\mathbb{P}\Bigg{[}\textbf{MED}_{k}\bigg{(}[-Ak^{1/3},Ak^{1/3}];\frac{k^{2/3}}{15000};1600k^{2/3}\bigg{)}^{\complement}\cap\textbf{TOP}\bigg{(}&[-10Ak^{1/3},10Ak^{1/3}];\displaystyle\frac{k^{2/3}}{30000}\bigg{)}\Bigg{]}\\\
&\qquad\qquad\leq CAe^{-(\log k)^{3}}.\end{aligned}$

Indeed, the proposition then follows from taking a union bound of (6.10) (with
the $A$ there replaced by $AB$ here) over $j\in\llbracket
B^{-1}k,Bk\rrbracket$, using the facts that
$[-Ak^{1/3},Ak^{1/3}]\subseteq[-ABj^{2/3},ABj^{2/3}]$ for $j\in\llbracket
B^{-1}k,Bk\rrbracket$ and that

$\displaystyle\textbf{TOP}\Big{(}[-10AB^{2}k^{2/3},10AB^{2}k^{2/3}];\displaystyle\frac{k^{2/3}}{30000B}\Big{)}\subset\textbf{TOP}\Big{(}[-10ABj^{2/3},10ABj^{2/3}];\displaystyle\frac{j^{2/3}}{30000}\Big{)}.$

To establish (6.10), observe by (6.2) that it suffices to show

(6.11) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\textbf{HIGH}_{k}\big{(}[-Ak^{1/3},Ak^{1/3}];1600k^{2/3}\big{)}^{\complement}\cap\textbf{TOP}\big{(}[-10Ak^{1/3}&,10Ak^{1/3}];k^{2/3}\big{)}\Bigg{]}\leq
CAe^{-(\log k)^{3}},\end{aligned}$

and

(6.12) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\textbf{LOW}_{k}\bigg{(}[-Ak^{1/3},Ak^{1/3}];\displaystyle\frac{k^{2/3}}{15000}\bigg{)}^{\complement}\cap\textbf{TOP}\bigg{(}[-10Ak^{1/3}&,10Ak^{1/3}];\displaystyle\frac{k^{2/3}}{30000}\bigg{)}\Bigg{]}\leq
CAe^{-(\log k)^{3}}.\end{aligned}$

To establish (6.11), observe from 6.4, 6.5, and a union bound that there
exists a constant $C_{1}>1$ such that, for $S=32k^{1/3}$,

$\displaystyle\mathbb{P}\Big{[}$
$\displaystyle\textbf{HIGH}_{k}\big{(}[T-S;T+S];1600k^{2/3}\big{)}^{\complement}\cap\textbf{TOP}\big{(}[T-5S,T+5S];k^{2/3}\big{)}\Big{]}$
$\displaystyle\leq\mathbb{P}\Big{[}\textbf{LOW}_{k}\big{(}[T-3S;T-2S];1050k^{2/3}\big{)}\cap\textbf{TOP}\big{(}[T-5S,T+5S];k^{2/3}\big{)}\Big{]}$
$\displaystyle\qquad+\mathbb{P}\Big{[}\textbf{LOW}_{k}\big{(}[T+2S;T+3S];1050k^{2/3}\big{)}\cap\textbf{TOP}\big{(}[T-5S,T+5S];k^{2/3}\big{)}\Big{]}$
$\displaystyle\qquad+\mathbb{P}\bigg{[}\textbf{HIGH}_{k}\big{(}[T-S,T+S];1600k^{2/3}\big{)}^{\complement}\cap\textbf{LOW}_{k}\Big{(}\big{[}T-3S,T-2S\big{]};1050k^{2/3}\Big{)}^{\complement}$
$\displaystyle\qquad\qquad\cap\textbf{LOW}_{k}\Big{(}\big{[}T+2S,T+3S\big{]};1050k^{2/3}\Big{)}^{\complement}\bigg{]}\leq
C_{1}e^{-(\log k)^{3}}.$

Taking a union bound over a family (of at most $A$) intervals $[T-S,T+S]$ that
cover $[-Ak^{1/3},Ak^{1/3}]$ then yields (6.11). The proof of (6.12) is
similar. Indeed, observe from 6.4, 6.6, and a union bound that there exists a
constant $C_{1}>1$ such that, for $S=32k^{1/3}$,

$\displaystyle\mathbb{P}$
$\displaystyle\Bigg{[}\textbf{LOW}_{k}\bigg{(}[T-2S,T-S];\displaystyle\frac{k^{2/3}}{15000}\bigg{)}^{\complement}\cap\textbf{TOP}\bigg{(}[T-2S,T+2S];\displaystyle\frac{k^{2/3}}{30000}\bigg{)}\Bigg{]}$
$\displaystyle\leq\mathbb{P}\bigg{[}\textbf{LOW}_{k}\Big{(}[T-2S,T-S];\displaystyle\frac{k^{2/3}}{15000}\Big{)}^{\complement}\cap\textbf{LOW}_{k}\big{(}[T+S,T+2S];1050k^{2/3}\big{)}^{\complement}$
$\displaystyle\qquad\qquad\cap\textbf{TOP}\Big{(}[T-2S,T+2S];\displaystyle\frac{k^{2/3}}{30000}\Big{)}\bigg{]}$
$\displaystyle\quad+\mathbb{P}\bigg{[}\textbf{LOW}_{k}\Big{(}[T+S,T+2S];1050k^{2/3}\Big{)}\cap\textbf{TOP}\Big{(}[T+S,T+2S];\displaystyle\frac{k^{2/3}}{30000}\Big{)}\bigg{]}\leq
C_{2}e^{-(\log k)^{3}}.$

Again taking a union bound over a family (of at most $A$) intervals
$[T-2S,T-S]$ that cover $[-Ak^{1/3},Ak^{1/3}]$ then yields (6.12). This
verifies (6.10) and thus the proposition. ∎

####  6.3. Avoiding Low Intervals

In this section we establish 6.4, which is a quick consequence of the
following more precise variant.

######  Lemma 6.7.

Fix an integer $k\geq 1$; real numbers $T_{1},T_{2},B\in\mathbb{R}$ with
$T_{1}<T_{2}-1$; and real numbers $\varepsilon\in(0,1/4)$ and
$B,\mathfrak{C}\geq 0$, with

(6.13)
$\displaystyle\mathfrak{C}\geq(T_{2}-T_{1})^{2}+\big{(}2k(T_{2}-T_{1})\big{)}^{1/2};\qquad(T_{2}-T_{1})^{2}\geq
16B+8\big{(}k(T_{2}-T_{1})\big{)}^{1/2}.$

Setting $T=(T_{1}+T_{2})/2$, we have

$\displaystyle\mathbb{P}\Big{[}\textbf{\emph{LOW}}_{k}\big{(}[T_{1},T_{2}];\mathfrak{C}\big{)}\cap\textbf{\emph{TOP}}\big{(}\\{T_{1},T,T_{2}\\}$
$\displaystyle;B\big{)}\Big{]}\leq Ce^{-(\log k)^{3}}.$

###### Proof of 6.4.

This follows from applying 6.7 with the parameters
$(B,T_{2}-T_{1},\mathfrak{C})$ there equal to
$(k^{2/3},32k^{1/3},1050k^{2/3})$ here. ∎

Figure 2.5. Shown above is the coupling between $\bm{\mathsf{x}}$ and
$\bm{\mathsf{z}}$ used to show 6.7.

###### Proof of 6.7.

Throughout, we condition on $\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,k-1\rrbracket\times[T_{1},T_{2}]\big{)}$ and restrict to the event
$\mathscr{E}=\textbf{LOW}_{k}\big{(}[T_{1},T_{2}];\mathfrak{C}\big{)}\cap\textbf{TOP}\big{(}\\{T_{1},T_{2}\\};B\big{)}$;
we will then show that $\mathsf{x}_{1}(T)$ is likely lower than allowed by the
event $\textbf{TOP}(T;B)$. In what follows, we define the $(k-1)$-tuples
$\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,k-1\rrbracket}(T_{1})\in\overline{\mathbb{W}}_{k-1}$ and
$\bm{v}=\bm{\mathsf{x}}_{\llbracket
1,k-1\rrbracket}(T_{2})\big{)}\in\overline{\mathbb{W}}_{k-1}$ , and the
function $f=\mathsf{x}_{k}|_{[T_{1},T_{2}]}$. Then, the law of
$\big{(}\mathsf{x}_{j}(s)\big{)}$ over $(j,s)\in\llbracket
1,k-1\rrbracket\times[T_{1},T_{2}]$ is given by
$\mathsf{Q}_{f}^{\bm{u};\bm{v}}$.

Defining $u^{\prime}=B-2^{-1/2}T_{1}^{2}$ and
$v^{\prime}=B-2^{-1/2}T_{2}^{2}$, denote the $(k-1)$-tuples
$\bm{u}^{\prime}=(u^{\prime},u^{\prime},\ldots,u^{\prime})\in\overline{\mathbb{W}}_{k-1}$
and
$\bm{v}^{\prime}=(v^{\prime},v^{\prime},\ldots,v^{\prime})\in\overline{\mathbb{W}}_{k-1}$
(where the multiplicity of $u^{\prime}$ and $v^{\prime}$ are both $k-1$).
Further define the function $f^{\prime}:[T_{1},T_{2}]\rightarrow\mathbb{R}$ by
setting $f^{\prime}(s)=-2^{-1/2}s^{2}-\mathfrak{C}$ for $s\in[T_{1},T_{2}]$.
Then, sample two families of non-intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{k-1})\in\llbracket
1,k-1\rrbracket\times\mathcal{C}\big{(}[T_{1},T_{2}]\big{)}$ and
$\bm{\mathsf{z}}=(\mathsf{z}_{1},\mathsf{z}_{2},\ldots,\mathsf{z}_{k-1}\big{)}\in\llbracket
1,k-1\rrbracket\times\mathcal{C}\big{(}[T_{1},T_{2}]\big{)}$ from the measures
$\mathsf{Q}_{f^{\prime}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$ and
$\mathsf{Q}^{\bm{u}^{\prime};\bm{v}^{\prime}}$, respectively; see Figure 2.5
for the latter.

Observe that $f^{\prime}(s)=-2^{-1/2}s^{2}-\mathfrak{C}\geq f(s)$, since we
have restricted to
$\textbf{LOW}_{k}(\mathfrak{C};T_{1},T_{2})\subseteq\mathscr{E}$, and that
$u^{\prime}\geq\mathsf{x}_{1}(T_{1})\geq\mathsf{x}_{j}(T_{1})$ and
$v^{\prime}\geq\mathsf{x}_{1}(T_{2})\geq\mathsf{x}_{j}(T_{2})$ for each
$j\in\llbracket 1,k-1\rrbracket$, since we have restricted to
$\textbf{TOP}\big{(}\\{T_{1},T_{2}\\};B\big{)}\subseteq\mathscr{E}$. Thus, 4.6
yields a coupling between $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such that

(6.14) $\displaystyle\mathsf{x}_{j}(s)\leq\mathsf{y}_{j}(s),\qquad\text{for
each $(j,s)\in\llbracket 1,k-1\rrbracket\times[T_{1},T_{2}]$}.$

Next by 4.32 (with the $(n;a,b;u,v)$ there equal to
$(k-1;T_{1},T_{2};u^{\prime},v^{\prime})$ here, using the fact that
$(t-T_{1})(T_{2}-t)\leq(T_{2}-T_{1})^{2}/4$ for $t\in[T_{1},T_{2}]$) and the
fact that
$-2\leq\gamma_{\operatorname{sc};k-1}(k-1)\leq\gamma_{\operatorname{sc};k-1}(1)\leq
2$, there is a constant $C_{1}>1$ such that

(6.15) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\displaystyle\bigcup_{t\in[T_{1},T_{2}]}\Big{\\{}\mathsf{z}_{1}(t)\geq
B-2^{-1/2}(tT_{1}+tT_{2}-T_{1}T_{2})+\big{(}2k(&T_{2}-T_{1})\big{)}^{1/2}\Big{\\}}\Bigg{]}\leq
C_{1}e^{-(\log k)^{3}},\end{aligned}$

and

(6.16) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\displaystyle\bigcap_{t\in[T_{1},T_{2}]}\Big{\\{}\mathsf{z}_{k-1}(t)\geq
B-2^{-1/2}(tT_{1}+tT_{2}-T_{1}T_{2}&)-\big{(}2k(T_{2}-T_{1})\big{)}^{1/2}\Big{\\}}\Bigg{]}\geq
1-C_{1}e^{-(\log k)^{3}}.\end{aligned}$

Since the first bound in (6.13) (with the facts that $B\geq 0$ and
$(T_{2}-t)(t-T_{1})\leq(T_{2}-T_{1})^{2}/4$) implies for any
$t\in[T_{1},T_{2}]$ that

$\displaystyle
B-2^{-1/2}(tT_{1}+tT_{2}-T_{1}T_{2})-\big{(}2k(T_{2}-T_{1})\big{)}^{1/2}\geq-2^{-1/2}t^{2}-\mathfrak{C}=f^{\prime}(t),$

it follows from (6.16) that $\bm{\mathsf{z}}$ remains above $f^{\prime}$ with
probability at least $1-C_{1}e^{-(\log k)^{3}}$. Hence, we may couple
$\bm{\mathsf{y}}$ and $\bm{\mathsf{z}}$ to coincide with probability at least
$1-C_{1}e^{-(\log k)^{3}}$. Together with (6.14), this gives

(6.17)
$\displaystyle\mathbb{P}\big{[}\textbf{TOP}(T;B)\big{]}\leq\mathbb{P}\big{[}\mathsf{z}_{1}(T)\geq-2^{-1/2}T^{2}-B\big{]}+C_{1}e^{-(\log
k)^{3}}.$

Since the second bound in (6.13) implies

$\displaystyle
B-2^{-1/2}(T_{1}T+T_{2}T-T_{1}T_{2})+\big{(}2k(T_{2}-T_{1})\big{)}^{1/2}\leq-2^{-1/2}T^{2}-B,$

(6.15) yields
$\mathbb{P}\big{[}\mathsf{z}_{1}(T)\geq-2^{-1/2}T^{2}-B\big{]}\leq
C_{1}e^{-(\log k)^{3}}$, which upon insertion into (6.17) yields the lemma. ∎

####  6.4. Low Interval From a Low Point

In this section we establish 6.5, which is a quick consequence of the
following more precise variant; see Figure 2.6 for a depiction.

Figure 2.6. 6.8 states that, if $\mathsf{x}_{k}(t)$ fails to be consistently
below the blue parabola $-2^{-1/2}t^{2}-\mathfrak{C}$, both on the left blue
interval $[T-(a+1)S,T-aS]$ and the right one $[T+aS,T+(a+1)S]$, then it is
likely above the orange parabola $-2^{-1/2}t^{2}-B$ on the entire interval
$[T-S,T+S]$. The proof proceeds by coupling $\bm{\mathsf{x}}$ with the red
curves $\bm{\mathsf{y}}$, to likely satisfy
${\mathsf{x}}_{k}(t)\geq{\mathsf{y}}_{k}(t)\geq-2^{-1/2}t^{2}-B$ on
$[T-S,T+S]$.

######  Lemma 6.8.

There exists a constant $C>1$ such that the following holds. Let $k\geq 1$ be
an integer, and $B,T\in\mathbb{R}$, $S\geq 1$, and $a\geq 2$ be real numbers.
For any real number $\mathfrak{C}\geq 0$ with

(6.18) $\displaystyle\mathfrak{C}\leq
B-2(akS)^{1/2}-\displaystyle\frac{a^{2}S^{2}}{8},$

we have

$\displaystyle\mathbb{P}\bigg{[}\textbf{\emph{HIGH}}_{k}$
$\displaystyle\big{(}[T-S,T+S];B\big{)}^{\complement}\cap\textbf{\emph{LOW}}_{k}\Big{(}\big{[}T-(a+1)S,T-aS\big{]};\mathfrak{C}\Big{)}^{\complement}$
$\displaystyle\quad\cap\textbf{\emph{LOW}}_{k}\Big{(}\big{[}T+aS,T+(a+1)S\big{]};\mathfrak{C}\Big{)}^{\complement}\bigg{]}\leq
Ce^{-(\log k)^{3}}.$

###### Proof of 6.5.

This follows from applying 6.8, with the parameters $(a,B,\mathfrak{C})$ there
equal to $(2,1600k^{2/3},1050k^{2/3})$ here. ∎

###### Proof of 6.8.

Throughout, we condition on $\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,k\rrbracket\times[T-(a+1)S,T+(a+1)S]\big{)}$ and restrict to the event
$\textbf{LOW}_{k}\big{(}[T-(a+1)S,T-aS];\mathfrak{C}\big{)}^{\complement}\cap\textbf{LOW}_{k}\big{(}[T+aS,T+(a+1)S];\mathfrak{C}\big{)}^{\complement}$.
On this event, there exist times $T_{1}\in\big{[}T-(a+1)S,T-aS\big{]}$ and
$T_{2}\in\big{[}T+aS,T+(a+1)S\big{]}$ such that
$\mathsf{x}_{k}(T_{1})\geq-2^{-1/2}T_{1}^{2}-\mathfrak{C}$ and
$\mathsf{x}_{k}(T_{2})\geq 2^{-1/2}T_{2}^{2}-\mathfrak{C}$. Assume $T_{1}$ is
the smallest such time in $\big{[}T-(a+1)S,T-aS\big{]}$ and that $T_{2}$ is
the largest such time in $\big{[}T+aS,T+(a+1)S\big{]}$; then, $(T_{1},T_{2})$
is a $\llbracket 1,k\rrbracket$-stopping domain in the sense of Definition
4.1. Hence 4.2 implies that the law of $\big{(}\mathsf{x}_{j}(s)\big{)}$ for
$(j,s)\in\llbracket 1,k\rrbracket\times[T_{1},T_{2}]$, conditional on the
$k$-tuples $\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,k\rrbracket}(T_{1})\in\overline{\mathbb{W}}_{k}$ and
$\bm{v}=\mathsf{x}_{\llbracket
1,k\rrbracket}(T_{2})\in\overline{\mathbb{W}}_{k}$, and on the function
$f=\mathsf{x}_{k+1}|_{[T_{1},T_{2}]}$, is given by
$\mathsf{Q}_{f}^{\bm{u};\bm{v}}$. We will then show that $\mathsf{x}_{k}$ is
likely higher than allowed by the complement of the event
$\textbf{HIGH}_{k}\big{(}[T-S,T+S];B\big{)}$.

To this end, set $u^{\prime}=-2^{-1/2}T_{1}^{2}-\mathfrak{C}$ and
$v^{\prime}=-2^{-1/2}T_{2}^{2}-\mathfrak{C}$, and define the $k$-tuples
$\bm{u}^{\prime}=(u^{\prime},u^{\prime},\ldots,u^{\prime})\in\overline{\mathbb{W}}_{k}$
and
$\bm{v}^{\prime}=(v^{\prime},v^{\prime},\ldots,v^{\prime})\in\overline{\mathbb{W}}_{k}$
(where $u^{\prime}$ and $v^{\prime}$ both appear with multiplicity $k$). Then
sample non-intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{k})\in\llbracket
1,k\rrbracket\times\mathcal{C}\big{(}[T_{1},T_{2}]\big{)}$ from the measure
$\mathsf{Q}^{\bm{u}^{\prime};\bm{v}^{\prime}}$; see Figure 2.6. Since
$\mathsf{x}_{j}(T_{1})\geq\mathsf{x}_{k}(T_{1})\geq-2^{-1/2}T_{1}^{2}-\mathfrak{C}=u^{\prime}=\mathsf{y}_{j}(T_{1})$
(and similarly
$\mathsf{x}_{j}(T_{2})\geq-2^{-1/2}T_{2}^{2}-\mathfrak{C}=v^{\prime}=\mathsf{y}_{j}(T_{2})$)
for any $j\in\llbracket 1,k\rrbracket$, by 4.6 we may couple $\bm{\mathsf{x}}$
with $\bm{\mathsf{y}}$ such that $\mathsf{x}_{j}(t)\geq\mathsf{y}_{j}(t)$, for
each $(j,t)\in\llbracket 1,k\rrbracket\times[T_{1},T_{2}]$. Hence,

(6.19) $\displaystyle\begin{aligned}
\mathbb{P}\Big{[}&\textbf{HIGH}_{k}\big{(}[T-S,T+S];B\big{)}^{\complement}\cap\textbf{LOW}_{k}\big{(}[T-(a+1)S,T-aS];\mathfrak{C}\big{)}^{\complement}\\\
&\qquad\cap\textbf{LOW}_{k}\big{(}[T+aS,T+(a+1)S];\mathfrak{C}\big{)}^{\complement}\Big{]}\\\
&\leq\mathbb{P}\Bigg{[}\bigcup_{t\in[T-S,T+S]}\big{\\{}\mathsf{x}_{k}(t)\leq-2^{-1/2}t^{2}-B\big{\\}}\Bigg{]}\leq\mathbb{P}\Bigg{[}\bigcup_{t\in[T-S,T+S]}\big{\\{}\mathsf{y}_{k}(t)\leq-2^{-1/2}t^{2}-B\big{\\}}\Bigg{]}.\end{aligned}$

By the first part of 4.32 (with the $(n;a,b;u,v)$ there equal to
$(k;T_{1},T_{2};u^{\prime},v^{\prime})$ here) and the fact that
$\gamma_{\operatorname{sc};k}(k)\geq-2$, there exists a constant $C_{1}>1$
such that

(6.20) $\displaystyle\begin{aligned}
\mathbb{P}&\bigg{[}\bigcup_{t\in[T-S,T+S]}\Big{\\{}\mathsf{y}_{k}(t)\leq-2^{-1/2}(tT_{1}+tT_{2}-T_{1}T_{2})-\mathfrak{C}-\big{(}2k(T_{2}-T_{1})\big{)}^{1/2}\Big{\\}}\bigg{]}\leq
C_{1}e^{-(\log k)^{3}}.\end{aligned}$

Now observe for $t\in[T-S,T+S]$ that (since
$T_{1}\in\big{[}T-(a+1)S,T-aS\big{]}$, $T_{2}\in\big{[}T+aS,T+(a+1)S\big{]}$,
and $a\geq 2$)

$\displaystyle 2^{-1/2}(t-T_{1})(T_{2}-t)\leq 2^{-1/2}(a+1)^{2}S^{2}\leq
2a^{2}S^{2}$ $\displaystyle\leq B-2(akS)^{1/2}-\mathfrak{C}$
$\displaystyle\leq B-\big{(}2k(T_{2}-T_{1})\big{)}^{1/2}-\mathfrak{C},$

where in the third bound we used (6.18). Hence, for $t\in[T-S,T+S]$,

$\displaystyle-2^{-1/2}(tT_{1}+tT_{2}-T_{1}T_{2})-\mathfrak{C}-\big{(}2k(T_{2}-T_{1})\big{)}^{1/2}\geq
2^{-1/2}t^{2}-B.$

Inserting this into (6.20) gives

$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{t\in[T-S,T+S]}\big{\\{}\mathsf{y}_{k}(t)\leq-2^{-1/2}t^{2}-B\big{\\}}\Bigg{]}\leq
C_{1}e^{-(\log k)^{3}},$

which, together with (6.19), implies the lemma. ∎

####  6.5. Low Interval From TOP and a High Point

In this section we establish 6.6, which is a quick consequence of the
following more precise variant; see Figure 2.7 for a depcition.

