad\text{for
each $k\in\llbracket 0,m\rrbracket$ and $i\in\llbracket 0,k\rrbracket$}.$

Next, set $\bm{R}=\big{(}\bm{r}^{(k)}\big{)}$, where
$\bm{r}^{(k)}=\big{(}r_{0}^{(k)},r_{1}^{(k)},\ldots,r_{k}^{(k)}\big{)}\in\mathbb{R}^{k+1}$
for each $k\in\llbracket 1,m\rrbracket$. Due to (5.10) and a Taylor expansion,
we have

$\displaystyle\Bigg{|}\partial_{t}^{i}\partial_{x}^{j}\mathfrak{H}(\rho,0)-\displaystyle\sum_{i^{\prime}=0}^{m-i-j}\displaystyle\frac{\rho^{i^{\prime}}}{i^{\prime}!}r_{i+i^{\prime}}^{(i+i^{\prime}+j)}\Bigg{|}$
$\displaystyle=\Bigg{|}\partial_{t}^{i}\partial_{x}^{j}\mathfrak{H}(\rho,0)-\displaystyle\sum_{i^{\prime}=0}^{m-i-j}\displaystyle\frac{\rho^{i^{\prime}}}{i^{\prime}!}\partial_{t}^{i+i^{\prime}}\partial_{x}^{j}\mathfrak{H}(0,0)\Bigg{|}\leq
B\rho^{m-i-j+1}.$

Hence, by (5.17), it suffices to show

(5.18) $\displaystyle\big{|}r_{i}^{(k)}-\widetilde{r}_{i}^{(k)}\big{|}\leq
C\rho^{m-k+1},\qquad\text{for any $k\in\llbracket 1,m\rrbracket$ and
$i\in\llbracket 0,k\rrbracket$}.$

To that end, define $D:\mathbb{R}^{2}\rightarrow\mathbb{R}$ by, for any
$(t,x)\in\mathbb{R}^{2}$, setting

$\displaystyle
D(t,x)=\displaystyle\sum_{j,k\in\\{t,x\\}}\mathfrak{b}_{jk}\big{(}\nabla
A_{1}(t,x)\big{)}\partial_{j}\partial_{j}A_{1}(t,x)=\displaystyle\sum_{j,k\in\\{t,x\\}}\mathfrak{b}_{j,k}\big{(}\nabla
A_{0}(t-\rho,x)\big{)}\partial_{j}\partial_{k}A_{0}(t-\rho,x),$

were in the last equality we used (5.16). We will use a Taylor expansion to
view $D$ as an approximation of the left side of (3.25), where the $H$ there
is $A$ here. For each integer $k\in\llbracket 0,m\rrbracket$, set
$\bm{Q}^{(k)}=\big{(}\bm{q}^{(1)},\bm{q}^{(1)},\ldots,\bm{q}^{(k)}\big{)}\in\mathbb{R}^{2}\times\mathbb{R}^{3}\times\cdots\times\mathbb{R}^{k+1}$
and
$\widetilde{\bm{R}}^{(k)}=\big{(}\widetilde{\bm{r}}^{(1)},\widetilde{\bm{r}}^{(2)},\ldots,\widetilde{\bm{r}}^{(k)}\big{)}\in\mathbb{R}^{2}\times\mathbb{R}^{3}\times\cdots\times\mathbb{R}^{k+1}$.
Under this notation, we have from Definition 5.1 (and the fact from (5.15)
that $\bm{Q}_{A_{0}}^{(k)}(0,0)=\bm{Q}^{(k)}$ and
$\bm{Q}_{A_{1}}^{(k)}(0,0)=\widetilde{\bm{R}}^{(k)}$ for each $k\in\llbracket
1,m\rrbracket$), that

$\displaystyle\partial_{t}^{i}\partial_{x}^{j}D(0,0)=\mathfrak{v}_{i,j}\big{(}\widetilde{\bm{R}}^{(i+j+2)}\big{)};\qquad\partial_{t}^{i}\partial_{x}^{j}D(\rho,0)=\mathfrak{v}_{i,j}\big{(}\bm{Q}^{(i+j+2)}\big{)},$

for each $(i,j)\in\mathbb{R}_{\geq 0}^{2}$ with $i+j\leq m-2$. Since $\bm{Q}$
is consistent, we have from (5.2) and (5.11) (and the fact that all
derivatives of $D$ of order at most $2m$ are uniformly bounded on compact
sets) that there exists a constant $C_{4}=C_{4}(\varepsilon,B,m)>1$ with

$\displaystyle\partial_{t}^{i}\partial_{x}^{j}D(\rho,0)=0,\quad\text{for any
$(i,j)\in\mathbb{Z}_{\geq 0}^{2}$ with $i+j\leq
m-2$};\qquad\displaystyle\max_{i,j\in\llbracket
0,m\rrbracket}\displaystyle\sup_{|z|\leq
1}\big{|}\partial_{t}^{i}\partial_{x}^{j}D(z)\big{|}\leq C_{4}.$

Hence, it follows from a Taylor expansion that

$\displaystyle\Big{|}\mathfrak{v}_{i,j}\big{(}\widetilde{\bm{R}}^{(i+j+2)}\big{)}\Big{|}=\big{|}\partial_{t}^{i}\partial_{x}^{j}D(0,0)\big{|}\leq
C_{4}\rho^{m-i-j-1}.$

This, with 5.7, (5.12), (5.11), (5.13), and (5.14), implies (5.18) and thus
the lemma. ∎

######  Corollary 5.11.

Adopting the notation and assumptions of 5.9, there exist constants
$\nu=\nu(\varepsilon,B,m)\in(0,1)$, $c=c(\varepsilon,B,m)>0$, and
$C=C(\varepsilon,B,m)>1$; a strip domain $\mathfrak{S}\subset\mathbb{R}^{2}$;
and a function $\mathfrak{H}:\overline{\mathfrak{S}}\rightarrow\mathbb{R}$,
such that all statements but the first enumerated one in 5.9 hold. Moreover,
the following variant of that first statement holds, for any real number
$\rho\in(0,c)$.

  1. 1.

Denoting
$\bm{Q}_{\mathfrak{H}}^{(m)}(\rho,0)=\big{(}\bm{q}_{\mathfrak{H}}^{(k)}(\rho,0)\big{)}_{k\in\llbracket
1,m\rrbracket}$, we have
$\big{|}\bm{q}_{\mathfrak{H}}^{(k)}(\rho,0)-\bm{q}^{(k)}\big{|}\leq
C\rho^{m-k+1}$ for all $k\in\llbracket 1,m\rrbracket$.

###### Proof.

Define the $m$ real pairs
$\big{(}r_{0}^{(k)},r_{1}^{(k)}\big{)}\in(\mathbb{R}^{2})^{m}$ from $\bm{Q}$
through (5.9), and let $\bm{R}$ denote its consistent extension. By 5.9
(applied with the $\bm{Q}$ there equal to $\bm{R}$ here) and (5.12), there
exist constants $\nu=\nu(\varepsilon,B,m)\in(0,1)$, $c=c(\varepsilon,B,m)>1$,
and $C_{1}=C_{1}(\varepsilon,B,m)>1$; a strip domain
$\mathfrak{S}\subset\mathbb{R}^{2}$ of width $\nu$, such that
$[0,\nu]\times[-\nu,\nu]\subseteq\mathfrak{S}\subseteq[0,\nu]\times[-C_{1}\nu,C_{1}\nu]$;
and a function $\mathfrak{H}:\overline{\mathfrak{S}}\rightarrow\mathbb{R}$
with $\|\mathfrak{H}\|_{\mathcal{C}^{m+1}(\overline{\mathfrak{S}})}\leq C_{1}$
and $\inf_{z\in\mathfrak{S}}\partial_{x}\mathfrak{H}(z)\leq-C_{1}^{-1}$,
satisfying $\lim_{z\rightarrow(0,0)}\bm{Q}_{\mathfrak{H}}^{(m)}(z)=\bm{R}$ and
the following property. For any real numbers $\delta\in(0,1)$, there exists a
constant $C_{2}=C_{2}(\varepsilon,\delta,B,m)>1$ such that $\mathfrak{H}$ is
$(C_{2};c;\delta)$-concentrating. This $\mathfrak{H}$ satisfies the last
condition of the corollary, as 5.10 yields a constant
$C_{3}=C_{3}(\varepsilon,B,m)>1$ such that
$\big{|}\bm{q}_{\mathfrak{H}}^{(k)}(\rho,0)-\bm{q}^{(k)}\big{|}\leq
C_{3}\rho^{m-k+1}$, for each $k\in\llbracket 1,m\rrbracket$ and sufficiently
small $\rho$. ∎

##  6\. Proof of Concentration Bounds

In this section we establish 4.4.

###  6.1. Local Approximations of $\mathfrak{H}$

In this section we will establish various statements needed to show 4.4 in
Section 6.2 below; we only establish the bound in that statement on
$\Omega_{j;k}^{+}(z)$, as the proof of the bound on $\Omega_{j;k}^{-}(z)$ is
entirely analogous. Throughout, we recall the notation from Section 4.2; fix
$(j,k)\in\llbracket 0,J\rrbracket\times[0,K]$ and a point
$z\in\mathfrak{P}_{f;g}$; and set (recall (4.7))

$\displaystyle\varphi=\varphi_{j;k-1}^{+};\qquad\Omega=\bigcup_{w\in\mathfrak{P}}\Omega_{j;k-1}^{+};\qquad\kappa=1-(k-1)\rho^{2}n^{-\delta};\qquad\mathfrak{P}=\mathfrak{P}_{f;g}.$

Throughout this section, we further restrict to the event
$\Omega^{\complement}$.

Let us now briefly outline how we will establish 4.4. We will proceed by
locally comparing the Brownian bridges $\bm{x}$ in a neighborhood of $z$ with
a family $\bm{y}$ of non-intersecting Brownian bridges, whose associated
normalized height function $n^{-1}\mathsf{H}^{\bm{y}}$ is known to concentrate
around its limit shape $\mathfrak{H}$. By 5.11, there are many choices for
this limit shape $\mathfrak{H}$; we will make its second derivatives slightly
smaller than those of $\varphi$ (while making its gradient match that of
$\varphi$), so as to ensure that $\varphi_{j;k}^{+}(z)>\mathfrak{H}(z)$ while
$\varphi|_{\partial\mathfrak{B}_{\rho}(z)}<\mathfrak{H}|_{\partial\mathfrak{B}_{\rho}(z)}$.
Since the normalized height function $n^{-1}H$ of $\bm{x}$ is bounded above by
$\varphi$ on the event $\Omega^{\complement}$, comparing $\bm{x}$ to $\bm{y}$
using monotonicity (2.3) and applying the concentration bound for $\bm{y}$
(5.11) will yield
$n^{-1}\mathsf{H}^{\bm{x}}(z)\leq\mathfrak{H}(z)<\varphi_{j;k}^{+}(z)$ with
high probability, establishing 4.4.

To implement this, we must fix the derivatives of the function $\mathfrak{H}$,
to which end we require some additional notation. Recalling the
$\mathfrak{b}_{ij}$ from (3.26), define for any
$(u,v)\in\mathbb{R}\times\mathbb{R}_{<0}$ the vector

$\displaystyle\bm{p}(u,v)=\big{(}\mathfrak{b}_{xx}(u,v),2\mathfrak{b}_{tx}(u,v),\mathfrak{b}_{tt}(u,v)\big{)}\in\mathbb{R}^{3}.$

Observe under this notation, and recalling $\bm{q}_{F}^{(2)}$ for any function
$F\in\mathcal{C}^{2}(\mathfrak{P})$ from (5.1), the constraint (3.25) is
equivalent to $\bm{p}\big{(}\nabla F(z)\big{)}\cdot\bm{q}_{F}^{(2)}(z)=0$.

######  Lemma 6.1.

The following two statements hold for $n$ sufficiently large.

  1. (1)

For any $z_{0}\in\mathfrak{P}$, we have

$\displaystyle\partial_{x}\varphi(z_{0})\leq-\frac{\varepsilon}{2};\qquad\|\varphi\|_{\mathcal{C}^{m+1}(\overline{\mathfrak{P}})}\leq
2B.$

  2. (2)

For any $z_{0}\in\mathfrak{P}$, we have

$\displaystyle\bigg{|}\displaystyle\frac{\bm{p}\big{(}\nabla\varphi(z_{0})\big{)}\cdot\bm{q}_{H}^{(2)}(z_{0})}{\bm{p}\big{(}\nabla\varphi(z_{0})\big{)}\cdot\bm{q}_{\psi}^{(2)}(z_{0})}\bigg{|}\leq\displaystyle\frac{\kappa\omega}{2}.$

###### Proof.

From the definition (4.7) of $\varphi=\varphi_{j;k-1}^{+}$, we have for any
$z_{0}=(t,x)\in\mathfrak{P}$ that

(6.1) $\displaystyle\big{|}\nabla\varphi(z_{0})-\nabla
H(z_{0})\big{|}\leq\kappa\omega\big{|}\nabla\psi(z_{0})\big{|}\leq
2\kappa\omega\zeta\max\\{e^{\zeta t},e^{\zeta x}\\}.$

where in the second inequality we used the definition (4.6) of $\psi$.
Together with the fact that any $z_{0}=(t,x)\in\mathfrak{P}$ satisfies
$|x|+|t|\leq 3B_{0}$ (as
$|x|+|t|\leq\|f\|_{\mathcal{C}^{m+1}}+\|g\|_{\mathcal{C}^{m+1}}+L^{-1}\leq
2\|G\|_{\mathcal{C}^{m+1}}+B_{0}\leq 3B_{0}$) and the definition (4.6) of
$\omega$, this implies for sufficiently large $n$ that

(6.2)
$\displaystyle\big{|}\partial_{x}\varphi(z_{0})-\partial_{x}H(z_{0})\big{|}\leq
2\kappa\omega\zeta e^{3B_{0}\zeta}\leq\frac{\varepsilon}{2}.$

Since (4.1) gives $\partial_{x}H(z_{0})\leq-\varepsilon$, this yields the
first part of the first statement of the lemma; the proof of the second part
of the first statement is entirely analogous.

To establish the second, first observe from (6.2) and (4.2) that
$\big{|}\nabla\varphi(z_{0})\big{|}\leq\big{|}\nabla
H(z_{0})\big{|}+\varepsilon\leq 2B$ for each $z_{0}\in\mathfrak{P}$. Moreover,
by (3.26), each $\mathfrak{b}_{ij}(u,v)$ is $100B^{3}$-Lipschitz on the region
where $\max\big{\\{}|u|,|v|\big{\\}}\leq 2B$. Thus, since $\bm{p}\big{(}\nabla
H(z_{0})\big{)}\cdot\bm{q}_{H}^{(2)}(z_{0})=0$ (by the fact that $H$ solves
(3.25)), we have for any $z_{0}\in\mathfrak{P}$ that

(6.3) $\displaystyle\begin{aligned}
\Big{|}\bm{p}\big{(}\nabla\varphi(z_{0})\big{)}\cdot\bm{q}_{H}^{(2)}(z_{0})\Big{|}&=\bigg{|}\Big{(}\bm{p}\big{(}\nabla\varphi(z_{0})\big{)}-\bm{p}\big{(}\nabla
H(z_{0})\big{)}\Big{)}\cdot\bm{q}_{H}^{(2)}(z_{0})\bigg{|}\\\ &\leq
200B^{3}\big{|}\nabla\varphi(z_{0})-\nabla
H(z_{0})\big{|}\big{|}\bm{q}_{H}^{(2)}(z_{0})\big{|}\leq
200B^{4}\big{|}\nabla\varphi(z_{0})-\nabla H(z_{0})\big{|},\end{aligned}$

where in the last bound we used (4.2). Again by (3.26) and the first part of
the lemma, we have

$\displaystyle\inf_{z_{0}\in\mathfrak{P}}\min\big{\\{}\mathfrak{b}_{tt}\big{(}\nabla\varphi(z_{0})\big{)},\mathfrak{b}_{xx}\big{(}\nabla\varphi(z_{0})\big{)}\big{\\}}\geq\frac{\varepsilon^{4}}{16}.$

Since (4.6) gives for $z_{0}=(t,x)\in\mathfrak{P}$ that
$\bm{q}_{\psi}^{(2)}(z_{0})=(-\zeta^{2}e^{\zeta x},0,-\zeta^{2}e^{\zeta t})$,
this implies

$\displaystyle\Big{|}\bm{p}\big{(}\nabla\varphi(z_{0})\big{)}\cdot\bm{q}_{\psi}^{(2)}(z_{0})\Big{|}\geq\displaystyle\frac{\zeta^{2}\varepsilon^{4}}{16}\displaystyle\max\\{e^{\zeta
t},e^{\zeta x}\\}.$

Together with (6.3) and (6.1), this yields

$\displaystyle\displaystyle\sup_{z_{0}\in\mathfrak{P}}\displaystyle\frac{\Big{|}\bm{p}\big{(}\nabla\varphi(z_{0})\big{)}\cdot\bm{q}_{H}^{(2)}(z_{0})\Big{|}}{\Big{|}\bm{p}\big{(}\nabla\varphi(z_{0})\big{)}\cdot\bm{q}_{\psi}^{(2)}(z_{0})\Big{|}}\leq\displaystyle\frac{6400B^{4}}{\varepsilon^{4}\zeta}\kappa\omega,$

which by (4.6) yields the lemma. ∎

Now let us proceed to prescribe the derivatives of $\mathfrak{H}$ at $z$,
which will be a higher order variant of what was done in [52, Equation (72)].
They will be set to be near an admissible, consistent profile
$\bm{Q}=\big{(}\bm{q}^{(k)}\big{)}\in\mathbb{R}^{2}\times\mathbb{R}^{3}\times\cdots\times\mathbb{R}^{m+1}$,
defined as follows. Denoting
$\bm{q}^{(k)}=\big{(}q_{0}^{(k)},q_{1}^{(k)},\ldots,q_{k}^{(k)}\big{)}\in\mathbb{R}^{k+1}$
for each $k\in\llbracket 3,m\rrbracket$, set

(6.4)
$\displaystyle\bm{q}^{(1)}=\nabla\varphi(z);\quad\bm{q}^{(2)}=\bm{q}_{H}^{(2)}(z)-\displaystyle\frac{\bm{p}\big{(}\nabla\varphi(z)\big{)}\cdot\bm{q}_{H}^{(2)}(z)}{\bm{p}\big{(}\nabla\varphi(z)\big{)}\cdot\bm{q}_{\psi}^{(2)}(z)}\bm{q}_{\psi}^{(2)}(z);\quad
q_{0}^{(k)}=\partial_{x}^{k}\varphi(z);\quad
q_{1}^{(k)}=\partial_{t}\partial_{x}^{k-1}\varphi(z).$

Observe from the first part of 6.1 that $q_{0}^{(1)}<0$. Thus, the sequence of
$m$ pairs $\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}\in(\mathbb{R}^{2})^{m}$
admits a consistent extension,999Recall in Definition 5.5 that consistent
extensions were defined on sequences of pairs, while $\bm{q}^{(2)}$ above is a
triple. Our choice of $\bm{q}^{(2)}$ ensures that this introduces no conflict
in notation. Indeed, (6.4) implies that
$\bm{p}\big{(}\bm{q}^{(1)}\big{)}\cdot\bm{q}^{(2)}=0$, which by (3.25) and
(5.2) gives $\mathfrak{v}_{0,0}\big{(}\bm{q}^{(1)},\bm{q}^{(2)}\big{)}=0$. In
particular, the second vector of the admissible extension $\bm{Q}$ of
$\big{(}q_{0}^{(k)},q_{1}^{(k)}\big{)}$ must indeed be $\bm{q}^{(2)}$. which
we denote by $\bm{Q}=\big{(}\bm{q}^{(k)}\big{)}$.

