k^{-2/3}\cdot\mathsf{x}_{j}(tk^{1/3})\\\
&\leq\gamma_{j-\mathfrak{m}_{j}}+C_{3}(L^{\prime})^{\zeta(2\beta-7/8)}\big{\\}}\Bigg{]}\geq
1-C_{4}e^{-c_{1}(\log k)^{2}}.\end{aligned}$

###### Proof.

Throughout this proof, we abbreviate
$\textbf{ICE}_{n^{\prime}}=\textbf{ICE}_{n^{\prime}}^{\bm{\mathsf{x}}}(A;B;D;\beta;\delta;R;S;k)$.
Define the $n^{\prime}$-tuples $\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}\rrbracket}(-Ak^{1/3})\in\overline{\mathbb{W}}_{n^{\prime}}$ and
$\bm{v}=\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}\rrbracket}(Ak^{1/3})\in\overline{\mathbb{W}}_{n^{\prime}}$, and
sample $n^{\prime}$ non-intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{n^{\prime}})\in\llbracket
1,n^{\prime}\rrbracket\times\mathcal{C}\big{(}[-Ak^{1/3},Ak^{1/3}]\big{)}$
from $\mathsf{Q}^{\bm{u};\bm{v}}$.

Observe from Definition 16.8 (and Definition 16.1 for the LOC event) that on
$\textbf{ICE}_{n^{\prime}}$ we have for each $j\in\llbracket
1,n^{\prime}\rrbracket$ that

(16.15) $\displaystyle\begin{aligned} &-Bj^{2/3}-Bk^{2/3}\leq u_{j}\leq
Bk^{2/3}-B^{-1}j^{2/3};\qquad-Bj^{2/3}-Bk^{2/3}\leq v_{j}\leq
Bk^{2/3}-B^{-1}j^{2/3}.\end{aligned}$

Since we have conditioned on
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$, and since (16.9)
gives $\mathbb{P}[\textbf{ICE}_{n^{\prime}}]>0$ for sufficiently large $n$, it
follows (from the LOC event in Definition 16.8 for
$\textbf{ICE}_{n^{\prime}}$) that (16.15) holds. In particular,
$\bm{\mathsf{y}}$ satisfies 15.1 with the $(n;A;B)$ there equal to
$(n^{\prime};2A;B+b^{-1})$ here. Hence, by 15.3, there exist constants
$c_{2}=c_{2}(b,A,B)>0$, $C_{3}=C_{3}(b,A,B)>1$, and $C_{4}=C_{4}(b,A,B,D)>1$,
and a measure $\mu_{t}\in\mathscr{P}_{\operatorname{fin}}$ with
$\mu_{t}(\mathbb{R})=(L^{\prime})^{3/2}$, such that the following hold if
$L\geq C_{4}$. First, we have
$\operatorname{supp}\mu_{t}\subseteq\big{[}-C_{3}L^{\prime},C_{3}(L^{\prime})^{3/4}\big{]}$,
and $\mu_{t}$ admits a density $\varrho_{t}\in L^{1}(\mathbb{R})$ with respect
to Lebesgue measure, satisfying $\varrho_{t}(x)\leq C_{3}\max\\{1,-x\\}^{3/4}$
(by (15.6)). Second, we have

(16.16)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{n}\big{\\{}\gamma_{j+\mathfrak{m}_{j}}\leq
k^{-2/3}\cdot\mathsf{y}_{j}(tk^{1/3})\leq\gamma_{j-\mathfrak{m}_{j}}\big{\\}}\Bigg{]}\geq
1-C_{4}e^{-c_{2}(\log k)^{2}},$

where we have used the fact that $\log n^{\prime}\geq\log k$ (as
$n^{\prime}=(L^{\prime})^{3/2}k\geq k$). The first statement confirms the
first part of the corollary. The second (16.16), together with the two
couplings of 16.10, implies the second part. ∎

####  16.4. Proof of 16.4

In this section we establish 16.4. This will be done using 16.9, to which end
we must verify (16.9) there. To this end, we have the following lemma,
indicating that this estimate holds if the boundary tall rectangle event
likely implies the improved Hölder one (see (16.17)). Here, we recall the
events BTR, IHR, and ICE from Definition 16.2, Definition 16.7, and Definition
16.8, respectively.

######  Lemma 16.12.

Adopting 16.3, for any real numbers $\xi_{0}>0$ and $\Xi_{0}\geq 1$, there
exist constants $c=c(A,B,\xi_{0})>0$ and $C=C(A,B,D,\Xi_{0})$ such that the
following holds. Let $A^{\prime}\in[0,A]$; $R,S>1$; and $\beta\in[0,1)$ be
real numbers, with $\delta\in(0,S^{-1})$, and let
$n^{\prime},n^{\prime\prime},n^{\prime\prime\prime}\in\llbracket
L^{3S\delta/2}k,n\rrbracket$ be integers with $n^{\prime\prime}\leq
n^{\prime}$ and $n^{\prime\prime\prime}\leq n^{\prime}$. Assume

(16.17)
$\displaystyle\mathbb{P}\big{[}\textbf{\emph{BTR}}_{n}^{\bm{\mathsf{x}}}(A;B)\cap\textbf{\emph{IHR}}_{n^{\prime}}^{\bm{\mathsf{x}}}(A^{\prime};\beta;R;S;k;\delta;D)^{\complement}\big{]}\leq\Xi_{0}e^{-\xi_{0}(\log
k)^{2}}.$

For any real numbers $\xi>0$ and $\Xi>1$, define the event
$\mathscr{G}_{0}(\xi;\Xi)$, measurable with respect to
$\mathcal{F}_{\operatorname{ext}}=\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime\prime\prime}\rrbracket\times[-A^{\prime}k^{1/3},A^{\prime}k^{1/3}]\big{)}$,
by

(16.18)
$\displaystyle\mathscr{G}_{0}(\xi;\Xi)=\Big{\\{}\mathbb{P}\big{[}\textbf{\emph{ICE}}_{n^{\prime\prime}}^{\bm{\mathsf{x}}}(A^{\prime},12A^{2}B^{3},D;\beta,\delta;R,S;k)\big{|}\mathcal{F}_{\operatorname{ext}}\big{]}\geq
1-\Xi e^{-\xi(\log k)^{2}}\Big{\\}},$

where we conditioned on $\mathcal{F}_{\operatorname{ext}}$ in the probability.
Then,
$\mathbb{P}[\textbf{\emph{BTR}}_{n}^{\bm{\mathsf{x}}}(A;B)\cap\mathscr{G}_{0}(c,C)^{\complement}\big{]}\leq
Ce^{-c(\log k)^{2}}$.

###### Proof.

Set $\widetilde{B}=12A^{3}B^{3}$; abbreviate
$\textbf{ICE}_{m}(A_{0})=\textbf{ICE}_{m}^{\bm{\mathsf{x}}}(A_{0},\widetilde{B},D;\beta,\delta;R,S;k)$
for any integer $m\in\llbracket L^{3S\delta/2}k,n\rrbracket$ and real number
$A_{0}\in[0,A]$; and abbreviate
$\textbf{ICE}_{m}=\textbf{ICE}_{m}(A^{\prime})$. We will also assume that
$\xi\leq 1$ (by replacing $\xi$ with $\min\\{\xi,1\\}$ if necessary). First
observe that there exist constants $c_{1}=c_{1}(A,B)\in(0,1)$ and
$C_{1}=C_{1}(A,B,D)>1$ such that

$\displaystyle\mathbb{P}\Bigg{[}\textbf{BTR}_{n}(A;B)\cap\bigcup_{j=1}^{n}\textbf{LOC}_{j}\big{(}[-Ak^{1/3},$
$\displaystyle
Ak^{1/3}];\widetilde{B}^{-1}j^{2/3}-\widetilde{B}k^{2/3};\widetilde{B}j^{2/3}+\widetilde{B}k^{2/3}\big{)}^{\complement}\Bigg{]}$
$\displaystyle\leq\mathbb{P}\big{[}\textbf{BTR}_{n}(A;B)\cap\textbf{CTR}_{n}(A;\widetilde{B})^{\complement}\big{]}\leq
C_{1}e^{-c_{1}(\log n)^{2}},$

where the first inequality follows from (16.3) and the second from 16.6.
Hence, by (16.17), Definition 16.8 and a union bound, we have
$\mathbb{P}\big{[}\textbf{BTR}_{n}(A;B)\cap\textbf{ICE}_{n^{\prime}}(A)^{\complement}\big{]}\leq
C_{1}e^{-c_{1}(\log n)^{2}}+\Xi_{0}e^{-\xi_{0}(\log n)^{2}}\leq
C_{1}\Xi_{0}e^{-c_{1}\xi_{0}(\log n)^{2}}$. Since
$\textbf{ICE}_{n^{\prime}}(A)\subseteq\textbf{ICE}_{n^{\prime\prime}}(A^{\prime})=\textbf{ICE}_{n^{\prime\prime}}$
(by Definition 16.8, as $n^{\prime\prime}\leq n^{\prime}$ and $A^{\prime}\leq
A$), we deduce

$\displaystyle\mathbb{P}\big{[}\textbf{BTR}_{n}(A;B)\cap\textbf{ICE}_{n^{\prime\prime}}^{\complement}\big{]}\leq(C_{1}+\Xi_{0})e^{-c_{1}\xi_{0}(\log
k)^{2}},$

By 16.3, we have $\mathbb{P}\big{[}\textbf{BTR}_{n}(A;B)\big{]}\geq 1/2$, and
so it follows that

$\displaystyle\mathbb{P}\big{[}\textbf{ICE}_{n^{\prime\prime}}\big{|}\textbf{BTR}_{n}(A;B)\big{]}\geq
1-2(C_{1}+\Xi_{0})e^{-c_{1}\xi_{0}(\log k)^{2}},$

where on the left side we conditioned on the event $\textbf{BTR}_{n}(A;B)$. It
thus follows by (16.18) and a Markov bound that for $c_{2}=c_{1}\xi_{0}/2$ and
$C_{2}=2(C_{1}+\Xi_{0})$ we have
$\mathbb{P}\big{[}\textbf{BTR}_{n}(A;B)\cap\mathscr{G}_{0}(c_{2};C_{2})^{\complement}\big{]}\leq
C_{2}e^{-c_{2}(\log k)^{2}}$, confirming the lemma. ∎

Although 16.12 can be used to verify (16.9) in 16.9, we must then confirm that
(16.17) holds. This will be done through the following proposition, to be
established in Section 18 below. It indicates the boundary tall rectangle
event of Definition 16.2 likely implies the improved Hölder regularity one of
Definition 16.7 (on a smaller time interval); we establish it in Section 18.1
below.

######  Proposition 16.13.

Adopting 16.3 and recalling Definition 16.7, there exist constants
$c=c(A,B)>0$, $C_{1}=C_{1}(A,B)>1$, and $C_{2}=C_{2}(A,B,D,\delta)>1$ such
that the following holds for $L>C_{2}$ and $\delta\in(0,2^{-4000})$. Letting
$n^{\prime}=\lceil L^{1/2^{3200}}k\rceil$, we have

(16.19)
$\displaystyle\mathbb{P}\bigg{[}\textbf{\emph{BTR}}_{n}(A;B)\cap\textbf{\emph{IHR}}_{n^{\prime}}\Big{(}\displaystyle\frac{A}{2};\displaystyle\frac{3}{8};C_{1};2^{14}\Big{)}^{\complement}\bigg{]}\leq
C_{2}e^{-c(\log k)^{2}}.$

Given this result, we can now establish 16.4.

###### Proof of 16.4.

Set $\widehat{n}=\lceil L^{1/2^{3200}}n\rceil\geq n^{\prime}$. By 16.13, there
exist constants $c_{1}=c_{1}(A,B)>0$, $C_{1}=C_{1}(A,B)>2A$, and
$C_{2}=C_{2}(A,B,D,\delta)>1$ such that

$\displaystyle\mathbb{P}\bigg{[}\textbf{BTR}_{n}(A;B)\cap\textbf{IHR}_{\widehat{n}}\Big{(}\displaystyle\frac{A}{2};\displaystyle\frac{3}{8};C_{1};2^{14}\Big{)}^{\complement}\bigg{]}\leq
C_{2}e^{-c_{1}(\log k)^{2}}.$

This verifies (16.17), with the integers
$(n^{\prime},n^{\prime\prime},n^{\prime\prime\prime})$ there equal to
$(\widehat{n},n^{\prime},n^{\prime})$ here and the real numbers
$(\beta;A^{\prime},R,S;\xi_{0};\Xi_{0})$ there equal to
$(3/8;A/2,C_{1},2^{14};c_{1},C_{2})$ here (observing that $n^{\prime}\geq
L^{2^{20}\delta}k$, since $\delta<2^{-5000}$ and $n^{\prime}\geq
L^{1/2^{4000}}k$). Hence, by 16.12, there exist constants $c_{2}=c_{2}(A,B)>0$
and $C_{3}(A,B,D,\delta)>1$, and an event
$\mathscr{A}\subseteq\textbf{BTR}_{n}(A;B)$ (obtained by intersecting the
event $\mathscr{G}_{0}$ in (16.18) with $\textbf{BTR}_{n}(A;B)$) measurable
with respect to
$\mathcal{F}_{\operatorname{ext}}=\mathcal{F}_{\operatorname{ext}}\big{(}\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3}/2,Ak^{1/3}/2]\big{)}$ such that the
following holds. First, we have

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

Second, conditioning on $\mathcal{F}_{\operatorname{ext}}$ and restricting to
the event $\mathscr{A}$, we have

(16.20)
$\displaystyle\mathbb{P}\bigg{[}\textbf{ICE}_{n^{\prime}}^{\bm{\mathsf{x}}}\Big{(}\displaystyle\frac{A}{2},12A^{2}B^{3},D;\displaystyle\frac{3}{8},\delta;C_{1},2^{14};k\Big{)}\bigg{]}\geq
1-C_{3}e^{-c_{1}(\log k)^{2}}.$

Now we apply 16.10, with the $(\beta;R,S)$ there equal to $(3/8;C_{1},2^{14})$
here; recall $\zeta\in(0,2^{-512})$ from that corollary and denote
$L^{\prime}=(k^{-1}n^{\prime})^{2/3}\geq L^{1/2^{4100}}$, so that
$n^{\prime}=(L^{\prime})^{3/2}k$. The assumption (16.9) is verified by
(16.20), and (16.8) is confirmed by the fact that $L^{\prime}\geq
L^{1/2^{4100}}\geq L^{2^{16}\delta/\zeta}$ (as $\delta\leq 2^{-5000}$ and
$\zeta\leq 2^{-512}$). Hence, 16.10 applies (with the $(A,B;\beta;R,S)$ there
given by $(A/2,12A^{2}B^{3};3/8;C_{1},2^{14})$ here); its second part gives
the second part of 16.4. Its first part yields constants $c_{2}=c_{2}(A,B)>0$,
$C_{4}=C_{4}(A,B)>1$, and $C_{5}=C_{5}(A,B,D,\delta)>1$, and a coupling
between $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{\lceil(L^{\prime})^{3\zeta/2}k\rceil}\bigcap_{|s|\leq
Ak^{1/3}/2}\big{\\{}\mathsf{y}_{j}(s)\geq\mathsf{x}_{j}(s)-C_{4}(L^{\prime})^{-\zeta/8}k^{2/3}\big{\\}}\Bigg{]}\geq
1-C_{5}e^{-c_{2}(\log k)^{2}}.$

This, together with the facts that $n^{\prime\prime}=\lceil
L^{1/2^{5000}}k\rceil\leq(L^{\prime})^{3\zeta/2}k$ and
$C_{4}(L^{\prime})^{-\zeta/8}\leq L^{-1/2^{5000}}$ for sufficiently large $L$
(both of which hold since $L^{\prime}\geq L^{1/2^{4100}}$ and $\zeta\geq
2^{-512}$), yields 16.4. ∎

###  17\. Existence of Preliminary Couplings

In this section we establish the preliminary couplings from Section 16.3. We
begin by showing 16.10 using 16.9 in Section 17.1. In Section 17.2 we
establish 16.9 assuming an additional result, which is proven in Section 17.3.
Throughout this section, we let
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)$ denote a
$\mathbb{Z}_{\geq 1}\times\mathbb{R}$ indexed line ensemble satisfying the
Brownian Gibbs property; we also recall the $\sigma$-algebra
$\mathcal{F}_{\operatorname{ext}}$ from Definition 2.2, the location events
from Definition 16.1, and the boundary tall rectangle event BTR from
Definition 16.2.

####  17.1. Proof of 16.10

In this section we establish 16.10. We use the notation of that corollary
throughout, and we also abbreviate
$\textbf{ICE}_{m}=\textbf{ICE}_{m}^{\bm{\mathsf{x}}}(A;B;D;\beta;\delta;R;S;k)$
for any integer $m\in\llbracket L^{3S\delta/2}k,n\rrbracket$.

Define the (random) function $f:[-Ak^{1/3},Ak^{1/3}]$ by setting
$f(s)=\mathsf{y}_{n^{\prime}}(s)$ for each $s\in[-Ak^{1/3},Ak^{1/3}]$, and
define
$\bm{\mathsf{z}}=(\mathsf{z}_{1},\mathsf{z}_{2},\ldots,\mathsf{z}_{n^{\prime}-1})\in\llbracket
1,n^{\prime}-1\rrbracket\times\mathcal{C}\big{(}[-Ak^{1/3},Ak^{1/3}]\big{)}$
by setting $\mathsf{z}_{j}(s)=\mathsf{y}_{j}(s)$ for each $(j,s)\in\llbracket
1,n^{\prime}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$. We will apply 16.9, with
the $\bm{\mathsf{y}}$ there equal to $\bm{\mathsf{z}}$ here, to which end we
must verify the assumptions of that proposition. To do this, define the event

$\displaystyle\mathscr{E}=\bigcap_{|s|\leq
Ak^{1/3}}\Big{\\{}\big{|}\mathsf{x}_{n^{\prime}}(s)-f(s)\big{|}\leq
9R^{2}(L^{\prime})^{7/8}k^{2/3}\Big{\\}}.$

######  Lemma 17.1.

There exist constants $c=c(\xi)>0$ and $C=C(A,R)>1$ such that, if
$L^{\prime}>C$, then $\mathbb{P}[\mathscr{E}]\geq 1-(C+\Xi)e^{-c(\log
k)^{2}}$.

###### Proof.

Define the events

(17.1) $\displaystyle\mathscr{E}_{1}=\bigcap_{|s|\leq
Ak^{1/3}}\bigg{\\{}\Big{|}f(s)-\displaystyle\frac{Ak^{1/3}-s}{2Ak^{1/3}}\cdot
u_{n^{\prime}}-\displaystyle\frac{Ak^{1/3}+s}{2Ak^{1/3}}\cdot
v_{n^{\prime}}\Big{|}\leq(L^{\prime})^{7/8}k^{2/3}\bigg{\\}};\quad\mathscr{E}_{2}=\mathscr{E}_{1}\cap\textbf{ICE}_{n^{\prime}}.$

We first claim that $\mathscr{E}_{2}\subseteq\mathscr{E}$. To show this
observe that, by Definition 16.8 (with the definitions (16.7) and (16.6) of
the IHR and SHR events), the fact that $n^{\prime}=(L^{\prime})^{3/2}k$, and
the fact that $t\big{|}\log(R|t|^{-1})\big{|}^{2}\leq R$ for $|t|\leq R$, we
have on $\textbf{ICE}_{n^{\prime}}$ that

$\displaystyle\displaystyle\sup_{|s|\leq
Ak^{1/3}}\big{|}\mathsf{x}_{n^{\prime}}(s)-u_{n^{\prime}}\big{|}\leq
Rk^{2/3}\big{(}R(L^{\prime})^{3/4}+(L^{\prime})^{\beta}(2A)^{1/2}+k^{-D}\big{)}\leq
2R^{2}k^{2/3}\big{(}(L^{\prime})^{3/4}+(L^{\prime})^{\beta}\big{)},$

where we have used the facts that $R\geq 2A$, that
$u_{j}=\mathsf{x}_{n^{\prime}}(-Ak^{1/3})$, and $|s+Ak^{1/3}|\leq 2Ak^{1/3}$.
Together with the definition (17.1) of $\mathscr{E}_{1}$ and the facts that
$v_{n^{\prime}}=\mathsf{x}_{n^{\prime}}(Ak^{1/3})$ and $\beta\leq 7/8$, it
follows that on $\mathscr{E}_{2}$ we have for any $s\in[-Ak^{1/3},Ak^{1/3}]$
that

$\displaystyle\big{|}\mathsf{x}_{n^{\prime}}(s)-f(s)\big{|}$
$\displaystyle\leq\big{|}\mathsf{x}_{n^{\prime}}(s)-u_{n^{\prime}}\big{|}+\big{|}u_{n^{\prime}}-v_{n^{\prime}}\big{|}+\Big{|}f(s)-\displaystyle\frac{Ak^{1/3}-s}{2Ak^{1/3}}\cdot
u_{n^{\prime}}-\displaystyle\frac{Ak^{1/3}+s}{2Ak^{1/3}}\cdot
v_{n^{\prime}}\Big{|}$ $\displaystyle\leq
4R^{2}k^{2/3}\big{(}(L^{\prime})^{3/4}+(L^{\prime})^{\beta}\big{)}+(L^{\prime})^{7/8}k^{2/3}\leq
9R^{2}(L^{\prime})^{7/8}k^{2/3},$

meaning that $\mathscr{E}$ holds.

Thus, it remains to show that $\mathbb{P}[\mathscr{E}_{2}]\geq
1-(C+\Xi)e^{-c(\log k)^{2}}$. To this end, by (16.9) and a union bound, it
suffices to show for sufficiently large $L^{\prime}$ that

(17.2) $\displaystyle\mathbb{P}[\mathscr{E}_{1}]\geq 1-Ce^{-c(\log k)^{2}}.$

Set $W=(L^{\prime})^{1/20}(n^{\prime})^{1/2}$ and apply 4.8, with the $B$
there equal to $W$ here. Since

$\displaystyle\displaystyle\sup_{s\in[0,2Ak^{1/3}]}s^{1/2}\log(4Ak^{1/3}s^{-1})\leq
2A^{1/2}k^{1/6},$

that lemma yields a constant $C_{1}>0$ such that

(17.3) $\displaystyle\mathbb{P}\Bigg{[}\displaystyle\sup_{|s|\leq
Ak^{1/3}}\Big{|}\mathsf{y}_{n^{\prime}}(s)-\displaystyle\frac{Ak^{1/3}-s}{2Ak^{1/3}}\cdot
u_{n^{\prime}}-\displaystyle\frac{Ak^{1/3}+s}{2Ak^{1/3}}\cdot
v_{n^{\prime}}\Big{|}\geq
2A^{1/2}k^{1/6}(L^{\prime})^{1/20}(n^{\prime})^{1/2}\Bigg{]}\leq
Ce^{-n^{\prime}},$

for sufficiently large $L^{\prime}$ (so that $(L^{\prime})^{1/10}>2c^{-1}C$
for the constants $c$ and $C$ in 4.8). Since $n^{\prime}=(L^{\prime})^{3/2}k$,
we have for sufficiently large $L^{\prime}$ that
$2A^{1/2}k^{1/6}(L^{\prime})^{1/20}(n^{\prime})^{1/2}=2A^{1/2}(L^{\prime})^{4/5}k^{2/3}\leq(L^{\prime})^{7/8}k^{2/3}$,
and so (17.3) (with the definition (17.1) of $\mathscr{E}_{1}$ and the facts
that $f=\mathsf{y}_{n^{\prime}}$ and $e^{-n^{\prime}}\leq e^{-(\log k)^{2}}$,
since $n^{\prime}\geq k$) verifies (17.2) and thus the lemma. ∎

Now we can establish 16.10.