Figure 2.7. Shown above is a depiction of 6.9, indicating that if
$\mathsf{x}_{k}$ is not (entirely) below
$-2^{-1/2}t^{2}-\mathfrak{{\mathfrak{c}}}$ (the left orange parabola) on the
interval $[T-2S,T-S]$ (the left blue interval), and is also not below
$-2^{-1/2}t^{2}-\mathfrak{{\mathfrak{C}}}$ (the right orange parabola) on
$[T+S,T+2S]$ (the right blue interval), then $\mathsf{x}_{1}$ is likely above
$-2^{-1/2}t^{2}+B$ (the blue parabola) on $[T-2S,T+2S]$. The proof proceeds by
coupling $\bm{\mathsf{x}}$ with the red curves $\bm{\mathsf{y}}$ to likely
satisfy ${\mathsf{x}}_{k}(t)\geq{\mathsf{y}}_{k}(t)\geq-2^{-1/2}t^{2}+B$.

######  Lemma 6.9.

There exists a constant $C>1$ such that the following holds. Fix an integer
$k\geq 1$, and fix real numbers $B,T\in\mathbb{R}$, and $S\geq 1$. If
$\mathfrak{c},\mathfrak{C}\geq 0$ are real numbers such that

(6.21) $\displaystyle 4(16S^{2}+\mathfrak{C})(\mathfrak{c}+B+2S^{1/2})\leq
kS\leq(4S^{2}+\mathfrak{C})^{2}.$

then we have

$\displaystyle\mathbb{P}\bigg{[}\textbf{\emph{LOW}}_{k}\Big{(}\big{[}T-2S,T-S\big{]};\mathfrak{c}\Big{)}^{\complement}\cap\textbf{\emph{LOW}}_{k}\big{(}[T+S,T+2S];\mathfrak{C}\big{)}^{\complement}$
$\displaystyle\cap\textbf{\emph{TOP}}\big{(}[T-2S,T+2S];B\big{)}\bigg{]}$
$\displaystyle\qquad\qquad\qquad\leq Ce^{-(\log k)^{3}}.$

###### Proof of 6.6.

This follows from applying 6.9 with the parameters $2S\in[k^{1/3},32k^{1/3}]$
and $(B,\mathfrak{c},\mathfrak{C})$ there equal to
$(k^{2/3}/30000,k^{2/3}/15000,1050k^{2/3})$ here (which satisfies (6.21) if
$k$ is sufficiently large, which we may assume by increasing the constant $C$
from the lemma, if necessary). ∎

###### Proof of 6.9.

Throughout, we condition on $\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,k\rrbracket\times[T-2S-S,T+2S]\big{)}$ and restrict to the event
$\mathscr{E}=\textbf{LOW}_{k}\big{(}[T-2S,T-S];\mathfrak{c}\big{)}^{\complement}\cap\textbf{LOW}_{k}\big{(}[T+S,T+2S];\mathfrak{C}\big{)}^{\complement}$;
we will then show that $\mathsf{x}_{1}$ is at some point likely larger than
allowed by $\textbf{TOP}\big{(}[T-2S,T+2S];B\big{)}$. On $\mathscr{E}$, there
exist times $T_{1}\in[T-2S,T-S]$ and $T_{2}\in[T+S,T+2S]$ such that
$\mathsf{x}_{k}(T_{1})\geq-2^{-1/2}T_{1}^{2}-\mathfrak{c}$ and
$\mathsf{x}_{k}(T_{2})\geq-2^{-1/2}T_{2}^{2}-\mathfrak{C}$, respectively.
Assume $T_{1}$ is the smallest such time in $[T-2S,T-S]$ and that $T_{2}$ is
the largest such time in $[T+S,T+2S]$; then, $(T_{1},T_{2})$ is a $\llbracket
1,k\rrbracket$-stopping time in the sense of Definition 4.1. Then 4.2 implies
that the law of $\big{(}\mathsf{x}_{j}(s)\big{)}$ for $(j,t)\in\llbracket
1,k\rrbracket\times[T_{1},T_{2}]$, conditional on the $k$-tuples
$\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,k\rrbracket}(T_{1})\in\overline{\mathbb{W}}_{k}$ and
$\bm{v}=\bm{\mathsf{x}}_{\llbracket
1,k\rrbracket}(T_{2})\in\overline{\mathbb{W}}_{k}$, and the function
$f=\mathsf{x}_{k+1}|_{[T_{1},T_{2}]}$, is given by the non-intersecting
Brownian bridge measure $\mathsf{Q}_{f}^{\bm{u};\bm{v}}$.

Set $u^{\prime}=\mathsf{x}_{k}(T_{1})$ and $v^{\prime}=\mathsf{x}_{k}(T_{2})$.
Further define the $k$-tuples
$\bm{u}^{\prime}=(u^{\prime},u^{\prime},\ldots,u^{\prime})\in\overline{\mathbb{W}}_{k}$
and
$\bm{v}^{\prime}=(v^{\prime},v^{\prime},\ldots,v^{\prime})\in\overline{\mathbb{W}}_{k}$
(where $u^{\prime}$ and $v^{\prime}$ both appear with multiplicity $k$), and
sample non-intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{k})\in\llbracket
1,k\rrbracket\times\mathcal{C}\big{(}[T_{1},T_{2}]\big{)}$ according to the
measure $\mathsf{Q}^{\bm{u}^{\prime};\bm{v}^{\prime}}$; see Figure 2.7. Since
$\bm{u}^{\prime}\leq\bm{u}$ and $\bm{v}^{\prime}\leq\bm{v}$, 4.6 indicates
that we may couple $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such that

(6.22) $\displaystyle\mathsf{x}_{j}(t)\geq\mathsf{y}_{j}(t),\qquad\text{for
each $(j,t)\in\llbracket 1,k\rrbracket\times[T_{1},T_{2}]$}.$

By the first statement in 4.32 with $j=1$, and the fact that
$\gamma_{\operatorname{sc};n}(1)\geq 2^{1/2}$ for sufficiently large $n$ (by
4.31), there exists a constant $C_{1}>1$ such that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{t\in[T_{1},T_{2}]}\bigg{\\{}\mathsf{y}_{1}(t)\geq\displaystyle\frac{T_{2}-t}{T_{2}-T_{1}}\cdot
u^{\prime}+\displaystyle\frac{t-T_{1}}{T_{2}-T_{1}}\cdot
v^{\prime}+\Big{(}2k\cdot\displaystyle\frac{(T_{2}-t)(t-T_{1})}{T_{2}-T_{1}}$
$\displaystyle\Big{)}^{1/2}-(T_{2}-T_{1})^{1/2}\bigg{\\}}\Bigg{]}$
$\displaystyle\qquad\geq 1-C_{1}e^{-(\log k)^{3}}.$

Fixing $\beta\in(0,1)$ and taking $t=R=(1-\beta)T_{1}+\beta T_{2}$, we deduce
for sufficiently large $k$ that

(6.23)
$\displaystyle\mathbb{P}\Big{[}\mathsf{y}_{1}(R)\geq(1-\beta)u^{\prime}+\beta
v^{\prime}+\big{(}\beta
k(T_{2}-T_{1})\big{)}^{1/2}-(T_{2}-T_{1})^{1/2}\Big{]}\geq 1-C_{1}e^{-(\log
k)^{3}}.$

Now take

$\displaystyle
0<\beta=\displaystyle\frac{k(T_{2}-T_{1})}{4\big{(}(T_{2}-T_{1})^{2}+\mathfrak{C}\big{)}^{2}}<\frac{k4S}{4((2S)^{2}+\mathfrak{C})}\leq
1,$

where we used the facts that $T_{1}\in[T-2S,T-S]$ and $T_{2}\in[T+S,T+2S]$ (so
$2S\leq T_{2}-T_{1}\leq 4S$) in the second inequality, and the upper bound in
(6.21) for the last inequality. Moreover, using the lower bound in (6.21), and
again the relation $2S\leq T_{2}-T_{1}\leq 4S$, we get

$\displaystyle\big{(}\beta k(T_{2}-T_{1}$
$\displaystyle)\big{)}^{1/2}-2^{-1/2}(1-\beta)\beta(T_{2}-T_{1})^{2}-\beta\mathfrak{C}$
$\displaystyle\geq\big{(}\beta
k(T_{2}-T_{1})\big{)}^{1/2}-\beta\cdot\big{(}(T_{2}-T_{1})^{2}+\mathfrak{C}\big{)}$
$\displaystyle=\displaystyle\frac{k(T_{2}-T_{1})}{4\big{(}(T_{2}-T_{1})^{2}+\mathfrak{C}\big{)}}\geq\displaystyle\frac{kS}{4(16S^{2}+\mathfrak{C})}\geq\mathfrak{c}+B+2S^{1/2}\geq\mathfrak{c}+B+(T_{2}-T_{1})^{1/2}.$

Combining this with the facts that
$u^{\prime}\geq-2^{-1/2}T_{1}^{2}-\mathfrak{c}$ and
$v^{\prime}\geq-2^{-1/2}T_{2}^{2}-\mathfrak{C}$, we deduce

$\displaystyle(1-\beta$ $\displaystyle)u^{\prime}+\beta
v^{\prime}+\big{(}\beta k(T_{2}-T_{1})\big{)}^{1/2}-(T_{2}-T_{1})^{1/2}$
$\displaystyle\geq\big{(}\beta
k(T_{2}-T_{1})\big{)}^{1/2}-2^{-1/2}(1-\beta)T_{1}^{2}-2^{-1/2}\beta
T_{2}^{2}-(1-\beta)\mathfrak{c}-\beta\mathfrak{C}-(T_{2}-T_{1})^{1/2}$
$\displaystyle\geq B-2^{-1/2}\big{(}(1-\beta)T_{1}+\beta
T_{2})^{2}=B-2^{-1/2}R^{2}.$

Together with (6.22) and (6.23), this yields (since $R\in[T-2S,T+2S]$, as
$\beta\in(0,1)$, $T_{1}\in[T-2S,T-S]$, and $T_{2}\in[T+S,T+2S]$)

$\displaystyle\mathbb{P}\Big{[}\textbf{TOP}\big{(}[T-2S,T+2S];B\big{)}\cap\mathscr{E}\Big{]}\leq\mathbb{P}\big{[}\mathsf{x}_{1}(R)\leq
B-2^{-1/2}R^{2}\big{]}$
$\displaystyle\leq\mathbb{P}\big{[}\mathsf{y}_{1}(R)\leq
B-2^{-1/2}R^{2}\big{]}$ $\displaystyle\leq C_{1}e^{-(\log k)^{3}},$

which establishes the lemma. ∎

###  7\. Likelihood of On-Scale and Improved Medium Events

In this section we establish 3.8, showing that the on-scale event SCL (from
Definition 3.7) is likely upon restricting to the TOP event. We also define an
“improved variant” (see Definition 7.4 below) of the MED part of that event,
which considerably extends the range of the index $k$ appearing there, and
show it is likely (see 7.5 below). Throughout this section, we let
$\bm{\mathsf{x}}$ denote a $\mathbb{Z}_{\geq 1}\times\mathbb{R}$ indexed line
ensemble satisfying the Brownian Gibbs property and recall the notation of
Section 3.1.

####  7.1. Proof of 3.8

In this section we establish 3.8. That the MED part of that event is likely
was shown as 6.3, so we must show that the REG and GAP parts of that event are
also likely. This is done through the first and second propositions below,
which are established in Section 7.2 and Section 7.3, respectively.

######  Proposition 7.1.

For any real number $D>1$, there exists a constant $C=C(D)>1$ such that, for
any integer $k\geq 1$ and real numbers $A,B\geq 1$ with $k\geq A+B$, we have

$\displaystyle\mathbb{P}\bigg{[}\textbf{\emph{REG}}_{k}\big{(}[-Ak^{1/3},Ak^{1/3}$
$\displaystyle];3(A+B);n^{-D};k\big{)}^{\complement}\cap\textbf{\emph{TOP}}\big{(}[-3Ak^{1/3},3Ak^{1/3}];k^{2/3}\big{)}$
$\displaystyle\qquad\cap\textbf{\emph{MED}}_{k+1}\big{(}[-3Ak^{1/3},3Ak^{1/3}];Bk^{2/3}\big{)}\bigg{]}\
\leq Ce^{-(\log k)^{2}}.$

######  Proposition 7.2.

For any real number $B\geq 1$, there exist constants $c=c(B)>0$, $A=A(B)>B\geq
1$, and $C=C(B)>1$ such that, for any integer $k\geq 1$, we have

$\displaystyle\mathbb{P}\Bigg{[}\textbf{\emph{TOP}}\big{(}\\{-Ak^{1/3},Ak^{1/3}\\};Bk^{2/3}$
$\displaystyle\big{)}\cap\textbf{\emph{MED}}_{k}\big{(}\\{-Ak^{1/3},Ak^{1/3}\\};Bk^{2/3}\big{)}$
$\displaystyle\cap\textbf{\emph{GAP}}_{\lfloor
k/2\rfloor}\bigg{(}\Big{[}-\displaystyle\frac{Ak^{1/3}}{2},\displaystyle\frac{Ak^{1/3}}{2}\Big{]};C\bigg{)}^{\complement}\Bigg{]}\leq
c^{-1}e^{-c(\log k)^{2}}.$

###### Proof of 3.8.

Throughout this proof, we will repeatedly use the facts (which are quick
consequences of Definition 3.2, Definition 3.5, and Definition 3.6) that

(7.1) $\displaystyle\begin{aligned}
&\textbf{MED}_{k}(\mathcal{T};b;B)\subseteq\textbf{MED}_{k}(\mathcal{T}^{\prime};b^{\prime};B^{\prime})\subseteq\textbf{MED}_{k}(\mathcal{T}^{\prime\prime};B^{\prime\prime});\\\
\textbf{GAP}_{n}(\mathcal{T}&;B)\subseteq\textbf{GAP}_{n}(\mathcal{T}^{\prime};B^{\prime});\qquad\textbf{REG}_{k}(\mathcal{T};B;\varsigma;n)\subseteq\textbf{REG}_{k}(\mathcal{T}^{\prime};B^{\prime};\varsigma^{\prime};n^{\prime}),\end{aligned}$

if
$\mathcal{T}^{\prime\prime}\subseteq\mathcal{T}^{\prime}\subseteq\mathcal{T}$,
$b\geq b^{\prime}\geq 0$, $0\leq B\leq B^{\prime}\leq B^{\prime\prime}$,
$\varsigma\leq\varsigma^{\prime}$, and $n\leq n^{\prime}$. Then, applying 6.3
with the $(k,A,B)$ there equal to $(n,6AB,2B)$ here (using (7.1) with the fact
that $[-3AB^{2/3}n^{1/3},3AB^{2/3}n^{1/3}]\subseteq[-6ABk^{1/3},6ABk^{1/3}]$
for each $k\in\big{\llbracket}(2B)^{-1}n,2Bn\big{\rrbracket}$) yields a
constant $C_{3}>1$ such that

(7.2) $\displaystyle\begin{aligned} \mathbb{P}\Bigg{[}\bigcup_{k=\lceil
n/(2B)\rceil}^{\lfloor
2Bn\rfloor}\textbf{MED}_{k}&\Big{(}[-3AB^{2/3}n^{1/3},3AB^{2/3}n^{1/3}];\displaystyle\frac{k^{2/3}}{15000};1500k^{2/3}\Big{)}^{\complement}\\\
&\cap\textbf{TOP}\Big{(}[-240AB^{3}n^{2/3},240AB^{3}n^{2/3}];\displaystyle\frac{n^{2/3}}{10^{5}B^{2}}\Big{)}\Bigg{]}\leq
C_{3}e^{-(\log n)^{2}}.\end{aligned}$

Next using 7.1, with the $(k,A,B,D)$ there equal to $(n,AB^{1/3},1600,D)$ here
and a union bound (with (7.1) and the facts that
$[-An^{1/3},An^{1/3}]\subseteq[-AB^{1/3}k^{1/3},AB^{1/3}k^{1/3}]$ and
$[-3AB^{1/3}k^{1/3},3AB^{1/3}k^{1/3}]\subseteq[-3AB^{2/3}n^{2/3},3AB^{2/3}n^{2/3}]$
whenever $k\in\llbracket B^{-1}n,Bn\rrbracket$) yields a constant
$C_{4}=C_{4}(D)>1$ such that

(7.3) $\displaystyle\begin{aligned} \mathbb{P}\Bigg{[}&\bigcap_{k=\lceil
n/B\rceil}^{\lfloor
Bn\rfloor}\textbf{REG}_{k}\big{(}[-An^{1/3},An^{1/3}];3(AB^{1/3}+1600);n^{-D};Bn\big{)}^{\complement}\\\
&\qquad\cap\textbf{TOP}\big{(}[-3AB^{2/3}n^{1/3},3AB^{2/3}n^{1/3}];B^{-2/3}n^{2/3}\big{)}\\\
&\qquad\cap\bigcap_{k=\lceil n/(2B)\rceil}^{\lfloor
2Bn\rfloor}\textbf{MED}_{k}\Big{(}[-3AB^{2/3}n^{1/3},3AB^{2/3}n^{1/3}];\displaystyle\frac{k^{2/3}}{15000};1500k^{2/3}\Big{)}\Bigg{]}\leq
C_{4}Bne^{-(\log n)^{2}}.\end{aligned}$

Now using 7.2 with the $(B,k)$ there equal to $(2A+1500,2n)$ here (with (7.1))
yields constants $c_{1}=c_{1}(A)>0$, $A_{0}=A_{0}(A)\geq 2A+1500$, and
$C_{5}=C_{5}(A)>1$ such that

(7.4) $\displaystyle\begin{aligned}
\mathbb{P}\Big{[}\textbf{GAP}_{n}\big{(}[-An^{1/3},An^{1/3}&];C_{5})^{\complement}\cap\textbf{TOP}\big{(}[-A_{0}n^{1/3},A_{0}n^{1/3}];n^{2/3}\big{)}\\\
&\cap\textbf{MED}_{n}\big{(}[-A_{0}n^{1/3},A_{0}n^{1/3}];1500n^{2/3}\big{)}\Big{]}\leq
c_{1}^{-1}e^{-c_{1}(\log n)^{2}}.\end{aligned}$

Again applying 6.3 with the $(A,B)$ there equal to $(A_{0},1)$ here (and
(7.1)) with

(7.5) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\textbf{MED}_{n}\big{(}[-A_{0}n^{1/3},A_{0}n^{1/3}&];1500n^{2/3}\big{)}^{\complement}\\\
&\cap\textbf{TOP}\Big{(}[-10A_{0}n^{2/3},10A_{0}n^{2/3}];\displaystyle\frac{n^{2/3}}{30000}\Big{)}\Bigg{]}\leq
C_{3}e^{-(\log n)^{2}}.\end{aligned}$

Applying (7.2), (7.3), (7.4), (7.5), (7.1), a union bound, and Definition 3.7
for the event SCL yields

$\displaystyle\mathbb{P}\bigg{[}\textbf{SCL}_{n}(A_{0};B;D;C_{5})^{\complement}\cap\textbf{TOP}\Big{(}[-C_{6}n^{2/3},C_{6}n^{2/3}];\displaystyle\frac{n^{2/3}}{10^{5}B^{2}}\Big{)}\bigg{]}\leq
c_{2}^{-1}e^{-c_{2}(\log n)^{2}}.$

for some constants $C_{6}=C_{6}(A,B)>1$ (for instance, one can take
$C_{6}=\max\\{10A_{0},240AB^{3}\\}$) and $c_{2}=c_{2}(A,B,D)>0$; this (with
(7.1)) yields the theorem. ∎

####  7.2. Likelihood of REG

In this section we establish 7.1, which will be a quick consequence of the
following more precise variant.

######  Lemma 7.3.

For any real number $D>1$, there exists a constant $C(D)>1$ such that the
following holds. For any real numbers $A,\mathfrak{C}\geq 1$ and
$s,t\in\mathbb{R}$ with $-Ak^{1/3}\leq t\leq t+s\leq Ak^{1/3}$, we have

$\displaystyle\mathbb{P}\bigg{[}\Big{\\{}\big{|}\mathsf{x}_{k}(t+s)-\mathsf{x}_{k}(t)\big{|}$
$\displaystyle\geq 4(ks)^{1/2}+3(A+\mathfrak{C})k^{1/3}s+k^{-D}\Big{\\}}$
$\displaystyle\qquad\quad\cap\textbf{\emph{MED}}_{k}\big{(}[t-k^{1/3},t+s+k^{1/3}];\mathfrak{C}k^{2/3}\big{)}\bigg{]}\leq
Ce^{-(\log k)^{3}}.$

###### Proof of 7.1.

Condition on $\mathcal{F}_{\exp}\big{(}\llbracket
1,k\rrbracket\times[-3Ak^{1/3},3Ak^{1/3}]\big{)}$ and restrict to the event
$\mathscr{E}=\textbf{TOP}\big{(}\\{-3Ak^{1/3},3Ak^{1/3}\\};k^{2/3}\big{)}\cap\textbf{MED}_{k+1}\big{(}[-3Ak^{1/3},3Ak^{1/3}];Bk^{2/3}\big{)}$;
observe that
$\mathscr{E}\subseteq\textbf{MED}_{k}\big{(}\\{-3Ak^{1/3},3Ak^{1/3}\\};Bk^{2/3}\big{)}$,
since $\mathsf{x}_{k+1}\leq\mathsf{x}_{k}\leq\mathsf{x}_{1}$ and $B\geq 1$.
Then, 4.9 (where the $(n;A,B;a,b;\mathsf{T};\kappa)$ there is
$(k;Ak^{2/3},k^{4};-3Ak^{1/3},3Ak^{1/3};6Ak^{1/3};1/2)$ here) gives constants
$c>0$ and $C_{1}>1$ such that

(7.6) $\displaystyle\mathbb{P}\Bigg{[}\bigcap_{\begin{subarray}{c}|t|\leq
2Ak^{1/3}\\\ |t+s|\leq
2Ak^{1/3}\end{subarray}}\bigg{\\{}\Big{|}\mathsf{x}_{k}(t+s)-\mathsf{x}_{k}(t)-s\cdot\displaystyle\frac{\mathsf{x}_{k}(3Ak^{1/3})-\mathsf{x}_{k}(-3Ak^{1/3})}{6Ak^{1/3}}\Big{|}\bigg{\\}}\geq
k^{5}|s|^{1/3}\Bigg{]}\leq C_{1}e^{-ck^{4}}.$

if $k\geq A+B$. Since we have restricted to the event
$\textbf{MED}_{k}\big{(}\\{-3Ak^{1/3},3Ak^{1/3}\\};Bk^{2/3}\big{)}\subseteq\mathscr{E}$,
we have for $t,s+t\in[-3Ak^{1/3},3Ak^{1/3}]$ that

$\displaystyle\big{|}\mathsf{x}_{k}(3Ak^{1/3})-\mathsf{x}_{k}(3Ak^{1/3})\big{|}\leq
2Bk^{2/3}\leq 2k^{2}.$

Inserting this into (7.6) (and using the fact that, for sufficiently large
$k$, we have $6k^{2}s\leq k^{5}s^{1/3}$, if $s\leq 4Ak^{1/3}$ and $k\geq A+B$)
gives

(7.7) $\displaystyle\mathbb{P}\Bigg{[}\bigcap_{\begin{subarray}{c}|t|\leq
2Ak^{1/3}\\\ |t+s|\leq
2Ak^{1/3}\end{subarray}}\Big{\\{}\big{|}\mathsf{x}_{k}(t+s)-\mathsf{x}_{k}(t)\big{|}\geq
2k^{5}|s|^{1/3}\Big{\\}}\Bigg{]}\leq C_{1}e^{-ck^{4}}.$

Next, define the set
$\mathcal{S}=[-2Ak^{1/3},2Ak^{1/3}]\cap(k^{-6D-60}\cdot\mathbb{Z})$. Applying
7.3 and a union bound over all $s,t\in\mathcal{S}$, we deduce the existence of
a constant $C_{2}=C_{2}(D)>1$ such that

(7.8) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}&\bigcap_{t,t+s\in\mathcal{S}}\Big{\\{}\big{|}\mathsf{x}_{k}(t+s)-\mathsf{x}_{k}(t)\big{|}\geq
4(ks)^{1/2}+3(A+B)k^{1/3}s+k^{-2D}\Big{\\}}\\\
&\qquad\qquad\qquad\cap\textbf{MED}_{k}\big{(}[-3Ak^{1/3},-3Ak^{1/3}];Bk^{2/3}\big{)}\Bigg{]}\leq
C_{2}Ak^{6D+65}e^{-(\log k)^{3}}.\end{aligned}$

Further observe for any real numbers $t,t+s\in[-2Ak^{1/3},2Ak^{1/3}]$ that,
for sufficiently large $k$,

$\displaystyle 4($ $\displaystyle
ks)^{1/2}+3(A+B)k^{1/3}s+k^{-2D}+2k^{5}|t+s-t^{\prime}-s^{\prime}|^{1/3}+2k^{5}|t-t^{\prime}|^{1/3}$
$\displaystyle\leq 4(ks)^{1/2}+3(A+B)k^{1/3}s+k^{-2D}+4k^{-2D-5}\leq
4(ks)^{1/2}+3(A+B)k^{1/3}s+k^{-D},$

where $t^{\prime}$ and $t^{\prime}+s^{\prime}$ are the closest elements in
$\mathcal{S}$ to $t^{\prime}$ and $t^{\prime}+s^{\prime}$, respectively.
Inserting this into (7.8) and (7.7) yields the proposition. ∎

###### Proof of 7.3.

Throughout this proof, we set $r_{1}=t-k^{1/3}$ and $r_{2}=t+s+k^{1/3}$, and
we also denote the event
$\mathscr{E}=\textbf{MED}_{k}\big{(}\\{r_{1},r_{2}\\};\mathfrak{C}k^{2/3}\big{)}$.
It suffices to show that

(7.9) $\displaystyle\begin{aligned}
&\mathbb{P}\Big{[}\big{\\{}\mathsf{x}_{k}(t+s)-\mathsf{x}_{k}(t)\leq-4(ks)^{1/2}-3(A+\mathfrak{C})k^{1/3}s-k^{-D}\big{\\}}\cap\mathscr{E}\Big{]}\leq
Ce^{-(\log k)^{2}};\\\
&\mathbb{P}\Big{[}\big{\\{}\mathsf{x}_{k}(t+s)-\mathsf{x}_{k}(t)\geq
4(ks)^{1/2}+3(A+\mathfrak{C})k^{1/3}s+k^{-D}\big{\\}}\cap\mathscr{E}\Big{]}\leq
Ce^{-(\log k)^{2}}.\end{aligned}$

We only show the former bound in (7.9), as the proof of the latter is entirely
analogous (obtained by taking $r=r_{1}=t-k^{1/3}$ below, instead of
$r=r_{2}=t+s+k^{1/3}$). To this end, set $r=t+s+k^{1/3}$; condition on
$\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,k\rrbracket\times[t,r]\big{)}$; restrict to the event $\mathscr{E}$; and
define the $k$-tuples $\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,k\rrbracket}(t)\in\overline{\mathbb{W}}_{k}$ and
$\bm{v}=\bm{\mathsf{x}}_{\llbracket
1,k\rrbracket}(t)\in\overline{\mathbb{W}}_{k}$, as well as the function
$f=\mathsf{x}_{k+1}|_{[t,r]}$. Then the law of
$\big{(}\mathsf{x}_{j}(t^{\prime})\big{)}$, for $(j,t^{\prime})\in\llbracket
1,k\rrbracket\times[t,r]$, is given by the non-intersecting Brownian bridge
measure $\mathsf{Q}_{f}^{\bm{u};\bm{v}}$.

