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

######  Lemma 10.8 ([8, Lemma 3.23(1)]).

Fix real numbers $a<b$ and $A>0$, and compactly supported measures
$\mu_{a},\mu_{b}\in\mathscr{P}_{\operatorname{fin}}$, satisfying
$\mu_{a}(\mathbb{R})=A=\mu_{b}(\mathbb{R})$. Let
$\bm{\mu}^{\star}=(\mu_{t}^{\star})_{t\in[a,b]}\in\mathcal{C}\big{(}[a,b];\mathscr{P}_{\operatorname{fin}}\big{)}$
denote the bridge-limiting measure process on $[a,b]$ with boundary data
$(\mu_{a};\mu_{b})$. Further let
$H^{\star}:[a,b]\times\mathbb{R}\rightarrow\mathbb{R}$ and
$G^{\star}:[a,b]\times[0,A]\rightarrow\mathbb{R}$ denote the associated height
and inverted height functions, respectively. Then, $H^{\star}(t,x)$ is smooth
for $(t,x)\in\Omega$ and $G^{\star}(t,y)$ is smooth for
$(t,y)\in\Omega^{\operatorname{inv}}$.

We next define a complex slope associated with a limit shape; its imaginary
part is given by the associated density $\varrho^{\star}$, and its real part
is given by the $t$-derivative of the inverted height function, which we
denote by $u^{\star}:\Omega\rightarrow\mathbb{R}$.

######  Definition 10.9.

Adopt the notation and assumptions of 10.8. Define the function
$u^{\star}:\Omega\rightarrow\mathbb{R}$ as follows. For any point
$(t,x)\in\Omega$, let $(t,y)\in\Omega^{\operatorname{inv}}$ be the unique
point such that $G^{\star}(t,y)=x$ (which is guaranteed to exist since the map
$(t,y)\mapsto(t,G^{\star}(t,y))$ is a bijection from
$\Omega^{\operatorname{inv}}$ to $\Omega$). Then, define
$u^{\star}(t,x)=u_{t}^{\star}(x)$ by setting

(10.6) $\displaystyle u_{t}^{\star}(x)=\partial_{t}G^{\star}(t,y),$

and observe that

(10.7)
$\displaystyle\varrho_{t}^{\star}(x)=\varrho_{t}^{\star}\big{(}G(t,y)\big{)}=-\partial_{x}H^{\star}(t,x)=-\big{(}\partial_{y}G^{\star}(t,y)\big{)}^{-1},$

where the last equality holds by 10.8 and the fact from (10.4) that
$H^{\star}\big{(}t,G^{\star}(t,y)\big{)}=y$ (see also [8, Remark 3.15]).
Further define associated _complex slope_
$f=f^{(\mu_{a};\mu_{b})}:\Omega\rightarrow\overline{\mathbb{H}}$ by setting

(10.8) $\displaystyle
f(t,x)=u_{t}^{\star}(x)+\pi\mathrm{i}\varrho_{t}^{\star}(x),\qquad\text{for
each $(t,x)\in\Omega$}.$

The following lemma from [18] (implicitly due to the earlier work [66]; see
also [8, Remark 3.15 and Lemma 3.23(2)]) indicates that this function $f$
satisfies a complex variant of the Burgers equation.

######  Lemma 10.10 ([18, Lemma 3.23(2)]).

Adopting the notation and assumptions of 10.8, the associated complex slope
$f$ satisfies the complex Burgers equation,

(10.9) $\partial_{t}f(x,t)+f(x,t)\cdot\partial_{x}f(x,t)=0,\qquad\text{for all
$(t,x)\in\Omega$}.$

####  10.2. Examples of Bridge-Limiting Measure Processes

In this section we describe several examples of the bridge-limiting measure
processes from Section 10.1. The first concerns the case when $\mu_{0}$ and
$\mu_{1}$ are delta measures, in which the associated non-intersecting
Brownian bridges form a Brownian watermelon (recall Section 4.8); see the left
side of Figure 3.6 for a depiction.

Figure 3.6. Shown on the left is a depiction for Example 10.11 at
$(a,b,u,v,A)=(0,6,0,0,2)$. Shown on the right is a depiction for Example 10.12
at $(a,b,d,A)=(0,10,2,2)$. In both, the entire shaded region is the associated
liquid region $\Omega$.

######  Example 10.11.

Fix real numbers $a<b$; $u,v\in\mathbb{R}$; and $A>0$. Assume that
$(\mu_{a},\mu_{b})=(A\cdot\delta_{u},A\cdot\delta_{v})$, where
$\delta_{x}\in\mathscr{P}_{0}$ denotes the delta measure at $x\in\mathbb{R}$.
Then, it follows from 4.32 (multiplying its results by $(An^{-1})^{1/2}$ to
account for the fact that the Brownian motions have variance $An^{-1}$ here)
and the second statement of 10.7 (and also the continuity of
$\gamma_{\operatorname{sc}}(y)$ below in $y$) that the inverted height
function $G^{\star}:[0,1]\times[0,A]\rightarrow\mathbb{R}$ associated with
boundary data $(\mu_{0};\mu_{1})$ is given by

$\displaystyle
G^{\star}(t,y)=\bigg{(}\displaystyle\frac{A(b-t)(t-a)}{b-a}\bigg{)}^{1/2}\cdot\gamma_{\operatorname{sc}}\Big{(}\displaystyle\frac{y}{A}\Big{)}+\displaystyle\frac{b-t}{b-a}\cdot
u+\displaystyle\frac{t-a}{b-a}\cdot v,$

where $\gamma_{\operatorname{sc}}(y)$ is the classical location of the
semicircle distribution given by (4.23). Together with (10.3) and (10.4), it
follows that the associated density process $(\varrho_{t}^{\star})$ and the
height function $H^{\star}:[0,1]\times\mathbb{R}$ are given by

$\displaystyle\varrho_{t}^{\star}(x)=A\cdot\varrho_{\operatorname{sc}}^{(\frac{A(b-t)(t-a)}{(b-a)})}\bigg{(}x-\displaystyle\frac{b-t}{b-a}\cdot
u-\displaystyle\frac{t-a}{b-a}\cdot v\bigg{)},$

and $H^{\star}(t,x)=\int_{x}^{\infty}\varrho_{t}^{\star}(y)dy$, where we
recall the rescaled semcircle density $\varrho_{\operatorname{sc}}^{(t)}$ from
(4.6).

The second example from [8] concerns the case when $\mu_{a}$ and $\mu_{b}$ are
rescaled semicircle distributions (recall (4.6)), which can be obtained by
restricting a watermelon to a smaller time interval; see the right side of
Figure 3.6 for a depiction.

######  Example 10.12 ([8, Corollary 3.10]).

Fix real numbers $a<b$ and $d,A>0$; assume that
$\mu_{a}=A\cdot\mu_{\operatorname{sc}}^{(d)}=\mu_{b}$. Then, the inverted
height function $G^{\star}:[a,b]\times[0,A]\rightarrow\mathbb{R}$ and density
process $(\varrho_{t}^{\star})$ associated with boundary data
$(\mu_{a};\mu_{b})$ are given by

(10.10)
$\displaystyle\varrho_{t}^{\star}(x)=A\cdot\varrho_{\operatorname{sc}}^{(d+\frac{A(b-t)(t-a)}{b-a+2\kappa})}(x);\qquad
G^{\star}(t,y)=\bigg{(}d+\displaystyle\frac{A(b-t)(t-a)}{b-a+2\kappa}\bigg{)}^{1/2}\cdot\gamma_{\operatorname{sc}}\Big{(}\displaystyle\frac{y}{A}\Big{)},$

where $\kappa=\kappa(a,b,d)>0$ is defined by

(10.11)
$\displaystyle\kappa=\displaystyle\frac{d}{A}+\displaystyle\frac{a-b}{2}+\bigg{(}\Big{(}\displaystyle\frac{b-a}{2}\Big{)}^{2}+\Big{(}\displaystyle\frac{d}{A}\Big{)}^{2}\bigg{)}^{1/2}.$

######  Remark 10.13.

Let us consider the limiting profile associated with affine shifts of the
parabolic Airy line ensemble $\bm{\mathcal{R}}$. Fix real numbers
$\mathfrak{a},\mathfrak{b},\mathfrak{c}$ with $\mathfrak{c}>0$, and set
$\sigma=2^{1/6}\mathfrak{c}^{1/3}$. For any integer $n\geq 1$, define the
affine shift
$\bm{\mathcal{R}}^{(\sigma;\mathfrak{a},\mathfrak{b};n)}=\big{(}\mathcal{R}_{1}^{(\sigma;\mathfrak{a},\mathfrak{b};n)},\mathcal{R}_{2}^{(\sigma;\mathfrak{a},\mathfrak{b};n)},\ldots\big{)}\in\mathbb{Z}_{\geq
1}\times\mathcal{C}(\mathbb{R})$ of the rescaled parabolic Airy line ensemble
$\bm{\mathcal{R}}^{(\sigma)}$ (recall from (2.3)), by for each
$(j,t)\in\mathbb{Z}_{\geq 1}\times\mathbb{R}$ setting

$\displaystyle\mathcal{R}_{j}^{(\sigma;a,b;n)}(t)=2^{-1/2}\cdot\mathcal{R}_{1}^{(\sigma)}(t)+\mathfrak{a}n^{2/3}+\mathfrak{b}n^{1/3}t.$

Observe from 2.10 and 4.3 that
$\bm{\mathcal{R}}^{(\sigma;\mathfrak{a},\mathfrak{b};n)}$ satisfies the
Brownian Gibbs property. Define the _limiting Airy profile_ to be the function
$\mathfrak{G}_{\operatorname{Ai}}=\mathfrak{G}_{\operatorname{Ai};\mathfrak{a},\mathfrak{b},\mathfrak{c}}:\mathbb{R}\times\mathbb{R}_{\geq
0}\rightarrow\mathbb{R}$ by setting

(10.12)
$\displaystyle\mathfrak{G}_{\operatorname{Ai}}(t,y)={\mathfrak{a}}+{\mathfrak{b}}t-{\mathfrak{c}}t^{2}-\left(\frac{3\pi}{4{\mathfrak{c}}^{1/2}}\right)^{2/3}y^{2/3},$

for each $(t,y)\in\mathbb{R}\times\mathbb{R}_{\geq 0}$. By 4.34, a union
bound, and the definition $\sigma=2^{1/6}\mathfrak{c}^{1/3}$, we have for any
real numbers $a<b$ and $\varepsilon>0$ that

$\displaystyle\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\Bigg{[}\displaystyle\sup_{t\in[a,b]}\displaystyle\sup_{y\in[0,1]}\Big{|}n^{-2/3}\cdot\mathcal{R}_{\lfloor
yn\rfloor}^{(\sigma;\mathfrak{a},\mathfrak{b};n)}(tn^{1/3})-\mathfrak{G}_{\operatorname{Ai}}(t,y)\Big{|}\leq\varepsilon\Bigg{]}=1.$

Define the process
$\bm{\mu}_{\operatorname{Ai}}=\bm{\mu}_{\operatorname{Ai};\mathfrak{a},\mathfrak{b},\mathfrak{c}}=(\mu_{t})=(\mu_{t}^{\operatorname{Ai};\mathfrak{a},\mathfrak{b},\mathfrak{c}})$
(over $t\in\mathbb{R})$; the density process
$(\varrho_{t})=(\varrho_{t}^{\operatorname{Ai};\mathfrak{a},\mathfrak{b},\mathfrak{c}})$;
and the function
$\mathfrak{H}_{\operatorname{Ai}}=\mathfrak{H}_{\operatorname{Ai};\mathfrak{a},\mathfrak{b},\mathfrak{c}}:\mathbb{R}^{2}\rightarrow\mathbb{R}$
by setting

$\displaystyle\varrho_{t}(x)=\displaystyle\frac{2\mathfrak{c}^{1/2}}{\pi}(\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}t^{2}-x)^{1/2}\cdot\textbf{1}_{x\leq\mathfrak{a}+\mathfrak{b}t-\mathfrak{c}t^{2}};\quad\mu_{t}(dx)=\varrho_{t}(x)dx;\quad\mathfrak{H}_{\operatorname{Ai}}(t,x)=\displaystyle\int_{x}^{\infty}\varrho_{t}(w)dw.$

By (10.3) and (10.4), it is quickly confirmed that
$\mathfrak{H}_{\operatorname{Ai}}$ and $\mathfrak{G}_{\operatorname{Ai}}$ are
the height and inverted height functions associated with the process
$\bm{\mu}_{\operatorname{Ai}}$ (as in Definition 10.4, whose notions are also
well-defined if $\bm{\mu}$ has infinite mass). In this way, one can view
$\mathfrak{H}_{\operatorname{Ai}}$ and $\mathfrak{G}_{\operatorname{Ai}}$ as
the large scale limits of the parabolic Airy line ensemble. Since each
$\mu_{t}$ has infinite mass, it is not a bridge-limiting measure process in
the sense of Definition 10.2, but we will see that it will satisfies many of
the same properties as one.

####  10.3. Continuous Variants of Monotonicity

In this section we discuss continuous variants of both height monotonicity
(4.6) and gap monotonicity (5.1), which apply to bridge-limiting measure
processes (Definition 10.2). They are given by the first and second lemmas
below, respectively. The first statements of these lemmas assume some type of
(either inverted height or gap) comparison between two families of boundary
data, along their entire west and east boundaries, and deduce that the
comparison continues to hold in the interior of the domain. The second
statements assume a comparison between these boundary data, but only along the
parts of their west and east boundaries that lie above a given “level line.”
It then shows the comparison continues to hold in the interior of the domain
above this level line, if one further assumes a certain comparison between the
level lines of the two processes (one lies above the other for height
monotonicity, and one is “more concave” than the other for gap monotonicity,
parallel to 4.6 and 5.1, respectively). The proofs of these two lemmas, which
are quick consequences of the discrete variants of monotonicity (4.6 and 5.1),
with the convergence of non-intersecting Brownian bridges to their limit
shapes (10.7), are provided in Section 23.1 below. In what follows, we recall
the inverted height function and inverted liquid region associated with a
measure-valued process from Definition 10.4 and LABEL:omega12, respectively.

Figure 3.7. Shown to the left is a depiction for the continuous variant of
height monotonicity; shown to the right is a depiction for the continuous
variant of gap monotonicity.

