k)^{2}}$. Next, by the $\mathsf{t}=tk^{1/3}$ case of 19.6 (and altering the
constants $c_{1}>0$ and $C_{3}>1$ if necessary), there exists an
$\mathcal{F}_{\operatorname{ext}}$-measurable event
$\mathscr{A}_{2}\subseteq\textbf{BTR}_{n}$ satisfying
$\mathbb{P}[\textbf{BTR}_{n}\setminus\mathscr{A}_{2}]\leq C_{3}e^{-c_{1}(\log
k)^{2}}$ and the following. Conditioning on $\mathcal{F}_{\operatorname{ext}}$
and restricting to $\mathscr{A}_{2}$, there exists a coupling between
$\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such that

(19.23)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k^{2}}\Big{\\{}\big{|}\mathsf{x}_{j}(tk^{1/3})-\mathsf{y}_{j}(tk^{1/3})\big{|}\geq
k^{-2}\Big{\\}}\Bigg{]}\leq C_{3}k^{-200}.$

Define the $\mathcal{F}_{\operatorname{ext}}$-measurable event
$\mathscr{A}=\mathscr{A}_{1}\cap\mathscr{A}_{2}\subseteq\textbf{BTR}_{n}$,
which by a union bound satisfies

(19.24) $\displaystyle\mathbb{P}[\textbf{BTR}_{n}\setminus\mathscr{A}]\leq
2C_{3}e^{-c_{1}(\log k)^{2}}.$

Condition on $\mathcal{F}_{\operatorname{ext}}$ and restrict to the event
$\mathscr{A}$. By (19.24), it suffices to show for some constants
$C_{1}=C_{1}(A,B,D)>1$ and $C_{2}=C_{2}(A,B,D,R,\delta)>1$ that

(19.25)
$\displaystyle\mathbb{P}\bigg{[}\textbf{PFL}^{\bm{x}}\Big{(}t;\displaystyle\frac{(\log
k)^{6}}{2k};C_{1}\Big{)}\cap\textbf{GAP}_{n}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}\bigg{]}\leq
C_{2}k^{-200}.$

This will follow from 15.2; we must first verify 15.1 for that proposition.
Denote the $n$-tuples $\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,n\rrbracket}(-Ak^{1/3}/2)\in\overline{\mathbb{W}}_{n}$ and
$\bm{v}=\bm{\mathsf{x}}_{\llbracket
1,n\rrbracket}(Ak^{1/3}/2)\in\overline{\mathbb{W}}_{n}$. Observe (by (19.22)
and Definition 16.1) since we have restricted to
$\mathscr{A}\subseteq\mathscr{A}_{1}$ that, for each integer $j\in\llbracket
1,n^{\prime}\rrbracket$,

(19.26) $\displaystyle-\widetilde{B}j^{2/3}-\widetilde{B}k^{2/3}\leq
u_{j}\leq\widetilde{B}k^{2/3}-\widetilde{B}^{-1}j^{2/3};\quad-\widetilde{B}j^{2/3}-\widetilde{B}k^{2/3}\leq
v_{j}\leq\widetilde{B}k^{2/3}-\widetilde{B}^{-1}j^{2/3}.$

This verifies (15.1) of 15.1. Since we moreover have $t\in[-A/4,A/4]$, 15.1
holds with the $(\bm{\mathsf{x}};k,L,n;A,B,D;t)$ there equal to
$\big{(}\bm{\mathsf{y}};k,(k^{-1}n^{\prime})^{2/3},n^{\prime};A,\widetilde{B},D;t+A/2\big{)}$
here (with the arguments of the paths in $\bm{\mathsf{y}}$ shifted by
$Ak^{1/3}/2$). Thus, 15.2 applies; its first and third parts will be the ones
of relevance for us.

Its first part yields constants $c_{2}=c_{2}(A,B)>0$, $C_{4}=C_{4}(A,B)>1$,
and $C_{5}=C_{5}(A,B,D)>1$ and an event $\mathscr{E}$, with

(19.27) $\displaystyle\mathbb{P}[\mathscr{E}]\geq 1-C_{5}e^{-c_{2}(\log
k)^{2}},$

on which there exists a (random) measure $\mu$ with
$\mu(\mathbb{R})=k^{-1}n^{\prime}$, satisfying the following property.
Denoting the classical locations of $\mu$ (recall Definition 4.21) by
$\gamma_{j}=\gamma_{j;n^{\prime}}^{\mu}$ for each $j\in\llbracket
1,n^{\prime}\rrbracket$, we have

(19.28) $\displaystyle\gamma_{j+\lfloor(3D\log k)^{5}\rfloor}-k^{-2}\leq
k^{-2/3}\cdot\mathsf{y}_{j}(tk^{1/3})\leq\gamma_{j-\lfloor(3D\log
k)^{5}\rfloor}+k^{-2},\qquad\text{for each $j\in\llbracket
1,n^{\prime}\rrbracket$},$

where we have used the facts that $(n^{\prime})^{-D}\leq k^{-D}\leq k^{-2}$
and that $3D\log k=\log k^{3D}\geq\log k^{3D/2+1}\geq\log(L^{3/2}k)=\log
n\geq\log n^{\prime}$.

We would also like to use the third statement in Proposition 15.2, to which
end we must verify its hypothesis, which we will do this upon further
restricting to the event

(19.29)
$\displaystyle\mathscr{E}^{\prime}=\bigcap_{j=1}^{k^{2}}\Big{\\{}\big{|}\mathsf{x}_{j}(tk^{1/3})-\mathsf{y}_{j}(tk^{1/3})\big{|}\leq
k^{-2}\Big{\\}}\cap\textbf{GAP}_{n}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}.$

So, as in (15.2), define the function
$\gamma:[0,k^{-1}n^{\prime}]\rightarrow\mathbb{R}$ by

(19.30)
$\displaystyle\gamma(y)=\sup\Bigg{\\{}x\in\mathbb{R}:\displaystyle\int_{x}^{\infty}\mu(dx)\geq
y\Bigg{\\}},\qquad\text{so
that}\qquad\gamma_{j}=\gamma\Big{(}\displaystyle\frac{2j-1}{2k}\Big{)},\quad\text{for
each $j\in\llbracket 1,n^{\prime}\rrbracket$}.$

On the event $\mathscr{E}^{\prime}$, (19.28), the first event in (19.29), and
the second statement in (19.30) give for sufficiently large $k$ (using the
facts that $\gamma$ is non-increasing and that $(6D\log k)^{5}\geq
2\big{\lfloor}(3D\log k)^{5}\big{\rfloor}+1$) that

(19.31) $\displaystyle\gamma\Big{(}\displaystyle\frac{2j+(6D\log
k)^{5}}{2k}\Big{)}-2k^{-2}\leq
k^{-2/3}\cdot\mathsf{x}_{j}(tk^{1/3})\leq\gamma\Big{(}\displaystyle\frac{j-(6D\log
k)^{5}}{k}\Big{)}+2k^{-2},\quad\text{for $j\in\llbracket 1,k^{2}\rrbracket$}.$

On the event
$\mathscr{E}^{\prime}\subseteq\textbf{GAP}_{n}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}$,
we further have by Definition 3.5 that (as $tk^{1/3}\in[-Ak^{1/3},Ak^{1/3}]$),
for any integers $1\leq i\leq j\leq n$,

(19.32)
$\displaystyle\big{|}{\mathsf{x}}_{i}(tk^{1/3})-{\mathsf{x}}_{j}(tk^{1/3})\big{|}\leq
R(j^{2/3}-i^{2/3})+i^{-1/3}(\log k)^{25}.$

Hence, for $k$ sufficiently large and any real numbers $\widetilde{B}^{-1}\leq
y\leq y^{\prime}\leq\widetilde{B}$ with $y^{\prime}-y\geq 10k^{-1}(\log
n^{\prime})^{50}$, it follows that

(19.33) $\displaystyle\begin{aligned}
\big{|}\gamma(y^{\prime})-\gamma(y)\big{|}=\gamma(y)-\gamma(y^{\prime})&\leq
k^{-2/3}\cdot\big{(}{\mathsf{x}}_{\lfloor yk\rfloor-(6D\log
k)^{5}}(tk^{1/3})-{\mathsf{x}}_{\lceil y^{\prime}k\rceil+(6D\log
k)^{5}}\big{)}(tk^{1/3})+4k^{-2}\\\ &\leq
R\Big{(}\big{(}y^{\prime}+k^{-1}(12D\log
k)^{5}\big{)}^{2/3}-\big{(}y-k^{-1}(12D\log k)^{5}\big{)}^{2/3}\Big{)}\\\
&\qquad+\displaystyle\frac{(\log k)^{25}}{k^{2/3}\big{(}yk-(6D\log
k)^{5}-1\big{)}^{1/3}}+4k^{-2}\\\ &\leq
R\big{(}(y^{\prime})^{2/3}-y^{2/3}\big{)}+10\widetilde{B}k^{-1}(12D\log
k)^{25}\leq 2R\big{(}(y^{\prime})^{2/3}-y^{2/3}\big{)}.\end{aligned}$

Here, in the first statement, we used the fact that $\gamma$ is non-
increasing; in the second we used (19.31) (which applies since $\lceil
yk\rceil+(6D\log k)^{5}\leq\lceil y^{\prime}k\rceil+(6D\log
k)^{5}\leq\lceil\widetilde{B}k\rceil+(6D\log k)^{5}<k^{2}$ for sufficiently
large $k$); and, in the third, we used (19.32). In the fourth we used the
facts that $a^{2/3}-b^{2/3}\leq b^{-1/3}(a-b)$ for any real numbers $a\geq
b>0$ (applied for $(a,b)=\big{(}y^{\prime}+k^{-1}(12D\log
k)^{5},y^{\prime}\big{)}$ and $(a,b)=\big{(}y,y-k^{-1}(12D\log
k)^{5}\big{)}$), that $(2\widetilde{B})^{-1}\leq y-k^{-1}(12D\log
k)^{5}\leq\widetilde{B}$ for sufficiently large $k$, and that $4k^{-2}\leq
4\widetilde{B}k^{-1}(12D\log k)^{25}$. In the fifth, we used the bound
$(y^{\prime})^{2/3}-y^{2/3}\geq 2|y^{\prime}-y|/3\widetilde{B}^{1/3}\geq
20(3\widetilde{B}k)^{-1}(\log n^{\prime})^{50}\geq
4\widetilde{B}k^{-1}(12D\log k)^{25}$ for sufficiently large $k$ (as
$n^{\prime}\geq k$).

The estimate (19.33) verifies the assumptions in the third statement in
Proposition 15.2 (with the $R$ there equal to $2R$ here). Since
$\widetilde{B}=12A^{2}B^{3}\geq 12$, we have
$[1/6,6]\subseteq[2/\widetilde{B},\widetilde{B}/2]$, and so it follows from
the third part of 15.2 that there exists a constant $C_{6}=C_{6}(A,B,D,R)>1$
such that

(19.34)
$\displaystyle\big{\|}\gamma|_{[1/6,6]}-\gamma(0)\big{\|}_{{\mathcal{C}}^{\ell}}\leq
C_{6}$

We now claim for sufficiently large $k$ that
$\textbf{PFL}^{\bm{x}}\big{(}t;(2k)^{-1}(\log k)^{6};C_{6}\big{)}$ holds on
$\mathscr{E}\cap\mathscr{E}^{\prime}$, with the associated function
$\gamma_{t}:[0,1]\rightarrow\mathbb{R}$ of Definition 3.11 given by
$\gamma_{t}(x)=\gamma(x+1)$ for each $x\in[0,1]$. That this choice of
$\gamma_{t}$ satisfies the second bound in (3.7) follows from (19.34). To
verify that it satisfies the first, observe for sufficiently large $k$ and any
$j\in\llbracket 1,k\rrbracket$ that

$\displaystyle\big{|}x_{j}(t)-\gamma_{t}(k^{-1}j)\big{|}=\bigg{|}k^{-2/3}\cdot\mathsf{x}_{j+k}(tk^{1/3})-\gamma\Big{(}\displaystyle\frac{j+k}{k}\Big{)}\bigg{|}\leq
C_{6}\cdot\displaystyle\frac{(6D\log
k)^{5}}{2k}+2k^{-2}\leq\displaystyle\frac{(\log k)^{6}}{2k},$

where in the first statement we used the facts that
$x_{j}(t)=k^{-2/3}\cdot\mathsf{x}_{j+k}(tk^{1/3})$ and
$\gamma_{t}(x)=\gamma(x+1)$; in the second we used (19.31) and (19.34); and in
the third we used that $k$ is sufficiently large. Thus, $\gamma_{t}$ satisfies
the first bound in (3.7), and so $\textbf{PFL}^{\bm{x}}\big{(}t;(2k)^{-1}(\log
k)^{6};C_{6}\big{)}$ holds.

Hence,
$\mathscr{E}\cap\mathscr{E}^{\prime}\subseteq\textbf{PFL}^{\bm{x}}\big{(}t;(2k)^{-1}(\log
k)^{6};C_{6}\big{)}$. With the bound $\mathbb{P}[\mathscr{E}]\geq
1-C_{5}e^{-c_{2}(\log k)^{2}}$ from (19.27), the definition (19.29) of
$\mathscr{E}^{\prime}$, (19.23), and a union bound, this gives (19.25) and
thus the proposition. ∎

####  19.4. Proof of 19.7

In this section we establish 19.7, which will follow as an application of 4.8
and 4.6.

###### Proof of 19.7.

Throughout this proof, we condition on the $\sigma$-algebra given by
$\mathcal{F}_{\operatorname{ext}}=\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$, and we abbreviate the event
$\textbf{BTR}_{n}=\textbf{BTR}_{n}^{\bm{\mathsf{x}}}(A;B)$. Observe that it
suffices to show that there exists constants $c=c(A,B)>0$ and $C=C(A,B,D)>1$
such that, for any integers $i\geq 1$ and $j\in\llbracket
1,n^{\prime}\rrbracket$, we have

(19.35)
$\displaystyle\mathbb{P}\bigg{[}\Big{\\{}\big{|}\mathsf{x}_{j}(\mathsf{t})\big{|}\geq
c^{-1}ik^{4D}\Big{\\}}\cap\textbf{BTR}_{n}\bigg{]}\leq
Ce^{-4ik};\qquad\mathbb{P}\bigg{[}\Big{\\{}\big{|}\mathsf{y}_{j}(\mathsf{t})\big{|}\geq
c^{-1}ik^{4D}\Big{\\}}\cap\textbf{BTR}_{n}\bigg{]}\leq Ce^{-4ik}.$

Indeed, (19.35), together with a union bound and a Markov estimate, yields for
each integer $i\geq 1$ an event
$\mathscr{A}(i)=\mathscr{A}_{t}(i)\subseteq\textbf{BTR}_{n}$, measurable with
respect to
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3}/2,Ak^{1/3}/2]\big{)}$, satisfying the
following two properties. First,
$\mathbb{P}\big{[}\textbf{BTR}_{n}\setminus\mathscr{A}(i)\big{]}\leq
2Cn^{\prime}e^{-2ik}$. Second, conditioning on
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3}/2,Ak^{1/3}/2]\big{)}$ and restricting
to $\mathscr{A}(i)$, we have

(19.36)
$\displaystyle\mathbb{P}\Bigg{[}\bigg{\\{}\displaystyle\max_{j\in\llbracket
1,n^{\prime}\rrbracket}\big{|}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\big{|}\geq
2c^{-1}ik^{4D}\bigg{\\}}\cap\textbf{BTR}_{n}\Bigg{]}\leq
2Cn^{\prime}e^{-2ik}.$

The lemma then follows from taking
$\mathscr{A}=\bigcap_{i=1}^{\infty}\mathscr{A}(i)$, which by a union bound
satisfies

$\displaystyle\mathbb{P}[\textbf{BTR}_{n}\setminus\mathscr{A}]\leq\sum_{i=1}^{\infty}\mathbb{P}\big{[}\textbf{BTR}_{n}\setminus\mathscr{A}(i)\big{]}\leq
2Cn\sum_{i=1}^{\infty}e^{-2ik}\leq 4n^{\prime}Ce^{-2k}\leq C^{\prime}e^{-k},$

for some constant $C^{\prime}=C^{\prime}(A,B,D)>1$, where in the last bound we
used the fact that $n^{\prime}\leq n=L^{3/2}k\leq k^{3D/2+1}$. By (19.36)
satisfies (19.16) for any integer $i\geq 1$, with the $(c,C)$ there equal to
$(c/2,C^{\prime})$ here; observe then that (19.16) also holds for $i=0$ (as
$C^{\prime}>1$), establishing the lemma.

It therefore remains to establish (19.35); in what follows, we fix integers
$i\geq 1$ and $j\in\llbracket 1,n^{\prime}\rrbracket$. Let us only verify the
first bound in (19.35), as the proof of the second is entirely analogous.
Since $\mathsf{x}_{1}\geq\mathsf{x}_{2}\geq\cdots\geq\mathsf{x}_{n}$, we must
then show for sufficiently small $c=c(A,B)>0$ and large $C=C(A,B,D)>1$ that

(19.37) $\displaystyle\mathbb{P}\Big{[}\big{\\{}\mathsf{x}_{1}(\mathsf{t})\geq
c^{-1}ik^{4D}\big{\\}}\cap\textbf{BTR}_{n}\Big{]}\leq
Ce^{-4ik};\quad\mathbb{P}\Big{[}\\{\mathsf{x}_{n}(\mathsf{t})\leq-c^{-1}ik^{4D}\big{\\}}\cap\textbf{BTR}_{n}\Big{]}\leq
Ce^{-4ik}.$

We only confirm the first bound in (19.37), as the proof of the second is
again very similar. Recall that we have conditioned on
$\mathcal{F}_{\operatorname{ext}}=\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$; we further restrict to the
event $\textbf{BTR}_{n}$. Set $u=C_{0}ik^{4D}$, for a constant
$C_{0}=C_{0}(A,B)>B\geq 1$ to be fixed later. Denote the $n$-tuple
$\bm{u}=(u,u,\ldots,u)\in\overline{\mathbb{W}}_{n}$ (where $u$ appears with
multiplicity $n$), and define the function
$f:[-Ak^{1/3},Ak^{1/3}]\rightarrow\mathbb{R}$ by setting
$f(s)=\mathsf{x}_{n+1}(s)$ for each $s\in[-Ak^{1/3},Ak^{1/3}]$. Then, sample
two families of non-intersecting Brownian bridges
$\bm{\mathsf{z}}=(\mathsf{z}_{1},\mathsf{z}_{2},\ldots,\mathsf{z}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-Ak^{1/3},Ak^{1/3}]\big{)}$ and
$\widetilde{\bm{\mathsf{z}}}=(\widetilde{\mathsf{z}}_{1},\widetilde{\mathsf{z}}_{2},\ldots,\widetilde{\mathsf{z}}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-Ak^{1/3},Ak^{1/3}]\big{)}$, from the
measures $\mathsf{Q}^{\bm{u};\bm{u}}$ and $\mathsf{Q}_{f}^{\bm{u};\bm{u}}$,
respectively.

We will compare $\bm{\mathsf{x}}$ to $\bm{\mathsf{z}}$ through
$\widetilde{\bm{\mathsf{z}}}$, and then use 4.8 to analyze $\bm{\mathsf{z}}$.
To implement this, first observe since we have restricted to the event
$\textbf{BTR}_{n}$ that Definition 16.2 implies for each
$r\in\\{-Ak^{1/3},Ak^{1/3}\\}$ that
$\mathsf{x}_{j}(r)\leq\mathsf{x}_{1}(r)\leq Bk^{2/3}\leq C_{0}ik^{3D}=u$.
Hence, since the law of $\bm{\mathsf{x}}$ is given by
$\mathsf{Q}_{f}^{\bm{\mathsf{x}}(-Ak^{1/3});\bm{\mathsf{x}}(Ak^{1/3})}$, it
follows from 4.6 that we may couple $\bm{\mathsf{x}}$ and
$\widetilde{\bm{\mathsf{z}}}$ such that
$\mathsf{x}_{1}(\mathsf{t})\leq\widetilde{\mathsf{z}}_{1}(\mathsf{t})$.

Next, we compare $\bm{\mathsf{z}}$ and $\widetilde{\bm{\mathsf{z}}}$ by
showing that the paths in $\bm{\mathsf{z}}$ are with high probability already
above the lower boundary $f$. To do this, observe by 4.8 (applied with the
$(t,s;a,b;n;B)$ there given by $(\mathsf{s},Ak^{1/3};-Ak^{1/3},Ak^{1/3};n;R)$
here) that there exist constants $c_{1}\in(0,1)$ and $C_{1}>2$ such that, for
any real number $R\geq 1$, we have

(19.38) $\displaystyle\mathbb{P}\Bigg{[}\displaystyle\sup_{|s|\leq
Ak^{1/3}}\mathsf{z}_{n}(s)\leq u-2AnR\Bigg{]}\leq
C_{1}e^{C_{1}n-c_{1}R^{2}};\qquad\mathbb{P}\Bigg{[}\mathsf{z}_{1}(\mathsf{t})\geq
u+2AnR\Bigg{]}\leq C_{1}e^{C_{1}n-c_{1}R^{2}},$

where we used the fact that
$(Ak^{1/3}-s)^{1/2}\log\big{(}4Ak^{1/3}(Ak^{1/3}-s)^{-1}\big{)}\leq
2A^{1/2}k^{1/6}\leq 2An$ for each $s\in[-Ak^{1/3},Ak^{1/3}]$. Now set
$R=3c_{1}^{-1}C_{1}ik^{3D/2}$ and fix $C_{0}=7c_{1}^{-1}C_{1}AB$. Observe that

(19.39) $\displaystyle\begin{aligned}
u-2AnR&=7c_{1}^{-1}C_{1}ABik^{4D}-6c_{1}^{-1}C_{1}Aik^{3D/2}n\\\ &\geq
7c_{1}^{-1}C_{1}ABik^{4D}-6c_{1}^{-1}C_{1}Aik^{4D}\geq
ABik^{4D}\geq\displaystyle\sup_{|s|\leq Ak^{1/3}}f(s),\end{aligned}$

where in the first statement we used the definitions of $u=C_{0}ik^{4D}$, of
$C_{0}$, and of $R$; in the second we used the fact that $n=L^{3/2}k\leq
L^{3D/2}k\leq k^{5D/2}$; in the third we used the facts that $c_{1}\in(0,1)$
and $A,B,C_{1}\geq 1$; and in the fourth we used the fact that
$f(s)=\mathsf{x}_{n+1}(s)\leq Bk^{2/3}\leq ABik^{4D}$, which holds (by
Definition 16.2) since we have restricted to the event $\textbf{BTR}_{n}$.
Inserting this into the first bound in (19.38) (and using the bound
$C_{1}n-c_{1}R^{2}\leq-4ik$, since $C_{1}n=C_{1}L^{3/2}k\leq
C_{1}k^{3D/2+1}\leq C_{1}k^{5D/2}$ and
$c_{1}R^{2}=9c_{1}^{-1}C_{1}^{2}i^{2}k^{3D}\geq 9C_{1}(i+1)k^{5D/2}$) yields

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{|s|\leq
Ak^{1/3}}\big{\\{}\mathsf{z}_{n}(s)\geq f(s)\big{\\}}\Bigg{]}\leq
C_{1}e^{-4ik}.$

Thus, the paths in $\bm{\mathsf{z}}$ are above the lower boundary $f$ with
probability at least $1-C_{1}e^{-4ik}$, and so we may couple $\bm{\mathsf{z}}$
and $\widetilde{\bm{\mathsf{z}}}$ so that they coincide with probability at
least $1-C_{1}e^{-4ik}$. Together with the second bound in (19.38), the fact
that $u+2AnR=7c_{1}^{-1}C_{1}ABik^{4D}+6c_{1}^{-1}C_{1}Aikn\leq
13c_{1}^{-1}C_{1}ABik^{4D}$ (by following the same reasoning as used to deduce
(19.39)), and the bound $C_{1}n-c_{1}R^{2}\leq-4ik$, this gives

$\displaystyle\mathbb{P}\Bigg{[}\widetilde{\mathsf{z}}_{1}(\mathsf{t})\geq
13c_{1}^{-1}C_{1}ABik^{4D}\Bigg{]}\leq 2C_{1}e^{-4ik}.$

Combining this with the existence of a coupling between $\bm{\mathsf{x}}$ and
$\bm{\mathsf{z}}$ such that
$\mathsf{x}_{1}(\mathsf{t})\leq\widetilde{\mathsf{z}}_{1}(\mathsf{t})$, this
yields (19.37) and thus the lemma. ∎

###  20\. Proof of the Global Law

In this section we establish 3.10, indicating that a line ensemble satisfying
2.8 satisfies a global law. As in Section 19.2, we first in Section 20.1
reduce it to a general statement, given by 20.1 below, about line ensembles
$\bm{\mathsf{x}}$ that likely satisfy a boundary tall rectangle and gap event.
To show the latter, we will restrict $\bm{\mathsf{x}}$ to a tall rectangle,
and use 16.4 to couple it to a family of non-intersecting Brownian bridges
without lower boundary; we will then analyze the latter using 10.1 (stating it
converges to a limit shape) and 14.1 (to analyze the edge behavior of this
limit shape). To implement this, we require a variant of 10.1 that has uniform
rates of convergence; we show such a statement in Section 20.2 using
compactness arguments. We then establish 20.1 in Section 20.3.