Next set $u^{\prime}=\mathsf{x}_{k}(t)$ and $v^{\prime}=\mathsf{x}_{k}(r)$,
and denote the $k$-tuples
$\bm{u}^{\prime}=(u^{\prime},u^{\prime},\ldots,u^{\prime})\in\overline{\mathbb{W}}_{k}$
and
$\bm{v}^{\prime}=(v^{\prime},v^{\prime},\ldots,v^{\prime})\in\overline{\mathbb{W}}_{k}$
(where $u^{\prime}$ and $v^{\prime}$ both appear with multiplicity $k$).
Sample non-intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{k})\in\llbracket
1,k\rrbracket\times\mathcal{C}\big{(}[t,r]\big{)}$ from the measure
$\mathsf{Q}^{\bm{u};\bm{v}}$. Since $\bm{u}\geq\bm{u}^{\prime}$ and
$\bm{v}\geq\bm{v}^{\prime}$, 4.6 gives a coupling between $\bm{\mathsf{x}}$
and $\bm{\mathsf{y}}$ such that
$\mathsf{x}_{j}(t^{\prime})\geq\mathsf{y}_{j}(t^{\prime})$, for each
$(j,t^{\prime})\in\llbracket 1,k\rrbracket\times[t,r]$.

From the second part of 4.32 (applied with the $(n;a,b;u,v;t)$ there equal to
$(k;t,r;u^{\prime},v^{\prime};t+s)$ here), there exists a constant $C=C(D)>1$
such that

$\displaystyle\mathbb{P}\bigg{[}\mathsf{y}_{k}(t+s)\leq
u^{\prime}+\displaystyle\frac{s}{k^{1/3}+s}\cdot(v^{\prime}-u^{\prime})-4(ks)^{1/2}-k^{-D}\bigg{]}\leq
Ce^{-(\log k)^{3}},$

which together with the above coupling between $\bm{\mathsf{x}}$ and
$\bm{\mathsf{y}}$ (with the facts that $u^{\prime}=\mathsf{x}_{k}(t)$ and
$v^{\prime}=\mathsf{x}_{k}(r)$) yields

(7.10)
$\displaystyle\mathbb{P}\bigg{[}\mathsf{x}_{k}(t+s)-\mathsf{x}_{k}(t)\leq-\displaystyle\frac{s}{k^{1/3}+s}\cdot\big{|}\mathsf{x}_{k}(r)-\mathsf{x}_{k}(t)\big{|}-4(ks)^{1/2}-k^{-D}\bigg{]}\leq
Ce^{-(\log k)^{3}}.$

Since we are restricting to $\mathscr{E}$ and since
$|t^{2}-r^{2}|\leq|r-t|\big{(}|t|+|r|\big{)}\leq 3Ak^{1/3}(k^{1/3}+s)$ (which
holds since $r-t=k^{1/3}+s$ and $|t|+|r|\leq 2Ak^{1/3}+k^{1/3}\leq
3Ak^{1/3}$), we have

$\displaystyle\big{|}\mathsf{x}_{k}(r)-\mathsf{x}_{k}(t)\big{|}\leq
2^{-1/2}(t^{2}-r^{2})+2\mathfrak{C}k^{2/3}$ $\displaystyle\leq
2\mathfrak{C}k^{2/3}+3Ak^{1/3}(k^{1/3}+s).$

Inserting this into (7.10), we deduce the first bound in (7.9). As mentioned
previously, the proof of the second is very similar and thus omitted; this
yields the lemma. ∎

####  7.3. Likelihood of GAP

In this section we establish 7.2, which will be a quick consequence of 5.1 and
4.30.

###### Proof of 7.2.

Recall the constant $C_{1}(B)>1$ from 4.30, and let $A=B\cdot C(B)>B\geq 1$.
We then condition on
$\mathcal{F}=\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,k\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$ and restrict to the event
$\textbf{TOP}\big{(}\\{-Ak^{1/3},Ak^{1/3}\\};Bk^{2/3}\big{)}\cap\textbf{MED}_{k}\big{(}\\{-Ak^{1/3},Ak^{1/3}\\};Bk^{2/3}\big{)}$.
It suffices to show that for some constants $c=c(B)>0$ and $C=C(B)>1$ we have

(7.11) $\displaystyle\mathbb{P}\Bigg{[}\bigcap_{|t|\leq
Ak^{1/3}/2}\bigcap_{1\leq i<j\leq\lfloor
k/2\rfloor}\big{\\{}\mathsf{x}_{i}(t)-\mathsf{x}_{j}(t)\leq
C(j^{2/3}-i^{2/3})+(\log k)^{25}i^{-1/3}\big{\\}}\Bigg{]}\geq
1-c^{-1}e^{-c(\log k)^{2}}.$

To this end, define the $k$-tuples $\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,k\rrbracket}(-Ak^{1/3})\in\overline{\mathbb{W}}_{k}$ and
$\bm{v}=\bm{\mathsf{x}}_{\llbracket
1,k\rrbracket}(Ak^{1/3})\in\overline{\mathbb{W}}_{k}$. Then sample $k$ non-
intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{k})$ from
the measure $\mathsf{Q}^{\bm{u};\bm{v}}$. Since the law of
$\bm{\mathsf{x}}_{\llbracket 1,k\rrbracket}$ on $[-Ak^{1/3},Ak^{1/3}]$ is
given by $\mathsf{Q}_{\mathsf{x}_{k+1}}^{\bm{u};\bm{v}}$, it follows from gap
monotonicity 5.1 that we may couple $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$
such that

(7.12)
$\displaystyle\mathsf{x}_{i}(s)-\mathsf{x}_{j}(s)\leq\mathsf{y}_{i}(s)-\mathsf{y}_{j}(s),\qquad\text{for
any $1\leq i<j\leq k$ and $s\in[-An^{1/3},An^{1/3}]$}.$

Further observe since we have restricted to
$\textbf{TOP}(-Ak^{1/3};Bk^{2/3})\cap\textbf{MED}_{k}(-Ak^{1/3},Bk^{2/3})$
that
$|u_{1}+2^{1/2}A^{2}k^{2/3}|=\big{|}\mathsf{x}_{1}(-Ak^{1/3})+2^{1/2}A^{2}k^{2/3}\big{|}\leq
Bk^{2/3}$; by similar reasoning, we have $|u_{k}+2^{1/2}A^{2}k^{2/3}|\leq
Bk^{2/3}$, $|v_{1}+2^{1/2}A^{2}k^{2/3}|\leq Bk^{2/3}$, and
$|v_{k}+2^{1/2}A^{2}k^{2/3}|\leq Bk^{1/3}$. Hence, 4.30 (applied with the
$\bm{\mathsf{x}}$ there equal to $\bm{\mathsf{y}}$ here, translated vertically
by $2^{1/2}A^{2}k^{2/3}$ and horizontally by $Ak^{1/3}$; the $A$ there equal
to $2B$ here; and the $T$ there equal to $2A=2B\cdot C_{1}(B)$ here) yields
constants $c=c(B)>0$ and $C=C(B)>1$ such that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{|t|\leq Ak^{1/3}/2}\bigcap_{1\leq
i<j\leq\lfloor
k/2\rfloor}\big{\\{}\mathsf{y}_{i}(tn^{1/3})-\mathsf{y}_{j}(tn^{1/3})\leq
C(j^{2/3}-i^{2/3})+($ $\displaystyle\log n)^{25}i^{-1/3}\big{\\}}\Bigg{]}$
$\displaystyle\geq 1-c^{-1}e^{-c(\log k)^{2}}.$

This, together with (7.12) verifies (7.11) and thus the proposition. ∎

####  7.4. Improved Medium Position Events

Observe that the SCL event from Definition 3.7 is the intersection of the
medium position events $\textbf{MED}_{k}$ only for $k$ within a constant
multiple of $n$. It will later be useful to have $\mathsf{x}_{k}$ be of order
$-k^{2/3}$, for much larger values of $k$ (say, for $k\in[B^{-1}n,n^{100}]$).
To this end, we define the following improvement of the medium position event.

######  Definition 7.4.

For any integer $n\geq 1$ and real numbers $A\geq 0$ and $B,C,R\geq 1$, define
the _improved medium position event_
$\textbf{IMP}_{n}(A;B;C;R)=\textbf{IMP}_{n}^{\bm{\mathsf{x}}}(A;B;C;R)$ by
setting

$\displaystyle\textbf{IMP}_{n}(A;B;C;R)=\bigcap_{|t|\leq
An^{1/3}}\bigcap_{j=\lceil n/B\rceil}^{\lfloor
Rn\rfloor}\big{\\{}C^{-1}n^{2/3}-Cj^{2/3}\leq\mathsf{x}_{j}(t)\leq
Cn^{2/3}-C^{-1}j^{2/3}\big{\\}}.$

What will later (in Section 19.1 below) be relevant for us is to have
$\textbf{IMP}_{n}(A;B;C;R)$ hold when $R=n^{D}$ for some large (but uniformly
bounded) $D>1$. Observe, by 3.8, that $\textbf{MED}_{k}$ is very likely if we
restrict to the event
$\textbf{TOP}\big{(}[-Ck^{1/3},Ck^{1/3}];C^{-1}k^{2/3}\big{)}$ for some
sufficiently large constant $C>1$. Thus IMP would be very likely if we
restricted to the intersection over, say $\log n$, many of these TOP events
(for example, for any $k$ equal to power of $2$ in $\llbracket
n,n^{D+1}\rrbracket$); this would require us to take a union bound over $\log
n$ many such events. Unfortunately, 2.8 only indicates that for any given
$k\in\llbracket B^{-1}n,n^{D+1}\rrbracket$, such a TOP event holds with
probability $1-\delta_{k}$ satisfying $\lim_{k\rightarrow\infty}\delta_{k}=0$,
but without an effective rate. Thus, it is unclear if one can efficiently take
a union bound over them.

The following proposition indicates that IMP is very likely, upon restricting
to only a uniformly bounded number (with respect to the index $k$) of TOP and
MED events (for which a union bound can be taken).

######  Proposition 7.5.

For any real numbers $b\in(0,1/2)$ and $A,B,D\geq 3$, there exist constants
$c=c(A,b,B,D)>0$, $C_{1}=C_{1}(B)>1$, and $C_{2}=C_{2}(A,b,B)>1$ such that

(7.13) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\textbf{\emph{IMP}}_{n}(&A;B;C_{2};n^{D})^{\complement}\cap\bigcap_{|t|\leq
An^{1/3}}\big{(}\textbf{\emph{MED}}_{\lfloor
n/4B\rfloor}(t;2bn^{2/3};Bn^{2/3})\cap\textbf{\emph{TOP}}(t;bn^{2/3})\big{)}\\\
&\qquad\cap\bigcap_{t\in\\{-C_{1}n^{10D},C_{1}n^{10D}\\}}\big{(}\textbf{\emph{MED}}_{n^{30D}}(t;Bn^{20D})\cap\textbf{\emph{TOP}}(t;Bn^{20D})\big{)}\Bigg{]}\leq
c^{-1}e^{-c(\log n)^{2}}.\end{aligned}$

To show (7.13), we must show (on the TOP and MED events there) a lower and an
upper bounds on $\mathsf{x}_{k}$, which amount to an upper and a lower bounds
on $\mathsf{x}_{1}-\mathsf{x}_{k}$, for any $k\in\llbracket
B^{-1}n,n^{D+1}\rrbracket$. The upper bound on this difference will eventually
follow from a GAP event, which will guaranteed by 7.2. To lower bound this
difference, we will show that if $\mathsf{x}_{1}-\mathsf{x}_{k}$ is too small,
then a $\textbf{GAP}_{k}(t;\omega)$ event holds for a very small value of
$\omega$ (equivalently, $\mathsf{x}_{\llbracket 1,k/2\rrbracket}(t)$ has a
very high density), which will contradict the $\textbf{MED}_{\lfloor
n/4B\rfloor}$ event in (7.13).

To make this precise, we begin with the following definition indicating when
an $n$-tuple has a high density near its top entries, after applying Dyson
Brownian motion for some time.

######  Definition 7.6.

Fix an integer $n\geq 1$; a bounded interval $(a,b)\subset\mathbb{R}_{\geq
0}$; and real numbers $\omega\geq 0$ and $\xi\in(0,1)$. An $n$-tuple
$\bm{u}=(u_{1},u_{2},\ldots,u_{n})\in\overline{\mathbb{W}}_{n}$ is called
_$(a,b;\omega;\xi)$ -packed_ if the following holds. Defining
$\bm{\lambda}=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}(\mathbb{R}_{\geq 0})$ by letting
$\bm{\lambda}(s)$ denote Dyson Brownian motion, run for time $s$, with initial
data $\bm{\lambda}(0)=\bm{u}$, we have

(7.14) $\displaystyle\mathbb{P}\Bigg{[}\bigcup_{s\in[a,b]}\bigcup_{1\leq
j<k\leq\lfloor
n/2\rfloor}\big{\\{}\lambda_{j}(sn^{1/3})-\lambda_{k}(sn^{1/3})\geq\omega(k^{2/3}-j^{2/3})+(\log
n)^{25}j^{-1/3}\big{\\}}\Bigg{]}\leq\xi^{-1}e^{-\xi(\log n)^{2}}.$

Given a line ensemble
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ and real number $t\in\mathbb{R}$, let
$\textbf{PAC}_{n}(t;a,b;\omega;\xi)=\textbf{PAC}_{n}^{\bm{\mathsf{x}}}(t;a,b;\omega;\xi)$
denote the event that $\bm{\mathsf{x}}_{\llbracket
1,n\rrbracket}(tn^{1/3})\in\overline{\mathbb{W}}_{n}$ is
$(a,b;\omega;\xi)$-packed. For any subset $\mathcal{T}\subset\mathbb{R}$,
further define
$\textbf{PAC}_{n}(\mathcal{T};a,b;\omega;\xi)=\textbf{PAC}_{n}^{\bm{\mathsf{x}}}(\mathcal{T};a,b;\omega;\xi)$
by setting

$\displaystyle\textbf{PAC}_{n}(\mathcal{T};a,b;\omega;\xi)=\bigcap_{t\in\mathcal{T}}\textbf{PAC}_{n}(t;a,b;\omega;\xi).$

######  Remark 7.7.

Observe for any subset $\mathcal{T}\subseteq\mathbb{R}$, and real numbers
$\omega,b\geq 0$ and $\xi\in(0,1)$, that
$\textbf{PAC}_{n}(\mathcal{T};0,b;\omega,\xi)\subseteq\textbf{GAP}_{\lfloor
n/2\rfloor}(n^{1/3}\cdot\mathcal{T};\omega)$, for sufficiently large $n$.
Indeed, on $\textbf{PAC}_{n}(\mathcal{T};0,b;\omega;\xi)$, we have for any
$t\in n^{1/3}\cdot\mathcal{T}$ that $\bm{\mathsf{x}}_{\llbracket
1,n\rrbracket}(t)$ is $(0,b;\omega;\xi)$-packed. Since the $s=0\in[0,b]$ case
of the event in (7.14) is deterministic for a given $\bm{u}$, it follows (for
$n$ sufficiently large so that $\xi^{-1}e^{-\xi(\log n)^{2}}<1$) that
$\mathsf{x}_{j}(tn^{1/3})-\mathsf{x}_{k}(tn^{1/3})\leq\omega(k^{2/3}-j^{2/3})+(\log
n)^{25}j^{-1/3}$ for each $1\leq j<k\leq\lfloor n/2\rfloor$, and hence
$\textbf{GAP}_{\lfloor n/2\rfloor}(n^{1/3}\cdot\mathcal{T};\omega)$ holds.

The next lemma, which is a quick consequence of 4.23, indicates that an
$n$-tuple $\bm{u}=(u_{1},u_{2},\ldots,u_{n})$ with $u_{1}-u_{n}$ sufficiently
small is packed.

######  Lemma 7.8.

For any real number $\omega>0$, there exist real numbers $a=a(\omega)>0$,
$c=c(\omega)>0$, and $\xi=\xi(\omega)\in(0,1)$ such that the following holds.
Let $n\geq 1$ be an integer and
$\bm{u}=(u_{1},u_{2},\ldots,u_{n})\in\overline{\mathbb{W}}_{n}$ be an
$n$-tuple with $u_{1}-u_{n}<cn^{2/3}$. Then, $\bm{u}$ is
$(a,100a;\omega;\xi)$-packed.

###### Proof.

Let $C_{0}>1$ denote the constant $C$ from 4.23; fix a real number $B>1$ such
that $C_{0}B^{-1/2}\leq\omega$; let $c_{0}=c_{0}(B)>1$ be the constant
$c(100B)$ from 4.23; and set $a=(100B)^{-1}$ and $\xi=c=c_{0}$. If
$u_{1}-u_{n}<cn^{2/3}$, then defining
$\bm{\lambda}=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}(\mathbb{R}_{\geq 0})$ by letting
$\bm{\lambda}(s)$ denote Dyson Brownian motion with initial data
$\bm{\lambda}(0)=\bm{u}$, 4.23 yields

$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{t\in[a,100a]}\bigcup_{1\leq
j<k\leq\lfloor
n/2\rfloor}\big{\\{}\lambda_{j}(tn^{1/3})-\lambda_{k}(tn^{1/3})\geq\omega(k^{2/3}-j^{2/3})+(\log
n$ $\displaystyle)^{-20}j^{1/3}\big{\\}}\Bigg{]}$
$\displaystyle\leq\xi^{-1}e^{-\xi(\log n)^{2}},$

where we used the facts that
$[a,100a]=\big{[}(100B)^{-1},B^{-1}\big{]}\subseteq\big{[}(100B)^{-1},100B\big{]}$
and that $C_{0}t^{1/2}\leq C_{0}B^{-1/2}\leq\omega$ if $t\leq B^{-1}$; this
verifies that $\bm{u}$ is $(a,100a;\omega;\xi)$-packed. ∎

The following proposition indicates that, if the line ensemble
$\bm{\mathsf{x}}$ is packed at some time $t_{0}$ (and some weak GAP event
holds), then $\bm{\mathsf{x}}$ is packed on certain other time intervals; see
the left side of Figure 2.8. We establish it in Section 7.5 below.

Figure 2.8. Shown to the left is a depiction of 7.9, which states that if
$\bm{\mathsf{x}}$ is packed at some time $t_{0}n^{1/3}$ (the blue line) and
some weak GAP event holds at times $\pm n^{4}$ (the orange lines), then
$\bm{\mathsf{x}}$ is packed on certain other time intervals (the blue
regions). Shown on the right is the setup for the proof of 7.5.

######  Proposition 7.9.

For any real numbers $b\geq a\geq 0$ with $b\geq 2a$; $\omega>0$; and
$\xi\in(0,1)$, there exists a constant $c=c(a,b,\omega,\xi)>0$ such that the
following holds. Let $n\geq 1$ be an integer and $t_{0}\in[-n,n]$ be a real
number. Defining the interval
$\mathcal{I}=\mathcal{I}(t_{0};a)=[t_{0}-2a,t_{0}-a]\cup[t_{0}+a,t_{0}+2a]$
and denoting $\xi_{0}=\xi/2$, we have

$\displaystyle\mathbb{P}\Bigg{[}$
$\displaystyle\textbf{\emph{PAC}}_{n}(t_{0};a,b;\omega;\xi)\cap\textbf{\emph{GAP}}_{n}\big{(}\\{-n^{4},n^{4}\\};n\big{)}\cap\textbf{\emph{PAC}}_{n}(\mathcal{I};0,b-2a;2\omega;\xi_{0})^{\complement}\Bigg{]}\leq
c^{-1}e^{-c(\log n)^{2}}.$

Now we can establish 7.5.