###### Proof of 16.10.

Since the laws of $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ are given by
$\mathsf{Q}_{\mathsf{x}_{n+1}}^{\bm{u};\bm{v}}$ and
$\mathsf{Q}^{\bm{u};\bm{v}}$, respectively, the second statement of the
corollary follows from height monotonicity 4.6. So, it remains to establish
the first. We recall $\bm{\mathsf{z}}$ from the beginning of (17.1).

To this end, observe by (16.9), 17.1, a union bound, and a Markov inequality
that there exist constants $c_{1}=c_{1}(\xi)>0$ and $C_{3}=C_{3}(A,R)>1$, and
an event $\mathscr{E}_{0}$, measurable with respect to the $\sigma$-algebra
generated by
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$ and
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{y}}}\big{(}\llbracket
1,n^{\prime}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$, such that the
following properties hold. First, we have

(17.4) $\displaystyle\mathbb{P}[\mathscr{E}_{0}]\geq 1-C_{3}e^{-c_{1}(\log
k)^{2}}.$

Second, conditioning on the two $\sigma$-algebras generated by
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$ and
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{y}}}\big{(}\llbracket
1,n^{\prime}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$, and restricting to
the event $\mathscr{E}_{0}$, we have

(17.5) $\displaystyle
f(s)=\mathsf{y}_{n^{\prime}}(s)\geq\mathsf{x}_{n^{\prime}}(s)-9R^{2}(L^{\prime})^{7/8}k^{2/3}.$

Third, under the same conditioning and restriction, we have

(17.6) $\displaystyle\mathbb{P}[\textbf{ICE}_{n^{\prime}}]\geq
1-C_{3}e^{-c_{2}(\log k)^{2}}.$

By (17.4) and a Markov estimate, it follows that there exists an event
$\mathscr{E}_{1}$ measurable with respect to the $\sigma$-algebra
$\mathcal{F}_{\operatorname{ext}}$ generated by
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$ and
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{y}}}\big{(}\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$ such that, denoting
$c_{3}=c_{2}/2$, the following two properties hold. First, we have

(17.7) $\displaystyle\mathbb{P}[\mathscr{E}_{1}]\geq 1-C_{3}e^{-c_{3}(\log
k)^{2}}.$

Second, conditioning on $\mathcal{F}_{\operatorname{ext}}$ and restricting to
$\mathscr{E}_{1}$, we have

(17.8) $\displaystyle\mathbb{P}[\mathscr{E}_{0}]\geq 1-C_{3}e^{-c_{3}(\log
k)^{2}}.$

Now, condition on $\mathcal{F}_{\operatorname{ext}}$, and restrict to the
event $\mathscr{E}_{1}$. If $\mathscr{E}_{0}$ holds, then apply 16.9 with the
$(\alpha,\omega;\xi,\Xi;\bm{\mathsf{x}};\bm{\mathsf{y}})$ there equal to
$(7/8,1/8;c_{1},C_{3};\bm{\mathsf{x}},\bm{\mathsf{z}})$ here, which yields
constants $\zeta$, $C_{1}$, and $C_{2}$ satisfying the conditions stated
there. Observe that the hypotheses (16.9), and (16.10) of 16.9 hold by (17.6)
and (17.5), respectively (and we assumed (16.8) to hold).

Hence, 16.9 applies and yields on $\mathscr{E}_{0}\cap\mathscr{E}_{1}$ a
coupling between $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such that (16.12)
holds (where here we use the fact that $\mathsf{z}_{j}=\mathsf{y}_{j}$ for
$j\in\llbracket 1,n-1\rrbracket$). Together with the probability bounds (17.7)
and (17.8) for $\mathscr{E}_{1}$ and $\mathscr{E}_{0}$, this establishes the
first statement of the corollary. ∎

####  17.2. Proof of the Preliminary Coupling

In this section we prove the preliminary coupling from Section 16.3, given by
16.9. To this end, we require the following proposition, which provides a
weaker coupling than the one stated in 16.9. It will be established in Section
17.3 below.

######  Proposition 17.2.

Adopt the notation and assumptions of 16.9, except for (16.8), assuming
instead that $L^{\prime}\geq L^{3S\delta}$. Fix a real number
$\mathfrak{d}>0$, and set ${\mathfrak{C}}=36BP(R^{2}+B)$. There exists a
constant $C_{3}=C_{3}(A,B,P,\omega,\mathfrak{d})>1$ and a coupling between
$\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$, such that the following holds if
$L^{\prime}>C_{3}$.

  1. (1)

If $\alpha>2\beta-1/2-{\mathfrak{d}}$ then, letting
$\widetilde{L}=(L^{\prime})^{3/4}$ and
$\widetilde{n}=\lceil\widetilde{L}^{3/2}k\rceil$, we have

(17.9)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{\tilde{n}}\bigcap_{|s|\leq
Ak^{1/3}}\big{\\{}\mathsf{y}_{j}(s)\geq\mathsf{x}_{j}(s)-{\mathfrak{C}}\widetilde{L}^{(4\alpha-1+4{\mathfrak{d}})/3}k^{2/3}\big{\\}}\Bigg{]}\geq
1-2\Xi e^{-\xi(\log k)^{2}}.$

  2. (2)

If $\alpha\leq 2\beta-1/2-{\mathfrak{d}}$ then, letting
$\widetilde{L}=(L^{\prime})^{\beta+(1-\alpha)/2}$ and
$\widetilde{n}=\lceil\widetilde{L}^{3/2}k\rceil$, we have

(17.10)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{\tilde{n}}\bigcap_{|s|\leq
Ak^{1/3}}\big{\\{}\mathsf{y}_{j}(s)\geq\mathsf{x}_{j}(s)-{\mathfrak{C}}\widetilde{L}^{(2\beta+\alpha-1)/(2\beta-\alpha+1)}k^{2/3}\big{\\}}\Bigg{]}\geq
1-2\Xi e^{-\xi(\log k)^{2}}.$

We will establish 16.9 by repeated application of 17.2. To this end, we first
require the following lemma indicating the behavior of a certain family of
recursions related to the exponents appearing in 17.2.

######  Lemma 17.3.

Fix real numbers $\omega\in(0,1/4)$; $\beta\in[3/8,7/8]$;
$\alpha\in[2\beta-1,1-\omega]$; and $\mathfrak{d}\in(0,\omega/8)$. Define the
sequence of real numbers $\bm{\alpha}=(\alpha_{0},\alpha_{1},\ldots)$ and
$\bm{d}=(d_{0},d_{1},\ldots)$ as follows. First set $\alpha_{0}=\alpha$ and
$d_{0}=1$. For any integer $\ell\geq 0$, if
$\alpha_{\ell}>2\beta-1/2+{\mathfrak{d}}$, then define

(17.11) $\displaystyle
d_{\ell+1}=\frac{3d_{\ell}}{4};\qquad\alpha_{\ell+1}=\frac{4}{3}\Big{(}\alpha_{\ell}-\frac{1}{4}+{\mathfrak{d}}\Big{)}.$

If instead $\alpha_{\ell}\leq 2\beta-1/2+{\mathfrak{d}}$, then define

(17.12) $\displaystyle
d_{\ell+1}=d_{\ell}\left(\beta-\frac{\alpha_{\ell}-1}{2}\right);\qquad\alpha_{\ell+1}=\frac{2\beta+\alpha_{\ell}-1}{2\beta-\alpha_{\ell}+1}.$

Then, for any integer $j>32\omega^{-1}$, we have $\alpha_{j}\leq 2\beta-7/8$.
Moreover, for any integer $j\geq 0$, we have $d_{j+1}\leq d_{j}$ and
$2^{-j}\leq d_{j}\leq 1$.

###### Proof.

First observe for any integer $\ell\geq 0$ that $d_{\ell}\geq 0$ and
$d_{\ell+1}\geq d_{\ell}/2$. Indeed, if
$\alpha_{\ell}>2\beta-1/2+\mathfrak{d}$, then this follows from (17.11). If
instead $\alpha_{\ell}\leq 2\beta-1/2+\mathfrak{d}$, then

$\displaystyle\displaystyle\frac{d_{\ell+1}}{d_{\ell}}=\beta-\displaystyle\frac{\alpha_{\ell}-1}{2}\geq\displaystyle\frac{3}{4}-\displaystyle\frac{\mathfrak{d}}{2}\geq\displaystyle\frac{1}{2},$

where the first statement follows from (17.12); the second from the fact that
$\alpha_{\ell}\leq 2\beta-1/2+\mathfrak{d}$; and the third from the fact that
$\mathfrak{d}<\omega/8<1/32$. Hence, $d_{\ell+1}\geq d_{\ell}/2$, which
verifies by induction on $j$ that $d_{j}\geq 2^{-j}d_{0}\geq 2^{-j}$ for any
integer $j\geq 0$; this confirms the first bound on $d_{j}$.

Next let us show the lower bound on $\alpha_{j}$. To this end, first observe
by induction on $\ell$ that

(17.13) $\displaystyle\alpha_{\ell}\geq 2\beta-1,\qquad\text{for each integer
$\ell\geq 0$}.$

Indeed, this holds at $\ell=0$, as $\alpha_{0}=\alpha>2\beta-1$. Now fix an
integer $\ell\geq 0$ and assume that $\alpha_{\ell}\geq 2\beta-1$. If
$\alpha_{\ell}>2\beta-1/2+\mathfrak{d}$, then

$\displaystyle\alpha_{\ell+1}=\displaystyle\frac{4\alpha_{\ell}}{3}-\displaystyle\frac{1}{3}+\displaystyle\frac{4\mathfrak{d}}{3}\geq\displaystyle\frac{8}{3}(\beta+\mathfrak{d})-1\geq
2\beta-1,$

where in the first statement we used (17.11), in the second we used the fact
that $\alpha_{\ell}>2\beta-1/2+\mathfrak{d}$, and in the third we used the
fact that $\beta>0$. If instead $\alpha_{\ell}\leq 2\beta-1/2+\mathfrak{d}$,
then

$\displaystyle\alpha_{\ell+1}-2\beta+1=\displaystyle\frac{2\beta}{2\beta-\alpha_{\ell}+1}\cdot(\alpha_{\ell}-2\beta+1)\geq
0,$

where in the first statement we used (17.12) and in the second we used the
inductive hypothesis (with the fact that $2\beta-\alpha_{\ell}+1\geq 0$, since
$\alpha\leq 2\beta-1/2+\mathfrak{d}$ and $\mathfrak{d}<\omega/8<1/32$). This
verifies (17.13). Together with by (17.11) and (17.12), this implies that
$d_{\ell+1}\leq d_{\ell}$ for each integer $\ell\geq 0$, so in particular
$d_{j}\leq d_{0}=1$ for each integer $j\geq 0$. This verifies the second bound
on the $d_{j}$.

It remains to show the upper bound on the $\alpha_{j}$, to which end we next
claim that

(17.14) $\displaystyle\alpha_{\ell+1}\leq\alpha_{\ell},\quad\text{for each
$\ell\geq-1$};\qquad\qquad\alpha_{\ell+1}\leq\alpha_{\ell}-\frac{\omega}{16},\quad\text{for
each $\ell\geq-1$ with $\alpha_{\ell}\geq 2\beta-\displaystyle\frac{7}{8}$},$

with both statements being by definition empty if $\ell=-1$. To this end, we
induct on $\ell\geq-1$. Fix some integer $\ell\geq 0$, and assume that (17.14)
holds for each integer $\ell^{\prime}\leq\ell-1$; we will show it holds for
$\ell$. To this end, observe that if $\alpha_{\ell}>2\beta-1/2+\mathfrak{d}$
then

$\displaystyle\alpha_{\ell+1}-\alpha_{\ell}=\displaystyle\frac{1}{3}(\alpha_{\ell}+4\mathfrak{d}-1)\leq-\displaystyle\frac{\omega}{6},$

where in the first statement we used (17.11) and in the second we used the
facts that $4\mathfrak{d}<\omega/2$ and that
$\alpha_{\ell}\leq\alpha_{0}=\alpha\leq 1-\omega$ (where the first bound holds
by the inductive hypothesis). This verifies both statements of (17.14). If
instead $\alpha_{\ell}\leq 2\beta-1/2+\mathfrak{d}$, then

$\displaystyle\alpha_{\ell+1}-\alpha_{\ell}=\displaystyle\frac{(\alpha_{\ell}-1)(\alpha_{\ell}-2\beta+1)}{2\beta-\alpha_{\ell}+1}\leq\displaystyle\frac{\omega(2\beta-\alpha_{\ell}-1)}{2},$

where in the first statement we used (17.12) and in the second we used the
facts that $\alpha_{\ell}\leq\alpha_{0}=\alpha\leq 1-\omega$ (where the first
bound holds by the inductive hypothesis), that $\alpha_{\ell}\geq 2\beta-1$
(by (17.13)), and that $2\beta-\alpha_{\ell}+1\leq 2$ (again by (17.13)). This
again implies both bounds in (17.14) (the first since $\alpha_{\ell}\geq
2\beta-1$ by (17.13)), verifying the two inequalities there.

Then, for any integer $j\geq 32\omega^{-1}$ we have

$\displaystyle\alpha_{j}\leq\max\bigg{\\{}2\beta-\frac{7}{8},\alpha_{j-\lceil
32/\omega\rceil}-\displaystyle\frac{\omega}{16}\cdot\Big{\lceil}\displaystyle\frac{32}{\omega}\Big{\rceil}\bigg{\\}}\leq\max\bigg{\\{}2\beta-\displaystyle\frac{7}{8},\alpha-2\bigg{\\}}\leq
2\beta-\displaystyle\frac{7}{8},$

where the first bound holds by (17.14), the second holds since
$\alpha_{j-\lceil 32/\omega\rceil}\leq\alpha_{0}=\alpha$ (by the first
statement in (17.14)), and the third holds since $\alpha\leq 1$ and $\beta>0$.
This establishes the lemma. ∎

Now we can establish 16.9.

Figure 5.4.  Shown above is a depiction of the inductive argument used in the
proof of 16.9.

###### Proof of 16.9.

Throughout this proof, for each integer $m\in\llbracket
L^{3S\delta/2}k,n\rrbracket$, we abbreviate
$\textbf{ICE}_{m}=\textbf{ICE}_{m;n}^{\bm{\mathsf{x}}}(A,B,D;\beta;\delta;R;S;k)$,
and recall $\xi^{\prime}=\xi/2$ from the statement of 16.9. Set
$\mathfrak{d}=\omega/16$ and $M=\lceil 32\omega^{-1}\rceil\leq 64\omega^{-1}$.
Define the sequences $\bm{\alpha}=(\alpha_{0},\alpha_{1},\ldots)$ and
$\bm{d}=(d_{0},d_{1},\ldots)$ as in 17.3. Also inductively define the sequence
$\bm{P}=(P_{0},P_{1},\ldots)\subset\mathbb{R}_{>0}$ by setting $P_{0}=P$ and
$P_{\ell+1}=36BP_{\ell}(R^{2}+B)$ for each integer $\ell\geq 0$. For each
integer $i\geq 0$, set

(17.15) $\displaystyle L_{i}=(L^{\prime})^{d_{i}};\qquad n_{i}=\lceil
L_{i}^{3/2}k\rceil;\qquad\Xi_{i}=3^{i}\Xi.$

Further fix $\zeta=d_{M}$; since $M\leq 64\omega^{-1}$, we have by 17.3 that
$2^{-64/\omega}\leq\zeta\leq 1$.

In what follows, we omit the ceilings in the definition (17.15) of $n_{i}$,
assuming that $n_{i}=L_{i}^{3/2}k$, as this will barely affect the proofs
below. We claim for each integer $i\in\llbracket 0,M\rrbracket$ that it is
possible to couple $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such that

(17.16) $\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{n_{i}}\bigcap_{|s|\leq
Ak^{1/3}}\big{\\{}\mathsf{y}_{j}(s)\geq\mathsf{x}_{j}(s)-P_{i}L_{i}^{\alpha_{i}}k^{2/3}\big{\\}}\Bigg{]}\geq
1-\Xi_{i}e^{-\xi^{\prime}(\log k)^{2}}.$

To this end, we induct on $i\in\llbracket 0,M\rrbracket$, beginning with the
case $i=0$. Since the laws of $\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}\rrbracket}$ and $\bm{\mathsf{y}}$ are given by
$\mathsf{Q}_{\mathsf{x}_{n^{\prime}+1}}^{\bm{u};\bm{v}}$ and
$\mathsf{Q}_{f}^{\bm{u};\bm{v}}$, respectively, (16.10) and 4.6 together yield
a coupling between $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such that
$\mathsf{y}_{j}(s)\geq\mathsf{x}_{j}(s)-P(L^{\prime})^{\alpha}k^{2/3}$ for
each $(j,s)\in\llbracket 1,n^{\prime}\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$.
Since $(n_{0},L_{0},\alpha_{0},P_{0})=(n^{\prime},L^{\prime},\alpha,P)$, this
verifies (17.16) at $i=0$.

Thus, fix some integer $i_{0}\in\llbracket 0,M-1\rrbracket$, and assume that
(17.16) holds whenever $i\leq i_{0}$; we will show it holds for $i=i_{0}+1$.
By (16.9), the fact that
$\textbf{ICE}_{n^{\prime}}\subseteq\textbf{ICE}_{n_{i_{0}}}$ (by Definition
16.8, (16.7), and the fact that $n=n_{0}\geq n_{i_{0}}$) and a Markov
inequality, there exists an event $\mathscr{E}_{i_{0}}$, measurable with
respect to
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n_{i_{0}}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$, satisfying the
following two properties. First, recalling that $\xi^{\prime}=\xi/2$, we have
$\mathbb{P}[\mathscr{E}_{i_{0}}]\geq 1-\Xi^{1/2}e^{-\xi^{\prime}(\log
k)^{2}}$. Second, conditioning on
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n_{i_{0}}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$ and restricting to
$\mathscr{E}_{i_{0}}$, we have

(17.17) $\displaystyle\mathbb{P}\big{[}\textbf{ICE}_{n_{i_{0}}}\big{]}\geq
1-\Xi^{1/2}e^{-\xi^{\prime}(\log k)^{2}}.$

This, together with the $i=i_{0}$ case of (17.16) and a union bound, yields an
event $\mathscr{E}_{i_{0}}^{\prime}$, measurable with respect to the
$\sigma$-algebras generated by
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n_{i_{0}}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$ and
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{y}}}\big{(}\llbracket
1,n_{i_{0}}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$, such that the
following two statements hold. First,

$\displaystyle\mathbb{P}[\mathscr{E}_{i_{0}}^{\prime}]\geq
1-(\Xi^{1/2}+\Xi_{i_{0}})e^{-\xi^{\prime}(\log k)^{2}}.$

Second, conditional on the two $\sigma$-algebras generated by
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n_{i_{0}}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$ and
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{y}}}\big{(}\llbracket
1,n_{i_{0}}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$, and restricting to
$\mathscr{E}_{i_{0}}^{\prime}$, we have (17.17) and

$\displaystyle\mathsf{y}_{n_{i_{0}}}(s)\geq\mathsf{x}_{n_{i_{0}}}(s)-P_{i_{0}}L_{i_{0}}^{\alpha_{i_{0}}}k^{2/3},\qquad\text{for
each $s\in[-Ak^{1/3},Ak^{1/3}]$}.$

Under the same conditioning and restriction, this verifies on the event
$\mathscr{E}_{i_{0}}^{\prime}$ the bounds (16.9) and (16.10), with the
$(n^{\prime};L^{\prime};f;\alpha;P)$ there equal to
$(n_{i_{0}};L_{i_{0}};\mathsf{y}_{n_{i_{0}}};\alpha_{i_{0}};P_{i_{0}})$ here.
Thus, since $L_{i_{0}}\geq(L^{\prime})^{d_{i_{0}}}\geq(L^{\prime})^{d_{M}}\geq
L^{3S\delta}$ is sufficiently large (using 17.3 and the facts that
$L^{\prime}\geq L^{4S\delta/\zeta}$ by (16.8), that $\zeta=d_{M}\geq
2^{-64/\omega}$, and that $L\geq C_{2}$ is sufficiently large), 17.2 applies,
with the $(\widetilde{n},\widetilde{L},\mathfrak{C})$ there equal to
$(n_{i_{0}+1},L_{i_{0}+1},P_{i_{0}+1})$ here, by (17.11) and (17.12) (with the
recursive definition of $P_{j}$). This proposition yields a coupling between
$\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{n_{i_{0}+1}}$
$\displaystyle\bigcap_{|s|\leq
Ak^{1/3}}\big{\\{}\mathsf{y}_{j}(s)\geq\mathsf{x}_{j}(s)-P_{i_{0}+1}L_{i_{0}+1}^{\alpha_{i_{0}+1}}k^{2/3}\big{\\}}\Bigg{]}$
$\displaystyle\geq 1-2(\Xi^{1/2}+\Xi_{i_{0}})e^{-\xi^{\prime}(\log k)^{2}}\geq
1-3\Xi_{i_{0}}e^{-\xi^{\prime}(\log
k)^{2}}=1-\Xi_{i_{0}+1}e^{-\xi^{\prime}(\log k)^{2}},$

where we again used (17.11) and (17.12) to equate the upper bounds in
$\mathsf{x}_{j}-\mathsf{y}_{j}$ from (17.9) and (17.10) with
$P_{i_{0}+1}L_{i_{0}+1}^{d_{i_{0}+1}}k^{2/3}$ (additionally using the
definition (17.15) of $\Xi_{j}$). See Figure 5.4 for a depiction. This
verifies (17.16).