Throughout this section, we recall from the beginning of this chapter that
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)$ is a
$\mathbb{Z}_{\geq 1}\times\mathbb{R}$ indexed line ensemble satisfying the
Brownian Gibbs property.

####  20.1. Likelihood of the Global Law Event

In this section we prove 3.10 as a consequence of the following general
result, to be established in Section 20.3 below. It states that, if
$\bm{\mathsf{x}}$ satisfies a boundary tall rectangle event BTR and a gap
event GAP then, for small $\theta$ and sufficiently large $k$, its top
$\lceil\theta^{3}k\rceil$ curves on the time interval $[-\theta k^{1/3},\theta
k^{1/3}]$ approximate a limiting Airy profile
$\mathfrak{G}_{\operatorname{Ai};\mathfrak{a},\mathfrak{b},\mathfrak{c}}$ from
(10.12) (with random coefficients $\mathfrak{a},\mathfrak{b},\mathfrak{c}$),
up to error $\mathcal{O}(\theta^{3}k^{2/3})$. Observe that, unlike in 19.4
where $L$ was growing faster than $k$, in the below proposition $L$ is bounded
independently of $k$ (though the constant $C$ prescribing the error111However,
the lower bound $K_{0}$ on $k$ depends on all parameters
$(\theta,\varpi,B,R,L)$ involved below does not depend on $L$). In what
follows, we recall the events GAP and BTR from Definition 3.5 and Definition
16.2, respectively.

######  Proposition 20.1.

For any real numbers $\theta,\varpi\in(0,1/2)$ and $B,R,L\geq 1$, there exist
constants $C=C(B,R)>1$ and $K_{0}=K_{0}(\theta,\varpi,B,R,L)>1$ such that the
following holds for any integer $k\geq K_{0}$. Fix integers $n\geq k$; assume
that $n=L^{3/2}k$, that $L\geq C+\theta^{-2^{6000}}$, and that
$\theta<C^{-1}$. If

(20.1)
$\displaystyle\mathbb{P}\Big{[}\textbf{\emph{BTR}}_{n}^{\bm{\mathsf{x}}}(4,B;k,L;2^{-6000})\cap\textbf{\emph{GAP}}_{n}^{\bm{\mathsf{x}}}\big{(}[-4k^{1/3},4k^{1/3}];R\big{)}\Big{]}\geq
1-\varpi,$

then there exist random variables $\mathfrak{a},\mathfrak{b}\in\mathbb{R}$ and
$\mathfrak{c}\in[C^{-1},C]$ such that

(20.2)
$\displaystyle\mathbb{P}\left[\bigcap_{j=1}^{\lfloor\theta^{3}k\rfloor}\bigcap_{|t|\leq\theta}\Bigg{\\{}\bigg{|}\mathsf{x}_{j}(tk^{1/3})-k^{2/3}\cdot(\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}t^{2})+\Big{(}\displaystyle\frac{3\pi}{4\mathfrak{c}^{1/2}}\Big{)}^{2/3}j^{2/3}\bigg{|}\leq
C\theta^{3}k^{2/3}\Bigg{\\}}\right]\geq 1-3\varpi.$

###### Proof of 3.10.

Throughout this proof, we recall the events TOP, GAP, and BTR from Definition
3.2, Definition 3.5, and Definition 16.2, respectively. Let $B_{0},R_{0}>1$
denote the constants $B(4)$ and $R(4)$ from 19.1 (with the parameters
$(A,D,\delta)$ there given by $(4,4,2^{-6000})$ here). Further let
$\mathfrak{C}>1$ denote the constant $C(B_{0},R_{0})$ from 20.1, and set

(20.3)
$\displaystyle\varpi=\displaystyle\frac{\delta}{4};\quad\theta=\displaystyle\frac{\delta}{90\mathfrak{C}B^{3}};\quad
k=\Big{(}\displaystyle\frac{B}{\theta}\Big{)}^{3}n;\quad
L=B+\mathfrak{C}+\theta^{-2^{6000}};\quad N=L^{3/2}k,$

assuming in what follows that $k$ and $N$ are integers (for otherwise we may
increase $B$ and $\mathfrak{C}$, and decrease $\theta$, to ensure this to
hold).

By (the $(A,\delta)=(4,2^{-6000})$ case of) 19.1, we have for sufficiently
large $k\geq n$ that
$\mathbb{P}\big{[}\textbf{BTR}_{N}^{\bm{\mathcal{L}}}(4,B_{0};k,L;2^{-6000})\cap\textbf{GAP}_{N}^{\bm{\mathcal{L}}}([-4k^{1/3},4k^{1/3}];R_{0})\big{]}\geq
1-\varpi$, verifying (20.1) (with the $(\bm{\mathsf{x}},n)$ there given by
$(\bm{\mathcal{L}},N)$ here). Hence, 20.1 applies and yields random variables
$\mathfrak{a},\mathfrak{b}\in\mathbb{R}$ and
$\mathfrak{c}\in[\mathfrak{C}^{-1},\mathfrak{C}]$ such that, for sufficiently
large $k$, we have $\mathbb{P}[\mathscr{E}_{1}]\geq 1-3\varpi$, where we have
defined the event

(20.4)
$\displaystyle\mathscr{E}_{1}=\bigcap_{j=1}^{\lfloor\theta^{3}k\rfloor}\bigcap_{|t|\leq\theta}\Bigg{\\{}\bigg{|}\mathsf{x}_{j}(tk^{1/3})-k^{2/3}\cdot(\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}t^{2})+\Big{(}\displaystyle\frac{3\pi}{4\mathfrak{c}^{1/2}}\Big{)}^{2/3}j^{2/3}\bigg{|}\leq\mathfrak{C}\theta^{3}k^{2/3}\Bigg{\\}}.$

Moreover, from 3.4 (with the $(B,\vartheta)$ there given by
$(B\theta^{-1},\theta)$ here), we have for sufficiently large $k\geq n$ that
$\mathbb{P}[\mathscr{E}_{2}]\geq 1-\varpi$, where

(20.5)
$\displaystyle\mathscr{E}_{2}=\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-B\theta^{-1}n^{1/3},B\theta^{-1}n^{1/3}];\theta
n^{2/3}\big{)}.$

Hence, denoting $\mathscr{E}=\mathscr{E}_{1}\cap\mathscr{E}_{2}$, we have by a
union bound that $\mathbb{P}[\mathscr{E}]\geq 1-4\varpi=1-\delta$. It
therefore suffices to show that
$\textbf{GBL}_{n}^{\bm{\mathcal{L}}}(\delta;B)$ holds on $\mathscr{E}$, for
sufficiently large $n$.

To this end, restrict to the event $\mathscr{E}$; by Definition 3.9 (and
(3.5)), we must show that

(20.6)
$\displaystyle\big{|}\mathsf{x}_{j}(tn^{1/3})+2^{-1/2}t^{2}n^{2/3}+2^{-7/6}(3\pi)^{2/3}j^{2/3}\big{|}\leq\delta
n^{2/3},\quad\text{for all $(j,t)\in\llbracket
1,Bn\rrbracket\times[-Bn^{1/3},Bn^{1/3}]$}.$

We will show that
$\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}^{2}t\approx-2^{-1/2}t^{2}$ and
$(3\pi/4\mathfrak{c}^{1/2})^{2/3}\approx 2^{-7/6}(3\pi)^{2/3}$ in (20.4), by
comparing the $j=1$ case of (20.4) with (20.5). First observe for sufficiently
large $n$ that

$\displaystyle\mathfrak{C}\theta^{3}k^{2/3}+\Big{(}\displaystyle\frac{3\pi}{4\mathfrak{c}^{1/2}}\Big{)}^{2/3}\leq\mathfrak{C}\theta^{3}k^{2/3}+3\mathfrak{C}\leq
2\mathfrak{C}\theta^{3}k^{2/3}\leq\displaystyle\frac{\delta n^{2/3}}{45},$

where in the first bound we used the fact that
$\mathfrak{c}\in[\mathfrak{C}^{-1},\mathfrak{C}]$; in the second we used the
fact that $k\geq n$ is sufficiently large; and in the third we used (20.3).
Thus, since we have restricted to $\mathscr{E}\subseteq\mathscr{E}_{1}$,
applying (20.4) at $j=1$ gives for $t\in[-\theta,\theta]$ that

(20.7)
$\displaystyle\big{|}\mathsf{x}_{1}(tk^{1/3})-k^{2/3}(\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}t^{2})\big{|}\leq\displaystyle\frac{\delta
n^{2/3}}{45}.$

Since we have also restricted to $\mathscr{E}\subseteq\mathscr{E}_{2}$, (20.5)
(with Definition 3.2), (20.3), and the fact that
$B\theta^{-1}n^{1/3}=k^{1/3}\geq\theta k^{1/3}$ together imply for each
$t\in[-\theta,\theta]$ that

$\displaystyle\big{|}\mathsf{x}_{1}(tk^{1/3})+2^{-1/2}t^{2}k^{2/3}\big{|}\leq\theta
n^{2/3}<\displaystyle\frac{\delta n^{2/3}}{90}.$

Together with (20.7) (and (20.3)), this gives

(20.8)
$\displaystyle\displaystyle\sup_{|t|\leq\theta}\big{|}\mathfrak{a}+\mathfrak{b}t-t^{2}(\mathfrak{c}-2^{-1/2})\big{|}\leq\displaystyle\frac{\delta}{30}\cdot(k^{-1}n)^{2/3}=\displaystyle\frac{\delta\theta^{2}}{30B^{2}}.$

Adding (20.8) at $t\in\\{-\theta,\theta\\}$ and subtracting twice of it at
$t=0$ yields $2\theta^{2}|\mathfrak{c}+2^{1/2}|\leq
2\delta\theta^{2}/15B^{2}$, so that

$\displaystyle\displaystyle\frac{3}{5}\leq
2^{-1/2}-\displaystyle\frac{\delta}{15B^{2}}\leq\mathfrak{c}\leq
2^{-1/2}+\displaystyle\frac{\delta}{15B^{2}}\leq\displaystyle\frac{4}{5}.$

In particular, since
$|a^{-1/3}-b^{-1/3}|\leq\big{(}|a-b|/3\big{)}\cdot\max\\{a^{-4/3},b^{-4/3}\\}$
for $a,b>0$, it follows that

(20.9)
$\displaystyle\bigg{|}2^{-7/6}(3\pi)^{2/3}-\Big{(}\displaystyle\frac{3\pi}{4\mathfrak{c}^{1/2}}\Big{)}^{2/3}\bigg{|}\leq\Big{(}\displaystyle\frac{3\pi}{4}\Big{)}^{2/3}\cdot\displaystyle\frac{1}{3}\Big{(}\displaystyle\frac{3}{5}\Big{)}^{-4/3}\cdot\displaystyle\frac{\delta}{15B^{2}}\leq\displaystyle\frac{\delta}{5B^{2}}.$

Since by (20.3) we have $\lfloor\theta^{3}k\rfloor=\lfloor B^{3}n\rfloor$ it
follows that, for any $(j,t)\in\llbracket
1,Bn\rrbracket\times[-\theta,\theta]$,

$\displaystyle\big{|}\mathsf{x}_{j}$
$\displaystyle(tk^{1/3})+2^{-1/2}t^{2}k^{2/3}+2^{-7/6}(3\pi)^{2/3}j^{2/3}\big{|}$
$\displaystyle\leq\bigg{|}\mathsf{x}_{j}(tk^{1/3})-k^{2/3}\cdot(\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}t^{2})+\Big{(}\displaystyle\frac{3\pi}{4\mathfrak{c}^{1/2}}\Big{)}^{2/3}j^{2/3}\bigg{|}+j^{2/3}\cdot\bigg{|}2^{-7/6}(3\pi)^{2/3}-\Big{(}\displaystyle\frac{3\pi}{4\mathfrak{c}}\Big{)}^{2/3}\bigg{|}$
$\displaystyle\qquad+k^{2/3}\cdot\big{|}\mathfrak{a}+\mathfrak{b}t-t^{2}(\mathfrak{c}-2^{-1/2})\big{|}$
$\displaystyle\leq\mathfrak{C}\theta^{3}k^{2/3}+\displaystyle\frac{\delta
j^{2/3}}{5B^{2}}+\displaystyle\frac{\delta\theta^{2}k^{2/3}}{30B^{2}}\leq\displaystyle\frac{\delta
n^{2/3}}{90B}+\displaystyle\frac{\delta n^{2/3}}{5B}+\displaystyle\frac{\delta
n^{2/3}}{30}\leq\delta n^{2/3},$

In the second bound we used the fact that we are restricting to
$\mathscr{E}\subseteq\mathscr{E}_{1}$ (with (20.4)), (20.9), and (20.8); in
the third we used (20.3) and the fact that $j\leq Bn$; and in the fourth we
used the fact that $B>1$. Since $[-\theta k^{1/3},\theta
k^{1/3}]=[-Bn^{1/3},Bn^{1/3}]$, this (upon replacing $t$ by $B^{-1}\theta
t=(k^{-1}n)^{1/3}t$) establishes (20.6) and thus the theorem. ∎

####  20.2. Uniform Convergence to Bridge-Limiting Measure Processes

To show 20.1, we will use some form of 10.1, stating convergence of non-
intersecting Brownian bridges without upper and lower boundaries to a limit
shape. Observe that 10.1 assumes that the limiting starting and ending data
for this family (denoted there by $\mu_{a}$ and $\mu_{b}$) have been fixed in
advance; this will not be the case in our context. So, in this section we
provide a variant of that result applying to all boundary data, subject to
certain conditions, uniformly.

To state this result, we first require the following set of measures. The
second condition below may be viewed as a continuum analog for the first
intersection of LOC events (recall Definition 16.1) appearing in the BTR event
of Definition 16.2), and the third as one for the GAP event from Definition
3.5. Observe that these conditions also serve as reformulations of those in
13.7 and 13.8, which will enable us to use results from Chapter 4 (such as
14.1).

######  Definition 20.2.

For any real numbers $U>0$ and $L\geq 1$, let
$\mathscr{P}(L;U)\subset\mathscr{P}_{\operatorname{fin}}$ denote the set of
measures $\mu$ satisfying the following three properties.

  1. (1)

We have $\mu(\mathbb{R})=L^{3/2}$ and
$\operatorname{supp}\mu\subseteq[-UL,U]$.

  2. (2)

For any real number $x\leq-1$, we have $\mu\big{(}[x,\infty)\big{)}\leq
U|x|^{3/2}$.

  3. (3)

Define (analogously to (10.4)) the function
$G=G^{\mu}:[0,L^{3/2}]\rightarrow\mathbb{R}$ by setting

(20.10) $\displaystyle
G(y)=\displaystyle\sup\Big{\\{}x\in\mathbb{R}:\mu\big{(}[x,U]\big{)}\geq
y\Big{\\}},\qquad\text{for each $y\in[0,L^{3/2}]$}.$

Then, for any real numbers $0\leq x\leq y\leq L^{3/2}$, we have $G(x)-G(y)\leq
U(y^{2/3}-x^{2/3})$.

Now we can state the following proposition, to be established at the end of
this section, which indicates the following. Given some $n$-tuples $\bm{u}$
and $\bm{v}$ satisfying variants of the gap event (from Definition 3.5) and of
the boundary tall rectangle event (from Definition 16.2), one can find a
limiting bridge-limiting measure process $\bm{\mu}$, with boundary data in
some $\mathscr{P}(L;U)$, that approximates non-intersecting Brownian bridges
with starting data $\bm{u}$ and ending data $\bm{v}$. In the below, we recall
the notation $\operatorname{emp}$ from (1.18), the Lévy metric
$d_{\operatorname{L}}$ from (10.1), and notation on measure-valued processes
and bridge-limiting measure processes from Section 10.1 (and Definition 10.2).

######  Proposition 20.3.

For any real numbers $\theta>0$ and $A,B,L,R\geq 1$, there exists a constant
$C=C(A,B,L,R,\theta)>1$ such that the following holds. Let $n\geq k\geq C$ be
integers with $n=L^{3/2}k$. Also let
$\bm{u},\bm{v}\in\overline{\mathbb{W}}_{n}$ be $n$-tuples such that, for any
index $a\in\\{u,v\\}$ and integers $1\leq i\leq j\leq n$, we have

(20.11) $\displaystyle-Bj^{2/3}-Bk^{2/3}\leq a_{j}\leq
Bk^{2/3}-B^{-1}j^{2/3},\quad\text{and}\quad a_{i}-a_{j}\leq
R(j^{2/3}-i^{2/3})+(\log n)^{30}i^{-1/3}.$

Sample $n$ non-intersecting Brownian bridges
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots,\mathsf{x}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[-Ak^{1/3},Ak^{1/3}]\big{)}$ from the
measure $\mathsf{Q}^{\bm{u};\bm{v}}$, and define the measure-valued process
$\bm{\mu}=(\mu_{t})_{t\in[0,1]}\in\mathcal{C}\big{(}[0,1];\mathscr{P}_{\operatorname{fin}}\big{)}$
by setting

(20.12)
$\displaystyle\mu_{t}=L^{3/2}\cdot\operatorname{emp}\Big{(}(2A)^{-1/2}k^{-2/3}\cdot\bm{\mathsf{x}}\big{(}A(2t-1)\cdot
k^{1/3}\big{)}\Big{)},\qquad\text{for each $t\in[0,1]$}.$

Then, there exist measures $\nu_{0},\nu_{1}\in\mathscr{P}(L;16B^{3}+12R)$ such
that
$\mathbb{P}\big{[}d_{\operatorname{L}}(\bm{\mu},\bm{\nu})\leq\theta\big{]}\geq
1-\theta$, where
$\bm{\nu}\in\mathcal{C}\big{(}[0,1];\mathscr{P}_{\operatorname{fin}}\big{)}$
denotes the bridge-limiting measure process on $[0,1]$ with boundary data
$(\nu_{0};\nu_{1})$.

The proof of 20.3 proceeds by combining 10.1 with the following two lemmas.
The first indicates that the set of measures $\mathscr{P}(L;U)$ from
Definition 20.2 is compact. The second indicates that, given any sequence
$\bm{a}$ satisfying (20.11), there exists a measure in
$\mathscr{P}(L;16B^{3}+12R)$ approximating the (shifted and rescaled)
empirical measure associated with $\bm{a}$.

######  Lemma 20.4.

The set $\mathscr{P}(L;U)$ from Definition 20.2 is compact under the Lévy
metric.

###### Proof.

Fix some sequence of measures $\mu_{1},\mu_{2},\ldots\in\mathscr{P}(L;U)$.
Since each $\operatorname{supp}\mu_{j}\subseteq[-UL,U]$, this sequence is
tight and therefore admits a weak limit
$\mu\in\mathscr{P}_{\operatorname{fin}}$ with
$\lim_{j\rightarrow\infty}d_{\operatorname{L}}(\mu_{j},\mu)=0$. To show
$\mathscr{P}(L;U)$ is compact, we must verify that $\mu\in\mathscr{P}(L;U)$.
That $\mu$ satisfies Item 1 of Definition 20.2 follows from the fact that each
$\mu_{j}$ does. Moreover, by weak convergence, we have

$\displaystyle\mu\big{(}[x,\infty)\big{)}=\lim_{x^{\prime}\rightarrow
x^{-}}\mu\big{(}(x^{\prime},U]\big{)}\leq\lim_{x^{\prime}\rightarrow
x^{-}}\bigg{(}\lim_{m\rightarrow\infty}\mu_{m}\big{(}(x^{\prime},U]\big{)}\bigg{)}\leq
U\cdot\displaystyle\lim_{x^{\prime}\rightarrow
x^{-}}|x^{\prime}|^{3/2}=U|x|^{3/2},$

and so $\mu$ also satisfies Item 2 of Definition 20.2. Defining
$G_{m}=G^{\mu_{m}}:[0,L^{3/2}]\rightarrow\mathbb{R}$ and
$G=G^{\mu}:[0,L^{3/2}]\rightarrow\mathbb{R}$ as in (20.10), we have by weak
convergence that, for any $0\leq x\leq y\leq L^{3/2}$,

$\displaystyle G(x)-G(y)$ $\displaystyle\leq\lim_{\varepsilon\rightarrow
0}\bigg{(}\lim_{m\rightarrow\infty}\big{(}G_{m}(x-\varepsilon)-G_{m}(y+\varepsilon)\big{)}\bigg{)}$
$\displaystyle\leq U\cdot\lim_{\varepsilon\rightarrow
0}\big{(}(y+\varepsilon)^{2/3}-(x+\varepsilon)^{2/3}\big{)}\leq
U(y^{2/3}-x^{2/3}),$

Thus, $\mu$ also satisfies Item 3 of Definition 20.2, and we conclude that
$\mu\in\mathscr{P}(L;U)$. ∎

######  Lemma 20.5.

For any real numbers $A\geq 1/2$; $B,L,R\geq 1$; and $\theta>0$, the following
holds for any sufficiently large integer $k\geq 1$. Let $n\geq k$ be an
integer with $n=L^{3/2}k$, and let
$\bm{a}=(a_{1},a_{2},\ldots,a_{n})\in\overline{\mathbb{W}}_{n}$ be an
$n$-tuple such that (20.11) holds for any integers $1\leq i\leq j\leq n$.
Defining the measure
$\mu=L^{3/2}\cdot\operatorname{emp}\big{(}(2A)^{-1/2}k^{-2/3}\cdot\bm{a}\big{)}\in\mathscr{P}_{\operatorname{fin}}$,
there exists a measure $\nu\in\mathscr{P}\big{(}L;16B^{3}+12R\big{)}$
satisfying $d_{{\rm L}}(\mu,\nu)\leq\theta$.