###### Proof of 7.5.

Let $C_{1}\geq B\geq 2$ denote the constant $A(B)$ from 7.2, and define the
events

(7.15) $\displaystyle\begin{aligned}
&\quad\mathscr{E}_{1}=\textbf{MED}_{n^{30D}}\big{(}\\{-C_{1}n^{10D},C_{1}n^{10D}\\};Bn^{20D}\big{)}\cap\textbf{TOP}\big{(}\\{-C_{1}n^{10D},C_{1}n^{10D}\\};Bn^{20D}\big{)};\\\
&\mathscr{E}_{2}=\textbf{TOP}\big{(}[-An^{1/3},An^{1/3}];bn^{2/3}\big{)};\quad\mathscr{E}_{3}=\textbf{MED}_{\lfloor
n/4B\rfloor}\big{(}\\{-An^{1/3},An^{1/3}\\};2bn^{2/3};Bn^{2/3}\big{)}.\end{aligned}$

Condition on $\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,n^{30D}\rrbracket\times[-C_{1}n^{10D},C_{1}n^{10D}]\big{)}$ and restrict to
the event $\mathscr{E}_{1}$. By Definition 7.4 and a union bound, it suffices
to show that there exist constants $c=c(A,b,B,D)>0$ and $C_{2}=C_{2}(A,b,B)>1$
such that for each $j\in\llbracket B^{-1}n,n^{D+1}\rrbracket$ we have

(7.16) $\displaystyle\begin{aligned} &\mathbb{P}\Bigg{[}\bigcap_{|t|\leq
An^{1/3}}\big{\\{}\mathsf{x}_{j}(t)\leq
C_{2}^{-1}n^{2/3}-C_{2}j^{2/3}\big{\\}}\cap\mathscr{E}_{2}\Bigg{]}\leq
c^{-1}e^{-c(\log n)^{2}};\\\ &\mathbb{P}\Bigg{[}\bigcap_{|t|\leq
An^{1/3}}\big{\\{}\mathsf{x}_{j}(t)\geq
C_{2}n^{2/3}-C_{2}^{-1}j^{2/3}\big{\\}}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\Bigg{]}\leq
c^{-1}e^{-c(\log n)^{2}};\end{aligned}$

To this end, first observe since we have restricted to $\mathscr{E}_{1}$ that
7.2 (with the $k$ there equal to $n^{30D}$ here) yields $c_{1}=c_{1}(B)>0$ and
$C_{3}=C_{3}(B)>1$ such that

(7.17) $\displaystyle\mathbb{P}[\mathscr{E}_{4}^{\complement}\big{]}\leq
c_{1}^{-1}e^{-c_{1}(\log
n)^{2}},\qquad\text{where}\quad\mathscr{E}_{4}=\textbf{GAP}_{\lfloor
n^{30D}/2\rfloor}\big{(}[-n^{10D},-An^{1/3}]\cup[An^{1/3},n^{10D}];C_{3}\big{)}.$

By Definition 3.5 of the GAP event we have, for each $(k,t)\in\llbracket
1,n^{D+1}\rrbracket\times\\{-An^{1/3},An^{1/3}\\}$, that

(7.18) $\displaystyle\mathsf{x}_{1}(t)-\mathsf{x}_{k}(t)\leq
C_{3}k^{2/3}+(\log n)^{25}\leq 2C_{3}n^{2/3},\qquad\text{on the event
$\mathscr{E}_{4}$},$

where in the second inequality we used that $n$ is sufficiently large (which
can be stipulated by decreasing the constant $c$ on the right side of (7.13)).

Next, on the event
$\mathscr{E}_{2}=\textbf{TOP}\big{(}[-An^{1/3},An^{1/3}];bn^{2/3}\big{)}$, we
have for $t\in[-An^{1/3},An^{1/3}]$ that

(7.19) $\displaystyle-(A^{2}+1)n^{2/3}\leq-
bn^{2/3}-2^{-1/2}t^{2}\leq\mathsf{x}_{1}(t)\leq
bn^{2/3}-2^{-1/2}t^{2}\leq\displaystyle\frac{n^{2/3}}{4},$

where in the first and last inequalities we used the fact that $b<1/4$.
Together with (7.18), it follows for each $(k,t)\in\llbracket
B^{-1}n,n^{D+1}\rrbracket\times\\{-An^{1/3},An^{1/3}\\}$ that

$\displaystyle\mathsf{x}_{k}(t)\geq-(2C_{3}+A^{2}+1)n^{2/3}\geq-(2C_{3}+A^{2}+1)B^{2/3}k^{2/3},\qquad\text{on
the event $\mathscr{E}_{2}\cap\mathscr{E}_{4}$},$

which together with (7.17) establishes the first bound in (7.16).

To establish the second bound there, observe from the upper bound in (7.19)
that it suffices to show that there exist constants $c=c(A,b,B,D)>0$ and
$C_{2}=C_{2}(A,b,B)>1$ such that for each $(j,t)\in\llbracket
B^{-1}n,n^{D+1}\rrbracket\times[-An^{1/3},An^{1/3}]$ we have

(7.20)
$\displaystyle\mathbb{P}\Big{[}\big{\\{}\mathsf{x}_{1}(t)-\mathsf{x}_{j}(t)\leq
C_{2}^{-1}j^{2/3}\big{\\}}\cap\mathscr{E}_{2}\cap\mathscr{E}_{3}\Big{]}\leq
c^{-1}e^{-c(\log n)^{2}}.$

Let us briefly outline how we will proceed. First, we will apply 7.8 to show
that if $\mathsf{x}_{1}(t)-\mathsf{x}_{j}(t)\leq C_{2}^{-1}j^{2/3}$, then
$\bm{\mathsf{x}}_{\llbracket 1,j\rrbracket}(t)$ is packed. We will then use
7.9 to deduce that $\bm{\mathsf{x}}_{\llbracket 1,j\rrbracket}(t+aj^{1/3})$ is
packed for some constant $a>0$, and then apply 7.9 again (with 7.7) to deduce
that $\bm{\mathsf{x}}_{\llbracket 1,j/2\rrbracket}(t)$ has small gaps. The
latter will contradict the $\textbf{TOP}\cap\textbf{MED}$ event defining
$\mathscr{E}_{2}\cap\mathscr{E}_{3}$ (recall (7.15)). See the right side of
Figure 2.8.

Now let us implement this in more detail. Define $C_{2}=c(b)^{-1}$, where
$c(b)$ is the constant from 7.8 (with $\omega$ there taken to be $b$ here). We
further fix some $(j,t)\in\llbracket
B^{-1}n,n^{D+1}\rrbracket\times[-An^{1/3},An^{1/3}]$ and define the event

(7.21)
$\displaystyle\mathscr{E}_{5}=\mathscr{E}_{5}(j,t)=\big{\\{}\mathsf{x}_{1}(t)-\mathsf{x}_{j}(t)\leq
C_{2}^{-1}j^{2/3}\big{\\}}.$

Then, by 7.8, there exist constants $a=a(b)>0$ and $\xi=\xi(b)>0$ such that
$\mathscr{E}_{5}\subseteq\textbf{PAC}_{j}(t;a,100a;b;\xi)$. Hence, applying
7.9 with the $(n;t_{0};a,b;\omega;\xi)$ there equal to
$(j;j^{-1/3}t;a,100a;b;\xi)$ here (using the fact that
$\mathscr{E}_{4}\subseteq\textbf{GAP}_{j}\big{(}\\{-j^{4},j^{4}\\};j\big{)}$
for $j\in\llbracket B^{-1}n,n^{D+1}\rrbracket$ and sufficiently large $n$, due
to (7.17), the fact that
$\\{-j^{4},j^{4}\\}\in[-n^{10D},-An^{1/3}]\cup[An^{1/3},n^{10D}]$, and the
fact that $C_{1}(k^{2/3}-i^{2/3})+(30D\log n)^{25}i^{-1/3}\leq
j(k^{2/3}-i^{2/3})+(\log j)^{25}i^{-1/3}$ for any $1\leq i<k\leq j$) yields
the existence of a constant $c_{2}=c_{2}(A,b,B,D)>0$ such that

(7.22)
$\displaystyle\mathbb{P}\bigg{[}\mathscr{E}_{5}\cap\mathscr{E}_{4}\cap\textbf{PAC}_{j}\Big{(}tj^{-1/3}+a;0,98a;2b;\displaystyle\frac{\xi}{2}\Big{)}^{\complement}\bigg{]}\leq
c_{2}^{-1}e^{-c_{2}(\log n)^{2}}.$

Applying 7.9 again with the $(n;t_{0};a,b;\omega;\xi)$ there equal to
$(j;tj^{-1/3}+a;0,98a;2b;\xi/2)$ here (again using the fact that
$\mathscr{E}_{4}\subseteq\textbf{GAP}_{j}\big{(}\\{-j^{4},j^{4}\\};j\big{)}$)
yields a constant $c_{3}=c_{3}(A,b,B,D)>0$ such that

$\displaystyle\mathbb{P}\bigg{[}\textbf{PAC}_{j}\Big{(}tj^{-1/3}+a;0,98a;2b;\displaystyle\frac{\xi}{2}\Big{)}\cap\mathscr{E}_{4}\cap\textbf{PAC}_{j}\Big{(}tj^{-1/3};0,96a;4b;\displaystyle\frac{\xi}{4}\Big{)}^{\complement}\bigg{]}\leq
c_{3}^{-1}e^{-c_{3}(\log n)^{2}}.$

Together with (7.22), (7.21), (7.17), and a union bound, this yields a
constant $c_{4}=c_{4}(A,b,B,D)>0$ such that

(7.23)
$\displaystyle\mathbb{P}\bigg{[}\big{\\{}\mathsf{x}_{1}(t)-\mathsf{x}_{j}(t)\leq
C_{2}^{-1}j^{2/3}\big{\\}}\cap\mathscr{E}_{4}\cap\textbf{PAC}_{j}\Big{(}tj^{-1/3};0,96a;4b;\displaystyle\frac{\xi}{4}\Big{)}^{\complement}\bigg{]}\leq
c_{4}^{-1}e^{-c_{4}(\log n)^{2}}.$

By 7.7, we have for sufficiently large $n$ that

(7.24) $\displaystyle\textbf{GAP}_{\lfloor
j/2\rfloor}(t;4b)^{\complement}\subseteq\textbf{PAC}_{j}\Big{(}tj^{-1/3};0,96a;4b;\displaystyle\frac{\xi}{4}\Big{)}^{\complement}.$

We then claim that
$\mathscr{E}_{2}\cap\mathscr{E}_{3}\subseteq\textbf{GAP}_{\lfloor
j/2\rfloor}(t;4b)^{\complement}$. To this end, first observe on
$\textbf{GAP}_{\lfloor j/2\rfloor}(t;4b)$ that

$\displaystyle\mathsf{x}_{1}(t)-\mathsf{x}_{\lfloor n/4B\rfloor}(t)\leq
4b\Big{(}\displaystyle\frac{n}{4B}\Big{)}^{2/3}+(\log n)^{25}\leq bn^{2/3},$

where we used the facts that $B\geq 3$ and $n$ is sufficiently large. However,
by Definition 3.2 and (7.15), we have $\mathsf{x}_{1}(t)-\mathsf{x}_{\lfloor
n/4B\rfloor}(t)\geq bn^{2/3}$ on $\mathscr{E}_{2}\cap\mathscr{E}_{3}$, meaning
that $\mathscr{E}_{2}\cap\mathscr{E}_{3}\subseteq\textbf{GAP}_{\lfloor
j/2\rfloor}(t;4b)^{\complement}$. This, together with (7.23) and (7.24),
establishes (7.20) and thus the proposition. ∎

####  7.5. Deriving 7.9

We begin with the following lemma, which is a quick consequence of 4.20 and
shows that any $n$-tuple that is close to a packed one is also packed.

######  Lemma 7.10.

Adopt the notation of Definition 7.6; fix a real number $\varsigma\geq 0$, and
suppose that $\bm{u}\in\overline{\mathbb{W}}_{n}$ is
$(a,b;\omega;\xi)$-packed. Then any $n$-tuple
$\widetilde{\bm{u}}\in\overline{\mathbb{W}}_{n}$ such that
$\big{|}u_{j}-\widetilde{u}_{j}\big{|}\leq\varsigma/(4n^{1/3})$ for each
integer $j\in\llbracket 1,n\rrbracket$ is $(a,b;\omega+\varsigma;\xi)$-packed.

###### Proof.

Define $\bm{\lambda}=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}(\mathbb{R}_{\geq 0})$ and
$\widetilde{\bm{\lambda}}=\big{(}\widetilde{\lambda}_{1},\widetilde{\lambda}_{2},\ldots,\widetilde{\lambda}_{n}\big{)}\in\llbracket
1,n\rrbracket\times\mathcal{C}(\mathbb{R}_{\geq 0})$ by letting
$\bm{\lambda}(s)$ and $\widetilde{\bm{\lambda}}(s)$ denote Dyson Brownian
motions with initial data $\bm{\lambda}(0)=\bm{u}$ and
$\widetilde{\bm{\lambda}}(0)=\widetilde{\bm{u}}$, respectively. By 4.20, there
is a coupling between $\bm{\lambda}$ and $\widetilde{\bm{\lambda}}$ such that
$\big{|}\lambda_{j}(s)-\widetilde{\lambda}_{j}(s)\big{|}\leq\varsigma/(4n^{1/3})$
for each $(j,s)\in\llbracket 1,n\rrbracket\times\mathbb{R}_{\geq 0}$. It
follows for any $1\leq j<k\leq n$ and $s\in\mathbb{R}_{\geq 0}$ that

$\displaystyle\widetilde{\lambda}_{k}(s)-\widetilde{\lambda}_{j}(s)\leq\lambda_{k}(s)-\lambda_{j}(s)+\displaystyle\frac{\varsigma}{2n^{1/3}}\leq\lambda_{k}(s)-\lambda_{j}(s)+\varsigma(k^{2/3}-j^{2/3}),$

where in the last inequality we used the bound $k^{2/3}-j^{2/3}\geq
1/(2n^{1/3})$ for $1\leq j<k\leq n$. Together with (7.14), this implies that
$\widetilde{\bm{u}}$ is $(a,b;\omega+\varsigma;\xi)$-packed. ∎

The following lemma indicates that applying Dyson Brownian motion to a packed
$n$-tuple likely yields a packed $n$-tuple.

######  Lemma 7.11.

Adopt the notation of Definition 7.6, assuming that $\bm{u}$ is
$(a,b;c;\omega)$-packed; set $\xi_{0}=\xi/2$. For any real number
$s_{0}\in[0,b]$ the $n$-tuple $\bm{\lambda}(s_{0}n^{1/3})$ is
$\big{(}\min\\{a-s_{0},0\\},b-s_{0};\xi_{0};\omega\big{)}$-packed with
probability at least $1-\xi_{0}^{-1}e^{-\xi_{0}(\log n)^{2}}$.

###### Proof.

Let us assume in what follows that $s_{0}\leq a$, as the proof when
$s_{0}\in(a,b]$ is entirely analogous. Since $\bm{u}$ is
$(a,b;c;\omega)$-packed, we have

$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{t\in[a-s_{0},b-s_{0}]}\bigcup_{1\leq
j<k\leq\lfloor n/2\rfloor}\Big{\\{}\lambda_{j}\big{(}$
$\displaystyle(t+s_{0})n^{1/3}\big{)}-\lambda_{k}\big{(}(t+s_{0})n^{1/3}\big{)}$
$\displaystyle\geq\omega(k^{2/3}-j^{2/3})+(\log
n)^{25}j^{-1/3}\Big{\\}}\Bigg{]}\leq\xi^{-1}e^{-\xi(\log n)^{2}}.$

Together with a Markov estimate, this yields the existence of an event
$\mathscr{E}$ measurable with respect to $\bm{\lambda}(s_{0})$, with
$\mathbb{P}[\mathscr{E}]\geq 1-(\xi^{-1}e^{-\xi(\log n)^{2}})^{1/2}\geq
1-\xi_{0}^{-1}e^{-\xi_{0}(\log n)^{2}}$, such that the following holds.
Conditioning on $\bm{\lambda}(s_{0}n^{1/3})$ and restricting to $\mathscr{E}$,
we have

$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{t\in[a-s_{0},b-s_{0}]}$
$\displaystyle\bigcap_{1\leq j<k\leq\lfloor
n/2\rfloor}\Big{\\{}\lambda_{1}\big{(}(t+s_{0})n^{1/3}\big{)}-\lambda_{\lfloor
n/2\rfloor}\big{(}(t+s_{0})n^{1/3}\big{)}$
$\displaystyle\geq\omega(k^{2/3}-j^{2/3})+(\log
n)^{25}j^{-1/3}\Big{\\}}\Bigg{]}\leq(\xi^{-1}e^{-\xi(\log
n)^{2}})^{1/2}\leq\xi_{0}^{-1}e^{-\xi_{0}(\log n)^{2}}.$

This, together with the fact that $\bm{\lambda}(s+s_{0}n^{1/3})$ has the same
law as Dyson Brownian motion run for time $s$ with initial data
$\bm{\lambda}(s_{0}n^{1/3})$ for $s\geq 0$ (and Definition 7.6), implies the
lemma. ∎

Figure 2.9. Shown above is the setup for the proof of 7.9.

Now we can establish 7.9.

###### Proof of 7.9.

Denote $\mathcal{I}_{1}=\mathcal{I}_{1}(t_{0};a;n)=[t_{0}-2a,t_{0}-a]$ and
$\mathcal{I}_{2}=\mathcal{I}_{2}(t_{0};a)=[t_{0}+a,t_{0}+2a]$. Then
$\mathcal{I}=\mathcal{I}_{1}\cup\mathcal{I}_{2}$ and so by a union bound it
suffices to show that

(7.25) $\displaystyle\begin{aligned}
&\mathbb{P}\Bigg{[}\textbf{PAC}_{n}(t_{0};a,b;\omega;\xi)\cap\textbf{GAP}_{n}\big{(}\\{-n^{4},n^{4}\\};n\big{)}\cap\textbf{PAC}_{n}(\mathcal{I}_{1};0,b-2a;\xi_{0};2\omega)^{\complement}\Bigg{]}\leq
c^{-1}e^{-c(\log n)^{2}};\\\
&\mathbb{P}\Bigg{[}\textbf{PAC}_{n}(t_{0};a,b;\omega;\xi)\cap\textbf{GAP}_{n}\big{(}\\{-n^{4},n^{4}\\};n\big{)}\cap\textbf{PAC}_{n}(\mathcal{I}_{2};0,b-2a;\xi_{0};2\omega)^{\complement}\Bigg{]}\leq
c^{-1}e^{-c(\log n)^{2}}.\end{aligned}$

We only show the second bound in (7.25), as the first then follows from
reflecting the line ensemble $\bm{\mathsf{x}}$ through the line
$\\{t=t_{0}n^{1/3}\\}$. To this end, we condition on
$\mathcal{F}=\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,k\rrbracket\times[t_{0}n^{1/3},n^{4}]\big{)}$; and restrict to the event
$\textbf{PAC}_{n}(t_{0};a,b;\omega;\xi)\cap\textbf{GAP}_{n}\big{(}\\{-n^{4},n^{4}\\};n\big{)}$.
It then suffices to show that

(7.26)
$\displaystyle\mathbb{P}\Big{[}\textbf{PAC}_{n}\big{(}[t_{0}+a,t_{0}+2a];0,b-2a;\xi_{0};2\omega\big{)}\Big{]}\geq
1-c^{-1}e^{-c(\log n)^{2}}.$

To this end, denote the $n$-tuples $\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,n\rrbracket}(t_{0}n^{1/3})\in\overline{\mathbb{W}}_{n}$ and
$\bm{v}=\bm{\mathsf{x}}_{\llbracket
1,n\rrbracket}(n^{4})\in\overline{\mathbb{W}}_{n}$. Define the line ensemble
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[t_{0}n^{1/3},n^{4}]\big{)}$ by letting
$\bm{\mathsf{y}}(s)$ denote Dyson Brownian motion (recall Section 4.4) run for
time $s-t_{0}n^{1/3}$ with initial data
$\bm{\mathsf{y}}(t_{0}n^{1/3})=\bm{u}$. Condition on $\bm{\mathsf{y}}(n^{4})$,
and define the $n$-tuples
$\bm{v}^{\prime},\bm{v}^{\prime\prime}\in\mathbb{W}_{n}$ by setting
$v_{j}^{\prime}=\mathsf{y}_{j}(n^{4})$ and
$v_{j}^{\prime\prime}=v_{j}^{\prime}+(n-j)n$ for each $j\in\llbracket
1,n\rrbracket$. Then sample $n$ non-intersecting Brownian bridges
$\bm{\mathsf{z}}=(\mathsf{z}_{1},\mathsf{z}_{2},\ldots,\mathsf{z}_{n})\in\mathcal{C}\big{(}[t_{0}n^{1/3},n^{4}]\big{)}$
from the measure $\mathsf{Q}^{\bm{u};\bm{v}^{\prime\prime}}$. See Figure 2.9.

We will first use gap and height monotonicity to compare the gaps of
$\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$, through $\bm{\mathsf{z}}$. To do
this, first observe that the law of $\bm{\mathsf{x}}$ is given by
$\mathsf{Q}_{\mathsf{x}_{n+1}}^{\bm{u};\bm{v}}$. For any $j\in\llbracket
1,n-1\rrbracket$, we have $v_{j}-v_{j+1}=x_{j}(n^{4})-x_{j+1}(n^{4})\leq n\leq
v_{j}^{\prime\prime}-v_{j+1}^{\prime\prime}$, where the first statement holds
by the definition of $\bm{v}$; the second by the fact that we have restricted
to the event $\textbf{GAP}_{n}(n^{4};n)$ (and the fact that
$n\big{(}(j+1)^{2/3}-j^{2/3}\big{)}+(\log n)^{25}\leq n$ for sufficiently
large $n$); and the third by the definition of $\bm{v}^{\prime\prime}$. Hence,
it follows from gap monotonicity 5.1 that we may couple $\bm{\mathsf{x}}$ and
$\bm{\mathsf{z}}$ such that

(7.27)
$\displaystyle\mathsf{x}_{j}(t)-\mathsf{x}_{j+1}(t)\leq\mathsf{z}_{j}(t)-\mathsf{z}_{j+1}(t),\qquad\text{for
$(j,t)\in\llbracket 1,n-1\rrbracket\times[t_{0}n^{1/3},n^{4}]$}.$

Moreover observe from the second part of 4.17 that the law of
$\bm{\mathsf{y}}$ is given by $\mathsf{Q}^{\bm{u};\bm{v}^{\prime}}$. Since
that of $\bm{\mathsf{z}}$ is given by
$\mathsf{Q}^{\bm{u};\bm{v}^{\prime\prime}}$ and since $0\leq
v_{j}^{\prime\prime}-v_{j}^{\prime}\leq n^{2}$ for each $j\in\llbracket
1,n\rrbracket$, it follows from the second part of 4.7 that we may couple
$\bm{\mathsf{y}}$ and $\bm{\mathsf{z}}$ in such a way that

$\displaystyle\big{|}\mathsf{y}_{j}(t)-\mathsf{z}_{j}(t)\big{|}\leq\displaystyle\frac{bn^{1/3}}{n^{4}-t_{0}n^{1/3}}\cdot
n^{2}\leq 2bn^{-5/3},\qquad\text{for $(j,t)\in\llbracket
1,n\rrbracket\times\big{[}t_{0}n^{1/3},(t_{0}+b)n^{1/3}\big{]}$},$

where we used the fact that $n^{4}-t_{0}n^{1/3}\geq\frac{n^{4}}{2}$ for $n\geq
2$ (as $|t_{0}|\leq n$) for sufficiently large $n$. Combining this bound with
(7.27) yields

(7.28)
$\displaystyle\mathsf{x}_{j}(t)-\mathsf{x}_{k}(t)\leq\mathsf{y}_{j}(t)-\mathsf{y}_{k}(t)+4bn^{-5/3},\qquad\text{for
$1\leq j<k\leq n$ and $t\in\big{[}t_{0}n^{1/3},(t_{0}+b)n^{1/3}\big{]}$}.$

Next, by 7.11, for any real number $s\in[an^{1/3},2an^{1/3}]$ the $n$-tuple
$\bm{\mathsf{y}}(t_{0}n^{1/3}+s)$ is $(0,b-s;\omega;\xi_{0})$-packed with
probability at least $1-\xi_{0}^{-1}e^{-\xi_{0}(\log n)^{2}}$; in particular,
it is $(0,b-2a;\omega;\xi_{0})$-packed with at least the same probability
$1-\xi_{0}^{-1}e^{-\xi_{0}(\log n)^{2}}$. This, (7.28), Definition 7.6, and
the fact that $8bn^{-5/3}\leq\omega n^{-1/3}$ for sufficiently large $n$
together imply that (7.26) holds, establishing 7.9. ∎

##  Chapter 3 Airy Statistics From Non-Intersecting Brownian Bridges

###  8\. Gap Convergence to the Airy Point Process

In this section we establish the Airy gaps 3.18. We first stochastically bound
the gaps of $\bm{\mathsf{x}}$ below by that of an Airy point process in
Section 8.1; then, after recalling a result for edge statistics of Dyson
Brownian motion in Section 4.6, we provide the complementary stochastic upper
bound for the gaps in Section 8.2.