######  Lemma 10.14.

Fix real numbers $a<b$ and $A,\widetilde{A}>0$; set
$A_{0}=\min\\{A,\widetilde{A}\\}$; and fix compactly supported measures
$\mu_{a},\widetilde{\mu}_{a},\mu_{b},\widetilde{\mu}_{b}\in\mathscr{P}_{\operatorname{fin}}$
such that $\mu_{a}(\mathbb{R})=A=\mu_{b}(\mathbb{R})$ and
$\widetilde{\mu}_{a}(\mathbb{R})=\widetilde{A}=\widetilde{\mu}_{b}(\mathbb{R})$.
Let $\bm{\mu}^{\star}$ and $\widetilde{\bm{\mu}}^{\star}$ denote the bridge-
limiting measure processes on $[a,b]$ with boundary data $(\mu_{a};\mu_{b})$
and $(\widetilde{\mu}_{a};\widetilde{\mu}_{b})$, respectively. Also denote the
associated inverted height functions by
$G^{\star}:[a,b]\times[0,A]\rightarrow\mathbb{R}$ and
$\widetilde{G}^{\star}:[a,b]\times[0,\widetilde{A}]\rightarrow\mathbb{R}$ and
the associated inverted liquid regions by $\Omega^{\operatorname{inv}}$ and
$\widetilde{\Omega}^{\operatorname{inv}}$, respectively.

  1. (1)

Assume $\widetilde{A}\geq A$ and for each $(t,y)\in\\{a,b\\}\times[0,A_{0}]$
that $G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y)$. Then,
$G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y)$ holds for each
$(t,y)\in[a,b]\times[0,A_{0}]$.

  2. (2)

Fix a real number $w\in(0,A_{0})$ such that
$(t,w)\in\Omega^{\operatorname{inv}}\cap\widetilde{\Omega}^{\operatorname{inv}}$
for each $t\in(a,b)$. Assume for each $(t,y)\in\\{a,b\\}\times[0,w]$ that
$G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y)$, and further assume for each
$t\in[a,b]$ that $G^{\star}(t,w)\leq\widetilde{G}^{\star}(t,w)$. Then,
$G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y)$ holds for all
$(t,y)\in[a,b]\times[0,w]$.

######  Lemma 10.15.

Adopt the notation and assumptions of 10.14.

  1. (1)

Assume that $A\geq\widetilde{A}$ and
$G^{\star}(t,y)-G^{\star}(t,y^{\prime})\leq\widetilde{G}^{\star}(t,y)-\widetilde{G}^{\star}(t,y^{\prime})$
holds for each $t\in\\{a,b\\}$ and $y,y^{\prime}\in[0,A_{0}]$ with
$y<y^{\prime}$. Then,
$G^{\star}(t,y)-G^{\star}(t,y^{\prime})\leq\widetilde{G}^{\star}(t,y)-\widetilde{G}^{\star}(t,y^{\prime})$
holds for each $t\in[a,b]$ and $y,y^{\prime}\in[0,A_{0}]$ with $y<y^{\prime}$.

  2. (2)

Fix a real number $w\in(0,A_{0})$ so that
$(t,w)\in\Omega^{\operatorname{inv}}\cap\widetilde{\Omega}^{\operatorname{inv}}$
for each $t\in(a,b)$. Assume that

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

for all $(t,y),(t,y^{\prime})\in\\{a,b\\}\times[0,w]$ Further assume that

(10.13) $\displaystyle\begin{aligned} r\cdot
G^{\star}(t_{1},w&)-G^{\star}\big{(}rt_{1}+(1-r)t_{2},w\big{)}+(1-r)\cdot
G^{\star}(t_{2},w)\\\ &\leq
r\cdot\widetilde{G}^{\star}(t_{1},w)-\widetilde{G}^{\star}\big{(}rt_{1}+(1-r)t_{2},w\big{)}+(1-r)\cdot\widetilde{G}^{\star}(t_{2},w),\end{aligned}$

for all real numbers $t_{1},t_{2}\in[a,b]$ and $r\in[0,1]$. Then,
$\big{|}G^{\star}(t,y)-G^{\star}(t,y^{\prime})\big{|}\leq\big{|}\widetilde{G}^{\star}(t,y)-\widetilde{G}^{\star}(t,y^{\prime})\big{|}$
holds for all $(t,y),(t,y^{\prime})\in[a,b]\times[0,w]$.

See the left and right sides of Figure 3.7 for depictions of 10.14 and 10.15,
respectively.

While the limiting Airy profiles of 10.13 are not quite bridge-limiting
measure processes in the sense of Definition 10.2 (as they have infinite
mass), the following analog of 10.14 provides a height comparison between
bridge-limiting measure processes and limiting Airy profiles. Its proof is
very similar to that of 10.14 (using the concentration bound 4.34 for the
parabolic Airy line ensemble in place of 10.7) and is therefore omitted.

######  Lemma 10.16.

Fix real numbers $a<b$ and $A>0$; and fix measures $\mu_{a},\mu_{b}$ such that
$\mu_{a}(\mathbb{R})=A=\mu_{b}(\mathbb{R})$. Let $\bm{\mu}^{\star}$ denote the
bridge-limiting measure processes on $[a,b]$ with boundary data
$(\mu_{a};\mu_{b})$. Denote the associated inverted height function by
$G^{\star}:[a,b]\times[0,A]\rightarrow\mathbb{R}$ and the associated inverted
liquid region by $\Omega^{\operatorname{inv}}$. Let
$\widetilde{G}^{\star}:[a,b]\times[0,\infty]\rightarrow{\mathbb{R}}$ be a
limiting Airy profile of the form (10.12).

  1. (1)

Assume for each $(t,y)\in\\{a,b\\}\times[0,A]$ that
$G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y)$. Then,
$G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y)$ holds for each
$(t,y)\in[a,b]\times[0,A]$.

  2. (2)

Fix a real number $w\in(0,A)$ such that $(t,w)\in\Omega^{\operatorname{inv}}$
for each $t\in(a,b)$.

    1. (a)

Assume for each $(t,y)\in\\{a,b\\}\times[0,w]$ that
$G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y)$, and for each $t\in[a,b]$ that
$G^{\star}(t,w)\leq\widetilde{G}^{\star}(t,w)$. Then,
$G^{\star}(t,y)\leq\widetilde{G}^{\star}(t,y)$ holds for all
$(t,y)\in[a,b]\times[0,w]$.

    2. (b)

Assume for each $(t,y)\in\\{a,b\\}\times[0,w]$ that
$G^{\star}(t,y)\geq\widetilde{G}^{\star}(t,y)$, and for each $t\in[a,b]$ that
$G^{\star}(t,w)\geq\widetilde{G}^{\star}(t,w)$. Then,
$G^{\star}(t,y)\geq\widetilde{G}^{\star}(t,y)$ holds for all
$(t,y)\in[a,b]\times[0,w]$.

It is also possible to state and prove a variant of 10.16 that compares the
gaps between limiting Airy profiles and those of inverted height function
associated with bridge-limiting measure processes. However, we will not pursue
this here, since we will not need it.

####  10.4. Elliptic Partial Differential Equations for the Height Function

In this section we state an elliptic partial differential equation satisfied
by the inverted height function associated with a bridge-limiting measure
process, and related results. The former is provided through the following
lemma, shown as stated below in [8] (though implicitly due to the earlier
works [68, 66]).

######  Lemma 10.17 ([8, Lemma 3.23(3)]).

Adopting the notation and assumptions of 10.8, we have

(10.14)
$\displaystyle\partial_{t}^{2}G^{\star}(t,y)+\pi^{2}\big{(}\partial_{y}G^{\star}(t,y)\big{)}^{-4}\cdot\partial_{y}^{2}G^{\star}(t,y)=0,\qquad\text{for
each $(t,y)\in\Omega^{\operatorname{inv}}$}.$

It will be useful to make use of invariances of the equation (10.14) under the
following (linear and multiplicative) transformations. We first require an
additional definition.

######  Definition 10.18.

For any bounded, open subset $\mathfrak{R}\subset\mathbb{R}^{2}$, we let
$\operatorname{Adm}(\mathfrak{R})$ denote the set of locally Lipschitz
functions $F\in\mathcal{C}(\overline{\mathfrak{R}})$ such that
$\partial_{y}F(t,y)<0$, for almost all $(t,y)\in\mathfrak{R}$ (with respect to
Lebesgue measure); we call such functions _admissible_.

######  Lemma 10.19 ([8, Lemma 3.21]).

Fix a bounded, open subset $\mathfrak{R}\subset\mathbb{R}^{2}$ and a function
$G\in\operatorname{Adm}(\mathfrak{R})\cap\mathcal{C}^{2}(\mathfrak{R})$;
assume on $\mathfrak{R}$ that $G$ satisfies (10.14). Fix nonzero real numbers
$\alpha$ and $\beta$, and denote
$\widetilde{\mathfrak{R}}=\widetilde{\mathfrak{R}}_{\alpha;\beta}=\big{\\{}(t,y)\in\mathbb{R}^{2}:(\alpha
t,\beta y)\in\mathfrak{R}\big{\\}}$.

  1. (1)

Assuming $\alpha>0$ and $\beta>0$, define
$\widetilde{G}\in\mathcal{C}^{2}(\widetilde{\mathfrak{R}})$ by
$\widetilde{G}(t,y)=(\alpha\beta)^{-1/2}G(\alpha t,\beta y)$. Then
$\widetilde{G}$ satisfies (10.14) on $\widetilde{\mathfrak{R}}$.

  2. (2)

Define $\widehat{G}\in\mathcal{C}^{2}(\mathfrak{R})$ by
$\widehat{G}(t,y)=G(t,y)+\alpha t$. Then, $\widehat{G}$ satisfies (10.14) on
$\mathfrak{R}$.

The $\alpha=\beta$ case of Item 1 in 10.19 would have held for a solution $G$
to the equation $\sum_{i,j\in\\{t,y\\}}\mathfrak{a}_{ij}(\nabla
G)\cdot\partial_{i}\partial_{j}G=0$, for any measurable coefficients
$\mathfrak{a}_{ij}$. However, that this remains true for all $(\alpha,\beta)$
is special to the specific choice of these coefficients appearing in (10.14).
This more general scaling invariance will be useful in analyzing solutions to
(10.14) in Chapter 4 (see, for example, the proof of 13.13).

####  10.5. Regularity Estimates

In this section we recall from [8] various estimates for solutions to the
partial differential equation (10.14); thorughout, we recall the norms defined
in (1.17). We first require the following definition.

######  Definition 10.20.

For any real number $\varepsilon\in(0,1)$ and bounded, open subset
$\mathfrak{R}\subset\mathbb{R}^{2}$, we let
$\operatorname{Adm}_{\varepsilon}(\mathfrak{R})\subset\operatorname{Adm}(\mathfrak{R})$
denote the set of functions $F\in\operatorname{Adm}(\mathfrak{R})$ such that
$\varepsilon<-\partial_{y}F(t,y)<\varepsilon^{-1}$ for almost all
$(t,y)\in\mathfrak{R}$ (with respect to Lebesgue measure).

Next, we state the maximum principle for solutions of (10.14).

######  Lemma 10.21 ([8, Lemma 9.1]).

Fix some open set $\mathfrak{R}\subset\mathbb{R}$, and let
$F_{1},F_{2},F\in\operatorname{Adm}(\mathfrak{R})\cap\mathcal{C}^{2}(\mathfrak{R})$
denote solutions to (10.14) on $\mathfrak{R}$.

  1. (1)

If $F_{1}(z)\leq F_{2}(z)$ for each $z\in\partial\mathfrak{R}$, then
$F_{1}(z)\leq F_{2}(z)$ for each $z\in\mathfrak{R}$.

  2. (2)

We have
$\sup_{z\in\mathfrak{R}}\big{|}F_{1}(z)-F_{2}(z)\big{|}\leq\sup_{z\in\partial\mathfrak{R}}\big{|}F_{1}(z)-F_{2}(z)\big{|}$.
In particular,
$\sup_{z\in\mathfrak{R}}\big{|}F(z)\big{|}=\sup_{z\in\partial\mathfrak{R}}\big{|}F(z)\big{|}$.

Figure 3.8. Shown to the left is a depiction of $\mathfrak{D}_{r}$ in 10.22.
Shown in the middle is a depiction for 10.24 stating, if $F_{1}(z)=F_{2}(z)$
for each $z$ on the blue sides of $\mathfrak{R}$, then $F_{1}-F_{2}$ is
exponentially small in the shaded region. Shown to the right is a depiction
for 10.25.

We next have the following lemma indicating boundedness of the interior
derivatives for a solution to (10.14); see the left side of Figure 3.8.

######  Lemma 10.22 ([8, Lemma 9.2]).

For any integer $m\geq 1$, and real numbers $r>0$; $\varepsilon\in(0,1)$; and
$B>1$, there exists a constant $C=C(\varepsilon,r,B,m)>1$ such that the
following holds. Let $\mathfrak{R}\subset\mathbb{R}^{2}$ be a bounded open
set, let $f\in\mathcal{C}(\partial\mathfrak{R})$ be a function satisfying
$\|f\|_{0}\leq B$, and let
$F\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})\cap\mathcal{C}^{2}(\mathfrak{R})$
be a solution to (10.14) on $\mathfrak{R}$ such that
$F|_{\partial\mathfrak{R}}=f$. Letting
$\mathfrak{D}_{r}=\big{\\{}z\in\mathfrak{R}:\operatorname{dist}(z,\partial\mathfrak{R})>r\big{\\}}$,
we have $\|F\|_{\mathcal{C}^{m}(\overline{\mathfrak{D}}_{r})}\leq C$.

The following lemma states that the solutions of (10.14) are real analytic.

######  Lemma 10.23 ([8, Lemma 9.3]).

Fix a real number $\varepsilon\in(0,1)$, some open set
$\mathfrak{R}\subset\mathbb{R}^{2}$, and let
$F\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})\cap\mathcal{C}^{2}(\mathfrak{R})$
denote a solution to (10.14) on $\mathfrak{R}$. Then, $F$ is real analytic on
$\mathfrak{R}$.

The following result states that, given two solutions $F_{1},F_{2}$ to (10.14)
on a tall rectangle of aspect ratio $2L$, whose boundary data match on its
west and east boundaries, $|F_{1}-F_{2}|$ decays exponentially in $L$ in the
middle of the rectangle; see the middle of Figure 3.8.

######  Lemma 10.24 ([8, Proposition 9.5]).

For any real numbers $\varepsilon,r\in(0,1/4)$ and $B>1$, there exists a
constant $c=c(\varepsilon,r,B)>0$ such that the following holds. Fix a real
number $L>0$, and define the open rectangle
$\mathfrak{R}=(0,L^{-1})\times(-1,1)$. Let
$F_{1},F_{2}\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})\cap\mathcal{C}^{5}(\overline{\mathfrak{R}})$
be two solutions to (10.14) on $\mathfrak{R}$ such that
$\|F_{i}\|_{\mathcal{C}^{5}(\mathfrak{R})}\leq B$ for each $i\in\\{1,2\\}$.
Assume that $F_{1}(t,x)=F_{2}(t,x)$ for any $(t,x)\in\partial\mathfrak{R}$
with $t\in\\{0,L^{-1}\\}$. Then,

$\displaystyle\big{|}F_{1}(t,x)-F_{2}(t,x)\big{|}\leq
c^{-1}e^{-cL^{1/8}},\qquad\text{for any $(t,x)\in[0,L^{-1}]\times[r-1,1-r]$}.$

We conclude this section with the next lemma, which states the following. Fix
a solution $F$ to (10.14), bounded in $\mathcal{C}^{m}$ for some integer $m$,
on a rectangle $\mathfrak{R}$, as well some boundary data $g_{0}$ and $g_{1}$
on the two vertical sides on the rectangle that are close to $F$. Then, it is
possible to find a solution $G$ to (10.14) on a slightly shorter rectangle
$\mathfrak{S}$, whose boundary data on the vertical sides of the rectangle are
given by $g_{0}$ and $g_{1}$ (the first condition in the lemma), and that is
close to $F$ (quantified through the second and third conditions of the
lemma). The second part of the lemma states that $F$ and $G$ are close in any
$\mathcal{C}^{k}$ norm in the interior of $\mathfrak{S}$, and the third part
states that $F$ and $G$ are close in $\mathcal{C}^{m-5}$ (that is, fewer
derivatives than the original assumed bound on $F$) up to the boundary of
$\mathfrak{S}$. See the right side of Figure 3.8 for a depiction.