###### Proof.

Throughout this proof, we set $z_{j}=k^{-2/3}\cdot a_{j}$ for each integer
$j\in\llbracket 1,n\rrbracket$. Also denote $\ell=\lfloor k^{1/10}\rfloor$ and
$m=\big{\lceil}\ell^{-1}(n-1)\big{\rceil}$; we will omit the floors and
ceilings in what follows, assuming that $\ell=n^{1/10}$ and that $n=\ell m+1$,
as this will have litle effect on the proofs. We will also assume for
notational simplicity that $A=1/2$ (so that
$\mu=L^{3/2}\operatorname{emp}(k^{2/3}\cdot\bm{a})$), as the general scenario
is entirely analogous (and can be recovered from the $A=1/2$ case by scaling,
since we have imposed $A\geq 1/2$).

We will define the restriction $\nu_{m}$ of $\nu$ to the interval
$[z_{(i+1)\ell+1},z_{i\ell+1}]$, for each integer $i\in\llbracket
0,m-1\rrbracket$. First observe by the second bound in (20.11) that for
$\ell/2\leq j-i\leq 2\ell$ and sufficiently large $k$ we have

(20.13) $\displaystyle z_{i}-z_{j}\leq
R\cdot\displaystyle\frac{j^{2/3}-i^{2/3}}{k^{2/3}}+(\log
n)^{30}k^{-2/3}i^{-1/3}\leq
2R\cdot\displaystyle\frac{j^{2/3}-i^{2/3}}{k^{2/3}},$

where in the last inequality we used the fact that $j^{2/3}-i^{2/3}\geq(\log
k)^{30}i^{-1/3}$; the latter holds since

$\displaystyle
j^{2/3}-i^{2/3}\geq\frac{2(j-i)}{3j^{1/3}}\geq\frac{\ell}{3(3\ell
i)^{1/3}}\geq\displaystyle\frac{k^{1/15}}{5i^{1/3}}\geq(\log k)^{30}i^{-1/3},$

where in the first bound we used the fact that $j\geq i$; in the second that
$j-i\geq\ell/2$ and $j\leq i+2\ell\leq 3i\ell$; in the third that
$\ell=k^{1/10}$; and in the fourth that $k$ is sufficiently large.

Define the real numbers $\sigma_{0},\sigma_{1},\ldots,\sigma_{m-1}>0$ by

(20.14)
$\displaystyle\sigma_{0}=\displaystyle\frac{3n}{2mk}\cdot(z_{1}-z_{\ell+1})^{-3/2};\qquad\quad\sigma_{j}=\displaystyle\frac{n}{mk}\cdot(z_{j\ell+1}-z_{(j+1)\ell+1})^{-1},\quad\text{if
$j\in\llbracket 1,m-1\rrbracket$},$

assuming they are well-defined (that is, if $z_{j\ell+1}\neq
z_{(j+1)\ell+1}$). Then, we define the measure $\nu_{0}$ by

(20.15) $\displaystyle\nu_{0}=n(mk)^{-1}\cdot\delta_{z_{1}},\quad\text{if
$z_{1}=z_{\ell+1}$};\qquad\nu_{0}=\textbf{1}_{x\in[z_{\ell+1},z_{1}]}\cdot\sigma_{0}(z_{1}-x)^{1/2}dx,\quad\text{if
$z_{1}>z_{\ell+1}$},$

and the measures $\nu_{1},\nu_{2},\ldots,\nu_{m-1}$ by for each integer
$j\in\llbracket 1,m-1\rrbracket$ setting

(20.16)
$\displaystyle\nu_{j}=n(mk)^{-1}\cdot\delta_{z_{j\ell+1}},\quad\text{if
$z_{j\ell+1}=z_{(j+1)\ell+1}$};\quad\nu_{j}=\textbf{1}_{x\in[z_{(j+1)\ell+1},z_{j\ell+1}]}\cdot\sigma_{j}dx,\quad\text{if
$z_{j\ell+1}>z_{(j+1)\ell+1}$}.$

Observe in this way that $\nu_{j}(\mathbb{R})=n(mk)^{-1}$ for each
$j\in\llbracket 0,m-1\rrbracket$ (by the choice (20.14) of each $\sigma_{j}$).
Then, define $\nu=\sum_{j=0}^{m-1}\nu_{j}$, which satisfies

(20.17)
$\displaystyle\nu(\mathbb{R})=nk^{-1}=L^{3/2};\qquad\operatorname{supp}\nu=[z_{n},z_{1}]\subseteq[-2BL,2B],$

where the last inclusion follows from the facts that $-2Bn^{2/3}\leq-
Bn^{2/3}-Bk^{2/3}\leq a_{n}\leq a_{1}\leq Bk^{2/3}$ (by the first bound in
(20.11)), with the facts that $n=L^{3/2}k$ and that $z_{j}=k^{-2/3}a_{j}$.

We claim that $\nu$ satisfies $d_{\operatorname{L}}(\mu,\nu)\leq
n(mk)^{-1}\leq 2\ell k^{-1}\leq\theta$ (where the second inequality follows
from the fact that $n=\ell m+1$, and the third holds for sufficiently large
$k$ since $\ell=k^{1/10}$). To show this, observe by (10.1) that it suffices
to verify for any real number $x\in\mathbb{R}$ that
$\nu\big{(}[x,\infty)\big{)}-n(mk)^{-1}\leq\mu\big{(}[x,\infty)\big{)}<\nu\big{(}[x,\infty)\big{)}+n(mk)^{-1}$.
Since $\mu(\mathbb{R})=L^{3/2}=\nu(\mathbb{R})$,
$\operatorname{supp}\mu\subseteq[z_{n},z_{1}]$, and
$\operatorname{supp}\nu\subseteq[z_{n},z_{1}]$, this holds if
$x\notin[z_{n},z_{1}]$. If $x\in[z_{n},z_{1}]$, then let $j\in\llbracket
1,m\rrbracket$ be an integer such that $z_{j\ell+1}\leq x\leq
z_{(j-1)\ell+1}$. We have

(20.18)
$\displaystyle\mu\big{(}[x,\infty)\big{)}\geq\mu\big{(}[z_{(j-1)\ell+1},\infty)\big{)}=\displaystyle\frac{(j-1)n}{mk};\qquad\nu\big{(}[x,\infty)\big{)}\leq\nu\big{(}[z_{j\ell+1},\infty)\big{)}\leq\displaystyle\frac{jn}{mk},$

showing the lower bound
$\mu\big{(}[x,\infty)\big{)}\geq\nu\big{(}[x,\infty)\big{)}-n(mk)^{-1}$. The
proof of the corresponding upper bound
$\mu\big{(}[x,\infty)\big{)}\leq\nu\big{(}[x,\infty)\big{)}+n(mk)^{-1}$ is
very similar and thus omitted.

It remains to confirm that $\nu\in\mathscr{P}(L;16B^{3}+12R)$. By (20.17),
$\nu$ satisfies Item 1 of Definition 20.2. Next let us verify Item 2 of
Definition 20.2, indicating that
$\nu\big{(}[x,\infty)\big{)}\leq(16B^{3}+12R)|x|^{3/2}$ for each $x\leq-1$.
First observe that if $x>z_{1}$ then $\nu\big{(}[x,\infty)\big{)}=0$, and so
this holds. Similarly, if $x<z_{n}$, then
$\nu\big{(}[x,\infty)\big{)}=\nu(\mathbb{R})=L^{3/2}\leq(2B)^{3/2}|x|^{3/2}$.
In the last inequality, we used the facts that $x\leq-1$ to address the case
when $L\leq 2B$, and that $|x|\geq(2B)^{-1}L$ if $L\geq 2B$, as then
$x<z_{n}\leq 1-B^{-1}k^{-2/3}n^{2/3}=1-B^{-1}L\leq-(2B)^{-1}L$ (by the first
bound in (20.11), with the facts that $n=L^{3/2}k$ and $z_{j}=k^{-2/3}a_{j}$).

Now suppose that $z_{n}\leq x\leq z_{1}$, and let $j\in\llbracket
1,m\rrbracket$ be an integer such that $z_{j\ell+1}\leq x\leq
z_{(j-1)\ell+1}$. Then, (20.18) indicates that
$\nu\big{(}[x,\infty)\big{)}\leq jn(mk)^{-1}$. Moreover, since $x\leq
z_{(j-1)\ell+1}\leq B-B^{-1}k^{-2/3}\big{(}(j-1)\ell+1\big{)}^{2/3}$ (by
(20.11)), we have

(20.19) $\displaystyle|x|\geq
B^{-1}k^{-2/3}\big{(}(j-1)\ell+1\big{)}^{2/3}-B\geq
B^{-1}\Big{(}\displaystyle\frac{j\ell}{2k}\Big{)}^{2/3}-B\geq
B^{-1}\Big{(}\displaystyle\frac{jn}{4mk}\Big{)}^{2/3}-B,$

where we have used the facts that $1\leq j\leq k$ and that $n=\ell m+1\leq
2\ell m$. It follows that $\nu\big{(}[x,\infty)\big{)}\leq jn(mk)^{-1}\leq
16B^{3}|x|^{3/2}$ (by the fact that $x\leq-1$ if $jn(mk)^{-1}\leq 16B^{3}$,
and that $B^{-1}(jn/4mk)^{2/3}-B\geq(2B)^{-1}(jn/4mk)^{2/3}$ with (20.19)
otherwise), confirming Item 2 of Definition 20.2.

It remains to verify that $\nu$ satisfies Item 3 of Definition 20.2. To do
this, define $G=G^{\nu}:[0,L^{3/2}]\rightarrow\mathbb{R}$ as in (20.10); we
must show for any real numbers $0\leq x\leq y\leq L^{3/2}$ that
$G(x)-G(y)\leq(16B^{3}+12R)(y^{2/3}-x^{2/3})$. We may assume that there exist
an integer $j\in\llbracket 1,m\rrbracket$ such that $z_{j\ell+1}\leq G(y)\leq
G(x)\leq z_{(j-1)\ell+1}$ (for in general, the result would follow from
summing the bound over a sequence of pairs in the same such intervals), and
also that $z_{j\ell+1}\neq z_{(j-1)\ell+1}$ (for otherwise $G(x)-G(y)=0$). If
$j\in\llbracket 2,m\rrbracket$, then we have

$\displaystyle G(x)-G(y)=\displaystyle\frac{y-x}{\sigma_{j-1}}$
$\displaystyle=n^{-1}mk(z_{(j-1)\ell+1}-z_{j\ell+1})(y-x)$ $\displaystyle\leq
2Rn^{-1}mk^{1/3}\Big{(}(j\ell+1)^{2/3}-\big{(}(j-1)\ell+1\big{)}^{2/3}\Big{)}(y-x)$
$\displaystyle\leq 4Rn^{-1}mk^{1/3}\ell^{2/3}j^{-1/3}(y-x)$ $\displaystyle\leq
6Rn^{-1}mk^{1/3}\ell^{2/3}j^{-1/3}y^{1/3}(y^{2/3}-x^{2/3})\leq
6R(y^{2/3}-x^{2/3}),$

where in the first statement we used (20.16); in the second we used (20.14);
in the third we used (20.13); in the fourth we used the fact that
$(j\ell+1)^{2/3}-\big{(}(j-1)\ell+1\big{)}^{2/3}\leq(j\ell)^{2/3}-\big{(}(j-1)\ell\big{)}^{2/3}\leq\ell^{2/3}(j-1)^{-1/3}\leq
2\ell^{2/3}j^{-1/3}$; in the fifth we used the fact that $y^{2/3}-x^{2/3}\geq
2(y-x)/(3y^{1/3})$; and in the sixth we used the bounds $\ell\leq nm^{-1}$ and
$y\leq jn(mk)^{-1}$ (by (20.10), the fact that $z_{j\ell+1}\leq G(y)$, and the
fact that $\nu\big{(}[z_{j\ell+1},\infty)\big{)}=jn(mk)^{-1}$). If instead
$j=1$, then observe by (20.15) and (20.14) that for any real number
$z\in[z_{\ell+1},z_{1}]$,

$\displaystyle\mu\big{(}[z,\infty)\big{)}=\displaystyle\frac{2\sigma_{0}}{3}\cdot(z_{1}-z)^{3/2}=\displaystyle\frac{n}{mk}\cdot\Big{(}\displaystyle\frac{z_{1}-z}{z_{1}-z_{\ell+1}}\Big{)}^{3/2}.$

Hence, by (20.10) we have for any $r$ with $G(r)\in[z_{\ell+1},z_{1}]$ that
$G(r)=z_{1}-(rn^{-1}mk)^{2/3}(z_{1}-z_{\ell+1})$. This, together with the fact
that $z_{1}-z_{\ell+1}\leq 2Rk^{-2/3}\ell^{2/3}$ by (20.13) (and that $n=\ell
m+1\geq\ell m$), implies that

$\displaystyle G(x)-G(y)\leq 2R(n^{-1}m\ell)^{2/3}(y^{2/3}-x^{2/3})\leq
2R(y^{2/3}-x^{2/3})\leq(16B^{3}+12R)(y^{2/3}-x^{2/3}).$

Thus, $\nu$ satisfies Item 3 of Definition 20.2, and so
$\nu\in\mathscr{P}(L;U)$, establishing the lemma. ∎

Now we can establish 20.3.

###### Proof of 20.3.

We first set some notation. Define $U=16B^{3}+12R$. Since $\mathscr{P}(L;U)$
is compact by 20.4, there exists an integer $K=K(B,R,L,\theta)\geq 1$ and
measures $\nu^{(1)},\nu^{(2)},\ldots,\nu^{(K)}\in\mathscr{P}(L;U)$ so that,
for any measure $\nu\in\mathscr{P}(L;U)$, there exists some
$i=i(\nu)\in\llbracket 1,K\rrbracket$ with
$d_{\operatorname{L}}\big{(}\nu,\nu^{(i)}\big{)}<\theta/4$. For any integers
$i_{1},i_{2}\in\llbracket 1,K\rrbracket$, let
$\bm{\nu}^{(i_{1},i_{2})}\in\mathcal{C}\big{(}[0,1];\mathscr{P}_{\operatorname{fin}}\big{)}$
denote the bridge-limiting measure process on $[0,1]$ with boundary data
$\big{(}\nu^{(i_{1})};\nu^{(i_{2})}\big{)}$.

Given any integers $j\geq 1$ and $i\in\llbracket 1,K\rrbracket$, set $J=\lceil
L^{3/2}j\rceil$ and fix a $J$-tuple
$\bm{a}^{(i;j)}\in\overline{\mathbb{W}}_{J}$ such that the following holds.
Defining the measure
$\mu^{(i;j)}=L^{3/2}\cdot\operatorname{emp}\big{(}(2A)^{-1/2}j^{-2/3}\cdot\bm{a}^{(i;j)}\big{)}$,
we have

(20.20)
$\displaystyle\lim_{j\rightarrow\infty}d_{\operatorname{L}}\big{(}\mu^{(i;j)},\nu^{(i)}\big{)}=0.$

We will omit the ceilings in what follows, assuming that $J=L^{3/2}j$, as this
will barely affect the proof. Next, for any integers $i_{1},i_{2}\in\llbracket
1,K\rrbracket$, sample a family of $J$ non-intersecting Brownian bridges
$\bm{\mathsf{y}}^{(i_{1},i_{2};j)}=\big{(}\mathsf{y}_{1}^{(i_{1},i_{2};j)},\mathsf{y}_{2}^{(i_{1},i_{2};j)},\ldots,\mathsf{y}_{J}^{(i_{1},i_{2};j)}\big{)}\in\llbracket
1,J\rrbracket\times\mathcal{C}\big{(}[-Aj^{1/3},Aj^{1/3}]\big{)}$ from the
measure $\mathsf{Q}^{\bm{a}^{(i_{1};j)};\bm{a}^{(i_{2};j)}}$. Then the
rescaled paths
$\big{(}y_{1}^{(i_{1},i_{2};j)},y_{2}^{(i_{1},i_{2};j)},\ldots,y_{J}^{(i_{1},i_{2};j)}\big{)}\in\llbracket
1,J\rrbracket\times\mathcal{C}\big{(}[0,1]\big{)}$, defined by setting
$y_{h}^{(i_{1},i_{2};j)}(t)=(2A)^{-1/2}j^{-2/3}\cdot\mathsf{y}_{h}^{(i_{1},i_{2};j)}\big{(}(2A-1)tj^{1/3}\big{)}$
for each $(h,t)\in\llbracket 1,J\rrbracket\times[0,1]$, are non-intersecting
Brownian bridges with variances $j^{-1}=L^{3/2}J^{-1}$. So, by (20.20), 10.1
yields

$\displaystyle\lim_{j\rightarrow\infty}\mathbb{P}\Big{[}d_{\operatorname{L}}\big{(}\bm{\mu}^{(i_{1},i_{2};j)},\bm{\nu}^{(i_{1},i_{2})}\big{)}<\displaystyle\frac{\theta}{4}\Big{]}=1,$

where we have defined the measure-valued process
$\bm{\mu}^{(i_{1},i_{2};j)}=\big{(}\mu_{t}^{(i_{1},i_{2};j)}\big{)}\in\mathcal{C}\big{(}[0,1];\mathscr{P}_{\operatorname{fin}}\big{)}$
by setting

(20.21)
$\displaystyle\mu_{t}^{(i_{1},i_{2};j)}=L^{3/2}\cdot\operatorname{emp}\Big{(}(2A)^{-1/2}j^{-2/3}\cdot\bm{\mathsf{y}}\big{(}A(2t-1)j^{1/3}\big{)}\Big{)},\qquad\text{for
each $t\in[0,1]$}.$

This yields a constant $C_{1}=C_{1}(A,B,R,L,\theta)>1$ such that, for any
integers $i_{1},i_{2}\in\llbracket 1,K\rrbracket$,

(20.22)
$\displaystyle\mathbb{P}\Big{[}d_{\operatorname{L}}\big{(}\bm{\mu}^{(i_{1},i_{2};j)},\bm{\nu}^{(i_{1},i_{2})}\big{)}<\displaystyle\frac{\theta}{4}\Big{]}\geq
1-\displaystyle\frac{\theta}{2},\qquad\text{whenever $j\geq C_{1}$}.$

Now, recall from (20.12) that $\mu_{0}$ and $\mu_{1}$ are given by
$\mu_{0}=L^{3/2}\cdot\operatorname{emp}\big{(}(2A)^{-1/2}k^{-2/3}\cdot\bm{u}\big{)}$
and
$\mu_{1}=L^{3/2}\cdot\operatorname{emp}\big{(}(2A)^{-1/2}k^{-2/3}\cdot\bm{v}\big{)}$.
By 20.5, there exists a constant $C_{2}=C_{2}(A,B,R,L,\theta)>1$ and measures
$\nu_{0}^{\prime},\nu_{1}^{\prime}\in\mathscr{P}(L;U)$ such that

(20.23) $\displaystyle
d_{\operatorname{L}}(\mu_{0},\nu_{0}^{\prime})<\displaystyle\frac{\theta}{4},\quad\text{and}\quad
d_{\operatorname{L}}(\mu_{1},\nu_{1}^{\prime})<\displaystyle\frac{\theta}{4},$

whenever $k\geq C_{2}$. Setting $C=\max\\{C_{1},C_{2}\\}$, we assume for the
remainder of this proof that $k\geq C$.

Fix integers $i_{1},i_{2}\in\llbracket 1,K\rrbracket$ satisfying

(20.24) $\displaystyle
d_{\operatorname{L}}\big{(}\nu_{0}^{\prime},\nu^{(i_{1})}\big{)}<\displaystyle\frac{\theta}{4};\qquad
d_{\operatorname{L}}\big{(}\nu_{1}^{\prime},\nu^{(i_{2})}\big{)}<\displaystyle\frac{\theta}{4}.$

Also observe that, by (20.22) and (10.2) (taken at $t\in\\{0,1\\}$), we have
$d_{\operatorname{L}}\big{(}\mu^{(i_{1};k)},\nu^{(i_{1})}\big{)}<\theta/4$ and
$d_{\operatorname{L}}\big{(}\mu^{(i_{2};k)},\nu^{(i_{2})}\big{)}<\theta/4$
(where these events hold deterministically, as
$\big{(}\mu^{(i_{1};k)};\mu^{(i_{2};k)}\big{)}$ constitutes the deterministic
boundary data for $\bm{\mathsf{y}}^{(i_{1},i_{2};k)}$). Together with (20.23)
and (20.24), this gives

(20.25) $\displaystyle
d_{\operatorname{L}}\big{(}\mu_{0},\mu^{(i_{1};k)}\big{)}<\displaystyle\frac{3\theta}{4},\qquad\text{and}\qquad
d_{\operatorname{L}}\big{(}\mu_{1},\mu^{(i_{2};k)}\big{)}<\displaystyle\frac{3\theta}{4}.$

It then suffices to show that it is possible to couple $\bm{\mathsf{x}}$ and
$\bm{\mathsf{y}}^{(i_{1},i_{2};k)}$ in two ways, so that under the first
coupling we have for each $(x,t)\in\mathbb{R}\times[0,1]$ that

(20.26)
$\displaystyle\displaystyle\int_{x}^{\infty}\mu_{t}(dr)\leq\displaystyle\int_{x-3\theta/4}^{\infty}\mu_{t}^{(i_{1},i_{2};k)}(dr)+\displaystyle\frac{3\theta}{4}.$

and under the second we have for each $(x,t)\in\mathbb{R}\times[0,1]$ that

(20.27)
$\displaystyle\displaystyle\int_{x}^{\infty}\mu_{t}(dr)\geq\displaystyle\int_{x+3\theta/4}^{\infty}\mu_{t}^{(i_{1},i_{2};k)}(dr)-\displaystyle\frac{3\theta}{4}.$

Indeed, assuming (20.26) and (20.27), set
$\nu_{0}=\nu^{(i_{1})}\in\mathscr{P}(L;U)$ and
$\nu_{1}=\nu^{(i_{2})}\in\mathscr{P}(L;U)$, so that
$\bm{\nu}=\bm{\nu}^{(i_{1},i_{2})}\in\mathcal{C}\big{(}[0,1];\mathscr{P}_{\operatorname{fin}}\big{)}$
is the bridge-limiting measure process on $[0,1]$ with boundary data
$(\nu_{0};\nu_{1})$. Then, (20.22) (with the definition (10.1) of
$d_{\operatorname{L}}$), (20.26), and (20.27) together imply that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{x\in\mathbb{R}}\bigcap_{t\in[0,1]}\bigg{\\{}\displaystyle\int_{x}^{\infty}\mu_{t}(dr)\leq\displaystyle\int_{x-\theta}^{\infty}\nu_{t}^{(i_{1},i_{2})}(dr)+\theta\bigg{\\}}\Bigg{]}\geq
1-\displaystyle\frac{\theta}{2};$
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{x\in\mathbb{R}}\bigcap_{t\in[0,1]}\bigg{\\{}\displaystyle\int_{x}^{\infty}\mu_{t}(dr)\geq\displaystyle\int_{x+\theta}^{\infty}\nu_{t}^{(i_{1},i_{2})}(dr)-\theta\bigg{\\}}\Bigg{]}\geq
1-\displaystyle\frac{\theta}{2}.$

Together with a union bound, the fact that
$\bm{\nu}=\bm{\nu}^{(i_{1},i_{2})}$, and the definition (10.1) of
$d_{\operatorname{L}}$, this implies the proposition.