####  8.1. Gap Lower Bound

In this section we establish a lower bound on the gaps considered in 3.18.
This will follow from a suitable application of gap monotonicity 5.1, together
with the following result indicating convergence to the Airy line ensemble for
non-intersecting Brownian bridges whose lower boundary is given by a rescaled
semicircle; see the left side of Figure 3.1 for a depiction. In what follows,
we recall the classical locations $\gamma_{\operatorname{sc};n}$ of the
semicircle distribution from (4.24), which in the next lemma will be related
to the stretching factor for the semicircle (when we later apply the lemma, it
will be close to $2$).

######  Lemma 8.1.

Let $\bm{\sigma}=(\sigma_{1},\sigma_{2},\ldots)$ and
$\bm{\mathsf{T}}=(\mathsf{T}_{1},\mathsf{T}_{2},\ldots)$ be two sequences of
positive real numbers, so that $\lim_{k\rightarrow\infty}\sigma_{k}=1$ and
$\mathsf{T}_{k}\in[2k^{1/3},k^{1/2}]$, for each integer $k\geq 64$. For any
integer $n\geq 64$, define
$f=f_{n}:[-\mathsf{T}_{n},\mathsf{T}_{n}]\rightarrow\mathbb{R}$ by setting

(8.1) $\displaystyle
f_{n}(t)=\sigma_{n}\mathsf{T}_{n}\Big{(}\displaystyle\frac{\mathsf{T}_{n}^{2}-t^{2}}{2}\Big{)}^{1/2}\cdot\gamma_{\operatorname{sc};\lfloor\sigma_{n}^{2}\mathsf{T}_{n}^{3}\rfloor}(n+1),\qquad\text{for
each $t\in[-\mathsf{T}_{n},\mathsf{T}_{n}]$}.$

Sample non-intersecting Brownian bridges
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots,\mathsf{x}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\mathsf{T}_{n},\mathsf{T}_{n}]\big{)}$
under the measure $\mathsf{Q}_{f}^{\bm{0}_{n};\bm{0}_{n}}$, and define

$\displaystyle\bm{\mathsf{X}}^{n}=(\mathsf{X}_{1}^{n},\mathsf{X}_{2}^{n},\ldots,\mathsf{X}_{n}^{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\mathsf{T},\mathsf{T}]\big{)},\quad\text{where}\quad\mathsf{X}_{j}^{n}(t)=2^{1/2}\big{(}\mathsf{x}_{i}(t)-2^{1/2}\sigma_{n}\mathsf{T}_{n}^{2}\big{)}.$

Then, $\bm{\mathsf{X}}^{n}$ converges to $\bm{\mathcal{R}}$ on compact subsets
of $\mathbb{Z}_{\geq 1}\times\mathbb{R}$, as $n$ tends to $\infty$.

Figure 3.1. Shown on the left is a depiction of the setup for 8.1, whose proof
follows from coupling $\bm{\mathsf{x}}$ with the Brownian watermelon
$\widetilde{\bm{\mathsf{x}}}$ as shown to the right.

###### Proof.

Throughout this proof, we abbreviate $\mathsf{T}=\mathsf{T}_{n}$ and
$\sigma=\sigma_{n}$, and we set
$m=m_{n}=\lfloor\sigma^{2}\mathsf{T}^{3}\rfloor\geq 2n$. To establish the
lemma, we will first use 4.32 to approximate the ensemble $\bm{\mathsf{x}}$ by
the first $n$ curves of a watermelon $\widetilde{\bm{\mathsf{x}}}$ with $m$
bridges (where its $(n+1)$-th curve closely mimics the shape of $f(t)$); then,
we will apply the first part of 4.33 (with $(n,a,b,j)$ there equal to
$(m,-\mathsf{T},\mathsf{T},n+1)$ here) to show the latter converges to
$\bm{\mathcal{R}}$, implying the same for $\bm{\mathsf{x}}$.

To this end, sample $m$ non-intersecting Brownian bridges
$\widetilde{\bm{\mathsf{x}}}=(\widetilde{\mathsf{x}}_{1},\widetilde{\mathsf{x}}_{2},\ldots,\widetilde{\mathsf{x}}_{m})\in\llbracket
1,m\rrbracket\times\mathcal{C}\big{(}[-\mathsf{T},\mathsf{T}]\big{)}$ from the
measure $\mathsf{Q}^{\bm{0}_{m};\bm{0}_{m}}$. Then, 4.32 gives for
sufficiently large $n$ that

$\displaystyle\mathbb{P}\Bigg{[}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\widetilde{\mathsf{x}}_{n+1}(t)-\sigma\mathsf{T}\Big{(}\displaystyle\frac{\mathsf{T}^{2}-t^{2}}{2}\Big{)}^{1/2}\cdot\gamma_{\operatorname{sc};m}(n+1)\big{|}\geq
n^{-1/6}\Bigg{]}\leq n^{-5}.$

So, defining the (random) function
$\widetilde{f}=\widetilde{f}_{n}:[-\mathsf{T},\mathsf{T}]\rightarrow\mathbb{R}$
by $\widetilde{f}(t)=\widetilde{\mathsf{x}}_{n+1}(t)$, we have

(8.2) $\displaystyle\mathbb{P}[\mathscr{E}_{n}]\leq
n^{-5},\qquad\text{where}\qquad\mathscr{E}_{n}=\Bigg{\\{}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\widetilde{f}(t)-f(t)\big{|}\geq
n^{-1/6}\Bigg{\\}}.$

Now condition on $\widetilde{f}=\widetilde{\mathsf{x}}_{n+1}$. Then, the law
of the first $n$ curves
$\big{(}\widetilde{\mathsf{x}}_{j}(t)\big{)}\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\mathsf{T},\mathsf{T}]\big{)}$ of
$\widetilde{\bm{x}}$ is given by
$\mathsf{Q}_{\tilde{f}}^{\bm{0}_{n};\bm{0}_{n}}$. By the first part of 4.7, it
follows that we can couple $\bm{\mathsf{x}}$ and $\widetilde{\bm{\mathsf{x}}}$
such that

(8.3) $\displaystyle\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\mathsf{x}_{j}(t)-\widetilde{\mathsf{x}}_{j}(t)\big{|}\leq
n^{-1/6},\qquad\text{on the event $\mathscr{E}_{n}^{\complement}$}.$

Moreover, by 4.33 (applied with the $(n,T,\sigma)$ there equal to
$(m,\sigma_{n}^{-2/3},\sigma_{n}^{-1/3}$ here), the ensemble
$\widetilde{\bm{\mathsf{X}}}^{n}=(\widetilde{\mathsf{X}}_{1}^{n},\widetilde{\mathsf{X}}_{2}^{n},\ldots,\widetilde{\mathsf{X}}_{n}^{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\mathsf{T},\mathsf{T}]\big{)}$ defined
by
$\widetilde{\mathsf{X}}_{j}^{n}(t)=2^{1/2}\sigma_{n}^{1/3}\big{(}\widetilde{\mathsf{x}}_{j}(\sigma_{n}^{-2/3}t)-2^{1/2}\sigma_{n}\mathsf{T}_{n}^{2}\big{)}$
converges to $\bm{\mathcal{R}}$ on compact subsets of $\mathbb{Z}_{\geq
1}\times\mathbb{R}$, as $n$ tends to $\infty$. This, together with (8.2),
(8.3), and the fact that $\lim_{n\rightarrow\infty}\sigma_{n}=1$, implies the
same for $\bm{\mathsf{X}}^{n}$, thus implying the lemma. ∎

Using 8.1, we can lower bound the gaps of the bridges from 3.16.

######  Proposition 8.2.

Adopt 3.16. Fix an integer $k\geq 1$; a real number $t\in\mathbb{R}$; and
nonnegative real numbers $r_{1},r_{2},\ldots,r_{k}\geq 0$. Then, recalling the
Airy point process
$\bm{\mathfrak{a}}=(\mathfrak{a}_{1},\mathfrak{a}_{2},\ldots)$ from Definition
3.15, we have

$\displaystyle\displaystyle\liminf_{n\rightarrow\infty}\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\Big{\\{}2^{1/2}\big{(}\mathsf{x}_{j}(t)-\mathsf{x}_{j+1}(t)\big{)}\geq
r_{j}\Big{\\}}\Bigg{]}\geq\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\\{\mathfrak{a}_{j}-\mathfrak{a}_{j+1}\geq
r_{j}\\}\Bigg{]}.$

###### Proof.

Set $\sigma_{n}=1+\delta_{n}^{1/2}$, and denote
$m_{n}=\lfloor\sigma_{n}^{2}\mathsf{T}_{n}^{3}\rfloor$. By the first part of
4.31 (and the fact that $n\mathsf{T}^{-3}\leq\delta^{3}$ by 3.16), there
exists a constant $C>1$ such that

$\displaystyle 0\leq 2-\gamma_{\operatorname{sc};m_{n}}(n+1)\leq
C\delta_{n}^{2},\qquad\text{so}\qquad\sigma_{n}\cdot\gamma_{\operatorname{sc};m_{n}}(n+1)\geq(1+\delta_{n}^{1/2})(2-C\delta_{n}^{2})\geq
2+2^{1/2}\delta_{n},$

for sufficiently large $n$. Thus, letting $\widetilde{f}_{n}$ denote the
function $f_{n}$ from (8.1), for each $s\in[-\mathsf{T}_{n},\mathsf{T}_{n}]$
we have

$\displaystyle\partial_{s}^{2}\widetilde{f}_{n}(s)=-\displaystyle\frac{\sigma_{n}\mathsf{T_{n}}}{(2\mathsf{T}_{n}^{2}-2s^{2})^{1/2}}\bigg{(}1+\displaystyle\frac{s^{2}}{\mathsf{T}_{n}^{2}-s^{2}}\bigg{)}\cdot\gamma_{\operatorname{sc};m_{n}}(n+1)\leq-\displaystyle\frac{\sigma_{n}}{2^{1/2}}\cdot\gamma_{\operatorname{sc};m_{n}}(n+1)\leq-2^{1/2}-\delta_{n}.$

In particular, by (3.12) it follows for sufficiently large $n$ that

(8.4)
$\displaystyle\partial_{s}^{2}\widetilde{f}_{n}(s)\leq\partial_{s}^{2}h_{n}(s),\qquad\text{for
each $s\in[-\mathsf{T}_{n},\mathsf{T}_{n}]$},$

which will enable us to apply the gap monotonicity 5.1.

To implement this, condition on
$\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,n\rrbracket\times[-\mathsf{T}_{n},\mathsf{T}_{n}]\big{)}$ and let
$\bm{w}=\bm{\mathsf{x}}(\mathsf{T})\in\overline{\mathbb{W}}_{n}$. Identifying
$f_{n}$ with $f_{n}|_{[-\mathsf{T}_{n},\mathsf{T}_{n}]}$, the law of
$\bm{\mathsf{x}}|_{[-\mathsf{T}_{n},\mathsf{T}_{n}]}$ is then given by
$\mathsf{Q}_{f_{n}}^{\bm{u};\bm{w}}$. Sample non-intersecting Brownian bridges
$\breve{\bm{\mathsf{x}}}=\breve{\bm{\mathsf{x}}}^{n}=(\breve{\mathsf{x}}_{1},\breve{\mathsf{x}}_{2},\ldots,\breve{\mathsf{x}}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\mathsf{T}_{n},\mathsf{T}_{n}]\big{)}$
and
$\widetilde{\bm{\mathsf{x}}}=\widetilde{\bm{\mathsf{x}}}^{n}=(\widetilde{\mathsf{x}}_{1},\widetilde{\mathsf{x}}_{2},\ldots,\widetilde{\mathsf{x}}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\mathsf{T}_{n},\mathsf{T}_{n}]\big{)}$
from the measures $\mathsf{Q}_{h_{n}}^{\bm{u};\bm{w}}$ and
$\mathsf{Q}_{\tilde{f}_{n}}^{\bm{0}_{n};\bm{0}_{n}}$, respectively. Due to the
second bound in (3.12), the first part of 4.7 yields a coupling between
$\bm{\mathsf{x}}$ and $\breve{\bm{\mathsf{x}}}$ such that

(8.5)
$\displaystyle\big{|}\mathsf{x}_{j}(t)-\breve{\mathsf{x}}_{j}(t)\big{|}\leq\delta_{n},\qquad\text{for
each $(j,t)\in\llbracket
1,n\rrbracket\times[-\mathsf{T}_{n},\mathsf{T}_{n}]$}.$

Moreover, (8.4) and gap monotonicity 5.1 together yield a coupling between
$\breve{\bm{\mathsf{x}}}$ and $\widetilde{\bm{\mathsf{x}}}$ so that
$\breve{\mathsf{x}}_{j}(t)-\breve{\mathsf{x}}_{j+1}(t)\geq\widetilde{\mathsf{x}}_{j}(t)-\widetilde{\mathsf{x}}_{j+1}(t)$,
for each $t\in[-\mathsf{T}_{n},\mathsf{T}_{n}]$ and $j\in\llbracket
1,n-1\rrbracket$. Together with (8.5), this implies

(8.6)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\big{\\{}\mathsf{x}_{j}(t)-\mathsf{x}_{j+1}(t)\geq
2^{-1/2}r_{j}\\}\Bigg{]}\geq\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\big{\\{}\widetilde{\mathsf{x}}_{j}(t)-\widetilde{\mathsf{x}}_{j+1}(t)\geq
2^{-1/2}r_{j}-2\delta_{n}\big{\\}}\Bigg{]}.$

Moreover, 8.1, Definition 3.15, and the fact that
$\lim_{n\rightarrow\infty}\delta_{n}=0$ together imply that

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\Big{\\{}2^{1/2}\big{(}\widetilde{\mathsf{x}}_{j}(t)-\widetilde{\mathsf{x}}_{j+1}(t)\big{)}\geq
r_{j}-2^{3/2}\delta_{n}\Big{\\}}\Bigg{]}=\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\\{\mathfrak{a}_{j}-\mathfrak{a}_{j+1}\geq
r_{j}\\}\Bigg{]}.$

This and (8.6) together yield the lemma. ∎

####  8.2. Gap Upper Bound

In this section we establish the following upper bound for the gaps between
the bridges in $\bm{\mathsf{x}}$ satisfying 3.16, which (together with 8.2)
quickly implies the Airy gaps 3.18.

Figure 3.2. The proof of 8.3 is illustrated above.

######  Proposition 8.3.

Adopting the notation of 8.2 and also 3.17, we have

$\displaystyle\displaystyle\limsup_{n\rightarrow\infty}\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\Big{\\{}2^{1/2}\big{(}\mathsf{x}_{j}(t)-\mathsf{x}_{j+1}(t)\big{)}\geq
r_{k}\Big{\\}}\Bigg{]}\leq\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\\{\mathfrak{a}_{j}-\mathfrak{a}_{j+1}\geq
r_{j}\\}\Bigg{]}.$

###### Proof of 3.18.

This follows from 8.2 and 8.3. ∎

To establish 8.3, we will use 4.26, to which end we require the following
estimate on the constant $\sigma$ appearing there.

######  Lemma 8.4.

There exists a constant $C>1$ such that the following holds. Fix a real number
$\varepsilon\in(0,1)$; adopt the notation of 4.26; and assume that

(8.7)
$\displaystyle\nu(dy)=\textbf{\emph{1}}_{y\in[-2^{-7/6}(3\pi)^{2/3},0]}\cdot
2^{3/4}\pi^{-1}|y|^{1/2}dy.$

For any real number $t\in(0,2\varepsilon)$, we have
$|\sigma_{\nu;t}-2^{1/2}|\leq C\varepsilon$.

###### Proof.

Recall the real number $z_{0}=z_{0}(\nu;t)>0$ from (4.16), and denote
$\mathfrak{c}=2^{-7/6}(3\pi)^{2/3}$. Let us begin by verifying the
approximation $z_{0}\approx 2^{-1/2}t^{2}$. Changing variables from $y$ to
$-z_{0}w^{2}$ in the first integral in (4.16), we deduce

(8.8) $\displaystyle 2^{-3/4}\pi
t^{-1}z_{0}^{1/2}=z_{0}^{1/2}\displaystyle\int_{-\mathfrak{c}}^{0}\displaystyle\frac{|y|^{1/2}dy}{(y-z_{0})^{2}}=2\displaystyle\int_{0}^{(\mathfrak{c}/z_{0})^{1/2}}\displaystyle\frac{w^{2}dw}{(w^{2}+1)^{2}}\leq
2\displaystyle\int_{0}^{\infty}\displaystyle\frac{w^{2}dw}{(w^{2}+1)^{2}}=\displaystyle\frac{\pi}{2},$

from which it follows that $z_{0}\leq 2^{-1/2}t^{2}$. Inserting this into
(8.8), it follows that

$\displaystyle 2^{-3/4}\pi
t^{-1}z_{0}^{1/2}=\displaystyle\frac{\pi}{2}-2\displaystyle\int_{(\mathfrak{c}/z_{0})^{1/2}}^{\infty}\displaystyle\frac{w^{2}dw}{(w^{2}+1)^{2}}\geq\displaystyle\frac{\pi}{2}-\displaystyle\int_{(\mathfrak{c}/z_{0})^{1/2}}^{\infty}\frac{w^{-2}dw}{2}\geq\displaystyle\frac{\pi}{2}-\frac{z_{0}^{1/2}}{2}\geq\displaystyle\frac{\pi}{2}-\displaystyle\frac{t}{2},$

from which we deduce $z_{0}^{1/2}\geq 2^{-1/4}t-t^{2}/(2^{1/4}\pi)$ and thus

$\displaystyle
2^{-1/2}t^{2}-z_{0}=(2^{-1/4}t-z_{0}^{1/2})(2^{-1/4}t+z_{0}^{1/2})\leq\displaystyle\frac{t^{2}}{2^{1/4}\pi}\cdot
2^{3/4}t\leq t^{3},$

and so

(8.9) $\displaystyle 0\leq 2^{-1/2}t^{2}-z_{0}\leq t^{3}.$

Next, changing variables from $y$ to $-z_{0}w^{2}$ in the second integral in
(4.16) yields

$\displaystyle
2^{-3/4}\pi\sigma_{\nu;t}^{-3}=t^{3}\displaystyle\int_{-\mathfrak{c}}^{0}\displaystyle\frac{|y|^{1/2}dy}{(y-z_{0})^{3}}$
$\displaystyle=2t^{3}z_{0}^{-3/2}\displaystyle\int_{0}^{(\mathfrak{c}/z_{0})^{1/2}}\displaystyle\frac{w^{2}dw}{(w^{2}+1)^{3}}$
$\displaystyle=t^{3}z_{0}^{-3/2}\bigg{(}\displaystyle\frac{\pi}{8}-2\displaystyle\int_{(\mathfrak{c}/z_{0})^{1/2}}^{\infty}\displaystyle\frac{w^{2}dw}{(w^{2}+1)^{3}}\bigg{)}.$

Together with (8.9) and the fact

$\displaystyle\displaystyle\int_{(\mathfrak{c}/z_{0})^{1/2}}^{\infty}\displaystyle\frac{w^{2}dw}{(w^{2}+1)^{3}}\leq\displaystyle\int_{(\mathfrak{c}/z_{0})^{1/2}}^{\infty}w^{-4}dw\leq\displaystyle\frac{z_{0}^{3/2}}{3}\leq\frac{t^{3}}{3},$

it follows that

$\displaystyle\sigma_{\nu;t}^{3}=\displaystyle\frac{2^{9/4}z_{0}^{3/2}}{t^{3}+\mathcal{O}(t^{6})}=2^{3/2}+\mathcal{O}(t)=2^{3/2}+\mathcal{O}(\varepsilon),$

from which we deduce the lemma. ∎

We now establish 8.3 using 4.26 and gap monotonicity 5.1 (in a way broadly
analogous to the proof of 7.9).

###### Proof of 8.3.

We will prove 8.3 for $t\geq 0$; the case when $t\leq 0$ can be proven in the
same way by symmetry. Fix some $\varepsilon\in(0,1/4)$, condition on
$\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,n\rrbracket\times[-\varepsilon n^{1/3},\mathsf{R}]\big{)}$, and restrict to
the event $\mathscr{F}_{n}(-\varepsilon n^{1/3})$ of (3.13). Denote the
$n$-tuple $\bm{u}^{\prime}=\bm{\mathsf{x}}^{n}(-\varepsilon
n^{1/3})\in\overline{\mathbb{W}}_{n}$; the law of $\bm{\mathsf{x}}^{n}$ is
then given by $\mathsf{Q}_{f}^{\bm{u}^{\prime};\bm{v}}$. Further define the
process
$\bm{\mathsf{z}}^{n}=(\mathsf{z}_{1}^{n},\mathsf{z}_{2}^{n},\ldots,\mathsf{z}_{n}^{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[0,n^{2/3}\mathsf{R}+\varepsilon
n]\big{)}$ so that $\bm{\mathsf{z}}(s)$ is obtained by running Dyson Brownian
motion for time $s$ with initial data $n^{1/3}\cdot\bm{u}^{\prime}$. Then
define
$\bm{\mathsf{y}}^{n}=(\mathsf{y}_{1}^{n},\mathsf{y}_{2}^{n},\ldots,\mathsf{y}_{n}^{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\varepsilon n^{1/3},\mathsf{R}]\big{)}$
by scaling, so that
$\mathsf{y}_{j}^{n}(s)=n^{-1/3}\cdot\mathsf{z}_{j}^{n}(n^{2/3}s+\varepsilon
n)$ for each $(j,s)\in\llbracket 1,n\rrbracket\times[-\varepsilon
n^{1/3},\mathsf{R}]$. Denoting
$\bm{v}^{\prime}=\bm{\mathsf{y}}^{n}(\mathsf{R})\in\overline{\mathbb{W}}_{n}$,
observe from the second part of 4.17 with 4.4 that $\bm{y}^{n}$ has law
$\mathsf{Q}^{\bm{u}^{\prime};\bm{v}^{\prime}}$. See Figure 3.2.