Taking $j=M$ in (17.16) and using the facts that

$\displaystyle\Xi_{M}=3^{M}\Xi\leq
3^{64/\omega}\Xi;\qquad\qquad\qquad\qquad\zeta=d_{M}\geq 2^{-M};$
$\displaystyle
n_{M}=\big{\lceil}(L^{\prime})^{3d_{M}/2}k\big{\rceil}=\big{\lceil}(L^{\prime})^{3\zeta/2}k\big{\rceil};\qquad
L_{M}^{\alpha_{M}}=(L^{\prime})^{d_{M}\alpha_{M}}\leq(L^{\prime})^{\zeta(2\beta-7/8)},$

which hold by 17.3 (and the facts that $M=\lceil 32\omega^{-1}\rceil\leq
64\omega^{-1}$ and $\zeta=d_{M}$), this verifies (16.11) and thus the
proposition. ∎

####  17.3. Proof of 17.2

In this section we establish 17.2; we adopt the notation and assumptions of
that proposition throughout. First observe that, since $L^{\prime}\geq C_{3}$,
we may assume that $L^{\prime}$ is sufficiently large; in particular,
$L^{\prime}\geq(6BP)^{1/\omega}\geq(6BP)^{1/(1-\alpha)}$. Throughout this
section, we fix the real number

(17.18)
$\displaystyle\vartheta=2BP(L^{\prime})^{\alpha-1}\leq\displaystyle\frac{1}{3}.$

To prove 17.2, we will first couple two ensembles on the interval
$\big{[}-(1-\vartheta)^{2}Ak^{1/3},(1-\vartheta)^{2}Ak^{1/3}\big{]}$; the
first is the restriction of $\bm{\mathsf{x}}$ to this interval, and the second
is a rescaled variant of $\bm{\mathsf{y}}$.

To make the latter more explicit, define the line ensemble
$\bm{\mathsf{z}}=(\mathsf{z}_{1},\mathsf{z}_{2},\ldots,\mathsf{z}_{n^{\prime}-1})\in\llbracket
1,n^{\prime}-1\rrbracket\times\mathcal{C}\big{(}[-(1-\vartheta)^{2}Ak^{1/3},(1-\vartheta)^{2}Ak^{1/3}]\big{)}$
(see Figure 5.5) by for each $(j,s)\in\llbracket
1,n^{\prime}-1\rrbracket\times\big{[}-(1-\vartheta)^{2}Ak^{1/3},(1-\vartheta)^{2}Ak^{1/3}\big{]}$
setting

(17.19)
$\displaystyle\mathsf{z}_{j}(s)=(1-\vartheta)\cdot\mathsf{y}_{j}\big{(}(1-\vartheta)^{-2}s\big{)}+2k^{2/3}(R^{2}+B)\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}.$

See Figure 5.5 for a depiction.

Figure 5.5. Shown above is a depiction of Section 17.3, where
$\bm{\mathsf{z}}$ is a rescaled variant of $\bm{\mathsf{y}}$, and with high
probability we can couple $\bm{\mathsf{z}}\geq\bm{\mathsf{x}}$.

######  Lemma 17.4.

There exists a constant $C=C(A,B,P,R)>1$ such that, if $L^{\prime}>C$, then
there exists a coupling between $\bm{\mathsf{x}}$ and $\bm{\mathsf{z}}$ such
that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{n^{\prime}-1}\bigcap_{|s|\leq(1-\vartheta)^{2}Ak^{1/3}}\big{\\{}\mathsf{z}_{j}(s)\geq\mathsf{x}_{j}(s)\big{\\}}\Bigg{]}\geq
1-\Xi e^{-\xi(\log k)^{2}}.$

To establish 17.4, we require the following lemma. In the below, we abbreviate
the event
$\textbf{ICE}_{n^{\prime}}=\textbf{ICE}_{n^{\prime}}^{\bm{\mathsf{x}}}(A;B;D;\beta;\delta;R;S;k)$.

######  Lemma 17.5.

There exists a constant $C=C(A,B,P,R)>1$ such that the following hold on the
event $\textbf{\emph{ICE}}_{n^{\prime}}$ if $L^{\prime}>C$.

  1. (1)

For each $s\in[-Ak^{1/3},Ak^{1/3}]$, we have $(1-\vartheta)\cdot
f(s)\geq\mathsf{x}_{n^{\prime}}(s)$.

  2. (2)

For each $(j,s)\in\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$, we have

(17.20)
$\displaystyle\Big{|}{\mathsf{x}}_{j}\big{(}(1-\vartheta)^{2}s\big{)}-{\mathsf{x}}_{j}(s)\Big{|}\leq
2R^{2}k^{2/3}\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)},$

###### Proof.

First observe by Definition 16.8 (and Definition 16.1 for the LOC event) that,
on $\textbf{ICE}_{n^{\prime}}$, we have

(17.21) $\displaystyle\mathsf{x}_{n^{\prime}}(s)\leq
Bk^{2/3}-B^{-1}(n^{\prime})^{2/3}=(B-B^{-1}L^{\prime})k^{2/3},\qquad\text{for
each $s\in[-Ak^{1/3},Ak^{1/3}]$}.$

Hence, on $\textbf{ICE}_{n^{\prime}}$ we have that

$\displaystyle(1-\vartheta)\cdot f(s)$
$\displaystyle\geq(1-\vartheta)\cdot\big{(}{\mathsf{x}}_{n^{\prime}}(s)-P(L^{\prime})^{\alpha}k^{2/3}\big{)}$
$\displaystyle\geq{\mathsf{x}}_{n^{\prime}}(s)-2BP(L^{\prime})^{\alpha-1}\cdot{\mathsf{x}}_{n^{\prime}}(s)-P(L^{\prime})^{\alpha}k^{2/3}$
$\displaystyle\geq{\mathsf{x}}_{n^{\prime}}(s)+2BP(L^{\prime})^{\alpha-1}k^{2/3}\cdot(B^{-1}L^{\prime}-B)-P(L^{\prime})^{\alpha}k^{2/3}\geq{\mathsf{x}}_{n^{\prime}}(s),$

where the first bound follows from (16.10), the second from (17.18), the third
from (17.21), and the fourth from the fact that $L^{\prime}\geq 2B^{2}$ is
sufficiently large. This verifies the first statement of the lemma.

By Definition 16.8 (and (16.7) and (16.6) for the IHR and SHR events), we have
on $\textbf{ICE}_{n^{\prime}}$ that, for each $(j,s)\in\llbracket
L^{3S\delta/2}k,n^{\prime}\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$,

$\displaystyle\Big{|}{\mathsf{x}}_{j}\big{(}(1-\vartheta)^{2}s\big{)}-{\mathsf{x}}_{j}(s)\Big{|}$
$\displaystyle\leq
Rk^{2/3}\cdot\bigg{(}\Big{(}\displaystyle\frac{j}{k}\Big{)}^{1/2}t_{0}\big{|}\log(Rt_{0}^{-1})\big{|}^{2}+\Big{(}\displaystyle\frac{j}{k}\Big{)}^{2\beta/3}t_{0}^{1/2}+k^{-D}\bigg{)}$
$\displaystyle\leq
2R^{2}k^{2/3}\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)},$

where we have denoted $t_{0}=k^{-1/3}\big{|}s-(1-\vartheta)^{2}s\big{|}$.
Here, we used the facts that $jk^{-1}\leq(L^{\prime})^{3/2}$, that
$t_{0}k^{1/3}=\big{|}s-(1-\vartheta)^{2}s\big{|}\leq 2\vartheta|s|\leq
2\vartheta Ak^{1/3}\leq\vartheta Rk^{1/3}$, that
$t_{0}\big{|}\log(Rt_{0}^{-1})\big{|}^{2}\leq\vartheta R|\log\vartheta|^{2}$
for $t_{0}\in[0,\vartheta R]$ and $L^{\prime}$ sufficiently large (so that
$\vartheta$ is sufficiently small), and that
$(L^{\prime})^{\beta}\vartheta^{1/2}\geq\vartheta^{1/2}\geq(L^{\prime})^{-1/2}\geq
k^{-D}$ (as $L^{\prime}\leq L\leq k^{D}$). This verifies (17.20) for
$j\in\llbracket
L^{3S\delta/2}k,n^{\prime}\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$. Similarly, by
Definition 16.8 (and (16.7) and (16.6) for the IHR and FHR events), we have on
$\textbf{ICE}_{n^{\prime}}$ that, for each $(j,s)\in\llbracket
1,L^{3S\delta/2}k\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$,

$\displaystyle\Big{|}{\mathsf{x}}_{j}\big{(}(1-\vartheta)^{2}s\big{)}-{\mathsf{x}}_{j}(s)\Big{|}\leq
2Ak^{2/3}\cdot\big{(}L^{3S\delta/2}\vartheta+5L^{S\delta}\vartheta^{1/2}\big{)}\leq
Rk^{2/3}\cdot\big{(}(L^{\prime})^{3/4}\vartheta+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}.$

Here, in the first estimate we used the facts that we have
$\big{|}(1-\vartheta)^{2}s-s\big{|}\leq 2A\vartheta k^{1/3}$; that
$L^{\delta}\cdot\max\\{j^{1/3}k^{-1/3},1\\}\leq L^{(S/2+1)\delta}\leq
L^{3S\delta/2}$ (since $j\leq L^{3S\delta/2}k$ and $S\geq 1$); that
$\max\\{j^{1/2}k^{-1/2},1\\}\leq(L^{\prime})^{S\delta}$; and that
$L^{S\delta}\vartheta^{1/2}\geq\vartheta^{1/2}\geq(L^{\prime})^{-1/2}\geq
k^{-D}$. In the second, we used the facts that $R\geq 2A$, that $\beta\geq
3/8$, and that $L^{\prime}\geq L^{3S\delta}\geq L^{S\delta/\beta}$ (with the
fact that $L^{\prime}$ is sufficiently large). It follows that (17.20) holds
on $\textbf{ICE}_{n^{\prime}}$ for each $(j,s)\in\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$, verifying the second part
of the lemma. ∎

###### Proof of 17.4.

Set $\mathsf{T}=(1-\vartheta)^{2}Ak^{1/3}$, and condition on
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}-1\rrbracket\times[-\mathsf{T},\mathsf{T}]\big{)}$. Define the
$(n^{\prime}-1)$-tuples
$\bm{u}^{\prime},\bm{v}^{\prime},\widetilde{\bm{u}},\widetilde{\bm{v}}\in\overline{\mathbb{W}}_{n^{\prime}-1}$
and the function $\widetilde{f}:[-\mathsf{T},\mathsf{T}]\rightarrow\mathbb{R}$
by for each $s\in[-\mathsf{T},\mathsf{T}]$ setting

(17.22) $\displaystyle\begin{aligned}
&\bm{u}^{\prime}=\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}-1\rrbracket}(-\mathsf{T});\qquad\bm{v}^{\prime}=\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}-1\rrbracket}(\mathsf{T});\qquad\widetilde{\bm{u}}=\bm{\mathsf{z}}(-\mathsf{T});\qquad\widetilde{\bm{v}}=\bm{\mathsf{z}}(\mathsf{T});\\\
&\widetilde{f}(s)=(1-\vartheta)\cdot
f\big{(}(1-\vartheta)^{-2}s\big{)}+2k^{2/3}(R^{2}+B)\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}.\end{aligned}$

Then, by (17.19) and 4.4, the law of $\bm{\mathsf{z}}$ is given by
$\mathsf{Q}_{\tilde{f}}^{\tilde{\bm{u}};\tilde{\bm{v}}}$. Hence, by (16.9) and
height monotonicity 4.6 (with the fact that the law of
$\bm{\mathsf{x}}_{\llbracket 1,n^{\prime}-1\rrbracket}$ is given by
$\mathsf{Q}_{\mathsf{x}_{n^{\prime}}}^{\bm{u}^{\prime};\bm{v}^{\prime}}$), it
suffices to show for sufficiently large $L^{\prime}$ that

(17.23)
$\displaystyle\bm{u}^{\prime}\leq\widetilde{\bm{u}},\quad\bm{v}^{\prime}\leq\widetilde{\bm{v}},\quad\text{and}\quad\mathsf{x}_{n}\leq\widetilde{f},\qquad\text{all
hold on the event $\textbf{ICE}_{n^{\prime}}$}.$

The proofs of the first and second statements in (17.23) are entirely
analogous, so we only provide that of the former. To this end, observe on
$\textbf{ICE}_{n^{\prime}}$ that, for each $j\in\llbracket
1,n^{\prime}\rrbracket$,

$\displaystyle\widetilde{u}_{j}$
$\displaystyle=(1-\vartheta)\cdot{\mathsf{y}}_{j}(-Ak^{1/3})+2k^{2/3}(R^{2}+B)\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}$
$\displaystyle=(1-\vartheta)\cdot\mathsf{x}_{j}(-Ak^{1/3})+2k^{2/3}(R^{2}+B)\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}$
$\displaystyle\geq{\mathsf{x}}_{j}(-Ak^{1/3})+2R^{2}k^{2/3}\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}\geq{\mathsf{x}}_{j}(-\mathsf{T}),$

where the first statement holds by (17.22) and (17.19); the second by the fact
${\mathsf{y}}_{j}(-Ak^{1/3})=u_{j}={\mathsf{x}}_{j}(-Ak^{1/3})$; the third by
the bound $\vartheta\cdot{\mathsf{x}}_{j}(-Ak^{1/3})\leq\vartheta Bk^{2/3}\leq
B(L^{\prime})^{\beta}\vartheta^{1/2}k^{2/3}$ on $\textbf{ICE}_{n^{\prime}}$
(where the former is due to the LOC events in Definition 16.8, and the latter
is due to the facts that $\vartheta\leq 1$ and $B,L^{\prime}\geq 1$); and the
fourth by (the $s=-Ak^{1/3}$ case of) (17.20). This verifies the first bound
in (17.23); the proof of the second is entirely analogous and is thus omitted.

To confirm the third, observe on $\textbf{ICE}_{n^{\prime}}$ that, for
$s\in[-Ak^{1/3},Ak^{1/3}]$,

$\displaystyle\widetilde{f}\big{(}(1-\vartheta)^{2}s\big{)}$
$\displaystyle=(1-\vartheta)\cdot
f(s)+2k^{2/3}(R^{2}+B)\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}$
$\displaystyle\geq{\mathsf{x}}_{n^{\prime}}(s)+2k^{2/3}(R^{2}+B)\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}\geq{\mathsf{x}}_{n^{\prime}}\big{(}(1-\vartheta)^{2}s\big{)},$

where the first statement holds by (17.22); the second by the first statement
of 17.5; and the third by (17.20). This establishes (17.23) and thus the
lemma. ∎

Now we can establish 17.2.

###### Proof of 17.2.

First observe that we may couple $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such
that the following holds with probability at least $1-\Xi e^{-\xi(\log
k)^{2}}$. For each $(j,s)\in\llbracket
1,n^{\prime}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$, we have

$\displaystyle(1-\vartheta)\cdot{\mathsf{y}}_{j}($ $\displaystyle
s)+2k^{2/3}(R^{2}+B)\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}$
$\displaystyle={\mathsf{z}}_{j}\big{(}(1-\vartheta)^{2}s\big{)}\geq{\mathsf{x}}_{j}\big{(}(1-\vartheta)^{2}s\big{)}\geq{\mathsf{x}}_{j}(s)-2R^{2}k^{2/3}\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)},$

where the first statement holds by (17.19); the second by 17.4; and the third
by (17.20). Hence, with probability at least $1-2\Xi e^{-\xi(\log k)^{2}}$,
for each $(j,s)\in\llbracket
1,n^{\prime}-1\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$ we have

(17.24)
$\displaystyle\begin{split}{\mathsf{y}}_{j}(s)&\geq(1-\vartheta)^{-1}\cdot{\mathsf{x}}_{j}(s)-2(1-\vartheta)^{-1}R^{2}k^{2/3}\cdot\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}\\\
&\geq{\mathsf{x}}_{j}(s)-6(R^{2}+B)\Big{(}k^{2/3}\big{(}(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}+(L^{\prime})^{\beta}\vartheta^{1/2}\big{)}+\vartheta(k^{2/3}+j^{2/3})\Big{)},\end{split}$

where for the second inequality, we used the facts that $\vartheta\leq 1/3$
(by (17.18)) and that
$(1-\vartheta)^{-1}\cdot{\mathsf{x}}_{j}(s)\geq\mathsf{x}_{j}(s)-2\vartheta
B(k^{2/3}+j^{2/3})$ holds with probability at least $1-\Xi e^{-\xi(\log
k)^{2}}$ (and a union bound). Indeed, the latter follows from the bound
$(1-\vartheta)^{-1}\leq 1+2\vartheta$ for $\vartheta\leq 1/3$, the fact that
$\mathsf{x}_{j}(s)\geq-B(k^{2/3}+j^{2/3})$ on $\textbf{ICE}_{n^{\prime}}$ (by
the LOC events in Definition 16.8), and (16.9).

Now, if $\alpha>2\beta-1/2-{\mathfrak{d}}$ then for sufficiently large
$L^{\prime}$ we have by (17.18) that

(17.25)
$\displaystyle\begin{split}&(L^{\prime})^{\beta}\vartheta^{1/2}=(2BP)^{1/2}(L^{\prime})^{\beta+(\alpha-1)/2}\leq(2BP)^{1/2}(L^{\prime})^{\alpha-1/4+{\mathfrak{d}}/2}\leq(L^{\prime})^{\alpha-1/4+{\mathfrak{d}}};\\\
&(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}=2BP(L^{\prime})^{\alpha-1/4}|\log\vartheta|^{2}\leq(L^{\prime})^{\alpha-1/4+{\mathfrak{d}}}.\end{split}$

Taking $\widetilde{L}=(L^{\prime})^{3/4}$ and
$\widetilde{n}=\lceil\widetilde{L}^{3/2}k\rceil$, we also have

$\displaystyle\vartheta(k^{2/3}+j^{2/3})\leq
2BP(L^{\prime})^{\alpha-1}\cdot\big{(}(L^{\prime})^{3/4}+2\big{)}k^{2/3}\leq(L^{\prime})^{\alpha-1/4+\mathfrak{d}}k^{2/3},$

for sufficiently large $L^{\prime}$ and any integer $j\in\llbracket
1,\widetilde{n}\rrbracket$. This, together with (17.24) and (LABEL:e:someup),
gives for each $(j,s)\in\llbracket
1,\widetilde{n}\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$ that

(17.26)
$\displaystyle{\mathsf{y}}_{j}(s)\geq{\mathsf{x}}_{j}(s)-18(R^{2}+B)(L^{\prime})^{\alpha-1/4+{\mathfrak{d}}}k^{2/3}=\mathsf{x}_{j}(s)-18(R^{2}+B)\widetilde{L}^{(4\alpha-1+4\mathfrak{d})/3}k^{2/3},$

which (together with the fact that $\mathfrak{C}=36BP(R^{2}+B)\geq
18(R^{2}+B)$) finishes the proof of (17.9).

If instead $\alpha\leq 2\beta-1/2-{\mathfrak{d}}$, then for sufficiently large
$L^{\prime}$ we have by (17.18) that

(17.27)
$\displaystyle(L^{\prime})^{\beta}\vartheta^{1/2}=(2BP)^{1/2}(L^{\prime})^{\beta+(\alpha-1)/2}\geq(2BP)^{1/2}(L^{\prime})^{\alpha-1/4+{\mathfrak{d}}/2}\geq(L^{\prime})^{3/4}\vartheta|\log\vartheta|^{2}.$

Taking $\widetilde{L}=(L^{\prime})^{\beta+(1-\alpha)/2}$ and
$\widetilde{n}=\lceil\widetilde{L}^{3/2}k\rceil$, we also have

$\displaystyle\vartheta(k^{2/3}+j^{2/3})\leq
2BP(L^{\prime})^{\alpha-1}\cdot\big{(}(L^{\prime})^{\beta+(1-\alpha)/2}+2\big{)}k^{2/3}\leq
3BP(L^{\prime})^{\beta+(\alpha-1)/2}k^{2/3},$

for sufficiently large $L^{\prime}$ and any integer $j\in\llbracket
1,\widetilde{n}\rrbracket$. This, together with (17.24) and (17.27), gives for
each $(j,s)\in\llbracket 1,\widetilde{n}\rrbracket\times[-Ak^{1/3},Ak^{1/3}]$
that

(17.28)
$\displaystyle{\mathsf{y}}_{j}(s)\geq{\mathsf{x}}_{j}(s)-36BP(R^{2}+B)(L^{\prime})^{\beta+(\alpha-1)/2}k^{2/3}=\mathsf{x}_{j}(s)-\mathfrak{C}\widetilde{L}^{(2\beta+\alpha-1)/(2\beta-\alpha+1)}.$

This finishes the proof of (17.10) and thus of the proposition. ∎

###  18\. Improved Hölder Estimates

In this section we establish the improved Hölder estimate 16.13, which will be
based on three results. The first indicates, under 16.3, that the improved
Hölder event IHR (from Definition 16.7) likely holds at $\beta=3/4$; to
establish 16.13, we must improve this value of $\beta$ to $3/8$. To this end,
we define a “density regularity event” DEN, on which the paths in the line
ensemble $\bm{\mathsf{x}}$ are well approximated by a measure with regular
density; this event will also involve a parameter $\beta$, prescribing the
error in the approximation. The second result we will show indicates that DEN
likely implies IHR with an improved value of $\beta$; the third indicates that
IHR likely implies DEN with an improved value of $\beta$. By inductively
applying the latter two statements, we will improve the $\beta$ in IHR from
$3/4$ to $3/8$, establishing 16.13.

We begin in Section 18 by defining the regular density event DEN, formulating
these three statements, and establishing 16.13 assuming them. We then
establish the first, second, and third results mentioned above in Section
18.2, Section 18.3, and Section 18.4, respectively. Throughout this section,
we let $\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)$ denote a
$\mathbb{Z}_{\geq 1}\times\mathbb{R}$ indexed line ensemble satisfying the
Brownian Gibbs property; we also recall the $\sigma$-algebra
$\mathcal{F}_{\operatorname{ext}}$ from Definition 2.2, and the location event
LOC from Definition 16.1 and the boundary tall rectangle event BTR from
Definition 16.2.

####  18.1. Proof of the Improved Hölder Estimate

In this section we establish 16.13. We begin with the following lemma, to be
established in Section 18.2 below, indicating that the boundary tall rectangle
event BTR of Definition 16.2 likely implies the first Hölder events FHR of
(16.6).

######  Lemma 18.1.

Adopting 16.3, there exist constants $c=c(A,B)>0$ and $C=C(A,B,D)>1$ such that
the following holds if $L\geq(2B)^{2/\delta}$. For any real number
$A^{\prime}\in[0,A-k^{-1/3}]$, we have (recalling Definition 16.7) that

$\displaystyle\mathbb{P}\Bigg{[}\textbf{\emph{BTR}}_{n}^{\bm{\mathsf{x}}}(A;B)\cap\bigcup_{j=1}^{n}\textbf{\emph{FHR}}_{j}^{\bm{\mathsf{x}}}(A^{\prime};L^{\delta};D;k)^{\complement}\Bigg{]}\leq
Ce^{-c(\log k)^{2}}.$

The following lemma indicates that intersections of the FHR events are equal
to the $\beta=3/4$ cases of the improved Hölder events IHR from (16.7). So,
18.1 can be viewed as the inital case of the induction outlined at the
beginning of this section.