It therefore remains to find a coupling between $\bm{\mathsf{x}}$ and
$\bm{\mathsf{y}}^{(i_{1},i_{2};k)}$ such that (20.26) holds, and one such that
(20.27) does. Both follow in a very similar way from 4.6, so let us only
implement the former. To this end, observe since
$\mu_{0}=L^{3/2}\cdot\operatorname{emp}\big{(}(2A)^{-1/2}k^{-2/3}\cdot\bm{u}\big{)}$
and
$\mu_{0}^{(i_{1},i_{2};k)}=\mu^{(i_{1};k)}=L^{3/2}\cdot\operatorname{emp}\big{(}(2A)^{-1/2}k^{-2/3}\cdot\bm{a}^{(i_{1};k)}\big{)}$
that (20.25) (with (10.1)) yields

$\displaystyle u_{h}\leq a_{h-\lfloor 3\theta
L^{3/2}/4n\rfloor}^{(i_{1};k)}+\displaystyle\frac{3\theta}{4}\cdot(2A)^{1/2}k^{2/3}=a_{h-\lfloor
3\theta
k/4\rfloor}^{(i_{1};k)}+\displaystyle\frac{3\theta}{4}\cdot(2A)^{1/2}k^{2/3},$

for any integer $h\in\llbracket 1,n\rrbracket$ (where we also used the fact
that $L^{-3/2}n=k$). Similarly, $v_{h}\leq a_{h-\lfloor 3\theta
k/4\rfloor}^{(i_{2};k)}+3(2A)^{1/2}\theta k^{2/3}/4$ for any $h\in\llbracket
1,k\rrbracket$. Thus, since the laws of $\bm{\mathsf{x}}$ and
$\bm{\mathsf{y}}$ are given by $\mathsf{Q}^{\bm{u};\bm{v}}$ and
$\mathsf{Q}^{\bm{a}^{(i_{1};k)};\bm{a}^{(i_{2};k)}}$, respectively, 4.6 yields
a coupling between these two ensembles such that
$\mathsf{x}_{h}(tk^{1/3})\leq\mathsf{y}_{h-\lfloor 3\theta
k/4\rfloor}^{(i_{1},i_{2};k)}(tk^{1/3})+3\theta(2A)^{1/2}k^{2/3}/4$, or
equivalently

$\displaystyle(2A)^{-1/2}k^{-2/3}\cdot\mathsf{x}_{h}(tk^{1/3})\leq(2A)^{-1/2}k^{-2/3}\cdot\mathsf{y}_{h-\lfloor
3\theta
k/4\rfloor}^{(i_{1},i_{2};k)}(tk^{1/3})+\displaystyle\frac{3\theta}{4},\quad\text{for
all $(h,t)\in\llbracket 1,n\rrbracket\times[0,1]$}.$

This, together with (20.12) and (20.21), establishes (20.26) and thus the
proposition. ∎

####  20.3. Proof of 20.1

In this section we establish 20.1.

###### Proof of 20.1.

Throughout this proof, we recall the notation $\operatorname{emp}$ from
(1.18); that on measure-valued processes, bridge-limiting measures, and
inverted height functions from Section 10.1; the set of measures
$\mathscr{P}(L;U)$ from Definition 20.2; and the completed rectangle event CTR
from Definition 16.5. Let us briefly outline how we will proceed. First, using
16.4, we will exhibit a coupling between $\bm{\mathsf{x}}$ and an ensemble
$\bm{\mathsf{y}}$ of non-intersecting Brownian bridges without lower boundary,
so that their upper paths are close with high probability. Using 20.3, we will
show that the empirical measure of the latter converges under the Lévy metric
to a bridge limiting measure $\bm{\nu}$, with boundary data in some
$\mathscr{P}(L;U)$. By restricting to the gap event GAP, we will show that
this implies the paths in $\bm{\mathsf{x}}$ are approximated by the inverted
height function $G=G^{\bm{\nu}}$ associated with $\bm{\nu}$; we will then use
14.1 to show that the edge behavior of the latter behaves as the function
appearing in (20.2).

To implement this, set $\widetilde{B}=192B^{3}$; denote the integers
$n^{\prime}=\lceil L^{1/2^{4000}}k\rceil$ and $n^{\prime\prime}=\lceil
L^{2^{1/5000}}\rceil$. Also abbreviate the events
$\textbf{BTR}_{n}=\textbf{BTR}_{n}(4,B;k,L;2^{-6000})$,
$\textbf{CTR}_{n}=\textbf{CTR}_{n}(4,\widetilde{B};k,L;2^{-6000})$, and
$\textbf{GAP}_{n}=\textbf{GAP}_{n}\big{(}[-4k^{1/3},4k^{1/3}];R\big{)}$;
further abbreviate the $\sigma$-algebra
$\mathcal{F}_{\operatorname{ext}}=\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}\rrbracket\times[-2k^{1/3},2k^{1/3}]\big{)}$. Analogously to
(19.22), define the $\mathcal{F}_{\operatorname{ext}}$-measurable event

(20.28)
$\displaystyle\mathscr{A}_{1}=\textbf{BTR}_{n}\cap\bigcap_{j=1}^{n^{\prime}}\textbf{LOC}_{j}\big{(}\\{-2k^{1/3},2k^{1/3}\\};-\widetilde{B}j^{2/3}-\widetilde{B}k^{2/3};\widetilde{B}k^{2/3}-\widetilde{B}j^{2/3}\big{)}.$

By Definition 16.5, we have $\textbf{CTR}_{n}\subseteq\mathscr{A}_{1}$, and so
16.6 (applied at $D=1$) yields constants $c_{1}=c_{1}(B)>0$ and
$C_{1}=C_{1}(B)>1$ such that
$\mathbb{P}[\textbf{BTR}_{n}\setminus\mathscr{A}_{1}]\leq C_{1}e^{-c_{1}(\log
k)^{2}}$. Next, by the $A=4$ case of 16.4 (after altering $c_{1}$ and $C_{1}$
if necessary), there exists an $\mathcal{F}_{\operatorname{ext}}$-measurable
event $\mathscr{A}_{2}\subseteq\textbf{BTR}_{n}$ satisfying
$\mathbb{P}[\textbf{BTR}_{n}\setminus\mathscr{A}_{2}]\leq C_{1}e^{-c_{1}(\log
k)^{2}}$ and the following. Condition on $\mathcal{F}_{\operatorname{ext}}$
and restrict to $\mathscr{A}_{2}$. Denote the $n^{\prime}$-tuples
$\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}\rrbracket}(-2k^{1/3})\in\overline{\mathbb{W}}_{n^{\prime}}$ and
$\bm{v}=\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}\rrbracket}(2k^{1/3})\in\overline{\mathbb{W}}_{n^{\prime}}$, and
sample a family of $n^{\prime}$ non-intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{n^{\prime}})\in\llbracket
1,n^{\prime}\rrbracket\times\mathcal{C}\big{(}[-2k^{1/3},2k^{1/3}]\big{)}$.
There exist two couplings between $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such
that, under the first we have

(20.29)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{n^{\prime\prime}}\bigcap_{|t|\leq
2k^{1/3}}\big{\\{}\mathsf{x}_{j}(t)\leq\mathsf{y}_{j}(t)-L^{-1/2^{5000}}k^{2/3}\big{\\}}\Bigg{]}\geq
1-C_{1}e^{-c_{1}(\log k)^{2}},$

and under the second we almost surely have

(20.30) $\displaystyle\mathsf{x}_{j}(t)\geq\mathsf{y}_{j}(t),\qquad\text{for
each $(j,t)\in\llbracket 1,n^{\prime}\rrbracket\times[-2k^{1/3},2k^{1/3}]$}.$

Denote the $\mathcal{F}_{\operatorname{ext}}$-measurable event
$\mathscr{A}=\mathscr{A}_{1}\cap\mathscr{A}_{2}\cap\textbf{GAP}_{n}\big{(}\\{-2k^{1/3},2k^{1/3}\\};R\big{)}$.
In view of the inclusion
$\textbf{GAP}_{n}\subseteq\textbf{GAP}_{n}\big{(}\\{-2k^{1/3},2k^{1/3}\\};R\big{)}$,
(20.1) and a union bound together indicate for sufficiently large $k$ that

(20.31) $\displaystyle\begin{aligned}
\mathbb{P}[\mathscr{A}]&\geq\mathbb{P}[\textbf{BTR}_{n}\cap\textbf{GAP}_{n}]-\mathbb{P}[\textbf{BTR}_{n}\setminus\mathscr{A}_{1}]-\mathbb{P}[\textbf{BTR}_{n}\setminus\mathscr{A}_{2}]\\\
&\geq 1-\varpi-2C_{1}e^{-c_{1}(\log k)^{2}}\geq
1-\displaystyle\frac{3\varpi}{2}.\end{aligned}$

For the remainder of this proof, we condition on
$\mathcal{F}_{\operatorname{ext}}$ and restrict to the event $\mathscr{A}$. By
(20.31), the fact that $\mathbb{P}[\textbf{GAP}_{n}]\geq 1-\varpi$ (by
(20.1)), and a union bound, it then suffices to show for some constant
$C=C(B,R)>1$ and sufficiently large $k$ that there exist real numbers
$\mathfrak{a},\mathfrak{b}\in\mathbb{R}$ and $\mathfrak{c}\in[C^{-1},C]$ such
that

(20.32)
$\displaystyle\mathbb{P}[\textbf{GAP}_{n}\cap\mathscr{E}^{\complement}]\leq\displaystyle\frac{\varpi}{2},$

where we have defined the event

(20.33)
$\displaystyle\mathscr{E}=\bigcap_{j=1}^{\lfloor\theta^{3}k\rfloor}\bigcap_{|t|\leq\theta}\Bigg{\\{}\bigg{|}\mathsf{x}_{j}(tk^{1/3})-k^{2/3}\cdot(\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}t^{2})+\Big{(}\displaystyle\frac{3\pi}{4\mathfrak{c}^{1/2}}\Big{)}^{2/3}j^{2/3}\bigg{|}\leq
C\theta^{3}k^{2/3}\Bigg{\\}}.$

To do this, set $U=16\widetilde{B}^{3}+12R$; let
$L^{\prime}=(k^{-1}n^{\prime})^{2/3}\geq L^{1/2^{4500}}$, so that
$n^{\prime}=(L^{\prime})^{3/2}k$; and define the measure-valued process
$\bm{\mu}=(\mu_{t})_{t\in[0,1]}\in\mathcal{C}\big{(}[0,1];\mathscr{P}_{\operatorname{fin}}\big{)}$
by setting

(20.34)
$\displaystyle\mu_{t}=(L^{\prime})^{3/2}\cdot\operatorname{emp}\Big{(}(2k^{2/3})^{-1}\cdot\bm{\mathsf{y}}\big{(}(4t-2)k^{1/3})\big{)}\Big{)},\qquad\text{for
each $t\in[0,1]$}.$

Since we have restricted to
$\mathscr{A}\subseteq\textbf{GAP}_{n}\big{(}\\{-2k^{1/3},2k^{1/3}\\};R\big{)}\cap\mathscr{A}_{1}$,
we have by Definition 3.5 (with the fact that $(\log n)^{25}\leq(\log
n^{\prime})^{30}$ for sufficiently large $k$, as $n=L^{3/2}k\leq
L^{3/2}n^{\prime}$) and (20.28) that $\bm{u}$ and $\bm{v}$ satisfy (20.11),
with the $B$ there equal to $\widetilde{B}$ here.

Thus, 20.3 applies, with the $(\bm{\mathsf{x}};n;A,B)$ there equal to
$(\bm{\mathsf{y}};n^{\prime};2,\widetilde{B})$ here. Setting
$U=16\widetilde{B}^{3}+12R$, it yields measures
$\nu_{0},\nu_{1}\in\mathscr{P}(L^{\prime};U)$ such that
$\mathbb{P}\big{[}d_{\operatorname{L}}(\bm{\mu},\bm{\nu})\leq\theta^{6}/2\big{]}\geq
1-\varpi/8$, where
$\bm{\nu}\in\mathcal{C}\big{(}[0,1];\mathscr{P}_{\operatorname{fin}}\big{)}$
is the bridge-limiting measure process on $[0,1]$ with boundary data
$(\mu_{0};\mu_{1})$. Denote the inverted height function associated with
$\bm{\nu}$ (recall Definition 10.4) by
$G=G^{\bm{\nu}}:\big{[}0,(L^{\prime})^{3/2}\big{]}\rightarrow\mathbb{R}$. By
(20.34), the definitions (10.1) of $d_{\operatorname{L}}$ and (10.4) of $G$,
the bound
$\mathbb{P}\big{[}d_{\operatorname{L}}(\bm{\mu},\bm{\nu})\leq\theta^{6}/2\big{]}\geq
1-\varpi/8$ is equivalent to

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{n^{\prime}}\bigcap_{|t|\leq
2}\bigg{\\{}k^{-2/3}\cdot\mathsf{y}_{j+\lfloor\theta^{6}k\rfloor}(tk^{1/3})-\theta^{6}\leq
2G\Big{(}\displaystyle\frac{t+2}{4},\displaystyle\frac{j}{k}\Big{)}\leq
k^{-2/3}\cdot\mathsf{y}_{j-\lfloor\theta^{6}k\rfloor}(tk^{1/3}$
$\displaystyle)+\theta^{6}\bigg{\\}}\Bigg{]}$ $\displaystyle\geq
1-\displaystyle\frac{\varpi}{8},$

where we have denoted $\mathsf{y}_{j}=\infty$ if $j<1$ and
$\mathsf{y}_{j}=-\infty$ if $j>n^{\prime}$. Together with the couplings
(20.29) and (20.30), and a union bound, it follows for $k$ sufficiently large
(so that $C_{1}e^{-c_{1}(\log k)^{2}}\leq\varpi/4$) that
$\mathbb{P}[\mathscr{E}_{0}]\geq 1-\varpi/2$, where we have defined the event

$\displaystyle\mathscr{E}_{0}=\bigcap_{j=1}^{n^{\prime\prime}-\lfloor\theta^{6}k\rfloor}\bigcap_{|t|\leq
2}\bigg{\\{}k^{-2/3}\cdot\mathsf{x}_{j+\lfloor\theta^{6}k\rfloor}($
$\displaystyle tk^{1/3})-\theta^{6}-L^{-1/2^{5000}}$ $\displaystyle\leq
2G\Big{(}\displaystyle\frac{t+2}{4},\displaystyle\frac{j}{k}\Big{)}\leq
k^{-2/3}\cdot\mathsf{x}_{j-\lfloor\theta^{6}k\rfloor}(tk^{1/3})+\theta^{6}\bigg{\\}}.$

Thus, to show (20.32), it suffices to show that there exists a constant
$C=C(B,R)>1$ and real numbers $\mathfrak{a},\mathfrak{b}\in\mathbb{R}$ and
$\mathfrak{c}\in[C^{-1},C]$ such that

(20.35)
$\displaystyle\mathscr{E}_{0}\cap\textbf{GAP}_{n}\subseteq\mathscr{E}.$

To this end, restrict to the event $\mathscr{E}_{0}\cap\textbf{GAP}_{n}$.
Then, for any $(j,t)\in\llbracket
1,n^{\prime\prime}-\theta^{6}k\rrbracket\times[-2,2]$,

(20.36) $\displaystyle
k^{-2/3}\cdot\mathsf{x}_{j}(tk^{1/3})\geq^{-2/3}\cdot\mathsf{x}_{j+\lfloor\theta^{6}k\rfloor}(tk^{1/3})\geq
2G\Big{(}\displaystyle\frac{t+2}{4},\displaystyle\frac{j}{k}+2\theta^{6}\Big{)}-\theta^{6},$

where in the first bound we used the fact that
$\mathsf{x}_{j}\geq\mathsf{x}_{j^{\prime}}$ whenever $j\leq j^{\prime}$, and
in the second we used the fact that we are restricting to $\mathscr{E}_{0}$.
Similarly, for any $(j,t)\in\llbracket
1,n^{\prime\prime}-\theta^{6}k\rrbracket\times[-2,2]$, we have for
sufficiently large $k$ that

(20.37) $\displaystyle\begin{aligned}
k^{-2/3}\cdot\mathsf{x}_{j}(tk^{1/3})&\leq
k^{-2/3}\cdot\mathsf{x}_{j+\lfloor\theta^{6}k\rfloor}(tk^{1/3})+k^{-2/3}\big{(}R(j+\theta^{6}k)^{2/3}-Rj^{2/3}+(\log
n)^{25}\big{)}\\\ &\leq
2G\Big{(}\displaystyle\frac{t+2}{4},\displaystyle\frac{j}{k}\Big{)}+\theta^{6}+L^{-1/2^{5000}}+(R+1)\theta^{4}\leq
2G\Big{(}\displaystyle\frac{t+2}{4},\displaystyle\frac{j}{k}\Big{)}+(R+3)\theta^{4}.\end{aligned}$

Here, in the first bound we used the fact that we have restricted to
$\textbf{GAP}_{n}$; in the second, we used the fact that we have restricted to
$\mathscr{E}_{0}$, as well as the bounds
$(j+\theta^{6}k)^{2/3}-j^{2/3}\leq(\theta^{6}k)^{2/3}=\theta^{4}k^{2/3}$ and
$(\log n)^{25}\leq\theta^{4}k^{2/3}$ for sufficiently large $k$ (as
$n^{\prime}\leq n\leq L^{3/2}k$); and in the third we used the fact that
$L^{-1/2^{5000}}\leq\theta^{4}$ (as $L\geq\theta^{-2^{6000}}$). Together with
the fact that $n^{\prime\prime}-\theta^{6}k\geq
L^{1/2^{5000}}k-\theta^{6}k\geq\theta^{3}k$ (as
$L^{1/2^{5000}}\geq\theta^{-2^{1000}}$), we find from (20.36) and (20.37)
that, for sufficiently large $k$,

(20.38) $\displaystyle
2G\Big{(}\displaystyle\frac{t+2}{4},\displaystyle\frac{j}{k}+2\theta^{6}\Big{)}-\theta^{6}\leq
k^{-2/3}\cdot\mathsf{x}_{j}(tk^{1/3})\leq
2G\Big{(}\displaystyle\frac{t+2}{4},\displaystyle\frac{j}{k}\Big{)}+(R+3)\theta^{4},$

for any $(j,t)\in\llbracket 1,\theta^{3}k\rrbracket\times[-2,2]$.

Now, observe that $\bm{\nu}$ is the bridge-limiting measure associated with
boundary data $\nu_{0},\nu_{1}\in\mathscr{P}(L;U)$. We will assume in what
follows that $G(0,0)=0=G(1,0)$ (as we may otherwise apply an affine
transformation to $G(t,y)$, using the second part of 10.19 to replace it by
$G(t,y)-(1-t)\cdot G(0,0)-t\cdot G(1,0)$; such an affine transformation will
only affect the constants $\mathfrak{a}$ and $\mathfrak{b}$ that appear
below). Then, by Definition 20.2, $\nu_{0}$ and $\nu_{1}$ satisfy 13.7 and
13.8 (with the $B$ there equal to $U$ here). Thus, 14.1 applies and yields a
constant $C_{2}=C_{2}(B,R)>1$ and real numbers
$\mathfrak{a}_{0},\mathfrak{b}_{0}\in\mathbb{R}$ and
$\mathfrak{c}\in[C_{2}^{-1},C_{2}]$ such that for $L>C_{2}$ and
$\theta<C_{2}^{-1}$ we have

$\displaystyle\displaystyle\sup_{|y|\leq
2\theta^{3}}\displaystyle\sup_{|s|\leq\theta}\Bigg{|}G\Big{(}\displaystyle\frac{1}{2}+s,y\Big{)}-(\mathfrak{a}_{0}+\mathfrak{b}_{0}s-\mathfrak{c}_{0}s^{2})+\bigg{(}\displaystyle\frac{3\pi}{4\mathfrak{c}_{0}^{1/2}}\bigg{)}^{1/2}y^{2/3}\Bigg{|}\leq
C_{2}\theta^{3}.$

Setting $s=t/4$, it follows for
$(\mathfrak{a},\mathfrak{b})=(2\mathfrak{a}_{0},\mathfrak{b}_{0}/2)$ and
$\mathfrak{c}=\mathfrak{c}_{0}/8\in\big{[}(8C_{2})^{-1},C_{2}\big{]}$ that

(20.39) $\displaystyle\displaystyle\sup_{|y|\leq
2\theta^{3}}\displaystyle\sup_{|t|\leq\theta}\bigg{|}2G\Big{(}\displaystyle\frac{1}{2}+\displaystyle\frac{t}{4},y\Big{)}-(\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}t^{2})+\Big{(}\displaystyle\frac{3\pi}{4\mathfrak{c}^{1/2}}\Big{)}^{2/3}y^{2/3}\bigg{|}\leq
2C_{2}\theta^{3}.$

In particular,

$\displaystyle\displaystyle\sup_{|y|\leq
2\theta^{3}-2\theta^{6}}\displaystyle\sup_{|t|\leq\theta}$
$\displaystyle\bigg{|}2G\Big{(}\displaystyle\frac{1}{2}+\displaystyle\frac{t}{4},y\Big{)}-2G\Big{(}\displaystyle\frac{1}{2}+\displaystyle\frac{t}{4},y+2\theta^{6}\Big{)}\bigg{|}$
$\displaystyle\leq
4C_{2}\theta^{3}+\Big{(}\displaystyle\frac{3\pi}{4\mathfrak{c}^{1/2}}\Big{)}^{2/3}\big{(}(y+\theta^{6})^{2/3}-y^{2/3}\big{)}\leq
4C_{2}\theta^{3}+4C_{2}^{1/3}\theta^{4}\leq 8C_{2}\theta^{3},$

where in the first inequality we applied (20.39) twice, and in the second we
used the facts that $\mathfrak{c}=\mathfrak{c}_{0}/8\geq(8C_{2})^{-1}$ and
that $(y+\theta^{6})^{2/3}-y^{2/3}\leq\theta^{4}$. Inserting this with (20.39)
into (20.38) yields

$\displaystyle\displaystyle\max_{j\in\llbracket
1,\theta^{3}k\rrbracket}\displaystyle\sup_{|t|\leq\theta}\bigg{|}k^{-2/3}\cdot\mathsf{x}_{j}(tk^{1/3})-(\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}t^{2})+\Big{(}\displaystyle\frac{3\pi}{4\mathfrak{c}^{1/2}}\Big{)}^{2/3}\Big{(}\displaystyle\frac{j}{k}\Big{)}^{2/3}\bigg{|}\leq(8C_{2}+R+3)\theta^{3}.$

This verifies that $\mathscr{E}$ holds (by its definition (20.33)) at
$C=8C_{2}+R+3$, thereby confirming (20.35) and establishing the proposition. ∎

##  Chapter 7 Appendices

###  21\. Proofs of Results From Chapter 1

####  21.1. Proofs of 4.28 and 4.31

In this section we establish first 4.28 and then 4.31.

###### Proof of 4.28.

By the second part of 4.17, the law of $\bm{\mathsf{x}}(t)$ is given by Dyson
Brownian motion run for time $t$, with initial data $\bm{u}$, conditioned to
end at $\bm{v}$ at time $\mathsf{T}$. Denoting by $\bm{H}(s)=\bm{H}_{n}(s)$ an
$n\times n$ Hermitian Brownian motion, the first part of 4.17 implies that the
latter process is given by $\operatorname{eig}\big{(}\bm{U}+\bm{H}(s)\big{)}$,
where $\bm{U}+\bm{H}(s)$ is conditioned to be of the form
$\bm{W}\bm{V}\bm{W}^{*}$ at time $s=\mathsf{T}$, for some unitary matrix
$\bm{W}\in\mathrm{U}(n)$.