Define the $n$-tuple $\bm{v}^{\prime\prime}\in\mathbb{W}_{n}$ by setting
$v_{j}^{\prime\prime}=v_{j}^{\prime}+(n-j)n$, for each $j\in\llbracket
1,n\rrbracket$. Then, sample non-intersecting Brownian bridges
$\widetilde{\bm{\mathsf{y}}}^{n}=(\widetilde{\mathsf{y}}_{1}^{n},\widetilde{\mathsf{y}}_{2}^{n},\ldots,\widetilde{\mathsf{y}}_{n}^{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-\varepsilon n^{1/3},\mathsf{R}]\big{)}$
from the measure $\mathsf{Q}^{\bm{u}^{\prime};\bm{v}^{\prime\prime}}$.
Applying the second part of 4.7 (with the
$(\bm{u},\bm{v},\widetilde{\bm{v}},B)$ there equal to the
$(\bm{u}^{\prime},\bm{v}^{\prime},\bm{v}^{\prime\prime},n^{2})$ here), yields
a coupling between $\mathsf{y}^{n}$ and $\widetilde{\mathsf{y}}^{n}$ such that
for each $j\in\llbracket 1,n\rrbracket$ we have

(8.10) $\displaystyle\displaystyle\max_{s\in[-\varepsilon
n^{1/3},n^{1/3}]}\big{|}\mathsf{y}_{j}^{n}(s)-\widetilde{\mathsf{y}}_{j}^{n}(s)\big{|}\leq\frac{(1+\varepsilon)n^{1/3}}{\mathsf{R}+\varepsilon
n^{1/3}}\cdot n^{2}\leq 2n^{-2},$

where in the last bound we used the facts that $\varepsilon\leq 1/2$ and
$\mathsf{R}=n^{20}$. Moreover, since $\bm{\mathsf{x}}^{n}$ has law
$\mathsf{Q}_{f}^{\bm{u}^{\prime};\bm{v}}$, applying gap monotonicity 5.1 to
the measures $\mathsf{Q}_{f}^{\bm{u}^{\prime};\bm{v}}$ and
$\mathsf{Q}^{\bm{u};\bm{v}^{\prime\prime}}$ (using the fact that
$v_{j}^{\prime\prime}-v_{j+1}^{\prime\prime}\geq n\geq v_{1}-v_{n}\geq
v_{j}-v_{j+1}$ for each $j\in\llbracket 1,n-1\rrbracket$, where in the third
bound we used 3.16), there is a coupling between $\bm{\mathsf{x}}$ and
$\widetilde{\bm{\mathsf{y}}}$ satisfying
$\mathsf{x}_{j}^{n}(s)-\mathsf{x}_{j+1}^{n}(s)\leq\widetilde{\mathsf{y}}_{j}^{n}(s)-\widetilde{\mathsf{y}}_{j+1}^{n}(s)$,
for each $(j,s)\in\llbracket 1,n-1\rrbracket\times[-\varepsilon
n^{1/3},\mathsf{R}]$. Together with (8.10), this yields a coupling between
$\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such that

(8.11)
$\displaystyle\mathsf{x}_{j}^{n}(t)-\mathsf{x}_{j+1}^{n}(t)\leq\mathsf{y}_{j}^{n}(t)-\mathsf{y}_{j+1}^{n}(t)+4n^{-2},\qquad\text{for
each $(j,t)\in\llbracket 1,n\rrbracket\times[-\varepsilon
n^{1/3},\mathsf{R}]$}.$

Next define the probability measure
$\nu=\nu_{n}=n^{-1}\sum_{j=1}^{n}\delta_{u_{j}^{\prime}/n}\in\mathscr{P}_{0}$.
Then (3.13) implies on the event $\mathscr{F}_{n}(-\varepsilon n^{1/3})$ that
$\nu_{n}$, translated by $-u_{n}^{\prime}/n$, converges weakly to the measure
$\nu$ from (8.7), as $n$ tends to $\infty$. Hence 4.26 (with the $t$ there
equal to $\varepsilon+n^{-1/3}t$ here. We also notice that for $n$ large
enough, $\varepsilon+n^{-1/3}t<2\varepsilon$), 4.27, and 4.4, yield a
uniformly bounded real number $\sigma_{n;t}>0$ such that

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\Big{\\{}\sigma_{n;t}\big{(}\mathsf{y}_{j}^{n}(t)-\mathsf{y}_{j+1}^{n}(t)\big{)}\geq
r_{j}\Big{\\}}\Bigg{]}$
$\displaystyle\quad=\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\Big{\\{}\sigma_{n;t}n^{-1/3}\big{(}\mathsf{z}_{t}^{n}(tn^{2/3}+\varepsilon
n)-\mathsf{z}_{j+1}^{n}(tn^{2/3}+\varepsilon
n)\big{)}\Big{\\}}\Bigg{]}=\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\\{\mathfrak{a}_{j}-\mathfrak{a}_{j+1}\geq
r_{j}\\}\Bigg{]}.$

Together with (8.11), this yields

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\Big{\\{}\sigma_{n;t}\big{(}\mathsf{x}_{j+1}^{n}(t)-\mathsf{x}_{j}^{n}(t)\big{)}\geq
r_{j}\Big{\\}}\Bigg{]}\leq\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k}\\{\mathfrak{a}_{j}-\mathfrak{a}_{j+1}\geq
r_{j}\\}\Bigg{]}.$

Since 8.4 yields a constant $C>1$ such that
$\sigma_{n;t}\in[1-C\varepsilon,1+C\varepsilon]$, this implies the lemma upon
letting $\varepsilon$ tend to $0$. ∎

###  9\. Airy Line Ensembles From Airy Point Processes

####  9.1. Proof of 3.19

In this section we establish 3.19, which will follow as a consequence of the
following proposition, to be established in Section 9.3 below. It states that
the edge statistics of a family of $N$ non-intersecting Brownian bridges on a
shorter interval $[-n^{1/3},n^{1/3}]$ (where $n$ is much smaller than $N$),
whose boundary data is close to the expected values of the Airy point process
(and whose lower boundary is not too irregular), converges to the Airy line
ensemble. See the left side of Figure 3.3 for a depiction.

######  Proposition 9.1.

Let $n\geq 1$ denote an integer, and set $N=n^{15}$. Let
$\bm{u}=\bm{u}^{n}\in\overline{\mathbb{W}}_{N}$ and
$\bm{v}=\bm{v}^{n}\in\overline{\mathbb{W}}_{N}$ denote two $n$-tuples such
that

(9.1)
$\displaystyle\big{|}u_{j}+2^{-1/2}n^{2/3}+2^{-7/6}(3\pi)^{2/3}j^{2/3}\big{|}+\big{|}v_{j}+2^{-1/2}n^{2/3}+2^{-7/6}(3\pi)^{2/3}j^{2/3}\big{|}\leq(\log
n)^{3}j^{-1/3},$

for each integer $j\in\llbracket 1,N\rrbracket$. Also let
$f=f_{n}:[-n^{1/3},n^{1/3}]\rightarrow\mathbb{R}$ denote a function such that

(9.2) $\displaystyle\displaystyle\sup_{|s|\leq
n^{1/3}}\big{|}f(s)-u_{N}\big{|}\leq n^{8}.$

Sample $N$ non-intersecting Brownian bridges
$\bm{\mathsf{x}}^{n}=(\mathsf{x}_{1}^{n},\mathsf{x}_{2}^{n},\ldots,\mathsf{x}_{N}^{n})\in\llbracket
1,N\rrbracket\times\mathcal{C}\big{(}[-n^{1/3},n^{1/3}]\big{)}$ from the
measure $\mathsf{Q}_{f}^{\bm{u};\bm{v}}$. Then,

$\displaystyle\bm{\mathsf{X}}^{n}=(\mathsf{X}_{1}^{n},\mathsf{X}_{2}^{n},\ldots,\mathsf{X}_{N}^{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-n^{1/3},n^{1/3}]\big{)},\qquad\text{where}\qquad\mathsf{X}_{j}^{n}(t)=2^{1/2}\cdot\mathsf{x}_{j}^{n}(t),$

converges to $\bm{\mathcal{R}}$, uniformly on compact subsets of
$\mathbb{Z}_{\geq 1}\times\mathbb{R}$, as $n$ tends to $\infty$.

Figure 3.3. Shown to the left is a depiction of 9.1; shown to the right is a
depiction of its proof.

###### Proof of 3.19.

Let $n\geq 1$ be an integer; set $N=N_{n}=n^{15}$ and
$\mathsf{T}=\mathsf{T}_{n}=n^{1/3}$; and abbreviate
$\mathcal{F}_{\operatorname{ext}}^{n}=\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,N\rrbracket\times[-\mathsf{T},\mathsf{T}]\big{)}$ (recall Definition 2.2).
Define the $\mathcal{F}_{\operatorname{ext}}^{n}$-measurable random variables

(9.3) $\displaystyle\begin{aligned}
\xi_{n}&=(2\mathsf{T})^{-1}\cdot\big{(}\mathcal{L}_{N}(\mathsf{T})-\mathcal{L}_{N}(-\mathsf{T})\big{)};\\\
\zeta_{n}&=\displaystyle\frac{1}{2}\cdot\big{(}\mathcal{L}_{N}(-\mathsf{T})+\mathcal{L}_{N}(\mathsf{T})\big{)}+2^{-1/2}n^{2/3}+2^{-7/6}(3\pi)^{2/3}N^{2/3}.\end{aligned}$

Also define the family of non-intersecting curves
$\bm{\mathsf{L}}^{n}=(\mathsf{L}_{1}^{n},\mathsf{L}_{2}^{n},\ldots,\mathsf{L}_{N}^{n})\in\llbracket
1,N\rrbracket\times\mathcal{C}\big{(}[-\mathsf{T},\mathsf{T}]\big{)}$; the
$N$-tuples $\bm{u}=\bm{u}^{n}\in\overline{\mathbb{W}}_{N}$ and
$\bm{v}=\bm{v}^{n}\in\overline{\mathbb{W}}_{N}$; and the function
$f=f_{n}:[-\mathsf{T},\mathsf{T}]\rightarrow\mathbb{R}$, by setting

(9.4)
$\displaystyle\mathsf{L}_{j}^{n}(s)=\mathcal{L}_{j}(s)-\xi_{n}s-\zeta_{n};\qquad
u_{j}=\mathsf{L}_{j}^{n}(-\mathsf{T});\qquad
v_{j}=\mathsf{L}_{j}^{n}(\mathsf{T});\qquad f(s)=\mathsf{L}_{N+1}^{n}(s),$

for each integer $j\geq 1$ and real number $s\in\mathbb{R}$ (so these
quantities are still defined for $(j,s)\notin\llbracket
1,N\rrbracket\times[-\mathsf{T},\mathsf{T}]$). This in particular guarantees
that

(9.5)
$\displaystyle\mathsf{L}_{N}^{n}(-\mathsf{T})=-2^{-1/2}n^{2/3}-2^{-7/6}(3\pi)^{2/3}N^{2/3}=\mathsf{L}_{N}^{n}(\mathsf{T}).$

Moreover, conditional on $\mathcal{F}_{\operatorname{ext}}^{n}$, the ensemble
$\bm{\mathsf{L}}^{n}$ is a family of $N$ non-intersecting Brownian bridges
sampled from the measure $\mathsf{Q}_{f}^{\bm{u};\bm{v}}$ (by 4.3). To apply
9.1 to this ensemble, we must restrict to an event on which its conditions
(9.1) and (9.2) hold.

So, define the $\mathcal{F}_{\operatorname{ext}}^{n}$-measurable event
$\mathscr{E}(n)=\mathscr{E}_{1}(n)\cap\mathscr{E}_{2}(n)\cap\mathscr{E}_{3}(n)$,
where

$\displaystyle\mathscr{E}_{1}(n)$ $\displaystyle=\bigcap_{j\in\llbracket
1,N\rrbracket}\Big{\\{}\big{|}u_{j}+2^{-1/2}n^{2/3}+2^{-7/6}(3\pi)^{2/3}j^{2/3}\big{|}\leq(\log
n)^{5/2}j^{-1/3}\Big{\\}};$ $\displaystyle\mathscr{E}_{2}(n)$
$\displaystyle=\bigcap_{j\in\llbracket
1,N\rrbracket}\Big{\\{}\big{|}v_{j}+2^{-1/2}n^{2/3}+2^{-7/6}(3\pi)^{2/3}\big{|}\leq(\log
n)^{5/2}j^{-1/3}\Big{\\}};$ $\displaystyle\mathscr{E}_{3}(n)$
$\displaystyle=\Bigg{\\{}\displaystyle\sup_{|t|\leq\mathsf{T}}\big{|}f(t)-u_{N}\big{|}\leq
n^{8}\Bigg{\\}}.$

Let us show that
$\lim_{n\rightarrow\infty}\mathbb{P}\big{[}\mathscr{E}(n)\big{]}=1$; it
suffices by a union bound to show that

(9.6)
$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\big{[}\mathscr{E}_{1}(n)^{\complement}\big{]}=0;\qquad\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\big{[}\mathscr{E}_{2}(n)^{\complement}\big{]}=0;\qquad\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\big{[}\mathscr{E}_{3}(n)^{\complement}\big{]}=0.$

We begin by confirming the first bound in (9.6). Observe since the gaps
$\big{(}\mathcal{L}_{j}(-\mathsf{T})-\mathcal{L}_{j+1}(-\mathsf{T})\big{)}_{j\geq
1}$ of $\bm{\mathcal{L}}$ have the same law as those
$(\mathfrak{a}_{j}-\mathfrak{a}_{j+1})_{j\geq 1}$ of the Airy point process
$\bm{\mathfrak{a}}$, so do those of the gaps
$\big{(}\mathsf{L}_{j}^{n}(-\mathsf{T})\big{)}_{j\geq 1}$ of
$\bm{\mathsf{L}}^{n}$ (by (9.4)). Next, by 4.34, we have

$\displaystyle\mathbb{P}\Big{[}\big{|}\mathfrak{a}_{N}+2^{-7/6}(3\pi)^{2/3}N^{2/3}\big{|}\leq(\log
n)^{2}N^{-1/3}\Big{]}\geq 1-c_{1}^{-1}e^{-c_{1}(\log n)^{2}},$

for some constant $c_{1}>0$. Together with (9.5) and the fact that the law of
the gaps of $\bm{\mathsf{L}}^{n}$ coincides with that of $\bm{\mathfrak{a}}$,
this gives a coupling between $\bm{\mathsf{L}}^{n}$ and $\bm{\mathfrak{a}}$
such that

$\displaystyle\mathbb{P}\Bigg{[}\displaystyle\max_{j\in\llbracket
1,N\rrbracket}\big{|}\mathfrak{a}_{j}-\mathsf{L}_{j}^{n}(-\mathsf{T})-2^{-1/2}n^{2/3}\big{|}\leq(\log
n)^{2}N^{-1/3}\Bigg{]}\geq 1-c_{1}^{-1}e^{-c_{1}(\log n)^{2}}.$

Together with the fact (from 4.34 with $\sigma=1$ and
$\mathfrak{a}_{j}=2^{-1/2}\mathcal{R}^{(\sigma)}_{j}$ and a union bound) that

$\displaystyle\mathbb{P}\Bigg{[}\displaystyle\max_{j\in\llbracket
1,N\rrbracket}\big{|}\mathfrak{a}_{j}+2^{-7/6}(3\pi)^{2/3}j^{2/3}\big{|}\leq(\log
n)^{2}j^{-1/3}\Bigg{]}\geq 1-c_{2}^{-1}e^{-c_{2}(\log n)^{2}},$

for some constant $c_{2}\in(0,c_{1}]$, this yields

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j\in\llbracket
1,N\rrbracket}\Big{\\{}\big{|}\mathsf{L}_{j}^{n}(-\mathsf{T})+2^{-1/2}n^{2/3}+2^{-7/6}(3\pi)^{2/3}j^{2/3}\big{|}\leq(\log
n)^{2}($ $\displaystyle j^{-1/3}+N^{-1/3})\Big{\\}}\Bigg{]}$
$\displaystyle\geq 1-2c_{2}^{-1}e^{-c_{2}(\log n)^{2}},$

which implies that $\mathbb{P}\big{[}\mathscr{E}_{1}(n)\big{]}\geq
1-2c_{2}^{-1}e^{-c_{2}(\log n)^{2}}$ (as
$u_{j}=\mathsf{L}_{j}^{n}(-\mathsf{T})$ and $(\log
n)^{2}(j^{-1/3}+N^{-1/3})\leq 2(\log n)^{2}j^{-1/3}\leq(\log
n)^{5/2}j^{-1/3}$, for sufficiently large $n$). This verifies the first
statement in (9.6); the proof of the second is entirely analogous and is thus
omitted.

We next confirm the third statement in (9.6). Let $\mathfrak{c}$,
$\mathfrak{C}_{1}$, and $\mathfrak{C}_{2}$ denote the constants $c$, $C_{1}$,
and $C_{2}$ from 3.8 at $(A,B,D,R)=(2,2,10,\mathfrak{C}_{2})$. By 3.4 (with
the $(n,B,\vartheta)$ there equal to
$(N,2\mathfrak{C}_{2},\mathfrak{C}_{1}^{-1})$ here), there exists a non-
increasing sequence $\bm{\delta}=(\delta_{1},\delta_{2},\ldots)$ of real
numbers with $\lim_{j\rightarrow\infty}\delta_{j}=0$ such that
$\delta_{j}\geq(\log j+1)^{-1}$ and

(9.7)
$\displaystyle\mathbb{P}\Big{[}\textbf{TOP}\big{(}[-2\mathfrak{C}_{2}N^{1/3},2\mathfrak{C}_{2}N^{1/3}];\mathfrak{C}_{1}^{-1}N^{2/3}\big{)}\Big{]}\geq
1-\delta_{n}.$

Hence, for sufficiently large $n$, we have

(9.8) $\displaystyle\begin{aligned}
\mathbb{P}&\Big{[}\textbf{REG}_{N,N+1}\big{(}[-2N^{1/3},2N^{1/3}];12;N^{-10};2N+2\big{)}\cap\textbf{GAP}_{N+1}\big{(}[-2N^{1/3},2N^{1/3}];\mathfrak{C}_{2}\big{)}\Big{]}\\\
&\geq\mathbb{P}\Big{[}\textbf{SCL}_{N}(2;2;10;\mathfrak{C}_{2})\Big{]}\geq\mathbb{P}\Big{[}\textbf{TOP}\big{(}[-\mathfrak{C}_{2}N^{1/3},\mathfrak{C}_{2}N^{1/3}];\mathfrak{C}_{1}^{-1}N^{2/3}\big{)}\Big{]}-\mathfrak{c}^{-1}e^{-\mathfrak{c}(\log
n)^{2}}\geq 1-2\delta_{n},\end{aligned}$

where here we set
$\textbf{REG}_{N,N+1}=\textbf{REG}_{N}\cap\textbf{REG}_{N+1}$. Here, to obtain
the first bound we used the inclusion of events (recall Definition 3.7)

$\displaystyle\textbf{SCL}_{N}(2;2;10;\mathfrak{C}_{2})$
$\displaystyle\subseteq\textbf{REG}_{N,N+1}\big{(}[-2N^{1/3},2N^{1/3}];12;N^{-10};2N+2)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\cap\textbf{GAP}_{N+1}\big{(}[-2N^{1/3},2N^{1/3}];\mathfrak{C}_{2}\big{)};$

to obtain the second we used 3.8; and the third follows we used (9.7) and the
fact that $\delta_{n}\geq(\log n+1)^{-1}$. Observe on the event in the left
side of (9.8) that

$\displaystyle\displaystyle\sup_{|t|\leq\mathsf{T}}\big{|}f(t)-u_{N}\big{|}$
$\displaystyle\leq\displaystyle\sup_{|t|\leq\mathsf{T}}\big{|}\mathsf{L}_{N+1}^{n}(t)-\mathsf{L}_{N+1}^{n}(-\mathsf{T})\big{|}+\mathsf{L}_{N}(-\mathsf{T})-\mathsf{L}_{N+1}(-\mathsf{T})$
$\displaystyle\leq\displaystyle\sup_{|t|\leq\mathsf{T}}\big{|}\mathcal{L}_{N+1}(t)-\mathcal{L}_{N+1}(-\mathsf{T})\big{|}+(t+\mathsf{T})|\xi_{n}|+2\mathfrak{C}_{2}N^{-1/3}(\log
n)^{30}$ $\displaystyle\leq
4\big{(}4(N+1)\mathsf{T}\big{)}^{1/2}+24\mathsf{T}+n^{-10}+2\mathfrak{C}_{2}N^{-1/3}(\log
n)^{30}+2\mathsf{T}|\xi_{n}|$ $\displaystyle\leq
10n^{1/6}N^{1/2}+\big{|}\mathcal{L}_{N}(\mathsf{T})-\mathcal{L}_{N}(-\mathsf{T})\big{|}\leq
20n^{1/6}N^{1/2}\leq n^{8}.$

Here, in the first statement we used the definitions (9.4) of $\bm{u}$ and
$f$; in the second we used Definition 3.5 for the $\textbf{GAP}_{N+1}$ event,
the fact that
$\mathfrak{C}_{2}\big{(}(N+1)^{2/3}-N^{2/3}\big{)}+N^{-1/3}\big{(}\log(N+1)\big{)}^{25}\leq\mathfrak{C}_{2}N^{-1/3}(\log
n)^{30}$ for sufficiently large $n$, and the definition (9.4) of
$\bm{\mathsf{L}}^{n}$; in the third we used Definition 3.6 for the
$\textbf{REG}_{N}$ event (with the fact that
$\mathsf{T}+t\leq\mathsf{T}+|t|\leq 2\mathsf{T}$ for $|t|\leq\mathsf{T}$); in
the fourth we used the definition (9.3) of $\xi_{n}$ and the facts that
$\mathsf{T}=n^{1/3}$, that $N=n^{15}$, and that $n$ is sufficiently large; and
in the fifth and sixth we again used Definition 3.6 for the
$\textbf{REG}_{N+1}$ event with the fact that $n$ is sufficiently large (and
that $N=n^{15}$). Hence, $\sup_{|t|\leq\mathsf{T}}\big{|}f(t)-u_{N}\big{|}\leq
n^{8}$ holds in the event on the left side of (9.8). Together with (9.8) and
the fact that $\lim_{j\rightarrow\infty}\delta_{j}=0$, this yields the third
statement of (9.6).

Hence, (9.6) holds and thus
$\lim_{n\rightarrow\infty}\mathbb{P}[\mathscr{E}]=1$. Now let us condition on
$\mathcal{F}_{\operatorname{ext}}^{n}$ and restrict to the event
$\mathscr{E}$. Then apply 9.1, with the $\bm{\mathsf{x}}^{n}$ there equal to
$\bm{\mathsf{L}}^{n}$ here; observe that its condition (9.1) is verified by
$\mathscr{E}_{1}\cap\mathscr{E}_{2}$ and (9.2) by $\mathscr{E}_{3}$. Thus,
this proposition implies that the conditional law of $\bm{\mathsf{L}}^{n}$ on
$\mathscr{E}$ converges as $n$ tends to $\infty$ to
$2^{-1/2}\cdot\bm{\mathcal{R}}$, uniformly on compact subsets of
$\mathbb{Z}_{\geq 1}\times\mathbb{R}$.

This, together with the fact (from (9.4)) that
$\mathsf{L}_{j}^{n}(t)=\mathcal{L}_{j}(t)-\xi_{n}t-\zeta_{n}$ and the
tightness of the random variables
$\big{\\{}\mathcal{L}_{1}(-1),\mathcal{L}_{1}(1),\mathcal{R}_{1}(-1),\mathcal{R}_{1}(1)\big{\\}}$,
implies that the pair of random variables $(\xi_{n},\zeta_{n})$ is also tight;
hence, there exists a sequence $n_{1}<n_{2}<\cdots$ of integers such that
$(\xi_{n_{j}},\zeta_{n_{j}})$ converges to a pair of real-valued random
variables $(\mathfrak{l},\mathfrak{c})$, as $j$ tends to $\infty$. Applying
the convergence of $\bm{\mathsf{L}}^{n}$ to $2^{-1/2}\cdot\bm{\mathcal{R}}$,
it follows that the ensemble

(9.9) $\displaystyle\big{(}\mathcal{L}_{j}(t)\big{)}_{(j,t)\in\llbracket
1,N_{n_{j}}\rrbracket\times[-\mathsf{T}_{n_{j}},\mathsf{T}_{n_{j}}]}\qquad\text{converges
to}\qquad\big{(}2^{-1/2}\cdot\mathcal{R}_{j}(t)+\mathfrak{l}t+\mathfrak{c}\big{)}_{(j,t)\in\mathbb{Z}_{\geq
1}\times\mathbb{R}},$

uniformly on compact subsets of $\mathbb{Z}_{>0}\times\mathbb{R}$, as $j$
tends to $\infty$. Here, the pair $(\mathfrak{l},\mathfrak{c})$ is independent
from $\bm{\mathcal{R}}$, since by (9.3) the pair $(\xi_{n},\zeta_{n})$ is
measurable with respect to $\mathcal{F}_{\operatorname{ext}}^{n}$, which was
conditioned on in the convergence of $\bm{\mathsf{L}}^{n}$ to
$2^{-1/2}\cdot\bm{\mathcal{R}}$. The theorem then follows from (9.9), together
with the fact that the left side of (9.9) converges to $\bm{\mathcal{L}}$ as
$j$ tends to $\infty$. ∎

####  9.2. Approximate Parabolicity of Paths

To establish 9.1, we will use the following lemma indicating that the paths in
$\bm{\mathsf{x}}^{n}$ closely approximate parabolas; see the left side of
Figure 3.4 for a depiction.