######  Lemma 18.2.

Fix integers $n\geq k\geq 1$, and real numbers $A,B,D,L\geq 1$; $S\geq 4$;
$\delta\in(0,S^{-1})$; and $R\geq\max\\{2A,5\\}$, such that $n=L^{3/2}k$ and
$L\in[1,k^{D}]$. Recalling Definition 16.7, we have for any integer
$n^{\prime}\in\llbracket L^{3S\delta/2}k,n\rrbracket$ that

$\displaystyle\textbf{\emph{IHR}}_{n^{\prime}}\Big{(}A;\displaystyle\frac{3}{4};R;S\Big{)}=\bigcap_{j=1}^{n^{\prime}}\textbf{\emph{FHR}}_{j}(A;L^{\delta};D).$

###### Proof.

By (16.7), it suffices to show that
$\textbf{FHR}_{j}(A;L^{\delta};D)\subseteq\textbf{SHR}_{j}(A;3/4;R;4)$ for
each integer $j\in\llbracket L^{3S\delta/2}k,n\rrbracket$. Setting
$\beta=3/4$, by (16.6), this follows from the facts that $R\geq 5$ and that,
for any $(j,t)\in\llbracket L^{3S\delta/2}k,n\rrbracket\times[-A,A]$, we have

$\displaystyle
L^{\delta}\Big{(}\displaystyle\frac{j}{k}\Big{)}^{1/3}t+4\Big{(}\displaystyle\frac{j}{k}\Big{)}^{1/2}t^{1/2}+k^{-D}$
$\displaystyle\leq
5\Big{(}\displaystyle\frac{j}{k}\Big{)}^{1/2}t^{1/2}+k^{-D}$
$\displaystyle\leq
5\Bigg{(}\bigg{(}\displaystyle\frac{j}{k}\Big{)}^{1/2}t\Big{(}\log(5|t|^{-1}\big{)}\Big{)}^{2}+\Big{(}\displaystyle\frac{j}{k}\Big{)}^{2\beta/3}t^{1/2}+k^{-D}\Bigg{)},$

where the first bound holds since
$L^{\delta}(k^{-1}j)^{1/3}\leq(k^{-1}j)^{1/2}$ for $j\geq L^{3S\delta/2}\geq
L^{6\delta}k$ (recall $S\geq 4$). ∎

We next introduce the following event on which the
$\bm{\mathsf{x}}_{\llbracket 1,i\rrbracket}(tk^{1/3})$ are, for each $(t,i)$,
well-approximated by the classical locations with respect to a measure with a
regular density (in a form similar to what is guaranteed by 16.11). In what
follows, we recall the classical locations with respect to a measure from
Definition 4.21.

######  Definition 18.3.

Fix integers $n\geq k\geq 1$ and real numbers $A,D,L,R,S\geq 1$;
$\delta\in(0,S^{-1})$; and $\beta\in[-1,3/4]$, with $n=L^{3/2}k$ and
$L\in[1,k^{D}]$. For any integer $i\in\llbracket L^{3S\delta/2}k,n\rrbracket$
and real number $t\in[-A,A]$, define the _regular density event_
$\textbf{DEN}_{i}(t;\beta;R)=\textbf{DEN}_{i}^{\bm{\mathsf{x}}}(t;\beta;R;k;\delta;D)$
to be that on which the following holds. There exists a measure
$\mu=\mu_{t}^{(i)}$ with $\mu(\mathbb{R})=k^{-1}i$, satisfying the following
properties. In the below, we denote the classical locations of $\mu$ by
$\gamma_{j}=\gamma_{j;i}^{\mu}$ and set
$\mathfrak{m}_{j}=\mathfrak{m}_{j}(R)=\big{\lceil}R\log
n\cdot\max\\{j^{1/2},k^{1/2}\\}\big{\rceil}$, for each $j\in\llbracket
1,i\rrbracket$.

  1. (1)

The measure $\mu$ admits a density $\varrho\in L^{1}(\mathbb{R})$ with respect
to Lebesgue measure satisfying $\varrho(x)\leq R\cdot\max\\{1,-x\\}^{3/4}$ for
each $x\in\mathbb{R}$.

  2. (2)

For each integer $j\in\llbracket 1,i\rrbracket$, we have

(18.1)
$\displaystyle\gamma_{j+\mathfrak{m}_{j}}-R\left(\frac{i}{k}\right)^{2\beta/3}\leq
k^{-2/3}\cdot{\mathsf{x}}_{j}(tk^{1/3})\leq\gamma_{j-\mathfrak{m}_{j}}+R\left(\frac{i}{k}\right)^{2\beta/3}.$

For any integer $n^{\prime}\in\llbracket L^{3S\delta/2}k,n\rrbracket$, also
define
$\textbf{DEN}_{n^{\prime}}(A;\beta;R;S)=\textbf{DEN}_{n^{\prime}}^{\bm{\mathsf{x}}}(A;\beta;R;S;k;\delta;D)$
by

$\displaystyle\textbf{DEN}_{n^{\prime}}(A;\beta;R;S)=\bigcap_{i=\lceil
L^{3S\delta/2}k\rceil}^{n^{\prime}}\bigcap_{|t|\leq
A}\textbf{DEN}_{i}(t;\beta;R).$

The following two propositions provide implications between the regular
density event DEN and improved Hölder one IHR. The first, to be established in
Section 18.3 below (and eventually amounting from 15.7), indicates (upon
restricting to BTR) that DEN likely implies IHR with a different value of
$\beta$, on a slightly thinner rectangle. The second, to be established in
Section 18.4 below (and eventually amounting from 16.11), indicates that IHR
likely implies DEN also with a different value of $\beta$ (but does so by
stating that, if BTR likely implies IHR, then it also likely implies DEN), on
a slightly thinner rectangle.

######  Proposition 18.4.

Adopting 16.3 and letting $R\geq 1$ be a real number, there exist constants
$c_{1}=c_{1}(A,B,R)>0$, $C_{1}=C_{1}(A,B,R)>1$, and
$C_{2}=C_{2}(A,B,R,D,\delta)>1$ such that the following holds if $L\geq
C_{2}$. Fix real numbers $\beta\in[0,3/4]$, $A^{\prime}\in[0,A-k^{-1/3}]$, and
$S\geq 4$ with $\delta\in(0,S^{-1})$. For any integer $n^{\prime}\in\llbracket
L^{3S\delta/2}k,n\rrbracket$, we have (recalling Definition 16.7 and
Definition 18.3) that

(18.2)
$\displaystyle\begin{split}\mathbb{P}\bigg{[}\textbf{\emph{BTR}}_{n}(A;B)\cap\textbf{\emph{DEN}}_{n^{\prime}}(A^{\prime};\beta;R;S)\cap\textbf{\emph{IHR}}_{n^{\prime}}\Big{(}A^{\prime};\displaystyle\frac{\beta}{2}+\displaystyle\frac{3}{8};C_{1};S\Big{)}^{\complement}\bigg{]}\leq
C_{2}e^{-c(\log k)^{2}}.\end{split}$

######  Proposition 18.5.

Adopting 16.3 and letting $A^{\prime}\in[1,A-k^{-1/3}]$; $b,\xi\in(0,1/4)$;
and $R,\Xi\geq 1$ be real numbers, there exist constants
$\zeta\in[2^{-512},1]$, $c=c(b,A,A^{\prime},B,\xi)>0$,
$C_{1}=C_{1}(b,A,A^{\prime},B,R)>1$, and
$C_{2}=C_{2}(b,A,A^{\prime},B,D,R,\delta,\xi,\Xi)>0$ such that the following
holds if $L\geq C_{2}$. Fix real numbers $\beta\in[3/8,3/4]$ and $S\geq 1$
with $\delta\in(0,1/2^{520}S)$. Assume for some integer
$n^{\prime}\in\llbracket L^{3S\delta/2}k,n\rrbracket$ and real number
$L^{\prime}\in[C_{2},L]$, such that $n^{\prime}=(L^{\prime})^{3/2}k$ and
$(L^{\prime})^{3\zeta/2}\geq L^{6S\delta}$, that we have

(18.3)
$\displaystyle\mathbb{P}\big{[}\textbf{\emph{BTR}}_{n}(A;B)\cap\textbf{\emph{IHR}}_{n^{\prime}}(A^{\prime};\beta;R;S)^{\complement}\big{]}\leq\Xi
e^{-\xi(\log k)^{2}}.$

Then, denoting
$\widetilde{n}=\big{\lceil}(L^{\prime})^{3\zeta/2}k\big{\rceil}$, we have
(recalling Definition 16.7 and Definition 18.3) that

$\displaystyle\mathbb{P}\bigg{[}\textbf{\emph{BTR}}_{n}(A;B)\cap\textbf{\emph{DEN}}_{\widetilde{n}}\Big{(}(1-b)A^{\prime};2\beta-\displaystyle\frac{7}{8};C_{1};4S\Big{)}^{\complement}\bigg{]}\leq
C_{2}e^{-c(\log k)^{2}}.$

Given the above results, we can establish 16.13.

###### Proof of 16.13.

Let $\zeta\in[2^{-512},1]$ be as in 18.5, and set $b_{0}=1-2^{-1/7}$. For each
integer $i\in\llbracket 0,7\rrbracket$, set

$\displaystyle
A_{i}=(1-b_{0})^{i}A;\qquad\beta_{i}=\displaystyle\frac{3}{4}-\displaystyle\frac{i-1}{16};\qquad
S_{i}=4^{i};\qquad L_{i}=L^{\zeta^{i-1}};\qquad n_{i}=\lceil
L_{i}^{3/2}k\rceil.$

We will omit the ceilings in what follows, assuming that $n_{i}=L_{i}^{3/2}k$,
as this will barely affect the proofs; we may also suppose that $k$ is
sufficiently large so that $A_{i}-k^{-1/3}\geq A_{i+1}$ for each
$i\in\llbracket 0,6\rrbracket$. Observe that $n_{i}\geq n_{7}\geq
L^{3S_{7}\delta/2}k$, since $2S_{7}\delta=2^{15}\delta<2^{-3100}\leq\zeta^{6}$
(as $\delta<2^{-4000}$) for each $i\in\llbracket 1,7\rrbracket$.

We claim for each integer $i\in\llbracket 1,7\rrbracket$ that there exist
constants $\xi_{i}=\xi_{i}(A,B)>0$, $R_{i}=R_{i}(A,B)>1$, and
$\Xi_{i}=\Xi_{i}(A,B,D,\delta)>1$ such that for $L>\Xi_{i}$ we have

(18.4)
$\displaystyle\mathbb{P}\bigg{[}\textbf{BTR}_{n}(A;B)\cap\textbf{IHR}_{n_{i}}(A_{i};\beta_{i};R_{i};S_{i}\Big{)}^{\complement}\bigg{]}\leq\Xi
e^{-\xi_{i}(\log k)^{2}}.$

The proposition would then follow from taking $i=7$ in (18.4) and using the
inclusion of events
$\textbf{IHR}_{n_{7}}(A_{7};\beta_{7};R_{7};S_{7})\subseteq\textbf{IHR}_{n^{\prime}}(A/2;3/8;R_{7};2^{14})$,
which holds since $(A_{7},\beta_{7},S_{7})=(A/2,3/8,2^{14})$, since
$\textbf{IHR}_{m}\subseteq\textbf{IHR}_{j}$ whenever $j\leq m$ (by Definition
16.7), and since $n_{7}\leq n^{\prime}$ (as $\zeta^{7}\leq 2^{-3200}$).

It therefore remains to verify (18.4), which we do by induction on $i$. We
begin with the case $i=1$. To this end, first observe by 18.1 that there exist
constants $\xi_{1}=c_{1}(A,B)>0$ and $\Xi_{1}=\Xi_{1}(A,B,D,\delta)>1$ such
that for $L>\Xi_{1}$ we have

$\displaystyle\mathbb{P}\Bigg{[}\textbf{BTR}_{n}(A;B)\cap\bigcup_{j=1}^{n}\textbf{FHR}_{j}(A_{1};L^{\delta};D)^{\complement}\Bigg{]}\leq\Xi_{1}e^{-\xi_{1}(\log
k)^{2}},$

which together with 18.2 yields (18.4) at $i=1$ (with $R_{1}=\max\\{2A,5\\}$).

Now, assume (18.4) holds for some integer $i\in\llbracket 1,6\rrbracket$, and
we will show it continues to hold upon replacing $i$ with $i+1$. By 18.5 (with
the parameters $(n^{\prime},L^{\prime};\widetilde{n})$ there equal to
$(n_{i},L_{i};n_{i+1})$ here; the $(b;\beta)$ there equal to
$(b_{0};\beta_{i})$ here; and the
$\big{(}A^{\prime},(1-b)A^{\prime};R,S,4S;\xi,\Xi\big{)}$ there equal to
$(A_{i},A_{i+1};R_{i},S_{i},S_{i+1};\xi_{i},\Xi_{i})$ here, observing that
$L_{i}^{3\zeta/2}\geq L^{\zeta^{6}}\geq L^{-2^{1/3200}}\geq
L^{2^{20}\delta}\geq L^{6S_{i}\delta}$ as $\delta\in(0,2^{-4000})$), there
exist constants $c_{1}=c_{1}(A,B,\xi_{i})>0$, $C_{1}=C_{1}(A,B,R_{i})>1$, and
$C_{2}=C_{2}(A,B,D,R_{i},\delta,\xi_{i},\Xi_{i})>1$ such that, for $L>C_{2}$,
we have

(18.5)
$\displaystyle\mathbb{P}\bigg{[}\textbf{BTR}_{n}(A;B)\cap\textbf{DEN}_{n_{i+1}}\Big{(}A_{i+1};2\beta_{i}-\displaystyle\frac{7}{8};C_{1};S_{i+1}\Big{)}^{\complement}\bigg{]}\leq
C_{2}e^{-c_{1}(\log k)^{2}}.$

Moreover, 18.4 (with the $(n^{\prime};\beta;A^{\prime},R,S)$ there equal to
$(n_{i+1};2\beta-7/8;A_{i+1},C_{1},S_{i+1})$ here, where we observe that
$2\beta_{i}-7/8\geq 2\beta_{6}-7/8=0$) yields constants
$c_{2}=c_{2}(A,B,R_{i})>0$, $R_{i+1}=R_{i+1}(A,B,C_{1})>1$, and
$C_{3}=C_{3}(A,B,C_{1},D,\delta)>1$ such that, for $L>C_{3}$, we have

$\displaystyle\mathbb{P}\bigg{[}\textbf{BTR}_{n}(A;B)$
$\displaystyle\cap\textbf{DEN}_{n_{i+1}}(A_{i+1};2\beta_{i}-\displaystyle\frac{7}{8};C_{1};S_{i+1})$
$\displaystyle\cap\textbf{IHR}_{n_{i+1}}\Big{(}A_{i+1};\beta_{i}-\displaystyle\frac{1}{16};R_{i+1};S_{i+1}\Big{)}^{\complement}\bigg{]}\leq
C_{3}e^{-c_{2}(\log k)^{2}}.$

This together with (18.5) and union bound (with the fact that
$\beta_{i+1}=\beta_{i}-1/16$) yields (18.4) with the $i$ there given by $i+1$.
This establishes (18.4) and thus the propostion. ∎

####  18.2. Likelihood of FHR Restricted to BTR

In this section we establish 18.1, which a consequence of the below
“pointwise” variant of it.

######  Lemma 18.6.

Adopting 16.3, there exist constants $c=c(A,B)>0$ and $C_{1}=C(A,B,D)>1$ such
that the following holds if $L\geq(2B)^{2/\delta}$. For any integer
$j\in\llbracket 1,n\rrbracket$ and real numbers
$\mathsf{s},\mathsf{s}+tk^{1/3}\in[-Ak^{1/3},Ak^{1/3}]$, we have

$\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\textbf{\emph{BTR}}_{n}(A;B)\cap\bigg{\\{}\frac{{\mathsf{x}}_{j}(\mathsf{s}+tk^{1/3})-{\mathsf{x}}_{j}(\mathsf{s})}{k^{2/3}}\leq-L^{\delta}|t|\Big{(}\frac{j\vee
k}{k}\Big{)}^{1/3}-4|t|^{1/2}\Big{(}\frac{j\vee
k}{k}&\Big{)}^{1/2}-k^{-D}\bigg{\\}}\Bigg{]}\\\ &\leq Ce^{-c(\log
k)^{2}}.\end{aligned}$

###### Proof.

The proof of this lemma will be similar to that of 7.3. In what follows, we
will assume that $t\geq 0$, as we may by symmetry under reflection through the
line $\\{t=0\\}$.

Let ${\mathsf{T}}=2A(j\vee k)^{1/3}$ and $\widetilde{B}=12A^{2}B^{3}$, and
define the event

$\displaystyle\mathscr{E}=\big{\\{}\mathsf{x}_{j}(\mathsf{s})\leq\widetilde{B}k^{2/3}-\widetilde{B}^{-1}j^{2/3}\big{\\}}\cap\big{\\{}\mathsf{x}_{j}(\mathsf{T})\geq-L^{\delta/2}(j\vee
k)^{2/3}\big{\\}},\qquad\text{so
that}\qquad\textbf{CTR}_{n}(A;\widetilde{B})\subseteq\mathscr{E},$

where the last inclusion follows from Definition 16.2 and Definition 16.5
(with the fact that $\mathsf{x}_{j}(\mathsf{T})\geq\mathsf{x}_{k}(\mathsf{T})$
if $j\leq k$). Further recall by 16.6 that there exist constants
$c_{1}=c_{1}(A,B)>0$ and $C_{1}=C_{1}(A,B)>1$ such that

$\displaystyle\mathbb{P}\big{[}\textbf{BTR}_{n}(A;B)\cap\textbf{CTR}_{n}(A;\widetilde{B})^{\complement}\big{]}\leq
C_{1}e^{-c_{1}(\log k)^{2}},$

Thus, by a union bound, it suffices to show

(18.6)
$\displaystyle\mathbb{P}\Bigg{[}\mathscr{E}\cap\bigg{\\{}\frac{{\mathsf{x}}_{j}(\mathsf{s}+tk^{1/3})-{\mathsf{x}}_{j}(\mathsf{s})}{k^{2/3}}\leq-L^{\delta}t\Big{(}\frac{j\vee
k}{k}\Big{)}^{1/3}-4t^{1/2}\Big{(}\frac{j\vee k}{k}$
$\displaystyle\Big{)}^{1/2}-k^{-D}\bigg{\\}}\Bigg{]}\leq Ce^{-c(\log k)^{2}}.$

To this end, condition on
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,j\rrbracket\times[\mathsf{s},\mathsf{T}]\big{)}$ and restrict to the event
$\mathscr{E}$. Let $u=\mathsf{x}_{j}(\mathsf{s})$ and
$v=\mathsf{x}_{j}(\mathsf{T})$, and set $j_{0}=j\vee k$. Sample $j_{0}$ non-
intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{j_{0}})\in\llbracket
1,j_{0}\rrbracket\times\mathcal{C}\big{(}[\mathsf{s},\mathsf{T}]\big{)}$ from
the measure $\mathsf{Q}^{\bm{u};\bm{v}}$, where $\bm{u}=(u,u,\ldots,u)$ and
$\bm{v}=(v,v,\ldots,v)$ (with each entry appearing with multiplicity $j_{0}$).
Then,
$\mathsf{x}_{i+j-j_{0}}(\mathsf{s})\geq\mathsf{x}_{j}(\mathsf{s})=u=\mathsf{y}_{i}(\mathsf{s})$
and
$\mathsf{x}_{i+j-j_{0}}(\mathsf{T})\geq\mathsf{x}_{j}(\mathsf{T})=v=\mathsf{y}_{i}(\mathsf{T})$
for each $i\in\llbracket 1,j\rrbracket$, where we have set
$\mathsf{x}_{m}=\infty$ if $m<1$. Hence, by 4.6, we may couple
$\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ in such a way that

(18.7)
$\displaystyle\mathsf{x}_{j}(\mathsf{s}+tk^{1/3})\geq\mathsf{y}_{j_{0}}(\mathsf{s}+tk^{1/3}).$

Next, by the second part of 4.32 (and using the facts that $\log j_{0}\geq\log
k$ and that
$\big{(}(tk^{1/3})(\mathsf{T}-\mathsf{s}-tk^{1/3})(\mathsf{T}-\mathsf{s})^{-1}\big{)}^{1/2}\leq
t^{1/2}k^{1/6}$), there exists constants $c_{2}=c_{2}(A)>0$ and
$C_{2}=C_{2}(A,D)>1$ such that

$\displaystyle\mathbb{P}\bigg{[}\mathsf{y}_{j_{0}}(\mathsf{s}+tk^{1/3})-u\leq\displaystyle\frac{tk^{1/3}}{\mathsf{T}-\mathsf{s}}\cdot(v-u)-(8j_{0}t)^{1/2}k^{1/6}-k^{-D}\bigg{]}\leq
C_{2}e^{-c_{2}(\log k)^{5}},$

Since $\mathscr{E}$ holds, we have
$v-u=\mathsf{x}_{j}(\mathsf{T})-\mathsf{x}_{j}(\mathsf{s})\geq\widetilde{B}^{-1}j^{2/3}-\widetilde{B}k^{2/3}-L^{\delta/2}j_{0}^{2/3}\geq-
j_{0}^{2/3}(L^{\delta/2}+B)$. Thus, since $\mathsf{T}-\mathsf{s}\geq
2Aj_{0}^{1/3}-Ak^{1/3}\geq Aj_{0}^{1/3}$ and $u=\mathsf{x}_{j}(\mathsf{s})$,
we have

$\displaystyle\mathbb{P}\bigg{[}\mathsf{y}_{j_{0}}(\mathsf{s}+tk^{1/3})-\mathsf{x}_{j}(\mathsf{s})\leq-A^{-1}j_{0}^{1/3}k^{1/3}(L^{\delta/2}+B)\cdot
t-4j_{0}^{1/2}k^{1/6}t^{1/2}-k^{-D}\bigg{]}\leq C_{2}e^{-c_{2}(\log k)^{5}}.$

Together with (18.7) and the facts that $j_{0}=j\vee k$ and
$L^{\delta/2}+B\leq L^{\delta}A$ (as $L^{\delta/2}\geq 2B\geq B(A^{-1}+1)$),
this yields (18.6) and thus the lemma. ∎

###### Proof of 18.1.

The proof of this lemma given 18.6 is similar to that of 7.1 given 7.3. In
particular, by Definition 16.7, it suffices to show that

(18.8) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\bigcup_{j=1}^{n^{\prime}}\bigcup_{\begin{subarray}{c}|s|\leq
A^{\prime}k^{1/3}\\\ |s+tk^{1/3}|\leq
A^{\prime}k^{1/3}\end{subarray}}\bigg{\\{}\displaystyle\frac{\mathsf{x}_{j}(s+tk^{1/3})-\mathsf{x}_{j}(s)}{k^{2/3}}<-L^{\delta}\Big{(}\displaystyle\frac{j\vee
k}{k}&\Big{)}^{1/3}|t|-4\Big{(}\displaystyle\frac{j\vee
k}{k}\Big{)}^{1/2}|t|^{1/2}-k^{-D}\bigg{\\}}\\\
&\cap\textbf{BTR}_{n}(A;B)\Bigg{]}\leq C_{2}e^{-c_{1}(\log
k)^{2}},\end{aligned}$

observing that $t$ can be either positive or negative above (and for any
$M\geq 0$ that $\big{|}\mathsf{x}_{j}(s+tk^{1/3})-\mathsf{x}_{j}(s)\big{|}\leq
M$ holds if and only if we have both
$\mathsf{x}_{j}(s+tk^{1/3})-\mathsf{x}_{j}(s)\geq-M$ and
$\mathsf{x}_{j}(s)-\mathsf{x}_{j}(s+tk^{1/3})\geq-M$).