######  Lemma 10.25 ([8, Lemma 9.6]).

For any integers $m,k\geq 7$, and real numbers $\varepsilon>0$; $r\in(0,1/4)$;
and $B>1$, there exist constants $\delta=\delta(\varepsilon,B)>0$,
$C_{1}=C_{1}(\varepsilon,r,B,k)>1$, and $C_{2}=C_{2}(\varepsilon,B,m)>1$ such
that the following holds. Fix a real number $L>2$, and define the open
rectangles

$\displaystyle\mathfrak{R}=\Big{(}0,\displaystyle\frac{1}{L}\Big{)}\times(-1,1);\qquad\mathfrak{S}_{r}=\Big{(}\displaystyle\frac{r}{L},\displaystyle\frac{1-r}{L}\Big{)}\times\Big{(}\displaystyle\frac{r+1}{L}-1,1-\displaystyle\frac{r+1}{L}\Big{)};\qquad\mathfrak{S}=\mathfrak{S}_{0}.$

Let
$F\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})\cap\mathcal{C}^{m}(\overline{\mathfrak{R}})$
denote a solution to (10.14) on $\mathfrak{R}$ such that
$\|F\|_{\mathcal{C}^{m}(\mathfrak{R})}\leq B$, and define the functions
$f_{0},f_{1}:[-1,1]\rightarrow\mathbb{R}$ by setting $f_{i}(x)=F(iL^{-1},x)$
for each $(i,x)\in\\{0,1\\}\times[-1,1]$. Further let
$g_{0},g_{1}:[-1,1]\rightarrow\mathbb{R}$ denote two functions such that
$\|g_{i}\|_{\mathcal{C}^{m}(-1,1)}\leq B$ and
$\big{|}g_{i}(x)-f_{i}(x)\big{|}\leq\delta L^{-1}$ for each
$(i,x)\in\\{0,1\\}\times[-1,1]$. Then, there exists a solution
$G\in\operatorname{Adm}_{\varepsilon/2}(\mathfrak{S})\cap\mathcal{C}^{m-5}(\overline{\mathfrak{S}})$
to (10.14) on $\mathfrak{S}$ satisfying the following three properties.

  1. (1)

For each $i\in\\{0,1\\}$ and $x\in\big{[}L^{-1}-1,1-L^{-1}\big{]}$, we have
$G(iL^{-1},x)=g_{i}(x)$.

  2. (2)

We have $\|F-G\|_{\mathcal{C}^{k}(\mathfrak{S}_{r})}\leq
C_{1}L^{k}\cdot\big{(}\|f_{0}-g_{0}\|_{\mathcal{C}^{0}}+\|f_{1}-g_{1}\|_{\mathcal{C}^{0}}\big{)}$.

  3. (3)

We have $\|F-G\|_{\mathcal{C}^{m-5}(\mathfrak{S})}\leq
C_{2}L^{m-5}\cdot\big{(}\|f_{0}-g_{0}\|_{\mathcal{C}^{0}}^{3/m}+\|f_{1}-g_{1}\|_{\mathcal{C}^{0}}^{3/m}\big{)}$.

####  10.6. Concentration Estimates for Non-Intersecting Brownian Bridges

In this section we state a concentration bound from [8] (stronger than 4.11
but requiring more stringent hypotheses) for families of non-intersecting
Brownian bridges sampled from the measure $\mathsf{Q}_{f;g}^{\bm{u};\bm{v}}$
of Definition 2.1, which we will use in the proof of 11.1. We begin by
specifying the regularity assumption to which we will subject our boundary
data (namely, the starting and ending data, $\bm{u}$ and $\bm{v}$, and the
lower and upper boundaries, $f$ and $g$, of the paths).

######  Assumption 10.26.

Fix an integer $m\geq 4$ and three real numbers $\varepsilon\in(0,1/2)$,
$\delta\in\big{(}0,1/(5m^{2})\big{)}$, and $B>1$. Let $n\geq 1$ be an integer,
let $L>0$ be a real number, define the open rectangle
$\mathfrak{R}=(0,L^{-1})\times(0,1)\subset\mathbb{R}^{2}$, and let
$G\in\mathcal{C}^{m+1}(\overline{\mathfrak{R}})$ be a function. Assume that

(10.15) $\displaystyle L\in(B^{-1},n^{\delta});\qquad
G\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R});\qquad\big{\|}G-G(0,0)\big{\|}_{\mathcal{C}^{m+1}(\mathfrak{R})}\leq
B,$

and that $G$ solves (10.14) on $\mathfrak{R}$. Define
$f,g:[0,L^{-1}]\rightarrow\mathbb{R}$ by setting $f(s)=G(s,1)$ and
$g(s)=G(s,0)$, for each $s\in[0,L^{-1}]$. Further let $\varkappa>0$ be a real
number, and let $\bm{u},\bm{v}\in\overline{\mathbb{W}}_{n}$ be $n$-tuples with

(10.16) $\displaystyle\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}u_{j}-G(0,jn^{-1})\big{|}\leq\varkappa;\qquad\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}v_{j}-G(L^{-1},jn^{-1})\big{|}\leq\varkappa.$

Sample non-intersecting Brownian bridges
$\bm{x}=(x_{1},x_{2},\ldots,x_{n})\in\llbracket
1,n\rrbracket\times\mathcal{C}\big{(}[0,L^{-1}]\big{)}$ from the measure
$\mathsf{Q}_{f;g}^{\bm{u};\bm{v}}(n^{-1})$.

Let us briefly explain 10.26. The function $G$ will eventually be the limit
shape for the family $\bm{x}$ of non-intersecting Brownian bridges, in the
sense that we will have $x_{j}(t)\approx G(t,jn^{-1})$. The conditions that
$f(s)=G(s,0)$ and $g(s)=G(s,1)$ ensure that this holds for the upper and lower
boundaries of the model (formally, when $j\in\\{0,n+1\\}$), and (10.16)
ensures that this holds (up to an error of $\varkappa$) when $t\in\\{0,1\\}$.
The constraint that $G\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})$
ensures that $G$ has no “frozen facets” (macroscopic regions containing no
curves), and the constraint that $\|G\|_{\mathcal{C}^{m+1}(\mathfrak{R})}\leq
B$ ensures that $G$ has some regularity.

We then have the following concentration bound, stating that $x_{j}(t)\approx
G(t,jn^{-1})+\mathcal{O}(n^{\delta+2/m-1}+\varkappa)$. Thus, the error can be
made smaller by increasing $m$, which is the parameter accounting for the
regularity of the boundary data.

######  Lemma 10.27 ([8, Theorem 1.5]).

Adopt 10.26. There is a constant $c=c(\varepsilon,\delta,B,m)>0$ such that

$\displaystyle\mathbb{P}\Bigg{[}\displaystyle\sup_{s\in[0,L^{-1}]}\bigg{(}\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}x_{j}(s)-G(s,jn^{-1})\big{|}\bigg{)}>\varkappa+c^{-1}n^{2/m+\delta-1}\Bigg{]}\leq
c^{-1}e^{-c(\log n)^{2}}.$

###  11\. Second Derivative Approximations for Paths

####  11.1. Proof of 3.14

In this section we establish 3.14, which follows from the following
generalization of it that replaces the function $G$ from (3.11) with a nearly
arbitrary one satisfying (10.14).

######  Proposition 11.1.

Letting $\varepsilon\in(0,1/2)$ be a real number and adopting 3.13, there
exist constants $c=c(\varepsilon,B)>$ and $C=C(\varepsilon,B)>1$ such that the
following holds with probability at least $1-Cn^{-10}$, whenever $\delta<c$.
Assume that $G\in\mathcal{C}^{50}(\overline{\mathfrak{R}})$ satisfies the
equation (10.14) on $\mathfrak{R}$, and that

(11.1) $\displaystyle
G\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R});\qquad\|G\|_{\mathcal{C}^{50}(\mathfrak{R})}\leq
B;\qquad\displaystyle\max_{(t,x)\in\mathfrak{R}}\big{|}\partial_{t}^{2}G(t,x)+2^{-1/2}\big{|}\leq\delta.$

Then, for each integer $j\in\llbracket n/3,2n/3\rrbracket$, there is a
(random) twice-differentiable function
$h_{j}:[-\xi/2,\xi/2]\rightarrow\mathbb{R}$ with

(11.2)
$\displaystyle\displaystyle\sup_{|s|\leq-\xi/2}\big{|}\partial_{s}^{2}h_{j}(s)+2^{-1/2}\big{|}\leq\delta^{1/8}+(\log
n)^{-1/4},\qquad\text{and}\qquad\|h_{j}\|_{\mathcal{C}^{1}}\leq 20B,$

such that

(11.3)
$\displaystyle\displaystyle\sup_{|s|\leq-\mathsf{T}/2}\big{|}\mathsf{x}_{j}(s)-n^{2/3}\cdot
h_{j}(n^{-1/3}s)\big{|}\leq n^{-1/5}.$

###### Proof of 3.14.

Observe that the function $G$ given by (3.11) satisfies (11.1) for
$\varepsilon=1/3$ and sufficiently large $B$ by its definition, and satisfies
(10.14) on $\mathfrak{R}$ (either by 10.13 and 10.17, or by direct
verification). Thus, 3.14 follows as a special case of 11.1. ∎

To establish 11.1 we will first “locally” produce the functions $h_{j}$, on
time scales of length $2n^{1/3}e^{-\sqrt{\log n}}$, that satisfy the required
properties; we will then “glue” these local functions together to form a
global one; see Figure 3.9 for a depiction. The following proposition
implements the first task; its proof is given in Section 11.3 below.

Figure 3.9. Shown above in red are the local approximations $h_{j_{0};s_{0}}$
to the black curve $\mathsf{x}_{j_{0}}$, which “glue” together to form the
blue global approximation $h_{j_{0}}$.

######  Proposition 11.2.

Adopting the notation and assumptions of 11.1, there exist constants
$c=c(\varepsilon,B)>0$ and $C=C(\varepsilon,B)>1$ such that the following
holds whenever $\delta\in(0,c)$ and $n>C$. Denote $\mathfrak{w}=e^{-\sqrt{\log
n}}$; fix a real number $s_{0}\in[-\mathsf{T}/2,\mathsf{T}/2]$; and fix an
integer $j_{0}\in\llbracket n/3,2n/3\rrbracket$. Then, with probability at
least $1-n^{-15}$, there exists a (random) twice-differentiable function
$h_{j_{0};s_{0}}:[-\mathfrak{w},\mathfrak{w}]\rightarrow\mathbb{R}$ with

(11.4)
$\displaystyle\displaystyle\sup_{|s|\leq\mathfrak{w}}\big{|}\partial_{s}^{2}h_{j_{0};s_{0}}(s)+2^{-1/2}\big{|}\leq\delta^{1/6}+(\log
n)^{-1/3},\qquad\text{and}\qquad\|h_{j_{0};s_{0}}\|_{\mathcal{C}^{1}}\leq
10B,$

such that

(11.5) $\displaystyle\displaystyle\sup_{|s-s_{0}|\leq
n^{1/3}\mathfrak{w}}\Big{|}\mathsf{x}_{j_{0}}(s)-n^{2/3}\cdot
h_{j_{0};s_{0}}\big{(}n^{-1/3}(s-s_{0})\big{)}\Big{|}\leq n^{-1/5}.$

The next lemma explains how to combine such local functions with almost
constant second derivative to form a new one with a similar property.

######  Lemma 11.3.

Fix an integer $K\geq 3$; positive real numbers $\mathfrak{a},\varpi,\theta>0$
and $B>1$; and a real number $\mathfrak{q}\in\mathbb{R}$. Let
$h:[-\mathfrak{a},\mathfrak{a}]\rightarrow\mathbb{R}$ be a function, and
assume for each integer $k\in\llbracket-K,K-2\rrbracket$ that there exists a
twice-differentiable function
$\mathfrak{h}_{k}:\big{[}kK^{-1}\mathfrak{a},(k+2)K^{-1}\mathfrak{a}\big{]}\rightarrow\mathbb{R}$
such that

(11.6)
$\displaystyle\displaystyle\sup_{s\in[k\mathfrak{a}/K,(k+2)\mathfrak{a}/K]}\big{|}\mathfrak{h}_{k}(s)-h(s)\big{|}\leq\varpi;\qquad\displaystyle\sup_{s\in[k\mathfrak{a}/K,(k+2)\mathfrak{a}/K]}\big{|}\mathfrak{h}^{\prime\prime}_{k}(s)+\mathfrak{q}\big{|}\leq\theta;\qquad\|\mathfrak{h}_{k}\|_{\mathcal{C}^{1}}\leq
B.$

Then, there exists a twice-differentiable function
$\mathfrak{h}:[-\mathfrak{a},\mathfrak{a}]\rightarrow\mathbb{R}$ such that

(11.7) $\displaystyle\begin{aligned}
\displaystyle\sup_{|s|\leq\mathfrak{a}}\big{|}\mathfrak{h}(s)-h&(s)\big{|}\leq\varpi;\qquad\|\mathfrak{h}\|_{\mathcal{C}^{1}}\leq
B+50K\mathfrak{a}^{-1}\varpi+5K^{-1}\mathfrak{a}\theta;\\\
&\displaystyle\sup_{|s|\leq\mathfrak{a}}\big{|}\mathfrak{h}^{\prime\prime}(s)+\mathfrak{q}\big{|}\leq
600(K^{2}\mathfrak{a}^{-2}\varpi+\theta).\end{aligned}$

###### Proof.

For each integer $k\in\llbracket-K,K-1\rrbracket$, define the intervals
$\mathfrak{I}_{k}^{-},\mathfrak{I}_{k},\mathfrak{I}_{k}^{+}\subset[-\mathfrak{a},\mathfrak{a}]$
by setting