######  Lemma 9.2.

Adopt the notation and assumptions of 9.1. There exists a constant $c>1$ such
that, for any integer $k\in\llbracket n^{1/6},n^{1/5}\rrbracket$, we have

$\displaystyle\mathbb{P}\Bigg{[}\displaystyle\sup_{|t|\leq
n^{1/3}}\big{|}\mathsf{x}_{k}^{n}(t)+2^{-1/2}t^{2}+2^{-7/6}(3\pi)^{2/3}k^{2/3}\big{|}\geq
k^{-1/30}\Bigg{]}\leq c^{-1}e^{-c(\log n)^{2}}.$

Figure 3.4. Shown to the left is a depiction of 9.2, indicating that
$\mathsf{x}_{k}$ is close to the orange parabola. Shown to the right is a
depiction of its proof.

###### Proof.

Throughout this proof, we abbreviate $\bm{\mathsf{x}}=\bm{\mathsf{x}}^{n}$ and
$\mathsf{x}_{j}=\mathsf{x}_{j}^{n}$ for each integer $j\in\llbracket
1,N\rrbracket$; we also set $\mathsf{T}=n^{1/3}$. We establish the lemma by
bounding $\bm{\mathsf{x}}$ between two parabolic Airy line ensembles with
approximately equal parameters. To this end, define the line ensembles
$\bm{\mathsf{y}}^{\prime}=(\mathsf{y}_{1}^{\prime},\mathsf{y}_{2}^{\prime},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ and
$\bm{\mathsf{y}}^{\prime\prime}=(\mathsf{y}_{1}^{\prime\prime},\mathsf{y}_{2}^{\prime\prime},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ by

(9.10)
$\displaystyle\bm{\mathsf{y}}^{\prime}=2^{-1/2}\cdot\bm{\mathcal{R}}^{(\sigma^{\prime})},\quad\text{and}\quad\bm{\mathsf{y}}^{\prime\prime}=2^{-1/2}\cdot\bm{\mathcal{R}}^{(\sigma^{\prime\prime})},\qquad\text{where}\quad\sigma^{\prime}=1-n^{-1},\quad\text{and}\quad\sigma^{\prime\prime}=1+n^{-1}.$

where we recall the rescaled parabolic Airy line ensemble
$\bm{\mathcal{R}}^{(\sigma)}$ from (2.3); we also set
$\bm{\mathsf{y}}_{j}^{\prime}=\bm{\mathsf{y}}_{j}^{\prime\prime}=\infty$ if
$j\leq 0$. Then, $\bm{\mathsf{y}}^{\prime}$ and $\bm{\mathsf{y}}^{\prime}$
both satisfy the Brownian Gibbs property by 2.5. We further define the line
ensembles
$\bm{\mathsf{x}}^{\prime}=(\mathsf{x}_{1}^{\prime},\mathsf{x}_{2}^{\prime},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ and
$\bm{\mathsf{x}}^{\prime\prime}=(\mathsf{x}_{1}^{\prime\prime},\mathsf{x}_{2}^{\prime\prime},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ by shifting the indices of
$\bm{\mathsf{y}}^{\prime}$ and $\bm{\mathsf{y}}^{\prime\prime}$ respectively,
namely, by setting for any $j\in\mathbb{Z}$

(9.11)
$\displaystyle\mathsf{x}_{j}^{\prime}=\mathsf{y}_{j+n_{0}}^{\prime}-n^{-1/4},\quad\text{and}\quad\mathsf{x}_{j}^{\prime\prime}=\mathsf{y}_{j-n_{0}}^{\prime\prime}+n^{-1/4},\qquad\text{where}\quad
n_{0}=\lfloor n^{1/50}\rfloor.$

Then $\bm{\mathsf{x}}^{\prime}$ and $\bm{\mathsf{x}}^{\prime\prime}$ satisfy
the Brownian Gibbs property, since $\bm{\mathsf{y}}^{\prime}$ and
$\bm{\mathsf{y}}^{\prime\prime}$ do. We will show that it is with high
probability possible to couple $\bm{\mathsf{x}}$ to lie between
$\bm{\mathsf{x}}^{\prime}$ and $\bm{\mathsf{x}}^{\prime\prime}$. See the right
side of Figure 3.4.

To this end, we define the event
$\mathscr{E}=\mathscr{E}_{1}\cap\mathscr{E}_{2}$, where
$\mathscr{E}_{1}=\mathscr{E}_{1}^{\prime}\cap\mathscr{E}_{1}^{\prime\prime}$
and
$\mathscr{E}_{2}=\mathscr{E}_{2}^{\prime}\cap\mathscr{E}_{2}^{\prime\prime}$.
Here,

(9.12) $\displaystyle\begin{aligned}
\mathscr{E}_{1}^{\prime}&=\bigcap_{j=1}^{N}\Bigg{\\{}\displaystyle\sup_{t\in\\{-\mathsf{T},\mathsf{T}\\}}\big{|}\mathsf{x}_{j}^{\prime}(t)+\mathfrak{p}(j+n_{0};t;\sigma^{\prime})+n^{-1/4}\big{|}\leq(\log
n)^{3}j^{-1/3}\Bigg{\\}};\\\
\mathscr{E}_{1}^{\prime\prime}&=\bigcap_{j=n_{0}+1}^{N}\Bigg{\\{}\displaystyle\sup_{t\in\\{-\mathsf{T},\mathsf{T}\\}}\big{|}\mathsf{x}_{j}^{\prime\prime}(t)+\mathfrak{p}(j-n_{0};t;\sigma^{\prime\prime})-n^{-1/4}\big{|}\leq(\log
n)^{3}j^{-1/3}\Bigg{\\}};\\\
\mathscr{E}_{2}^{\prime}&=\Bigg{\\{}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\mathsf{x}_{N+1}^{\prime}(t)+\mathfrak{p}(N+n_{0}+1;t;\sigma^{\prime})+n^{-1/4}\big{|}\leq(\log
n)^{3}N^{-1/3}\Bigg{\\}};\\\
\mathscr{E}_{2}^{\prime\prime}&=\Bigg{\\{}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\mathsf{x}_{N+1}^{\prime\prime}(t)+\mathfrak{p}(N-n_{0}+1;t;\sigma^{\prime\prime})-n^{-1/4}\big{|}\leq(\log
n)^{3}N^{-1/3}\Bigg{\\}}.\end{aligned}$

where we have denoted

(9.13)
$\displaystyle\mathfrak{p}(j;t;\sigma)=2^{-1/2}\sigma^{3}t^{2}+2^{-7/6}(3\pi)^{2/3}\sigma^{-1}j^{2/3}.$

Applying the definitions (9.11) and (9.10) of $\bm{\mathsf{x}}$ and
$\bm{\mathsf{y}}$ in terms of rescaled parabolic Airy line ensembles; the
concentration estimate 4.34 for the latter (with the fact that $N-n_{0}\geq
N/2$ for sufficiently large $n$); and a union bound yields a constant
$c_{1}>0$ such that

(9.14)
$\displaystyle\max\Big{\\{}\mathbb{P}\big{[}\mathscr{E}_{1}^{\prime\complement}\big{]},\mathbb{P}\big{[}\mathscr{E}_{1}^{\prime\prime\complement}\big{]},\mathbb{P}\big{[}\mathscr{E}_{2}^{\prime\complement}\big{]},\mathbb{P}\big{[}\mathscr{E}_{2}^{\prime\prime\complement}\big{]}\Big{\\}}\leq(4c_{1})^{-1}e^{-c_{1}(\log
n)^{2}},\qquad\text{so}\qquad\mathbb{P}[\mathscr{E}^{\complement}]\leq
c_{1}^{-1}e^{-c_{1}(\log n)^{2}}.$

Now condition on the curves $\big{(}\mathsf{x}_{j}^{\prime}(t)\big{)}$ and
$\big{(}\mathsf{x}_{j}^{\prime\prime}(t)\big{)}$ for $(j,t)\notin\llbracket
1,N\rrbracket\times(-\mathsf{T},\mathsf{T})$, and restrict to the event
$\mathscr{E}$. We claim that

(9.15) $\displaystyle\begin{aligned}
&\mathsf{x}_{j}^{\prime}(t)\leq\mathsf{x}_{j}(t)\leq\mathsf{x}_{j}^{\prime\prime}(t),\qquad\qquad\text{for
each $(j,t)\in\llbracket 1,N\rrbracket\times\\{-\mathsf{T},\mathsf{T}\\}$};\\\
&\mathsf{x}_{N+1}^{\prime}(t)\leq
f(t)\leq\mathsf{x}_{N+1}^{\prime\prime}(t),\qquad\text{for each
$t\in[-\mathsf{T},\mathsf{T}]$}.\end{aligned}$

To this end, observe for any $(j,t)\in\llbracket
1,N\rrbracket\times\\{-\mathsf{T},\mathsf{T}\\}$ and sufficiently large $n$
that

$\displaystyle\mathsf{x}_{j}(t)-\mathsf{x}_{j}^{\prime}(t)$
$\displaystyle\geq\big{(}-(\log
n)^{3}j^{-1/3}-2^{-1/2}n^{2/3}-2^{-7/6}(3\pi)^{2/3}j^{2/3}\big{)}$
$\displaystyle\qquad-\big{(}(\log n)^{3}j^{-1/3}-2^{-1/2}\sigma^{\prime
3}n^{2/3}-2^{-7/6}(3\pi)^{2/3}\sigma^{\prime-1}(j+n_{0})^{2/3}-n^{-1/4}\big{)}$
$\displaystyle=2^{-7/6}(3\pi)^{2/3}\big{(}(j+n_{0})^{2/3}-j^{2/3}\big{)}-2(\log
n)^{3}j^{-1/3}$ $\displaystyle\qquad+2^{-1/2}(\sigma^{\prime
3}-1)n^{2/3}+2^{-7/6}(3\pi)^{2/3}(\sigma^{\prime-1}-1)(j+n_{0})^{2/3}+n^{-1/4}$
$\displaystyle\geq n_{0}^{2/3}j^{-1/3}-2(\log
n)^{3}j^{-1/3}-2^{3/2}n^{-1/3}+n^{-1/4}\geq 0,$

where in the first statement we used (9.1), the fact that we are restricting
to the event $\mathscr{E}_{1}^{\prime}$ from (9.12), and the definition (9.13)
of $\mathfrak{p}$; in the second we performed the subtraction; in the third we
used the facts that $\sigma^{\prime 3}-1\geq-4n^{-1}$ and
$\sigma^{\prime-1}-1\geq n^{-1}\geq 0$ (by the definition (9.10) of
$\sigma^{\prime}=1-n^{-1}$), that $(j+n_{0})^{2/3}-j^{2/3}\geq
n_{0}^{2/3}/(2j^{1/3})$, and that $2^{-7/6}(3\pi)^{2/3}\geq 2$; and in the
fourth we used the definition (9.11) of $n_{0}$ and the fact that $n$ is
sufficiently large. This verifies the first bound in the first statement of
(9.15); the proof of the second part is entirely analogous (upon taking into
account the fact that $\mathsf{x}_{j}^{\prime\prime}=\infty$ for
$j\in\llbracket 1,n_{0}\rrbracket$) and is therefore omitted.

To verify the second statement in (9.15), observe for any
$t\in[-\mathsf{T},\mathsf{T}]$ that

$\displaystyle f(t$ $\displaystyle)-\mathsf{x}_{N+1}^{\prime}(t)$
$\displaystyle\geq\big{(}-2^{-1/2}n^{2/3}-2^{-7/6}(3\pi)^{2/3}N^{2/3}-(\log
n)^{3}N^{-1/3}-n^{8}\big{)}$ $\displaystyle\quad-\big{(}(\log
n)^{3}(N+n_{0}+1)^{-1/3}-2^{-1/2}\sigma^{\prime
3}t^{2}-2^{-7/6}(3\pi)^{2/3}\sigma^{\prime-1}(N+n_{0}+1)^{2/3}-n^{-1/4}\big{)}$
$\displaystyle\geq
2^{-7/6}(3\pi)^{2/3}\big{(}(1+n^{-1})(N+n_{0}+1)^{2/3}-N^{2/3}\big{)}-n^{8}-2^{-1/2}n^{2/3}-2(\log
n)^{3}N^{-1/3}$ $\displaystyle\geq n^{-1}N^{2/3}-2n^{8}\geq n^{9}-2n^{8}\geq
0,$

where in the first inequality we used (9.2), (9.1) (to bound $u_{N}$), the
fact that we are restricting to the event $\mathscr{E}_{2}^{\prime}$ from
(9.12), and the definition (9.13) of $\mathfrak{p}$; in the second we used the
fact that $\sigma^{\prime-1}\geq 1+n^{-1}$ (by the definition (9.11) of
$\sigma^{\prime}=1-n^{-1}$); in the third we used the facts that
$2^{-7/6}(3\pi)^{2/3}\geq 1$, that $(1+n^{-1})(N+n_{0}+1)^{2/3}\geq
N^{2/3}+n^{-1}N^{2/3}$, and that $n$ is sufficiently large; in the fourth we
used the fact that $N=n^{15}$; and in the fifth we used the fact that $n\geq
2$. This verifies the first part of the second bound in (9.15); the proof of
the second part is entirely analogous and is therefore omitted.

Thus, (9.15) holds. Denote the four $N$-tuples
$\bm{u}^{\prime}=\bm{\mathsf{x}}^{\prime}_{\llbracket
1,N\rrbracket}(-\mathsf{T})\in\overline{\mathbb{W}}_{n}$,
$\bm{v}^{\prime}=\bm{\mathsf{x}}_{\llbracket
1,N\rrbracket}^{\prime}(\mathsf{T})\in\overline{\mathbb{W}}_{n}$,
$\bm{u}^{\prime\prime}=\bm{\mathsf{x}}^{\prime\prime}_{\llbracket
1,N\rrbracket}(-\mathsf{T})\in\overline{\mathbb{W}}_{n}$, and
$\bm{v}^{\prime\prime}=\bm{\mathsf{x}}^{\prime\prime}_{\llbracket
1,N\rrbracket}(\mathsf{T})\in\overline{\mathbb{W}}_{n}$. Then, the laws of
$\big{(}\mathsf{x}_{j}^{\prime}(s)\big{)}$ and
$\big{(}\mathsf{x}_{j}^{\prime\prime}(s)\big{)}$ for $(j,s)\in\llbracket
1,N\rrbracket\times[-\mathsf{T},\mathsf{T}]$ are given by
$\mathsf{Q}_{\mathsf{x}_{N+1}^{\prime}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$ and
$\mathsf{Q}_{\mathsf{x}_{N+1}^{\prime\prime}}^{\bm{u}^{\prime\prime};\bm{v}^{\prime\prime}}$,
respectively. Thus, by (9.15) and 4.6, we may couple $\bm{\mathsf{x}}$,
$\bm{\mathsf{x}}^{\prime}$, and $\bm{\mathsf{x}}^{\prime\prime}$ so that

(9.16)
$\displaystyle\mathsf{x}_{j}^{\prime}(s)\leq\mathsf{x}_{j}(s)\leq\mathsf{x}_{j}^{\prime\prime}(s),\qquad\text{for
each $(j,s)\in\llbracket 1,N\rrbracket\times[-\mathsf{T},\mathsf{T}]$}.$

Now fix an integer $k\in\llbracket n^{1/6},n^{1/5}\rrbracket$, and define the
event
$\mathscr{E}_{3}=\mathscr{E}_{3}^{\prime}\cap\mathscr{E}_{3}^{\prime\prime}$,
where

$\displaystyle\mathscr{E}_{3}^{\prime}$
$\displaystyle=\Bigg{\\{}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\mathsf{x}_{k}^{\prime}(t)+2^{-1/2}t^{2}+2^{-7/6}(3\pi)^{2/3}k^{2/3}\big{|}\leq
k^{-1/30}\Bigg{\\}};$ $\displaystyle\mathscr{E}_{3}^{\prime\prime}$
$\displaystyle=\Bigg{\\{}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\mathsf{x}_{k}^{\prime\prime}(t)+2^{-1/2}t^{2}+2^{-7/6}(3\pi)^{2/3}k^{2/3}\big{|}\leq
k^{-1/30}\Bigg{\\}}.$

We claim that

(9.17)
$\displaystyle\max\Big{\\{}\mathbb{P}\big{[}\mathscr{E}_{3}^{\prime\complement}\big{]},\mathbb{P}\big{[}\mathscr{E}_{3}^{\prime\prime\complement}\big{]}\Big{\\}}\leq
c_{1}^{-1}e^{-c_{1}(\log
n)^{2}},\quad\text{so}\quad\mathbb{P}\big{[}\mathscr{E}_{3}^{\complement}\big{]}\leq
2c_{1}^{-1}e^{-c_{1}(\log n)^{2}}.$

Together with (9.14), (9.17) would imply that
$\mathbb{P}\big{[}(\mathscr{E}\cap\mathscr{E}_{3})^{\complement}\big{]}\leq
3c_{1}^{-1}e^{-c_{1}(\log n)^{2}}$. Since
$\mathsf{x}_{k}(t)+2^{-1/2}t^{2}+2^{-7/6}(3\pi)^{2/3}k^{2/3}\big{|}\leq
k^{-1/30}$ holds on $\mathscr{E}\cap\mathscr{E}_{3}$ by the definitions of
$\mathscr{E}_{3}^{\prime}$ and $\mathscr{E}_{3}^{\prime\prime}$ and (9.16),
this would imply the lemma. Hence, it suffices to verify (9.17).

We only establish the first bound there, as the proof of the second is
entirely analogous. To this end, observe from 4.34 (and the definitions (9.11)
and (9.10) of $\mathsf{x}_{j}^{\prime}$ and $\mathsf{y}_{j}^{\prime}$) that

$\displaystyle\mathbb{P}\Bigg{[}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\mathsf{x}_{k}^{\prime}(t)+2^{-1/2}\sigma^{\prime
3}t^{2}+2^{-7/6}(3\pi)^{2/3}\sigma^{\prime-1}(k+n_{0})^{2/3}\big{|}\leq(\log
k)^{2}k^{-1/3}\Bigg{]}\leq c_{1}^{-1}e^{-c_{1}(\log n)^{2}}.$

This, together with the fact that, for any $t\in[-\mathsf{T},\mathsf{T}]$,

$\displaystyle\Big{|}$ $\displaystyle\big{(}2^{-1/2}\sigma^{\prime
3}t^{2}+2^{-7/6}(3\pi)^{2/3}\sigma^{\prime-1}(k+n_{0})^{2/3}\big{)}-\big{(}2^{-1/2}t^{2}+2^{-7/6}(3\pi)^{2/3}k^{2/3}\big{)}\Big{|}$
$\displaystyle\leq|1-\sigma^{\prime
3}|n^{2/3}+5\big{(}(k+n_{0})^{2/3}-k^{2/3}\big{)}+5|\sigma^{\prime-1}-1|(k+n_{0})^{2/3}$
$\displaystyle\leq
5\big{(}(k+n_{0})^{2/3}-k^{2/3}\big{)}+4n^{-1/3}+10n^{-1}(k+n_{0})^{2/3}\leq
5n_{0}k^{-1/3}+25n^{-1/3}\leq n^{-1/30},$

implies the first bound in (9.17) and thus the lemma. To establish the first
statement above, we used the facts that $t^{2}\leq n^{2/3}$ and that
$2^{-7/6}(3\pi)^{2/3}\leq 5$; to establish the second we used the facts that
$|1-\sigma^{\prime 3}|\leq 4n^{-1}$ and $|\sigma^{\prime-1}-1|\leq 2n^{-1}$
for sufficiently large $n$ (by the definition (9.10) of
$\sigma^{\prime}=1-n^{-1}$); to establish the third we used the facts that
$(k+n_{0})^{2/3}-k^{2/3}\leq n_{0}k^{-1/3}$ and $k+n_{0}\leq n$; and to
establish in the fourth we used the facts that $k\geq n^{1/6}$ and that
$n_{0}\leq n^{1/50}$ (and that $n$ is sufficiently large). ∎

####  9.3. Proof of 9.1

In this section we establish 9.1. Given 9.2, its proof is similar to that of
[7, Proposition 3.18], obtained by locally sandwiching $\bm{\mathsf{x}}^{n}$
between two parabolic Airy line ensembles with slightly different curvatures.

###### Proof of 9.1.

Throughout this proof, we abbreviate $\bm{\mathsf{x}}=\bm{\mathsf{x}}^{n}$ and
$\mathsf{x}_{j}=\mathsf{x}_{j}^{n}$ for each integer $j\in\llbracket
1,N\rrbracket$. We also set $\mathsf{T}=n^{1/3}$, abbreviate
$n^{\prime}=\lceil n^{1/6}\rceil-1$, and define the real numbers

(9.18)
$\displaystyle\sigma^{\prime}=1+n^{-1/4};\qquad\sigma^{\prime\prime}=1-n^{-1/4}.$

Further define the line ensembles
$\bm{\mathsf{x}}^{\prime}=(\mathsf{x}_{1}^{\prime},\mathsf{x}_{2}^{\prime},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ and
$\bm{\mathsf{x}}^{\prime\prime}=(\mathsf{x}_{1}^{\prime\prime},\mathsf{x}_{2}^{\prime\prime},\ldots)\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ by for each integer $j\geq 1$ setting

(9.19)
$\displaystyle\mathsf{x}_{j}^{\prime}=2^{-1/2}\cdot\mathcal{R}_{j}^{(\sigma^{\prime})}-n^{-1/75},\quad\text{and}\quad\mathsf{x}^{\prime\prime}=2^{-1/2}\cdot\mathcal{R}_{j}^{(\sigma^{\prime\prime})}+n^{-1/75}.$

where we recall the rescaled parabolic Airy line ensemble
$\bm{\mathcal{R}}^{(\sigma)}=\big{(}\mathcal{R}_{1}^{(\sigma)},\mathcal{R}_{2}^{(\sigma)},\ldots\big{)}$
from (2.3). Then, $\bm{\mathsf{x}}^{\prime}$ and $\bm{\mathsf{x}}^{\prime}$
both satisfy the Brownian Gibbs property by 2.5. We will show that it is with
high probability possible to couple $\bm{\mathsf{x}}$ to lie between
$\bm{\mathsf{x}}^{\prime}$ and $\bm{\mathsf{x}}^{\prime\prime}$; see the right
side of Figure 3.3.