By 5.11 (and the first statement of 6.1), there exist constants
$\nu=\nu(\varepsilon,B,m)>0$ and $C_{1}=C_{1}(\varepsilon,B,m)>1$; a strip
domain
$\mathfrak{S}=\mathfrak{P}_{\mathfrak{f};\mathfrak{g}}\subset\mathbb{R}^{2}$
with
$[0,\nu]\times[-\nu,\nu]\subseteq\mathfrak{S}\subseteq[0,\nu]\times[-C_{1}\nu,C_{1}\nu]$;
and a function $\mathfrak{H}:\overline{\mathfrak{S}}\rightarrow\mathbb{R}$
satisfying (recall (5.1))

(6.5)
$\displaystyle\|\mathfrak{H}\|_{\mathcal{C}^{m+1}(\overline{\mathfrak{S}})}\leq
C_{1};\quad\displaystyle\inf_{z\in\mathfrak{S}}\partial_{x}\mathfrak{H}(z)\leq-\displaystyle\frac{1}{C_{1}};\quad\text{$\big{|}\bm{q}_{\mathfrak{H}}^{(k)}(\rho,0)-\bm{q}^{(k)}\big{|}\leq
C_{1}\rho^{m-k+1}$, for each $k\in\llbracket 1,m\rrbracket$},$

and the below property. There exist constants
$c_{0}=c_{0}(\varepsilon,\delta,B,m)>0$ and
$C_{2}=C_{2}(\varepsilon,\delta,B,m)>1$ so that $\mathfrak{H}$ is
$\big{(}C_{2};c_{0};\delta\big{)}$-concentrating, meaning (recall Definition
5.8) that $\mathfrak{H}$ satisfies (3.25) and the following. There exist an
integer $n^{\prime}\in[c_{0}n,n]$, entrance data
$\overline{\bm{u}}\in\overline{\mathbb{W}}_{n^{\prime}}$ and ending data
$\overline{\bm{v}}\in\mathbb{W}_{n^{\prime}}$ on $\mathfrak{S}$ such that,
sampling non-intersecting Brownian bridges
$\bm{y}=(y_{1},y_{2},\ldots,y_{n^{\prime}})$ from
$\mathfrak{Q}_{\mathfrak{f};\mathfrak{g}}^{\overline{\bm{u}};\overline{\bm{v}}}(n^{-1})$
and denoting the associated height function by
$\mathsf{H}^{\bm{y}}:\overline{\mathfrak{S}}\rightarrow\mathbb{Z}$, we have

$\displaystyle\mathbb{P}\Bigg{[}\displaystyle\sup_{z\in\mathfrak{S}}\big{|}n^{-1}\mathsf{H}^{\bm{y}}(z)-\mathfrak{H}(z)\big{|}\geq
n^{\delta-1}\Bigg{]}\leq C_{2}e^{-c_{0}(\log n)^{2}}.$

The following lemma bounds the distance between $\bm{Q}_{\varphi}(z)$ and
$\bm{Q}_{\mathfrak{H}}(z)$ under this notation.

######  Lemma 6.2.

There exists a constant $C=C(\varepsilon,B,m)>1$ such that

(6.6)
$\displaystyle\big{|}\bm{q}_{\varphi}^{(k)}(z)-\bm{q}_{\mathfrak{H}}^{(k)}(\rho,0)\big{|}\leq
C(\rho^{m-k+1}+\omega),\qquad\text{for any $k\in\llbracket 1,m\rrbracket$}.$

###### Proof.

The bound (6.6) holds at $k=1$ by the first statement of (6.4) and the second
statement of (6.5). So, we may assume in what follows that $k\in\llbracket
2,m\rrbracket$. To establish (6.6) in that case, it suffices to show that

(6.7)
$\displaystyle\big{|}\bm{q}_{\varphi}^{(k)}(z)-\bm{q}_{H}^{(k)}(z)\big{|}\leq
C\omega;\qquad\big{|}\bm{q}^{(k)}-\bm{q}_{\mathfrak{H}}^{(k)}(\rho,0)\big{|}\leq
C\rho^{m-k+1};\qquad\big{|}\bm{q}_{H}^{(k)}(z)-\bm{q}^{(k)}\big{|}\leq
C\omega.$

To verify the first, observe from (4.7) and (4.6) that

(6.8)
$\displaystyle\big{|}\bm{q}_{\varphi}^{(k)}(z)-\bm{q}_{H}^{(k)}(z)\big{|}\leq\omega\big{|}\bm{q}_{\psi}^{(k)}(z)\big{|}\leq(m+1)\zeta^{m}e^{\zeta|z|}\omega\leq
C_{3}\omega,$

for some constant $C_{3}=C_{3}(\varepsilon,B,m)>1$ (where in the last
inequality we implicitly used the fact that $|z|\leq 3B_{0}$, which holds
since $|z|\leq\|f\|_{\mathcal{C}^{m+1}}+\|g\|_{\mathcal{C}^{m+1}}+L^{-1}\leq
3B_{0}$). The second follows from the last statement of (6.5).

To establish the third statement of (6.7), observe since $\bm{Q}_{H}^{(m)}(z)$
and $\bm{Q}$ are consistent that it suffices by 5.6 (and the first part of
6.1) to show the existence of a constant $C_{4}=C_{4}(\varepsilon,B,m)>1$ such
that

(6.9) $\displaystyle\big{|}\partial_{x}^{k}H(z)-q_{0}^{(k)}\big{|}\leq
C_{4}\omega;\qquad\big{|}\partial_{t}\partial_{x}^{k-1}H(z)-q_{1}^{(k)}\big{|}\leq
C_{4}\omega,\qquad\text{for each $k\in\llbracket 1,m\rrbracket$}.$

For $k\neq 2$, this follows from (6.4) and (6.8), and for $k=2$ it follows
from the fact that

$\displaystyle\big{|}\bm{q}_{H}^{(2)}(z)-\bm{q}^{(2)}\big{|}=\Bigg{|}\displaystyle\frac{\bm{p}\big{(}\nabla\varphi(z)\big{)}\cdot\bm{q}_{H}^{(2)}(z)}{\bm{p}\big{(}\nabla\varphi(z)\big{)}\cdot\bm{q}_{\psi}^{(2)}(z)}\Bigg{|}\big{|}\bm{q}_{\psi}^{(2)}(z)\big{|}\leq\big{|}\bm{q}_{\psi}^{(2)}(z)\big{|}\frac{\omega\kappa}{2}\leq
2\zeta^{2}e^{3B_{0}\zeta}\omega\kappa,$

where the first statement holds by (6.4), the second by the second part of
6.1; and the third from the explicit form (4.6) for $\psi$, with the fact that
$|z|\leq 3B_{0}$ for $z\in\mathfrak{P}$. This confirms (6.9) and thus the
lemma. ∎

In the below, for any point $w\in\mathbb{R}^{d}$ and real number $r>0$, we
denote the open disk
$\mathcal{B}_{r}(w)=\big{\\{}w^{\prime}\in\mathbb{R}^{d}:|w^{\prime}-w|\leq
r\big{\\}}$. We next have the following lemma, which is similar to [52,
Equation (88)], indicating the relative height of $\varphi(z)$ with respect to
$\varphi|_{\partial\mathcal{B}_{\rho}(z)}$ is larger than that of
$\mathfrak{H}(\rho,0)$ with respect to that of
$\mathfrak{H}|_{\partial\mathcal{B}_{\rho}(\rho,0)}$.

######  Lemma 6.3.

There exists a constant $c=c(\varepsilon,B,m)>0$ such that the following
holds. Fix $w\in\mathfrak{P}$ with $|z-w|\leq n^{1/2m^{2}}\rho$; set
$z_{0}=(\rho,0)\in\mathfrak{S}$; and set $w_{0}=z_{0}-z+w$. If
$w_{0}\in\mathfrak{S}$, then

$\displaystyle\varphi(z)-\varphi(w)\geq\mathfrak{H}(z_{0})-\mathfrak{H}(w_{0})+c\omega|z-w|^{2}-c^{-1}\rho^{m}|z-w|.$

###### Proof.

For any continuously twice-differentiable function $F$ and point $z^{\prime}$
in its domain, let the $2\times 2$ matrix $\bm{H}_{F}(z^{\prime})$ denote its
Hessian at $z^{\prime}$, given by

$\displaystyle\bm{H}_{F}(z^{\prime})=\left[\begin{array}[]{cc}\partial_{t}^{2}F(z^{\prime})&\partial_{t}\partial_{x}F(z^{\prime})\\\
\partial_{t}\partial_{x}F(z^{\prime})&\partial_{x}^{2}F(z^{\prime})\end{array}\right].$

Then, by (6.4), (6.5), a Taylor expansion, the fact that $w_{0}-z_{0}=w-z$,
the first part of 6.1, and 6.2, there is a constant
$C_{3}=C_{3}(\varepsilon,B,m)>1$ such that

(6.10) $\displaystyle\begin{aligned}
\Big{|}\varphi(w)-\varphi(z)-\big{(}&\mathfrak{H}(w_{0})-\mathfrak{H}(z_{0})\big{)}-(w-z)\cdot\big{(}\bm{H}_{\varphi}(z)-\bm{H}_{\mathfrak{H}}(z_{0})\big{)}(w-z)\Big{|}\\\
&\leq|z-w|\cdot\big{|}\bm{q}_{\varphi}^{(1)}(z)-\bm{q}_{\mathfrak{H}}^{(1)}(z_{0})\big{|}+\displaystyle\sum_{k=3}^{m}|z-w|^{k}\big{|}\bm{q}_{\varphi}^{(k)}(z)-\bm{q}_{\mathfrak{H}}^{(k)}(z_{0})\big{|}\\\
&\qquad\qquad\qquad\qquad\qquad+|z-w|^{m+1}\big{(}\|\varphi\|_{\mathcal{C}^{m+1}(\overline{\mathfrak{S}})}+\|\mathfrak{H}\|_{\mathcal{C}^{m+1}(\overline{\mathfrak{S}})}\big{)}\\\
&\leq
C_{3}\rho^{m}|z-w|+C_{3}|z-w|^{m+1}+C_{3}\displaystyle\sum_{k=3}^{m}(\rho^{m-k+1}+\omega)|z-w|^{k}.\end{aligned}$

Next, observe that

(6.11) $\displaystyle\begin{aligned}
(w-z)&\cdot\big{(}\bm{H}_{\varphi}(z)-\bm{H}_{\mathfrak{H}}(z_{0})\big{)}(w-z)\\\
&=(w-z)\cdot\big{(}\bm{H}_{H}(z)-\bm{H}_{\mathfrak{H}}(z_{0})+\kappa\omega\bm{H}_{\psi}(z)\big{)}(w-z)\\\
&\leq
C_{1}\rho^{m-1}|z-w|^{2}+\Bigg{(}\kappa\omega-\displaystyle\frac{\bm{p}\big{(}\nabla\varphi(z)\big{)}\cdot\bm{q}_{H}^{(2)}(z)}{\bm{p}\big{(}\nabla\varphi(z)\big{)}\cdot\bm{q}_{\psi}^{(2)}(z)}\Bigg{)}(w-z)\cdot\big{(}\bm{H}_{\psi}(z)\big{)}(w-z).\end{aligned}$

where for the first statement we applied the definition (4.7) of $\varphi$ and
for the second we applied (6.4) and (6.5). By the second part of 6.1, and the
fact that $\bm{q}_{\psi}^{(2)}$ has all negative entries (by the explicit form
(4.6) for $\psi$), we have

$\displaystyle\Bigg{(}\kappa\omega-\displaystyle\frac{\bm{p}\big{(}\nabla\varphi(z)\big{)}\cdot\bm{q}_{H}^{(2)}(z)}{\bm{p}\big{(}\nabla\varphi(z)\big{)}\cdot\bm{q}_{\psi}^{(2)}(z)}\Bigg{)}\bm{q}_{\psi}^{(2)}(z)\leq\frac{\kappa\omega}{2}\bm{q}_{\psi}^{(2)}(z),\qquad\text{entrywise}.$

Since $\bm{q}_{\psi}^{(2)}(t,x)=(-\zeta^{2}e^{\zeta t},0,-\zeta^{2}e^{\zeta
x})$, since $\kappa\geq\frac{1}{4}$, and since $|z|\leq 3B_{0}$ for any
$z\in\mathfrak{P}$ (as
$|z|\leq\|f\|_{\mathcal{C}^{m+1}}+\|g\|_{\mathcal{C}^{m+1}}+L^{-1}\leq
3B_{0}$), it follows that there is a constant $c_{1}=c_{1}(\varepsilon,m)>0$
such that

$\displaystyle\Bigg{(}\kappa\omega-\displaystyle\frac{\bm{p}\big{(}\nabla\varphi(z)\big{)}\cdot\bm{q}_{H}^{(2)}(z)}{\bm{p}\big{(}\nabla\varphi(z)\big{)}\cdot\bm{q}_{\psi}^{(2)}(z)}\Bigg{)}(w-z)\cdot\big{(}\bm{H}_{\psi}(z)\big{)}(w-z)\leq-
c_{1}\omega|z-w|^{2}.$

Together with (6.10) and (6.11), this gives

$\displaystyle\varphi(w)-\varphi(z)-\big{(}\mathfrak{H}(w_{0})-\mathfrak{H}(z_{0})\big{)}$
$\displaystyle\leq
C_{3}\displaystyle\sum_{k=1}^{m+1}\rho^{m-k+1}|z-w|^{k}+C_{3}m\omega|z-w|^{3}-c_{1}\omega|z-w|^{2}.$

This, since $\rho^{m-k+1}|z-w|^{k}+\omega|z-w|^{3}\leq
2n^{-1/m^{2}}\omega|z-w|^{2}$ for each $k\in\llbracket 2,m+1\rrbracket$ (which
holds by (4.6), the fact that $|z-w|\leq n^{1/2m^{2}}\rho$, and the fact that
$\delta\leq\frac{1}{5m^{2}}$), implies the lemma. ∎

###  6.2. Proof of 4.4

In this section we establish 4.4. As indicated in Section 6.1, we only show
the bound in that proposition on $\Omega_{j;k}^{+}(z)$, as the proof of the
bound on $\Omega_{j;k}^{-}(z)$ is entirely analogous; recall we assume that
$\Omega^{\complement}$ holds. We proceed by comparing the bridges $\bm{x}(t)$
with those $\bm{y}(t)$, concentrating around $\mathfrak{H}$, in a local
neighborhood (of diameter about $\rho$) around $z$. To implement this, we must
reindex the bridges in $\bm{x}$ and $\bm{y}$ to form $\bm{x}^{\prime}$ and
$\bm{y}^{\prime}$, defined as follows.

Figure 6. For the proof of 4.4, we compare the non-intersecting Brownian
bridges $\bm{x}(t)$ in a neighborhood of $z$ with bridges $\bm{y}(t)$ in one
of $z_{0}$.

Let $z=(t,x)\in\mathfrak{P}$ and $z_{0}=(\rho,0)\in\mathfrak{S}$; see Figure 6
for a depiction. Letting $c_{1}=c_{1}(\varepsilon,B,m)>0$ denote the constant
$c$ from 6.3 (and recalling $C_{1}$ from (6.5)), set

(6.12) $\displaystyle\begin{aligned}
&A=\bigg{\lfloor}n\varphi(z)-\displaystyle\frac{c_{1}\rho^{2}\omega
n}{2}\bigg{\rfloor};\quad
B=\big{\lfloor}n\mathfrak{H}(z_{0})+n^{\delta}\big{\rfloor};\quad\varsigma_{1}=\min\\{\rho,t\\};\quad\varsigma_{2}=\min\\{\rho,L^{-1}-t\\};\\\
&d_{1}=\min\big{\\{}\lceil 5C_{1}^{3}\rho n\rceil,A-1\big{\\}};\quad
d_{2}=\min\big{\\{}\lceil 5C_{1}^{3}\rho
n\rceil,n-A\big{\\}};\quad\varsigma=\varsigma_{1}+\varsigma_{2};\quad
d=d_{1}+d_{2}.\end{aligned}$

In what follows, we omit the floors and ceilings, as they do not affect the
proofs; we may also assume that $A\leq n-5n^{\delta}$, for otherwise
$\varphi(z)\geq 1+n^{-\delta}\rho^{2}\omega$ and so
$n^{-1}\mathsf{H}^{x}(z)\leq
1\leq\varphi(z)-n^{-\delta}\rho^{2}\omega=\varphi_{j;k}(z)$ holds
deterministically. For each integer $j\in\llbracket 0,d+2\rrbracket$ and real
number $s\in[0,\varsigma]$, set

(6.13) $\displaystyle
x_{j}^{\prime}(s)=x_{A+j-d_{1}-1}(s+t-\varsigma_{1})-x;\qquad
y_{j}^{\prime}(s)=y_{B+j-d_{1}-1}(s+\rho-\varsigma_{1}),$

where $x_{0}=g$ and $x_{n+1}=f$. Further denote
$\bm{x}^{\prime}=(x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{d+1}^{\prime})\in\llbracket
1,d+1\rrbracket\times\mathcal{C}\big{(}[0,\varsigma]\big{)}$ and
$\bm{y}^{\prime}=(y_{1}^{\prime},y_{2}^{\prime},\ldots,y_{d+1}^{\prime})\in\llbracket
1,d+1\rrbracket\times\mathcal{C}\big{(}[0,\varsigma]\big{)}$. Also define the
functions
$\mathfrak{x}^{-},\mathfrak{x}^{+},\mathfrak{y}^{-},\mathfrak{y}^{+}:[0,\varsigma]\rightarrow\mathbb{R}$
by for each $s\in[0,\varsigma]$ setting

$\displaystyle\mathfrak{x}^{-}(s)=x_{d+2}^{\prime}(s);\qquad\mathfrak{x}^{+}(s)=x_{0}^{\prime}(s);\qquad\mathfrak{y}^{-}(s)=y_{d+2}^{\prime}(s);\qquad\mathfrak{y}^{+}(s)=y_{0}^{\prime}(s).$

Further define the entrance data
$\bm{u}^{\prime},\bm{u}^{\prime\prime}\in\overline{\mathbb{W}}_{d+1}$ and
ending data
$\bm{v}^{\prime},\bm{v}^{\prime\prime}\in\overline{\mathbb{W}}_{d+1}$ by
setting $\bm{u}^{\prime}=\bm{x}^{\prime}(0)$,
$\bm{v}^{\prime}=\bm{x}^{\prime}(\varsigma)$,
$\bm{u}^{\prime\prime}=\bm{y}^{\prime}(0)$, and
$\bm{v}^{\prime\prime}=\bm{y}^{\prime}(\varsigma)$.