Since the entries of $\bm{H}(s)$ are complex Gaussian random variables of
variance $s$, the density of $\bm{U}+\bm{H}(\mathsf{T})$ is proportional to

$\displaystyle\exp\Big{(}-\displaystyle\frac{1}{2\mathsf{T}}\operatorname{Tr}\bm{H}(\mathsf{T})^{2}\Big{)}d\bm{H}(\mathsf{T})=\exp\Big{(}-\displaystyle\frac{1}{2\mathsf{T}}\operatorname{Tr}(\bm{W}\bm{V}\bm{W}^{*}-\bm{U})^{2}\Big{)}d(\bm{W}\bm{V}\bm{W}^{*}).$

Upon conditioning on the eigenvalues of $\bm{W}\bm{V}\bm{W}^{*}$ (and dividing
by the constant
$e^{\mathsf{T}^{-1}\operatorname{Tr}(\bm{U}^{2}+\bm{V}^{2})}$), the above
density is proportional to (4.20), which therefore prescribes the law of the
unitary matrix $\bm{W}$.

Hence, denoting the $(i,j)$ entry of any matrix $\bm{M}$ by $(\bm{M})_{ij}$,
the law of the upper triangular entries $\big{(}\bm{U}+\bm{H}(s)\big{)}_{ij}$
(for $1\leq i\leq j\leq n$) are given by Brownian bridges conditioned to start
at $(\bm{U})_{ij}$ (at time $t=0$) and end at $(\bm{W}\bm{V}\bm{W}^{*})_{ij}$
(at time $s=\mathsf{T})$. Since any Brownian bridge
$B:[0,\mathsf{T}]\rightarrow\mathbb{R}$ with $B(0)=a$ and $B(\mathsf{T})=b$
can be represented as

$\displaystyle B(s)=\displaystyle\frac{\mathsf{T}-s}{\mathsf{T}}\cdot
a+\displaystyle\frac{s}{\mathsf{T}}\cdot
b+\displaystyle\frac{\mathsf{T}-s}{\mathsf{T}^{1/2}}\cdot
Y\Big{(}\displaystyle\frac{s}{\mathsf{T}-s}\Big{)},$

for some Brownian motion $Y:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$, it
follows that $\bm{\mathsf{x}}(t)$ has the same law as

$\displaystyle\operatorname{eig}\bigg{(}\displaystyle\frac{\mathsf{T}-t}{\mathsf{T}}\cdot\bm{U}+\displaystyle\frac{s}{\mathsf{T}}\cdot\bm{W}\bm{V}\bm{W}^{*}+\displaystyle\frac{\mathsf{T}-t}{\mathsf{T}^{1/2}}\cdot\bm{G}\Big{(}\displaystyle\frac{t}{\mathsf{T}-t}\Big{)}\bigg{)},$

where $\bm{W}$ is sampled under (4.20) and
$\bm{G}\big{(}t/(\mathsf{T}-t)\big{)}$ is an independent Hermitian Brownian
motion run for time $t(\mathsf{T}-t)^{-1}$. The lemma then follows from the
fact that $\bm{G}\big{(}t/(\mathsf{T}-t)\big{)}$ has the same law as
$t^{1/2}(\mathsf{T}-t)^{-1/2}\cdot\bm{G}$. ∎

###### Proof of 4.31.

Observe from (4.23) that, for $y\in[0,1]$, we have

(21.1)
$\displaystyle(2\pi)^{-1}\displaystyle\int_{\gamma_{\operatorname{sc}}(y)}^{2}(4-w^{2})^{1/2}dw=y.$

By (21.1) and the symmetry of the integrand $(4-w^{2})^{1/2}$ there in $w$, we
have $0\leq\gamma_{\operatorname{sc}}(y)\leq 2$ for $y\in[0,1/2]$ and
$-2\leq\gamma_{\operatorname{sc}}(y)\leq 0$ for $y\in[1/2,1]$. The latter
verifies the first statement of the lemma when $y\in[1/2,1]$, so let assume
that $y\in[0,1/2)$ so $0\leq\gamma_{\operatorname{sc}}(y)\leq 2$. Then the
first part of the lemma follows from the fact that, for any real number
$\theta\in[0,2]$ we have

$\displaystyle\Big{(}\displaystyle\frac{\theta}{8}\Big{)}^{3/2}\leq\displaystyle\frac{2^{1/2}\theta^{3/2}}{3\pi}$
$\displaystyle=2^{-1/2}\pi^{-1}\displaystyle\int_{2-\theta}^{2}(2-w)^{1/2}dw$
$\displaystyle\leq(2\pi)^{-1}\displaystyle\int_{2-\theta}^{2}(4-w^{2})^{1/2}dw\leq\pi^{-1}\displaystyle\int_{2-\theta}^{2}(2-w)^{1/2}dw=\displaystyle\frac{2\theta^{3/2}}{3\pi}\leq\Big{(}\displaystyle\frac{\theta}{2}\Big{)}^{3/2},$

where we used the bound $2(2-w)\leq 4-w^{2}\leq 4(2-w)$ for $w\in[0,2]$.

To establish the second part of the lemma, we differentiate (21.1) with
respect to $y$ to obtain

(21.2)
$\displaystyle-\gamma^{\prime}_{\operatorname{sc}}(y)=2\pi\big{(}4-\gamma_{\operatorname{sc}}(y)^{2}\big{)}^{-1/2}.$

Since $0\leq\gamma_{\operatorname{sc}}(y)\leq 2$ for $y\in[0,1/2]$, by the
first part of the lemma we have $4y^{2/3}\leq
4-\gamma^{2}_{\operatorname{sc}}(y)\leq 32y^{2/3}$ for $0\leq y\leq 1/2$ and
$4-\gamma^{2}_{\operatorname{sc}}(y)\leq 32y^{2/3}$ for $0\leq y\leq 1$.
Together with (21.2), these estimates yield the second part of the lemma. ∎

####  21.2. Proofs of Corollary 4.30 and Lemma 4.35

In this section we establish first 4.30 and then 4.35.

###### Proof of 4.30.

We claim that there exist constants $c=c(A,B)>0$ and $C=C(A,B)>1$ such that,
for any fixed real number $t\in[T/4,3T/4]$, we have

(21.3) $\displaystyle\mathbb{P}\Bigg{[}\bigcup_{1\leq j<k\leq\lfloor
n/2\rfloor}\Big{\\{}\mathsf{x}_{j}(tn^{1/3})-\mathsf{x}_{k}(tn^{1/3})\leq
C(k^{2/3}-j^{2/3})+(\log n)^{24}j^{-1/3}\Big{\\}}\Bigg{]}\leq c^{-1}e^{-c(\log
n)^{2}}.$

We first establish the corollary assuming (21.3). To this end, define the set
$\mathcal{T}=(n^{-9}\cdot\mathbb{Z})\cap[0,Tn^{1/3}]$ and the events

$\displaystyle\mathscr{E}_{1}=\bigcap_{j=1}^{k}\bigcap_{0\leq r<r+s\leq
Tn^{1/3}}\Big{\\{}\big{|}\mathsf{x}_{j}(r+s)-\mathsf{x}_{j}(r)\big{|}\leq
2n^{2}s^{1/3}\Big{\\}};$
$\displaystyle\mathscr{E}_{2}=\bigcap_{t\in\mathcal{T}}\bigcap_{1\leq
j<k\leq\lfloor
n/2\rfloor}\big{\\{}\mathsf{x}_{j}(tn^{1/3})-\mathsf{x}_{k}(tn^{1/3})\leq
C(k^{2/3}-j^{2/3})+(\log n)^{24}j^{-1/3}\big{\\}}.$

We then claim that there exists a constant $c_{0}=c_{0}(A,B)>0$ such that

(21.4) $\displaystyle\mathbb{P}[\mathscr{E}_{1}^{\complement}]\leq
c_{0}^{-1}e^{-c_{0}(\log
n)^{2}};\qquad\mathbb{P}\big{[}\mathscr{E}_{2}^{\complement}\big{]}\leq
c_{0}^{-1}e^{-c_{0}(\log n)^{2}}.$

Indeed, the first bound in (21.4) follows from the $B=n$ case of 4.8, together
with the facts that for sufficiently large $n$ we have that $|v_{j}-u_{j}|\leq
2Bn^{2/3}\leq n$ (by (4.21)), that $s(Tn^{1/3})^{-1}\leq s^{1/3}n$ (for
$s\in[0,Tn^{1/3}]$), and that $ns^{1/2}\log(2s^{-1}Tn^{1/3})\leq
n^{2}s^{1/3}$. The second bound in (21.4) follows from taking a union bound in
(21.3) over $t\in\mathcal{T}$ (and using the fact that $|\mathcal{T}|\leq
3An^{10}$).

Now restrict to the event $\mathscr{E}_{1}\cap\mathscr{E}_{2}$. Fix
$s\in[Tn^{1/3}/4,3Tn^{1/3}/4]$ and let $s^{\prime}\in\mathcal{T}$ be the
closest number in $\mathcal{T}$ to $s$ (if more than one exists, we select one
arbitrarily). Then, for any integers $1\leq j<k\leq n$, we have

$\displaystyle\mathsf{x}_{j}(s)-\mathsf{x}_{k}(s)$
$\displaystyle\leq\big{|}\mathsf{x}_{j}(s)-\mathsf{x}_{j}(s^{\prime})\big{|}+\big{|}\mathsf{x}_{j}(s^{\prime})-\mathsf{x}_{k}(s^{\prime})\big{|}+\big{|}\mathsf{x}_{k}(s^{\prime})-\mathsf{x}_{k}(s)\big{|}$
$\displaystyle\leq C(k^{2/3}-j^{2/3})+(\log
n)^{24}j^{-1/3}+4n^{2}|s-s^{\prime}|^{1/3}$ $\displaystyle\leq
C(k^{2/3}-j^{2/3})+(\log n)^{24}j^{-1/3}+4n^{-1}\leq C(k^{2/3}-j^{2/3})+(\log
n)^{25}j^{-1/3},$

where in the second bound we used the fact that we are restricting to
$\mathscr{E}_{1}\cap\mathscr{E}_{2}$; in the third we used the fact that
$|s-s^{\prime}|\leq n^{-9}$ (since $s\in[0,Tn^{1/3}]$ and
$\mathcal{T}=(n^{9}\cdot\mathbb{Z})\cap[0,Tn^{1/3}]$); and in the fourth we
used the fact that $(\log n)^{24}j^{-1/3}+4n^{-1}\leq(\log n)^{25}j^{-1/3}$
for $j\in\llbracket 1,n\rrbracket$ and sufficiently large $n$. This, together
with the fact that $\mathbb{P}[\mathscr{E}_{1}\cap\mathscr{E}_{2}]\geq
1-2c_{0}^{-1}e^{-c_{0}(\log n)^{2}}$ (by (21.4) and a union bound), implies
(4.22) and thus the corollary.

It therefore remains to establish (21.3). To this end, define the real numbers
$t_{0},T_{0}>0$; the $n$-tuples
$\bm{u}^{\prime},\bm{v}^{\prime}\in\overline{\mathbb{W}}_{n}$; and the
ensemble
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[0,T_{0}n^{1/3}]\big{)}$ by for any
$s\in[0,T_{0}n^{1/3}]$ setting

(21.5) $\displaystyle\begin{aligned}
t_{0}=\bigg{(}t\Big{(}1-\frac{t}{T}\Big{)}\bigg{)}^{1/2};\quad&T_{0}=t_{0}^{-2}T;\quad\bm{u}^{\prime}=t_{0}^{-1}\cdot\bm{u};\quad\bm{v}^{\prime}=t_{0}^{-1}\cdot\bm{v};\quad\mathsf{y}_{j}(s)=t_{0}^{-1}\cdot\mathsf{x}_{j}(t_{0}^{2}s).\end{aligned}$

By 4.4, the law of $\bm{\mathsf{y}}$ is given by
$\mathsf{Q}^{\bm{u}^{\prime};\bm{v}^{\prime}}$. Next, denote the $n\times n$
diagonal matrices $\bm{U}=\operatorname{diag}(\bm{u}^{\prime})$ and
$\bm{V}=\operatorname{diag}(\bm{v}^{\prime})$; letting the unitary random
matrix $\bm{W}\in\mathrm{U}(n)$ have law (4.20), set
$\bm{A}=(T_{0}-t_{0})T_{0}^{-1}\cdot\bm{U}+t_{0}T_{0}^{-1}\cdot\bm{W}\bm{V}\bm{W}^{*}$
and $\bm{a}=\operatorname{eig}(\bm{A})$. Then, by 4.29, the law of
$\bm{\mathsf{y}}(t_{0}^{-2}tn^{1/3})=t_{0}^{-1}\cdot\bm{\mathsf{x}}(tn^{1/3})$
is given by $\bm{\lambda}(n^{1/3})$, where
$\bm{\lambda}(s)=\big{(}\lambda_{1}(s),\lambda_{2}(s),\ldots,\lambda_{n}(s)\big{)}\in\overline{\mathbb{W}}_{n}$
denotes Dyson Brownian motion run for time $s$ with initial data
$\bm{\lambda}(0)=\bm{a}$.

We analyze $\bm{\lambda}(n^{1/3})$ using 4.23. By the Weyl interlacing
inequality, we have

$\displaystyle\max\bm{a}\leq\max\operatorname{eig}(\bm{U})+\max\operatorname{eig}(\bm{V})=t_{0}^{-1}(\max\bm{u}+\max\bm{v})\leq
2Bt_{0}^{-1}n^{2/3},$

and similarly $\min\bm{a}\geq-2Bt_{0}^{-1}n^{2/3}$. Let $c_{1}$ denote the
constant $c(2)>0$ from 4.23. Observe since $t\in[T/4,3T/4]$ that $t_{0}\geq
T^{1/2}/4$ and $T\geq C_{1}$ that we can make $4Bt_{0}^{-1}<c_{1}$ by taking
$C_{1}=C_{1}(B)>1$ sufficiently large. Then, $\max\bm{a}-\min\bm{a}\leq
4Bt_{0}^{-1}n^{2/3}<c_{1}n^{2/3}$, and so 4.23 applies and (since
$1\in(1/2,2)$) yields constants $c_{2}>0$ and $C_{2}>0$ such that

$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{1\leq j<k\leq\lfloor
n/2\rfloor}\Big{\\{}\big{|}\lambda_{j}(n^{1/3})-\lambda_{k}(n^{1/3})\geq
C_{2}(k^{2/3}-j^{2/3})+(\log n)^{20}j^{-1/3}\Big{\\}}\Bigg{]}\leq
c_{2}^{-1}e^{-c_{2}(\log n)^{2}}.$

Since $\bm{\lambda}(n^{1/3})$ has the same law as
$\bm{\mathsf{y}}(t_{0}^{-2}tn^{1/3})=t_{0}^{-1}\bm{\mathsf{x}}(tn^{1/3})$, and
since $t_{0}<T^{1/2}\leq(AC_{1})^{1/2}$, it follows that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{1\leq j<k\leq\lfloor
n/2\rfloor}\big{\\{}\mathsf{x}_{j}(tn^{1/3})-\mathsf{x}_{k}(tn^{1/3})\geq(AC_{1})^{1/2}C_{2}(k^{2/3}-j^{2/3})+(AC_{1})^{1/2}($
$\displaystyle\log n)^{20}j^{-1/3}\big{\\}}\Bigg{]}$ $\displaystyle\leq
c_{2}^{-1}e^{-c_{2}(\log n)^{2}},$

from which (21.3) follows, as $(AC_{1})^{1/2}(\log n)^{20}\leq(\log n)^{24}$
for sufficiently large $n$. ∎

###### Proof of 4.35.

We will establish the lemma by comparing the non-intersecting Brownian bridges
$\bm{\mathsf{x}}$ with certain (rescaled) parabolic Airy line ensembles and
Brownian watermelons; we will prove the first part of the lemma in detail and
only outline the proof for the second part, as it is fairly similar. In what
follows, for any real number $\sigma>0$, 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). For any integer $n\geq 1$, 4.34 and a union bound (with the $u$
there equal to $(\log n)^{2}$ here) together yield a constant
$c_{3}=c_{3}(\sigma,D)>0$ such that

(21.6)
$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{j=1}^{2n}\bigcup_{t\in[-n^{D},n^{D}]}\bigg{|}2^{-1/2}\cdot\mathcal{R}_{j}^{(\sigma)}(s)+2^{-1/2}\sigma^{3}s^{2}+\displaystyle\frac{(3\pi)^{2/3}j^{2/3}}{2^{7/6}\sigma}\bigg{|}\geq(\log
n)^{2}j^{-1/3}\Bigg{]}\leq c_{3}^{-1}e^{-c_{3}(\log n)^{2}}.$

We begin by verifying the first part of the lemma. To this end, take
$\sigma_{1}=2^{-7/6}(3\pi)^{2/3}d^{-1}$, and denote the line ensemble
$\widetilde{\bm{\mathcal{R}}}=\big{(}\widetilde{\mathcal{R}}_{1},\widetilde{\mathcal{R}}_{2},\ldots\big{)}\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ by for each $(j,t)\in\mathbb{Z}_{\geq
1}\times\mathbb{R}$ setting

(21.7)
$\displaystyle\widetilde{\mathcal{R}}_{j}(t)=2^{-1/2}\cdot\mathcal{R}^{(\sigma_{1})}_{j}\Big{(}t-\displaystyle\frac{a+b}{2}\Big{)}+2^{-5/2}\sigma_{1}^{3}(b-a)^{2}+M+(\log
n)^{2},$

which satisfies the Brownian Gibbs property by 2.10. Now observe, due to the
upper bounds assumed on $\bm{u}$, $\bm{v}$, and $f$, that for each
$(j,t)\in\llbracket 1,n\rrbracket\times\\{a,b\\}$ we have

$\displaystyle
2^{-5/2}\sigma_{1}^{3}(b-a)^{2}+M-2^{-1/2}\sigma_{1}^{3}\Big{(}t-\displaystyle\frac{b+a}{2}\Big{)}^{2}-\displaystyle\frac{(3\pi)^{2/3}j^{2/3}}{2^{7/6}\sigma_{1}}=M-dj^{2/3}\geq\max\\{u_{j},v_{j}\\},$

and for each $t\in[a,b]$ (using the bound
$\big{(}t-(a+b)/2\big{)}^{2}\leq(b-a)^{2}/4$ for $t\in[a,b]$) that

$\displaystyle
2^{-5/2}\sigma_{1}^{3}(b-a)^{2}+M-2^{-1/2}\sigma_{1}^{3}\Big{(}t-\displaystyle\frac{b+a}{2}\Big{)}^{2}-\displaystyle\frac{(3\pi)^{2/3}(n+1)^{2/3}}{2^{7/6}\sigma_{1}}\geq
M-d(n+1)^{2/3}\geq f(t).$

Thus, from (21.7) and (21.6) (with the translation-invariance of
$\bm{\mathcal{R}}^{(\sigma_{1})}$, which holds by 2.6, to shift the interval
$[-n^{D},n^{D}]$ in (21.6) to one containing $[a,b]$ here), there exists a
constant $c_{4}=c_{4}(d,D)>0$ such that

(21.8) $\displaystyle\mathbb{P}[\mathscr{E}_{1}]\geq
1-c_{4}^{-1}e^{-c_{4}(\log n)^{2}},$

where we have defined the event

$\displaystyle\mathscr{E}_{1}=\bigcap_{j=1}^{n}\big{\\{}\widetilde{\mathcal{R}}_{j}(a)\geq
u_{j}\big{\\}}\cap\big{\\{}\widetilde{\mathcal{R}}_{j}(b)\geq
v_{j}\big{\\}}\cap\bigcap_{t\in[a,b]}\big{\\{}\widetilde{\mathcal{R}}_{j}(t)\geq
f(t)\big{\\}}.$

Conditioning on $\mathcal{R}_{j}(t)$ for $(j,t)\notin\llbracket
1,n\rrbracket\times[a,b]$, it follows from 4.6 that on $\mathscr{E}_{1}$ we
may couple $\bm{\mathsf{x}}$ and $\widetilde{\bm{\mathcal{R}}}$ such that
$\mathsf{x}_{j}(t)\leq\widetilde{\mathcal{R}}_{j}(t)$ for each
$(j,t)\in\llbracket 1,n\rrbracket\times[a,b]$. This, (21.7), (21.6) (again
with the translation-invariance of $\bm{\mathcal{R}}^{(\sigma_{1})}$), (21.8),
and the bound

$\displaystyle
2^{-5/2}\sigma_{1}^{3}(b-a)^{2}+M-2^{-1/2}\sigma_{1}^{3}\Big{(}t-\displaystyle\frac{a+b}{2}\Big{)}^{2}-\displaystyle\frac{(3\pi)^{2/3}j^{2/3}}{2^{7/6}\sigma_{1}}\leq
2^{-5/2}\sigma_{1}^{3}(b-a)^{2}+M-dj^{2/3},$

yields a constant $c_{1}=c_{1}(d,D)>0$ such that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{n}\bigcap_{t\in[a,b]}\bigg{\\{}\mathsf{x}_{j}(t)\leq
M+2^{-5/2}\sigma_{1}^{3}(b-a)^{2}-dj^{2/3}+2(\log n)^{2}\bigg{\\}}\Bigg{]}\geq
1-c_{1}^{-1}e^{-c_{1}(\log n)^{2}},$

which with the definition of $\sigma_{1}$ gives (4.25).

To establish the second part of the lemma, first observe that we may assume
$f=-\infty$, by 4.6. Next define
$\bm{u}^{\prime},\bm{v}^{\prime}\in\overline{\mathbb{W}}_{n}$ by setting
$u_{j}^{\prime}=u_{n}$ and $v_{j}^{\prime}=v_{n}$ for each $j\in\llbracket
1,n\rrbracket$. Denote the associated Brownian watermelon
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$, given by $n$ non-
intersecting Brownian bridges sampled from the measure
$\mathsf{Q}^{\bm{u}^{\prime};\bm{v}^{\prime}}$. Since
$\bm{u}^{\prime}\leq\bm{u}$ and $\bm{v}^{\prime}\leq\bm{v}$, we may by 4.6
couple $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ in such a way that
$\mathsf{x}_{j}(t)\geq\mathsf{y}_{j}(t)$ for each $(j,t)\in\llbracket
1,n\rrbracket\times[a,b]$. Hence there exists a constant $C=C(A)>1$ such that,
with probability at least $1-Ce^{-(\log n)^{5}}$, we have

$\displaystyle\mathsf{y}_{n}(t)$ $\displaystyle\geq\frac{t-a}{b-a}\cdot
v_{n}+\frac{b-t}{b-a}\cdot u_{n}-(b-a)^{1/2}n^{1/2}-(A+1)(\log n)^{9}$
$\displaystyle\geq-Bn^{2/3}-M-(b-a)^{1/2}n^{1/2}-(A+1)(\log
n)^{9}\geq-(B+A^{1/2}+1)n^{2/3}-M,$

where in the first inequality we used the first part of 4.32 (with the facts
that $b-a\leq An^{1/3}$, that $(b-t)(t-a)\leq(b-a)^{2}/4$, and that
$\gamma_{\operatorname{sc};n}(n)\geq-2$, by the first part of 4.31); in the
second we used the fact that $\min\\{u_{j},v_{j}\\}\geq-Bj^{2/3}-M$ for each
$j\in\llbracket 1,n\rrbracket$; and in the third we used the fact that
$b-a\leq An^{1/3}$. Together with the coupling
$\mathsf{x}_{j}(t)\geq\mathsf{y}_{j}(t)$, this implies that

(21.9)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{t\in[a,b]}\big{\\{}\mathsf{x}_{n}(t)\geq-(A^{1/2}+B+1)n^{2/3}-M\big{\\}}\Bigg{]}\geq
1-Ce^{-(\log n)^{5}}.$

The claim (4.26) follows by using (21.9) to compare $\bm{\mathsf{x}}$ to
another rescaled parabolic Airy line ensemble
$\widehat{\bm{\mathcal{R}}}=\big{(}\widehat{\mathcal{R}}_{1},\widehat{\mathcal{R}}_{2},\ldots\big{)}\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$, defined by for any $(j,t)\in\mathbb{Z}_{\geq
1}\times\mathbb{R}$ setting

$\displaystyle\widehat{\mathcal{R}}_{j}(t)=2^{-1/2}\cdot\mathcal{R}^{(\sigma_{2})}_{j}\Big{(}t-\displaystyle\frac{a+b}{2}\Big{)}+2^{-5/2}\sigma_{2}^{3}(b-a)^{2}-M-(\log
n)^{2},$

where we have denoted $\sigma_{2}=2^{-7/6}(3\pi)^{2/3}(2A^{2}+B+3)^{-1}$.
Using (21.6) and the facts that for $(j,t)\in\llbracket
1,n-1\rrbracket\times\\{a,b\\}$ we have

$\displaystyle
2^{-5/2}\sigma_{2}^{3}(b-a)^{2}-M-2^{-1/2}\sigma_{2}^{3}\Big{(}t-\displaystyle\frac{a+b}{2}\Big{)}^{2}-\displaystyle\frac{(3\pi)^{2/3}j^{2/3}}{2^{7/6}\sigma_{2}}\leq-
Bj^{2/3}-M\leq\min\\{u_{j},v_{j}\\}$

and for $t\in[a,b]$ we have (since $\sigma_{2}\leq 1$, $b-a\leq An^{1/3}$, and
$A^{2}+B+3\geq A^{1/2}+B+1$)

$\displaystyle 2^{-5/2}\sigma_{2}^{3}($ $\displaystyle
b-a)^{2}-M-2^{-1/2}\sigma_{2}^{3}\Big{(}t-\displaystyle\frac{a+b}{2}\Big{)}^{2}-\displaystyle\frac{(3\pi)^{2/3}n^{2/3}}{2^{7/6}\sigma_{2}}$
$\displaystyle\leq(b-a)^{2}-M-(2A^{2}+B+3)n^{2/3}\leq-(A^{1/2}+B+1)n^{2/3}-M,$

the proof of (4.26) closely follows that of (4.25), so further details are
omitted. ∎

###  22\. Proofs of Results From Chapter 2

####  22.1. Convergence of the Alternating Dynamics

Let $\Omega$ be a measurable space with $\sigma$-algebra $\mathcal{F}$; let
$\mathscr{P}(\Omega)$ denote the space of probability measures on
$(\Omega,\mathcal{F})$.