$\displaystyle\mathfrak{I}_{k}^{-}=\Big{(}\displaystyle\frac{k\mathfrak{a}}{K},\displaystyle\frac{(3k+1)\mathfrak{a}}{3K}\Big{]};\qquad\mathfrak{I}_{k}=\Big{(}\displaystyle\frac{(3k+1)\mathfrak{a}}{3K},\displaystyle\frac{(3k+2)\mathfrak{a}}{3K}\Big{]};\qquad\mathfrak{I}_{k}^{+}=\Big{(}\displaystyle\frac{(3k+2)\mathfrak{a}}{3K},\displaystyle\frac{(k+1)\mathfrak{a}}{K}\Big{]},$

and also denote
$\mathfrak{J}_{k}=\mathfrak{I}_{k}^{-}\cup\mathfrak{I}_{k}\cup\mathfrak{I}_{k}^{+}\subset[-\mathfrak{a},\mathfrak{a}]$.
Fix a twice-differentiable function $\psi:[0,1]\rightarrow[0,1]$ such that

$\displaystyle\psi(s)=1,\quad\text{for
$s\in\Big{[}0,\displaystyle\frac{1}{3}\Big{]}$};\qquad\psi(s)=0,\quad\text{for
$s\in\Big{[}\displaystyle\frac{2}{3},1\Big{]}$};\qquad\|\psi\|_{\mathcal{C}^{1}}\leq
20;\qquad\|\psi\|_{\mathcal{C}^{2}}\leq 200.$

For each integer $k\in\llbracket-K,K-1\rrbracket$, define
$\psi_{k}:\mathfrak{J}_{k}\rightarrow\mathbb{R}$ by for each
$s\in\mathfrak{J}_{k}$ setting

(11.8) $\displaystyle\psi_{k}(s)=\psi(K\mathfrak{a}^{-1}s-k),\qquad\text{so
that}\quad[\psi_{k}]_{1}\leq
20K\mathfrak{a}^{-1},\quad\text{and}\quad[\psi_{k}]_{2}\leq
200K^{2}\mathfrak{a}^{-2}.$

Then, define $\mathfrak{h}:[-\mathfrak{a},\mathfrak{a}]\rightarrow\mathbb{R}$
by setting

$\displaystyle\mathfrak{h}(s)=\displaystyle\sum_{k=-K}^{k=K-1}\textbf{1}_{s\in\mathfrak{J}_{k}}\cdot\Big{(}\psi_{k}(s)\cdot\mathfrak{h}_{k-1}(s)+\big{(}1-\psi_{k}(s)\big{)}\cdot\mathfrak{h}_{k}(s)\Big{)},\qquad\text{for
each $s\in[-\mathfrak{a},\mathfrak{a}]$},$

where we have set $\mathfrak{h}_{-K-1}=\mathfrak{h}_{-K}$ and
$\mathfrak{h}_{K-1}=\mathfrak{h}_{K-2}$. Observe in this way that
$\mathfrak{h}(s)=\mathfrak{h}_{k}(s)$ for each $s$ in a neighborhood of
$(k+1)K^{-1}\mathfrak{a}$, since $\psi(0^{+})=1$ and $\psi(1^{-})=0$. By the
facts that $0\leq\psi_{k}\leq 1$ (as $0\leq\psi\leq 1$) and that the intervals
$\\{\mathfrak{J}_{k}\\}$ are disjoint, we then have

$\displaystyle\displaystyle\sup_{|s|\leq\mathfrak{a}}\big{|}\mathfrak{h}(s)-h(s)\big{|}\leq\displaystyle\max_{k\in\llbracket-K,K-1\rrbracket}\Bigg{(}\displaystyle\sup_{s\in\mathfrak{J}_{k}}\bigg{(}\max\Big{\\{}\big{|}\mathfrak{h}_{k-1}(s)-h(s)\big{|},\big{|}\mathfrak{h}_{k}(s)-h(s)\big{|}\Big{\\}}\bigg{)}\Bigg{)}\leq\varpi,$

which verifies the first bound in (11.7). Moreover, for any
$s\in[-\mathfrak{a},\mathfrak{a}]$, letting
$k=k(s)\in\llbracket-K,K-1\rrbracket$ be the unique integer such that
$s\in\mathfrak{J}_{k}$, we have

(11.9) $\displaystyle\begin{aligned}
&\big{|}\mathfrak{h}^{\prime}(s)\big{|}=\Big{|}\big{(}\mathfrak{h}_{k-1}(s)-\mathfrak{h}_{k}(s)\big{)}\cdot\psi_{k}^{\prime}(s)+\big{(}\mathfrak{h}_{k-1}^{\prime}(s)-\mathfrak{h}_{k}^{\prime}(s)\big{)}\cdot\psi_{k}(s)+\mathfrak{h}_{k}^{\prime}(s)\Big{|};\\\
&\big{|}\mathfrak{h}^{\prime\prime}(s)\big{|}=\Big{|}\big{(}\mathfrak{h}_{k-1}(s)-\mathfrak{h}_{k}(s)\big{)}\cdot\psi_{k}^{\prime\prime}(s)+2\big{(}\mathfrak{h}_{k-1}^{\prime}(s)-\mathfrak{h}_{k}^{\prime}(s)\big{)}\cdot\mathfrak{\psi}_{k}^{\prime}(s)\\\
&\qquad\qquad\qquad+\big{(}\mathfrak{h}_{k-1}^{\prime\prime}(s)-\mathfrak{h}_{k}^{\prime\prime}(s)\big{)}\cdot\psi_{k}(s)+\mathfrak{h}_{k}^{\prime\prime}(s)\Big{|}.\end{aligned}$

To bound the right side of (11.9), first observe by (11.6) that

(11.10)
$\displaystyle\displaystyle\sup_{s\in\mathfrak{J}_{k}}\big{|}\mathfrak{h}_{k-1}(s)-\mathfrak{h}_{k}(s)\big{|}\leq\displaystyle\sup_{s\in\mathfrak{J}_{k}}\Big{(}\big{|}\mathfrak{h}_{k-1}(s)-h(s)\big{|}+\big{|}\mathfrak{h}_{k}(s)-h(s)\big{|}\Big{)}\leq
2\varpi.$

Applying (11.10) at $s=kK^{-1}\mathfrak{a}$ and $s=(k+1)K^{-1}\mathfrak{a}$,
and using the continuity of
$\mathfrak{h}_{k-1}^{\prime}-\mathfrak{h}_{k}^{\prime}$, we find that there
exists an $s_{0}\in\mathfrak{J}_{k}$ such that

(11.11)
$\displaystyle\big{|}\mathfrak{h}_{k-1}^{\prime}(s_{0})-\mathfrak{h}_{k}^{\prime}(s_{0})\big{|}\leq
4K\mathfrak{a}^{-1}\varpi.$

Again by (11.6), we have

(11.12)
$\displaystyle\displaystyle\sup_{s\in\mathfrak{J}_{k}}\big{|}\mathfrak{h}_{k-1}^{\prime\prime}(s^{\prime})-\mathfrak{h}_{k}^{\prime\prime}(s)\big{|}\leq
2\theta,$

which with (11.11) yields

(11.13)
$\displaystyle\displaystyle\sup_{s\in\mathfrak{J}_{k}}\big{|}\mathfrak{h}_{k-1}^{\prime}(s)-\mathfrak{h}_{k}^{\prime}(s)\big{|}\leq
4K\mathfrak{a}^{-1}\varpi+4K^{-1}\mathfrak{a}\theta.$

Inserting (11.10), (11.12), (11.13), and (11.8) (with the fact that
$\|\mathfrak{h}_{k}\|_{\mathcal{C}^{1}}\leq B$ by (11.6)) into (11.9) yields

$\displaystyle\|\mathfrak{h}\|_{\mathcal{C}^{1}}\leq
44K\mathfrak{a}^{-1}\varpi+4K^{-1}\mathfrak{a}\theta+B;$
$\displaystyle\big{|}\mathfrak{h}^{\prime\prime}(s)+\mathfrak{q}\big{|}\leq
560K^{2}\mathfrak{a}^{-2}\varpi+162\theta+\big{|}\mathfrak{h}_{k}^{\prime\prime}(s)+\mathfrak{q}\big{|}\leq
560K^{2}\mathfrak{a}^{-2}\varpi+163\theta,$

where in the last inequality we used the second statement of (11.6). This
establishes the second and third bounds in (11.7) and thus the lemma. ∎

We can now establish 11.1.

###### Proof of 11.1.

Throughout this proof, set $\mathfrak{a}=\xi/2$, and let $K\geq 1$ denote the
minimal integer such that $\mathfrak{a}K^{-1}\leq\mathfrak{w}$; observe that
$K\leq n^{1/10}$ for sufficiently large $n$, since
$(2B)^{-1}\leq\mathfrak{a}\leq B/2$ (by 3.13) and $\mathfrak{w}=e^{-\sqrt{\log
n}}\geq n^{-1/20}$. For each integer $k\in\llbracket-K,K-2\rrbracket$, denote
$s_{k}=(k+1)\mathfrak{w}-\mathfrak{a}$.

Then by 11.2 and a union bound, there exist constants $c=c(\varepsilon,B)>0$
and $C=C(\varepsilon,B)>1$ such that, if $\delta<c$, the following holds with
probability at least $1-Cn^{-10}$. For each integer $j\in\llbracket
n/3,2n/3\rrbracket$ and $k\in\llbracket-K,K-2\rrbracket$, there exists a
twice-differentiable (random) function
$\mathfrak{h}_{j;k}:[s_{k}-\mathfrak{a}K^{-1},s_{k}+\mathfrak{a}K^{-1}]\rightarrow\mathbb{R}$
with

$\displaystyle\displaystyle\sup_{|s-s_{k}|\leq\mathfrak{a}/K}\big{|}\mathfrak{h}^{\prime\prime}_{j;k}(s)+2^{-1/2}\big{|}\leq\delta^{1/6}+(\log
n)^{-1/3},\qquad\text{and}\qquad\|\mathfrak{h}_{j;k}\|_{\mathcal{C}^{1}}\leq
10B,$

such that (recalling $\bm{x}(s)=n^{-2/3}{\bm{\mathsf{x}}}(n^{1/3}s)$ from
(3.9))

$\displaystyle\displaystyle\sup_{|s-s_{k}|\leq\mathfrak{a}/K}\big{|}x_{j}(s)-\mathfrak{h}_{j;k}(s)\big{|}\leq
n^{-13/15}.$

Thus applying 11.3, with the parameters $(\mathfrak{q},\varpi,\theta,B)$ there
given by $\big{(}2^{-1/2},n^{-13/15},\delta^{1/6}+(\log n)^{-1/3},10B\big{)}$
here, yields for each integer $j\in\llbracket n/3,2n/3\rrbracket$ the
existence of a twice-differentiable function
$h_{j}:[-\mathfrak{a},\mathfrak{a}]\rightarrow\mathbb{R}$ such that

(11.14) $\displaystyle\begin{aligned}
&\displaystyle\sup_{|s|\leq\mathfrak{a}}\big{|}h_{j}(s)-x_{j}(s)\big{|}\leq
n^{-13/15};\\\
&\displaystyle\sup_{|s|\leq\mathfrak{a}}\big{|}\partial_{s}^{2}h_{j}(s)+2^{1/2}\big{|}\leq
600\big{(}K^{2}\mathfrak{a}^{2}n^{-13/15}+\delta^{1/6}+(\log
n)^{-1/3}\big{)}\leq\delta^{1/8}+(\log n)^{-1/4};\\\
&\|h_{j}\|_{\mathcal{C}^{1}}\leq
10B+50K\mathfrak{a}^{-1}n^{-13/15}+5K^{-1}\mathfrak{a}\big{(}\delta^{1/6}+(\log
n)^{-1/3}\big{)}\leq 20B,\end{aligned}$

where in the second and third statements we used the facts that
$(2B)^{-1}\leq\mathfrak{a}\leq B/2$ and that $1\leq K\leq n^{1/10}$ (and that
$n$ is sufficiently large and $\delta$ is sufficiently small). The first
statement of (11.14), together with (3.9), yields (11.3); moreover, the second
and third statements of (11.14) yield (11.2). This establishes the
proposition. ∎

####  11.2. Perturbations of Boundary Data for Limit Shapes

To establish 11.2, one might seek to apply 10.27 to the restriction of
$\bm{x}$ to a $2\mathfrak{w}\times 1$ rectangle centered at
$(s_{0},j_{0}n^{-1})$. To do this, one must verify the assumptions of 10.26
indicating that the boundary data of $\bm{x}$ along this rectangle are
sufficiently regular. That this holds for the starting and ending data would
be a consequence of (3.10) (indicating that the regular profile event
$\textbf{PFL}^{\bm{x}}$ likely holds), but no such guarantee holds for the
upper and lower boundaries.

To circumvent this issue, we will instead introduce two families $\bm{x}^{-}$
and $\bm{x}^{+}$ of non-intersecting Brownian bridges and sandwich $\bm{x}$
between $\bm{x}^{-}$ and $\bm{x}^{+}$. These families will be defined so that
their starting and ending data almost coincide with that of $\bm{x}$ near the
middle of the rectangle. However, around its top and bottom, the starting and
ending data of $\bm{x}$ will be higher than those of $\bm{x}^{-}$, and lower
than those of $\bm{x}^{+}$; we will also make the upper and lower boundaries
for $\bm{x}^{-}$ and $\bm{x}^{+}$ regular. Thus, 10.26 (and hence the
concentration bound 10.27) will apply to $\bm{x}^{-}$ and $\bm{x}^{+}$, giving
a bound on $\bm{x}$ due to the sandwiching; see the right side of Figure 3.10.
For this sandwiching to be effective, we must verify that it is possible to
introduce these boundary perturbations in such a way that they do not
substantially affect the model in the middle of the rectangle.

In this section we state the below lemma, showing that this holds for the
associated limit shape. Its proof largely follows from 10.24 and 10.25, and is
provided in Section 23.2 below.

Figure 3.10. Shown to the left is a depiction for 10.22, indicating that there
exist two inverted height functions $G^{\pm}$ on $\mathfrak{S}$ that are close
to $F$, such that their difference is exponentially small on the green region.
Shown to the right is a depiction for the sandwiching argument.