Then, the laws of $\bm{x}^{\prime}$ and $\bm{y}^{\prime}$ are given by non-
intersecting Brownian bridges sampled from the measures
$\mathfrak{Q}_{\mathfrak{x}^{-};\mathfrak{x}^{+}}^{\bm{u}^{\prime};\bm{v}^{\prime}}(n^{-1})$
and
$\mathfrak{Q}_{\mathfrak{y}^{-};\mathfrak{y}^{+}}^{\bm{u}^{\prime\prime};\bm{v}^{\prime\prime}}(n^{-1})$,
respectively. Denote their height functions (recall (3.6)) by
$\mathsf{H}^{\bm{x}^{\prime}}:\mathfrak{P}_{\mathfrak{x}^{-};\mathfrak{x}^{+}}\rightarrow\mathbb{R}$
and
$\mathsf{H}^{\bm{y}^{\prime}}:\mathfrak{P}_{\mathfrak{y}^{-};\mathfrak{y}^{+}}\rightarrow\mathbb{R}$,
respectively. Observe by (6.13) that for any
$(t_{0},x_{0})\in\mathfrak{P}_{\mathfrak{x}^{-},\mathfrak{x}^{+}}\cap\mathfrak{P}_{\mathfrak{y}^{-},\mathfrak{y}^{+}}$
we have for any $(t_{0},x_{0})\in[0,\varsigma]\times\mathbb{R}$ that

(6.14) $\displaystyle\begin{aligned}
&\mathsf{H}^{\bm{x}^{\prime}}(t_{0},x_{0})=\mathsf{H}^{\bm{x}}(t_{0}+t-\varsigma_{1},x_{0}+x)-A+d_{1}+1;\\\
&\mathsf{H}^{\bm{y}^{\prime}}(t_{0},x_{0})=\mathsf{H}^{\bm{y}}(t_{0}+\rho-\varsigma_{1},x_{0})-B+d_{1}+1.\end{aligned}$

######  Lemma 6.4.

There exists a constant $c=c(\varepsilon,B,m)>1$ and a coupling between
$\bm{x}^{\prime}$ and $\bm{y}^{\prime}$ such that the following holds. On an
event $\mathscr{F}$ with $\mathbb{P}[\mathscr{F}]\geq 1-c^{-1}e^{-c(\log
n)^{2}}$, we have $x_{j}^{\prime}(s)\leq y_{j}^{\prime}(s)$ for each
$j\in\llbracket 1,d+1\rrbracket$ and $s\in[0,\varsigma]$.

###### Proof.

By monotonicity (2.3), we must show on an event of probability at least
$1-c^{-1}e^{-c(\log n)^{2}}$ that

(6.15) $\displaystyle\begin{aligned} &x_{j}^{\prime}(0)\leq
y_{j}^{\prime}(0),\quad\text{and}\quad x_{j}^{\prime}(\varsigma)\leq
y_{j}^{\prime}(\varsigma),\qquad\quad\text{for each $j\in\llbracket
1,d+1\rrbracket$};\\\ &x_{0}^{\prime}(s)\leq
y_{0}^{\prime}(s),\quad\text{and}\quad x_{d+2}^{\prime}(s)\leq
y_{d+2}^{\prime}(s),\quad\text{for each $s\in[0,\varsigma]$}.\end{aligned}$

Defining the event

(6.16)
$\displaystyle\mathscr{E}=\Bigg{\\{}\displaystyle\sup_{z\in\mathfrak{S}}\big{|}n^{-1}\mathsf{H}^{\bm{y}}(z)-\mathfrak{H}(z)\big{|}\leq
n^{\delta-1}\Bigg{\\}},\qquad\text{we have}\qquad\mathbb{P}[\mathscr{E}]\geq
1-C_{2}e^{-c_{0}(\log n)^{2}},$

since $\mathfrak{H}$ is $(C_{2};c_{0};\delta)$-concentrating. On
$\mathscr{E}\cap\Omega^{\complement}$, we have

$\displaystyle\varphi\big{(}s,x_{j}(s)\big{)}\geq\displaystyle\frac{j}{n};\qquad\mathfrak{H}\big{(}s,y_{j-n^{\delta}}(s)\big{)}\leq\displaystyle\frac{j}{n},\qquad\text{for
any $j\in\llbracket 1,n\rrbracket$},$

where the first statement holds from $\Omega^{\complement}$ and the second
holds from $\mathscr{E}$. Replacing $(j,s)$ with
$(A+j-d_{1}-1,s+t-\varsigma_{1})$ in the former; replacing $(j,s)$ with
$(B+j+n^{\delta}-d_{1}-1,s)$ in latter; and using (6.13) and (6.12), we find
for $j\in\llbracket 1,n\rrbracket$ and $s\in[0,\varsigma]$ that

$\displaystyle\varphi\big{(}s+t-\varsigma_{1},x_{j}^{\prime}(s)+x\big{)}\geq\varphi(z)-\displaystyle\frac{c_{1}\rho^{2}\omega}{2}+\displaystyle\frac{j-d_{1}-1}{n};$
$\displaystyle\mathfrak{H}\big{(}s+\rho-\varsigma_{1},y_{j}^{\prime}(s)\big{)}\leq\mathfrak{H}(z_{0})+2n^{\delta-1}+\displaystyle\frac{j-d_{1}-1}{n}.$

It follows that

(6.17) $\displaystyle\begin{aligned}
\varphi(z)-\varphi\big{(}s+t-\varsigma_{1},x_{j}^{\prime}(s)+x\big{)}&\leq\displaystyle\frac{c_{1}\rho^{2}\omega}{2}+\displaystyle\frac{d_{1}-j+1}{n}\\\
&\leq\mathfrak{H}(z_{0})-\mathfrak{H}\big{(}s+\rho-\varsigma_{1},y_{j}^{\prime}(s)\big{)}+\displaystyle\frac{c_{1}\rho^{2}\omega}{2}+2n^{\delta-1}.\end{aligned}$

We next apply 6.3 with that
$w_{0}=\big{(}s+\rho-\varsigma_{1},y_{j}^{\prime}(s)\big{)}$ for either
$(j,s)\in\\{0,d+2\\}\times[0,\varsigma]$ or $(j,s)\in\llbracket
1,d+2\rrbracket\times\\{0,\varsigma\\}$. To that end, we must first verify
that $|z_{0}-w_{0}|\leq n^{1/2m^{2}}\rho$. In both cases, we have from the
event $\mathscr{E}$ from (6.16), the first two bounds in (6.5), and (6.12)
that

$\displaystyle|z_{0}-w_{0}|\leq\big{|}y_{j}^{\prime}(s)\big{|}+\rho$
$\displaystyle=\big{|}y_{B+j-d_{1}-1}(s+\rho-\varsigma_{1})\big{|}+\rho$
$\displaystyle\leq
C_{1}\bigg{|}\displaystyle\frac{B+j-d_{1}-1}{n}-\mathfrak{H}(s+\rho-\varsigma_{1},0)\bigg{|}+C_{1}n^{\delta-1}+\rho$
$\displaystyle\leq
C_{1}\bigg{|}\displaystyle\frac{B+j-d_{1}-1}{n}-\mathfrak{H}(z_{0})\bigg{|}+C_{1}n^{\delta-1}+2C_{1}^{2}\rho\leq
20C_{1}^{4}\rho\leq n^{1/2m^{2}}\rho,$

which establishes the bound $|w_{0}-z_{0}|\leq n^{1/2m^{2}}\rho$. We may thus
apply 6.3 with $w_{0}=\big{(}s+\rho-\varsigma_{1},y_{j}^{\prime}(s)\big{)}$.
By (6.17), this yields

(6.18) $\displaystyle\begin{aligned}
\varphi(z)-\varphi\big{(}s+t-\varsigma_{1},x_{j}^{\prime}(s)+x\big{)}&\leq\mathfrak{H}(z_{0})-\mathfrak{H}(w_{0})+\displaystyle\frac{c_{1}\rho^{2}\omega}{2}+2n^{\delta-1}\\\
&\leq\varphi(z)-\varphi\big{(}s+t-\varsigma_{1},y_{j}^{\prime}(s)+x\big{)}+\displaystyle\frac{c_{1}\rho^{2}\omega}{2}+2n^{\delta-1}\\\
&\qquad-
c_{1}\omega|w_{0}-z_{0}|^{2}+c_{1}^{-1}\rho^{m}|w_{0}-z_{0}|.\end{aligned}$

If $|w_{0}-z_{0}|\geq\rho$, then since
$c_{1}\rho^{2}\omega=c_{1}n^{3\delta-1}\geq 4n^{\delta-1}$ and
$c_{1}^{-1}\rho^{m}|w_{0}-z_{0}|\leq c_{1}^{-1}n^{1/2m^{2}}\rho^{m+1}\leq
c_{1}^{-1}n^{(m+1)\delta-1-1/m+1/2m^{2}}\leq n^{\delta-1}$ (by (4.6) and the
fact that $\delta\leq\frac{1}{5m^{2}}$) it follows that

$\displaystyle\varphi\big{(}s+t-\varsigma_{1},y_{j}^{\prime}(s)+x\big{)}\leq\varphi\big{(}s+t-\varsigma_{1},x_{j}^{\prime}(s)+x\big{)}.$

Since $\partial_{x}\varphi<0$ (by the first part of 6.1), this yields
$y_{j}^{\prime}(s)\geq x_{j}^{\prime}(s)$, confirming (6.15).

If instead $|w_{0}-z_{0}|<\rho$, then we can improve (6.18) using the fact
that $\varphi|_{\partial\mathfrak{P}}=H|_{\partial\mathfrak{P}}$. In
particular, assume first in (6.15) that $s\in\\{0,\varsigma\\}$; since the two
cases are entirely analogous, we let $s=0$. Since
$\varsigma_{1}\leq|w_{0}-z_{0}|<\rho$, it follows from (6.12) that $t<\rho$
and so $\varsigma_{1}=t$. Then,
$\varphi\big{(}s+t-\varsigma_{1},x_{j}^{\prime}(s)+x\big{)}=\varphi\big{(}0,x_{j}^{\prime}(s)+x\big{)}$.
We next claim that

(6.19)
$\displaystyle\varphi\big{(}0,y_{j}^{\prime}(s)+x\big{)}=H\big{(}0,y_{j}^{\prime}(s)+x\big{)}\leq\varphi\big{(}0^{+},y_{j}^{\prime}(s)+x\big{)}-c_{2}\omega,\qquad\text{where
$c_{2}=\frac{1}{8e^{3B_{0}\zeta}}$}.$

Indeed, the first statement follows from the definition of $\varphi$ (see
below (4.7)) with the fact that
$\big{(}0,y_{j}^{\prime}(s)+x\big{)}\in\partial\mathfrak{P}$. To see the
second, first observe that

(6.20)
$\displaystyle\sup_{z\in\mathfrak{P}}\psi(z)=\sup_{z\in\mathfrak{P}}(\Upsilon-e^{\zeta
t}-e^{\zeta x})\leq 2+\sup_{z\in\mathfrak{P}}(e^{\zeta t}+e^{\zeta x})\leq
2e^{3B_{0}\zeta}+2\leq 4e^{3B_{0}\zeta},$

where in the first statement we used the definition (4.6) of $\psi$; in the
second we used the fact that $\inf_{z\in\mathfrak{P}}\psi(z)=2$; in the third
we used the fact that
$|t|+|x|\leq\sup_{t\in[0,1/L]}\big{|}f(t)\big{|}+\sup_{t\in[0,1/L]}\big{|}g(t)\big{|}+L^{-1}\leq
3B_{0}$; and in the fourth we used the fact that $B_{0},\zeta\geq 0$. Next,
observe for any $z\in\mathfrak{P}$ that

$\displaystyle\varphi(z)-H(z)=3-(j+1)\omega+\kappa\omega\psi(z)$
$\displaystyle\geq\kappa\omega\psi(z)$
$\displaystyle\geq\displaystyle\frac{\omega}{4}\sup_{z\in\mathfrak{P}}\psi(z)\cdot\frac{\inf_{z\in\mathfrak{P}}\psi(z)}{\sup_{z\in\mathfrak{P}}\psi(z)}\geq\frac{\omega}{4}\cdot\frac{2}{4e^{3B_{0}\zeta}}\geq\frac{\omega}{8e^{3B_{0}\zeta}}=c_{2}\omega,$

where the first statement is from (4.7); the second is from the fact that
$j\leq J=3\omega^{-1}-1$; the third is from the fact $\kappa\geq\frac{1}{4}$;
the fourth is from the fact that $\inf_{z\in\mathfrak{P}}\psi(z)\geq 2$ (from
below equation (4.6)) and (6.20); and the fifth is from the definition of
$c_{2}$ as in (6.19). This verifies (6.19).

The estimate (6.19), together with the first bound in (6.18), (the
$w_{0}=\big{(}s^{+}-\varsigma_{1}+\rho,y_{j}^{\prime}(s)\big{)}$ case of) 6.3,
and (4.6), yields (recalling that $s=0$ and $t=\varsigma_{1}$)

$\displaystyle\varphi(z)-\varphi\big{(}$ $\displaystyle
0,x_{j}^{\prime}(s)+x\big{)}$
$\displaystyle\leq\mathfrak{H}(z_{0})-\mathfrak{H}\big{(}s^{+}+\rho-\varsigma_{1},y_{j}^{\prime}(s)\big{)}+\displaystyle\frac{c_{1}\rho^{2}\omega}{2}+2n^{\delta-1}$
$\displaystyle\leq\varphi(z)-\varphi\big{(}0^{+},y_{j}^{\prime}(s)+x\big{)}+\displaystyle\frac{c_{1}\rho^{2}\omega}{2}+2n^{\delta-1}-c_{1}\omega|w_{0}-z_{0}|^{2}+c_{1}^{-1}\rho^{m}|w_{0}-z_{0}|$
$\displaystyle\leq\varphi(z)-\varphi\big{(}0,y_{j}^{\prime}(s)+x\big{)}+\displaystyle\frac{c_{1}\rho^{2}\omega-2c_{2}\omega}{2}+2n^{\delta-1}+c_{1}^{-1}\rho^{m+1}\leq\varphi(z)-\varphi\big{(}0,y_{j}^{\prime}(s)+x\big{)},$

from which it again follows that $y_{j}^{\prime}(s)\geq x_{j}^{\prime}(s)$
since $\partial_{x}\varphi<0$.

Now assume that $j\in\\{0,d+2\\}$; the two cases are again entirely analogous,
so let $j=0$. Then,

$\displaystyle\rho>y_{0}^{\prime}(s)=y_{B-d_{1}-1}(s+\rho-\varsigma_{1})$
$\displaystyle\geq\displaystyle\frac{d_{1}}{C_{1}n}-C_{1}\big{|}n^{-1}B-\mathfrak{H}(s+\rho-\varsigma_{1},0)\big{|}-C_{1}n^{\delta-1}$
$\displaystyle\geq\displaystyle\frac{d_{1}}{C_{1}n}-C_{1}\big{|}\mathfrak{H}(\rho,0)-\mathfrak{H}(s+\rho-\varsigma_{1},0)\big{|}-2C_{1}n^{\delta-1}$
$\displaystyle\geq\displaystyle\frac{d_{1}}{C_{1}n}-C_{1}^{2}\rho-2C_{1}n^{\delta-1}.$

Here, the first statement follows from the fact that
$y_{0}^{\prime}(s)\leq|w_{0}-z_{0}|<\rho$; the second follows from (6.13); the
third follows from the facts that we are restricting to the event
$\mathscr{E}$ from (6.16), and also from the first two bounds in (6.5); the
fourth follows from the definition (6.12) of $B$; and the fifth follows from
the first bound in (6.5). Hence, $d_{1}<5C_{1}^{3}\rho n$, implying by (6.12)
that $d_{1}=A-1$.

We must show that $y_{0}^{\prime}(s)\geq x_{0}^{\prime}(s)$. Since
$x_{0}^{\prime}(s)=x_{A-d-1}(s+t-\varsigma_{1})-x=x_{0}(s+t-\varsigma)-x=g(s+t-\varsigma_{1})-x$
for each $s\in[0,\varsigma]$, this is equivalent to $y_{B-d_{1}-1}(s+\rho)\geq
g(s+t)-x$ for each $s\in[-\varsigma_{1},\varsigma_{2}]$. To that end, observe
that

(6.21) $\displaystyle\begin{aligned} \mathfrak{H}&(\rho,0)-\varphi(t,x)\\\
&\leq\mathfrak{H}\big{(}s+\rho-\varsigma_{1},g(t+s-\varsigma_{1})-x^{+}\big{)}-\varphi(t+s-\varsigma_{1},g(t+s-\varsigma_{1})-0^{+}\big{)}+c_{1}^{-1}n^{1/2m^{2}}\rho^{m+1}\\\
&\leq\mathfrak{H}\big{(}s+\rho-\varsigma_{1},g(s+t-\varsigma_{1})-x\big{)}-\displaystyle\frac{c_{2}\omega}{2}.\end{aligned}$

Here, in the first inequality, we applied 6.3; in the second, we used the fact
that $\varphi\big{(}t-\varsigma_{1},g(t+s-\varsigma_{1})-0^{+}\big{)}\geq
c_{2}\omega\geq 2c_{1}^{-1}n^{1/2m^{2}}\rho^{m+1}$ by (4.6) and (6.19) (since
$\mathfrak{H}\big{(}t+s-\varsigma_{1},g(t+s-\varsigma_{1})\big{)}=0$).
Moreover,

$\displaystyle\mathfrak{H}\big{(}s+\rho-\varsigma_{1},y_{0}^{\prime}(s)\big{)}$
$\displaystyle=\mathfrak{H}\big{(}s+\rho-\varsigma_{1},y_{B-d_{1}-1}(s+\rho-\varsigma_{1})\big{)}$
$\displaystyle\leq n^{-1}(B-d_{1}-1)+n^{\delta-1}=n^{-1}(B-A)+n^{\delta-1},$

where in the first statement we used the definition of $y_{j}^{\prime}$ from
(6.13); in the second we used the fact that we are restricting to
$\mathscr{E}$ (from (6.16)); and in the third we used the equality
$d_{1}=A-1$. Thus,

$\displaystyle\mathfrak{H}\big{(}s+\rho-\varsigma_{1},y_{0}^{\prime}(s)\big{)}$
$\displaystyle\leq n^{-1}(B-A)+n^{\delta-1}$
$\displaystyle\leq\mathfrak{H}(\rho,0)-\varphi(t,x)+\displaystyle\frac{c_{1}\rho^{2}\omega}{2}+2n^{\delta-1}$
$\displaystyle\leq\mathfrak{H}\big{(}s+\rho-\varsigma_{1},g(s+t-\varsigma_{1})-x\big{)}+\displaystyle\frac{c_{1}\rho^{2}\omega-
c_{2}\omega}{2}+2n^{\delta-1}$
$\displaystyle\leq\mathfrak{H}\big{(}s-\rho-\varsigma_{1},g(s+t-\varsigma_{1})-x\big{)}.$

where in the second bound we used the definitions (6.12) of $A$ and $B$; in
the third we applied (6.21); and in the fourth we used the fact that
$c_{2}\omega-c_{1}\rho^{2}\omega\geq\frac{c_{2}\omega}{2}>4n^{\delta-1}$ for
sufficiently large $n$. Since $\partial_{x}\mathfrak{H}<0$, it follows that
$y_{0}^{\prime}(s)\geq g(s+t-\varsigma_{1})-x$. In particular,
$x_{0}^{\prime}(s)\leq y_{0}^{\prime}(s)$, which again verifies (6.15) and
thus the lemma. ∎

We can now establish 4.4.