######  Assumption 22.1.

Adopting the above notation, let
$\mathsf{K}:\Omega\times\mathcal{F}\rightarrow\mathbb{R}_{\geq 0}$ be a Markov
transition kernel. For any function $\varphi:\Omega\rightarrow\mathbb{R}_{\geq
0}$ and measure $\mu$ on $\Omega$, define the function
$\mathfrak{\mathsf{K}}\varphi:\Omega\rightarrow\mathbb{R}_{\geq 0}$ and
measure $\mathsf{K}\mu$ on $\Omega$ by setting

$\displaystyle\mathsf{K}\varphi(x)=\displaystyle\int_{\Omega}\varphi(y)\mathsf{K}(x,dy);\qquad\mathsf{K}\mu(A)=\displaystyle\int_{\Omega}\mathsf{K}(x,A)\mu(dx),$

for any $x\in\Omega$ and measurable set $A\in\mathcal{F}$. Assume that there
exist constants $\alpha\in(0,1)$, $\gamma\in(0,1)$, $B\geq 0$, and
$R>\frac{2B}{1-\gamma}$; a potential function
$V:\Omega\rightarrow\mathbb{R}_{\geq 0}$; and a probability measure $\nu$ on
$\Omega$, such that the following two conditions hold.

  1. (1)

For each $x\in\Omega$, we have $\mathsf{K}V(x)\leq\gamma V(x)+B$, for each
$x\in\Omega$.

  2. (2)

For each $x\in\Omega$ with $V(x)\leq R$, and any measurable set
$A\in\mathcal{F}$, we have $\mathsf{K}(x,A)\geq\alpha\nu(A)$.

The following result provides a convergence theorem for Harris chains [75]. It
appears in [95], though it is stated as written below in [70].

######  Lemma 22.2 ([70, Theorem 1.2]).

Adopt 22.1, and fix some measure $\mu$ on $\Omega$. Then, the Markov process
defined by $\mathsf{K}$ has a unique stationary measure $\mu_{0}$, and
$\lim_{m\rightarrow\infty}\|\mathsf{K}^{m}\mu-\mu_{0}\|_{\operatorname{TV}}=0$.

We next apply 22.2 to the alternating dynamics of Definition 5.8. Throughout
the remainder of this section, we adopt the notation of that definition. These
include the function $f:\llbracket
0,T\rrbracket\rightarrow\overline{\mathbb{R}}$; the family $\bm{\mathsf{y}}$
of $n$ non-intersecting $(T+1)$-step walks
$\bm{\mathsf{y}}(t)=\big{(}\mathsf{y}_{1}(t),\mathsf{y}_{2}(t),\ldots,\mathsf{y}_{n}(t)\big{)}\in\overline{\mathbb{W}}_{n}$
(over $t\in\llbracket 0,T\rrbracket$), and the associated Markov operator
$\mathsf{P}$ (which we also interpret as a kernel) for the alternating
dynamics. We also set $\bm{u}=\bm{\mathsf{y}}(0)$ and
$\bm{v}=\bm{\mathsf{y}}(T)$, which are fixed throughout the dynamics. Then the
state space for the alternating dynamics $\mathsf{P}$ can be viewed as

$\displaystyle\Omega_{0}=\Big{\\{}\big{(}\bm{\mathsf{y}}(t)\big{)}_{t\in\llbracket
1,T-1\rrbracket}\in\mathbb{W}_{n}^{T-1}:\displaystyle\min_{t\in\llbracket
1,T-1\rrbracket}\big{(}\mathsf{y}_{n}(t)-f(t)\big{)}\geq 0\Big{\\}}.$

We define the associated potential function $V_{0}$ to be

(22.1) $\displaystyle V_{0}(\bm{\mathsf{y}})=\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\displaystyle\max_{t\in\llbracket
1,T-1\rrbracket}\Big{(}\big{|}\mathsf{y}_{j}(t)\big{|}+1\Big{)}.$

We then have the following two lemmas verifying 22.1 for the alternating
dynamics; the former is proven in Section 22.2 below, and the proof of the
latter is similar to that of [8, Lemma B.13].

######  Lemma 22.3.

There exist constants $\gamma=\gamma(f,\bm{u},\bm{v})\in(0,1)$ and
$B=B(f,\bm{u},\bm{v})\geq 0$ such that, for any family $\bm{\mathsf{y}}$ of
$n$ non-intersecting $(T+1)$-step walks with $\bm{\mathsf{y}}(0)=\bm{u}$ and
$\bm{\mathsf{y}}(T)=\bm{v}$, we have
$\mathsf{P}^{2}V_{0}(\bm{\mathsf{y}})\leq\gamma V_{0}(\bm{\mathsf{y}})+B$.

######  Lemma 22.4.

For any real number $R>1$, there exists a constant
$\alpha=\alpha(f,\bm{u},\bm{v},R)>0$ such that the following holds. Letting
$\nu_{0}$ denote the Lebesgue measure on the set

$\displaystyle\Omega_{1}=\bigg{\\{}\bm{\mathsf{x}}\in\Omega_{0}:V_{0}(\bm{\mathsf{x}})\leq\max_{t\in\llbracket
1,T-1\rrbracket}\max\big{\\{}f(t),0\big{\\}}+R+1\bigg{\\}},$

we have $\mathsf{P}^{2}(\bm{\mathsf{y}},A)\geq\alpha\nu_{0}(A)$, for each
$\bm{\mathsf{y}}\in\Omega_{1}$ and any measurable subset
$A\subseteq\mathbb{W}_{n}^{T-1}$.

###### Proof.

For any integer $T^{\prime}\geq 2$; two $n$-tuples
$\bm{u}^{\prime},\bm{v}^{\prime}\in\mathbb{W}_{n}$; and function
$f^{\prime}:\llbracket
0,T^{\prime}\rrbracket\rightarrow\overline{\mathbb{R}}$, the density of the
measure $\mathsf{G}_{f^{\prime}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$ on
sequences
$\bm{\mathsf{x}}(t)=\big{(}\mathsf{x}_{1}(t),\mathsf{x}_{2}(t),\ldots,\mathsf{x}_{n}(t)\big{)}$
is given by

(22.2) $\displaystyle
C\cdot\textbf{1}_{\bm{\mathsf{x}}\in\Omega_{0}}\cdot\displaystyle\prod_{j=1}^{n}\Bigg{(}\textbf{1}_{\mathsf{x}_{j}(0)=u_{j}^{\prime}}\textbf{1}_{\mathsf{x}_{j}(T^{\prime})=v_{j}^{\prime}}\displaystyle\prod_{t=1}^{T^{\prime}}\exp\bigg{(}-\displaystyle\frac{1}{2}\big{(}\mathsf{x}_{j}(t)-\mathsf{x}_{j}(t-1)\big{)}^{2}\bigg{)}\displaystyle\prod_{t=1}^{T^{\prime}-1}d\mathsf{x}_{j}(t)\Bigg{)},$

for some normalization constant
$C=C(f^{\prime},\bm{u}^{\prime},\bm{v}^{\prime})>0$. Observe that there exist
some constant $c_{1}=c_{1}(f^{\prime},\bm{u}^{\prime},\bm{v}^{\prime})>0$ such
that $C>c_{1}$, since the interior of $\Omega_{0}$ is nonempty. Further
observe that, for any fixed real number $R_{0}\geq\max_{t\in\llbracket
1,T-1\rrbracket}\max\big{\\{}f(t),0\big{\\}}+1$, when restricting to the set
of $\bm{\mathsf{x}}\in\Omega_{0}$ such that $V_{0}(\bm{\mathsf{x}})\leq
R_{0}$, the density (22.2) is uniformly bounded above and below (in a way
dependent on $R_{0}$). Thus, there exists a constant
$c_{1}=c_{1}(f^{\prime};\bm{u}^{\prime};\bm{v}^{\prime},R_{0})>0$ such that,
on $\big{\\{}\bm{\mathsf{x}}\in\Omega_{0}:V_{0}(\bm{\mathsf{x}})\leq
R_{0}\big{\\}}$, the measure
$\mathsf{G}_{f^{\prime}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$ is absolutely
continuous with respect to the Lebesgue measure on this set, and its
Radon–Nikodym derivative is bounded above by $c_{1}^{-1}$ and below by
$c_{1}$.

We use this twice, with
$(T^{\prime};\bm{u}^{\prime};\bm{v}^{\prime})=\big{(}2;\bm{u};\bm{\mathsf{y}}(2)\big{)}$
and then with
$(T^{\prime};\bm{u}^{\prime};\bm{v}^{\prime})=\big{(}T-1;\bm{\mathsf{y}}(1);\bm{v}\big{)}$;
by Definition 5.8, the former corresponds to the first application of
$\mathsf{P}$ and the latter to the second application of $\mathsf{P}$. The
former yields a constant $c_{1}=c_{1}(f,\bm{u},\bm{v},R)>0$ such that

(22.3)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{n}\big{\\{}\mathsf{P}\mathsf{y}_{j}(1)\in
S_{j}\big{\\}}\Bigg{]}\geq
c_{1}\displaystyle\prod_{j=1}^{n}\displaystyle\int_{S_{j}}dy,$

for any measurable subsets

(22.4) $\displaystyle
S_{1},S_{2},\ldots,S_{n}\subseteq\bigg{\\{}x\in\mathbb{R}:x\geq
f(1),|x|\leq\max_{t\in\llbracket
1,n\rrbracket}\max\big{\\{}f(t),0\big{\\}}+R+1\bigg{\\}}.$

The second application yields a constant $c_{2}=c_{2}(f,\bm{u},\bm{v},R)>0$
such that

(22.5)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{n}\bigcap_{t=2}^{T}\big{\\{}\mathsf{P}^{2}\mathsf{y}_{j}(t)\in
S_{t,j}\big{\\}}\Bigg{]}\geq
c_{2}\displaystyle\prod_{t=2}^{T-1}\displaystyle\prod_{j=1}^{n}\displaystyle\int_{S_{t,j}}dy,$

for any measurable subsets $S_{t,j}$ of the right side of (22.4). The lemma
then follows from combining (22.3) and (22.5). ∎

Given these lemmas, we can quicky establish 5.10.

###### Proof of 5.10.

The fact that $\mathsf{G}_{f}^{\bm{u};\bm{v}}$ is stationary for
$\mathsf{P}^{2}$ follows from 5.9. Thus, the lemma follows from 22.2, using
22.3 and 22.4 (normalizing $\nu_{0}$ in the latter so that it becomes a
probability measure) to verify 22.1. ∎

####  22.2. Proof of 22.3

In this section we establish 22.3. To this end, we first require the following
tail bound for non-intersecting Gaussian bridges.

######  Lemma 22.5.

For any integer $T^{\prime}\geq 2$; two $n^{\prime}$-tuples
$\bm{u}^{\prime},\bm{v}^{\prime}\in\overline{\mathbb{W}}_{n}$; and function
$f^{\prime}:\llbracket
0,T^{\prime}\rrbracket\rightarrow\overline{\mathbb{R}}$, there exists a
constant $c=c(f^{\prime},n)>0$ such that the following holds for any real
number $r\geq 0$. Sampling non-intersecting Gaussian bridges
$\bm{\mathsf{x}}^{\prime}(t)=\big{(}\mathsf{x}_{1}^{\prime}(t),\mathsf{x}_{2}^{\prime}(t),\ldots,\mathsf{x}_{n}^{\prime}(t)\big{)}$
from the measure $\mathsf{G}_{f^{\prime}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$,
we have

$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{t=1}^{T-1}\bigcup_{j=1}^{n}\bigg{\\{}\big{|}\mathsf{x}_{j}^{\prime}(t)\big{|}\geq\displaystyle\frac{T^{\prime}-t}{T^{\prime}}\cdot\displaystyle\max_{j\in\llbracket
1,n\rrbracket}|u_{j}^{\prime}|+\displaystyle\frac{t}{T^{\prime}}\cdot\displaystyle\max_{j\in\llbracket
1,n\rrbracket}|v_{j}^{\prime}|+r\bigg{\\}}\Bigg{]}\leq c^{-1}e^{-cr^{2}}.$

###### Proof.

First observe that there exists a constant $c_{1}=c_{1}(f^{\prime})>1$ such
that the following holds for any real number $r>0$. Given a
$(T^{\prime}+1)$-step Gaussian bridge
$\big{(}\mathsf{x}(0),\mathsf{x}(1),\ldots,\mathsf{x}(T^{\prime})\big{)}$
conditioned to start and end at some points $u^{\prime}\in\mathbb{R}$ and
$v^{\prime}\in\mathbb{R}$, respectively, and satisfy $\mathsf{x}(t)\geq f(t)$,
we have

(22.6) $\displaystyle\mathbb{P}\Bigg{[}\bigcup_{t\in\llbracket
0,T^{\prime}\rrbracket}\bigg{\\{}\big{|}\mathsf{x}(t)\big{|}\geq\displaystyle\frac{T^{\prime}-t}{T^{\prime}}\cdot|u^{\prime}|+\displaystyle\frac{t}{T^{\prime}}\cdot|v^{\prime}|+r\bigg{\\}}\Bigg{]}\leq
c_{1}^{-1}e^{-c_{1}r^{2}}.$

Now, let $\mathfrak{G}_{f^{\prime}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$ denote
the law on sequences
$\bm{\mathsf{x}}(t)=\big{(}\mathsf{x}_{1}(t),\mathsf{x}_{2}(t),\ldots,\mathsf{x}_{n}(t)\big{)}$,
with $t\in\llbracket 0,T^{\prime}\rrbracket$, of $n$ independent Gaussian
bridges starting at $\bm{u}^{\prime}$, ending at $\bm{v}^{\prime}$, and
conditioned to remain above $f^{\prime}$; as it does not impose the non-
intersecting condition, we may view it as the law of “free” Gaussian bridges.
Then, from (22.6) and a union bound, we deduce that there exists a constant
$c_{2}=c_{2}(f^{\prime},n)>0$ such that

(22.7)
$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{t=1}^{T-1}\bigcup_{j=1}^{n}\bigg{\\{}\big{|}x_{j}(t)\big{|}\geq\displaystyle\frac{T^{\prime}-t}{T^{\prime}}\cdot\displaystyle\max_{j\in\llbracket
1,n\rrbracket}|u_{j}^{\prime}|+\displaystyle\frac{t}{T^{\prime}}\cdot\displaystyle\max_{j\in\llbracket
1,n\rrbracket}|v_{j}^{\prime}|+r\bigg{\\}}\Bigg{]}\leq
c_{2}^{-1}e^{-c_{2}r^{2}}.$

Next, observe that there exists a constant
$c_{3}=c_{3}(f^{\prime},\bm{u}^{\prime},\bm{v}^{\prime})>0$ such that the
walks in $\bm{\mathsf{x}}$ sampled under
$\mathfrak{G}_{f^{\prime}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$ do not intersect
(that is, $\bm{\mathsf{x}}(t)\in\mathbb{W}_{n}$ for each $t\in\llbracket
1,T-1\rrbracket$), with probability at least $c_{3}$ under
$\mathfrak{G}_{f^{\prime}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$ (as this is an
open condition). Hence, the Radon–Nikodym derivative of the non-intersecting
measure $\mathsf{G}_{f^{\prime}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$ with
respect to the free one
$\mathfrak{G}_{f^{\prime}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$ is bounded above
by $c_{3}^{-1}$. Together with (22.7), this establishes the lemma. ∎

Now we can esetablish 22.3

###### Proof of 22.3.

This lemma will follow from two applications of 22.5. First taking the
$(T^{\prime};\bm{\mathsf{x}}^{\prime};\bm{u}^{\prime},\bm{v}^{\prime})$ there
to be $\big{(}2;\bm{\mathsf{y}}|_{\llbracket
0,2\rrbracket};\bm{u};\bm{\mathsf{y}}(2)\big{)}$ here yields (as
$\max_{j\in\llbracket 1,n\rrbracket}\big{|}\mathsf{y}_{j}(2)\big{|}\leq
V_{0}(\bm{\mathsf{y}})$) a constant $c_{1}=c_{1}(f,\bm{u})>0$ such that

(22.8)
$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{j=1}^{n}\bigg{\\{}\big{|}\mathsf{P}\mathsf{y}_{j}(1)\big{|}\geq\displaystyle\frac{V_{0}(\bm{\mathsf{y}})}{2}+r\bigg{\\}}\Bigg{]}\leq
c_{1}^{-1}e^{-c_{1}r^{2}},$

for any real number $r\geq 0$. Next applying 22.5 with the
$(T^{\prime};\bm{\mathsf{x}}^{\prime};\bm{u}^{\prime},\bm{v}^{\prime})$ there
to be $\big{(}T-1;\bm{\mathsf{y}}|_{\llbracket
1,T\rrbracket};\mathsf{P}\mathsf{y}_{j}(1);\bm{v}\big{)}$ here, yields a
constant $c_{2}=c_{2}(f,\bm{v})>0$ such that

$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{t=2}^{T-1}\bigcup_{j=1}^{n}\bigg{\\{}\big{|}\mathsf{y}_{j}(t)\big{|}\geq\displaystyle\frac{1}{2}\cdot\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}\mathsf{P}\mathsf{y}_{j}(1)\big{|}+r\bigg{\\}}\Bigg{]}\leq
c_{2}^{-1}e^{-c_{2}r^{2}},$

for any real number $r\geq 0$. This, together with (22.8), the definition
(22.1) of $V_{0}$, and a union bound, yields a constant
$c_{3}=c_{3}(f,\bm{u},\bm{v})>0$ such that

$\displaystyle\mathbb{P}\Big{[}V_{0}(\mathsf{P}^{2}\bm{\mathsf{y}})\leq\displaystyle\frac{1}{4}\cdot
V_{0}(\bm{\mathsf{y}})+\displaystyle\frac{3r}{2}\Big{]}\leq
c_{3}^{-1}e^{-c_{3}r^{2}}.$

Integrating this bound then establishes the lemma. ∎

###  23\. Proofs of Results From Chapter 3

####  23.1. Proofs of Continuum Monotonicity Results

In this section we first establish 10.14 and then outline the proof of 10.15.

###### Proof of 10.14.

We only establish the second part of the lemma, as the proof of the first is
entirely analogous. For each integer $n\geq 1$, define the $\lfloor
An\rfloor$-tuples and $\lfloor\widetilde{A}n\rfloor$-tuples111For notational
simplicity, we will omit the floors in what follows, assuming that $An=\lfloor
An\rfloor$, $\widetilde{A}n=\lfloor\widetilde{A}n\rfloor$, and $wn=\lfloor
wn\rfloor$; this will barely affect the analysis.
$\bm{u},\bm{v}\in\overline{\mathbb{W}}_{An}$ and
$\widetilde{\bm{u}},\widetilde{\bm{v}}\in\overline{\mathbb{W}}_{\tilde{A}n}$
by for each $j$ setting

$\displaystyle u_{j}=G^{\star}(a,n^{-1}j);\quad
v_{j}=G^{\star}(b,n^{-1}j);\quad\widetilde{u}_{j}=G^{\star}(a,n^{-1}j);\quad\widetilde{v}_{j}=\widetilde{G}^{\star}(b,n^{-1}j).$

Then sample the two families of non-intersecting Brownian bridges
$\bm{x}^{n}=(x_{1}^{n},x_{2}^{n},\ldots,x_{An}^{n})\in\llbracket
1,An\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$ and
$\widetilde{\bm{x}}^{n}=(\widetilde{x}_{1}^{n},\widetilde{x}_{2}^{n},\ldots,\widetilde{x}_{\tilde{A}n}^{n})\in\llbracket
1,\widetilde{A}n\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$ according to
the measures $\mathsf{Q}^{\bm{u};\bm{v}}(n^{-1})$ and
$\mathsf{Q}^{\tilde{\bm{u}};\tilde{\bm{v}}}(n^{-1})$, respectively. Further
fix a real number $\varepsilon\in(0,1)$, and define the functions
$f:[a,b]\rightarrow\mathbb{R}$ and $\widetilde{f}:[a,b]\rightarrow\mathbb{R}$
by setting

(23.1) $\displaystyle
f(t)=G^{\star}(t,w);\qquad\widetilde{f}(t)=\widetilde{G}^{\star}(t,w),\quad\qquad\text{for
each $t\in[a,b]$}.$

Define the $wn$-tuples
$\bm{u^{\prime}},\bm{v^{\prime}},\widetilde{\bm{u}}^{\prime},\widetilde{\bm{v}}^{\prime}\in\overline{\mathbb{W}}_{wn}$
to be the restriction of $\bm{u},\bm{v},\widetilde{\bm{u}},\widetilde{\bm{v}}$
on $\llbracket 1,wn-1\rrbracket$. Sample non-intersecting Brownian bridges
$\bm{y}^{n}=(y_{1}^{n},y_{2}^{n},\ldots,y_{wn-1}^{n})\in\llbracket
1,wn-1\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$ and
$\widetilde{\bm{y}}^{n}=(\widetilde{y}_{1}^{n},\widetilde{y}_{2}^{n},\ldots,\widetilde{y}_{wn-1}^{n})\in\llbracket
1,wn-1\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$ from
$\mathsf{Q}_{f}^{\bm{u}^{\prime};\bm{v}^{\prime}}(n^{-1})$ and
$\mathsf{Q}_{\tilde{f}}^{\tilde{\bm{u}}^{\prime};\tilde{\bm{v}}^{\prime}}(n^{-1})$,
respectively. By the second part of 10.7, we have with probability $1-o(1)$
(that is, tending to $1$ as $n$ tends to $\infty$) that
$\big{|}f(t)-x_{wn}(t)\big{|}\leq\varepsilon$ and
$\big{|}\widetilde{f}(t)-\widetilde{x}_{wn}(t)\big{|}\leq\varepsilon$ for all
$t\in(a,b)$. This, together with the Brownian Gibbs property for $\bm{x}$ and
$\widetilde{\bm{x}}$ and (the $B=\varepsilon$ case) of 4.7, yields a coupling
between $(\bm{x},\bm{y})$ and $(\widetilde{\bm{x}},\widetilde{\bm{y}})$ such
that with probability $1-o(1)$ we have

(23.2) $\displaystyle\displaystyle\max_{j\in\llbracket
1,wn-1\rrbracket}\displaystyle\sup_{t\in[a,b]}\big{|}y_{j}(t)-x_{j}(t)\big{|}\leq\varepsilon;\qquad\displaystyle\max_{j\in\llbracket
1,wn-1\rrbracket}\displaystyle\sup_{t\in[a,b]}\big{|}\widetilde{y}_{j}(t)-\widetilde{x}_{j}(t)\big{|}\leq\varepsilon.$

Moreover, since $\bm{u}\leq\widetilde{\bm{u}}$ and
$\bm{v}\leq\widetilde{\bm{v}}$ (as
$G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y)$ for
$(t,y)\in\\{a,b\\}\times[0,w]$), and $f\leq\widetilde{f}$ (by (23.1), as
$G^{\star}(t,w)\leq\widetilde{G}^{\star}(t,w)$), it follows from 4.6 that we
may couple $\bm{y}$ and $\widetilde{\bm{y}}$ in such a way that
$y_{j}(t)\leq\widetilde{y}_{j}(t)$ for each $(j,t)\in\llbracket
1,wn-1\rrbracket\times[a,b]$. Combining this with (23.2) induces a coupling
between $\bm{x}$ and $\widetilde{\bm{x}}$ with

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\Bigg{[}\bigcap_{j\in\llbracket
1,wn-1\rrbracket}\bigcap_{t\in[a,b]}\big{\\{}x_{j}(t)-2\varepsilon\leq\widetilde{x}_{j}(t)\big{\\}}\Bigg{]}=1.$

Together with the first statement of 10.7, this implies upon letting
$\varepsilon$ tend to $0$ that

(23.3) $\displaystyle
G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y),\qquad\text{whenever
$G^{\star}(t,y^{\prime})$ and $\widetilde{G}^{\star}(t,y^{\prime})$ are
continuous at $y^{\prime}=y$}.$

It thus remains to show that (23.3) continues for more general
$(t,y)\in[a,b]\times[0,w]$.