To this end, we define the event
$\mathscr{E}=\mathscr{E}_{1}\cap\mathscr{E}_{2}$, where
$\mathscr{E}_{1}=\mathscr{E}_{1}^{\prime}\cap\mathscr{E}_{1}^{\prime\prime}$
and
$\mathscr{E}_{2}=\breve{\mathscr{E}}_{2}\cap\mathscr{E}_{2}^{\prime}\cap\mathscr{E}_{2}^{\prime\prime}$.
Here,

(9.20) $\displaystyle\begin{aligned}
\mathscr{E}_{1}^{\prime}&=\bigcap_{j=1}^{n^{\prime}}\Bigg{\\{}\displaystyle\sup_{t\in\\{-\mathsf{T},\mathsf{T}\\}}\big{|}\mathsf{x}_{j}^{\prime}(t)+\mathfrak{p}(j;t;\sigma^{\prime})+n^{-1/75}\big{|}\leq(\log
n)^{3}j^{-1/3}\Bigg{\\}};\\\
\mathscr{E}_{1}^{\prime\prime}&=\bigcap_{j=1}^{n^{\prime}}\Bigg{\\{}\displaystyle\sup_{t\in\\{-\mathsf{T},\mathsf{T}\\}}\big{|}\mathsf{x}_{j}^{\prime\prime}(t)+\mathfrak{p}(j;t;\sigma^{\prime\prime})-n^{-1/75}\big{|}\leq(\log
n)^{3}j^{-1/3}\Bigg{\\}};\\\
\breve{\mathscr{E}}_{2}&=\Bigg{\\{}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\mathsf{x}_{n^{\prime}+1}(t)+2^{-1/2}t^{2}+2^{-7/6}(3\pi)^{2/3}(n^{\prime}+1)^{2/3}\big{|}\leq
n^{-1/30}\Bigg{\\}};\\\
\mathscr{E}_{2}^{\prime}&=\Bigg{\\{}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\mathsf{x}_{n^{\prime}+1}^{\prime}(t)+\mathfrak{p}(n^{\prime}+1;t;\sigma^{\prime})+n^{-1/75}\big{|}\leq(\log
n)^{3}n^{\prime-1/3}\Bigg{\\}};\\\
\mathscr{E}_{2}^{\prime\prime}&=\Bigg{\\{}\displaystyle\sup_{t\in[-\mathsf{T},\mathsf{T}]}\big{|}\mathsf{x}_{n^{\prime}+1}^{\prime\prime}(t)+\mathfrak{p}(n^{\prime}+1;t;\sigma^{\prime\prime})-n^{-1/75}\big{|}\leq(\log
n)^{3}n^{\prime-1/3}\Bigg{\\}}.\end{aligned}$

where we recall the function $\mathfrak{p}$ from (9.13). Applying the
definitions (9.19) of $\bm{\mathsf{x}}$ in terms of rescaled parabolic Airy
line ensembles; the concentration estimate 4.34 for the latter; and a union
bound yields a constant $c_{1}>0$ such that

$\displaystyle\max\Big{\\{}\mathbb{P}\big{[}\mathscr{E}_{1}^{\prime\complement}\big{]},\mathbb{P}\big{[}\mathscr{E}_{1}^{\prime\prime\complement}\big{]},\mathbb{P}\big{[}\mathscr{E}_{2}^{\prime\complement}\big{]},\mathbb{P}\big{[}\mathscr{E}_{2}^{\prime\prime\complement}\big{]}\Big{\\}}\leq(5c_{1})^{-1}e^{-c_{1}(\log
n)^{2}},$

Together with the bound
$\mathbb{P}\big{[}\breve{\mathscr{E}}_{2}^{\complement}\big{]}\leq(5c_{1})^{-1}e^{-c_{1}(\log
n)^{2}}$ (by 9.2) and a union bound, this yields

(9.21) $\displaystyle\mathbb{P}\big{[}\mathscr{E}^{\complement}\big{]}\leq
c_{1}^{-1}e^{-c_{1}(\log n)^{2}}.$

Now condition on the curves $\big{(}\mathsf{x}_{j}(t)\big{)}$,
$\big{(}\mathsf{x}_{j}^{\prime}(t)\big{)}$, and
$\big{(}\mathsf{x}_{j}^{\prime\prime}(t)\big{)}$ for $(j,t)\notin\llbracket
1,n^{\prime}\rrbracket\times(-\mathsf{T},\mathsf{T})$, and restrict to the
event $\mathscr{E}$. We claim that

(9.22) $\displaystyle\begin{aligned}
&\mathsf{x}_{j}^{\prime}(t)\leq\mathsf{x}_{j}(t)\leq\mathsf{x}_{j}^{\prime\prime}(t),\qquad\qquad\qquad\text{for
each $(j,t)\in\llbracket
1,n^{\prime}\rrbracket\times\\{-\mathsf{T},\mathsf{T}\\}$};\\\
&\mathsf{x}_{n^{\prime}+1}^{\prime}(t)\leq\mathsf{x}_{n^{\prime}+1}(t)\leq\mathsf{x}_{n^{\prime}+1}^{\prime\prime}(t),\qquad\text{for
each $t\in[-\mathsf{T},\mathsf{T}]$}.\end{aligned}$

To this end, observe for any $(j,t)\in\llbracket
1,N\rrbracket\times\\{-\mathsf{T},\mathsf{T}\\}$ and sufficiently large $n$
that

$\displaystyle\mathsf{x}_{j}(t)-\mathsf{x}_{j}^{\prime}(t)$
$\displaystyle\geq\big{(}-(\log
n)^{3}j^{-1/3}-2^{-1/2}n^{2/3}-2^{-7/6}(3\pi)^{2/3}j^{2/3}\big{)}$
$\displaystyle\qquad-\big{(}(\log n)^{3}j^{-1/3}-2^{-1/2}\sigma^{\prime
3}n^{2/3}-2^{-7/6}(3\pi)^{2/3}\sigma^{\prime-1}j^{2/3}-n^{-1/75}\big{)}$
$\displaystyle=2^{-1/2}(\sigma^{\prime 3}-1)n^{2/3}-2(\log
n)^{3}j^{-1/3}+2^{-7/6}(3\pi)^{2/3}(\sigma^{\prime-1}-1)j^{2/3}+n^{-1/75}$
$\displaystyle\geq-2(\log n)^{3}j^{-1/3}-5n^{-1/4}j^{2/3}+n^{-1/75}\geq 0,$

where in the first statement we used (9.1), the fact that we are restricting
to the event $\mathscr{E}_{1}^{\prime}$ from (9.20), and the definition (9.13)
of $\mathfrak{p}$; in the second we performed the subtraction; in the third we
used the facts that $\sigma^{\prime 3}-1\geq 0$, that
${\sigma^{\prime}}^{-1}-1\geq-n^{-1/4}$ (by the definition (9.18) of
$\sigma^{\prime}=1+n^{-1/4}$), and that $2^{-7/6}(3\pi)^{2/3}\leq 5$; and in
the fourth we used the fact that $j\leq n^{\prime}\leq n^{1/6}$ and that $n$
is sufficiently large. This verifies the first bound in the first statement of
(9.22); the proof of the second part is entirely analogous and is therefore
omitted.

To verify the second statement in (9.22), observe for any
$t\in[-\mathsf{T},\mathsf{T}]$ that

$\displaystyle\mathsf{x}_{n^{\prime}+1}(t)-\mathsf{x}_{n^{\prime}+1}^{\prime}(t)$
$\displaystyle\quad\geq\big{(}-2^{-1/2}t^{2}-2^{-7/6}(3\pi)^{2/3}(n^{\prime}+1)^{2/3}-n^{-1/50}\big{)}$
$\displaystyle\qquad-\big{(}(\log n)^{3}n^{\prime-1/3}-2^{-1/2}\sigma^{\prime
3}t^{2}-2^{-7/6}(3\pi)^{2/3}\sigma^{\prime-1}(n^{\prime}+1)^{2/3}-n^{-1/75}\big{)}$
$\displaystyle\quad=n^{1/75}+2^{-7/6}(3\pi)^{2/3}(\sigma^{\prime-1}-1)(n^{\prime}+1)^{2/3}+2^{-1/2}(\sigma^{\prime
3}-1)t^{2}-n^{-1/50}-(\log n)^{3}n^{\prime-1/3}$ $\displaystyle\quad\geq
n^{-1/75}-5n^{-1/4}(n^{\prime}+1)^{2/3}-n^{-1/50}-(\log
n)^{3}n^{\prime-1/3}\geq 0,$

where in the first statement we used the fact that we are restricting to the
events $\breve{\mathscr{E}}_{2}$ and $\mathscr{E}_{2}^{\prime}$ from (9.20),
as well as the definition (9.13) of $\mathfrak{p}$; in the second we performed
the subtraction; in the third we used the facts that $\sigma^{\prime}\geq 1$,
that $\sigma^{\prime-1}-1\leq-n^{-1/4}$ for $n$ sufficiently large (by the
definition (9.18) of $\sigma^{\prime}=1+n^{-1/4}$), and that
$2^{-7/6}(3\pi)^{2/3}\leq 5$; and in the fourth we used the facts that
$n^{\prime}+1\leq n^{1/6}+1\leq 2n^{1/6}$ and that $n$ is sufficiently large.
This verifies the first part of the second bound in (9.15); the proof of the
second part is entirely analogous and is therefore omitted.

Thus, (9.22) holds. As in the proof of 9.2, it follows from (9.22) and 4.6
that, on $\mathscr{E}$, we may couple $\bm{\mathsf{x}}$,
$\bm{\mathsf{x}}^{\prime}$, and $\bm{\mathsf{x}}^{\prime\prime}$ so that

$\displaystyle\mathsf{x}_{j}^{\prime}(s)\leq\mathsf{x}_{j}(s)\leq\mathsf{x}_{j}^{\prime\prime}(s),\qquad\text{for
each $(j,s)\in\llbracket
1,n^{\prime}\rrbracket\times[-\mathsf{T},\mathsf{T}]$}.$

Since (9.19) and (9.18) imply that $\mathsf{x}_{j}^{\prime}(s)$ and
$\mathsf{x}_{j}^{\prime\prime}(s)$ both converge to
$2^{-1/2}\cdot\bm{\mathcal{R}}$, uniformly on compact subsets on
$\mathbb{Z}_{\geq 1}\times\mathbb{R}$, as $n$ tends to $\infty$, this and
(9.21) together imply that $\bm{\mathsf{x}}^{n}$ converges to
$2^{-1/2}\cdot\bm{\mathcal{R}}$, establishing the proposition. ∎

###  10\. Limit Shapes for Non-intersecting Brownian Bridges

In this section we collect some results on limit shapes for families of non-
intersecting Brownian bridges, without upper and lower boundaries. We begin by
introducing them and their properties in Section 10.1; we then provide
examples of them in Section 10.2 and continuous variants of monotonicity for
them in Section 10.3. In Section 10.4 and Section 10.5 we recall an elliptic
partial differential equation and regularity results satisfied by these limit
shapes. In Section 10.6 we recall a concentration bound for non-intersecting
Brownian bridges, indicating conditions under which they are closely
approximated by their limit shapes.

####  10.1. Limit Shapes

In this section we recall results from [68, 66, 18] concerning the limiting
macroscopic behavior of non-intersecting Brownian bridges under given starting
and ending data, with no upper and lower boundaries. Throughout, we use
coordinates $(t,x)$ or sometimes $(t,y)$ for $\mathbb{R}^{2}$ (instead of
$(x,y)$).

Fix an interval $I\subseteq\mathbb{R}$. A _measure-valued process_ (on the
time interval $I$) is a family $\bm{\mu}=(\mu_{t})_{t\in I}$ of measures
$\mu_{t}\in\mathscr{P}_{\operatorname{fin}}$ for each $t\in I$. Given a real
number $A>0$, we say that $\bm{\mu}$ has _constant total mass $A$_ if
$\mu_{t}(\mathbb{R})=A$, for each $t\in I$. If $\bm{\mu}$ has constant total
mass $1$ (so each $\mu_{t}\in\mathscr{P}$), we call $\bm{\mu}$ a _probability
measure-valued process_. Measure-valued processes can be interpreted as
elements of $I\times\mathscr{P}_{\operatorname{fin}}$ and probability measure-
valued processes as ones of $I\times\mathscr{P}$. We let
$\mathcal{C}(I;\mathscr{P}_{\operatorname{fin}})$ and
$\mathcal{C}(I;\mathscr{P})$ denote the sets of measure-valued processes and
probability measure-valued processes that are continuous in $t\in I$, under
the topology of weak convergence on $\mathscr{P}_{\operatorname{fin}}$ and
$\mathscr{P}$, respectively.

Given two measures $\mu,\nu\in\mathscr{P}_{\operatorname{fin}}$ of finite
total mass, the Lévy distance between them is

(10.1) $\displaystyle
d_{\operatorname{L}}(\mu,\nu)=\inf\Bigg{\\{}a>0:\displaystyle\int_{-\infty}^{y-a}\mu(dx)-a\leq\displaystyle\int_{-\infty}^{y}\nu(dx)\leq\displaystyle\int_{-\infty}^{y+a}\mu(dx)+a,\quad\text{for
all $y\in\mathbb{R}$}\Bigg{\\}}.$

Given an interval $I\subseteq\mathbb{R}$ and two measure-valued processes
$\bm{\mu}=(\mu_{t})_{t\in I}\in I\times\mathscr{P}_{\operatorname{fin}}$ and
$\bm{\nu}=(\nu_{t})_{t\in I}\in I\times\mathscr{P}_{\operatorname{fin}}$ on
the time interval $I$, the Lévy distance between them is defined to be

(10.2) $\displaystyle d_{\operatorname{L}}(\bm{\mu},\bm{\nu})=\sup_{t\in
I}d_{\operatorname{L}}(\mu_{t},\nu_{t}).$

The following lemma from [18] (based on results from [66, 68]) states that, as
$n$ tends to $\infty$, the empirical measure (recall (1.18)) for $n$ non-
intersecting Brownian bridges (whose starting and ending data converge in a
certain way) has a limit. The following lemma was stated in [18] in the case
when $[a,b]=[0,1]$ and $A=1$ but, by the scaling invariance (4.4) for non-
intersecting Brownian bridges, it also holds for any interval $[a,b]$ and real
number $A>0$, as below. In what follows, we recall the notation
$\operatorname{emp}$ from (1.18).

######  Lemma 10.1 ([18, Claim 2.13]).

Fix real numbers $a<b$ and compactly supported measures
$\mu_{a},\mu_{b}\in\mathscr{P}_{\operatorname{fin}}$, both of total mass
$\mu_{a}(\mathbb{R})=A=\mu_{b}(\mathbb{R})$ for some real number $A>0$. There
is a measure-valued process
$\bm{\mu}^{\star}=(\mu_{t}^{\star})_{t\in[a,b]}\in\mathcal{C}\big{(}[a,b];\mathscr{P}_{\operatorname{fin}}\big{)}$
on $[a,b]$ of constant total mass $A$, which is continuous in the pair
$(\mu_{a},\mu_{b})\in\mathscr{P}_{\operatorname{fin}}^{2}$ under the Lévy
metric, such that the following holds. For each integer $n\geq 1$, let
$\bm{u}=\bm{u}^{n}\in\overline{\mathbb{W}}_{n}$ and
$\bm{v}=\bm{v}^{n}\in\overline{\mathbb{W}}_{n}$ denote sequences such that
$A\cdot\operatorname{emp}(\bm{u}^{n})$ and
$A\cdot\operatorname{emp}(\bm{v}^{n})$ converge to $\mu_{a}$ and $\mu_{b}$
under the Lévy metric as $n$ tends to $\infty$, respectively. Sample $n$ non-
intersecting Brownian bridges
$\bm{x}^{n}=(x_{1}^{n},x_{2}^{n},\ldots,x_{n}^{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$ from
$\mathsf{Q}^{\bm{u};\bm{v}}(An^{-1})$; for any $t\in[a,b]$, denote
$\nu_{t}^{n}=A\cdot\operatorname{emp}\big{(}\bm{x}^{n}(t)\big{)}\in\mathscr{P}$.
For any real number $\varepsilon>0$, we have
$\lim_{n\rightarrow\infty}\mathbb{P}\big{[}d_{\operatorname{L}}(\bm{\nu}^{n},\bm{\mu}^{\star})>\varepsilon\big{]}=0$.

Terminology for the limit shape provided by 10.1 is given by the following
definition.

Figure 3.5. Shown to the left is the liquid region $\Omega$ associated with a
bridge-limiting measure on $[a,b]$; restricting the latter to the (striped
blue) shorter interval $[\widetilde{a},\widetilde{b}]$ again gives a bridge-
limiting measure. The (red) curve $\gamma(t)$ traces the north boundary of
$\Omega$, called the arctic boundary. Shown to the right is the associated
inverted liquid region $\Omega^{\operatorname{inv}}$.

######  Definition 10.2.

Adopting the notation of 10.1, the measure-valued process
$\bm{\mu}^{\star}=(\mu_{t}^{\star})_{t\in[a,b]}$ is called the _bridge-
limiting measure process_ (on the interval $[a,b]$) with boundary data
$(\mu_{a};\mu_{b})$.

The following lemma from [8, Lemma 3.3] indicates how bridge-limiting measure
processes restrict to others; see the left side of Figure 3.5 for a depiction.

######  Lemma 10.3 ([8, Lemma 3.3]).

Adopt the notation and assumptions of 10.1, and let
$\widetilde{a},\widetilde{b}\in\mathbb{R}$ be real numbers such that
$a\leq\widetilde{a}<\widetilde{b}\leq b$. Then, the bridge-limiting measure
process on the interval $[\widetilde{a},\widetilde{b}]$ with boundary data
$(\mu_{\widetilde{a}}^{\star};\mu_{\tilde{b}}^{\star})$ is given by
$(\mu_{t}^{\star})_{t\in[\tilde{a},\tilde{b}]}$.

We will often make use of a height function and inverted height functions
associated with a measure-valued process, defined as follows.

######  Definition 10.4.

Fix an interval $I=[a,b]\subseteq\mathbb{R}$ and a measure-valued process
$\bm{\mu}=(\mu_{t})_{t\in I}$ of constant total mass $A>0$. The _height
function associated with $\bm{\mu}$_ is defined to be the function
$H=H^{\bm{\mu}}:I\times\mathbb{R}\rightarrow\mathbb{R}$ obtained by setting

(10.3) $\displaystyle
H(t,x)=\displaystyle\int_{x}^{\infty}\mu_{t}(dw),\qquad\text{for each
$(t,x)\in I\times\mathbb{R}$}.$

The _inverted height function_ associated with $\bm{\mu}$ is
$G=G^{\bm{\mu}}:I\times[0,A]\rightarrow\mathbb{R}$, defined by setting
$G(t,0)=\inf\big{\\{}x:H(t,x)=0\big{\\}}$ and

(10.4) $\displaystyle
G(t,y)=\sup\Bigg{\\{}x\in\mathbb{R}:H(t,x)=\displaystyle\int_{x}^{\infty}\mu_{t}(dw)\geq
y\Bigg{\\}},\qquad\text{for each $y\in(0,A]$}.$

Thus, in analogy with (4.23), we may view $G(t,y)$ as a classical location of
the measure $\mu_{t}\in\mathscr{P}$.

If $\mu_{t}=\varrho_{t}(x)dx$ has a density with respect to Lebesgue measure
for each $t\in(a,b)$, then we sometimes associate $H$ (or $G$) with
$\varrho=(\varrho_{t})$. Moreover, if $\bm{\mu}$ is the bridge-limiting
measure process with boundary data $(\mu_{a};\mu_{b})$ we say that $H$ (or
$G$) is associated with boundary data $(\mu_{a};\mu_{b})$.

The following lemma essentially due to [66] (but appearing as stated below in
[8]) indicates that the measures $\mu_{t}^{\star}$ have a density, and it also
discusses properties of this density. In what follows, we recall the free
convolution and semicircle distribution $\mu_{\operatorname{sc}}^{(t)}$ from
Section 4.3.

######  Lemma 10.5 ([8, Lemma 3.7 and Remark 3.14]).

Adopting the notation and assumptions of 10.1, the following statements hold
for each real number $t\in(a,b)$.

  1. (1)

There exists a measurable function
$\varrho_{t}^{\star}:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ such that
$\mu_{t}^{\star}(dx)=\varrho_{t}^{\star}(x)dx$.

  2. (2)

There exists some compactly supported measure
$\nu_{t}\in\mathscr{P}_{\operatorname{fin}}$ of total mass
$\nu_{t}(\mathbb{R})=A$, dependent on $\mu_{a}$ and $\mu_{b}$ with
$\operatorname{supp}\nu_{t}\subseteq\operatorname{supp}\mu_{a}+\operatorname{supp}\mu_{b}$,
such that
$\varrho_{t}^{\star}=\nu_{t}\boxplus\mu_{\operatorname{sc}}^{((t-a)(b-t)/(b-a))}$.

  3. (3)

We have $\varrho_{t}^{\star}(x)^{2}\leq A(b-a)\big{(}(t-a)(b-t)\big{)}^{-1}$,
for any $x\in\mathbb{R}$.

  4. (4)

The function $\varrho_{t}(x)$ is continuous on $(a,b)\times\mathbb{R}$.

The following definition provides notation for the region on which the density
$\varrho_{t}$ is positive (both in terms of the $(t,x)$ coordinates of the
height function and the $(t,y)$ coordinates of the inverted height function).
See Figure 3.5 for a depiction.

######  Definition 10.6.

Fix an interval $(a,b)\subseteq\mathbb{R}$ and a family of measures
$\bm{\mu}=(\mu_{t})_{t\in(a,b)}\in[a,b]\times\mathscr{P}_{\operatorname{fin}}$
of constant total mass $A>0$. Assume for each $t\in(a,b)$ that each $\mu_{t}$
has a density $\varrho_{t}$ with respect to Lebesgue measure, for some
continuous function $\varrho_{t}:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$
that is also continuous in $t$. Recalling the associated height and inverted
height functions $H=H^{\bm{\mu}}:[a,b]\times\mathbb{R}\rightarrow\mathbb{R}$
and $G=G^{\bm{\mu}}:[a,b]\times[0,A]\rightarrow\mathbb{R}$ from Definition
10.4, we define the associated _liquid region_
$\Omega\subset(a,b)\times\mathbb{R}$ and _inverted liquid region_
$\Omega^{\operatorname{inv}}\subseteq(a,b)\times(0,A)$ by@

(10.5)
$\displaystyle\begin{split}&\Omega=\big{\\{}(t,x)\in(a,b)\times\mathbb{R}:\varrho_{t}(x)>0\big{\\}};\\\
&\Omega^{\operatorname{inv}}=\big{\\{}(t,y)\in(a,b)\times(0,A):y=H^{\bm{\mu}}(t,x),(t,x)\in\Omega\big{\\}}.\end{split}$

Observe that the map $(t,x)\mapsto\big{(}t,H(t,x)\big{)}$ is a bijection to
from $\Omega$ to $\Omega^{\operatorname{inv}}$. Moreover, by the continuity of
$\varrho$ (which in our context will be verified by 10.5), the set $\Omega$ is
open, which implies that $\Omega^{\operatorname{inv}}$ is also open.

We next state two lemmas essentially due to [18, 68, 66] (but stated as below
in [8]). The first reformulates 10.1 through (inverted) height functions;
there, we recall the height function $\mathsf{H}^{\bm{x}}$ associated with a
line ensemble $\bm{x}$ from Definition 4.10. The second indicates that the
height and inverted height functions $H^{\star}$ and $G^{\star}$ are smooth on
$\Omega$ and $\Omega^{\operatorname{inv}}$, respectively.

######  Lemma 10.7 ([8, Corollary 3.6 and Corollary 3.8]).

Adopt the notation and assumptions of 10.1, and fix a real number
$\varepsilon>0$. Let $G^{\star}:[a,b]\times[0,A]\rightarrow\mathbb{R}$ denote
the inverted height function associated with $\bm{\mu}^{\star}$, respectively;
further denote the associated inverted liquid region by
$\Omega^{\operatorname{inv}}\subseteq(a,b)\times(0,A)$. Then the following two
statements hold.

  1. (1)

For any $y\in(0,1)$, we have

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\Bigg{[}\big{\\{}G^{\star}(t,Ay+\varepsilon)-\varepsilon\leq
x^{n}_{\lfloor yn\rfloor}(t)\leq
G^{\star}(t,Ay-\varepsilon)+\varepsilon\big{\\}}\Bigg{]}=1,$

  2. (2)

For any $y\in(0,1)$ such that $(t,Ay)\in\Omega^{\operatorname{inv}}$ holds for
each $t\in(a,b)$, we have that $G^{\star}(t,Ay)$ is continuous in $t\in[a,b]$,
and