###### Proof of 4.4.

By 6.4, there exists a constant $c_{2}=c_{2}(\varepsilon,B,m)>0$ and an event
$\mathscr{F}$ with $\mathbb{P}[\mathscr{F}]\geq 1-c_{2}^{-1}e^{-c_{2}(\log
n)^{2}}$ such that $x_{j}^{\prime}(s)\geq y_{j}^{\prime}(s)$ for each
$(j,s)\in\llbracket 1,d+1\rrbracket\times[0,\varsigma]$ on $\mathscr{F}$. By
(6.16) (and recalling the event $\mathscr{E}$ there), we may assume that
$\mathscr{F}\subseteq\mathscr{E}$, for otherwise we can replace $\mathscr{F}$
with $\mathscr{E}\cap\mathscr{F}$; we restrict to $\mathscr{F}$ in what
follows. Observe on $\mathscr{E}$ (and thus on $\mathscr{F}$) that
$(\varsigma_{1},0)\in\mathfrak{P}_{\mathfrak{y}^{-};\mathfrak{y}^{+}}$, since
by (6.14) and (6.12) we have (as $A\leq n-5n^{\delta}$)

$\displaystyle\mathsf{H}^{\bm{y}^{\prime}}(\varsigma_{1},0)=\mathsf{H}^{\bm{y}}(z_{0})-n\big{(}\mathfrak{H}(z_{0})+n^{\delta-1})+d_{1}+1\in[d_{1}-2n^{\delta}+1,d_{1}+2n^{\delta}+1]\subseteq[1,d+1].$

The above coupling thus implies that
$\mathsf{H}^{\bm{x}^{\prime}}(\varsigma_{1},0)\leq\mathsf{H}^{\bm{y}^{\prime}}(\varsigma_{1},0)$,
so by (6.14) we deduce

$\displaystyle\mathsf{H}^{\bm{x}}(z)=\mathsf{H}^{\bm{x}^{\prime}}(\varsigma_{1},0)+A-d_{1}-1$
$\displaystyle\leq\mathsf{H}^{\bm{y}^{\prime}}(\varsigma_{1},0)+A-d_{1}-1$
$\displaystyle=\mathsf{H}^{\bm{y}}(\rho,0)+A-B\leq
n\bigg{(}\varphi(z)-\displaystyle\frac{c_{1}\rho^{2}\omega}{2}\bigg{)}+\mathsf{H}^{\bm{y}}(\rho,0)-B.$

Since on the event $\mathscr{F}\subseteq\mathscr{E}$, we have
$\mathsf{H}^{\bm{y}}(\rho,0)\leq B$, so

$\displaystyle
n^{-1}\mathsf{H}^{\bm{x}}(z)\leq\varphi(z)-\displaystyle\frac{c_{1}\rho^{2}\omega}{2}\leq\varphi-n^{-\delta}\rho^{2}\omega\psi(z)\leq\varphi_{j;k}^{+}(z),$

where in the second bound we used the definition (4.7) of $\varphi_{j;k}^{+}$
(and the uniform boundedness of $\psi$, from (4.6)); this establishes the
proposition. ∎

##  7\. Concentrating Functions With Given Local Profiles

In this section we establish 5.9.

###  7.1. Taylor Approximations Through Dyson Brownian Motion

The following proposition produces measures whose Stieltjes transforms have
specified derivatives. In what follows, we recall the Stieltjes transform
$m_{\mu}$ of any measure $\mu\in\mathscr{P}$ from Section 2.2.

######  Proposition 7.1.

For any integer $m\geq 1$ and real numbers $\varpi>0$, $\varepsilon\in(0,1)$,
and $A,B>1$, there exists a constant $\delta=\delta(\varepsilon,\varpi,A,B)>0$
such that the following holds. Let
$\bm{r}=(r_{0},r_{1},r_{2},\ldots,r_{m})\in\mathbb{R}^{m+1}$ be an
$(m+1)$-tuple of real numbers with $|r_{j}|\leq A$ for each $j\in\llbracket
0,m\rrbracket$. Also let $\mu\in\mathscr{P}$ be a compactly supported
probability measure with density
$\varrho:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ that is $\mathcal{C}^{m}$
in a neighborhood of $0$, satisfying $\varepsilon\leq\varrho(x)\leq B$ for
each $x\in[-\varpi,\varpi]$; $\varrho(x)=0$ for $|x|>\varpi$; and
$\big{|}\operatorname{Re}m_{\mu}(0)\big{|}\leq A$. There exists a measure
$\widetilde{\mu}\in\mathscr{P}_{0}$ with density
$\widetilde{\varrho}:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ compactly
supported on $[-\varpi,\varpi]$, satisfying the following two properties.

  1. (1)

We have $\frac{\varepsilon}{2}\leq\widetilde{\varrho}(x)\leq 2B$ for each
$x\in[-\varpi,\varpi]$, and $\widetilde{\varrho}(x)=\varrho(x)$ for each
$x\in[-\delta,\delta]$.

  2. (2)

For each integer $k\in\llbracket 0,m\rrbracket$ we have

(7.1)
$\displaystyle\partial_{z}^{k}m_{\tilde{\mu}}(0)=r_{k}+\mathrm{i}\operatorname{Im}\partial_{z}^{k}m_{\mu}(0).$

###### Proof of 7.1.

If $\operatorname{Re}m_{\mu}(0)=r_{0}$, then we will establish the lemma with
the stronger bound on $\widetilde{\varrho}$ given by
$\frac{3\varepsilon}{4}\leq\widetilde{\varrho}(x)\leq
B+\frac{\varepsilon}{4}$. Let us first reduce to this statement. If
$\operatorname{Re}m_{\mu}(0)\neq r_{0}$, then denote
$\vartheta=r_{0}-\operatorname{Re}m_{\mu}(0)$; assume that $\vartheta>0$, as
the case when $\vartheta<0$ is entirely analogous. By our assumption,
$\vartheta\leq|r_{0}|+\big{|}\operatorname{Re}m_{\mu}(0)\big{|}\leq A+B$. Then
replace $\varrho(x)$ with $\varrho_{0}:\mathbb{R}\rightarrow\mathbb{R}$,
defined by for any $x\in\mathbb{R}$ setting

(7.2)
$\displaystyle\varrho_{0}(x)=\varrho(x)+\displaystyle\frac{\varepsilon}{4}\cdot\big{(}\bm{1}(d\leq
x\leq e^{8\vartheta/\varepsilon}d)-\bm{1}(e^{-4\vartheta/\varepsilon}d\leq
x\leq d)\big{)},$

for any real number $d<e^{-10(A+B)/\varepsilon}\varpi$. Then, $\varrho_{0}$
continues to be a probability measure supported in $[-\varpi,\varpi]$ and
satisfies $\frac{3\varepsilon}{4}\leq\varrho_{0}(x)\leq
B+\frac{\varepsilon}{4}$ for $x\in[-\varpi,\varpi]$. Moreover,
$\varrho_{0}=\varrho$ in a neighborhood of $0$, and thus
$\operatorname{Im}m(0)=\operatorname{Im}m_{\mu_{0}}(0)$ by the first statement
of (2.6), where $m_{\mu_{0}}$ denotes the Stieltjes transform of the measure
$\mu_{0}=\varrho_{0}(x)dx\in\mathscr{P}_{0}$. Additionally,

$\displaystyle\operatorname{Re}m_{\mu_{0}}(0)=\operatorname{Re}m(0)+\displaystyle\frac{\varepsilon}{4}\cdot(8\varepsilon^{-1}\vartheta-4\varepsilon^{-1}\vartheta)=\operatorname{Re}m(0)+\vartheta=r_{0}.$

Thus, $\mu_{0}$ and $\varrho_{0}$ continue to satisfy the hypotheses of the
lemma (after replacing $(\varepsilon,B)$ with
$\big{(}\frac{3\varepsilon}{4},B+\frac{\varepsilon}{4}\big{)}$ and altering
the $r_{j}$ for $j\in\llbracket 1,m\rrbracket$ to account for the difference
in (7.2)) but now with $\operatorname{Re}m_{\mu_{0}}(0)=r_{0}$. It follows
that there exists a measure $\widetilde{\mu}$ with density
$\widetilde{\varrho}$ satisfying the statement of the lemma (and in fact the
bounds
$\frac{\varepsilon}{2}\leq\frac{9\varepsilon}{16}\leq\widetilde{\varrho}(x)\leq
B+\frac{7\varepsilon}{16}\leq 2B$).

Thus, we will assume in what follows that $\operatorname{Re}m_{\mu}(0)=r_{0}$.
We will take $\widetilde{\varrho}(x)$ to be of the form

(7.3)
$\displaystyle\widetilde{\varrho}(x)=\varrho(x)+\sum_{\ell=1}^{m+2}c_{\ell}\cdot\bm{1}\big{(}a(y_{\ell}+y)\leq
x\leq a(y_{\ell}+y+b)\big{)},$

where the coefficients $c_{\ell}\in\mathbb{R}$ and positive real numbers
$a,b,y,y_{\ell}>0$ will be chosen later, in a way continuous in the parameters
$(\varepsilon,A,B)$, so that their uniform boundedness (fixing
$(\varepsilon,A,B)$) follows from compactness. In this way, for any complex
number $z\in\mathbb{C}$ in a neighborhood of $0$, we have

(7.4) $\displaystyle
m_{\tilde{\mu}}(z)=m_{\mu}(z)+\sum_{\ell=1}^{m+2}c_{\ell}\log\left(\frac{a(y_{\ell}+y+b)-z}{a(y_{\ell}+y)-z}\right).$

Thus, for any integer $k\geq 1$, we have

(7.5)
$\displaystyle\partial_{z}^{k}m_{\tilde{\mu}}(z)=\partial_{z}^{k}m_{\mu}(z)+\sum_{\ell=1}^{m+2}c_{\ell}(-1)^{k-1}(k-1)!\left(\frac{1}{\big{(}z-a(y_{\ell}+y+b)\big{)}^{k}}-\frac{1}{\big{(}z-a(y_{\ell}+y)\big{)}^{k}}\right).$

Since the parameters $a$, $b$, $y$, and $y_{\ell}$ are all positive, we have
$\widetilde{\varrho}(x)=\varrho(x)$ for $x\in\mathbb{R}$ in a sufficiently
small neighborhood of $0$; this verifies the second part of the first
statement of the proposition. Moreover, since by (2.6) (as $m_{\mu}$ and
$m_{\tilde{\mu}}$ are well-defined near $0$, by the above) we have
$\operatorname{Im}m_{\tilde{\mu}}(x)=\pi\widetilde{\varrho}(x)=\pi\varrho(x)=\operatorname{Im}m_{\mu}(x)$
for $x$ in a neighborhood of $0$. Differentiating this equation in $x$, it
follows that
$\partial_{z}^{k}\operatorname{Im}m_{\tilde{\mu}}(0)=\partial_{z}^{k}\operatorname{Im}m_{\mu}(0)$
for any integer $k\geq 1$, from which the imaginary part of (7.1) follows.

By (7.5), the real part of the second part of (7.1) holds for any
$k\in\llbracket 1,m\rrbracket$ if and only if

(7.6) $\displaystyle
a^{k}\big{(}\partial_{z}^{k}\operatorname{Re}m_{\mu}(0)-r_{k}\big{)}=\sum_{\ell=1}^{m+2}c_{\ell}(k-1)!\left(\frac{1}{(y_{\ell}+y+b)^{k}}-\frac{1}{(y_{\ell}+y)^{k}}\right)$

Moreover, $\widetilde{\varrho}$ has total mass $1$ (that is,
$\int_{-\infty}^{\infty}\widetilde{\varrho}(x)dx=1$) and the second part of
(7.1) holds at $k=0$ if and only if

(7.7)
$\displaystyle\displaystyle\sum_{\ell=1}^{m+2}c_{\ell}=0,\quad\text{and}\quad\displaystyle\sum_{j=1}^{m+2}c_{\ell}\log\bigg{(}\displaystyle\frac{y_{\ell}+y+b}{y_{\ell}+y}\bigg{)}=0,$

respectively, by (7.3) (since $\varrho$ has total mass $1$) and (7.4) (since
$\operatorname{Re}m_{\mu}=r_{0}$). We denote the $(m+2)$-dimensional vectors
$\bm{c}=(c_{1},c_{2},\cdots,c_{m+2})$ and
$\bm{p}=(p_{1},p_{2},\cdots,p_{m+2})$, where $p_{1}=0=p_{2}$ and

$\displaystyle
p_{k}=\partial_{z}^{k-2}\operatorname{Re}m_{\mu}(0)-r_{k-2},\qquad\text{for
each $k\in\llbracket 3,m+2\rrbracket$}.$

We also define the $(m+2)\times(m+2)$ matrices
$\bm{A}=\bm{A}(a)=\operatorname{diag}(a,a,a,a^{2},\ldots,a^{m})$ and
$\bm{W}(\bm{y},b)=\big{\\{}W_{k,\ell}(\bm{y},b)\big{\\}}_{\ell,k\in\llbracket
1,m+2\rrbracket}$, with entries given by

$\displaystyle
W_{k,\ell}(\bm{y},b)=(k-3)!\left(\frac{1}{(y_{\ell}+y+b)^{k-2}}-\frac{1}{(y_{\ell}+y)^{k-2}}\right),\quad\text{for
$k\in\llbracket 3,m+2\rrbracket$};$ $\displaystyle\qquad\qquad\qquad
W_{2,\ell}(\bm{y},b)=1,\qquad
W_{1,\ell}(\bm{y},b)=\log\bigg{(}\displaystyle\frac{y_{\ell}+y+b}{y_{\ell}+y}\bigg{)},$

for each $\ell\in\llbracket 1,m+2\rrbracket$. Then we can rewrite the
relations (7.6) and (7.7) as $\bm{A}\cdot\bm{p}=\bm{W}(\bm{y},b)\cdot\bm{c}$
or equivalently as $\bm{c}=\bm{W}(\bm{y},b)^{-1}\bm{A}\cdot\bm{p}$.

Next we show there exists some choice of $y,y_{\ell},b>0$ such that
$\bm{W}(\bm{y},b)$ is invertible. Observe that, for generic $b$, the first row
of $\bm{W}(\bm{y},b)$ is linearly independent from the bottom $m+1$ rows
(since the logarithmic function is not rational). It therefore suffices to
show that the left-bottom $(m+1)\times(m+1)$ block $\bm{W}_{0}(\bm{y},b)$ of
$\bm{W}(\bm{y},b)$ (constituting its $(k,\ell)\in\llbracket
2,m+2\rrbracket\times\llbracket 1,m+1\rrbracket$ entries) is invertible. Let
us write this block as $\bm{W}_{0}(\bm{y},b)=\bm{V}(\bm{y})-\bm{E}(\bm{y},b)$,
where the $(m+1)\times(m+1)$ matrices
$\bm{V}=\bm{V}(\bm{y})=\big{\\{}V_{k,\ell}(\bm{y})\big{\\}}$ and
$\bm{E}=\bm{E}(\bm{y},b)=\big{\\{}E_{k,\ell}(\bm{y},b)\big{\\}}$ are defined
by

$\displaystyle V_{1,\ell}(\bm{y})=1,\qquad\text{and}\qquad
V_{k,\ell}(\bm{y})=-(k-2)!(y_{\ell}+y)^{1-k},\quad\text{for $k\in\llbracket
2,m\rrbracket$};$ $\displaystyle E_{1,\ell}(\bm{y},b)=0,\qquad\text{and}\qquad
E_{k,\ell}(\bm{y},b)=-(k-2)!(y_{\ell}+y+b)^{1-k},\quad\text{for
$k\in\llbracket 2,m\rrbracket$},$

for each $\ell\in\llbracket 1,m+1\rrbracket$. Then, $\bm{V}(\bm{y})$ is a
Vandermonde matrix (up to multiplication by a diagonal matrix), and is
therefore invertible if we take $y_{1},y_{2},\ldots,y_{m+1}>0$ mutually
distinct. Moreover, by taking $b$ sufficiently large (and generic) the norm of
$\bm{E}(\bm{y},b)$ can be taken to be sufficiently small so as to guarantee
that $\bm{W}(\bm{y})=\bm{V}(\bm{y})-\bm{E}(\bm{y},b)$ remains invertible.

There thus exists a choice of the parameters $y,b,y_{1},y_{2},\ldots,y_{m+1}$
such that $\bm{W}_{0}(\bm{y},b)$ and thus $\bm{W}(\bm{y},b)$ is invertible;
then, $\bm{c}=\bm{W}(y)^{-1}\bm{A}\cdot\bm{p}$. By taking $a$ sufficiently
small, we can ensure that each entry of $\bm{c}$ is smaller than
$(4m+8)^{-1}\varepsilon$ and that $a(y_{\ell}+y+b)\leq\varpi$ for each
$\ell\in\llbracket 1,m+2\rrbracket$. Since $\varepsilon\leq\varrho(x)\leq B$
for $x\in[-\varpi,\varpi]$ and $\varrho(x)=0$ for $|x|>-\varpi$, it follows
from (7.3) that
$\frac{3\varepsilon}{4}\leq\big{|}\widetilde{\varrho}(x)\big{|}\leq
B+\frac{\varepsilon}{4}$ for $x\in[-\varpi,\varpi]$ and that
$\widetilde{\varrho}(x)=0$ for $|x|>\varpi$. In particular,
$\widetilde{\varrho}$ is a nonnegative measure satisfying the (previously
mentioned improvement of the) first statement of the proposition. That it is a
probability measure satisfying (7.1) follows from our choices of $y$, $b$, and
the $(y_{\ell})$. ∎

The next lemma, which is a quick consequence of 2.17, indicates that bridge-
limiting measures (recall Definition 3.2) can be obtained through free
convolution (recall Section 2.2). In what follows, we recall the rescaled
semicircle law $\mu_{\operatorname{semci}}^{(t)}$ from (2.9).

######  Lemma 7.2.

Fix a compactly supported measure $\nu_{0}\in\mathscr{P}_{\operatorname{fin}}$
and a real number $t>0$. For any real number $s>0$, denote the measure
$\nu_{s}=\nu\boxplus\mu_{\operatorname{semci}}^{(s)}\in\mathscr{P}_{\operatorname{fin}}$.
Then, $\bm{\nu}=(\nu_{s})_{s\in[0,t]}$ is the bridge-limiting measure process
on $[0,t]$ with boundary data $(\nu_{0};\nu_{t})$.

###### Proof.

By 2.12 and 3.5, we may assume that $\nu_{0}$ (and thus $\nu_{s}$ for each
$s\geq 0$) is a probability measure. Recalling (from Definition 2.16) the
classical locations $\gamma_{j;n}^{\mu}\in\mathbb{R}$ with respect to any
probability measure $\mu$, abbreviate
$\gamma_{j;n}(s)=\gamma_{j;n}^{(\nu_{s})}$, for each real number $s\geq 0$ and
integers $n\geq 1$ and $j\in\llbracket 1,n\rrbracket$. Further define the
$n$-tuple
$\bm{u}^{n}=(u_{1}^{n},u_{2}^{n},\ldots,u_{n}^{n})\in\overline{\mathbb{W}}_{n}$
by setting $u_{j}^{n}=\gamma_{j;n}(0)$, and let $\bm{u}^{n}(s)$ denote the
process obtained by running Dyson Brownian motion for time $s$ on $\bm{u}$;
set $\bm{v}^{n}=\bm{u}^{n}(t)$. Denote
$\mu_{s}^{n}=\operatorname{emp}\big{(}\bm{u}^{n}(s)\big{)}$ for each
$s\in[0,t]$ (recall (3.1)).