To this end, for $t\in(a,b)$ observe (since the densities $\varrho^{\star}$
and $\widetilde{\varrho}^{\star}$ associated with $\bm{\mu}^{\star}$ and
$\widetilde{\bm{\mu}}^{\star}$, respectively, are bounded by the third part of
10.5) that (10.3) and (10.4) together yield

(23.4) $\displaystyle
G^{\star}(t,y)\quad\text{and}\quad\widetilde{G}^{\star}(t,y)\qquad\text{are
lower semicontinuous and non-increasing in $y\in(0,A_{0}]$}.$

Thus, given any point $y\in[0,A_{0}]$ and real number $\delta>0$, there is a
point $y_{1}\in(y-\delta,y)$ such that $G^{\star}(t,y^{\prime})$ and
$\widetilde{G}^{\star}(t,y^{\prime})$ are continuous in its second variable at
$y^{\prime}=y_{1}$. Hence, letting $\delta$ tend to $0$, it follows from
(23.3) that $G^{\star}(t,y^{-})\leq\widetilde{G}^{\star}(t,y^{-})$, and so by
(23.4) we deduce that $G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y)$ for each
$(t,y)\in[a,b]\times[0,w]$, establishing the lemma. ∎

###### Proof of 10.15 (Outline).

We again only establish the second part of the lemma, as the proof of the
first is entirely analogous. Its proof will be similar to that of the second
part of 10.14, and so we only outline it. In what follows, we adopt the
notation of that lemma, recalling the entrance and exit data
$\bm{u},\bm{v}\in\overline{\mathbb{W}}_{An}$ and
$\widetilde{\bm{u}},\widetilde{\bm{v}}\in\overline{\mathbb{W}}_{\tilde{A}n}$;
the boundaries $f,\widetilde{f}:[a,b]\rightarrow\mathbb{R}$; and the non-
intersecting Brownian bridges $\bm{x}^{n}\in\llbracket
1,An\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$,
$\widetilde{\bm{x}}^{n}\in\llbracket
1,\widetilde{A}n\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$, and
$\bm{y}^{n},\widetilde{\bm{y}}^{n}\in\llbracket
1,wn-1\rrbracket\times\mathcal{C}\big{(}[a,b]\big{)}$. Following the proof of
10.14, we may couple $(\bm{x},\bm{y})$ and
$(\widetilde{\bm{x}},\widetilde{\bm{y}})$ such that with probability $1-o(1)$
(that is, tending to $1$ as $n$ tends to $\infty$) (23.2) holds.

Next, observe that $u_{j}-u_{j+1}\leq\widetilde{u}_{j}-\widetilde{u}_{j+1}$
and $v_{j}-v_{j+1}\leq\widetilde{v}_{j}-\widetilde{v}_{j+1}$, since
$\big{|}G^{\star}(t,y)-G^{\star}(t,y^{\prime})\big{|}\leq\big{|}\widetilde{G}^{\star}(t,y)-\widetilde{G}^{\star}(t,y^{\prime})\big{|}$
for each $(t,y),(t,y^{\prime})\in\\{a,b\\}\in[0,w]$. Moreover, by (23.1) and
(10.13), we have $r\cdot f(s)-f\big{(}rs+(1-r)t\big{)}+(1-r)\cdot f(t)\leq
r\cdot\widetilde{f}(s)-\widetilde{f}\big{(}rs+(1-r)t\big{)}+(1-r)\cdot\widetilde{f}(t)$
for any $s,t\in(a,b)$ and $r\in[0,1]$. Thus, it follows from gap monotonicity
5.1 that we may couple $\bm{y}$ and $\widetilde{\bm{y}}$ such that
$y_{j}(t)-y_{j+1}(t)\leq\widetilde{y}_{j}(t)-\widetilde{y}_{j+1}(t)$ for each
$(j,t)\in\llbracket 1,wn-1\rrbracket\times[a,b]$. Combining this with (23.2),
it follows that

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\Bigg{[}\bigcap_{j,j^{\prime}\in\llbracket
1,wn-1\rrbracket}\bigcap_{t\in[a,b]}\Big{\\{}\big{|}x_{j}(t)-x_{j^{\prime}}(t)-4\varepsilon\leq\big{|}\widetilde{x}_{j}(t)-\widetilde{x}_{j^{\prime}}(t)\big{|}\Big{\\}}\Bigg{]}=1.$

Together with the first statement of (10.7), this implies upon letting
$\varepsilon$ tend to $0$ that

$\displaystyle\big{|}G^{\star}(t,y)-G^{\star}(t,y^{\prime})\big{|}\leq\big{|}\widetilde{G}^{\star}(t,y)-\widetilde{G}^{\star}(t,y^{\prime})\big{|},$

if $G^{\star}(t,y^{\prime\prime})$ and
$\widetilde{G}^{\star}(t,y^{\prime\prime})$ are continuous at
$y^{\prime\prime}=y$ and $y^{\prime\prime}=y^{\prime}$. Extending this bound
to all $(t,y),(t,y^{\prime})\in(a,b)\times[0,w]$ then follows as in the proof
of 10.14, and so further details are omitted; this establishes the lemma. ∎

####  23.2. Proof of 11.4

In this section we establish 11.4; throughout, we recall the notation of that
lemma. By the scale invariance (Item 1 of 10.19) of solutions to (10.14), we
may assume that $\ell=1$ in what follows.222Indeed, given $G_{1}^{\pm}$
defined at $\ell=1$, we set $G^{\pm}(t,x)=\ell^{-1}\cdot G^{\pm}(\ell t,\ell
x)$, which continues to satisfy (10.14) (by 10.19) and the statements of 11.4
(possibly with different constants $c>0$ and $C>1$). We first define the open
rectangles

$\displaystyle\breve{\mathfrak{S}}=\Big{(}0,\displaystyle\frac{1}{L}\Big{)}\times\Big{(}\displaystyle\frac{1}{2L},1-\displaystyle\frac{1}{2L}\Big{)};\qquad\breve{\mathfrak{S}}^{\prime}=\Big{(}\displaystyle\frac{1}{8L},\displaystyle\frac{7}{8L}\Big{)}\times\Big{(}\displaystyle\frac{1}{L},1-\displaystyle\frac{1}{L}\Big{)}.$

We further fix functions333For example, fix a nonnegative, smooth function
$\psi:[0,1]\rightarrow\mathbb{R}$ with $\psi(x)=1$ if $x\in[1/5,4/5]$, with
$\psi(x)=0$ if $x\in[0,1/6]\cup[5/6,1]$, and with $0\leq\psi(x)\leq 1$ for
each $x\in[0,1]$. Then we may set $g_{i}^{\pm}(x)=\psi(x)\cdot
g_{i}(x)+\big{(}1-\psi(x)\big{)}\cdot\big{(}f_{i}(x)\pm\vartheta^{8/9}\big{)}$
for any indices $i\in\\{0,1\\}$ and $\pm\in\\{+,-\\}$, and any real number
$x\in[0,1]$.
$g_{0}^{-},g_{0}^{+},g_{1}^{-},g_{1}^{+}:[0,1]\rightarrow\mathbb{R}$ so that,
for some constant $C_{1}=C_{1}(\varepsilon,B,m)>1$ we have

(23.5) $\displaystyle\begin{aligned}
&g_{i}^{-}(x)=g_{i}(x)=g_{i}^{+}(x),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{for
each $x\in\Big{[}\displaystyle\frac{1}{5},\frac{4}{5}\Big{]}$};\\\
&g_{i}^{-}(x)\leq g_{i}(x)\leq
g_{i}^{+}(x),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{for each
$x\in[0,1]$};\\\ &g_{i}^{-}(x)=f_{i}(x)-\vartheta^{8/9},\quad\text{and}\quad
g_{i}^{+}(x)=f_{i}(x)+\vartheta^{8/9},\qquad\qquad\text{for each
$x\in\Big{[}0,\displaystyle\frac{1}{6}\Big{]}\cup\Big{[}\displaystyle\frac{5}{6},1\Big{]}$};\\\
&\|g_{i}^{-}-g_{-}\|_{\mathcal{C}_{0}}+\|g_{i}^{+}-g_{i}\|_{\mathcal{C}^{0}}\leq
C_{1}\vartheta^{8/9},\qquad\qquad\qquad\qquad\qquad\text{for each
$i\in\\{0,1\\}$};\\\
&\|g_{i}^{-}\|_{\mathcal{C}^{m}}+\|g_{i}^{+}\|_{\mathcal{C}^{m}}\leq
C_{1},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{for each
$i\in\\{0,1\\}$},\end{aligned}$

where the last three properties can be guaranteed since
$\|f_{i}-g_{i}\|_{\mathcal{C}^{0}}\leq\vartheta\leq\vartheta^{8/9}$ (with the
facts that
$\|f_{i}\|_{\mathcal{C}^{m}}\leq\|F_{i}\|_{\mathcal{C}^{m}(\mathfrak{R})}\leq
B$ and $\|g_{i}\|_{\mathcal{C}^{m}}\leq B$), for each $i\in\\{0,1\\}$.

Then, for $\vartheta$ sufficiently small, 10.25 yields (upon translating by
$(-1/2L,1)$; replacing the $L$ there by $L/2$ here; and scaling by a factor of
$2$, using 10.19; and taking $r=1/4$) solutions
$G^{-},G^{+}\in\operatorname{Adm}_{\varepsilon/2}(\breve{\mathfrak{S}})$ to
(10.14) on $\breve{\mathfrak{S}}$ and a constant
$C_{2}=C_{2}(\varepsilon,B,m)>1$ such that

(23.6) $\displaystyle\begin{aligned}
&G^{\pm}(iL^{-1},x)=g_{i}^{\pm}(x),\qquad\text{for each
$x\in\Big{[}\displaystyle\frac{1}{2L},1-\displaystyle\frac{1}{2L}\Big{]}$};\\\
&\|G^{\pm}-F\|_{\mathcal{C}^{m}(\breve{\mathfrak{S}}^{\prime})}\leq
C_{2}L^{m}\cdot\big{(}\|g_{0}^{\pm}-f_{0}\|_{\mathcal{C}^{0}}+\|g_{1}^{\pm}-f_{1}\|_{\mathcal{C}^{0}}\big{)}\\\
&\|G^{\pm}-F\|_{\mathcal{C}^{m-5}(\breve{\mathfrak{S}})}\leq
C_{2}L^{m-5}\cdot\big{(}\|g_{0}^{\pm}-f_{0}\|_{\mathcal{C}^{0}}^{3/m}+\|g_{1}^{\pm}-f_{1}\|_{\mathcal{C}^{0}}^{3/m}\big{)},\end{aligned}$

for any indices $i\in\\{0,1\\}$ and $\pm\in\\{+,-\\}$. This, together with the
fact that $\|g_{i}^{\pm}-f_{i}\|_{\mathcal{C}^{0}}\leq 2C_{1}\vartheta^{8/9}$
for each $i\in\\{0,1\\}$ (by (23.5) and the fact that
$\|f_{i}-g_{i}\|_{\mathcal{C}^{0}}\leq\vartheta$ for each $i$), implies that

(23.7) $\displaystyle\begin{aligned}
&\|G^{\pm}-F\|_{\mathcal{C}^{m}(\breve{\mathfrak{S}}^{\prime})}\leq
4C_{1}C_{2}L^{m}\vartheta^{8/9}\leq 4C_{1}C_{2}\vartheta^{4/5};\\\
&\|G^{\pm}-F\|_{\mathcal{C}^{m-5}(\breve{\mathfrak{S}})}\leq
4C_{1}C_{2}L^{m-5}\vartheta^{8/3m}\leq
4C_{1}C_{2}\vartheta^{1/m},\end{aligned}$

where in the last inequalities we also used the facts that
$L\leq\vartheta^{-1/2m^{2}}$ and that $m\geq 7$.

######  Lemma 23.1.

There exists a constant $c=c(\varepsilon,B,m)>0$ such that the following holds
if $\vartheta<c$.

  1. (1)

For each $(t,x)\in[0,L^{-1}]\times[1/4,3/4]$, we have
$\big{|}G^{+}(t,x)-G^{-}(t,x)\big{|}\leq c^{-1}e^{-cL^{1/8}}$.

  2. (2)

For each $(t,x)\in[0,L^{-1}]\times\big{(}[1/10,3/20]\cup[17/20,9/10]\big{)}$,
we have

(23.8) $\displaystyle\big{|}G^{-}(t,x)-F(t,x)+\vartheta^{8/9}\big{|}\leq
c^{-1}e^{-cL^{1/8}};\qquad\big{|}G^{+}(t,x)-F(t,x)-\vartheta^{8/9}\big{|}\leq
c^{-1}e^{-cL^{1/8}}.$

###### Proof.

This will follow from 10.24, together with an appropriate rescaling. To
establish the first statement of the lemma, define the function
$\widehat{G}^{\pm}:\big{[}0,2/(L-1)\big{]}\times[-1,1]\rightarrow\mathbb{R}$
by setting

(23.9) $\displaystyle\widehat{G}^{\pm}(t,x)=\displaystyle\frac{2L}{L-1}\cdot
G^{\pm}\bigg{(}\displaystyle\frac{L-1}{2L}\cdot
t,\displaystyle\frac{L-1}{2L}\cdot x+\displaystyle\frac{1}{2}\bigg{)},$

for any index $\pm\in\\{+,-\\}$ and pair
$(t,x)\in\big{[}0,2/(L-1)\big{]}\times[-1,1]$; these functions satisfy (10.14)
by Item 1 of 10.19. We next apply 10.24, with the
$(F_{1},F_{2};L;r;\varepsilon;B)$ there equal to
$\big{(}\widehat{G}^{-},\widehat{G}^{+};(L-1)/2;1/3;\varepsilon/2;4B\big{)}$
here, using the fact (from the second statement of (23.7) with the bound
$L>4$) that

$\displaystyle\|\widehat{G}^{\pm}\|_{\mathcal{C}^{5}(\breve{\mathfrak{S}})}\leq\frac{2L}{L-1}\cdot\|G^{\pm}\|_{\mathcal{C}^{5}(\breve{\mathfrak{S}})}\leq
3\|G^{\pm}-F\|_{\mathcal{C}^{5}(\breve{\mathfrak{S}})}+3\|F\|_{\mathcal{C}^{5}(\breve{\mathfrak{S}})}\leq
3B+12C_{1}C_{2}\vartheta^{1/m}\leq 4B,$

for sufficiently small $\vartheta$, to verify the bound on the
$\|F_{i}\|_{\mathcal{C}^{5}}$ assumed there. This yields a constant
$c_{1}=c_{1}(\varepsilon,B,m)>0$ such that

$\displaystyle\displaystyle\sup_{t\in[0,2/(L-1)]}\displaystyle\sup_{|x|\leq
2/3}\big{|}\widehat{G}^{+}(t,x)-\widehat{G}^{-}(t,x)\big{|}\leq
c_{1}^{-1}e^{-c_{1}L^{1/8}}.$

Together with (23.9), this implies for sufficiently small $\vartheta$ (and
thus sufficiently large $L$, as $L>|\log\vartheta|^{20}$) that

$\displaystyle\displaystyle\sup_{t\in[0,1/L]}\displaystyle\sup_{x\in[1/4,3/4]}\big{|}G^{+}(t,x)-G^{-}(t,x)\big{|}\leq\displaystyle\sup_{t\in[0,2/(L-1)]}\displaystyle\sup_{|x|\leq
2/3}\big{|}\widehat{G}^{+}(t,x)-\widehat{G}^{-}(t,x)\big{|}\leq
c_{1}^{-1}e^{-c_{1}L^{1/8}},$

and thus the first statement of the lemma.

Figure 7.1. Shown above is a depiction of the rescaling used in the proof of
the second part of 23.1.

Next we establish the second part of the lemma. Since the derivation of both
statements are entirely analogous, we only detail that of the first, namely,
of the bound $\big{|}G^{-}(t,x)-F(t,x)+\vartheta^{8/9}\big{|}\leq
c^{-1}e^{-cL^{1/8}}$ for $t\in[0,L^{-1}]$ and
$x\in[1/10,3/20]\cup[17/20,9/10]$; we also only address the case when
$x\in[17/20,9/10]$, as the proof in the complementary case is again very
similar. As before, we begin by rescaling, namely, we define the functions
$\widetilde{F},\widetilde{G}^{-}:[0,15L^{-1}]\times[-1,1]$ by setting

(23.10)
$\displaystyle\widetilde{F}(t,x)=15F\Big{(}\displaystyle\frac{t}{15},\displaystyle\frac{x+13}{15}\Big{)}-15\vartheta^{8/9};\qquad\widetilde{G}^{-}(t,x)=15G^{-}\Big{(}\displaystyle\frac{t}{15},\displaystyle\frac{x+13}{15}\Big{)}.$

for any pair $(t,x)\in[0,15L^{-1}]\times[0,1]$; see Figure 7.1. These
functions satisfy (10.14) by Item 1 of 10.19. We then apply 10.24, with the
parameters $(F_{1},F_{2};L;r;\varepsilon;B)$ there equal to
$(\widetilde{G}^{-},\widetilde{F};L/15;1/4;\varepsilon/2;20B)$ here (to verify
the bounds on the $\|F_{i}\|_{\mathcal{C}^{2}}$ assumed there, we used the
facts that $\|\widetilde{F}\|_{\mathcal{C}^{2}}\leq
15\|F\|_{\mathcal{C}^{2}}\leq 15B$ and
$\|\widetilde{G}^{-}\|_{\mathcal{C}^{2}}\leq 15\|G^{-}\|_{\mathcal{C}^{2}}\leq
15(B+4C_{2}\vartheta^{1/m})\leq 20B$, by (11.4) and taking $\vartheta$
sufficiently small), which yields a constant $c_{2}=c_{2}(\varepsilon,B,m)>0$
such that

$\displaystyle\displaystyle\sup_{t\in[0,15/L]}\displaystyle\sup_{x\in[-1/4,1/2]}\big{|}\widetilde{G}^{-}(t,x)-\widetilde{F}(t,x)\big{|}\leq\displaystyle\sup_{t\in[0,15/L]}\displaystyle\sup_{|x|\leq
3/4}\big{|}\widetilde{G}^{-}(t,x)-\widetilde{F}(t,x)\big{|}\leq
c_{2}^{-1}e^{-c_{2}L^{1/8}}.$

Together with (23.10), this yields

$\displaystyle\displaystyle\sup_{t\in[0,1/L]}\displaystyle\sup_{x\in[17/20,9/10]}\big{|}G^{-}(t,x)-F(t,x)+\vartheta^{8/9}\big{|}$
$\displaystyle\leq
15\displaystyle\sup_{t\in[0,15/L]}\displaystyle\sup_{|x|\leq
3/4}\big{|}\widetilde{G}^{-}(t,x)-\widetilde{F}(t,x)\big{|}$
$\displaystyle\leq 15c_{2}^{-1}e^{-c_{2}L^{1/8}},$

which verifies the first bound in (23.8) when $x\in[17/20,9/10]$. As mentioned
previously, the proof of this estimate when $x\in[1/10,3/20]$ is entirely
analogous, as is the proof of the second statement of (23.8); this establishes
the lemma. ∎

Now we can quickly establish 11.4.

###### Proof of 11.4.

As indicated above, we may assume that $\ell=1$. The $(G^{-},G^{+})$ of this
lemma will be taken to be $(G^{-}|_{\mathfrak{S}},G^{+}|_{\mathfrak{S}})$
here. Then the first statements of (23.6) and (23.5) together verify that
$(G^{-},G^{+})$ satisfy the first statement of the lemma; moreover, the first
statement of (23.6) with the second statement of (23.5) verify the second
statement of the lemma. The second bound in (23.7) (with the fact that
$\|F\|_{\mathcal{C}^{m-5}(\mathfrak{S})}\leq\|F\|_{\mathcal{C}^{m}(\mathfrak{R})}\leq
B$) verifies the third statement of the lemma, and the first bound in (23.7)
verifies the fourth. The first part of 23.1 verifies the fifth part of the
lemma, and its second part (together with the fact that
$\vartheta^{8/9}-c^{-1}e^{-cL^{1/8}}\geq\vartheta$ for sufficiently small
$\vartheta$, since $L>|\log\vartheta|^{20}$) verifies the sixth. ∎

###  24\. Proofs of Results From Chapter 4

####  24.1. Further Properties of Free Convolutions

In this section we collect some properties of free convolutions with a
rescaled semicircle distribution (which is essentially due to [19], but stated
as below in [6]) through the following lemma, which will be used repeatedly in
the below. In what follows, we recall the definitions related to free
convolutions from Section 4.3 (including the Stieltjes transform $m_{0}$ of
$\mu$ from (4.3), the function $M$ and set $\Lambda_{t}$ from (4.7), and the
density $\varrho_{t}\in L^{1}(\mathbb{R})$ of
$\mu_{t}=\mu\boxplus\mu_{\operatorname{sc}}^{(t)}$ with respect to Lebesgue
measure).