######  Lemma 11.4.

For any integer $m\geq 7$ and real numbers $\varepsilon>0$ and $B>1$, there
exist constants $c=c(\varepsilon,B,m)\in(0,1)$ and $C=C(\varepsilon,B,m)>1$
such that the following holds. Let $L>4$ and $\vartheta\in(0,c)$ be real
numbers with $|\log\vartheta|^{20}<L<\vartheta^{-1/2m^{2}}$; also let
$\ell\in(B^{-1},B)$ be a real number. Define the open rectangles

(11.15)
$\displaystyle\mathfrak{R}=\Big{(}0,\displaystyle\frac{\ell}{L}\Big{)}\times(0,\ell);\qquad\mathfrak{S}=\Big{(}0,\displaystyle\frac{\ell}{L}\Big{)}\times\Big{(}\displaystyle\frac{\ell}{8},\displaystyle\frac{7\ell}{8}\Big{)};\qquad\mathfrak{S}^{\prime}=\Big{(}\displaystyle\frac{\ell}{8L},\displaystyle\frac{7\ell}{8L}\Big{)}\times\Big{(}\displaystyle\frac{\ell}{8},\displaystyle\frac{7\ell}{8}\Big{)}.$

Let
$F\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})\cap\mathcal{C}^{m}(\overline{\mathfrak{R}})$
denote a solution to (10.14) such that
$\|F\|_{\mathcal{C}^{m}(\mathfrak{R})}\leq B$, and define the functions
$f_{0},f_{1}:[0,\ell]\rightarrow\mathbb{R}$ by setting $f_{i}(x)=F(i\ell
L^{-1},x)$ for each $i\in\\{0,1\\}$ and $x\in[0,\ell]$. Further fix functions
$g_{0},g_{1}:[0,\ell]\rightarrow\mathbb{R}$ such that
$\|f_{i}-g_{i}\|_{\mathcal{C}^{0}}\leq\vartheta$ and
$\|g_{i}\|_{\mathcal{C}^{m}}\leq B$ for each index $i\in\\{0,1\\}$. Then there
exist solutions
$G^{-},G^{+}\in\operatorname{Adm}_{\varepsilon/2}(\mathfrak{S})\cap\mathcal{C}^{m}(\overline{\mathfrak{S}})$
to (10.14) on $\mathfrak{S}$ satisfying the following properties.

  1. (1)

For each $i\in\\{0,1\\}$ and $x\in[\ell/5,4\ell/5]$, we have $G^{-}(i\ell
L^{-1},x)=g_{i}(x)=G^{+}(i\ell L^{-1},x)$.

  2. (2)

For each $i\in\\{0,1\\}$ and $x\in[\ell/8,7\ell/8]$, we have $G^{-}(i\ell
L^{-1},x)\leq g_{i}(x)\leq G^{+}(i\ell L^{-1},x)$.

  3. (3)

We have
$\|G^{-}\|_{\mathcal{C}^{m-5}(\mathfrak{S})}+\|G^{+}\|_{\mathcal{C}^{m-5}(\mathfrak{S})}\leq
C$.

  4. (4)

We have
$\|G^{-}-F\|_{\mathcal{C}^{m}(\mathfrak{S}^{\prime})}+\|G^{+}-F\|_{\mathcal{C}^{m}(\mathfrak{S}^{\prime})}\leq
C\vartheta^{3/4}$.

  5. (5)

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

  6. (6)

For each $(t,x)\in[0,\ell L^{-1}]\times\\{\ell/8,7\ell/8\\}$, we have
$G^{-}(t,x)\leq F(t,x)-\vartheta<F(t,x)+\vartheta\leq G^{+}(t,x)$.

Let us briefly explain 11.4; see the left side of Figure 3.10. One may view
$F$ as the “original” function and $G^{-}$ and $G^{+}$ as two perturbations of
it that have different boundary data along the two vertical sides of
$\mathfrak{R}$. The first part of the lemma indicates that the boundary data
of $G^{-}$ and $G^{+}$ are both given by $g_{i}$ around the middles of these
two sides; the second indicates that $G^{-}$ and $G^{+}$ are larger than
smaller than $g_{i}$ around the endpoints of these sides, respectively. The
third indicates that $G^{-}$ and $G^{+}$ are regular up to the boundary of
$\mathfrak{S}$; the fourth indicates that $G^{-}$ and $G^{+}$ (and their
derivatives) are close to the original function $F$ in the interior
$\mathfrak{S}^{\prime}\subset\mathfrak{S}$ of the rectangle. The fifth
indicates that $G^{+}$ and $G^{-}$ are quite close in the middle of
$\mathfrak{R}$, which will eventually make sandwiching between them effective.
The sixth indicates that the boundary data of $G^{-}$ and $G^{+}$ along the
two horizontal sides of $\mathfrak{R}$ are lower and higher than those of $F$,
by at least $\vartheta$, respectively.

####  11.3. Proof of 11.2

In this section we establish 11.2; we adopt the notation of that proposition
throughout. The content in Section 11.2 presented several elements of its
proof, but it simplified the discussion on the regularity for the starting and
ending data of $\bm{x}$ (along the $2\mathfrak{w}\times 1$ rectangle centered
at $(s_{0},j_{0}n^{-1})$ described there). Although the likelihood (3.10) of
the regular profile event indeed indicates that the starting and ending data
are each individually regular in the vertical direction, it does not directly
forbid the possibility that these data are far from each other; this would
make it impossible to find a regular profile (with uniformly bounded
$t$-derivative) that interpolates between them in the sense of (10.16) in
10.26. To preclude this possibility, we induct on scales, applying the
discussion of Section 11.2 on thinner rectangles, until eventually reaching
width around $2\mathfrak{w}$.

This requires some additional notation. Let $c_{0}=c_{0}(\varepsilon,B)>0$ and
$C_{0}=C_{0}(\varepsilon,B)>1$ denote the constants $c(\varepsilon/2,2B,50)>0$
and $C(\varepsilon/2,2B,50)>1$ from 11.4, respectively. For any integer $k\geq
0$, define the real numbers
$\delta_{0},\omega_{k},\varsigma_{k},\vartheta_{k},\Theta_{k}>0$ and
$L_{k}>1$, and integer $n_{k}\geq 1$, by setting

(11.16) $\displaystyle\begin{aligned} &\delta_{0}=\delta^{1/2}+(\log
n)^{-1};\qquad\omega_{k}=4^{-k-1};\qquad L_{k}=\delta_{0}^{-\sqrt{k+1}};\qquad
n_{k}=\lfloor\omega_{k}n\rfloor;\\\
&\varsigma_{k}=5C_{0}\exp\Big{(}-\displaystyle\frac{c_{0}}{2}L_{k}^{1/80}\Big{)};\qquad\vartheta_{k}=\varsigma_{k}+n^{-13/15};\qquad\Theta_{k}=\delta_{0}^{3/4}+\displaystyle\sum_{j=0}^{k}\omega_{j}^{-1}\vartheta_{j}^{3/4}.\end{aligned}$

Figure 3.11. Shown to the left is the rectangle $\mathsf{R}_{k}$. Shown to the
right is the rectangle $\mathfrak{R}_{k}$ obtained by “zooming in” around the
point $(s_{0}/n^{1/3},j_{0}/n)$.

Also let $K_{0}\geq 1$ denote the maximal integer such that
$\omega_{K_{0}}L_{K_{0}}^{-1}\geq 3\mathfrak{w}$. To ease notation, we will
omit the floors in what follows, assuming that each $\omega_{k}n$ is an
integer; this will barely affect the proofs. For each integer $k\in\llbracket
0,K_{0}\rrbracket$, define the set
$\mathsf{R}_{k}\subset\mathbb{Z}\times\mathbb{R}$ and open rectangle
$\mathfrak{R}_{k}\subset\mathbb{R}^{2}$ by

(11.17) $\displaystyle\begin{aligned}
&\mathsf{R}_{k}=\Big{\llbracket}j_{0}-\displaystyle\frac{n_{k}}{2}+1,j_{0}+\displaystyle\frac{n_{k}}{2}\Big{\rrbracket}\times\Big{(}s_{0}-\displaystyle\frac{\omega_{k}n^{1/3}}{2L_{k}},s_{0}+\displaystyle\frac{\omega_{k}n^{1/3}}{2L_{k}}\Big{)};\\\
&\mathfrak{R}_{k}=\Big{(}-\displaystyle\frac{1}{2L_{k}},\displaystyle\frac{1}{2L_{k}}\Big{)}\times\Big{(}-\displaystyle\frac{1}{2},\displaystyle\frac{1}{2}\Big{)}.\end{aligned}$

See the left side of Figure 3.11 for a depiction. For each $k\in\llbracket
0,K_{0}\rrbracket$, also define the function
$G_{k}:\overline{\mathfrak{R}}_{k}\rightarrow\mathbb{R}$ by rescaling $G$,
namely, by setting

(11.18) $\displaystyle\begin{aligned} G_{k}(s,x)=\omega_{k}^{-1}\cdot
G(n^{-1/3}s_{0}+\omega_{k}s,j_{0}n^{-1}+\omega_{k}x),\qquad\text{for each
$(s,x)\in\overline{\mathfrak{R}}_{k}$}.\end{aligned}$

See the right side of Figure 3.11 for a depiction. Observe that the $G_{k}$
satisfy (10.14) on $\mathfrak{R}_{k}$, by the first part (at
$\alpha=\beta=\omega_{k}$) of 10.19. Then, for each integer $k\in\llbracket
0,K_{0}\rrbracket$, define the sequence of functions
$\bm{x}^{(k)}=\big{(}x_{1}^{(k)},x_{2}^{(k)},\ldots,x_{n_{k}}^{(k)}\big{)}\in\llbracket
1,n_{k}\rrbracket\times\mathcal{C}\big{(}[-1/2L_{k},1/2L_{k}]\big{)}$ by
rescaling and reindexing $\bm{\mathsf{x}}$, namely, by setting

(11.19) $\displaystyle
x_{i}^{(k)}(s)=\omega_{k}^{-1}n^{-2/3}\cdot\mathsf{x}_{i+j_{0}-n_{k}/2}(s\omega_{k}n^{1/3}+s_{0}),\quad\text{for
each $(i,s)\in\llbracket
0,n_{k}+1\rrbracket\times\Big{[}-\displaystyle\frac{1}{2L_{k}},\displaystyle\frac{1}{2L_{k}}\Big{]}$}.$

In this way, $n_{k}$ will prescribe the number of curves in $\bm{x}^{(k)}$
(tracked by $\mathsf{R}_{k}$ and $\mathfrak{R}_{k}$); $L_{k}^{-1}$ will
prescribe the width of the rectangle $\mathfrak{R}_{k}$ (which becomes thinner
as $k$ increases, since $L_{k}$ is increasing); and $\varsigma_{k}$ is
analogous to (but larger than) the error in the fifth part of 11.4.

The following lemma bounds the quantities in (11.16); we establish it in
Section 12.1 below.

######  Lemma 11.5.

There exist constants $c=c(\varepsilon,B)>0$ and $C=C(\varepsilon,B)>1$ such
that the following hold if $n>C$ and $\delta<c$. First, we have
$\vartheta_{k}\leq\Theta_{k}\leq\varepsilon\delta_{0}^{1/2}$ for each integer
$k\in\big{\llbracket}0,(\log n)^{3/4}\big{\rrbracket}$. Second, we have

(11.20) $\displaystyle\begin{aligned} &(\log n)^{1/3}\leq K_{0}\leq(\log
n)^{1/2};\qquad e^{(\log n)^{1/6}}\leq L_{K_{0}}\leq e^{(\log n)^{1/2}};\\\
&\displaystyle\frac{n_{K_{0}}}{n}\geq
n^{-1/25000};\qquad\qquad\quad\qquad\varsigma_{K_{0}-1}\leq\vartheta_{K_{0}-1}\leq
2n^{-13/15}.\end{aligned}$

Third, for any integer $k\in\llbracket 1,K_{0}\rrbracket$, we have
$|\log\vartheta_{k-1}|^{20}\leq 4L_{k}/3\leq\vartheta_{k-1}^{-1/5000}$.

For each integer $k\in\llbracket 0,K_{0}\rrbracket$, we next inductively
define a sequence of events $\Omega_{k}$, measurable with respect to
$\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{k+1})$ (recall Definition 2.2),
and sequences of functions
$G_{k}^{-},G_{k}^{+}:\mathfrak{R}_{k}\rightarrow\mathbb{R}$ satisfying (10.14)
on $\mathfrak{R}_{k}$. At $k=0$, define the functions
$G_{0}^{-},G_{0}^{+}:\overline{\mathfrak{R}}_{0}\rightarrow\mathbb{R}$ and
event $\Omega_{0}$ by setting

(11.21) $\displaystyle\begin{aligned}
&G_{0}^{-}(s,x)=G_{0}(s,x)=G_{0}^{+}(s,x);\qquad\text{for each
$(s,x)\in\overline{\mathfrak{R}}_{0}$},\end{aligned}$

which (by the first part of 10.19) solve (10.14) on $\mathfrak{R}_{0}$, and

(11.22)
$\displaystyle\Omega_{0}=\Bigg{\\{}\displaystyle\sup_{(s,j/n_{0})\in\overline{\mathfrak{R}}_{0}\setminus\frac{1}{4}\cdot\overline{\mathfrak{R}}_{1}}\big{|}x_{j+n_{0}/2}^{(0)}(s)-G_{0}^{+}(s,jn_{0}^{-1})\big{|}\leq\delta_{0}^{3/4}\Bigg{\\}}.$

Observe that $\Omega_{0}$ is measurable with respect to
$\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{1})$ since (by (11.17), (11.19),
and the facts that $\omega_{0}=4\omega_{1}$ and $n_{0}=4n_{1}$) it amounts to
constraining $\mathsf{x}_{j}(t)$ for $(j,t)\notin\mathsf{R}_{1}$.

We now let $k\in\llbracket 1,K_{0}\rrbracket$ denote an integer; assume that
we have defined the functions $G_{k-1}^{-}$ and $G_{k-1}^{+}$ satsfying
(10.14) on $\mathfrak{R}_{k-1}$, and the event $\Omega_{k-1}$, measurable with
respect to $\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{k})$. We will set the
event $\Omega_{k}$ to be the intersection of $\Omega_{k-1}$ with certain
events $\Omega_{k}^{(i)}$ for $i\in\\{1,2\\}$ that we will define. Starting
with $i=1$, we first set

(11.23)
$\displaystyle\Omega_{k}^{(1)}=\textbf{PFL}^{\bm{x}^{(k-1)}}\Big{(}-\displaystyle\frac{1}{8L_{k}};\displaystyle\frac{1}{4n^{9/10}};2B\Big{)}\cap\textbf{PFL}^{\bm{x}^{(k-1)}}\Big{(}\displaystyle\frac{1}{8L_{k}};\displaystyle\frac{1}{4n^{9/10}};2B\Big{)}\cap\Omega_{k-1},$

where we recall the event PFL from Definition 3.11. Since $\Omega_{k-1}$ is
measurable with respect to $\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{k})$
and since by (11.19) the two regular profile events in (11.23) amount to
constraining $\bm{\mathsf{x}}(t)$ at
$t\in\big{\\{}s_{0}-\omega_{k-1}n^{1/3}/8L_{k},s_{0}+\omega_{k-1}n^{1/3}/8L_{k}\big{\\}}=\\{s_{0}-\omega_{k}n^{1/3}/2L_{k},s_{0}+\omega_{k}n^{1/3}/2L_{k}\\}$,
it follows from (11.17) that $\Omega_{k}^{(1)}$ is also measurable with
respect to $\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{k})$ (and thus to
$\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{k+1})$).