Now, denote the $n^{-50(D+1)}$-mesh
$\mathcal{S}=[-A^{\prime}k^{1/3},A^{\prime}k^{1/3}]\cap(n^{-50(D+1)}\cdot\mathbb{Z})$,
and take a union bound in 18.6 (with the $D$ there given by $2D$ here) over
all $i\in\llbracket 1,n\rrbracket$ and $s,t\in\mathcal{S}$; this consists of
at most $9(A^{\prime})^{2}k^{2/3}n^{100(D+1)+1}\leq 9A^{2}n^{300D}\leq
9A^{2}k^{750D^{2}}$ (as $n=L^{3/2}k\leq k^{3D/2+1}\leq k^{5D/2}$) triples
$(i,s,t)$. Hence, it yields constants $c_{1}=c_{1}(A,B)\in(0,1)$ and
$C_{1}=C_{1}(A,B,D)>1$ such that
$\mathbb{P}\big{[}\textbf{BTR}_{n}(A:B)\cap\mathscr{E}_{1}^{\complement}\big{]}\geq
1-C_{1}e^{-c_{1}(\log k)^{2}}$, where we have defined the event

$\displaystyle\mathscr{E}_{1}=\bigcap_{j=1}^{n}\bigcap_{s,s+tk^{1/3}\in\mathcal{S}}\bigg{\\{}\displaystyle\frac{\mathsf{x}_{j}(s+tk^{1/3})-\mathsf{x}_{j}(s)}{k^{2/3}}\geq-L^{\delta}\Big{(}\displaystyle\frac{j\vee
k}{k}$ $\displaystyle\Big{)}^{1/3}|t|-4\Big{(}\displaystyle\frac{j\vee
k}{k}\Big{)}^{1/2}|t|^{1/2}-k^{-2D}\bigg{\\}}.$

Also define the event

(18.9)
$\displaystyle\mathscr{E}_{2}=\bigcap_{j=1}^{n}\bigcap_{|s|,|s^{\prime}|\leq
A^{\prime}k^{1/3}}\Big{\\{}\big{|}\mathsf{x}_{j}(s)-\mathsf{x}_{j}(s^{\prime})\big{|}\leq
n^{5}|s-s^{\prime}|^{1/3}\Big{\\}}.$

We claim that there exist constants $c_{2}=c_{2}(A,B)\in(0,1)$ and
$C_{2}=C_{2}(A,B,D)>1$ such that

(18.10)
$\displaystyle\mathbb{P}\big{[}\mathscr{E}_{2}^{\complement}\cap\textbf{BTR}_{n}(A;B)\big{]}\leq
C_{2}e^{-c_{2}(\log k)^{2}}.$

Let us establish (18.8) assuming (18.10). First observe by (18.10), the
estimate
$\mathbb{P}\big{[}\textbf{BTR}_{n}(A,B)\cap\mathscr{E}_{1}^{\complement}\big{]}\geq
1-C_{1}e^{-c_{1}(\log k)^{2}}$, and a union bound that we have

(18.11)
$\displaystyle\mathbb{P}\big{[}\textbf{BTR}_{n}(A;B)\cap(\mathscr{E}_{1}^{\complement}\cup\mathscr{E}_{2}^{\complement})\big{]}\leq
2C_{1}C_{2}e^{-c_{1}c_{2}(\log k)^{2}}.$

Next, restrict to the event $\mathscr{E}_{1}\cap\mathscr{E}_{2}$; by (18.11),
it suffices to show that the event on the left side of (18.8) does not hold
for sufficiently large $k$. To do this, fix an integer $j\in\llbracket
1,n\rrbracket$ and real numbers
$s,s+tk^{1/3}\in[-A^{\prime}k^{1/3},A^{\prime}k^{1/3}]$. Set $r=s+tk^{1/3}$,
and let $s_{0},r_{0}\in\mathcal{S}$ be such that $|s-s_{0}|\leq n^{-50(D+1)}$
and $|r-r_{0}|\leq n^{-50(D+1)}$. Then,

$\displaystyle\displaystyle\frac{\mathsf{x}_{j}(s+tk^{1/3})-\mathsf{x}_{j}(s)}{k^{2/3}}$
$\displaystyle\geq\displaystyle\frac{\mathsf{x}_{j}(r_{0})-\mathsf{x}_{j}(s_{0})}{k^{2/3}}-k^{-2/3}\Big{(}\big{|}\mathsf{x}_{j}(r)-\mathsf{x}_{j}(r_{0})\big{|}-\big{|}\mathsf{x}_{j}(s)-\mathsf{x}_{j}(s_{0})\big{|}\Big{)}$
$\displaystyle\geq-L^{\delta}\Big{(}\displaystyle\frac{j\vee
k}{k}\Big{)}^{1/3}|t|-4\Big{(}\displaystyle\frac{j\vee
k}{k}\Big{)}^{1/2}|t|^{1/2}-k^{-2D}-2n^{5}k^{-15D-15}$
$\displaystyle\geq-L^{\delta}\Big{(}\displaystyle\frac{j\vee
k}{k}\Big{)}^{1/3}|t|-4\Big{(}\displaystyle\frac{j\vee
k}{k}\Big{)}^{1/2}|t|^{1/2}-k^{-D}.$

where the second bound follows from the facts that we have restricted to
$\mathscr{E}_{1}\cap\mathscr{E}_{2}$ and that
$|s-s_{0}|^{1/3}+|r-r_{0}|^{1/3}\leq 2k^{-15D-15}$, and the third holds since
$k^{-2D}+2n^{5}k^{-15D-15}\leq k^{-2D}+2n^{-5D-5}<k^{-D}$ for $k\geq 2$ (as
$k^{15D+15}\geq L^{15}k^{15}=n^{10}k^{5}$ and $n\geq k$). This confirms that
the event on the left side of (18.8) cannot hold on
$\mathscr{E}_{1}\cap\mathscr{E}_{2}$, which (as mentioned above) implies the
lemma.

It therefore suffices to verify (18.10), which will follow from 4.9. In
particular, condition on
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n\rrbracket\times[-Ak^{1/3},Ak^{1/3}]\big{)}$ and restrict to
$\textbf{BTR}_{n}(A;B)$. Then, for any $t_{0}\in\\{-Ak^{1/3},Ak^{1/3}\\}$ and
$s\in[-Ak^{1/3},Ak^{1/3}]$, we have
$\mathsf{x}_{n+1}(s)-\mathsf{x}_{1}(t_{0})\leq 2Bk^{2/3}+B$, due the LOC
events (recall Definition 16.1) in the definition (16.1) of BTR. Moreover,
since $|Ak^{1/3}-A^{\prime}k^{1/3}|\geq 1$ (as $A^{\prime}\in[0,A-k^{1/3}]$),
4.9 (with the $(a,b,\mathsf{T};A,B;\kappa)$ there equal to
$(-Ak^{1/3},Ak^{1/3},2Ak^{1/3};3Bk^{1/2},n^{2};1/4)$ here) applies and yields
constants $c_{3}=c_{3}(A,B)>0$ and $C_{3}=C_{3}(A,B)>1$ such that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{|s|,|s^{\prime}|\leq
A^{\prime}k^{1/3}}\bigg{\\{}\big{|}\mathsf{x}_{j}(s)-\mathsf{x}_{j}(s^{\prime})\big{|}$
$\displaystyle\leq|s-s^{\prime}|^{1/2}\Big{(}n^{2}\log\big{(}4Ak^{1/3}|s-s^{\prime}|^{-1}\big{)}+8Ak^{1/3}(n^{2}+3Bk^{1/2})\Big{)}^{2}$
$\displaystyle\qquad\qquad\qquad\qquad+4Bn^{2/3}|s-s^{\prime}|\bigg{\\}}\Bigg{]}\geq
1-C_{3}e^{C_{3}n-c_{3}n^{4}},$

where we also used the fact that
$\big{|}\mathsf{x}_{j}(Ak^{1/3})-\mathsf{x}_{j}(-Ak^{1/3})\big{|}\leq
2B(j^{2/3}+k^{2/3})\leq 4Bn^{2/3}$ (again due to the LOC events in BTR). This,
together with the definition (18.9) of $\mathscr{E}_{2}$ and the fact that for
sufficiently large $n\geq k$ we have

$\displaystyle|s-s^{\prime}$
$\displaystyle|^{1/2}\Big{(}n^{2}\log\big{(}4Ak^{1/3}|s-s^{\prime}|^{-1}\big{)}+8Ak^{1/3}(n^{2}+3Bk^{1/2})\Big{)}^{2}+4Bn^{2/3}|s-s^{\prime}|$
$\displaystyle\leq
2|s-s^{\prime}|^{1/2}\bigg{(}\Big{(}n^{2}\log\big{(}4Ak^{1/3}|s-s^{\prime}|^{-1}\big{)}\Big{)}^{2}+128A^{2}k^{2/3}(n^{4}+9B^{2}k)\bigg{)}+4ABn|s-s^{\prime}|^{1/3}$
$\displaystyle\leq|s-s^{\prime}|^{1/2}\Big{(}\big{(}20Ak^{1/3}n^{2}|s-s^{\prime}|^{-1/12}\big{)}^{2}+2560A^{3}B^{2}n^{29/6}|s-s^{\prime}|^{-1/6}\Big{)}+4ABn|s-s^{\prime}|^{1/3}$
$\displaystyle\leq|s-s^{\prime}|^{1/3}(400A^{2}n^{14/3}+2560A^{3}B^{2}n^{29/6}+4ABn)\leq
n^{5}|s-s^{\prime}|^{1/3}$

yields (18.10) and thus the lemma. Here, in the first inequality, we
repeatedly used the facts that $(x+y)^{2}\leq 2(x^{2}+y^{2})$ for any
$x,y\in\mathbb{R}$ and that $|s-s^{\prime}|\leq 2An^{1/3}|s-s^{\prime}|^{1/3}$
(as $|s-s^{\prime}|\leq 2Ak^{1/3}\leq 2An^{1/3}$); in the second used that
$w^{1/12}\log(Mw^{-1})\leq 12e^{-1}M^{1/12}\leq 5M$ for any real numbers
$M\geq 1$ and $w\in(0,M]$ (in particular, at
$(w,M)=\big{(}|s-s^{\prime}|,4Ak^{1/3}\big{)}$) and that
$128A^{2}k^{2/3}(n^{4}+9B^{2}k)\leq 128A^{2}n^{2/3}\cdot
10B^{2}n^{4}=1280A^{2}B^{2}n^{14/3}\leq
2560A^{2}B^{2}n^{29/6}|s-s^{\prime}|^{-1/6}$ (the last since
$|s-s^{\prime}|\leq 2Ak^{1/3}$); and in the third, we used the fact that
$n\geq k$; and in the fifth we used that $n$ is sufficiently large. ∎

####  18.3. Likelihood of IHR Restricted to DEN and BTR

In this section we establish 18.4, which will again be a consequence of its
below pointwise variant.

######  Lemma 18.7.

Adopt the notation and assumptions of 18.4. Setting
${\widetilde{\beta}}=\beta/2+3/8$, we have for any integer $i\in\llbracket
L^{3S\delta/2}k,n\rrbracket$ and real numbers
$\mathsf{s},\mathsf{s}+tk^{1/3}\in[-A^{\prime}k^{1/3},A^{\prime}k^{1/3}]$ that

(18.12) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\bigg{\\{}\frac{{\mathsf{x}}_{i}({\mathsf{s}}+tk^{1/3})-{\mathsf{x}}_{i}({\mathsf{s}})}{k^{2/3}}\leq-
C_{1}\bigg{(}\Big{(}&\displaystyle\frac{i}{k}\Big{)}^{1/2}|t|\big{|}\log(C_{1}|t|^{-1})\big{|}^{2}+\Big{(}\displaystyle\frac{i}{k}\Big{)}^{2{\widetilde{\beta}}/3}|t|^{1/2}\bigg{)}\bigg{\\}}\\\
&\cap\textbf{\emph{BTR}}_{n}(A;B)\cap\textbf{\emph{DEN}}_{n^{\prime}}(A^{\prime};\beta;R;S)\Bigg{]}\leq
C_{2}e^{-c(\log k)^{2}}.\end{aligned}$

###### Proof of 18.4 (Outline).

By 18.1, (16.7), (16.6), and a union bound, denoting
$\widetilde{\beta}=\beta/2+3\beta/8$, it suffices to show that

$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{i=\lceil
L^{3S\delta/2}k\rceil}^{n^{\prime}}\bigcup_{\begin{subarray}{c}|s|\leq
A^{\prime}k^{1/3}\\\ |s+tk^{1/3}|\leq A^{\prime}k^{1/3}\end{subarray}}$
$\displaystyle\bigg{\\{}\frac{{\mathsf{x}}_{i}({\mathsf{s}}+tk^{1/3})-{\mathsf{x}}_{i}({\mathsf{s}})}{k^{2/3}}<-C_{1}\bigg{(}\Big{(}\displaystyle\frac{i}{k}\Big{)}^{1/2}|t|\big{|}\log(C_{1}|t|^{-1})\big{|}^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\Big{(}\displaystyle\frac{i}{k}\Big{)}^{2{\widetilde{\beta}}/3}|t|^{1/2}+k^{-D}\bigg{)}\bigg{\\}}$
$\displaystyle\qquad\qquad\cap\textbf{BTR}_{n}(A;B)\cap\textbf{DEN}_{n^{\prime}}(A^{\prime};\beta;R;S)\Bigg{]}\leq
C_{2}e^{-c(\log k)^{2}}.$

Given 18.7, the proof of this bound is very similar to that of 18.1 given
18.6, by taking a union bound of 18.7 over all $i\in\llbracket
L^{3S\delta/2}k,n^{\prime}\rrbracket$ and $s,t$ in an $n^{-50(D+1)}$ mesh to
$[-A^{\prime}k^{1/3},A^{\prime}k^{1/3}]$, and then using the high probability
Hölder regularity of $\bm{\mathsf{x}}$ on
$[-A^{\prime}k^{1/3},A^{\prime}k^{1/3}]$ guaranteed by 4.9 to conclude. We
omit further details. ∎

Figure 5.6. Shown above is a depiction of the proof of 18.7.

###### Proof of 18.7.

The proof of this lemma will follow that of 18.6, replacing the use of 4.32
(when comparing to a Brownian watermelon) by one of 15.7. In what follows, we
will assume that $t\geq 0$, as we may by symmetry under reflection through the
line $\\{t=0\\}$; we also assume (by the scaling invariance 4.4) that $A=1$.

Set $L^{\prime\prime}=(ik^{-1})^{2/3}$ and $\mathsf{T}=2Ai^{1/3}=2i^{1/3}$.
Define the event

(18.13)
$\displaystyle\mathscr{E}=\textbf{DEN}_{i}(\mathsf{s}k^{-1/3};\beta;R)\cap\big{\\{}\mathsf{x}_{i}(\mathsf{T})\geq-L^{\delta/2}i^{2/3}\big{\\}},\quad\text{so}\quad\textbf{DEN}_{n^{\prime}}(A;\beta;R;S)\cap\textbf{BTR}_{n}(A;B)\subseteq\mathscr{E},$

where the last inclusion follows from Definition 18.3 and Definition 16.2.
Condition on
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,i\rrbracket\times[\mathsf{s},\mathsf{T}]\big{)}$, and restrict to the event
$\mathscr{E}$. Then, by (18.13), to verify (18.12) it suffices to show that

(18.14)
$\displaystyle\mathbb{P}\Bigg{[}\frac{{\mathsf{x}}_{i}({\mathsf{s}}+tk^{1/3})-{\mathsf{x}}_{i}({\mathsf{s}})}{k^{2/3}}\leq-
C_{1}\bigg{(}\Big{(}\displaystyle\frac{i}{k}\Big{)}^{1/2}\big{|}\log(2|t|^{-1})\big{|}^{2}t+\Big{(}\displaystyle\frac{i}{k}\Big{)}^{2{\widetilde{\beta}}/3}t^{1/2}\bigg{)}\Bigg{]}\leq
C_{2}e^{-c(\log k)^{2}}.$

To this end, let $v=-(B+R+2)(L^{\prime\prime})^{1/4}\mathsf{T}^{2}$, so that

(18.15) $\displaystyle
v=-(4B+4R+8)k^{-1/6}i^{5/6}\leq-L^{\delta/2}i^{2/3}\leq\mathsf{x}_{i}(\mathsf{T}),$

where the first statement used the facts that
$L^{\prime\prime}=(ik^{-1})^{2/3}$ and $\mathsf{T}=2i^{1/3}$; the second used
the facts that $B\geq 1$ and $ik^{-1}\geq L^{3S\delta/2}\geq L^{3\delta}$ (as
$S\geq 2$); and the third used the fact that $\mathscr{E}$ holds. Also define
the $i$-tuples $\bm{u}=\mathsf{x}_{\llbracket
1,i\rrbracket}\in\overline{\mathbb{W}}_{i}$ and
$\bm{v}=(v,v,\ldots,v)\in\overline{\mathbb{W}}_{i}$ (where $v$ appears with
multiplicty $i$). Sample $i$ non-intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{i})\in\llbracket
1,i\rrbracket\times\mathcal{C}\big{(}[\mathsf{s},\mathsf{T}]\big{)}$ from the
measure $\mathsf{Q}^{\bm{u};\bm{v}}$; see Figure 5.6. Since for any
$j\in\llbracket 1,i\rrbracket$ we have $\mathsf{x}_{j}(\mathsf{s})=u_{j}$ and
$\mathsf{x}_{j}(\mathsf{T})\geq\mathsf{x}_{i}(\mathsf{T})\geq v=v_{i}$ by
(18.15), 4.6 yields a coupling between $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$
such that
$\mathsf{x}_{i}(\mathsf{s}+tk^{1/3})\geq\mathsf{y}_{i}(\mathsf{s}+tk^{1/3})$.
Since $\mathsf{x}_{i}(\mathsf{s})=\mathsf{y}_{i}(\mathsf{s})$, to prove
(18.14) it therefore suffices to show

(18.16)
$\displaystyle\mathbb{P}\Bigg{[}\frac{\mathsf{y}_{i}({\mathsf{s}}+tk^{1/3})-\mathsf{y}_{i}({\mathsf{s}})}{k^{2/3}}\leq-
C_{1}\bigg{(}\Big{(}\displaystyle\frac{i}{k}\Big{)}^{1/2}\big{|}\log(2t^{-1})\big{|}^{2}t+\Big{(}\displaystyle\frac{i}{k}\Big{)}^{2{\widetilde{\beta}}/3}t^{1/2}\bigg{)}\Bigg{]}\leq
C_{2}e^{-c(\log k)^{2}}.$

This will follow from an application of 15.7, to which end we must verify that
$\bm{\mathsf{x}}$ satisfies 15.6. This will be a consequence of the fact that
we have restricted to the event
$\mathscr{E}\subseteq\textbf{DEN}_{i}(\mathsf{s}k^{-1/3};\beta;R)$. Indeed,
observe from Definition 18.3 that there exists a measure
$\mu\in\mathscr{P}_{\operatorname{fin}}$ with $\mu(\mathbb{R})=k^{-1}i$,
satisfying the following two properties. First, $\mu$ admits a density
$\varrho\in L^{1}(\mathbb{R})$ with respect to Lebesgue measure satisfying
$\varrho(x)\leq R\cdot\max\\{1,-x\\}^{3/4}$. Second, denoting its classical
locations by $\gamma_{j}=\gamma_{j;i}^{\mu}$ (recall Definition 4.21) and
setting $\mathfrak{m}_{j}=\big{\lceil}R\log
n\cdot\max\\{j^{1/2},k^{1/2}\\}\big{\rceil}$ for each $j\in\llbracket
1,i\rrbracket$, we have

(18.17)
$\displaystyle\gamma_{j+\mathfrak{m}_{j}}-R\Big{(}\frac{i}{k}\Big{)}^{2\beta/3}\leq
k^{-2/3}\cdot{\mathsf{x}}_{j}(\mathsf{s})\leq\gamma_{j+\mathfrak{m}_{j}}+R\Big{(}\frac{i}{k}\Big{)}^{2\beta/3}.$

This, together with the facts that
$\mathsf{T}=2i^{1/3}=2(L^{\prime\prime})^{1/2}k^{1/3}$ (as
$L^{\prime\prime}=(ik^{-1})^{2/3}$), $\mathsf{T}-\mathsf{s}\geq k^{1/3}$ (as
$\mathsf{T}\geq 2i^{1/3}\geq 2k^{1/3}\geq k^{1/3}\geq|\mathsf{s}|$), and
$v_{n}=v=-(B+R+2)(L^{\prime\prime})^{1/4}\mathsf{T}^{2}$, verifies 15.6 with
the $(n,k,L;A,B,D;K;M;\mathsf{T})$ there equal to
$\big{(}i,k,L^{\prime\prime};2,2B+R+2,D;\mathfrak{m}_{i};R(ik^{-1})^{2\beta/3};\mathsf{T}-\mathsf{s}\big{)}$
here (observing that $M\geq 1$ for $\beta\geq 0$ and $R\geq 1$).