By (the $A=2$ case of) 2.17, we have for sufficiently large $n$ that, with
probability at least $1-n^{-1}$,

(7.8) $\displaystyle\displaystyle\int_{-\infty}^{y-2(\log
n)^{5}/n}\nu_{s}(dx)\leq\displaystyle\int_{-\infty}^{y}\mu_{s}^{n}(dx)\leq\displaystyle\int_{-\infty}^{y+2(\log
n)^{5}/n}\nu_{s}(dx),\qquad\text{for all $(s,y)\in[0,t]\times\mathbb{R}$}.$

Moreover, the second part of 2.15 indicates that, conditional on
$\bm{v}^{n}=\bm{u}^{n}(t)$, the law of $\bm{u}^{n}(s)$ is given by
$\mathfrak{Q}^{\bm{u}^{n};\bm{v}^{n}}$. This, together with (7.8), 3.1, and
Definition 3.2, indicates that $\bm{\nu}=(\nu_{s})_{s\in[0,t]}$ is the bridge-
limiting measure process on $[0,t]$ with boundary data $(\nu_{0};\nu_{t})$. ∎

###  7.2. Proof of 5.9

In this section we establish 5.9. In what follows, we recall the notation from
that proposition; on free convolutions and Stieltjes and Hilbert transforms
from Section 2.2; on bridge-limiting measure processes from Definition 3.2;
and on the associated (inverted) height functions from Definition 3.4.

The function $\mathfrak{H}$ of the proposition will eventually be the height
function associated with a bridge-limiting measure process arising from a free
convolution (as in 7.2), to which end we must define the initial data for the
free convolution. This will proceed using 7.1, and so we must first define the
real numbers $\bm{r}$ and measure $\varrho$ appearing there; we begin with the
former. Define real numbers $r_{0},r_{1},\ldots,r_{m-1}\in\mathbb{R}$ through
the equations

(7.9)
$\displaystyle\displaystyle\sum_{j=0}^{k-1}\binom{k-1}{j}q_{0}^{(k-j)}r_{j}=q_{1}^{(k)},\qquad\text{for
each $k\in\llbracket 1,m\rrbracket$}.$

Since $q_{0}^{(1)}\leq-\varepsilon$ by (5.3), these equations uniquely define
the $r_{j}$ by induction on $j$. Furthermore, by (5.3) and induction on $j$,
there exists a constant $C_{2}=C_{2}(\varepsilon,B,m)>1$ such that
$|r_{j}|<C_{2}$ for each $j\in\llbracket 0,m-1\rrbracket$.

Next, observe that there exists a constant $C_{3}=C_{3}(\varepsilon,B,m)>1$
and a probability measure $\mu\in\mathscr{P}_{0}$ with density $\varrho\in
L^{2}(\mathbb{R})$ such that $\operatorname{supp}\varrho=[-1,1]$, such that
$\|\varrho\|_{\mathcal{C}^{m+3}([-1,1])}\leq C_{2}$ and such that

(7.10) $\displaystyle\partial_{x}^{k}\varrho(0)=-q_{0}^{(k+1)},\quad\text{for
each $k\in\llbracket 0,m-1\rrbracket$},\qquad\text{and}\qquad
C_{3}^{-1}<\varrho(x)\leq C_{3},\quad\text{for each $x\in[-1,1]$}.$

Moreover, after increasing $C_{3}$ if necessary, it follows from (2.6), the
fact that $\operatorname{supp}\varrho\subseteq[-1,1]$, and the bound
$\|\varrho\|_{\mathcal{C}^{m+3}([-1,1])}\leq C_{2}$, that

$\displaystyle\big{|}\operatorname{Re}m_{\mu}(0)\big{|}=\pi\big{|}H\varrho(0)\big{|}=\bigg{|}\displaystyle\int_{-1}^{1}\frac{\varrho(x)dx}{x}\bigg{|}=\bigg{|}\displaystyle\int_{-1}^{1}\displaystyle\frac{\big{(}\varrho(x)-\varrho(0)\big{)}dx}{x}\bigg{|}\leq
2\|\varrho\|_{\mathcal{C}_{1}([-1,1])}\leq 2C_{2}\leq C_{3}.$

This verifies the assumptions in 7.1. Thus, by 7.1, there is a constant
$c_{1}=c_{1}(\varepsilon,B,m)>1$ and a probability measure
$\mu_{0}\in\mathscr{P}_{0}$ with density $\varrho_{0}\in L^{1}(\mathbb{R})$,
such that $\operatorname{supp}\varrho_{0}=[-1,1]$ and

(7.11) $\displaystyle\begin{aligned} &(2C_{3})^{-1}\leq\varrho_{0}(x)\leq
2C_{3},\quad\text{for each
$x\in[-1,1]$};\qquad\varrho_{0}(x)=\varrho(x),\quad\text{for each
$x\in[-c_{1},c_{1}]$};\\\
&\partial_{z}^{k}m_{\mu_{0}}(0)=r_{k}+\mathrm{i}\operatorname{Im}\partial_{z}^{k}m_{\mu}(0),\quad\text{for
each $k\in\llbracket 0,m\rrbracket$}.\end{aligned}$

For each real number $t>0$, define the free convolution measure
$\mu_{t}=\mu_{0}\boxplus\mu_{\operatorname{semci}}^{(t)}\in\mathscr{P}_{0}$
and the associated density $\varrho_{t}\in L^{1}(\mathbb{R})$ satisfying
$\mu_{t}(dx)=\varrho_{t}(x)dx$. Define the constant
$c_{2}=\frac{c_{1}}{3C_{3}}$, and denote the probabilility measure-valued
process $\bm{\mu}=(\mu_{t})_{[0,c_{2}]}$; observe that it is the bridge-
limiting measure process on $[0,c_{2}]$ with boundary data
$(\mu_{0};\mu_{c_{2}})$, by 7.2. Denote its height function by
$H^{\star}:[0,c_{2}]\times\mathbb{R}\rightarrow[0,1]$. We will later define
$\mathfrak{H}$ as the restriction of $H^{\star}$ to a certain strip domain.

######  Lemma 7.3.

There exist constants $c_{3}=c_{3}(\varepsilon,B,m)>0$ and
$C_{4}=C_{4}(\varepsilon,B,m)>1$ such that the following statements hold. In
what follows, we denote $\mathfrak{U}_{0}=(0,c_{3})\times(-c_{3},c_{3})$.

  1. (1)

For each $z\in\overline{\mathfrak{U}}_{0}$, we have
$\partial_{x}H^{\star}\leq-c_{3}$ and
$\|H^{\star}\|_{\mathcal{C}^{m+2}(\mathfrak{U}_{0})}\leq C_{4}$.

  2. (2)

The function $H^{\star}$ satisfies the equation (3.25) on $\mathfrak{U}_{0}$.

  3. (3)

The full-$m$ local profile of $H^{\star}$ satisfies
$\lim_{z\rightarrow(0,0)}\bm{Q}_{H^{\star}}^{(m)}(z)=\bm{Q}$.

###### Proof.

The first statement of the lemma follows from 2.11, whose conditions are
verified by (7.10) and the fact that
$\|\varrho\|_{\mathcal{C}^{m+1}([-1,1])}\leq C_{3}$. This, together with 3.16,
implies the second statement of the lemma. To verify the third statement
observe, since $H^{\star}$ satisfies (3.25), its local profile at any point is
consistent in the sense of Definition 5.1. Hence, by 5.4, to show
$\lim_{z\rightarrow(0,0)}\bm{Q}_{H^{\star}}^{(m)}(z)=\bm{Q}$, it suffices to
show for each $k\in\llbracket 1,m\rrbracket$ that

(7.12)
$\displaystyle\displaystyle\lim_{z\rightarrow(0,0)}\partial_{x}^{k}H^{\star}(z)=q_{0}^{(k)},\qquad\text{and}\qquad\displaystyle\lim_{z\rightarrow(0,0)}\partial_{x}^{k-1}\partial_{t}H^{\star}(z)=q_{1}^{(k)}.$

The first statement in (7.12) holds since
$\lim_{z\rightarrow(0,0)}\partial_{x}^{k}H^{\star}(z)=-\partial_{x}^{k-1}\varrho_{0}(0)=-\partial_{x}^{k-1}\varrho(0)=q_{0}^{(k)}$,
where the first follows from (3.3); the second from (7.11); and the third from
(7.10).

To verify the second statement in (7.12), observe that

$\displaystyle\displaystyle\lim_{z\rightarrow(0,0)}\partial_{x}^{k-1}\partial_{t}H^{\star}(z)$
$\displaystyle=\bigg{(}\partial_{y}^{k-1}\partial_{s}\displaystyle\int_{y}^{\infty}\varrho_{s}(x)dx\bigg{)}\Bigg{|}_{s=0,y=0}$
$\displaystyle=\bigg{(}\pi\partial_{y}^{k-1}\displaystyle\int_{y}^{\infty}\partial_{x}\big{(}\varrho_{0}(x)H\varrho_{0}(x)\big{)}dx\bigg{)}\Bigg{|}_{y=0}$
$\displaystyle=-\pi\bigg{(}\partial_{y}^{k-1}\big{(}\varrho_{0}(y)H\varrho_{0}(y)\big{)}\bigg{)}\Bigg{|}_{y=0}=-\pi\displaystyle\sum_{j=0}^{k-1}\binom{k-1}{j}\partial_{x}^{j}H\varrho_{0}(0)\cdot\partial_{x}^{k-j-1}\varrho_{0}(0),$

where first statement follows from (3.3); the second from (2.12); the third
from performing the integration (and using the fact that
$\operatorname{supp}\varrho_{0}=[-1,1]$); and the fourth from taking the
$(k-1)$-fold derivative. Together with (7.10), the second statement of (2.6),
the last statement of (7.11), and (7.9), this yields

$\displaystyle\displaystyle\lim_{z\rightarrow(0,0)}\partial_{x}^{k-1}\partial_{t}H^{\star}(z)=\displaystyle\sum_{j=0}^{k-1}\binom{k-1}{j}q_{0}^{(k-j)}\partial_{x}^{j}\operatorname{Re}m_{\mu_{0}}(0)=\displaystyle\sum_{j=0}^{k-1}\binom{k-1}{j}q_{0}^{(k-j)}r_{j}=q_{1}^{(k)},$

confirming the second statement of (7.12) and thus the lemma. ∎

Now we can establish 5.9.

###### Proof of 5.9.

Let $G^{\star}$ denote the inverted height function associated with
$H^{\star}$ and, recalling the notation of 7.3, fix

(7.13) $\displaystyle
c_{0}=H^{\star}(0,0);\qquad\nu=\frac{c_{3}^{3}}{4C_{4}^{3}};\qquad
c_{4}=\displaystyle\frac{c_{3}^{2}}{2C_{4}}.$

For each real number $s\in[0,\nu]$, let $f(s)=G^{\star}(s,c_{0}+c_{4})$ and
let $g(s)=G^{\star}(s,c_{0}-c_{4})$. Observe that

$\displaystyle c_{0}=H^{\star}(0,0)\geq-
c_{3}\displaystyle\int_{0}^{c_{3}}\partial_{x}H^{\star}(0,y)dy\geq c_{3}^{2},$

by the first part of 7.3, so $c_{0}\geq c_{3}^{2}\geq 4c_{4}>c_{4}$. Then
define the strip domain $\mathfrak{S}=\mathfrak{P}_{f;g}\subset\mathbb{R}^{2}$
and the function $\mathfrak{H}:\overline{\mathfrak{S}}\rightarrow\mathbb{R}$
by setting $\mathfrak{H}(z)=H^{\star}(z)-c_{0}+c_{4}$ for each
$z\in\overline{\mathfrak{S}}$. See the left side of Figure 7. Let us verify
that the conditions of the proposition hold for this choice of
$(\mathfrak{S},\mathfrak{H})$.

To see that
$[0,\nu]\times[-\nu,\nu]\subseteq\mathfrak{P}_{f;g}\subseteq[0,\nu]\times[-C\nu,C\nu]$,
observe again by the first part of 7.3, and using $H^{\star}(0,0)=c_{0}$ and
$H^{\star}(0,f(0))=c_{0}+c_{4}$, we have
$-\frac{c_{3}}{2C_{4}}=-c_{3}^{-1}c_{4}\leq f(0)\leq-
C_{4}^{-1}c_{4}=-\frac{c_{3}^{2}}{2C_{4}^{2}}$. Since
$\big{|}f^{\prime}(s)\big{|}=\big{|}(\partial_{t}H^{\star})(s,f(s))\big{|}\cdot\big{|}(\partial_{x}H^{\star})(s,f(s))\big{|}^{-1}\leq
c_{3}^{-1}C_{4}$ for $s\in[0,\nu]$ by the first part of 7.3, it follows from
(7.13) that, for each $s\in[0,\nu]$,

$\displaystyle-\displaystyle\frac{4C^{2}_{4}\nu}{c^{2}_{3}}\leq-\frac{c_{3}}{2C_{4}}-\frac{C_{4}\nu}{c_{3}}\leq
f(s)\leq\frac{C_{4}\nu}{c_{3}}-\frac{c_{3}^{2}}{2C_{4}^{2}}=-\frac{C_{4}\nu}{c_{3}}\leq-\nu.$

where we also used $\nu=c_{3}^{3}(4C_{4}^{3})^{-1}$ from (7.13). Similar
reasoning implies that $\nu\leq g(s)\leq 4c_{3}^{-2}C^{2}_{4}\nu$, which
indicates that
$[0,\nu]\times[-\nu,\nu]\subseteq\mathfrak{P}_{f;g}\subseteq[0,\nu]\times[-C\nu,C\nu]$
for $C=4C^{2}_{4}c_{3}^{-2}$.

To establish the other parts of the proposition, observe that the bounds on
$-\partial_{x}\mathfrak{H}$ and on $\|\mathfrak{H}\|_{\mathcal{C}^{m+1}}$
follow from the first part of 7.3. Moreover, the fact that the full-$m$ local
profile of $\mathfrak{H}$ around $(0,0)$ is given by $\bm{Q}$ follows from the
third part of 7.3.

It therefore remains to show that $\mathfrak{H}$ is $(C_{1};c;\delta)$
concentrating in the sense of Definition 5.8. That it satisfies (3.25) follows
from the second part of 7.3. Now let $n\geq 1$ be an integer; set
$n^{\prime}=\lfloor 2c_{4}n\rfloor$ (in what follows, we again omit the floors
and ceilings, as they will not affect the proofs), and define the
$n^{\prime}$-tuples $\bm{u},\bm{v}\in\overline{\mathbb{W}}_{n^{\prime}}$ by
setting

(7.14) $\displaystyle
u_{j}=G^{\star}(0,c_{0}-c_{4}+jn^{-1}),\quad\text{and}\quad
v_{j}=G^{\star}(\nu,c_{0}-c_{4}+jn^{-1}),\quad\text{for each $j\in\llbracket
1,n^{\prime}\rrbracket$};$

and sample $n^{\prime}$ non-intersecting Brownian bridges
$\bm{y}=(y_{1},y_{2},\ldots,y_{n^{\prime}})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[0,\nu]\big{)}$ from the measure
$\mathfrak{Q}_{f;g}^{\bm{u};\bm{v}}(n^{-1})$. We must show that

(7.15)
$\displaystyle\mathbb{P}\Bigg{[}\displaystyle\sup_{z\in\mathfrak{S}}\big{|}n^{-1}\mathsf{H}^{\bm{y}}(z)-H^{\star}(z)+c_{0}-c_{4}\big{|}\leq
n^{\delta-1}\Bigg{]}\geq 1-C_{1}e^{-c(\log n)^{2}}.$

Figure 7. Depicted in the left panel is the strip domain
$\mathfrak{S}=\mathfrak{P}_{f;g}$. Depicted in the right panel is the coupling
such that $\widetilde{y}_{j}^{-}\leq y_{j}\leq\widetilde{y}_{j}^{+}$.

To that end, define the $n$-tuples
$\bm{u}^{-},\bm{u}^{+}\in\overline{\mathbb{W}}_{n}$ by for each
$j\in\llbracket 1,n\rrbracket$ setting

(7.16) $\displaystyle u_{j}^{-}=G^{\star}(0,jn^{-1}+n^{\delta/2-1});\qquad
u_{j}^{+}=G^{\star}(0,jn^{-1}-n^{\delta/2-1}).$

Let
$\bm{y}^{-}(t)=\big{(}y_{1}^{-}(t),y_{2}^{-}(t),\ldots,y_{n}^{-}(t)\big{)}$
and
$\bm{y}^{+}(t)=\big{(}y_{1}^{+}(t),y_{2}^{+}(t),\ldots,y_{n}^{+}(t)\big{)}$
denote Dyson Brownian motion run with initial data $\bm{u}^{-}$ and
$\bm{u}^{+}$, respectively. Since $H^{\star}$ is the height function
associated with the measure-valued process
$\big{(}\mu_{0}\boxplus\mu_{\operatorname{semci}}^{(t)}\big{)}_{t\in[0,c_{2}]}$,
2.17 gives constants $c=c(\varepsilon,\delta,B,m)>0$ and
$C_{1}=C_{1}(\varepsilon,\delta,B,m)>1$ such that, with probability at least
$1-C_{1}e^{-c(\log n)^{2}}$, we have

(7.17) $\displaystyle\begin{aligned}
&H^{\star}(t,x)-n^{\delta/2-1}-n^{\delta/4-1}\leq
n^{-1}\mathsf{H}^{\bm{y}^{-}}(t,x)\leq
H^{\star}(t,x)-n^{\delta/2-1}+n^{\delta/4-1};\\\
&H^{\star}(t,x)+n^{\delta/2-1}-n^{\delta/4-1}\leq
n^{-1}\mathsf{H}^{\bm{y}^{+}}(t,x)\leq
H^{\star}(t,x)+n^{\delta/2-1}+n^{\delta/4-1},\end{aligned}$

for any $(t,x)\in\overline{\mathfrak{S}}$. In the below, we restrict to this
event.