In what follows, we condition on
$\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{k})$ and restrict to the event
$\Omega_{k}^{(1)}$. Then, there exists for each real number
$t\in\big{\\{}-1/8L_{k},1/8L_{k}\big{\\}}$ a function
$\gamma_{t}^{(k-1)}:[-1/2,1/2]\rightarrow\mathbb{R}$ such that for each
$j\in\llbracket 1-n_{k-1}/2,n_{k-1}/2\rrbracket$ we have

(11.24)
$\displaystyle\big{|}x_{j+n_{k-1}/2}^{(k-1)}(t)-\gamma_{t}^{(k-1)}(jn_{k-1}^{-1})\big{|}\leq\displaystyle\frac{1}{4n^{9/10}};\qquad\big{\|}\gamma_{t}^{(k-1)}-\gamma_{t}^{(k-1)}(0)\big{\|}_{\mathcal{C}^{50}}\leq
2B;$

observe here that we have shifted the argument of $\gamma_{t}^{(k-1)}$ by
$1/2$ in comparison to Definition 3.11. Define for each
$t\in\\{-1/2L_{k},1/2L_{k}\\}$ the rescaled function
$\widetilde{\gamma}_{t}^{(k-1)}:[-2,2]\rightarrow\mathbb{R}$ by setting

$\displaystyle\widetilde{\gamma}_{t}^{(k-1)}(x)=4\gamma_{t/4}^{(k-1)}\Big{(}\displaystyle\frac{x}{4}\Big{)},\qquad\text{for
each $x\in[-2,2]$}.$

By (11.24) and the fact that
$x_{j+n_{k}/2}^{(k)}(s)=4x_{j+n_{k-1}/2}^{(k-1)}(s/4)$ (due to (11.19) and the
equality $\omega_{k-1}=4\omega_{k}$), we have for each
$t\in\\{-1/2L_{k},1/2L_{k}\\}$ and $j\in\llbracket
1-n_{k-1}/2,n_{k-1}/2\rrbracket=\llbracket 1-2n_{k},2n_{k}\rrbracket$ that, on
$\Omega_{k}^{(1)}$,

(11.25)
$\displaystyle\big{|}x_{j+n_{k}/2}^{(k)}(t)-\widetilde{\gamma}_{t}^{(k-1)}(jn_{k}^{-1})\big{|}\leq
n^{-9/10};\qquad\big{\|}\widetilde{\gamma}_{t}^{(k-1)}-\widetilde{\gamma}_{t}^{(k-1)}(0)\big{\|}_{\mathcal{C}^{50}}\leq
4B.$

Define the open rectangle
$\widetilde{\mathfrak{R}}_{k-1}=4\cdot\mathfrak{R}_{k-1}=(-2L_{k-1}^{-1},2L_{k-1}^{-1})\times(-2,2)$
and the function
$\widetilde{G}_{k-1}^{+}:\widetilde{\mathfrak{R}}_{k-1}\rightarrow\mathbb{R}$
by rescaling $G_{k-1}^{+}$, namely, by setting

(11.26)
$\displaystyle\widetilde{G}_{k-1}^{+}(s,x)=4G_{k-1}^{+}\Big{(}\displaystyle\frac{s}{4},\displaystyle\frac{x}{4}\Big{)},\qquad\text{for
each $(s,x)\in\widetilde{\mathfrak{R}}_{k-1}$},$

which satisfies (10.14) on $\widetilde{\mathfrak{R}}_{k-1}$ by the first part
(at $\alpha=\beta=1/4$) of 10.19. Further define the functions
$\mathfrak{y}_{0}^{(k)},\mathfrak{y}_{1}^{(k)},\mathfrak{z}_{0}^{(k)},\mathfrak{z}_{1}^{(k)},\widetilde{\mathfrak{z}}_{0}^{(k)},\widetilde{\mathfrak{z}}_{1}^{(k)}:[-2/3,2/3]\rightarrow\mathbb{R}$
by for each index $i\in\\{0,1\\}$ and real number $x\in[-2/3,2/3]$ setting

(11.27)
$\displaystyle\mathfrak{y}_{i}^{(k)}(x)=\widetilde{\gamma}_{(2i-1)/2L_{k}}^{(k-1)}(x);\qquad\mathfrak{z}_{i}^{(k)}(x)=G_{k}\Big{(}\displaystyle\frac{2i-1}{2L_{k}},x\Big{)};\qquad\widetilde{\mathfrak{z}}_{i}^{(k-1)}(x)=\widetilde{G}_{k-1}^{+}\Big{(}\displaystyle\frac{2i-1}{2L_{k}},x\Big{)}.$

We then have the following lemma, which states that $\widetilde{G}_{k-1}^{+}$
(with its derivatives) is close to $G_{k}$; it will be established in Section
12.2 below. In what follows, we recall the constant $C_{0}$ from above (11.16)
(as well as the norms $[f]_{k;\mathfrak{R}}$ from Section 1.7).

######  Lemma 11.6.

There exist constants $c=c(\varepsilon,B)>0$ and $C=C(\varepsilon,B)>1$ such
that the following holds for any integer $k\in\llbracket 1,K_{0}-1\rrbracket$.
If $n>C$ and $\delta<c$, then on the event $\Omega_{k}^{(1)}$ we have

(11.28)
$\displaystyle\big{[}\widetilde{G}_{k-1}^{+}-G_{k}\big{]}_{1;\mathfrak{R}_{k}}+\omega_{k-1}^{-1}\cdot\displaystyle\sum_{d=2}^{50}\big{[}\widetilde{G}_{k-1}^{+}-G_{k}\big{]}_{d;\mathfrak{R}_{k}}\leq
2C_{0}\Theta_{k-1}.$

Observe from 11.6; the bound
$\Theta_{k-1}\leq\varepsilon\delta_{0}^{1/2}\leq\varepsilon/4C_{0}$ for
sufficiently large $n$ and small $\delta$ (by 11.5); the fact that
$G_{k}\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R}_{k})$ (which holds
since $G\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})$, by (11.1),
together with the fact that the scaling (11.18) producing $G_{k}$ from $G$
preserves gradients); and the fact that
$\big{\|}G_{k}-G_{k}(0,0)\big{\|}_{\mathcal{C}^{50}(\mathfrak{R})}\leq 2B$ (as
$\big{\|}G-G(0,0)\big{\|}_{\mathcal{C}^{50}(\mathfrak{R})}\leq B$ by (11.1))
that

(11.29)
$\displaystyle\widetilde{G}_{k-1}^{+}\in\operatorname{Adm}_{\varepsilon/2}(\mathfrak{R}_{k}),\quad\text{and}\quad\big{\|}\widetilde{G}_{k-1}^{+}-\widetilde{G}_{k-1}^{+}(0,0)\big{\|}_{\mathcal{C}^{50}(\mathfrak{R}_{k-1})}\leq
4B+4C_{0}\Theta_{k-1}\leq 5B.$

The next lemma states that $\mathfrak{y}_{i}^{(k)}$ is very close to
$\widetilde{\mathfrak{z}}_{i}^{(k-1)}$, which will also be established in
Section 12.2 below.

######  Lemma 11.7.

There exist constants $c=c(\varepsilon,B)>0$ and $C=C(\varepsilon,B)>1$ such
that the following holds for any integer $k\in\llbracket 1,K_{0}-1\rrbracket$.
If $n>C$ and $\delta<c$, then on the event $\Omega_{k}^{(1)}$ we have

(11.30)
$\displaystyle\displaystyle\max_{i\in\\{0,1\\}}\big{\|}\mathfrak{y}_{i}^{(k)}-\widetilde{\mathfrak{z}}_{i}^{(k-1)}\big{\|}_{\mathcal{C}^{0}([-2/3,2/3])}\leq\vartheta_{k-1}.$

The next definition introduces the functions $G_{k}^{-}$ and $G_{k}^{+}$,
using 11.4. Here, we define the open rectangles

(11.31) $\displaystyle\begin{aligned}
&\widehat{\mathfrak{R}}_{k}=\Big{(}-\displaystyle\frac{1}{2L_{k}},\displaystyle\frac{1}{2L_{k}}\Big{)}\times\Big{(}-\displaystyle\frac{2}{3},\displaystyle\frac{2}{3}\Big{)};\qquad\mathfrak{S}_{k}^{\prime}=\Big{(}-\displaystyle\frac{3}{8L_{k}},\displaystyle\frac{3}{8L_{k}}\Big{)}\times\Big{(}-\displaystyle\frac{1}{2},\displaystyle\frac{1}{2}\Big{)}.\end{aligned}$

######  Definition 11.8.

Apply 11.4 (translated by $(-\ell/2L_{k},-\ell/2)$) with the
$(\ell,L,\vartheta)$ there equal to $(4/3,4L_{k}/3,\vartheta_{k-1})$ here; the
$(\mathfrak{R},\mathfrak{S},\mathfrak{S}^{\prime})$ there equal to
$\big{(}\widehat{\mathfrak{R}}_{k},\mathfrak{R}_{k},\mathfrak{S}_{k}^{\prime}\big{)}$
here; the $(\varepsilon,B,m)$ there equal to $(\varepsilon/4,5B,50)$ here; and
the $(F;g_{0},g_{1})$ there equal to
$\big{(}\widetilde{G}_{k-1}^{+}|_{\widehat{\mathfrak{R}}_{k}};\mathfrak{y}_{0}^{(k)};\mathfrak{y}_{1}^{(k)}\big{)}$
here (implicitly shifting all of these functions by the constant
$\widetilde{G}_{k-1}^{+}(0,0)$). The assumptions of this lemma are verified by
11.7, Equation 11.29, Equation 11.25, and 11.5. This yields solutions
$G_{k}^{-},G_{k}^{+}\in\operatorname{Adm}_{\varepsilon/4}(\mathfrak{R})\cap\mathcal{C}^{50}(\overline{\mathfrak{R}}_{k})$
to Equation 10.14, such that the following six properties hold.

  1. (1)

For each $i\in\\{0,1\\}$ and $x\in[-2/5,2/5]$, we have
$G_{k}^{-}\big{(}\frac{2i-1}{2L_{k}},x\big{)}=\mathfrak{y}_{i}^{(k)}(x)=G_{k}^{+}\big{(}\frac{2i-1}{2L_{k}},x\big{)}$.

  2. (2)

For each $i\in\\{0,1\\}$ and $x\in[-1/2,1/2]$, we have
$G_{k}^{-}\big{(}\frac{2i-1}{2L_{k}},x\big{)}\leq\mathfrak{y}_{i}^{(k)}(x)\leq
G_{k}^{+}\big{(}\frac{2i-1}{2L_{k}},x\big{)}$.

  3. (3)

We have
$\big{\|}G_{k}^{-}-G_{k}^{-}(0,0)\big{\|}_{\mathcal{C}^{45}(\mathfrak{S}_{k})}+\big{\|}G_{k}^{+}-G_{k}^{+}(0,0)\big{\|}_{\mathcal{C}^{45}(\mathfrak{S}_{k})}\leq
2C_{0}$.

  4. (4)

We have
$\|G_{k}^{-}-\widetilde{G}_{k-1}^{+}\|_{\mathcal{C}^{50}(\mathfrak{S}_{k}^{\prime})}+\|G_{k}^{+}-\widetilde{G}_{k-1}^{+}\|_{\mathcal{C}^{50}(\mathfrak{S}_{k}^{\prime})}\leq
C_{0}\vartheta_{k-1}^{3/4}$.

  5. (5)

For each $(t,x)\in[-1/2L_{k},1/2L_{k}]\times[-1/3,1/3]$, we have
$\big{|}G_{k}^{+}(t,x)-G_{k}^{-}(t,x)\big{|}\leq C_{0}e^{-c_{0}L_{k}^{1/8}}$.

  6. (6)

For each $(t,x)\in[-1/2L_{k},1/2L_{k}]\times\\{-1/2,1/2\\}$, we have
$G_{k}^{-}(t,x)\leq\widetilde{G}_{k-1}^{+}(t,x)-\vartheta_{k-1}<\widetilde{G}_{k-1}^{+}(t,x)+\vartheta_{k-1}\leq
G_{k}^{+}(t,x)$.

Then, define the event

(11.32) $\displaystyle\Omega_{k}^{(2)}$
$\displaystyle=\Bigg{\\{}\displaystyle\sup_{\begin{subarray}{c}(s,j/n_{k})\in\mathfrak{R}_{k}\setminus\frac{1}{4}\cdot\mathfrak{R}_{k+1}\\\
|j|\leq
n_{k}/4\end{subarray}}\big{|}x_{j+n_{k}/2}^{(k)}(s)-G_{k}^{+}(s,jn_{k}^{-1})\big{|}\leq\displaystyle\frac{\vartheta_{k}}{5}\Bigg{\\}},$

which is measurable with respect to
$\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{k+1})$ since (by (11.17),
(11.19), and the facts that $\omega_{k-1}=4\omega_{k}$ and $n_{k-1}=4n_{k}$)
it amounts to constraining the paths $\mathsf{x}_{j}(t)$ for
$(j,t)\notin\mathsf{R}_{k+1}$. Also define the event (not measurable with
respect to $\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{k+1})$)

$\displaystyle\Omega_{k}^{(3)}=\Bigg{\\{}\displaystyle\sup_{\begin{subarray}{c}(s,j/n_{k})\in\mathfrak{R}_{k}\\\
|j|\leq
n_{k}/4\end{subarray}}\Big{|}x_{j+n_{k}/2}^{(k)}(s)-G_{k}^{+}\big{(}s,jn_{k}^{-1}\big{)}\Big{|}\leq\displaystyle\frac{\vartheta_{k}}{5}\Bigg{\\}}.$

These events indicate that the random paths in $\bm{x}^{(k)}$ closely
approximate the limit shape $G_{k}^{+}$. That these two events likely hold
(stated as 12.2, and shown in Section 12.3) will follow the sandwiching scheme
outlined in the beginning of this section and Section 11.2.

Observing that $\Omega_{k}^{(3)}\subseteq\Omega_{k}^{(2)}$, further define the
events

(11.33)
$\displaystyle\Omega_{k}=\Omega_{k}^{(1)}\cap\Omega_{k}^{(2)};\qquad\Omega_{k}^{\prime}=\Omega_{k}^{(1)}\cap\Omega_{k}^{(3)}\subseteq\Omega_{k}.$

The next lemma, to be shown in Section 12.1, indicates that the final event
$\Omega_{K_{0}}^{\prime}$ is likely.

######  Lemma 11.9.

There exist constants $c=c(\varepsilon,B)>0$ and $C=C(\varepsilon,B)>1$ such
that $\mathbb{P}\big{[}\Omega_{K_{0}}^{\prime}\big{]}\geq 1-n^{-19}$, whenever
$n>C$ and $\delta<c$.

Given this result, we can establish 11.2.