######  Lemma 24.1 ([6, Lemma 2.3]).

The following statements hold, for any real number $t>0$.

  1. (1)

Define the function $v_{t}:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ by for
each $u\in\mathbb{R}$ setting

(24.1) $\displaystyle v_{t}(u)=\displaystyle\inf\bigg{\\{}v\geq
0:\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{\mu(dx)}{(u-x)^{2}+v^{2}}\leq
t^{-1}\bigg{\\}}.$

Then $v_{t}$ is continuous on $\mathbb{R}$. Moreover, the boundary of
$\Lambda_{t}$ is parameterized by
$\partial\Lambda_{t}=\\{E+\mathrm{i}v_{t}(E):E\in{\mathbb{R}}\\}$, and the set
$\big{\\{}E\in{\mathbb{R}}:v_{t}(E)>0\big{\\}}$ consists of countably many
open intervals $\bigcup_{i\geq 1}(a_{i},b_{i})$.

    1. (a)

For each $E\in\bigcup_{i\geq 1}(a_{i},b_{i})$, we have
$\int_{-\infty}^{\infty}\big{|}x-E-\mathrm{i}v_{t}(E)\big{|}^{-2}\mu(dx)=t^{-1}$.

    2. (b)

For each $E\in{\mathbb{R}}\setminus\bigcup_{i\geq 1}(a_{i},b_{i})$, we have
$\int_{-\infty}^{\infty}\big{|}x-E-\mathrm{i}v_{t}(E)\big{|}^{-2}\mu(dx)\leq
t^{-1}$.

  2. (2)

We have $\operatorname{supp}\mu\subseteq\bigcup_{i\geq 1}[a_{i},b_{i}]$, and
$\mu\big{(}\bigcup_{i\geq 1}\\{a_{i},b_{i}\\}\big{)}=0$.

  3. (3)

The function $M\big{(}E+\mathrm{i}v_{t}(E)\big{)}$ is (strictly) increasing in
$E\in{\mathbb{R}}$. Moreover, $v_{t}(E)$ and
$M\big{(}E+\mathrm{i}v_{t}(E)\big{)}$ are smooth for $E$ in the interior of
$\mathbb{R}\setminus\bigcup_{i\geq 1}\\{a_{i},b_{i}\\}$.

  4. (4)

A real number $y\in\mathbb{R}$ satisfies $\varrho_{t}(y)>0$ if and only if
$y=M(w)=w-tm_{0}(w)$ for some $w=w(y)\in\partial\Lambda_{t}\cap\mathbb{H}$.
Moreover, the function $w(y)$ is smooth in
$y\in\big{\\{}y^{\prime}\in\mathbb{R}:\varrho_{t}(y^{\prime})>0\big{\\}}$.

    1. (a)

We have $\varrho_{t}(y)=\pi^{-1}\operatorname{Im}m_{0}(w)=(\pi
t)^{-1}\operatorname{Im}w$.

    2. (b)

The Hilbert transform of $\varrho_{t}$ is given by
$H\varrho_{t}(y)=\pi^{-1}\operatorname{Re}m_{0}(w)=(\pi
t)^{-1}\operatorname{Re}(w-y)$.

  5. (5)

Denote $\mathfrak{e}_{+}=\mathfrak{e}_{+}(t)=\max\operatorname{supp}\mu_{t}$
and $\mathfrak{e}_{-}=\mathfrak{e}_{-}(t)=\min\operatorname{supp}\mu_{t}$, and
let

$\displaystyle
w_{+}=\sup\Bigg{\\{}w\in\mathbb{R}:\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{\mu(dx)}{(x-w)^{2}}>t^{-1}\Bigg{\\}};\qquad
w_{-}=\inf\Bigg{\\{}w\in\mathbb{R}:\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{\mu(dx)}{(x-w)^{2}}>t^{-1}\Bigg{\\}}.$

Then $\mathfrak{e}_{+}=w_{+}-tm_{0}(w_{+})$ and
$\mathfrak{e}_{-}=w_{-}-tm_{0}(w_{-})$.

####  24.2. Proof of 13.3

In what follows, we recall the notation from Section 4.3 and adopt 13.1. We
denote the Stieltjes transforms of $\nu$ and $\nu_{\tau}$ as
$m=m^{\nu}:\mathbb{H}\rightarrow\mathbb{H}$ and
$m_{\tau}=m^{\nu_{\tau}}:\mathbb{H}\rightarrow\mathbb{H}$, respectively. We
also recall the function $M=M^{\nu}$ and set $\Lambda_{\tau}$ from (4.7), as
well as the function $v_{\tau}$ from (24.1), which is continuous by the first
part of 24.1. In what follows, we abbreviate $v=v_{\tau}$ and further recall
from Item 1 of 24.1 that
$\partial\Lambda_{\tau}=\big{\\{}E+\mathrm{i}v(E):E\in\mathbb{R}\big{\\}}$.

By part (1a) of 24.1, we have

(24.2) $\displaystyle
1=\tau\int_{-\infty}^{\infty}\frac{\nu(dx)}{\big{|}x-E-\mathrm{i}v(E)\big{|}^{2}}=\tau\int_{-\infty}^{\infty}\frac{\nu(dx)}{(x-E)^{2}+v(E)^{2}},\qquad\text{if
$v(E)>0$}.$

Moreover, define the functions
$w,y:\mathbb{R}\rightarrow\overline{\mathbb{H}}$ by setting (here, we recall
$M$ from (4.7))

(24.3) $\displaystyle w(E)=E+\mathrm{i}v(E),\quad\text{and}\quad
y(E)=M\big{(}w(E)\big{)}=E+\mathrm{i}v(E)-\tau
m\big{(}E+\mathrm{i}v(E)\big{)}.$

By Item 1 of 24.1, we have $w(E)\in\partial\Lambda_{\tau}$. Together with
4.12, this implies that $y(E)\in\mathbb{R}$, so Item 4 of 24.1 gives

(24.4) $\displaystyle
y(E)=E-\tau\operatorname{Re}m\big{(}E+\mathrm{i}v(E)\big{)},\qquad\text{so}\quad
y(E)=E-\tau\operatorname{Re}m\big{(}E+\mathrm{i}v(E)\big{)}\in\operatorname{supp}\varrho_{\tau},\quad\text{if
$v(E)>0$},$

and also that

(24.5)
$\displaystyle\varrho_{\tau}\big{(}y(E)\big{)}=\pi^{-1}\operatorname{Im}m\big{(}w(E)\big{)}=(\pi\tau)^{-1}v(E).$

Moreover, for any $E\in\mathbb{R}$, Item 1 in 24.1 gives

(24.6)
$\displaystyle\int_{-\infty}^{\infty}\frac{\nu(dx)}{\big{|}x-E-\mathrm{i}v(E)\big{|}^{2}}\leq\tau^{-1}.$

Then (24.6) and the fact that $\nu(\mathbb{R})=L^{3/2}$ together imply for
each $E\in\mathbb{R}$ that

(24.7)
$\displaystyle\Big{|}m\big{(}E+\mathrm{i}v(E)\big{)}\Big{|}=\Bigg{|}\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{\nu(dx)}{x-E-\mathrm{i}v(E)}\Bigg{|}\leq\Bigg{(}\int_{-\infty}^{\infty}\frac{\nu(dx)}{|x-E-\mathrm{i}v(E)|^{2}}\int_{-\infty}^{\infty}\nu(dx)\Bigg{)}^{1/2}\leq\frac{L^{3/4}}{\tau^{1/2}}.$

Using (24.4) and (24.7), it follows for any $E\in\mathbb{R}$ that

(24.8)
$\displaystyle\big{|}y(E)-E\big{|}=\tau\Big{|}\operatorname{Re}m\big{(}E+\mathrm{i}v(E)\big{)}\Big{|}\leq\tau^{1/2}L^{3/4}.$

We next have the following lemma bounding the density $\varrho_{\tau}$ and the
endpoints of its support; in the below, we define

(24.9) $\displaystyle y_{-}=\inf(\operatorname{supp}\varrho_{\tau});\quad
y_{+}=\sup(\operatorname{supp}\varrho_{\tau}).$

######  Lemma 24.2.

The following two statements hold.

  1. (1)

For any $x\in\mathbb{R}$, we have
$\varrho_{\tau}(x)\leq\pi^{-1}B^{1/2}L^{3/4}$.

  2. (2)

We have $-BL-2B^{1/2}L^{3/4}\leq y_{-}\leq y_{+}\leq 2B^{1/2}L^{3/4}$.

###### Proof.

From (24.5), (24.3), and (24.7), we have

$\displaystyle\varrho_{\tau}(x)\leq\pi^{-1}\Big{|}m\big{(}x+\mathrm{i}v(x)\big{)}\Big{|}\leq\displaystyle\frac{L^{3/4}}{\pi\tau^{1/2}}\leq\displaystyle\frac{B^{1/2}L^{3/4}}{\pi},$

where in the last bound we used the fact that $\tau\geq B^{-1}$. This verifies
the first part of the lemma.

From Item 5 in 24.1, we have
$y_{-}=\min\operatorname{supp}\varrho_{\tau}=y(E_{-})$ and
$y_{+}=\max\operatorname{supp}\varrho_{\tau}=y(E_{+})$, where
$E_{-},E_{+}\in{\mathbb{R}}$ are supremum and infimum, respectively, over all
real numbers $E_{0}$ satisfying

$\displaystyle 1<\tau\int_{-\infty}^{\infty}\frac{\nu(dx)}{|x-E_{0}|^{2}}.$

Hence, for each index $\pm\in\\{+,-\\}$, we can estimate $E_{\pm}$ through the
bound

$\displaystyle
1\leq\frac{\tau\int_{-\infty}^{\infty}\nu(dx)}{\operatorname{dist}(E_{\pm},\operatorname{supp}\nu)^{2}}=\frac{\tau
L^{3/2}}{\operatorname{dist}(E_{\pm},\operatorname{supp}\nu)^{2}},\qquad\text{so
that}\qquad\operatorname{dist}(E_{\pm},\operatorname{supp}\nu)\leq\tau^{1/2}L^{3/4},$

where we used the fact that $\nu(\mathbb{R})=L^{3/2}$ in the second estimate
above. Since by 13.1 we have $\operatorname{supp}\nu\subseteq[-BL,0]$, it
follows that

$\displaystyle-BL-\tau^{1/2}L^{3/4}\leq E_{\pm}\leq\tau^{1/2}L^{3/4}.$

Together with (24.8) and the fact that $y_{\pm}=y(E_{\pm})$, this yields

$\displaystyle y_{-}\geq E_{-}-L^{3/4}\tau^{1/2}\geq-
BL-2\tau^{1/2}L^{3/4}\geq-BL-2B^{1/2}L^{3/4};$ $\displaystyle y_{+}\leq
E_{+}+L^{3/4}\tau^{1/2}\leq 2\tau^{1/2}L^{3/4}\leq 2B^{1/2}L^{3/4},$

where in the last inequalities of the above bounds we used the fact that
$\tau\leq B$; this implies the second part of the lemma. ∎

The below lemma shows that if $v$ is bounded above on $[a,b]$ then, up to a
multiplicative factor, $\nu_{\tau}\big{(}[y(a),y(b)]\big{)}$ is lower bounded
by $\nu\big{(}[a,b]\big{)}$; it is established in Section 24.3 below.

######  Lemma 24.3.

Fix real numbers $a<b$. If $v(E)\leq(b-a)/2$ for each real number $E\in[a,b]$,
then

(24.10)
$\displaystyle\nu_{\tau}\Big{(}\big{[}y(a),y(b)\big{]}\Big{)}\geq\frac{1}{8\pi}\cdot\nu\big{(}[a,b]\big{)}.$

We now fix positive real parameters $\mathfrak{M},r\in\mathbb{R}$ so that

(24.11) $\displaystyle{\mathfrak{M}}>2;\qquad
r=\big{\lceil}\log_{\mathfrak{M}}(BL+2B^{1/2}L^{3/4})\big{\rceil}.$

We further fix a sequence of numbers $y_{0}>y_{1}>\cdots>y_{r+1}$ defined by
setting

(24.12) $\displaystyle y_{0}=0,\qquad\text{and}\qquad
y_{i}-y_{i+1}={\mathfrak{M}}^{i},\quad\text{for each $i\in\llbracket
0,r\rrbracket$}.$

Then, recalling the endpoints $y_{-}$ and $y_{+}$ of
$\operatorname{supp}\varrho_{\tau}$ from (24.9), we have

(24.13) $\displaystyle
y_{r+1}\leq-{\mathfrak{M}}^{r}\leq-(BL+2B^{1/2}L^{3/4})\leq y_{-},$

where the last inequality follows from Item 2 of 24.2; hence,
$\operatorname{supp}\varrho_{\tau}\subseteq[y_{r+1},y_{+}]$. From (24.12) (and
(24.11)), we also have $-2{\mathfrak{M}}^{i-1}\leq y_{i}\leq 0$ for each
integer $i\in\llbracket 0,r+1\rrbracket$. Recalling the map $y(E)$ from (24.3)
(and (24.4)), Item 3 in 24.1 indicates $y:\mathbb{R}\rightarrow\mathbb{R}$ is
an increasing bijection. Therefore, there exist real numbers
$E_{0}>E_{1}>\cdots>E_{r+1}$ such that for each $i\in\llbracket
1,r+1\rrbracket$ we have

(24.14) $\displaystyle y_{i}=y(E_{i})=E_{i}+\mathrm{i}v(E_{i})-\tau
m\big{(}E_{i}+\mathrm{i}v(E_{i})\big{)}=E_{i}-\tau\operatorname{Re}m\big{(}E_{i}+\mathrm{i}v(E_{i})\big{)}.$

Defining the real numbers $E_{+},E_{-}\in\mathbb{R}$ by

$\displaystyle
E_{+}=\sup\Bigg{\\{}E\in\mathbb{R}:\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{\mu(dx)}{(x-w)^{2}}>t^{-1}\Bigg{\\}};\qquad
E_{-}=\sup\Bigg{\\{}E\in\mathbb{R}:\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{\mu(dx)}{(x-w)^{2}}>t^{-1}\Bigg{\\}},$

Item 5 of 24.1, (24.13), and the fact that $y$ is increasing together yield
$y(E_{-})=y_{-}\geq y_{r+1}=y(E_{r+1})$, so $E_{-}\geq E_{r+1}$. Since
$\operatorname{supp}\nu\subseteq[E_{-},E_{+}]$ by Item 2 of 24.1, we thus have

(24.15)
$\displaystyle\operatorname{supp}\nu\subseteq[E_{-},E_{+}]\subseteq[E_{r+1},E_{+}],$

and so by 13.1 (which also indicates that $B\geq 1$, so $\mathfrak{M}^{r}\geq
BL\geq L$) it follows that

(24.16)
$\displaystyle\nu\big{(}[E_{r+1},E_{+}]\big{)}=\nu({\mathbb{R}})=L^{3/2}\leq({\mathfrak{M}}^{r})^{3/2}={\mathfrak{M}}^{3r/2}.$

Using (24.8), we find for any integer $i\in\llbracket 0,r+1\rrbracket$ that
$|E_{i}-y_{i}|\leq\tau\big{|}m(E_{i}+\mathrm{i}v(E_{i}))\big{|}\leq\tau^{1/2}L^{3/4}$,
and so

(24.17) $\displaystyle E_{0}-E_{r+1}\geq
y_{0}-y_{r+1}-2\tau^{1/2}L^{3/4}\geq{\mathfrak{M}}^{r}-2\tau^{1/2}L^{3/4}\geq\displaystyle\frac{{\mathfrak{M}}^{r}}{3},$

where in the last bound we used that $\mathfrak{M}^{r}\geq BL+2B^{1/2}L^{3/4}$
and $BL\geq B^{1/2}L^{3/4}\geq\tau^{1/2}L^{3/4}$ (as $B,L>1$ and $\tau\leq
B$).

The following lemma bounds the $E_{i}$ and $\nu\big{(}[E_{i+1},E_{i}]\big{)}$
under 13.2.

######  Lemma 24.4.

Adopt 13.2. Fix two constants $c=(2^{9/2}\pi B)^{-1}>0$ and ${\mathfrak{K}}>1$
with ${\mathfrak{M}}^{{\mathfrak{K}}/2-4}\geq 72c^{-3}B$. For any integers
$k\in\llbracket{\mathfrak{K}},r+1\rrbracket$ and $i\in\llbracket
k,r+1\rrbracket$, we have

(24.18) $\displaystyle\begin{split}&E_{0}-E_{k}\geq
c\cdot{\mathfrak{M}}^{k-1};\qquad\qquad\qquad E_{i}-E_{i+1}\geq
c\cdot{\mathfrak{M}}^{i};\\\ &\nu\big{(}[E_{k},E_{+}]\big{)}\leq
c^{-1}\cdot{\mathfrak{M}}^{3(k-1)/2};\qquad\nu\big{(}[E_{i+1},E_{i}]\big{)}\leq
c^{-1}\cdot{\mathfrak{M}}^{3i/2},\end{split}$

where the second and fourth statements of (LABEL:e:ind) are empty if $k=r+1$.
Moreover,

(24.19) $\displaystyle
v(E)\leq\Big{(}\displaystyle\frac{2\tau}{c}\Big{)}^{1/2}{\mathfrak{M}}^{3k/4},\qquad\text{for
each real number $E\geq E_{k}$}.$

###### Proof.

We prove the lemma by induction on $r-k+1$. The statement (LABEL:e:ind) holds
for $k=r+1$ by (24.16) and (24.17). We therefore fix an integer
$k\in\llbracket\mathfrak{K},r+1\rrbracket$ and assume that the statement
(LABEL:e:ind) holds for $k+1$. We will then prove that (24.19) holds for $k+1$
and that (LABEL:e:ind) holds for $k$. We begin with the former.

Fix a real number $E\geq E_{k+1}$. If $v(E)=0$, then (24.19) holds. Otherwise,
$v(E)>0$, so

(24.20) $\displaystyle\begin{aligned}
\frac{1}{\tau}=\int_{-\infty}^{\infty}\frac{\nu(dx)}{(x-E)^{2}+v(E)^{2}}&\leq\int_{E_{k+2}}^{E_{+}}\frac{\nu(dx)}{v(E)^{2}}+\int_{E_{r+1}}^{E_{k+2}}\frac{\nu(dx)}{(x-E)^{2}}\\\
&\leq\frac{\nu\big{(}[E_{k+2},E_{+}]\big{)}}{v(E)^{2}}+\sum_{i=k+2}^{r}\frac{\nu\big{(}[E_{i+1},E_{i}]\big{)}}{(E_{k+1}-E_{i})^{2}}\\\
&\leq\frac{{\mathfrak{M}}^{3(k+1)/2}}{cv(E)^{2}}+\sum_{i=k+2}^{r}\frac{{\mathfrak{M}}^{3i/2}}{c(c{\mathfrak{M}}^{i-1})^{2}}=\frac{{\mathfrak{M}}^{3(k+1)/2}}{cv(E)^{2}}+\frac{4}{c^{3}{\mathfrak{M}}^{(k-2)/2}}.\end{aligned}$

where the first statement follows from (24.2); the second from (24.15); the
third from the bound $|x-E|\geq E_{k+1}-E_{i}$ whenever $E\geq E_{k+1}$ and
$x\in[E_{i+1},E_{i}]$ with $i\geq k+1$; the fourth from the inductive
hypothesis (the second, third and fourth statements of (LABEL:e:ind), applied
with $k$ there replaced by $i-1\geq k+1$, by $k+2$, and by $i\geq k+2$ here,
respectively); and the fifth from performing the sum (and using the fact that
$\mathfrak{M}>2$). It follows from (24.20) that, for $E\in[E_{k+1},E_{+}]$, we
have

(24.21) $\displaystyle
v(E)\leq\Big{(}\frac{2\tau}{c}\Big{)}^{1/2}{\mathfrak{M}}^{3(k+1)/4},$

since $c^{3}{\mathfrak{M}}^{(k-2)/2}\geq
c^{3}{\mathfrak{M}}^{({\mathfrak{K}}-2)/2}\geq 72B\geq 8\tau$. This verifies
(24.19) with its $k$ replaced by $k+1$.

We next show (LABEL:e:ind), beginning with the first two statements there.
From the defining relation (24.14), we have

$\displaystyle
y_{0}=E_{0}-\tau\operatorname{Re}m\big{(}E_{0}+\mathrm{i}v(E_{0})\big{)};\qquad
y_{k}=E_{k}-\tau\operatorname{Re}m\big{(}E_{k}+\mathrm{i}v(E_{k})\big{)}.$

By taking the difference and using (24.12), we get

(24.22) $\displaystyle{\mathfrak{M}}^{k-1}\leq
y_{0}-y_{k}\leq(E_{0}-E_{k})+\tau\Big{|}m\big{(}E_{0}+\mathrm{i}v(E_{0})\big{)}-m\big{(}E_{k}+\mathrm{i}v(E_{k})\big{)}\Big{|}$

To estimate the right side of this inequality, we bound $m^{\prime}$. To this
end, for any complex number of the form
$z=E+\mathrm{i}\eta\in\overline{\mathbb{H}}$ with $\eta\geq v(E)$, we have
from (4.3) that

(24.23) $\displaystyle\begin{aligned}
\big{|}m^{\prime}(z)\big{|}=\left|\int_{-\infty}^{\infty}\frac{\nu(dx)}{(x-z)^{2}}\right|\leq\int_{-\infty}^{\infty}\frac{\nu(dx)}{(x-E)^{2}+\eta^{2}}\leq\int_{-\infty}^{\infty}\frac{\nu(dx)}{(x-E)^{2}+v(E)^{2}}\leq\frac{1}{\tau},\end{aligned}$

where we used (24.6) for the last inequality.

Thus, to bound
$m\big{(}E_{0}+\mathrm{i}v(E_{0})\big{)}-m\big{(}E_{k}+\mathrm{i}v(E_{k})\big{)}$
we introduce the parameter

$\displaystyle\widetilde{\eta}=\Big{(}\frac{2\tau}{c}\Big{)}^{1/2}{\mathfrak{M}}^{3(k+1)/4}\geq
v(E),$

where the last inequality holds for any $E\geq E_{k+1}$ by (24.21). In
particular, the vertical segments from $E_{0}+\mathrm{i}v(E_{0})$ to
$E_{0}+\mathrm{i}\widetilde{\eta}$ and from $E_{k}+\mathrm{i}v(E_{k})$ to
$E_{k}+\mathrm{i}\widetilde{\eta}$, as well as the horizontal segment from
$E_{0}+\mathrm{i}\widetilde{\eta}$ to $E_{k}+\mathrm{i}\widetilde{\eta}$ are
all in the domain $\big{\\{}z=E+\mathrm{i}\eta:\eta\geq v(E)\big{\\}}$. Thus,

(24.24) $\displaystyle\begin{aligned}
\Big{|}m\big{(}&E_{0}+\mathrm{i}v(E_{0})\big{)}-m\big{(}E_{k}+\mathrm{i}v(E_{k})\big{)}\Big{|}\\\
&\leq\Big{|}m\big{(}E_{0}+\mathrm{i}v(E_{0})\big{)}-m(E_{0}+\mathrm{i}\widetilde{\eta})\Big{|}+\big{|}m(E_{0}+\mathrm{i}\widetilde{\eta})-m(E_{k}+\mathrm{i}\widetilde{\eta})\big{|}\\\
&\qquad+\Big{|}m(E_{k}+\mathrm{i}\widetilde{\eta})-m\big{(}E_{k}+\mathrm{i}v(E_{k})\big{)}\Big{|}\\\