Since $f(t)=G^{\star}(t,c_{0}-c_{4})$ and $g(t)=G^{\star}(t,c_{0}+c_{4})$,
applying (7.17) with $x$ equal to each of $y_{(c_{0}+c_{4})n+1}^{-}(t)$,
$y_{(c_{0}+c_{4})n+1}^{+}(t)$, $y_{(c_{0}-c_{4})n}^{-}(t)$,
$y_{(c_{0}-c_{4})n}^{+}(t)$, $f(t)$, and $g(t)$ (and using the fact that
$H^{\star}(t,x)$, $\mathsf{H}^{\bm{y}^{-}}(t,x)$, and
$\mathsf{H}^{\bm{y}^{+}}(t,x)$ are decreasing in $x$) yields

(7.18) $\displaystyle y_{(c_{0}+c_{4})n+1}^{-}(t)\leq f(t)\leq
y_{(c_{0}+c_{4})n+1}^{+}(t);\qquad y_{(c_{0}-c_{4})n}^{-}(t)\leq g(t)\leq
y_{(c_{0}-c_{4})n}^{+}(t).$

Now, denote $\mathfrak{c}=c_{0}-c_{4}$, and condition on the $y_{j}^{-}(s)$
and $y_{j}^{+}(s)$ with
$j\notin\llbracket\mathfrak{c}n+1,\mathfrak{c}n+n^{\prime}\rrbracket$. Define
the functions
$\widetilde{f}^{-},\widetilde{f}^{+},\widetilde{g}^{-},\widetilde{g}^{+}:[0,\nu]\rightarrow\mathbb{R}$
by setting

$\displaystyle\widetilde{f}^{\pm}(s)=y_{\mathfrak{c}n+n^{\prime}+1}^{\pm}(s);\qquad\widetilde{g}^{\pm}(s)=y_{\mathfrak{c}n}^{\pm}(s),$

for each index $\pm\in\\{+,-\\}$ and real number $s\in[0,\nu]$. Further define
$\widetilde{\bm{y}}^{-}=(\widetilde{y}_{1}^{-},\widetilde{y}_{2}^{-},\ldots,\widetilde{y}_{n^{\prime}}^{-})\in\llbracket
1,n^{\prime}\rrbracket\times\mathcal{C}\big{(}[0,\nu]\big{)}$ and
$\widetilde{\bm{y}}^{+}=(\widetilde{y}_{1}^{+},\widetilde{y}_{2}^{+},\ldots,\widetilde{y}_{n^{\prime}}^{+})\in\llbracket
1,n^{\prime}\rrbracket\times\mathcal{C}\big{(}[0,\nu]\big{)}$ by reindexing
$\bm{y}^{-}$ and $\bm{y}^{+}$, respectively, namely, by setting

(7.19)
$\displaystyle\widetilde{y}_{j}^{\pm}(s)=y_{j+\mathfrak{c}n}^{\pm}(s),\qquad\text{for
each $\pm\in\\{+,-\\}$ and $(j,s)\in\llbracket
1,n^{\prime}\rrbracket\times[0,\nu]$},$

and define the $n^{\prime}$-tuples
$\widetilde{\bm{u}}^{-},\widetilde{\bm{u}}^{+},\widetilde{\bm{v}}^{-},\widetilde{\bm{v}}^{+}\in\overline{\mathbb{W}}_{n^{\prime}}$
by setting

(7.20)
$\displaystyle\widetilde{u}_{j}^{\pm}=u_{j+\mathfrak{c}n}^{\pm}=\widetilde{y}_{j}^{\pm}(0),\quad\text{and}\quad\widetilde{v}_{j}^{\pm}=\widetilde{y}_{j}^{\pm}(\nu),\qquad\text{for
each $\pm\in\\{+,-\\}$ and $j\in\llbracket 1,n^{\prime}\rrbracket$.}$

Then, by the second part of 2.15, the laws of $\widetilde{\bm{y}}^{-}$ and
$\widetilde{\bm{y}}^{+}$ are given by
$\mathfrak{Q}_{\tilde{f}^{-};\tilde{g}^{-}}^{\tilde{\bm{u}}^{-};\tilde{\bm{v}}^{-}}(n^{-1})$
and
$\mathfrak{Q}_{\tilde{f}^{+};\tilde{g}^{+}}^{\tilde{\bm{u}}^{+};\tilde{\bm{v}}^{+}}(n^{-1})$,
respectively; see the right side of Figure 7. Observe from (7.14), (7.16), and
(7.19) that $\widetilde{\bm{u}}^{-}\leq\bm{u}\leq\widetilde{\bm{u}}^{+}$,
since $\partial_{y}G^{\star}\leq 0$. Moreover, (7.18) and (7.19) imply that
$\widetilde{f}^{-}=\widetilde{y}_{n^{\prime}+1}^{-}\leq
f\leq\widetilde{y}_{n^{\prime}+1}^{+}=\widetilde{f}^{+}$ and
$\widetilde{g}^{-}=\widetilde{y}_{0}^{-}\leq
g\leq\widetilde{y}_{0}^{+}=\widetilde{g}^{+}$. Additionally, by (7.14) and by
the $t=\nu$ (or $s=\nu$) case of (7.17), (7.19), (7.20), and the first
statement of 7.3, we find that
$\bm{\widetilde{v}}^{-}\leq\bm{v}\leq\bm{\widetilde{v}}^{+}$. Together with
the monotone coupling 2.3, these indicate that it is possible to couple
$\widetilde{\bm{y}}^{-}$, $\bm{y}$, and $\widetilde{\bm{y}}^{+}$ in such a way
that $\widetilde{y}_{j}^{-}(s)\leq y_{j}(s)\leq\widetilde{y}_{j}^{+}(s)$ for
each $(j,s)\in\llbracket 0,n^{\prime}\rrbracket\times[0,\nu]$. This, with
(7.17) and (7.19), yields (7.15) and thus the proposition. ∎

##  8\. Universality of Bulk Statistics

In this section we establish 1.5.

###  8.1. Proof of 1.5

In this section we establish 1.5. Given 1.4, its proof will be similar to what
was done in [2, Sections 3 and 4], by comparing non-intersecting Brownian
bridges to Dyson Brownian motion (though the affine invariance of our model
will simplify the arguments here). Throughout, we adopt the notation from 1.5,
and we set

(8.1) $\displaystyle\omega=90000(\delta+m^{-1});\qquad n^{\prime}=\lfloor
n^{1/2}\rfloor;\qquad n^{\prime\prime}=\lfloor n^{1/4}\rfloor,\qquad
s_{0}=n^{\omega/2-1},\qquad s_{1}=n^{3\omega-1},$

so that $\omega\leq\frac{1}{20}$ (as $m\geq 2^{25}$ and
$\delta\leq\frac{1}{5m^{2}}$). In what follows, we omit all floors and
ceilings, as they do not affect the proofs. We also assume (as we may by
shifting) that $G(t_{0}-s_{0},y_{0})=0$.

Define
$\breve{\bm{x}}=(\breve{x}_{1},\breve{x}_{2},\ldots,\breve{x}_{n^{\prime}})\in\llbracket
1,n^{\prime}\rrbracket\times\mathcal{C}\big{(}[0,s_{0}+s_{1}]\big{)}$ by
reindexing $\bm{x}$, specifically, by setting

$\displaystyle\breve{x}_{j}(s)=x_{y_{0}n-n^{\prime}/2+j}(t_{0}-s_{0}+s),\qquad\text{for
each $(j,s)\in\llbracket 1,n^{\prime}\rrbracket\times[0,s_{0}+s_{1}]$}.$

Also define the $n^{\prime}$-tuples
$\breve{\bm{u}}=(\breve{u}_{1},\breve{u}_{2},\ldots,\breve{u}_{n^{\prime}})\in\overline{\mathbb{W}}_{n^{\prime}}$
and
$\breve{\bm{v}}=(\breve{v}_{1},\breve{v}_{2},\ldots,\breve{v}_{n})\in\overline{\mathbb{W}}_{n^{\prime}}$,
and functions $\breve{f},\breve{g}:[0,s_{0}+s_{1}]\rightarrow\mathbb{R}$ by
for each $j\in\llbracket 1,n^{\prime}\rrbracket$ and $s\in[0,s_{0}+s_{1}]$
setting

$\displaystyle\breve{u}_{j}=\breve{x}_{j}(0);\qquad\breve{v}_{j}=\breve{x}_{j}(s_{0}+s_{1});\qquad\breve{f}(s)=\breve{x}_{n^{\prime}/2+n^{\prime\prime}+1}(s);\qquad\breve{g}(s)=\breve{x}_{n^{\prime}/2-n^{\prime\prime}-1}(s).$

Next, we condition on $\breve{x}_{j}(s)$ for
$(j,s)\notin\big{\llbracket}\frac{n^{\prime}}{2}-n^{\prime\prime},\frac{n^{\prime}}{2}+n^{\prime\prime}\rrbracket\times(0,s_{0}+s_{1})$.
Defining the event

(8.2) $\displaystyle\mathscr{A}_{0}=\bigcap_{(j,s)\notin\llbracket
n^{\prime}/2-n^{\prime\prime},n^{\prime}/2+n^{\prime\prime}\rrbracket\times(0,s_{0}+s_{1})}\Bigg{\\{}\bigg{|}\breve{x}_{j}(s)-G\Big{(}t_{0}-s_{0}+s,y_{0}-\displaystyle\frac{n^{\prime}}{2n}+\displaystyle\frac{j}{n}\Big{)}\bigg{|}\leq
n^{\omega/4-1}\Bigg{\\}},$

we have by 1.4 and the fact that
$n^{\omega/4-1}\geq\varkappa+n^{2/m+3\delta/2-1}$ (by (8.1) and the fact that
$\varkappa\leq n^{\delta-1}$) that there exists a constant
$c_{0}=c_{0}(\varepsilon,\delta,B_{0},m)>0$ such that

(8.3) $\displaystyle\mathbb{P}[\mathscr{A}_{0}]\geq 1-c_{0}^{-1}e^{-c_{0}(\log
n)^{2}}.$

We restrict to $\mathscr{A}_{0}$ in what follows.

Next, for each index $\pm\in\\{+,-\\}$, let
$G^{\pm}:[0,1]\rightarrow\mathbb{R}$ be a function satisfying the following
six properties.

  1. (1)

For all $y\in[0,1]$, we have $G(t_{0}-s_{0},y)-n^{\omega/2-1}\leq G^{-}(y)\leq
G(t_{0}-s_{0},y)$ and that $G(t_{0}-s_{0},y)\leq G^{+}(y)\leq
G(t_{0}-s_{0},y)+n^{\omega/2-1}$.

  2. (2)

For all $y\in[0,1]$ with $|y-y_{0}|\leq\frac{5n^{\prime\prime}}{6n}$, we have
$G(s_{0}-t_{0},y)-n^{3\omega/10-1}\leq G^{-}(y)\leq G(t_{0}-s_{0},y)$ and that
$G(t_{0}-s_{0},y)\leq G^{+}(y)\leq G(t_{0}-s_{0},y)+n^{3\delta/10-1}$.

  3. (3)

For $y\in[0,1]$ with $|y-y_{0}|\leq\frac{n^{\prime\prime}}{2n}$, we have
$G^{\pm}(y)=G(t_{0}-s_{0},y)$.

  4. (4)

For all $y\in[0,1]$ with
$|y-y_{0}|\in\big{[}\frac{2n^{\prime\prime}}{3n},\frac{5n^{\prime\prime}}{6n}\big{]}$,
we have $G^{\pm}(t_{0}-s_{0},y)=G(t_{0}-s_{0},y)\pm n^{3\omega/10-1}$.

  5. (5)

For all $y\in[0,1]$ with $|y-y_{0}|\geq\frac{n^{\prime\prime}}{n}$, we have
$G^{\pm}(y)=G(t_{0}-s_{0},y)\pm n^{\omega/2-1}$.

  6. (6)

For all $y\in[0,1]$, we have
$-\frac{2}{\varepsilon}\leq\partial_{y}G^{\pm}(y)\leq-\frac{\varepsilon}{2}$,
and $\|G^{\pm}\|_{\mathcal{C}^{m+1}}\leq 2B_{0}$.

The existence of such a function follows from the facts that $n^{\omega-1}\leq
n^{-3/4}=\frac{n^{\prime\prime}}{n}$, that
$G\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})$, and that
$\|G\|_{\mathcal{C}^{m+1}(\mathfrak{R})}\leq B_{0}$ (by 1.3). Then, define the
$n^{\prime}$-tuples
$\breve{\bm{u}}^{-},\breve{\bm{u}}^{+}\in\overline{\mathbb{W}}_{n^{\prime}}$
by, for each index $\pm\in\\{+,-\\}$ and integer $j\in\llbracket
1,n^{\prime}\rrbracket$, setting

(8.4) $\displaystyle\breve{u}_{j}^{\pm}=\breve{u}_{j},\quad\text{if
$j\in\Big{\llbracket}\displaystyle\frac{n^{\prime}}{2}-\displaystyle\frac{2n^{\prime\prime}}{3},\displaystyle\frac{n^{\prime}}{2}+\displaystyle\frac{2n^{\prime\prime}}{3}\Big{\rrbracket}$};\qquad\breve{u}_{j}^{\pm}=G^{\pm}\Big{(}y_{0}-\displaystyle\frac{n^{\prime}}{2n}+\displaystyle\frac{j}{n}\Big{)},\quad\text{otherwise},$

observing that $\bm{u}^{\pm}\in\overline{\mathbb{W}}_{n^{\prime}}$ on
$\mathscr{A}_{0}$ due to (8.2).

Let
$\breve{\bm{w}}(s)=\big{(}\breve{w}_{1}(s),\breve{w}_{2}(s),\ldots,\breve{w}_{n^{\prime}}(s)\big{)}$
denote Dyson Brownian motion with initial data $\breve{\bm{u}}$, as in (2.16),
but with the $d\breve{w}_{i}$ there multiplied by a factor of
$n^{-1}n^{\prime}$ and the $dB_{i}$ multiplied by a factor of
$(n^{-1}n)^{1/2}$; more specifically, they solve (instead of (2.16)) the
stochastic differential equation

$\displaystyle
d\breve{w}_{i}(s)=n^{-1}\displaystyle\sum_{\begin{subarray}{c}1\leq j\leq
n^{\prime}\\\ j\neq
i\end{subarray}}\displaystyle\frac{1}{\breve{w}_{i}^{\pm}(s)-\breve{w}_{j}^{\pm}(s)}+n^{-1/2}dB_{i}(s).$

Similarly, for each $\pm\in\\{+,-\\}$, let
$\breve{\bm{w}}^{\pm}(s)=\big{(}\breve{w}_{1}^{\pm}(s),\breve{w}_{2}^{\pm}(s),\ldots,\breve{w}_{n^{\prime}}^{\pm}(s)\big{)}$
denote Dyson Brownian motion with initial data $\breve{\bm{u}}^{\pm}$, but
with the same rescaling of the $d\breve{w}_{i}^{\pm}(s)$ as above; see Figure
8. Moreover define the functions
$\breve{f}^{-},\breve{f}^{+},\breve{g}^{-},\breve{g}^{+}:[0,s_{0}+s_{1}]\rightarrow\mathbb{R}$
by, for each index $\pm\in\\{+,-\\}$ and real number $s\in[0,s_{0}+s_{1}]$,
setting

$\displaystyle\breve{f}^{\pm}(s)=\breve{w}_{n^{\prime}/2+n^{\prime\prime}+1}^{\pm}(s);\qquad\breve{g}^{\pm}(s)=\breve{w}_{n^{\prime}/2-n^{\prime\prime}-1}^{\pm}(s).$

We then have the following proposition, to be established in Section 8.2
below; it indicates that the middle paths in $\breve{\bm{x}}$ are bounded
between those in $\breve{\bm{w}}^{-}$ and $\breve{\bm{w}}^{+}$ (after adding a
suitable drift).

Figure 8. Shown above are the paths of $\breve{\bm{w}}^{+}$ and
$\breve{\bm{w}}$ in the left panel, and of $\breve{\bm{w}}^{-}$ and
$\breve{\bm{w}}$ in the right panel.

######  Proposition 8.1.

Set $\mathfrak{a}=\partial_{t}G(t_{0},y_{0})$. There exist constants
$c=c(\varepsilon,\omega,m,B_{0})>0$ and $C=C(\varepsilon,\omega,m,B_{0})>1$; a
coupling between $\breve{\bm{x}}$, $\breve{\bm{w}}^{-}$, and
$\breve{\bm{w}}^{+}$; and an event $\mathscr{A}_{1}$, with
$\mathbb{P}[\mathscr{A}_{1}]\geq 1-Ce^{-c(\log n)^{2}}$, such that on
$\mathscr{A}_{1}$ we have

$\displaystyle\breve{w}_{j}^{-}(s)+(\mathfrak{a}-n^{-\omega})s\leq\breve{x}_{j}(s)\leq\breve{w}_{j}^{+}(s)+(\mathfrak{a}+n^{-\omega})s,$

for each
$(j,s)\in\big{\llbracket}\frac{n^{\prime}}{2}-n^{\prime\prime},\frac{n^{\prime}}{2}+n^{\prime\prime}\big{\rrbracket}\times[0,s_{0}+s_{1}]$.

We next have the following three lemmas, whose proofs are both consequences of
results in [50, 49] and are therefore deferred to Appendix D below. The first,
to be established in Section D.1 below, indicates that the local statistics of
$\breve{\bm{w}}$ are prescribed by the sine process. The second, to be
established in Section D.2 below, indicates that the middle paths of
$\breve{\bm{w}}^{-}$ and $\breve{\bm{w}}^{+}$ are close to those of
$\breve{\bm{w}}$. The third, to be established in Section D.3 below, states
that $\breve{\bm{w}}^{-}$ and $\breve{\bm{w}}^{+}$ are likely close in
expectation, under any coupling between them.

######  Lemma 8.2.

Let $\mathfrak{a}>0$ be a (bounded, uniformly in $n$) constant, and denote the
$n$-tuple $\bm{w}=\theta
n\cdot\big{(}\breve{\bm{w}}(s_{0})+\mathfrak{a}s_{0}\big{)}\in\overline{\mathbb{W}}_{n}$.
We have

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\bm{w}}^{(k)}(\bm{a})d\bm{a}=\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\sin}^{(k)}(\bm{a})d\bm{a}.$

######  Lemma 8.3.

There exist constants $c=c(\varepsilon,\delta,B_{0},m)>0$ and
$C=C(\varepsilon,\delta,B_{0},m)>1$; a coupling between $\breve{\bm{w}}^{-}$,
$\breve{\bm{w}}$; and $\breve{\bm{w}}^{+}$; and an event $\mathscr{A}_{2}$,
with $\mathbb{P}[\mathscr{A}_{2}]\geq 1-Cn^{-c}$, such that on
$\mathscr{A}_{2}$ we have

(8.5) $\displaystyle\begin{aligned} &\displaystyle\max_{j\in\llbracket
n^{\prime}/2-n^{\omega/15000},n^{\prime}/2+n^{\omega/15000}\rrbracket}\big{|}\breve{w}_{j}(s_{0})-\breve{w}_{j}^{-}(s_{0})\big{|}\leq
n^{-1-\omega/5};\\\ &\displaystyle\max_{j\in\llbracket
n^{\prime}/2-n^{\omega/15000},n^{\prime}/2+n^{\omega/15000}\rrbracket}\big{|}\breve{w}_{j}(s_{0})-\breve{w}_{j}^{+}(s_{0})\big{|}\leq
n^{-1-\omega/5}.\end{aligned}$

######  Lemma 8.4.

For any real number $\chi>0$, there exist constants
$c=c(\varepsilon,\delta,B_{0},m)>0$ and
$C=C(\chi,\varepsilon,\delta,B_{0},m)>1$ such that, under any coupling between
$\breve{\bm{w}}^{-}$ and $\breve{\bm{w}}^{+}$, we have

$\displaystyle\displaystyle\max_{j\in\llbracket
n^{\prime}/2-n^{\omega/15000},n^{\prime}/2+n^{\omega/15000}\rrbracket}\mathbb{E}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})-\breve{w}_{j}^{-}(s_{0})\big{|}\cdot\textbf{\emph{1}}_{|\breve{w}_{j}^{+}(s_{0})-\breve{w}_{j}^{-}(s_{0})|>n^{\chi-1}}\Big{]}<Ce^{-c(\log
n)^{2}}.$

Given the above four results, we can establish 1.5.