###### Proof of 11.2.

Throughout this proof, we abbreviate $K=K_{0}$ and assume in what follows that
the event $\Omega_{K}^{\prime}$ holds (which we may by 11.9). Then, define
$h_{j_{0};s_{0}}:[-\mathfrak{w},\mathfrak{w}]\rightarrow\mathbb{R}$ by setting

(11.34) $\displaystyle h_{j_{0};s_{0}}(s)=\omega_{K}\cdot
G_{K}^{+}(\omega_{K}^{-1}s,0).$

Let us first show that (11.5) holds. Observe that

$\displaystyle\displaystyle\sup_{|s-s_{0}|\leq
n^{1/3}\omega_{K}/2L_{K}}\bigg{|}\mathsf{x}_{j_{0}}(s+s_{0}$
$\displaystyle)-\omega_{K}n^{2/3}\cdot
G_{K}^{+}\Big{(}\displaystyle\frac{s}{\omega_{K}n^{1/3}},0\Big{)}\bigg{|}$
$\displaystyle=\omega_{K}n^{2/3}\cdot\displaystyle\sup_{|s|\leq
1/2L_{K}}\big{|}x_{n_{K}/2}^{(K)}(s)-G_{K}^{+}(s,0)\big{|}\leq\omega_{K}\vartheta_{K}n^{2/3},$

where the first statement follows from (11.19) and the second from the fact
that $\Omega_{K}^{(3)}\subseteq\Omega_{K}^{\prime}$ holds. Together with
(11.34), this yields

$\displaystyle\displaystyle\sup_{|s|\leq
n^{1/3}\omega_{K}/2L_{K}}\big{|}\mathsf{x}_{j_{0}}(s+s_{0})-n^{2/3}\cdot
h_{j_{0};s_{0}}(n^{-1/3}s)\big{|}\leq\omega_{K}\vartheta_{K}n^{2/3}\leq
2\omega_{K}n^{-1/5}\leq n^{-1/5},$

where in the third inequality we used (11.20), and in the fourth we used the
fact that $\omega_{K}\leq 1/4$. Since $\omega_{K}\geq 3\mathfrak{w}L_{K}$ by
the definition of $K=K_{0}$, this verifies (11.5).

Next let us confirm (11.4), starting with the first bound there. Observe from
(11.1) and (11.18) that

(11.35)
$\displaystyle\displaystyle\max_{|s|<1/2L_{K}}\big{|}\omega_{K}^{-1}\cdot\partial_{s}^{2}G_{K}(s,0)+2^{-1/2}\big{|}=\displaystyle\max_{|s|<\omega_{K}/2L_{K}}\big{|}\partial_{t}^{2}G(n^{-1/3}s_{0}+\omega_{K}s,0)+2^{-1/2}\big{|}\leq\delta.$

Therefore,

$\displaystyle\displaystyle\sup_{|s|\leq\mathfrak{w}}\big{|}\partial_{s}^{2}h_{j_{0};s_{0}}(s)+2^{-1/2}\big{|}$
$\displaystyle\leq\displaystyle\sup_{|s|<1/3L_{K}}\big{|}\omega_{K}^{-1}\cdot\partial_{s}^{2}G_{K}^{+}(s,0)+2^{-1/2}\big{|}$
$\displaystyle\leq\omega_{K}^{-1}\cdot\displaystyle\sup_{|s|<1/3L_{K}}\big{|}\partial_{s}^{2}G_{K}^{+}(s,0)-\partial_{s}^{2}G_{K}(s,0)\big{|}+\delta$
$\displaystyle\leq\omega_{K}^{-1}\cdot\displaystyle\sup_{|s|<1/3L_{K}}\big{|}\partial_{s}^{2}\widetilde{G}_{K-1}^{+}(s,0)-\partial_{s}^{2}G_{K}(s,0)\big{|}+C_{0}\omega_{K}^{-1}\vartheta_{K-1}^{3/4}+\delta$
$\displaystyle\leq
8C_{0}\Theta_{K-1}+4C_{0}\omega_{K-1}^{-1}\vartheta_{K-1}^{3/4}+\delta\leq
8C_{0}\delta_{0}^{1/2}+n^{-1/2}+\delta\leq\delta_{0}^{1/3}.$

Here, in the first bound we applied (11.34), replaced $s$ by
$\omega_{K}^{-1}s$, and used the fact that
$\omega_{K}^{-1}\mathfrak{w}\leq(3L_{K})^{-1}$; in the second we applied
(11.35); in the third we applied the fourth statement of Definition 11.8; in
the fourth we applied (11.28) (and that $\omega_{K-1}=4\omega_{K}$); in the
fifth we applied the facts that $\vartheta_{K-1}\leq 2n^{-13/15}$ and that
$\omega_{K-1}=4\omega_{K}\leq 4n^{-1/25000}$ (both as consequences of (11.16)
and 11.5); and in the sixth we used the fact that $n$ is sufficiently large
and $\delta$ is sufficiently small. Since $\delta_{0}=\delta^{1/2}+(\log
n)^{-1}$ by (11.16), this establishes the first statement of (11.4).

To establish the second statement of (11.4), observe that

$\displaystyle\|h_{j_{0};s_{0}}\|_{\mathcal{C}^{1}}\leq\|G_{K}^{+}\|_{\mathcal{C}^{1}(\mathfrak{S}_{K}^{\prime})}\leq\|\widetilde{G}_{K-1}^{+}\|_{\mathcal{C}^{1}(\mathfrak{R}_{K})}+C_{0}\vartheta_{K-1}^{3/4}\leq
5B+C_{0}\vartheta_{K-1}^{3/4}\leq 10B,$

where the first statement holds by (11.34), the second by the fourth part of
Definition 11.8 (with the fact that
$[-\mathfrak{w},\mathfrak{w}]\times\\{0\\}\subset\mathfrak{S}_{K}^{\prime}$),
the third by (11.29) (with the fact that $\Omega_{K}^{(1)}$ holds) and 11.5,
and the fourth by (11.20) and the fact that $n$ is sufficiently large. This
confirms the second part of (11.4), verifying the proposition. ∎

###  12\. Proofs of Results From Section 11.3

Throughout, we recall the notation from Section 11.3.

####  12.1. Proofs of 11.5 and 11.9

We begin by proving the 11.5, which will proceed inductively, showing that the
events $\Omega_{k}^{(i)}$ are likely if $\Omega_{k-1}$ holds. This is
summarized through the following two lemmas; the first will be shown in
Section 12.2 and the second in Section 12.3.

######  Lemma 12.1.

For sufficiently large $n$ and small $\delta$, we have
$\mathbb{P}\big{[}\Omega_{k-1}\cap(\Omega_{k}^{(1)})^{\complement}\big{]}\leq
2n^{-20}$ for any integer $k\in\llbracket 1,K_{0}\rrbracket$.

######  Lemma 12.2.

For sufficiently large $n$ and small $\delta$, we have
$\mathbb{P}\big{[}\Omega_{0}^{\complement}\big{]}\leq n^{-20}$. Moreover, for
any integer $k\in\llbracket 1,K_{0}\rrbracket$, we have

$\displaystyle\mathbb{P}\Big{[}\Omega_{k}^{(1)}\cap\big{(}\Omega_{k}^{(2)}\big{)}^{\complement}\Big{]}\leq\mathbb{P}\Big{[}\Omega_{k}^{(1)}\cap(\Omega_{k}^{(3)})^{\complement}\Big{]}\leq
n^{-20}.$

Given these results, we can quickly establish 11.9.

###### Proof of 11.9.

It suffices to show the bound

(12.1)
$\displaystyle\mathbb{P}\big{[}\Omega_{k}\big{]}\geq\mathbb{P}[\Omega_{k}^{\prime}]\geq
1-3(k+1)n^{-20},\qquad\text{for each integer $k\in\llbracket
0,K_{0}\rrbracket$},$

from which the lemma follows from taking $k=K_{0}\leq\log n$ (where the last
bound holds by (11.20)). To this end, we induct on $k\in\llbracket
0,K_{0}\rrbracket$; for $k=0$, (12.1) holds by the first statement of 12.2. We
then assume that (12.1) holds for some $k\in\llbracket 0,K_{0}-1\rrbracket$
and show that it continues to hold for $k$ replaced by $k+1$. Since
$\Omega_{k}^{\prime}\subseteq\Omega_{k}$, it suffices to show only the second
inequality in (12.1). This follows from the estimates

$\displaystyle\mathbb{P}[\Omega_{k+1}^{\prime}]$
$\displaystyle\geq\mathbb{P}[\Omega_{k}]-\mathbb{P}\Big{[}\Omega_{k}\cap\big{(}\Omega_{k+1}^{(1)}\big{)}^{\complement}\Big{]}-\mathbb{P}\Big{[}\Omega_{k}\cap\Omega_{k+1}^{(1)}\cap\big{(}\Omega_{k+1}^{(3)}\big{)}^{\complement}\Big{]}$
$\displaystyle\geq
1-3(k+1)n^{-20}-\mathbb{P}\Big{[}\Omega_{k}\cap\big{(}\Omega_{k+1}^{(1)}\big{)}^{\complement}\Big{]}-\mathbb{P}\Big{[}\Omega_{k+1}^{(1)}\cap\big{(}\Omega_{k+1}^{(3)}\big{)}^{\complement}\Big{]}\geq
1-3(k+4)n^{-20},$

where in the first inequality we applied a union bound and (11.33); in the
second we applied the inductive hypothesis; and in the third we applied 12.1
and 12.2. This yields the lemma. ∎

We next establish 11.5.

###### Proof of 11.5.

First observe from (11.16) that for sufficiently large $n$ and small $\delta$
we have $\varsigma_{k}\leq 1/4$ and $n^{-13/15}\leq 1/4$. Hence,
$\vartheta_{k}\leq 1/2$ and so the definition (11.16) of $\Theta_{k}$ yields
$\Theta_{k}\geq\omega_{k}^{-1}\vartheta_{k}^{3/4}\geq\vartheta_{k}$. Thus, for
any integer $k\geq 1$, we have

$\displaystyle\vartheta_{k}\leq\Theta_{k}=\delta_{0}^{3/4}+\displaystyle\sum_{j=0}^{k}\omega_{j}^{-1}\vartheta_{j}^{3/4}$
$\displaystyle\leq\delta_{0}^{3/4}+4\displaystyle\sum_{j=0}^{k}4^{j}(\varsigma_{j}^{3/4}+n^{-1/2})$
$\displaystyle\leq\delta_{0}^{3/4}+4^{k+2}n^{-1/2}+20C_{0}\displaystyle\sum_{j=0}^{k}4^{j}\exp\Big{(}-\displaystyle\frac{3c_{0}}{8}L_{j}^{1/80}\Big{)},$

where we have used the definitions (11.16) of $\Theta_{k}$, $\omega_{j}$,
$\vartheta_{j}$, and $\varsigma_{j}$, with the bound
$\vartheta_{j}^{3/4}=(\varsigma_{j}+n^{-13/15})^{3/4}\leq\varsigma_{j}^{3/4}+n^{-1/2}$.
Since, for sufficiently small $\delta_{0}>0$, we have
$L_{j}=\delta_{0}^{-\sqrt{j+1}}\geq(2c_{0}^{-1}(j+1))^{160}\delta_{0}^{-1/2}$
for each integer $j\geq 0$, it follows that

$\displaystyle\vartheta_{k}\leq\Theta_{k}\leq\delta_{0}^{3/4}+4^{k+2}n^{-1/2}+20C_{0}\displaystyle\sum_{j=0}^{k}\exp\big{(}2j-(j+1)^{2}\delta_{0}^{-1/200}\big{)}\leq
2\delta_{0}^{3/4}+4^{k+2}n^{-1/2},$

for sufficiently large $n$ and small $\delta$. Hence, since for $k\leq(\log
n)^{3/4}$ and sufficiently large $n$ we have $4^{k+2}n^{-1/2}\leq
n^{-1/4}\leq(\log n)^{-1}\leq\delta_{0}\leq\delta_{0}^{3/4}$, it follows that
$\vartheta_{k}\leq\Theta_{k}\leq
3\delta_{0}^{3/4}\leq\varepsilon\delta_{0}^{1/2}$, confirming the first
statement of the lemma.

We next verify (11.20). To establish the first bound there, on $K_{0}$,
observe for $k\leq(\log n)^{1/3}+1$ that for sufficiently large $n$ we have

$\displaystyle\omega_{k}L_{k}^{-1}=4^{-k-1}\delta_{0}^{\sqrt{k+1}}\geq
4^{-2(\log n)^{1/3}}\cdot(\log n)^{-2(\log n)^{1/3}}>e^{-(\log
n)^{2/5}}>3\mathfrak{w},$

which indicates that $K_{0}\geq(\log n)^{1/3}$, since $\omega_{k}L_{k}^{-1}$
is decreasing in $k$ (as $\omega_{k}$ and $L_{k}^{-1}$ are by (11.16)). Here
in the first statement we used the definition (11.16) of $\omega_{k}$ and
$L_{k}$; in the second we used the facts that $\delta_{0}\geq(\log n)^{-1}$
and that $\sqrt{k+1}\leq k+1\leq 2(\log n)^{1/3}$; in the third we used the
fact that $2(\log n)^{1/3}\cdot\log(4\log n)<(\log n)^{2/5}$ for sufficiently
large $n$; and in the fourth we recalled that $\mathfrak{w}=e^{-\sqrt{\log
n}}$. For $k\geq(\log n)^{1/2}$, we have that $\omega_{k}L_{k}^{-1}\leq
4^{-k-1}<e^{\sqrt{\log n}}<3\mathfrak{w}$, indicating that $K_{0}\leq(\log
n)^{1/2}$, verifying the first bound in (11.20).

The second bound in (11.20), given by $e^{(\log n)^{1/6}}\leq L_{K_{0}}\leq
e^{(\log n)^{1/2}}$, follows from the first bound in (11.20), together with
the facts that $L_{k}=\delta_{0}^{-\sqrt{k+1}}$ and that $e^{-(\log
n)^{1/5}}\leq\delta_{0}\leq e^{-1}$ for sufficiently large $n$ and small
$\delta$. The third follows from the fact that
$n_{K_{0}}n^{-1}=\omega_{K_{0}}=4^{-K_{0}-1}\geq e^{-2\sqrt{\log n}-1}\geq
n^{-1/25000}$. The fourth follows from the fact that

$\displaystyle\vartheta_{K_{0}-1}=\varsigma_{K_{0}-1}+n^{-13/15}$
$\displaystyle=n^{-13/15}+5C_{0}\exp\Big{(}-\displaystyle\frac{c_{0}}{2}L_{K_{0}-1}^{1/80}\Big{)}$
$\displaystyle\leq
n^{-13/15}+5C_{0}\exp\Big{(}-\displaystyle\frac{c_{0}}{2}e^{(\log
n)^{1/480}}\Big{)}$ $\displaystyle\leq n^{-13/15}+5C_{0}e^{-(\log n)^{2}}\leq
2n^{-13/15},$

where in the first and second statements we used the definitions (11.16) of
$\vartheta_{K_{0}-1}$ and $\varsigma_{K_{0}-1}$; in the third we used the fact
that $L_{K_{0}-1}=\delta_{0}^{-\sqrt{K_{0}}}\geq\delta_{0}^{(\log
n)^{1/6}}\geq e^{(\log n)^{1/6}}$ (by (11.16) and the first bound in (11.20));
and the fourth and fifth follow since $n$ is sufficiently large. This
establishes (11.20) and thus the second statement of the lemma.