Hence, 15.7 (with the facts that $(L^{\prime\prime})^{3/4}=(ik^{-1})^{1/2}$,
that $\mathfrak{m}_{i}=Ri^{1/2}\log n$, and that $|tk^{1/3}|\leq k^{1/3}$)
yields constants $c=c(A,B,R)>0$, $C_{3}=C_{3}(A,B,R)>1$, and
$C_{4}=C_{4}(A,B,R,D)>1$ such that

$\displaystyle\mathbb{P}\Bigg{[}$
$\displaystyle\frac{{\mathsf{y}}_{i}({\mathsf{s}}+tk^{1/3})-{\mathsf{y}}_{i}({\mathsf{s}})}{k^{2/3}}\leq-
C_{3}\Big{(}\displaystyle\frac{i}{k}\Big{)}^{1/2}\big{|}\log(2t^{-1})\big{|}^{2}t$
$\displaystyle\qquad\qquad\qquad\qquad-
C_{3}\bigg{(}\Big{(}\displaystyle\frac{i}{k}\Big{)}^{1/2+2\beta/3}t+i^{1/2}tk^{-1}\log
n+(\log n)^{2}k^{-1}t\bigg{)}^{1/2}\Bigg{]}\leq C_{4}e^{-c(\log k)^{2}}.$

Since $(ik^{-1})^{1/2+2\beta/3}\geq(ik^{-1})^{1/2}\geq 3Di^{1/2}k^{-1}\log
k\geq i^{1/2}k^{-1}\log n\geq k^{-1}(\log n)^{2}$ for sufficiently large $k$
(where we used the fact that $n=L^{3/2}k\leq k^{3D/2+1}\leq k^{3D}$ to bound
$3D\log k\geq\log n$, and the fact that $i\geq k\geq n^{1/3D}$), it follows
for $C_{1}=2C_{3}$ and sufficiently large $C_{2}=C_{2}(A,B,R,D)>1$ that

$\displaystyle\mathbb{P}\Bigg{[}$
$\displaystyle\frac{{\mathsf{y}}_{i}({\mathsf{s}}+tk^{1/3})-{\mathsf{y}}_{i}({\mathsf{s}})}{k^{2/3}}\leq-
C_{1}\bigg{(}\Big{(}\displaystyle\frac{i}{k}\Big{)}^{1/2}\big{|}\log(2t^{-1})\big{|}^{2}t+\Big{(}\displaystyle\frac{i}{k}\Big{)}^{1/4+\beta/3}t^{1/2}\bigg{)}\Bigg{]}\leq
C_{2}e^{-c(\log k)^{2}},$

which since $\widetilde{\beta}=\beta/2+3\beta/8$ establishes (18.16) and thus
the lemma. ∎

####  18.4. Likelihood of DEN Restricted to IHR and BTR

In this section we prove 18.5, which will be a consequence of its below
pointwise variant.

######  Lemma 18.8.

Adopting the notation and assumptions of 18.5, we have for any integer
$i\in\llbracket L^{6S\delta}k,\widetilde{n}\rrbracket$ and real number
$t\in\big{[}(b-1)A^{\prime},(1-b)A^{\prime}\big{]}$ that

(18.18)
$\displaystyle\mathbb{P}\bigg{[}\textbf{\emph{BTR}}_{n}(A;B)\cap\textbf{\emph{DEN}}_{i}\Big{(}t;2\beta-\displaystyle\frac{7}{8};C_{1}\Big{)}^{\complement}\bigg{]}\leq
C_{2}e^{-c(\log k)^{2}}.$

###### Proof of 18.5 (Outline).

The proof of this proposition given 18.8 is (as that of 18.4 given 18.7) very
similar to that of 18.1 given 18.6. In particular, we first take a union bound
in (18.18) over all $i\in\llbracket L^{6S\delta}k,\widetilde{n}\rrbracket$ and
$t\in\mathcal{S}$, for some $n^{-50}$-mesh $\mathcal{S}$ of
$\big{[}(b-1)A^{\prime},(1-b)A^{\prime}\big{]}$. For any integer
$i\in\llbracket L^{6S\delta}k,n^{\prime}\rrbracket$ and real number
$t\in\mathcal{S}$, this yields a measure
$\mu_{t}^{(i)}\in\mathscr{P}_{\operatorname{fin}}$ satisfying the properties
in Definition 18.3, with the $(\beta,R)$ there equal to $(2\beta-7/8,C_{1})$
here. For
$t\in\big{[}(b-1)A^{\prime},(1-b)A^{\prime}\big{]}\setminus\mathcal{S}$, set
$\mu_{t}^{(i)}=\mu_{t^{\prime}}^{(i)}$, where $t^{\prime}$ is an arbitrary
element of $\mathcal{S}$ such that $|t-t^{\prime}|\leq n^{-50}$; this
$\mu_{t}^{(i)}$ satisfies the first property in Definition 18.3, since
$\mu_{t^{\prime}}^{(i)}$ does. Using the high probability Hölder bound for
$\bm{\mathsf{x}}$ guaranteed by 4.9, it is then quickly verified that
$\mu_{t}^{(i)}$ likely satisfies the second property (18.1) in Definition
18.3, with the $(\beta,R)$ there equal to $(2\beta-7/8,2C_{1})$ here, for any
$t\in\big{[}(b-1)A^{\prime},(1-b)A^{\prime}\big{]}$; this confirms that
$\textbf{DEN}_{\tilde{n}}\big{(}(1-b)A^{\prime};2\beta-7/8;2C_{1};4S\big{)}$
holds with high probability. We omit further details. ∎

Figure 5.7. Shown above is a depiction of the proof of 18.8.

###### Proof of 18.8.

This lemma will follow from an application of 16.11. Let
$\zeta\in[2^{-512},1]$ denote the constant defined there (see also 16.10), and
define $\widehat{L}=(k^{-1}i)^{2/3\zeta}$ and
$\widehat{n}=\lceil\widehat{L}^{3/2}k\rceil$; we will omit the ceilings in
what follows, assuming that $\widetilde{n}=(L^{\prime})^{3\zeta/2}k$ and
$\widehat{n}=\widehat{L}^{3/2}k$, as this will barely affect the proofs.
Observe that
$\widehat{L}=(k^{-1}i)^{2/3\zeta}\leq(k^{-1}\widetilde{n})^{2/3\zeta}=L^{\prime}$
(as $i\leq\widetilde{n})$; in particular,
$\widehat{n}=\widehat{L}^{3/2}k\leq(L^{\prime})^{3/2}k=n^{\prime}$.
Throughout, we abbreviate the event
$\textbf{ICE}_{\widehat{n}}=\textbf{ICE}_{\widehat{n}}^{\bm{\mathsf{x}}}(A^{\prime},12A^{2}B^{3},D;\beta,\delta;R,S;k)$.

We will apply 16.11 with the $(n^{\prime},L^{\prime},k)$ there equal to
$(\widehat{n},\widehat{L},k)$ here, to which end we must verify the
assumptions imposed there. First observe since $i\geq L^{6S\delta}k$ that
$\widehat{L}=(k^{-1}i)^{2/3\zeta}\geq L^{4S\delta/\zeta}$, confirming (16.8).
To show (16.9), we apply 16.12 (with the
$(n^{\prime},n^{\prime\prime},n^{\prime\prime\prime})$ there given by
$(n^{\prime},\widehat{n},\widehat{n})$ here), whose hypothesis (16.17) is
verified by (18.3). That lemma yields constants $c_{1}=c_{1}(A,B,\xi)>0$ and
$C_{3}=C_{3}(A,B,D,\Xi)>1$, and an event
$\mathscr{G}\subseteq\textbf{BTR}_{n}(A;B)$ (obtained by intersecting the
$\mathscr{G}_{0}$ of (16.18) with $\textbf{BTR}_{n}(A;B)$) measurable with
respect to
$\mathcal{F}_{\operatorname{ext}}=\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,\widehat{n}\rrbracket\times[-A^{\prime}k^{1/3},A^{\prime}k^{1/3}]\big{)}$
satisfying the following two properties. First, we have

(18.19)
$\displaystyle\mathbb{P}\big{[}\textbf{BTR}_{n}(A;B)\setminus\mathscr{G}\big{]}\leq
C_{3}e^{-c_{1}(\log k)^{2}}.$

Second, conditioning on $\mathcal{F}_{\operatorname{ext}}$ and restricting to
$\mathscr{G}$, we have $\mathbb{P}[\textbf{ICE}_{\widehat{n}}]\geq
1-C_{3}e^{-c_{1}(\log k)^{2}}$. The latter verifies (16.9).

Thus, conditioning on $\mathcal{F}_{\operatorname{ext}}$ and restricting to
$\mathscr{G}$, 16.11 applies and (since
$t\in\big{[}(b-1)A^{\prime},(1-b)A^{\prime}\big{]}$) yields constants
$c_{2}=c_{2}(b,A^{\prime},B,\xi)>0$ and $C_{1}=C_{1}(b,A^{\prime},B,R)>1$, and
$C_{4}=C_{4}(b,A^{\prime},B,D,R,\delta,\xi,\Xi)>1$, and a measure
$\widehat{\mu}\in\mathscr{P}_{\operatorname{fin}}$ with
$\widehat{\mu}(\mathbb{R})=\widehat{L}^{3/2}$, satisfying the following two
properties. First, $\widehat{\mu}$ admits a density $\widehat{\varrho}\in
L^{1}(\mathbb{R})$ with respect to Lebesgue measure satisfying
$\widehat{\varrho}(x)\leq C_{1}\max\\{1,-x\\}^{3/4}$. Second, denoting the
classical locations of $\widehat{\mu}$ by
$\gamma_{j}=\gamma_{j;\widehat{n}}^{\widehat{\mu}}$ and
$\mathfrak{m}_{j}=\big{\lceil}C_{1}\log
n\cdot\max\\{j^{1/2},k^{1/2}\\}\big{\rceil}$ for each integer $j\in\llbracket
1,\widehat{n}\rrbracket$, we have

(18.20) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\bigcap_{j=1}^{\lceil\widehat{L}^{3\zeta/2}k\rceil}\big{\\{}\gamma_{j+\mathfrak{m}_{j}}-C_{1}\widehat{L}^{\zeta(2\beta-7/8)}\leq
k^{-2/3}\cdot\mathsf{x}_{j}(tk^{1/3})&\leq\gamma_{j-\mathfrak{m}_{j}}+C_{1}\widehat{L}^{\zeta(2\beta-7/8)}\big{\\}}\Bigg{]}\\\
&\qquad\qquad\quad\geq 1-C_{4}e^{-c_{2}(\log k)^{2}}.\end{aligned}$

Now, define $\gamma\in\mathbb{R}$ to be the minimal real number such that
$\widehat{\mu}\big{(}[\gamma,\infty)\big{)}=k^{-1}i$ (the existence of at
least one $\gamma$ follows from the fact that $\widehat{\varrho}$ is bounded
and that
$\widehat{\mu}(\mathbb{R})=\widehat{L}^{3/2}=(k^{-1}i)^{1/\zeta}>k^{-1}i$).
Define $\varrho\in L^{1}(\mathbb{R})$ by setting
$\varrho(x)=\widehat{\varrho}(x)\cdot\textbf{1}_{x\geq\gamma}$, and define the
measure $\mu=\varrho(x)dx\in\mathscr{P}_{\operatorname{fin}}$; then
$\mu(\mathbb{R})=k^{-1}i$, and the classical locations of $\mu$ are given by
$\gamma_{j}=\gamma_{j;i}^{\mu}=\gamma_{j;\widehat{n}}^{\widehat{\mu}}$, for
each $j\in\llbracket 1,i\rrbracket$. See Figure 5.7. Hence, the fact that
$\varrho(x)\leq\widehat{\varrho}(x)\leq C_{1}\max\\{1,-x\\}^{1/2}$; (18.20);
and the fact that $\widehat{L}^{\zeta}=(k^{-1}i)^{2/3}$ (so
$\widehat{L}^{3\zeta/2}k=i$), together imply by Definition 18.3 that

$\displaystyle\mathbb{P}\bigg{[}\textbf{DEN}_{i}^{\bm{\mathsf{x}}}\Big{(}t;2\beta-\displaystyle\frac{7}{8};C_{1}\Big{)}\bigg{]}\geq
1-C_{4}e^{-c_{2}(\log k)^{2}},$

after conditioning on $\mathcal{F}_{\operatorname{ext}}$ and restricting to
the event $\mathscr{G}\subseteq\textbf{BTR}_{n}(A;B)$. This, together with
(18.19) and a union bound, establishes the lemma. ∎

##  Chapter 6 Global Law and Regular Profiles

In this chapter we establish 3.10 and 3.12, indicating that a line ensemble
$\bm{\mathcal{L}}$ satisfying 2.8 likely satisfies the global law and regular
profile events. As mentioned in the beginning of Chapter 5, this will follow
from restricting $\bm{\mathcal{L}}$ to a tall rectangle, which gives rise to a
family of non-intersecting Brownian bridges with lower boundary. Using 16.4 to
couple this family to one without lower boundary, we will use previously
mentioned results on the latter (such as 15.2, 10.1, and 14.1) to establish
3.12 in Section 19 and 3.10 in Section 20.

Throughout this chapter, we let
$\bm{\mathsf{x}}=(\mathsf{x}_{1},\mathsf{x}_{2},\ldots)$ denote a
$\mathbb{Z}_{\geq 1}\times\mathbb{R}$ indexed line ensemble satisfying the
Brownian Gibbs property, and we recall the $\sigma$-algebra
$\mathcal{F}_{\operatorname{ext}}$ from Definition 2.2. We also recall the
events TOP, GAP, and BTR from Definition 3.2, Definition 3.5, and Definition
16.2, respectively.

###  19\. Likelihood of Regular Profile Events

In this section we prove 3.12, which indicates that regular profile events are
likely, upon restricting to the intersection of several TOP events (from
Definition 3.2). Recall from 16.4 that the existence of the boundary removal
coupling required 16.3, stating that the boundary tall rectangle event BTR
(from Definition 16.2) is likely; we verify that this holds upon restricting
to several TOP events in Section 19.1. We then establish 3.12 in Section 19.2
and Section 19.3; it will eventually amount to being a consequence of the
boundary removal coupling, together with 15.2.

####  19.1. Likelihood of BTR Restricted to TOP Events

In this section we verify 16.3, through the following proposition. In what
follows, we recall the events GAP from Definition 3.5, and BTR from Definition
16.2.

######  Proposition 19.1.

Adopt 2.8. For any real numbers $A,D\geq 3$ and
$\varepsilon,\delta\in(0,1/2)$, there exist constants $B=B(A)>1$,
$C=C(A,D,\delta)>1$, and $R=R(A)>1$, such that the following holds for any
sufficiently large integer $k\geq 1$. Let $n\geq k$ be an integer and
$L\in[C,k^{D}]$ be a real number, such that $n=L^{3/2}k$. Then,

$\displaystyle\mathbb{P}\Big{[}\textbf{\emph{BTR}}_{n}^{\bm{\mathcal{L}}}(A,B;k,L;\delta)\cap\textbf{\emph{GAP}}_{n}^{\bm{\mathcal{L}}}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}\Big{]}\geq
1-\varepsilon.$

We will quickly deduce 19.1 as a consequence of the following proposition
(together with 3.4), which states that the boundary tall rectangle event BTR
and gap event GAP are likely upon restricting to the intersection of several
TOP events; it applies to any $\mathbb{Z}_{\geq 1}\times\mathbb{R}$ indexed
line ensemble $\bm{\mathsf{x}}$ satisfying the Brownian Gibbs property (as
fixed at the beginning of this chapter). In what follows, we recall the TOP
event from Definition 3.2.

######  Proposition 19.2.

For any real numbers $A,D\geq 3$ and $\delta\in(0,1/2)$, there exist constants
$c=c(A,D,\delta)>0$, $\vartheta=\vartheta(A,D)>0$, $B=B(A)>1$, and $R=R(A)>1$,
such that the following holds. Set $\omega=\delta/8$, and fix integers $n\geq
k\geq 1$ and a real number $L\in[c^{-1},k^{D}]$, such that $n=L^{3/2}k$. For
each integer $j\geq 0$, set $K_{j}=\lceil L^{j\omega}k\rceil$. Then,

(19.1) $\displaystyle\begin{split}\mathbb{P}\Bigg{[}\bigcap_{j=0}^{\lfloor
3/\omega\rfloor}\textbf{\emph{TOP}}^{\bm{\mathsf{x}}}&\big{(}[-\vartheta^{-1}K_{j}^{1/3},\vartheta^{-1}K_{j}^{1/3}];\vartheta
K_{j}^{2/3}\big{)}\cap\textbf{\emph{TOP}}^{\bm{\mathsf{x}}}\big{(}[-\vartheta^{-1}K_{j}^{30D},\vartheta^{-1}K_{j}^{30D}];\vartheta
K_{j}^{60D}\big{)}\\\
&\cap\left(\textbf{\emph{BTR}}_{n}^{\bm{\mathsf{x}}}(A,B;k,L;\delta)\cap\textbf{\emph{GAP}}_{n}^{\bm{\mathsf{x}}}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}\right)^{\complement}\Bigg{]}\leq
c^{-1}e^{-c(\log k)^{2}}.\end{split}$

###### Proof of 19.1.

Fix $\omega=\delta/8$, denote $K_{j}=\lceil L^{j\omega}k\rceil$ for each
integer $j\geq 0$, and let $\vartheta=\vartheta(A,D)$ be as in 19.2. By 3.4,
we have for any sufficiently large real number $m>1$ that
$\mathbb{P}\big{[}\textbf{{TOP}}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}m^{1/3},\vartheta^{-1}m^{1/3}];\vartheta
m^{2/3}\big{)}\big{]}\geq 1-\omega\varepsilon/24$. Taking a union bound over
$m\in\\{K_{0},K_{1},\ldots,K_{\lfloor
3/\omega\rfloor}\\}\cup\\{K_{0}^{90D},K_{1}^{90D},\ldots,K_{\lfloor
3/\omega\rfloor}^{90D}\\}$ (observing that the size of this set is at most
$12\omega^{-1}$) and taking $k$ to be sufficiently large, we deduce that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=0}^{\lfloor
3/\omega\rfloor}\Big{(}\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}K_{j}^{1/3},\vartheta^{-1}K_{j}^{1/3}];\vartheta
K_{j}^{2/3}\big{)}\cap\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}K_{j}^{30D},\vartheta^{-1}K_{j}^{30D}$
$\displaystyle];\vartheta K_{j}^{60D}\big{)}\Big{)}\Bigg{]}$
$\displaystyle\geq 1-\displaystyle\frac{\varepsilon}{2}.$

This, together with 19.2 and a union bound, implies the proposition. ∎

To prove 19.2, we use the following lemma, stating that the intersection of
certain GAP and IMP events implies the BTR event. In the below, we recall the
improved medium position event IMP from Definition 7.4; observe that it can be
expressed through the LOC events of Definition 16.1 by

(19.2) $\displaystyle\textbf{IMP}_{n}(A;B;C;R)=\bigcap_{j=\lceil
n/B\rceil}^{\lfloor
Rn\rfloor}\textbf{LOC}_{j}\big{(}[-An^{1/3},An^{1/3}];C^{-1}n^{2/3}-Cj^{2/3};Cn^{2/3}-C^{-1}j^{2/3}\big{)}.$

######  Lemma 19.3.

Fix integers $n\geq k\geq 2^{500}$ and real numbers $\delta\in(0,1)$ and
$A,B,C,D,L,R\geq 2$, such that $n=L^{3/2}k$ and $L\geq(2C)^{4/\delta}$.
Defining $n_{j}=\lceil L^{3j\delta/8}k\rceil$ for each integer $j\geq 0$, we
have

$\displaystyle\textbf{\emph{GAP}}_{n}^{\bm{\mathsf{x}}}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}\cap\bigcap_{j=0}^{\lceil
4/\delta\rceil}\textbf{\emph{IMP}}_{n_{j}}^{\bm{\mathsf{x}}}(2A;B;C;n_{j}^{3D})\subseteq\textbf{\emph{BTR}}_{n}^{\bm{\mathsf{x}}}(A,C+R+2;k,L;\delta).$

###### Proof.

Set $B_{0}=C+R+2$. By Definition 16.2 (and the fact that $\llbracket
k,n\rrbracket\subseteq\bigcup_{j=1}^{\lceil 4/\delta\rceil}\llbracket
n_{j-1},n_{j}\rrbracket$, since $n_{0}=k$ and $n_{\lceil 4/\delta\rceil}\geq
L^{3/2}k=n$), to establish the lemma, it suffices to verify the following two
statements. First (recalling that $k=n_{0}$), for any integer $j\in\llbracket
1,n+1\rrbracket$, we have

(19.3) $\displaystyle\begin{aligned}
\textbf{{IMP}}_{k}(2A;B;C;k^{3D}&)\cap\textbf{{GAP}}_{n}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}\\\
&\subseteq\textbf{{LOC}}_{j}\big{(}[-Ak^{1/3},Ak^{1/3}];-B_{0}j^{2/3}-B_{0}k^{2/3};B_{0}k^{2/3}-B_{0}^{-1}j^{2/3}\big{)}.\end{aligned}$

Second, for any integers $i\in\big{\llbracket}1,\lceil
4\delta^{-1}\rceil\big{\rrbracket}$ and $j\in\llbracket
n_{i-1},n_{i}\rrbracket$, we have

(19.4)
$\displaystyle\textbf{{IMP}}_{n_{i}}(2A;B;C;n_{i}^{3D})\subseteq\big{\\{}{\mathsf{x}}_{j}(-2Aj^{1/3})\geq-L^{\delta/2}j^{2/3}\big{\\}}\cap\big{\\{}\mathsf{x}_{j}(2Aj^{1/3})\geq-L^{\delta/2}j^{2/3}\big{\\}}.$

For $j\in\llbracket k,n+1\rrbracket$, the inclusion (19.3) follows from
(19.2), the facts that $\llbracket k,n+1\rrbracket\subseteq\llbracket
B^{-1}k,k^{3D+1}\rrbracket$ (as $k^{3D+1}\geq 2k^{3D/2+1}\geq
L^{3/2}k+1=n+1$), that $B_{0}\geq C$, and that
$\textbf{LOC}_{j}(\mathcal{T};b;B)\subseteq\textbf{LOC}_{j}(\mathcal{T}^{\prime};b^{\prime};B^{\prime})$
whenever $\mathcal{T}^{\prime}\subseteq\mathcal{T}$ and $b^{\prime}\leq b\leq
B\leq B^{\prime}$. To confirm this inclusion when $j\in\llbracket
1,k-1\rrbracket$, restrict to the event
$\textbf{{IMP}}_{k}(2A;B;C;k^{3D})\cap\textbf{{GAP}}_{n}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}$.
We must verify that $-B_{0}j^{2/3}-B_{0}k^{2/3}\leq\mathsf{x}_{j}(t)\leq
B_{0}k^{2/3}-B_{0}^{-1}j^{2/3}$ for any $t\in[-Ak^{1/3},Ak^{1/3}]$. To
establish the lower bound on $\mathsf{x}_{j}$, observe for any such $t$ that

(19.5) $\displaystyle{\mathsf{x}}_{j}(t)$
$\displaystyle\geq{\mathsf{x}}_{k}(t)\geq-Ck^{2/3}\geq-
B_{0}j^{2/3}-B_{0}k^{2/3},$

where in the first bound we used the fact that
$\mathsf{x}_{j}\geq\mathsf{x}_{j^{\prime}}$ for $j\leq j^{\prime}$; in the
second we used the fact that we restricted to the $\textbf{LOC}_{k}$ event
contained in the $\textbf{IMP}_{k}$ one by (19.2); and in the third we used
the fact that $B_{0}\geq C$. To establish the upper bound on $\mathsf{x}_{j}$,
observe that

$\displaystyle\mathsf{x}_{j}(t)$
$\displaystyle\leq\big{|}{\mathsf{x}}_{j}(t)-{\mathsf{x}}_{k}(t)\big{|}+\mathsf{x}_{k}(t)\leq
Rk^{2/3}+(\log k)^{25}-C^{-1}k^{2/3}\leq(R+1)k^{2/3}\leq
B_{0}k^{2/3}-B_{0}^{-1}j^{2/3},$

where in the second bound we used our restriction to GAP event (recall
Definition 3.5) and the $\textbf{LOC}_{k}$ event contained in the
$\textbf{IMP}_{k}$ one (by (19.2)); in the third we used the fact that $(\log
k)^{25}\leq k^{2/3}$ for $k\geq 2^{500}$; and in the fourth we used the facts
that $B_{0}\geq R+2$ and $j\leq k$. This and (19.5) together show that the
event
$\textbf{LOC}_{j}\big{(}[-Ak^{1/3},Ak^{1/3}];-B_{0}j^{2/3}-B_{0}k^{2/3};B_{0}k^{2/3}-B_{0}^{-1}j^{2/3}\big{)}$
holds, verifying (19.3).