###### Proof of 1.5.

By a Taylor expansion (and the fact that $G(t_{0}-s_{0},y_{0})=0$), we have

(8.6)
$\displaystyle\big{|}G(t_{0},y_{0})-\mathfrak{a}s_{0}\big{|}=\big{|}G(t_{0},y_{0})-G(t_{0}-s_{0},y_{0})-\mathfrak{a}s_{0}\big{|}\leq
B_{0}s_{0}^{2}\leq n^{-1-\omega}.$

So, it suffices by the continuity of $F$ to show that

(8.7)
$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\theta
n\cdot(\breve{\bm{x}}(s_{0})-\mathfrak{a}s_{0})}^{(k)}(\bm{a})d\bm{a}=\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\sin}^{(k)}(\bm{a})d\bm{a}.$

We next claim that there exist constants
$c=c(\varepsilon,\delta,B_{0},m)\in\big{(}0,\frac{\omega}{30000}\big{)}$ and
$C=C(\varepsilon,\delta,B_{0},m)>1$, and a coupling between $\bm{x}$,
$\breve{\bm{w}}^{-}$, and $\breve{\bm{w}}^{+}$, such that

(8.8) $\displaystyle\mathbb{P}[\mathscr{A}]\geq
1-Cn^{-c},\qquad\text{where}\quad\mathscr{A}=\mathscr{A}_{0}^{\prime}\cap\mathscr{A}_{1}\cap\mathscr{A}_{3},$

where we have recalled $\mathscr{A}_{1}$ from 8.1 and defined the events

$\displaystyle\mathscr{A}_{0}^{\prime}=\bigcap_{|j-y_{0}n|\leq
n^{\omega/15000}}\Big{\\{}\big{|}x_{j}(s_{0})-G(s_{0},jn^{-1})\big{|}\leq
n^{\omega/30000-1}\Big{\\}};$
$\displaystyle\mathscr{A}_{3}=\bigcap_{|j-y_{0}n|\leq
n^{\omega/15000}}\Big{\\{}\big{|}\breve{w}_{j}^{-}(s_{0})-\breve{w}_{j}^{+}(s_{0})\big{|}\leq
n^{-1-c}\Big{\\}}.$

Let us establish the theorem assuming (8.8); throughout this proof, we set
$\omega^{\prime}=\frac{\omega}{30000}$. Since
$\lim_{n\rightarrow\infty}\mathbb{P}[\mathscr{A}]=1$, we may restrict to the
event $\mathscr{A}$ in what follows. Setting
$\mathfrak{a}=\partial_{t}G(t_{0},y_{0})$ and applying the definition of
$\mathscr{A}_{1}$ from 8.1 (at $s=s_{0}=n^{\omega/2-1}$) gives

(8.9)
$\displaystyle\breve{w}_{j}^{-}(s_{0})+\mathfrak{a}s_{0}-n^{-1-\omega/2}\leq\breve{x}_{j}(s_{0})\leq\breve{w}_{j}^{+}(s_{0})+\mathfrak{a}s_{0}+n^{-1-\omega/2},$

for each integer
$j\in\big{\llbracket}\frac{n^{\prime}}{2}-n^{\prime\prime},\frac{n^{\prime}}{2}+n^{\prime\prime}\big{\rrbracket}$.
Further observe on $\mathscr{A}_{0}^{\prime}$ that for
$j>\frac{n^{\prime}}{2}+n^{2\omega^{\prime}}$ we have
$\breve{x}_{j}(s_{0})-\mathfrak{a}s_{0}\leq
G(t_{0},y_{0}+n^{2\omega^{\prime}-1})-\mathfrak{a}s_{0}+n^{\omega^{\prime}-1}\leq
n^{-1-\omega}-\varepsilon
n^{2\omega^{\prime}-1}+n^{\omega^{\prime}-1}<-n^{\omega^{\prime}-1}$ (where we
have used (8.6) and the fact from 1.3 that $\partial_{y}G\leq-\varepsilon$).
By similar reasoning, we have on $\mathscr{A}_{0}$ that for
$j<\frac{n^{\prime}}{2}-n^{2\omega^{\prime}}$ that
$\breve{x}_{j}(s_{0})>n^{\omega^{\prime}-1}$. By these two bounds, together
with (8.9) and the fact that
$\big{|}\breve{w}_{j}^{-}(s_{0})-\breve{w}_{j}^{+}(s_{0})\big{|}\leq n^{-1-c}$
for
$j\in\llbracket\frac{n^{\prime}}{2}-n^{2\omega^{\prime}},\frac{n^{\prime}}{2}+n^{2\omega^{\prime}}\big{\rrbracket}$
on $\mathscr{A}$, it follows from the continuity of $F$ that, to verify (8.7),
it suffices to show

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\theta
n\cdot\breve{\bm{w}}^{-}(s_{0})}^{(k)}(\bm{a})d\bm{a}=\displaystyle\int_{\mathbb{R}^{k}}F(\bm{a})p_{\sin}^{(k)}(\bm{a})d\bm{a}.$

This follows from the first statement of (8.5) and 8.2 (with the continuity of
$F$).

It remains to confirm (8.8). To that end, it suffices by a union bound to show
that $\mathbb{P}[\mathscr{A}_{0}^{\prime}]\geq 1-Cn^{-c}$, that
$\mathbb{P}[\mathscr{A}_{1}]\geq 1-Cn^{-c}$, and that
$\mathbb{P}\big{[}\mathscr{A}_{1}\cap\mathscr{A}_{3}^{\complement}\big{]}\leq
Cn^{-c}$. The first follows by 1.4, since
$\omega^{\prime}=3(m^{-1}+\delta)>2m^{-1}+3\delta$, and the second follows
from 8.1.

The third will follow from (8.5), 8.4, and a Markov estimate. To implement
this, observe that (8.5) and 8.4 together imply the existence of constants
$c_{1}=c_{1}(\varepsilon,\delta,B_{0},m)>0$ and
$C_{1}=C_{1}(\varepsilon,\delta,B_{0},m)>1$, and a coupling between
$\breve{\bm{w}}^{-}$ and $\breve{\bm{w}}^{+}$ (possibly different from the one
above), such that

$\displaystyle\displaystyle\sum_{|j-n^{\prime}/2|\leq
n^{\omega/15000}}\mathbb{E}\Big{[}\big{|}\breve{w}_{j}^{+}(s_{0})-\breve{w}_{j}^{-}(s_{0})\big{|}\Big{]}\leq
C_{1}n^{-1-c_{1}},$

from which it follows that

(8.10) $\displaystyle\displaystyle\sum_{|j-n^{\prime}/2|\leq
n^{\omega/15000}}\Big{(}\mathbb{E}\big{[}\breve{w}_{j}^{+}(s_{0})\big{]}-\mathbb{E}\big{[}\breve{w}_{j}^{-}(s_{0})\big{]}\Big{)}\leq
C_{1}n^{-1-c_{1}}.$

Since this bound is independent of the coupling between $\breve{\bm{w}}^{+}$
and $\breve{\bm{w}}^{-}$, it also holds for our original one given by 8.1.
Next, restricting to the event $\mathscr{A}_{1}$, we have
$\breve{w}_{j}^{-}(s_{0})\leq\breve{w}_{j}^{+}(s_{0})+n^{-\omega}s_{0}=\breve{w}_{j}^{+}(s_{0})+n^{-1-\omega/2}$.
Together with (8.10), this implies under our original coupling that

$\displaystyle\displaystyle\sum_{|j-n^{\prime}/2|\leq n^{\omega/15000}}$
$\displaystyle\mathbb{E}\Big{[}\textbf{1}_{\mathscr{A}_{1}}\cdot\big{|}\breve{w}_{j}^{+}(s_{0})-\breve{w}_{j}^{-}(s_{0})\big{|}\Big{]}$
$\displaystyle\leq\displaystyle\sum_{|j-n^{\prime}/2|\leq
n^{\omega/15000}}\Big{(}\mathbb{E}\big{[}\breve{w}_{j}^{+}(s_{0})\big{]}-\mathbb{E}\big{[}\breve{w}_{j}^{-}(s_{0})\big{]}+2n^{-1-\omega/2}\Big{)}\leq
3C_{1}n^{-1-c_{2}},$

for some constant $c_{2}=c_{2}(\varepsilon,\delta,B_{0},m)>0$. By a Markov
estimate, we thus have for $c=\frac{c_{2}}{2}$ that

$\displaystyle\mathbb{P}\big{[}\mathscr{A}_{1}\cap\mathscr{A}_{3}^{\complement}\big{]}\leq
n^{1+c_{2}/2}\cdot\displaystyle\sum_{|j-n^{\prime}/2|\leq
n^{\omega/15000}}\mathbb{E}\Big{[}\textbf{1}_{\mathscr{A}_{1}}\cdot\big{|}\breve{w}_{j}^{+}(s_{0})-\breve{w}_{j}^{-}(s_{0})\big{|}\Big{]}$
$\displaystyle\leq 3C_{1}n^{-1-c_{2}/2}.$

This gives the third bound above,
$\mathbb{P}\big{[}\mathscr{A}_{1}\cap\mathscr{A}_{3}^{\complement}\big{]}\leq
Cn^{-c}$, and thus the theorem. ∎

###  8.2. Proof of 8.1

In this section we establish 8.1; throughout, we recall the notation of that
proposition. We first rescale $s_{0}$, $s_{1}$, $\breve{\bm{x}}(s)$,
$\breve{\bm{w}}^{-}(s)$, $\breve{\bm{w}}(s)$, $\breve{\bm{w}}^{+}(s)$, $G$,
$G^{-}$, and $G^{+}$ to form $\widetilde{s}_{0}$, $\widetilde{s}_{1}$,
$\widetilde{\bm{x}}(s)$, $\widetilde{\bm{w}}^{-}(s)$, $\widetilde{\bm{w}}(s)$,
$\widetilde{\bm{w}}^{+}(s)$, $\widetilde{G}$, $\widetilde{G}^{-}$, and
$\widetilde{G}^{+}$, respectively, defined by setting

(8.11) $\displaystyle\begin{aligned}
\widetilde{s}_{i}=\displaystyle\frac{ns_{i}}{n^{\prime}};\qquad\widetilde{z}_{j}(s)=\displaystyle\frac{n}{n^{\prime}}\cdot\breve{z}_{j}\Big{(}&\displaystyle\frac{n^{\prime}s}{n}\Big{)};\qquad\widetilde{G}(s,y)=\displaystyle\frac{n}{n^{\prime}}\cdot
G\Big{(}\displaystyle\frac{n^{\prime}s}{n}+t_{0}-s_{0},\displaystyle\frac{n^{\prime}y}{n}+y_{0}-\displaystyle\frac{n^{\prime}}{2n}\Big{)},\\\
\widetilde{G}^{\pm}(&y)=\displaystyle\frac{n}{n^{\prime}}\cdot
G^{\pm}\Big{(}\displaystyle\frac{n^{\prime}y}{n}+y_{0}-\displaystyle\frac{n^{\prime}}{2n}\Big{)},\end{aligned}$

for any integers $i\in\\{0,1\\}$ and $j\in\llbracket 1,n^{\prime}\rrbracket$;
indices $z\in\\{x,w^{-},w,w^{+}\\}$ and $\pm\in\\{+,-\\}$; and real numbers
$s\in[0,\widetilde{s}_{0}+\widetilde{s}_{1}]$ and
$y\in\big{[}1-y_{0}n^{1/2},(1-y_{0})n^{1/2}\big{]}$. In this way,
$\widetilde{\bm{w}}(s)$ is obtained by applying Dyson Brownian motion (2.16)
to the initial data $\widetilde{\bm{w}}(0)$ for time $s$ (without the
additional rescaling described in Section 8.1); similar statements are true
for $\bm{w}^{-}(s)$ and $\bm{w}^{+}(s)$. By (8.2), and the fact that
$(n^{\prime})^{2}=n$, we have

(8.12) $\displaystyle\mathscr{A}_{0}=\bigcap_{(j,s)\notin\llbracket
n^{\prime}/2-n^{\prime\prime},n^{\prime}/2+n^{\prime\prime}\rrbracket\times(0,\tilde{s}_{0}+\tilde{s}_{1})}\Bigg{\\{}\bigg{|}\widetilde{x}_{j}(s)-\widetilde{G}\Big{(}s,\displaystyle\frac{j}{n^{\prime}}\Big{)}\bigg{|}\leq(n^{\prime})^{\omega/2-1}\Bigg{\\}}.$

Further observe since $G\in\operatorname{Adm}_{\varepsilon/2}(\mathfrak{R})$;
since
$-\frac{2}{\varepsilon}\leq\partial_{y}G^{\pm}\leq\frac{\varepsilon}{2}$;
since $G(t_{0}-s_{0},y_{0})=0$; since
$\|G\|_{\mathcal{C}^{m+1}(\mathfrak{R})}\leq B_{0}$; and since
$\|G^{\pm}\|_{\mathcal{C}^{m+1}}\leq 2B_{0}$ that, for each $\pm\in\\{+,-\\}$
and $k\in\llbracket 1,m+1\rrbracket$,

(8.13) $\displaystyle\begin{aligned}
&\widetilde{G}\in\operatorname{Adm}_{\varepsilon/2}\Big{(}\displaystyle\frac{n}{n^{\prime}}\cdot\mathfrak{R}\Big{)};\qquad\qquad-\displaystyle\frac{2}{\varepsilon}\leq\partial_{y}\widetilde{G}^{\pm}(y)\leq-\displaystyle\frac{\varepsilon}{2},\quad\text{for
$y\in\big{[}1-y_{0}n^{1/2},(1-y_{0})n^{1/2}\big{]}$};\\\
&\widetilde{G}\Big{(}0,\displaystyle\frac{1}{2}\Big{)}=\widetilde{G}^{\pm}\Big{(}0,\displaystyle\frac{1}{2}\Big{)}=0;\qquad[\widetilde{G}]_{k},[\widetilde{G}^{\pm}]_{k}\leq
2B_{0}\Big{(}\displaystyle\frac{n^{\prime}}{n}\Big{)}^{k-1},\end{aligned}$

The proof of 8.1 will make use of the following lemma, to be established in
Section 8.3 below, that approximates the locations of the middle paths in
$\widetilde{\bm{w}}^{-}(s)$ and $\widetilde{\bm{w}}^{+}(s)$.

######  Lemma 8.5.

There exists a coupling between $\widetilde{\bm{w}}^{-}$,
$\widetilde{\bm{w}}$, and $\widetilde{\bm{w}}^{+}$, and a constant
$c=c(\varepsilon,\delta,B_{0},m)>0$ such that $\mathbb{P}[\mathscr{A}_{1}]\geq
1-c^{-1}e^{-c(\log n)^{2}}$, where we have defined the event

(8.14) $\displaystyle\begin{aligned}
\mathscr{A}_{1}&=\Bigg{\\{}\displaystyle\sup_{s\in[0,\tilde{s}_{0}+\tilde{s}_{1}]}\bigg{(}\displaystyle\max_{j\in\llbracket
n^{\prime}/2-2n^{\prime\prime},n^{\prime}/2+2n^{\prime\prime}\rrbracket}\big{|}\widetilde{w}_{j}^{-}(s)-\widetilde{w}_{j}^{-}(0)\big{|}\bigg{)}\leq(n^{\prime})^{2\omega/3-1}\Bigg{\\}}\\\
&\qquad\cap\Bigg{\\{}\displaystyle\sup_{s\in[0,\tilde{s}_{0}+\tilde{s}_{1}]}\bigg{(}\displaystyle\max_{j\in\llbracket
n^{\prime}/2-2n^{\prime\prime},n^{\prime}/2+2n^{\prime\prime}\rrbracket}\big{|}\widetilde{w}_{j}^{+}(s)-\widetilde{w}_{j}^{+}(0)\big{|}\bigg{)}\leq(n^{\prime})^{2\omega/3-1}\Bigg{\\}}\cap\mathscr{A}_{0}.\end{aligned}$

Using 8.5, we can establish 8.1.

###### Proof of 8.1.

Fix $\mathfrak{a}=\partial_{t}G(t_{0},y_{0})$, which is uniformly bounded in
$[-B_{0},B_{0}]$ by 1.3. We claim on the event $\mathscr{A}_{1}$ from (8.14)
that

(8.15) $\displaystyle\begin{aligned}
&\breve{w}_{j}^{-}(0)\leq\breve{u}_{j}\leq\breve{w}_{j}^{+}(0);\\\
&\breve{w}_{j}^{-}(s_{0}+s_{1})+(\mathfrak{a}-n^{-\omega/2})(s_{0}+s_{1})\leq\breve{v}_{j}\leq\breve{w}_{j}^{+}(s_{0}+s_{1})+(\mathfrak{a}+n^{-\omega/2})(s_{0}+s_{1});\\\
&\breve{f}^{-}(s)+(\mathfrak{a}-n^{-\omega/2})s\leq\breve{f}(s)\leq\breve{f}^{+}(s)+(\mathfrak{a}+n^{-\omega/2})s,\\\
&\breve{g}^{-}(s)+(\mathfrak{a}-n^{-\omega/2})s\leq\breve{g}(s)\leq\breve{g}^{+}(s)+(\mathfrak{a}+n^{-\omega/2})s,\end{aligned}$

for each integer
$j\in\big{\llbracket}\frac{n^{\prime}}{2}-n^{\prime\prime},\frac{n^{\prime}}{2}+n^{\prime\prime}\big{\rrbracket}$
and real number $s\in[0,s_{0}+s_{1}]$.

Let us quickly establish the proposition assuming (8.15). To that end, denote
the $(2n^{\prime\prime}+1)$-tuples
$\mathring{\bm{u}}=(\breve{u}_{n^{\prime}/2-n^{\prime\prime}},\ldots,\breve{u}_{n^{\prime}/2+n^{\prime\prime}})\in\overline{\mathbb{W}}_{2n^{\prime\prime}+1}$
and
$\mathring{\bm{v}}=(\breve{v}_{n^{\prime}/2-n^{\prime\prime}},\ldots,\breve{v}_{n^{\prime}/2+n^{\prime\prime}})\in\overline{\mathbb{W}}_{2n^{\prime\prime}+1}$.
Similarly, for each index $\pm\in\\{+,-\\}$ and real number
$s\in[0,s_{0}+s_{1}]$, denote

$\displaystyle\mathring{\bm{w}}^{\pm}(s)=\Big{(}\breve{w}_{n^{\prime}/2-n^{\prime\prime}}^{\pm}(s)+(\mathfrak{a}\pm
n^{-\omega/2})s,\ldots,\breve{w}_{n^{\prime}/2+n^{\prime\prime}}^{\pm}(s)+(\mathfrak{a}\pm
n^{-\omega/2})s\Big{)}\in\overline{\mathbb{W}}_{2n^{\prime\prime}+1};$
$\displaystyle\mathring{f}^{\pm}(s)=\breve{f}^{\pm}(s)+(\mathfrak{a}\pm
n^{-\omega/2})s;\qquad\mathring{g}^{\pm}(s)=\breve{g}^{\pm}(s)+(\mathfrak{a}\pm
n^{-\omega/2})s.$

Then, by the second part of 2.15 and 2.1, the laws of the processes
$\big{(}\breve{x}_{j}(s)\big{)}$ and $\big{(}\mathring{w}_{j}^{\pm}(s)\big{)}$
for each $\pm\in\\{+,-\\}$ and
$(j,s)\in\big{\llbracket}\frac{n^{\prime}}{2}-n^{\prime\prime},\frac{n^{\prime}}{2}+n^{\prime\prime}\big{\rrbracket}\times[0,s_{0}+s_{1}]$
are given by
$\mathfrak{Q}_{\breve{f};\breve{g}}^{\mathring{\bm{u}};\mathring{\bm{v}}}(n^{-1})$
and
$\mathfrak{Q}_{\mathring{f}^{\pm};\mathring{g}^{\pm}}^{\mathring{\bm{w}}^{\pm}(0);\mathring{\bm{w}}^{\pm}(s_{0}+s_{1})}(n^{-1})$,
respectively. This, together with (8.15) and 2.3 yields a coupling on
$\mathscr{A}_{1}$ that
$\breve{\bm{w}}^{-}(s)+(\mathfrak{a}-n^{-\omega/2})s=\mathring{\bm{w}}^{-}(s)\leq\breve{\bm{x}}(s)\leq\mathring{\bm{w}}^{+}(s)=\breve{\bm{w}}^{+}(s)+(\mathfrak{a}+n^{-\omega/2})s$
for each $s\in[s_{0},s_{1}]$, verifying the proposition.