The inclusion (19.4) follows from the fact that, fixing $j\in\llbracket
n_{i-1},n_{i}\rrbracket$ and restricting to the event
$\textbf{{IMP}}_{n_{i}}(2A;B;C;n_{i}^{3D})$, we have for any
$t\in\\{-2Aj^{1/3},2Aj^{1/3}\\}$ that

$\displaystyle{\mathsf{x}}_{j}(t)\geq{\mathsf{x}}_{n_{i}}(t)\geq-
Cn_{i}^{2/3}\geq-2CL^{\delta/4}n_{i-1}^{2/3}\geq-L^{\delta/2}j^{2/3}.$

Here, in the first bound we used the fact that
$\mathsf{x}_{j}\geq\mathsf{x}_{j^{\prime}}$ for $j\leq j^{\prime}$; in the
second we used our restriction to the $\textbf{LOC}_{n_{i}}$ event contained
in the $\textbf{IMP}_{n_{i}}$ one, by (19.2); in the third we used the fact
that $n_{i}=\lceil L^{3i\delta/8}k\rceil\leq 2L^{3\delta/8}\cdot\lceil
L^{3(i-1)\delta/8}k\rceil=2L^{3\delta/8}n_{i-1}$; and in the fourth we used
the facts that $j\geq n_{i-1}$ and $L\geq(2C)^{4/\delta}$. ∎

Now let us establish 19.2.

###### Proof of 19.2.

Throughout this proof, we recall the medium position event MED from Definition
3.2 and the on-scale event SCL from Definition 3.7; we will assume in the
below that $k\geq 2^{500}$ (as we may by altering the constant $c$ in the
proposition, if necessary). Denote the event
$\mathscr{E}=\mathscr{E}(\vartheta)$ by

(19.6) $\displaystyle\mathscr{E}=\bigcap_{j=0}^{\lfloor
3/\omega\rfloor}\textbf{TOP}\big{(}[-\vartheta^{-1}K_{j}^{1/3},\vartheta^{-1}K_{j}^{1/3}];\vartheta
K_{j}^{2/3}\big{)}\cap\textbf{TOP}\big{(}[-\vartheta^{-1}K_{j}^{30D},\vartheta^{-1}K_{j}^{30D}];\vartheta
K_{j}^{60D}\big{)}.$

Let us briefly outline how we will proceed; we wish to show that
$\textbf{GAP}\cap\textbf{BTR}$ is likely implied by $\mathscr{E}$ for
sufficiently small $\vartheta$. By 19.3, the former is likely implied by the
intersection of several IMP events and a GAP one. Next, by 7.5, each such IMP
event is likely implied by the intersection of several MED ones. By Definition
3.7, the MED and GAP events are implied by SCL; the latter is in turn likely
implied by TOP by 3.8, from which the proposition will follow.

To implement this, observe that $K_{3j}=\lceil L^{3j\delta/8}k\rceil$ for each
integer $j\geq 0$. Hence, it suffices to show for some constants
$c=c(A,D,\delta)>0$, $\vartheta=\vartheta(A,D)>0$, $B=B(A)>1500$, $M=M(A)>2$,
and $R=R(A)>2$ that

(19.7) $\displaystyle\begin{aligned}
&\mathbb{P}\Big{[}\textbf{TOP}\big{(}[-\vartheta^{-1}k^{1/3},\vartheta^{-1}k^{1/3}];\vartheta
k^{2/3}\big{)}\cap\textbf{GAP}_{n}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}^{\complement}\Big{]}\leq
c^{-1}e^{-c(\log k)^{2}};\\\
&\mathbb{P}\big{[}\mathscr{E}\cap\textbf{IMP}_{K_{3j}}(2A;B;M;K_{3j}^{3D})^{\complement}\big{]}\leq
c^{-1}e^{-c(\log k)^{2}},\end{aligned}$

for each integer $j\in\big{\llbracket}1,\lceil
4\delta^{-1}\rceil\big{\rrbracket}$. Indeed, since
$\mathscr{E}\subseteq\textbf{TOP}\big{(}[-\vartheta^{-1}k^{1/3},\vartheta^{-1}k^{1/3}];\vartheta
k^{2/3}\big{)}$ (as $K_{0}=k$), this would imply that

$\displaystyle\mathbb{P}\bigg{[}\mathscr{E}\cap\Big{(}$
$\displaystyle\textbf{BTR}_{n}^{\bm{\mathsf{x}}}(A,M+R+2;k,L;\delta)\cap\textbf{GAP}_{n}^{\bm{\mathsf{x}}}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}\Big{)}^{\complement}\bigg{]}$
$\displaystyle\leq\mathbb{P}\Bigg{[}\mathscr{E}\cap\bigcup_{j=0}^{\lceil
4/\delta\rceil}\textbf{IMP}_{K_{3j}}^{\lceil
4/\delta}(2A;B;M;K_{3j}^{3D})^{\complement}\cup\textbf{GAP}_{n}^{\bm{\mathsf{x}}}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}^{\complement}\Bigg{]}$
$\displaystyle\qquad+\mathbb{P}\Big{[}\mathscr{E}\cap\textbf{GAP}_{n}^{\bm{\mathsf{x}}}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}^{\complement}\Big{]}\leq
8\delta^{-1}c^{-1}e^{-c(\log k)^{2}},$

verifying the proposition (with the $B$ there equal to $M+R+2$ here). In the
first bound above, we used 19.3; in the second, we used (19.7) with a union
bound. Since for any real numbers $m\geq 0$ and
$\vartheta\leq\vartheta^{\prime}$ we have
$\textbf{TOP}\big{(}[-\vartheta^{-1}m,\vartheta^{-1}m];\vartheta
m^{2}\big{)}\subseteq\textbf{TOP}\big{(}[-(\vartheta^{\prime})^{-1}m,(\vartheta^{\prime})^{-1}m];\vartheta^{\prime}m\big{)}$
(by Definition 3.2), it suffices to verify the two bounds in (19.7) separately
(that is, with possibly different values $\vartheta^{\prime}$ and
$\vartheta^{\prime\prime}$ of $\vartheta$, as then we may set
$\vartheta=\min\\{\vartheta^{\prime},\vartheta^{\prime\prime}\\}$).

To confirm the first bound in (19.7), first observe from 3.8 that, for any
real number $B\geq 2$, there exist real numbers $c_{1}=c_{1}(A,B)>0$ and
$\vartheta_{1}=\vartheta_{1}(A,B)>0$ such that

(19.8)
$\displaystyle\mathbb{P}\Big{[}\textbf{TOP}\big{(}[-\vartheta_{1}^{-1}m^{1/3},\vartheta_{1}^{-1}m^{1/3}];\vartheta
m^{2/3}\big{)}\cap\textbf{SCL}_{m}(A;B;10;\vartheta_{1}^{-1})^{\complement}\Big{]}\leq
c_{1}^{-1}e^{-c_{1}(\log m)^{2}},$

for each integer $m\geq 1$. Since
$\textbf{SCL}_{k}(A;10;10;\vartheta_{1}^{-1})\subseteq\textbf{GAP}_{k}\big{(}[-Ak^{1/3},Ak^{1/3}];\vartheta_{1}^{-1}\big{)}$
by Definition 3.7, (19.8) at $(m,B)=(k,10)$ gives the first bound of (19.7)
(with $(\vartheta,R)=(\vartheta_{1},\vartheta_{1}^{-1})$).

To establish the second, fix an integer $j\in\big{\llbracket}1,\lceil
4\delta^{-1}\rceil\big{\rrbracket}$ and a real number $B>1500$; abbreviate
$K=K_{3j}$; and set $b=1/30000$. By 7.5 (with the $(A,D)$ there equal to
$(2A,3D)$ here), there exist constants $c_{2}=c_{2}(A,B,D)>0$,
$C_{1}=C_{1}(B)>1$, and $M=M(A,B)>1$ such that

(19.9) $\displaystyle\mathbb{P}\big{[}$
$\displaystyle\textbf{{IMP}}_{K}(2A;B;M;K^{3D})^{\complement}\cap\mathscr{F}\big{]}\leq
c_{2}^{-1}e^{-c_{2}(\log k)^{2}},$

for any integer $j\geq 0$. Here, we have defined the event
$\mathscr{F}=\mathscr{F}_{j}$ by
$\mathscr{F}=\mathscr{F}^{(1)}\cap\mathscr{F}^{(2)}$, where the events
$\mathscr{F}^{(1)}=\mathscr{F}_{j}^{(1)}$ and
$\mathscr{F}_{2}=\mathscr{F}_{2}^{(j)}$ are given by

$\displaystyle\mathscr{F}^{(1)}$ $\displaystyle=\bigcap_{|t|\leq
2AK^{1/3}}\textbf{{MED}}_{\lfloor
K/4B\rfloor}(t;2bK^{2/3};BK^{2/3})\cap\bigcap_{t\in\\{-C_{1}K^{30D},C_{1}K^{30D}\\}}\textbf{{MED}}_{K^{90D}}(t;BK^{60D})$
$\displaystyle\mathscr{F}^{(2)}$ $\displaystyle=\bigcap_{|t|\leq
2AK^{1/3}}\textbf{TOP}(t;bK^{2/3})\cap\bigcap_{t\in\\{-C_{1}K^{30D},C_{1}K^{30D}\\}}\textbf{{TOP}}(t;BK^{60D}),$

for any $j\geq 0$. Applying (19.8) for $m\in\\{K,K^{90D}\\}$, using the facts
(from Definition 3.7 and the bound $B>1500$) that

$\displaystyle\textbf{SCL}_{K}(A;8B;10;R)\cap\textbf{SCL}_{K^{90D}}(C_{1};10;10;R)\subseteq\mathscr{F}^{(1)},$

and observing that
$\textbf{TOP}\big{(}[-\vartheta^{-1}m^{1/3},\vartheta^{-1}m^{1/3}];\vartheta
m^{2/3}\big{)}\subseteq\textbf{TOP}\big{(}[-Cm^{1/3},Cm^{1/3}];cm^{1/3}\big{)}$
for $\vartheta\leq\min\\{c,C^{-1}\\}$, we deduce the existence of constants
$c_{3}=c_{3}(A,B)>0$ and $\vartheta_{2}=\vartheta_{2}(A,B)>0$ with

$\displaystyle\mathbb{P}\Big{[}\textbf{TOP}\big{(}[-\vartheta_{2}^{-1}K^{1/3},\vartheta_{2}^{-1}K^{1/3}];\vartheta_{2}K^{2/3}\big{)}\cap\textbf{TOP}\big{(}[-\vartheta_{2}^{-1}K^{30D},\vartheta_{2}^{-1}K^{30D}$
$\displaystyle];\vartheta_{2}K^{60D}\big{)}\cap\mathscr{F}^{\complement}\Big{]}$
$\displaystyle\quad\leq c_{3}^{-1}e^{-c_{3}(\log k)^{2}}.$

This, together with (19.9), the fact (from (19.6) and the bound $3j\leq
3\lceil 4\delta^{-1}\rceil\leq 24\delta^{-1}=3\omega^{-1}$) that for
$\vartheta\leq\vartheta_{2}$ we have

$\displaystyle\mathscr{E}\subseteq\textbf{TOP}\big{(}[-\vartheta_{2}^{-1}K^{1/3},\vartheta_{2}^{-1}K^{1/3}];\vartheta_{2}K^{2/3}\big{)}\cap\textbf{TOP}\big{(}[-\vartheta_{2}^{-1}K^{30D},\vartheta_{2}^{-1}K^{30D}];\vartheta_{2}K^{60D}\big{)},$

and a union bound, implies

$\displaystyle\mathbb{P}$
$\displaystyle\big{[}\mathscr{E}\cap\textbf{IMP}_{K}(2A;B;M;K^{3D})^{\complement}\big{]}$
$\displaystyle\leq\mathbb{P}\big{[}\mathscr{E}\cap\mathscr{F}^{\complement}\big{]}+\mathbb{P}\Big{[}\textbf{IMP}_{k}(2A;B;M;K^{3D})^{\complement}\cap\mathscr{F}\big{]}$
$\displaystyle\leq\mathbb{P}\Big{[}\textbf{TOP}\big{(}[-\vartheta_{2}^{-1}K^{1/3},\vartheta_{2}^{-1}K^{1/3}];\vartheta_{2}K^{2/3}\big{)}\cap\textbf{TOP}\big{(}[-\vartheta_{2}^{-1}K^{30D},\vartheta_{2}^{-1}K^{30D}];\vartheta_{2}K^{60D}\big{)}\cap\mathscr{F}^{\complement}\Big{]}$
$\displaystyle\qquad+c_{2}^{-1}e^{-c_{2}(\log k)^{2}}\leq
c_{2}^{-1}e^{-c_{2}(\log k)^{2}}+c_{3}^{-1}e^{-c_{3}(\log k)^{2}},$

which gives the second bound in (19.7) and thus the proposition. ∎

####  19.2. Likelihood of Regular Profile Events

In this section we establish 3.12, which will be a consequence of the below
proposition (together with 19.1 and 19.2). The latter is a general result
stating that, if $\bm{\mathsf{x}}$ is a $\mathbb{Z}_{\geq 1}\times\mathbb{R}$
indexed line ensemble satisfying the Brownian Gibbs property (as fixed at the
beginning of this chapter) for which both a boundary tall rectangle event BTR
and gap event GAP are likely, then $\bm{\mathsf{x}}$ also satisfies a regular
profile event PFL with high probability. In what follows, we recall the events
GAP, PFL, and BTR from Definition 3.5, Definition 3.11, and Definition 16.2,
respectively.

######  Proposition 19.4.

For any real numbers $A,B,R\geq 4$; $D\geq 2^{6000}$; and
$\delta\in(0,D^{-1})$, there exist constants $C_{1}=C_{1}(A,B,R)>1$ and
$C_{2}=C_{2}(A,B,D,R,\delta)>1$ such that the following holds. Let $n\geq
k\geq 1$ be integers and $L\in[k^{2^{6000}},k^{D}]$ be real numbers, such that
$n=L^{3/2}k$. Assume that
$\mathbb{P}\big{[}\textbf{\emph{BTR}}_{n}^{\bm{\mathsf{x}}}(A,B;k,L;\delta)\big{]}\geq
1/2$, and define $\bm{x}=(x_{1},x_{2},\ldots,x_{k})\in\llbracket
1,k\rrbracket\times\mathcal{C}\big{(}[-A/2,A/2]\big{)}$ from
$\bm{\mathsf{x}}\in\mathbb{Z}_{\geq 1}\times\mathcal{C}(\mathbb{R})$ by
setting $x_{j}(s)=k^{-2/3}\cdot\mathsf{x}_{j+k}(sk^{1/3})$ for each
$(j,s)\in\llbracket 1,k\rrbracket\times\big{[}-A/2,A/2\big{]}$. Then,

$\displaystyle\mathbb{P}\Bigg{[}\bigcup_{|t|\leq
A/4}\textbf{\emph{PFL}}^{\bm{x}}\big{(}t;k^{-1}(\log
k)^{6};C_{1}\big{)}^{\complement}$
$\displaystyle\cap\textbf{\emph{BTR}}_{n}^{\bm{\mathsf{x}}}(A,B;k,L;\delta)$
$\displaystyle\cap\textbf{\emph{GAP}}_{n}^{\bm{\mathsf{x}}}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}\Bigg{]}\leq
C_{2}k^{-100}.$

###### Proof of 3.12.

Throughout this proof, we recall the GAP and BTR events from Definition 3.5
and Definition 16.2, respectively. We further fix

$\displaystyle
D=2^{6000};\qquad\delta=2^{-7000};\qquad\omega=\displaystyle\frac{\delta}{8};\qquad
L=n^{D};\qquad N=L^{3/2}n.$

For each integer $j\geq 0$, also set $K_{j}=\lceil L^{j\omega}n\rceil$ and

(19.10) $\displaystyle m_{j}=jD+\omega^{-1};\quad
m_{j}^{\prime}=90m_{j},\qquad\text{so}\qquad K_{j}=\lceil
n^{m_{j}\omega}\rceil;\quad K_{j}^{90D}=\lceil
n^{m_{j}^{\prime}\omega}\rceil.$

By 19.2 (applied with the $(n,k,A)$ there equal to $(N,n,4A)$ here), there
exist constants $c=c(A)>0$, $\vartheta=\vartheta(A)>0$, $B=B(A)>1$, and
$R=R(A)>1$ such that

(19.11) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}&\bigcap_{j=0}^{3/\omega}\Big{(}\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}K_{j}^{1/3},\vartheta^{-1}K_{j}^{1/3}];\vartheta
K_{j}^{2/3}\big{)}\cap\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}K_{j}^{30D},\vartheta^{-1}K_{j}^{30D}];\vartheta
K_{j}^{60D}\big{)}\Big{)}\\\
&\qquad\qquad\cap\Big{(}\textbf{BTR}_{N}^{\bm{\mathcal{L}}}(4A,B;n,L;\delta)\cap\textbf{GAP}_{N}^{\bm{\mathcal{L}}}\big{(}[4An^{1/3},4An^{1/3}];R\big{)}\Big{)}^{\complement}\Bigg{]}\leq
c^{-1}e^{-c(\log n)^{2}}.\end{aligned}$

Moreover, increasing $B$ and $R$ if necessary (and using the facts that,
whenever $B\leq B^{\prime}$ and $R\leq R^{\prime}$, we have
$\textbf{BTR}_{N}^{\bm{\mathcal{L}}}(4A,B;n,L;\delta)\subseteq\textbf{BTR}_{N}^{\bm{\mathcal{L}}}(4A,B^{\prime};n,L;\delta)$
and
$\textbf{GAP}_{N}^{\bm{\mathcal{L}}}\big{(}[-4An^{1/3},4An^{1/3}];R\big{)}\subseteq\textbf{GAP}_{N}^{\bm{\mathcal{L}}}\big{(}[-4An^{1/3},4An^{1/3}];R^{\prime}\big{)}$,
by Definition 16.2 and Definition 3.5), 19.1 yields for sufficiently large $n$
that

$\displaystyle\mathbb{P}\Big{[}\textbf{BTR}_{N}^{\bm{\mathcal{L}}}(4A,B;n,L;\delta)\cap\textbf{GAP}_{N}^{\bm{\mathcal{L}}}\big{(}[-4An^{1/3},4An^{1/3}];R\big{)}\Big{]}\geq\displaystyle\frac{1}{2}.$

Thus, 19.4 (with the $(\bm{\mathsf{x}};k,n;A)$ there given by
$(\bm{\mathcal{L}};n,N;4A)$ here) applies. Together with (19.11) and a union
bound, it gives a constant $C_{0}=C_{0}(A)>1$ such that

(19.12) $\displaystyle\begin{aligned}
\mathbb{P}\Bigg{[}\bigcap_{j=0}^{3/\omega}\Big{(}\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}K_{j}^{1/3},\vartheta^{-1}K_{j}^{1/3}];\vartheta
K_{j}^{2/3}\big{)}&\cap\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}K_{j}^{30D},\vartheta^{-1}K_{j}^{30D}];\vartheta
K_{j}^{60D}\big{)}\Big{)}\\\ &\cap\bigcup_{|t|\leq
An^{1/3}}\textbf{PFL}^{\bm{l}}\big{(}t;n^{-1}(\log
n)^{6};C_{0}\big{)}^{\complement}\Bigg{]}\leq C_{0}n^{-100}.\end{aligned}$

Next, denoting $\vartheta_{0}=\vartheta/2$, we have by (19.10) (and the fact
that
$\textbf{TOP}^{\bm{\mathcal{L}}}(\mathcal{T};B)\subseteq\textbf{TOP}^{\bm{\mathcal{L}}}(\mathcal{T}^{\prime};B^{\prime})$
if $\mathcal{T}^{\prime}\subseteq\mathcal{T}$ and $B\leq B^{\prime}$, by
Definition 3.2), we have for each $j\in\llbracket 0,3\omega^{-1}\rrbracket$
that

$\displaystyle\textbf{TOP}^{\bm{\mathcal{L}}}$
$\displaystyle\big{(}[-\vartheta_{0}^{-1}n^{m_{j}\omega/3},\vartheta_{0}^{-1}n^{m_{j}\omega/3}];\vartheta_{0}n^{2m_{j}/3}\big{)}\cap\textbf{TOP}^{\bm{\mathcal{L}}}\Big{(}[-\vartheta_{0}^{-1}n^{m_{j}^{\prime}\omega/3},\vartheta_{0}^{-1}n^{m_{j}^{\prime}\omega/3}];\vartheta_{0}n^{2m_{j}^{\prime}\omega/3}\big{)}$
$\displaystyle\subseteq\textbf{TOP}^{\bm{\mathcal{L}}}\Big{(}\big{[}-\vartheta^{-1}\lceil
n^{m_{j}\omega}\rceil^{1/3},\vartheta^{-1}\lceil
n^{m_{j}\omega_{j}}\rceil^{1/3}\big{]};\vartheta\lceil
n^{m_{j}\omega}\rceil^{2/3}\Big{)}$
$\displaystyle\qquad\cap\textbf{TOP}^{\bm{\mathcal{L}}}\Big{(}\big{[}-\vartheta^{-1}\lceil
n^{m_{j}^{\prime}\omega}\rceil^{1/3},\vartheta^{-1}\lceil
n^{m_{j}^{\prime}\omega}\rceil^{1/3}\big{]};\vartheta\lceil
n^{m_{j}^{\prime}\omega}\rceil^{2/3}\Big{)}$
$\displaystyle=\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}K_{j}^{1/3},\vartheta^{-1}K_{j}^{1/3}];\vartheta
K_{j}^{2/3}\big{)}\cap\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}K_{j}^{30D},\vartheta^{-1}K_{j}^{30D}];\vartheta
K_{j}^{60D}\big{)}.$

Since $1\leq m_{j}\leq m_{j}^{\prime}\leq 270\omega^{-1}(D+1)\leq\omega^{-2}$
for each $j\in\llbracket 1,3\omega^{-1}\rrbracket$ (as $\omega D\leq\delta
D=2^{-1000}$), this implies

$\displaystyle\bigcap_{j=1}^{1/\omega^{2}}$
$\displaystyle\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta_{0}^{-1}n^{j\omega},\vartheta_{0}^{-1}n^{j\omega}];\vartheta_{0}n^{2j\omega/3}\big{)}$
$\displaystyle\subseteq\bigcap_{j=0}^{3/\omega}\Big{(}\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}K_{j}^{1/3},\vartheta^{-1}K_{j}^{1/3}];\vartheta
K_{j}^{2/3}\big{)}\cap\textbf{TOP}^{\bm{\mathcal{L}}}\big{(}[-\vartheta^{-1}K_{j}^{30D},\vartheta^{-1}K_{j}^{30D}];\vartheta
K_{j}^{60D}\big{)}\Big{)}.$

This, together with (19.12), yields the theorem (at
$(c,C)=(\vartheta_{0},C_{0}+\vartheta_{0}^{-1})$, using the facts that
$\textbf{PFL}^{\bm{\mathcal{L}}}(t;\delta;C_{0})\subseteq\textbf{PFL}^{\bm{\mathcal{L}}}(t;\delta;C)$
for $C_{0}\leq C$ and
$\textbf{TOP}^{\bm{\mathcal{L}}}(\mathcal{T};B)\subseteq\textbf{TOP}^{\bm{\mathcal{L}}}(\mathcal{T}^{\prime};B)$
for $\mathcal{T}\subseteq\mathcal{T}^{\prime}$, by Definition 3.2 and
Definition 3.11). ∎

The proof of 19.5 will be a quick consequence of the following proposition
(together with a high-probability Hölder bound on the paths in
$\bm{\mathsf{x}}$, guaranteed by 18.1), to be established in Section 19.3
below. Instead of showing that $\textbf{PFL}^{\bm{x}}$ holds for all
$t\in[-A/4,A/4]$ simultaneously, it shows this statement for a fixed time
$t\in[-A/4,A/4]$.

######  Proposition 19.5.

Adopt the notation and assumptions in 19.4. For any real number
$t\in[-A/4,A/4]$, we have

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

###### Proof of 19.4.

Throughout this proof, we abbreviate the events
$\textbf{BTR}_{n}=\textbf{BTR}_{n}^{\bm{\mathsf{x}}}(A,B;k,L;\delta)$ and
$\textbf{GAP}_{n}=\textbf{GAP}^{\bm{\mathsf{x}}}_{n}\big{(}[-Ak^{1/3},Ak^{1/3}];R\big{)}$.
The proposition will follow from applying 19.5 over $t$ in a $k^{-10}$-mesh of
the interval $[-Ak^{1/3}/4,Ak^{1/3}/4]$, and using a Hölder type bound for the
paths in $\bm{\mathsf{x}}$ guaranteed by 18.1. More specifically, define the
$k^{-10}$-mesh $\mathcal{S}=[-A/4,A/4]\cap(k^{-10}\cdot\mathbb{Z})$, which
satisfies $|\mathcal{S}|\leq Ak^{10}$, and let $C_{1}=C_{1}(A,B,R)>1$ denote
the constant $C_{1}$ from 19.5. Then, define event
$\mathscr{E}=\mathscr{E}_{1}\cap\mathscr{E}_{2}$, where

$\displaystyle\mathscr{E}_{1}=\bigcap_{s\in\mathcal{S}}\textbf{PFL}^{\bm{x}}\Big{(}t;\displaystyle\frac{(\log
k)^{6}}{2k};C_{1}\Big{)};\quad\mathscr{E}_{2}=\bigcap_{j=1}^{2k}\bigcup_{\begin{subarray}{c}|s|\leq
Ak^{1/3}/4\\\ |s+t|\leq
Ak^{1/3}/4\end{subarray}}\Big{\\{}\big{|}\mathsf{x}_{j}(s+t)-\mathsf{x}_{j}(t)\big{|}\leq
10Ak^{2}t^{1/2}+k^{-D}\Big{\\}}.$

Observe that there exist constants $c=c(A,B)>0$ and
$C_{2}=C_{2}(A,B,D,\delta)>0$ such that

(19.13)
$\displaystyle\mathbb{P}\big{[}\mathscr{E}_{1}^{\complement}\cap\textbf{BTR}_{n}\cap\textbf{GAP}_{n}\big{]}\leq
C_{2}k^{-150};\qquad\mathbb{P}\big{[}\mathscr{E}_{2}^{\complement}\cap\textbf{BTR}_{n}\big{]}\leq
C_{2}e^{-c(\log k)^{2}}.$

Indeed, the first follows from taking a union bound in 19.5 over
$t\in\mathcal{S}$. The second follows from the $A^{\prime}=A/4$ case of 18.1,
using the definition (16.6) of the event FHR; and the facts that
$L^{\delta}k^{1/3}(j\vee k)^{1/3}t\leq AL^{\delta}(j\vee k)t^{1/2}\leq
2Ak^{2}t^{1/2}$ (as $L^{\delta}\leq L^{1/D}\leq k$ for $\delta\in(0,D^{-1})$)
and $4k^{1/6}(j\vee k)^{1/2}t^{1/2}\leq 4(j\vee k)t^{1/2}\leq 8Ak^{2}t^{1/2}$
for $|t|\leq Ak^{1/3}/4$ and $j\in\llbracket 1,2k\rrbracket$.

By (19.13) and a union bound, it suffices to show for sufficiently large $k$
that we have the inclusion

(19.14) $\displaystyle\mathscr{E}\subseteq\bigcap_{|t|\leq
A/4}\textbf{PFL}^{\bm{x}}\big{(}t;k^{-1}(\log k)^{6};C_{1}\big{)}.$

To this end, restrict to the event $\mathscr{E}$ and fix a real number
$t_{0}\in[-A/4,A/4]$; it suffices to show that
$\textbf{PFL}^{\bm{x}}\big{(}t_{0};k^{-1}(\log k)^{6};C_{1}\big{)}$ holds. Fix
an arbitrary element $s\in\mathcal{S}$ such that $|s-t_{0}|\leq k^{-10}$.
Since we have restricted to the event $\mathscr{E}\subseteq\mathscr{E}_{1}$,
Definition 3.11 for the PFL event yields a function
$\gamma_{s}:[0,1]\rightarrow\mathbb{R}$ such that
$\big{|}x_{j}(s)-\gamma_{s}(jk^{-1})\big{|}\leq(2k)^{-1}(\log k)^{6}$ for each
$j\in\llbracket 1,k\rrbracket$ and
$\big{\|}\gamma_{s}-\gamma_{s}(0)\big{\|}_{\mathcal{C}^{50}}\leq C_{1}$. Set
$\gamma_{t_{0}}=\gamma_{s}$, which satisfies
$\big{|}\gamma_{t_{0}}-\gamma_{t_{0}}(0)\big{\|}\leq C_{1}$ since $\gamma_{s}$
does.

Moreover, for any integer $j\in\llbracket 1,k\rrbracket$, we have for
sufficiently large $k$ that

$\displaystyle\big{|}x_{j}(t_{0})-\gamma_{t_{0}}(jk^{-1})\big{|}$
$\displaystyle\leq\big{|}k^{-2/3}\cdot\mathsf{x}_{j+k}(sk^{1/3})-\gamma_{s}(jk^{-1})\big{|}+k^{-2/3}\cdot\big{|}\mathsf{x}_{j+k}(t_{0}k^{1/3})-\mathsf{x}_{j+k}(sk^{1/3})\big{|}$
$\displaystyle\leq\big{|}x_{j}(s)-\gamma_{s}(jk^{-1})\big{|}+10Ak^{2}|s-t_{0}|^{1/2}+k^{-D}$
$\displaystyle\leq(2k)^{-1}(\log k)^{6}+11Ak^{-3}\leq k^{-1}(\log k)^{6},$

where in the first statement we used the facts that
$x_{j}(t_{0})=k^{-2/3}\cdot\mathsf{x}_{j+k}(t_{0}k^{1/3})$ and
$\gamma_{s}=\gamma_{t_{0}}$; in the second we again used the fact that
$x_{j}(s)=k^{-2/3}\cdot\mathsf{x}_{j+k}(sk^{1/3})$ and also our restriction to
the event $\mathscr{E}\subseteq\mathscr{E}_{2}$; in the third we used the
facts that $|s-t_{0}|\leq k^{-10}$ and that
$\big{|}x_{j}(s)-\gamma_{s}(jk^{-1})\big{|}\leq(2k)^{-1}(\log k)^{6}$ (and
that $D\geq 2^{6000}$); and in the fourth we used the fact that $k$ is
sufficiently large.

This verifies that $\gamma_{t_{0}}$ satisfies the first bound in (3.7). Thus,
$\textbf{PFL}^{\bm{x}}\big{(}t_{0};k^{-1}(\log k)^{6};C_{1}\big{)}$ holds;
since $t_{0}\in[-A/4,A/4]$ was arbitrary, this verifies (19.14) and
establishes the proposition. ∎

####  19.3. Proof of 19.5

In this section we establish 19.5, guaranteeing that $\bm{x}$ likely satisfies
a regular profile event. The third part of 15.2 provides a way of ensuring
that families of non-intersecting Brownian bridges without lower boundary
satisfy these events. We therefore require a way of coupling $\bm{\mathsf{x}}$
with such a family $\bm{\mathsf{y}}$, in such a way that their upper paths are
close to each other; 16.4 does not quite do this, since the two couplings it
provides are not necessarily the same. The following lemma indicates that if
$L$ is sufficiently large with respect to $k$ (namely, $L\geq k^{2^{6000}}$),
then there exists a coupling between $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$
guaranteeing that their top $k^{2}$ paths are likely close.

######  Lemma 19.6.

Adopt the notation and assumptions of 16.4, and assume that $L\geq
k^{2^{6000}}$. For any real number $\mathsf{t}\in[-Ak^{1/3}/2,Ak^{1/3}/2]$,
there exists a coupling between $\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$ such
that

(19.15)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1}^{k^{2}}\Big{\\{}\big{|}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\big{|}\leq
k^{-2}\Big{\\}}\Bigg{]}\geq 1-Ck^{-200}.$

To prove 19.6, we will apply a Markov estimate to the quantity
$\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})$. This will require a
(weak) tail bound on the latter random variable, which is provided by the
following lemma, to be established in Section 19.4 below.

######  Lemma 19.7.

Adopting 16.3, there exist constants $c=c(A,B)>0$ and $C=C(A,B,D)>1$ such that
the following holds. Set $n^{\prime}=\big{\lceil}L^{1/2^{4000}}k\big{\rceil}$.
For any real number $\mathsf{t}\in[-Ak^{1/3}/2,Ak^{1/3}/2]$, there is an event
$\mathscr{A}=\mathscr{A}_{\mathsf{t}}\subseteq\textbf{\emph{BTR}}_{n}^{\bm{\mathsf{x}}}(A;B)$,
that is 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
$\mathbb{P}\big{[}\textbf{\emph{BTR}}_{n}^{\bm{\mathsf{x}}}(A;B)\setminus\mathscr{A}\big{]}\leq
c^{-1}e^{-k}$ and the following. Condition on
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}\rrbracket\times\big{(}[Ak^{1/3}/2,Ak^{1/3}/2]\big{)}$; restrict
to $\mathscr{A}$; and define the $n^{\prime}$-tuples
$\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}\rrbracket}(-Ak^{1/3}/2)\in\overline{\mathbb{W}}_{n^{\prime}}$ and
$\bm{v}=\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}\rrbracket}(Ak^{1/3}/2)\in\overline{\mathbb{W}}_{n^{\prime}}$.
Sample $n^{\prime}$ non-intersecting Brownian bridges
$\bm{\mathsf{y}}=(y_{1},y_{2},\ldots,y_{n^{\prime}})\in\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3}/2,Ak^{1/3}/2]$ from the measure
$\mathsf{Q}^{\bm{u};\bm{v}}$. Then, under any coupling between
$\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$, we have

(19.16) $\displaystyle\mathbb{P}\bigg{[}\displaystyle\max_{j\in\llbracket
1,n^{\prime}\rrbracket}\big{|}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\big{|}\leq
c^{-1}ik^{4D}\bigg{]}\leq Ce^{-ik},\qquad\text{for every integer $i\geq 0$}.$

###### Proof of 19.6.

Throughout this proof, we abbreviate
$\textbf{BTR}_{n}=\textbf{BTR}_{n}^{\bm{\mathsf{x}}}(A;B)$. By 19.7 and 16.4,
we deduce the existence of constants $c=c(A,B)\in(0,1)$ and
$C_{1}=C_{1}(A,B,D,\delta)>1$, and events
$\mathscr{A}^{\prime},\mathscr{A}^{\prime\prime}\subseteq\textbf{BTR}_{n}$,
both measurable with respect to
$\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3}/2,Ak^{1/3}/2]\big{)}$ (recalling
$n^{\prime}=\lceil L^{1/2^{4000}}k\rceil$, as we have adopted the notation of
16.4), satisfying the following three properties. First, we have

(19.17)
$\displaystyle\mathbb{P}[\textbf{BTR}_{n}\setminus\mathscr{A}^{\prime}]\leq
Ce^{-k};\qquad\mathbb{P}\big{[}\textbf{BTR}_{n}\setminus\mathscr{A}^{\prime\prime}]\leq
Ce^{-c(\log k)^{2}}.$

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 $\mathscr{A}^{\prime}$, we have for any coupling between $\bm{\mathsf{x}}$
and $\bm{\mathsf{y}}$ that

(19.18) $\displaystyle\mathbb{P}\bigg{[}\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\big{|}\geq
c^{-1}ik^{4D}\bigg{]}\leq C_{1}e^{-ik},\qquad\text{for each integer $i\geq
0$}.$

Third, again 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 now
restricting to $\mathscr{A}^{\prime\prime}$, there exist two coupling between
$\bm{\mathsf{x}}$ and $\bm{\mathsf{y}}$. Under the first, we have
$\mathsf{x}_{j}(s)\leq\mathsf{y}_{j}(s)$ for each $(j,s)\in\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3}/2,Ak^{1/3}/2]$, almost surely. Under
the second, we have (recalling $n^{\prime\prime}=\lceil
L^{1/2^{5000}}k\rceil$)

(19.19) $\displaystyle\mathbb{P}\big{[}\mathscr{E}^{\complement}\big{]}\leq
C_{1}e^{-c(\log
k)^{2}},\qquad\text{where}\qquad\mathscr{E}=\bigcap_{j=1}^{n^{\prime\prime}}\big{\\{}\mathsf{y}_{j}(\mathsf{t})\geq\mathsf{x}_{j}(\mathsf{t})-L^{-1/2^{5000}}k^{2/3}\big{\\}}.$

Set $\mathscr{A}=\mathscr{A}^{\prime}\cap\mathscr{A}^{\prime\prime}$, which by
(19.17) and a union bound satisfies
$\mathbb{P}[\textbf{BTR}\setminus\mathscr{A}]\geq
1-C_{1}e^{-k}-C_{1}e^{-c(\log k)^{2}}\geq 1-2C_{1}e^{-c(\log k)^{2}}$. As in
the statement of the lemma (see also that of 16.4), condition 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 restrict to
$\mathscr{A}$. We will exhibit a coupling between $\bm{\mathsf{x}}$ and
$\bm{\mathsf{y}}$ such that (19.15) holds. This will proceed by using a Markov
estimate.

In particular, we claim for sufficiently large $k$ that

(19.20)
$\displaystyle\mathbb{E}\big{[}\mathsf{x}_{j}(\mathsf{t})\big{]}-\mathbb{E}\big{[}\mathsf{y}_{j}(\mathsf{t})\big{]}\leq
k^{-250},\qquad\text{for each integer $j\in\llbracket 1,k^{2}\rrbracket$}.$

Let us establish the lemma assuming (19.20). Since we have restricted to
$\mathscr{A}$, there exists a coupling between $\bm{\mathsf{x}}$ and
$\bm{\mathsf{y}}$ such that $\mathsf{x}_{j}(s)\leq\mathsf{y}_{j}(s)$ for each
$(j,s)\in\llbracket 1,n^{\prime}\rrbracket\times[-Ak^{1/3}/2,Ak^{1/3}/2]$,
almost surely. Hence, under this coupling, we have for sufficiently large $k$
and any integer $j\in\llbracket 1,k^{2}\rrbracket$ that

$\displaystyle\mathbb{P}\Big{[}\big{|}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\big{|}\geq
k^{-2}\Big{]}=\mathbb{P}\big{[}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\geq
k^{-2}\big{]}\leq
k^{2}\cdot\mathbb{E}\big{[}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\big{]}\leq
k^{-210},$

where in the first statement we used the fact that
$\mathsf{x}_{j}\geq\mathsf{y}_{j}$; in the second we used a Markov bound; and
in the third we used (19.20). Taking a union bound over all $j\in\llbracket
1,k^{2}\rrbracket$ then yields the lemma.

It therefore remains to establish (19.20); in what follows, we fix an integer
$j\in\llbracket 1,k^{2}\rrbracket$. Since $L\geq k^{2^{6000}}$, we have
$L^{-1/2^{5000}}\leq k^{-2^{1000}}\leq k^{-300}$; in particular,
$n^{\prime\prime}\geq L^{1/2^{5000}}k\geq k^{300}\geq k^{2}$, so
$j\in\llbracket 1,n^{\prime\prime}\rrbracket$. Hence, for sufficiently large
$k$, (19.19) yields

(19.21)
$\displaystyle\mathbb{E}\Big{[}\textbf{1}_{\mathscr{E}}\cdot\big{(}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\big{)}\Big{]}\leq
L^{-1/2^{5000}}k\leq C_{1}k^{-299}\leq k^{-298}.$

It thus remains to bound the expectation of
$\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})$ off of $\mathscr{E}$,
which will make use of the tail bound (19.18). In particular, observe for
$c^{\prime}=c/2$ and $k$ sufficiently large that

$\displaystyle\mathbb{E}$
$\displaystyle\Big{[}\textbf{1}_{\mathscr{E}^{\complement}}\cdot\big{(}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\big{)}\Big{]}$
$\displaystyle\leq\mathbb{E}\big{[}\textbf{1}_{\mathscr{E}^{\complement}}\big{]}^{1/2}\cdot\mathbb{E}\Big{[}\big{|}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\big{|}^{2}\Big{]}^{1/2}$
$\displaystyle\leq C_{1}^{1/2}e^{-c^{\prime}(\log
k)^{2}}\cdot\Bigg{(}\displaystyle\sum_{i=0}^{\infty}c^{-2}(i+1)^{2}k^{8D}\cdot\mathbb{P}\Big{[}\big{|}\mathsf{x}_{j}(\mathsf{t})-\mathsf{y}_{j}(\mathsf{t})\big{|}\in\big{[}c^{-1}ik^{4D},c^{-1}(i+1)k^{4D}\big{]}\Big{]}\Bigg{)}^{1/2}$
$\displaystyle\leq c^{-1}C_{1}k^{4D}e^{-c^{\prime}(\log
k)^{2}}\cdot\bigg{(}\displaystyle\sum_{i=0}^{\infty}e^{-ik}(i+1)^{2}\bigg{)}^{1/2}\leq
3c^{-1}C_{1}k^{4D}e^{-c^{\prime}(\log k)^{2}}\leq k^{-298},$

where in the second inequality we used (19.19); in the third we used (19.18);
in the fourth we used the bound
$\sum_{i=0}^{\infty}e^{-ik}(i+1)^{2}\leq\sum_{i=0}^{\infty}e^{-i}(i+1)^{2}<9$;
and in the fifth we used the fact that $k$ is sufficiently large. This,
together with (19.21) and the fact that $2k^{-298}\leq k^{-250}$ confirms
(19.20). ∎

Now we can establish 19.5.

###### Proof of 19.5.

Recall the location event LOC from Definition 16.1 and the complete rectangle
event CTR from Definition 16.5. Throughout this proof, we define
$n^{\prime}=\lceil L^{1/4000}k\rceil$ and $\widetilde{B}=12A^{2}B^{3}$;
abbreviate the events
$\textbf{BTR}_{n}=\textbf{BTR}_{n}^{\bm{\mathsf{x}}}(A,B;k,L;\delta)$ and
$\textbf{CTR}_{n}=\textbf{CTR}_{n}^{\bm{\mathsf{x}}}(A,\widetilde{B};k,L;\delta)$;
abbreviate the $\sigma$-algebra
$\mathcal{F}_{\operatorname{ext}}=\mathcal{F}_{\operatorname{ext}}^{\bm{\mathsf{x}}}\big{(}\llbracket
1,n^{\prime}\rrbracket\times[-Ak^{1/3}/2,Ak^{1/3}/2]\big{)}$; and define the
$n^{\prime}$-tuples $\bm{u}=\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}\rrbracket}(-Ak^{1/3}/2)\in\overline{\mathbb{W}}_{n}$ and
$\bm{v}=\bm{\mathsf{x}}_{\llbracket
1,n^{\prime}\rrbracket}(Ak^{1/3}/2)\in\overline{\mathbb{W}}_{n}$. Further
sample $n^{\prime}$ non-intersecting Brownian bridges
$\bm{\mathsf{y}}=(\mathsf{y}_{1},\mathsf{y}_{2},\ldots,\mathsf{y}_{n^{\prime}})\in\llbracket
1,n^{\prime}\rrbracket\times\mathcal{C}\big{(}[-Ak^{1/3}/2,Ak^{1/3}/2]\big{)}$
from the measure $\mathsf{Q}^{\bm{u};\bm{v}}$.

Define the $\mathcal{F}_{\operatorname{ext}}$-measurable event

(19.22)
$\displaystyle\mathscr{A}_{1}=\textbf{BTR}_{n}\cap\bigcap_{j=1}^{n^{\prime}}\textbf{LOC}_{j}^{\bm{\mathsf{x}}}\Bigg{(}\Big{\\{}-\frac{Ak^{1/3}}{2},\displaystyle\frac{Ak^{1/3}}{2}\Big{\\}};-\widetilde{B}j^{2/3}-\widetilde{B}k^{2/3};\widetilde{B}k^{2/3}-\widetilde{B}^{-1}j^{2/3}\bigg{)}.$

Since $\textbf{BTR}_{n}\cap\textbf{CTR}_{n}\subseteq\mathscr{A}_{1}$ by
Definition 16.5, 16.6 yields constants $c_{1}=c_{1}(A,B)>0$ and
$C_{3}=C_{3}(A,B,D)>1$ such that
$\mathbb{P}[\textbf{BTR}_{n}\setminus\mathscr{A}_{1}]\leq C_{3}e^{-c_{1}(\log