The second bound in the first statement of (8.15) follows the fact that
$\breve{\bm{w}}^{\pm}(0)=\breve{\bm{u}}^{\pm}$; (8.4); the fact that
$\mathscr{A}_{1}\subseteq\mathscr{A}_{0}$; and the fact that on
$\mathscr{A}_{0}$ we have (by (8.2)) that $u_{j}\leq
G(t_{0}-s_{0},jn^{-1})+n^{\omega/4-1}\leq G^{+}(jn^{-1})=\breve{u}_{j}^{+}$
for
$j\notin\big{\llbracket}\frac{n^{\prime}}{2}-\frac{2n^{\prime\prime}}{3},\frac{n^{\prime}}{2}+\frac{2n^{\prime\prime}}{3}\big{\rrbracket}$.
The proof of the first bound there is entirely analogous.

For the second statement of (8.15), we only show that
$\breve{w}_{j}^{-}(s_{0}+s_{1})+(\mathfrak{a}-n^{-\omega/2})(s_{0}+s_{1})\leq\breve{v}_{j}$,
as the proof that
$\breve{v}_{j}\leq\breve{w}_{j}^{+}(s_{0}+s_{1})+(\mathfrak{a}+n^{-\omega/2})(s_{0}+s_{1})$
is entirely analogous. Since we have restricted to the event $\mathscr{A}_{0}$
(and since $n=(n^{\prime})^{2}$), we have that

$\displaystyle\frac{n}{n^{\prime}}\cdot\breve{v}_{j}\geq\frac{n}{n^{\prime}}\cdot\bigg{(}G\Big{(}t_{0}+s_{1},y_{0}-\frac{n^{\prime}}{2n}+\frac{j}{n}\Big{)}-n^{\omega/4-1}\bigg{)}=\widetilde{G}\Big{(}\widetilde{s}_{0}+\widetilde{s}_{1},\displaystyle\frac{j}{n^{\prime}}\Big{)}-(n^{\prime})^{\omega/2-1},$

and so it suffices to show that

$\displaystyle\widetilde{w}_{j}^{-}(\widetilde{s}_{0}+\widetilde{s}_{1})\leq\widetilde{G}\Big{(}\widetilde{s}_{0}+\widetilde{s}_{1},\displaystyle\frac{j}{n^{\prime}}\Big{)}-(\mathfrak{a}-n^{-\omega/2})(\widetilde{s}_{0}+\widetilde{s}_{1})-(n^{\prime})^{\omega-1}.$

Thus, since we have restricted to the event $\mathscr{A}_{1}$ from (8.14);
since for
$j\in\big{\llbracket}\frac{n^{\prime}}{2}-n^{\prime\prime},\frac{n^{\prime}}{2}+n^{\prime\prime}\big{\rrbracket}$
we have

$\displaystyle\widetilde{w}_{j}^{-}(0)=\frac{n}{n^{\prime}}\cdot\breve{u}_{j}^{-}\leq\widetilde{G}\Big{(}0,\frac{j}{n^{\prime}}\Big{)}+(n^{\prime})^{2\omega/3-1},$

by (8.4), (8.12), and the definition of $G^{-}$; and since
$\widetilde{s}_{0}=(n^{\prime})^{\omega-1}$ (by (8.1) and (8.11)), it suffices
to show that

$\displaystyle\widetilde{G}\Big{(}0,\frac{j}{n^{\prime}}\Big{)}\leq\widetilde{G}\Big{(}\widetilde{s}_{0}+\widetilde{s}_{1},\displaystyle\frac{j}{n^{\prime}}\Big{)}-\mathfrak{a}(\widetilde{s}_{0}+\widetilde{s}_{1})+(n^{\prime})^{-\omega}\widetilde{s}_{1}-2(n^{\prime})^{\omega-1}.$

This follows from the fact that

$\displaystyle\bigg{|}\widetilde{G}\Big{(}0,\displaystyle\frac{j}{n^{\prime}}\Big{)}-\widetilde{G}\Big{(}\widetilde{s}_{0}+\widetilde{s}_{1},\displaystyle\frac{j}{n^{\prime}}\Big{)}-\mathfrak{a}(\widetilde{s}_{0}+\widetilde{s}_{1})\bigg{|}\leq
B_{0}\Big{(}\widetilde{s}_{0}+\widetilde{s}_{1}+\displaystyle\frac{n^{\prime\prime}}{n^{\prime}}\Big{)}^{2}$
$\displaystyle\leq
9B_{0}\big{(}(n^{\prime})^{12\omega-2}+(n^{\prime})^{-1}\big{)}$
$\displaystyle\leq(n^{\prime})^{-\omega}\widetilde{s}_{1}-2(n^{\prime})^{\omega-1},$

for sufficiently large $n$. Here, in the first inequality we applied a Taylor
expansion, the fact that $\mathfrak{a}=\partial_{t}G(t_{0},y_{0})$, the fact
that $\big{|}j-\frac{n^{\prime}}{2}\big{|}\leq n^{\prime\prime}$, and the fact
that $\|G\|_{\mathcal{C}^{2}(\mathfrak{R})}\leq B_{0}$; in the second
inequality, we used the facts that $\widetilde{s}_{0}+\widetilde{s}_{1}\leq
2\widetilde{s}_{1}\leq 2(n^{\prime})^{6\omega-1}$ and that
$n^{\prime\prime}=(n^{\prime})^{1/2}$; and in the third inequality we used the
facts that $\omega\leq\frac{1}{20}$ (and again that
$\widetilde{s}_{1}=(n^{\prime})^{6\omega-1}$).

It therefore remains to establish the last two statements of (8.15); among
these, we only verify the bound $\breve{f}^{-}\leq\breve{f}$ there, as the
proofs of the remaining three are entirely analogous. To that end, observe on
$\mathscr{A}_{1}$ that for each $s\in[0,s_{0}+s_{1}]$ we have, denoting
$\widetilde{s}=\frac{ns}{n^{\prime}}$,

(8.16) $\displaystyle\begin{aligned}
\frac{n}{n^{\prime}}\cdot\mathring{f}^{-}(s)=\widetilde{w}_{n^{\prime}/2+n^{\prime\prime}+1}^{-}(\widetilde{s})+(\mathfrak{a}-n^{-\omega})\widetilde{s}&\leq\widetilde{w}_{n^{\prime}/2+n^{\prime\prime}+1}^{-}(0)+\mathfrak{a}\widetilde{s}+(n^{\prime})^{2\omega/3-1}\\\
&=\widetilde{G}^{-}\Big{(}\displaystyle\frac{1}{2}+\displaystyle\frac{n^{\prime\prime}+1}{n^{\prime}}\Big{)}+\mathfrak{a}\widetilde{s}+(n^{\prime})^{2\omega/3-1}\\\
&=\widetilde{G}\Big{(}0,\displaystyle\frac{1}{2}+\displaystyle\frac{n^{\prime\prime}+1}{n^{\prime}}\Big{)}+\mathfrak{a}\widetilde{s}-(n^{\prime})^{\omega-1}+(n^{\prime})^{2\omega/3-1},\end{aligned}$

where in the first statement we applied the definitions of $\mathring{f}$ and
$\breve{f}^{-}$ (as well as (8.11)); in the second we used the definition of
$\mathscr{A}_{1}$ from (8.14); in the third we used the fact that
$\breve{w}_{n^{\prime}/2+n^{\prime\prime}+1}^{-}(0)=G^{-}\big{(}y_{0}+\frac{n^{\prime\prime}+1}{n}\big{)}$
(by (8.4)) and the rescaling (8.11); and in the fourth we used the definition
of $\widetilde{G}^{-}$ in terms of $\widetilde{G}$. We further have that

$\displaystyle\widetilde{G}\Big{(}$ $\displaystyle
0,\displaystyle\frac{1}{2}+\displaystyle\frac{n^{\prime\prime}+1}{n^{\prime}}\Big{)}+\mathfrak{a}\widetilde{s}-(n^{\prime})^{\omega-1}+(n^{\prime})^{2\omega/3-1}$
$\displaystyle\leq\widetilde{G}\Big{(}\widetilde{s},\displaystyle\frac{1}{2}+\displaystyle\frac{n^{\prime\prime}+1}{n^{\prime}}\Big{)}-(n^{\prime})^{3\omega/4-1}+\displaystyle\frac{B_{0}n^{\prime}}{n}\Big{(}\widetilde{s}+\displaystyle\frac{n^{\prime\prime}+1}{n^{\prime}}\Big{)}^{2}$
$\displaystyle\leq\widetilde{x}_{n^{\prime}/2+n^{\prime\prime}+1}(\widetilde{s})-(n^{\prime})^{3\omega/4-1}+\displaystyle\frac{2B_{0}\widetilde{s}^{2}n^{\prime}}{n}+2(n^{\prime})^{\omega/2-1}\leq\displaystyle\frac{n}{n^{\prime}}\cdot\breve{x}_{n^{\prime}/2+n^{\prime\prime}+1}(s)=\displaystyle\frac{n}{n^{\prime}}\cdot\breve{f}(s),$

where in the first statement we applied a Taylor expansion for
$\widetilde{G}$, using the facts that
$\mathfrak{a}=\partial_{t}G(t_{0},y_{0})=\partial_{t}G^{-}(t_{0},y_{0})=\partial_{t}\widetilde{G}^{-}\big{(}0,\frac{1}{2}\big{)}$
and that
$[\widetilde{G}^{-}]_{2}=\frac{n^{\prime}}{n}\cdot[G^{-}]_{2}\leq\frac{2B_{0}n^{\prime}}{n}$;
in the second we used the fact that we are restricting to the event
$\mathscr{A}_{0}$, (8.12), and the fact that
$n^{\prime}n^{-1}(n^{\prime\prime}n^{\prime-1})^{2}=n^{-1}$; in the third we
used (8.11) and the fact that $\widetilde{s}^{2}\leq
4\widetilde{s}_{1}^{2}=4(n^{\prime})^{12\omega-2}\leq(n^{\prime})^{-1}$ (as
$\omega\leq\frac{1}{20}$); and in the fourth we used the definition of
$\breve{f}$. Combining this with (8.16) yields the first inequality in the
third statement of (8.15). As mentioned previously, the proof of the second
inequality is entirely analogous, as is the proof of the fourth statement of
(8.15); this establishes the proposition. ∎

###  8.3. Proof of 8.5

To establish 8.5, we set some additional notation, which will eventually more
precisely prescribe the limiting behaviors of the paths in
$\widetilde{\bm{w}}(s)$. To that end, let
$\widetilde{H}:[0,\widetilde{s}_{0}+\widetilde{s}_{1}]\times\mathbb{R}\rightarrow[0,1]$
denote the height function associated with the inverted height function
$\widetilde{G}$ (recall (3.4)), given by
$\widetilde{H}(t,x)=\sup\big{\\{}y\in\mathbb{R}:\widetilde{G}(t,y)\geq
x\big{\\}}$. In particular, for any
$(t,x)\in[0,\widetilde{s}_{0}+\widetilde{s}_{1}]\times[-n^{\omega},n^{\omega}]$,
we have

(8.17) $\displaystyle\widetilde{H}(t,x)\quad\text{is the unique value of
$y\in\big{[}-(n^{\prime})^{3\omega/2},(n^{\prime})^{3\omega/2}\big{]}$ such
that $\widetilde{G}(t,y)=x$},$

where this uniqueness follows from the fact that $\widetilde{G}(s,y)$ is
decreasing in $y$ (by (8.13)), and we implicitly used the fact that any
$(s,y)\in[0,\widetilde{s}_{0}+\widetilde{s}_{1}]\times\big{[}-(n^{\prime})^{3\omega/2},(n^{\prime})^{3\omega/2}\big{]}$
is in the domain of $\widetilde{G}$ since $0\leq
y_{0}-\frac{(n^{\prime})^{1+2\omega}}{n}-\frac{n^{\prime}}{2n}\leq
y_{0}+\frac{(n^{\prime})^{1+2\omega}}{n}-\frac{n^{\prime}}{2n}\leq 1$ (as
$\min\\{y_{0},1-y_{0}\\}\geq\operatorname{dist}\big{(}(t_{0},y_{0}),\partial\mathfrak{R}\big{)}\geq
n^{-\delta}$, as stipulated in 1.5). Next, define the density
$\widetilde{\varrho}_{0}\in L^{1}(\mathbb{R})$ by setting

(8.18) $\displaystyle\begin{aligned}
&\widetilde{\varrho}_{0}(x)=-\partial_{x}\widetilde{H}(0,x),\qquad\text{if
$\widetilde{G}(0,1)\leq x\leq\widetilde{G}(0,0)$};\\\
&\widetilde{\varrho}_{0}(x)=0,\qquad\qquad\qquad\quad\text{if
$x<\widetilde{G}(0,1)$ or if $x>\widetilde{G}(0,0)$.}\end{aligned}$

Observe in particular that
$\int_{-\infty}^{\infty}\widetilde{\varrho}_{0}(x)dx=\widetilde{H}\big{(}0,\widetilde{G}(0,1)\big{)}-\widetilde{H}\big{(}0,\widetilde{G}(0,0)\big{)}=1$.
Denoting the measure $\widetilde{\mu}_{0}=\widetilde{\varrho}_{0}(x)dx$, we
then find that $\widetilde{\mu}_{0}\in\mathscr{P}_{0}$, that is, it is a
probability measure. Denote its free convolution with the rescaled semicircle
law (recall Section 2.2) by
$\widetilde{\mu}_{t}=\widetilde{\mu}_{0}\boxplus\mu_{\operatorname{semci}}^{(t)}$
for any $t>0$; also denote the associated density by
$\widetilde{\varrho}_{t}\in L^{1}(\mathbb{R})$, which satisfies
$\widetilde{\mu}_{t}(dx)=\widetilde{\varrho}_{t}(x)dx$. Denote the classical
locations (recall Definition 2.16 with $n$ replaced by $n^{\prime}$) with
respect to $\widetilde{\mu}_{t}$ by
$\widetilde{\gamma}_{j}(t)=\gamma_{j;n^{\prime}}^{\tilde{\mu}_{t}}$ for each
$j\in\llbracket 1,n^{\prime}\rrbracket$.

We define the analogous quantities associated with $\widetilde{G}^{-}$ and
$\widetilde{G}^{+}$ very similarly. In particular, for each index
$\pm\in\\{+,-\\}$, we define
$\widetilde{H}^{\pm}(x)=\sup\big{\\{}y\in\mathbb{R}:\widetilde{G}^{\pm}(y)\geq
x\big{\\}}$, so that for any $x\in[-n^{\omega},n^{\omega}]$ we have that
$\widetilde{H}^{\pm}(x)$ is the unique value of
$y\in\big{[}-(n^{\prime})^{3\omega/2},(n^{\prime})^{3\omega/2}\big{]}$ such
that $\widetilde{G}^{\pm}(y)=x$. We further define the density
$\widetilde{\varrho}_{0}^{\pm}\in L^{1}(\mathbb{R})$ by setting
$\widetilde{\varrho}_{0}^{\pm}(x)=-\partial_{x}\widetilde{H}^{\pm}(x)$ if
$\widetilde{G}^{\pm}(1)\leq x\leq\widetilde{G}^{\pm}(0)$, and
$\widetilde{\varrho}_{0}^{\pm}(x)=0$ otherwise. Then,
$\widetilde{\mu}_{0}^{\pm}=\widetilde{\varrho}_{0}^{\pm}(x)dx$ is a
probability measure. Denote its free convolution with the semicircle law by
$\widetilde{\mu}_{t}^{\pm}=\widetilde{\mu}_{0}^{\pm}\boxplus\mu_{\operatorname{semci}}^{(t)}$
and the associated density by $\widetilde{\varrho}_{t}^{\pm}\in
L^{1}(\mathbb{R})$. Also denote the classical locations with respect to
$\widetilde{\mu}_{t}^{\pm}$ by
$\widetilde{\gamma}_{j}^{\pm}(t)=\gamma_{j;n^{\prime}}^{\tilde{\mu}_{t}^{\pm}}$.

The next lemma provides bounds on this density $\widetilde{\varrho}_{t}$, as
well as on the classical locations $\widetilde{\gamma}_{j}(t)$ (and their
analogs for $\widetilde{\varrho}_{t}^{\pm}$ and
$\widetilde{\gamma}_{j}^{\pm}(t)$); it is established in Section 8.4 below.

######  Lemma 8.6.

There exists a constant $C=C(\varepsilon,\delta,B_{0},m)>1$ such that the
following hold for each integer $n\geq C$ and index $\pm\in\\{+,-\\}$.

  1. (1)

If $(t,x)\in[0,n^{-\omega}]\times[-C^{-1},C^{-1}]$, then
$-4\varepsilon^{-1}\leq\widetilde{\varrho}_{t}(x)\leq-\frac{\varepsilon}{4}$
and
$-4\varepsilon^{-1}\leq\widetilde{\varrho}_{t}^{\pm}(x)\leq-\frac{\varepsilon}{4}$.

  2. (2)

If
$i\in\big{\llbracket}-(n^{\prime})^{1-\omega},(n^{\prime})^{1-\omega}\big{\rrbracket}$
and $t\in[0,n^{-\omega}]$, then
$\big{|}\widetilde{\gamma}_{n^{\prime}/2+i}(t)-\widetilde{\gamma}_{n^{\prime}/2+i}(0)\big{|}\leq
Ct\big{(}\frac{n^{\prime}}{n}+\frac{|i|}{n^{\prime}}+t\big{)}$ and
$\big{|}\widetilde{\gamma}_{n^{\prime}/2+i}^{\pm}(t)-\widetilde{\gamma}_{n^{\prime}/2+i}^{\pm}(0)\big{|}\leq
Ct\big{(}\frac{n^{\prime}}{n}+\frac{|i|}{n^{\prime}}+t\big{)}$.

We also require the below lemma indicating a monotone coupling for Dyson
Brownian motion, established in Section 8.4 below.

######  Lemma 8.7.

Let $M\geq 1$ and $k\geq 2$ be integers, and fix $k$ $M$-tuples
$\bm{\lambda}^{(1)},\bm{\lambda}^{(2)},\ldots,\bm{\lambda}^{(k)}\in\overline{\mathbb{W}}_{M}$
with
$\bm{\lambda}^{(1)}\leq\bm{\lambda}^{(2)}\leq\cdots\leq\bm{\lambda}^{(k)}$.
For each integer $j\in\llbracket 1,k\rrbracket$, let
$\bm{\lambda}^{(j)}(s)=\big{(}\lambda_{1}^{(j)}(s),\lambda_{2}^{(j)}(s),\ldots,\lambda_{M}^{(j)}(s)\big{)}$
denote Dyson Brownian motion run for time $s$ with initial data
$\bm{\lambda}^{(j)}$. Then it is possible to couple the
$\big{\\{}\bm{\lambda}^{(j)}(s)\big{\\}}$ such that
$\bm{\lambda}^{(1)}(s)\leq\bm{\lambda}^{(2)}(s)\leq\cdots\leq\bm{\lambda}^{(k)}(s)$,
for each real number $s\geq 0$.

Now we can establish 8.5.

###### Proof of 8.5.

By (8.3), we may restrict to the event $\mathscr{A}_{0}$ in what follows.
Define the two $n^{\prime}$-tuples
$\bm{\lambda}^{-}=(\lambda_{1}^{-},\lambda_{2}^{-},\ldots,\lambda_{n^{\prime}}^{-})\in\overline{\mathbb{W}}_{n^{\prime}}$
and
$\bm{\lambda}^{+}=(\lambda_{1}^{+},\lambda_{2}^{+},\ldots,\lambda_{n^{\prime}}^{+})\in\overline{\mathbb{W}}_{n^{\prime}}$
by setting

$\displaystyle\lambda_{j}^{\pm}=\widetilde{G}^{\pm}\Big{(}\frac{2j-1}{n^{\prime}}\Big{)},\qquad\text{for
each index $\pm\in\\{+,-\\}$ and integer $j\in\llbracket