To establish the third statement of the lemma, observe for any real numbers
$a,b\in(0,1/4)$ that $\big{|}\log(a+b)\big{|}\leq|\log a|$. Applying this with
$(a,b)=(\varsigma_{k},n^{-13/15})$ yields for sufficiently large $n$ and small
$\delta$ that

$\displaystyle|\log\vartheta_{k}|^{20}\leq|\log\varsigma_{k}|^{20}=\Big{(}\displaystyle\frac{c_{0}}{2}L_{k}^{1/80}-\log(5C_{0})\Big{)}^{20}\leq
2^{-20}c_{0}^{20}L_{k}^{1/4}\leq\displaystyle\frac{4L_{k}}{3},$

establishing the first bound of the third statement. To establish the second
bound there, observe for any real numbers $a,b\in(0,1/4)$ and $r\in(0,1)$ that
$(a+b)^{-r}\geq 2^{-r}\min\big{\\{}(2a)^{-r},(2b)^{-r}\big{\\}}$. Setting
$(a,b;r)=(\varsigma_{k},n^{-13/15};1/5000)$ yields

(12.2)
$\displaystyle\vartheta_{k}^{-1/5000}\geq\min\big{\\{}(2\varsigma_{k})^{-1/5000},n^{1/10000}\big{\\}}.$

By the definition (11.16), it is quickly verified that for sufficiently large
$L_{k}$ (and hence sufficiently large $n$ and small $\delta)$ we have
$(2\varsigma_{k})^{-1/5000}>4L_{k}/3$. Moreover, we have $4L_{k}/3\leq
2L_{k}\leq 2L_{K_{0}}\leq 2e^{\sqrt{\log n}}\leq n^{1/10000}$, where in the
first and second bounds we used the facts that $L_{k}$ is positive and
increasing in $k$ (by its definition (11.16)); in the third we used the second
statement of (11.20); and in the fourth we used the fact that $n$ is
sufficiently large. Together with (12.2), these two bounds verify the third
statement of the lemma. ∎

####  12.2. Proofs of 11.6, and 11.7, and 12.1

In this section we establish first 12.1, then 11.6, and next 11.7.

###### Proof of 12.1.

Since by (11.23) we have

$\displaystyle\Omega_{k-1}\cap\big{(}\Omega_{k}^{(1)}\big{)}^{\complement}\subseteq\textbf{PFL}^{\bm{x}^{(k-1)}}\Big{(}-\frac{1}{8L_{k}};\displaystyle\frac{1}{4n^{9/10}};2B\Big{)}^{\complement}\cup\textbf{PFL}^{\bm{x}^{(k-1)}}\Big{(}\frac{1}{8L_{k}};\displaystyle\frac{1}{4n^{9/10}};2B\Big{)}^{\complement},$

it suffices by (3.10) and a union bound to show that

(12.3) $\displaystyle\begin{aligned}
&\textbf{PFL}^{\bm{x}}\Big{(}\displaystyle\frac{s_{0}}{n^{1/3}}-\displaystyle\frac{\omega_{k-1}}{8L_{k}};n^{-19/20};B\Big{)}\subseteq\textbf{PFL}^{\bm{x}^{(k-1)}}\Big{(}-\displaystyle\frac{1}{8L_{k}};\displaystyle\frac{1}{4n^{9/10}};2B\Big{)};\\\
&\textbf{PFL}^{\bm{x}}\Big{(}\displaystyle\frac{s_{0}}{n^{1/3}}+\displaystyle\frac{\omega_{k-1}}{8L_{k}};n^{-19/20};B\Big{)}\subseteq\textbf{PFL}^{\bm{x}^{(k-1)}}\Big{(}\displaystyle\frac{1}{8L_{k}};\displaystyle\frac{1}{4n^{9/10}};2B\Big{)}.\end{aligned}$

We only show the first bound in (12.3), as the proof of the second is entirely
analogous. To this end, set $t_{1}=-1/8L_{k}$ and
$s_{1}=n^{-1/3}s_{0}+t_{1}\omega_{k}$, and observe on the event
$\textbf{PFL}^{\bm{x}}(s_{1};n^{-19/20};B)$ that there exists a function
$\gamma_{s_{1}}:[0,1]\rightarrow\mathbb{R}$ such that

(12.4) $\displaystyle\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}x_{j}(s_{1})-\gamma_{s_{1}}(jn^{-1})\big{|}\leq
n^{-19/20};\qquad\big{\|}\gamma_{s_{1}}-\gamma_{s_{1}}(0)\big{\|}_{\mathcal{C}^{50}}\leq
B.$

We then define $\gamma:[0,1]\rightarrow\mathbb{R}$ by rescaling
$\gamma_{s_{1}}$, namely, by setting

(12.5)
$\displaystyle\gamma(x)=\omega_{k-1}^{-1}\cdot\gamma_{s_{1}}\Big{(}\omega_{k-1}x+\displaystyle\frac{j_{0}}{n}-\displaystyle\frac{\omega_{k-1}}{2}\Big{)},$

so that

(12.6)
$\displaystyle\big{\|}\gamma-\gamma(0)\big{\|}_{\mathcal{C}^{50}}\leq\big{\|}\gamma_{s_{1}}-\gamma_{s_{1}}(0)\big{\|}_{\mathcal{C}^{50}}+\big{\|}\gamma_{s_{1}}-\gamma_{s_{1}}(0)\big{\|}_{\mathcal{C}^{1}}\leq
2B,$

where in the last inequalities we used the facts that
$\big{[}\gamma\big{]}_{m}=\omega_{k-1}^{m-1}\cdot[\gamma_{s_{1}}]$, for each
integer $m\geq 0$, and that $\omega_{k-1}\in[0,1]$. We further have for
sufficiently large $n$ that

$\displaystyle\displaystyle\max_{i\in\llbracket
1,n_{k-1}\rrbracket}\bigg{|}x_{i}^{(k-1)}(t_{1})-\gamma\Big{(}\displaystyle\frac{i}{n_{k-1}}\Big{)}\bigg{|}$
$\displaystyle=\omega_{k-1}^{-1}\cdot\displaystyle\max_{i\in\llbracket
1,n_{k-1}\rrbracket}\bigg{|}x_{i+j_{0}-n_{k-1}/2}(s_{1})-\gamma_{s_{1}}\Big{(}\displaystyle\frac{i+j_{0}}{n}-\displaystyle\frac{\omega_{k-1}}{2}\Big{)}\bigg{|}$
$\displaystyle\leq n^{-11/12}\leq\displaystyle\frac{1}{4n^{9/10}}.$

Here, in the first statement we used (12.5), with the facts that
$n_{k-1}=\omega_{k-1}n$ and that $x_{i}^{(k-1)}(t_{1})=\omega_{k-1}^{-1}\cdot
x_{i+j_{0}-n_{k-1}/2}(s_{1})$ (which holds by (11.19) and (3.9)); in the
second, we used the first statement of (12.4) and the fact that
$\omega_{k-1}^{-1}\leq\omega_{K_{0}}^{-1}\leq 4^{\sqrt{\log n}+1}<n^{1/30}$
(by (11.20)) for sufficiently large $n$; and in the third we used the fact
that $n$ is sufficiently large. This, together with (12.6) (and Definition
3.11) yields (12.3) and thus the lemma. ∎

###### Proof of 11.6.

We induct on $k\in\llbracket 1,K_{0}-1\rrbracket$. To verify the lemma in the
case $k=1$, observe by (11.26) and (11.21) that
$\widetilde{G}^{+}_{k-1}(s,x)=4G_{0}^{+}(s/4,x/4)=4G_{0}(s/4,x/4)$. Moreover,
by (11.18) (with the fact that $\omega_{0}=4\omega_{1}$), we have
$G_{1}(s,x)=4G_{0}(s/4,x/4)$. Thus, $G_{1}=\widetilde{G}_{0}$, which gives
(11.28) at $k=1$.

So, fix some integer $k\geq 2$ and assume that (11.28) holds for $k$ there
equal to $k-1$ here. In what follows, we restrict to the event
$\Omega_{k}^{(1)}$ and then must show that (11.28) holds. To this end, first
observe since we have restricted to the event
$\Omega_{k-1}^{(1)}\subseteq\Omega_{k-1}\subseteq\Omega_{k}^{(1)}$, the
inductive hypothesis (with the definition (11.16) of $\Theta_{k-2}$) yields

(12.7)
$\displaystyle\big{[}\widetilde{G}_{k-2}^{+}-G_{k-1}\big{]}_{1;\mathfrak{R}_{k-1}}+\omega_{k-2}^{-1}\cdot\displaystyle\sum_{d=2}^{50}\big{[}\widetilde{G}_{k-2}^{+}-G_{k-1}\big{]}_{d;\mathfrak{R}_{k-1}}\leq
2C_{0}\delta_{0}^{3/4}+2C_{0}\displaystyle\sum_{k=1}^{k-2}\omega_{j}^{-1}\vartheta_{j}^{3/4}.$

Now, define
$\widehat{G}_{k-2}^{+}:4\cdot\mathfrak{R}_{k-1}\rightarrow\mathbb{R}$ by
rescaling $\widetilde{G}_{k-2}^{+}$, namely, by setting

(12.8)
$\displaystyle\widehat{G}_{k-2}^{+}(s,x)=4\widetilde{G}_{k-2}^{+}\Big{(}\displaystyle\frac{s}{4},\displaystyle\frac{x}{4}\Big{)},$

for each $(s,x)\in 4\cdot\mathfrak{R}_{k-1}$. Since we have from (11.18) (with
the fact that $\omega_{k-1}=4\omega_{k}$) that $G_{k}(s,x)=4G_{k-1}(s/4,x/4)$,
this yields

$\displaystyle\nabla\widehat{G}_{k-2}^{+}(s,x)-\nabla
G_{k}(s,x)=\nabla\widetilde{G}_{k-2}^{+}\Big{(}\displaystyle\frac{s}{4},\displaystyle\frac{x}{4}\Big{)}-\nabla
G_{k-1}\Big{(}\displaystyle\frac{s}{4},\displaystyle\frac{x}{4}\Big{)},\qquad\qquad\text{for
each $(s,x)\in\mathfrak{R}_{k}$};$
$\displaystyle\big{[}\widehat{G}_{k-2}^{+}-G_{k}\big{]}_{d;\mathfrak{R}_{k}}=4^{1-d}\cdot\big{[}\widetilde{G}_{k-2}^{+}-G_{k-1}\big{]}_{d;\frac{1}{4}\cdot\mathfrak{R}_{k}},\qquad\qquad\qquad\qquad\text{for
each $d\geq 2$}.$

Together with (12.7) and the facts that
$\frac{1}{4}\cdot\mathfrak{R}_{k-1}\subseteq\mathfrak{R}_{k}$ and
$\omega_{k-2}=4\omega_{k-1}$, this gives

$\displaystyle\big{[}\widehat{G}_{k-2}^{+}-G_{k}\big{]}_{1;\mathfrak{R}_{k}}+\omega_{k-1}^{-1}\cdot\displaystyle\sum_{d=2}^{50}4^{d-2}\cdot\big{[}\widehat{G}_{k-2}^{+}-G_{k}\big{]}_{d;\mathfrak{R}_{k}}\leq
2C_{0}\delta_{0}^{3/4}+2C_{0}\displaystyle\sum_{k=1}^{k-2}\omega_{j}^{-1}\vartheta_{j}^{3/4}.$

Hence, to verify that (11.28) holds, it suffices to show that

(12.9)
$\displaystyle\big{[}\widetilde{G}_{k-1}^{+}-\widehat{G}_{k-2}^{+}\big{]}_{1;\mathfrak{R}_{k}}\leq
C_{0}\vartheta_{k-1}^{3/4};\qquad\displaystyle\sum_{d=2}^{50}4^{d-2}\cdot\big{[}\widetilde{G}_{k-1}^{+}-\widehat{G}_{k-2}^{+}\big{]}_{d;\mathfrak{R}_{k}}\leq
C_{0}\vartheta_{k-1}^{3/4}.$

By (12.8) and the fact from (11.26) that
$\widetilde{G}_{k-1}^{+}(s,x)=4G_{k-1}^{+}(s/4,x/4)$, we have

$\displaystyle\nabla\widehat{G}_{k-2}^{+}(s,x)-\nabla\widetilde{G}_{k-1}^{+}(s,x)=\nabla\widetilde{G}_{k-2}^{+}\Big{(}\displaystyle\frac{s}{4},\displaystyle\frac{x}{4}\Big{)}-\nabla
G_{k-1}^{+}\Big{(}\displaystyle\frac{s}{4},\displaystyle\frac{x}{4}\Big{)},\qquad\text{for
each $(s,x)\in 4\cdot\mathfrak{R}_{k-1}$};$
$\displaystyle\big{[}\widehat{G}_{k-2}^{+}-\widetilde{G}_{k-1}^{+}\big{]}_{d;\mathfrak{R}_{k}}=4^{1-d}\cdot\big{[}\widetilde{G}_{k-2}^{+}-G_{k-1}^{+}\big{]}_{d;\frac{1}{4}\cdot\mathfrak{R}_{k}},\qquad\qquad\qquad\quad\text{for
each $d\geq 2$}.$

and so (again since $\mathfrak{R}_{k}\subseteq\mathfrak{R}_{k-1}\subseteq
4\cdot\mathfrak{R}_{k-1}$) to confirm (12.9) we may show that

$\displaystyle\big{[}G_{k-1}^{+}-\widetilde{G}_{k-2}^{+}\big{]}_{1;\frac{1}{4}\cdot\mathfrak{R}_{k}}\leq
C_{0}\vartheta_{k-1}^{3/4};\qquad\displaystyle\sum_{d=2}^{50}\big{[}G_{k-1}^{+}-\widetilde{G}_{k-2}^{+}\big{]}_{d;\frac{1}{4}\cdot\mathfrak{R}_{k}}\leq
C_{0}\vartheta_{k-1}^{3/4}.$

Both follow from the fourth property in Definition 11.8 (with the $k$ there
equal to $k-1$ here), and the fact that $\mathfrak{R}_{k}\subseteq
4\cdot\mathfrak{S}_{k-1}^{\prime}$ (by (11.17) and (11.31)). This verifies
(12.9) and thus the lemma. ∎

###### Proof of 11.7.

Throughout this proof, we restrict to the event $\Omega_{k}^{(1)}$; we then
must show that (11.30) holds. We only verify the bound
$\big{\|}\mathfrak{y}_{i}^{(k)}-\widetilde{\mathfrak{z}}_{i}^{(k-1)}\big{\|}_{\mathcal{C}^{0}}\leq\vartheta_{k-1}$
at $i=0$, as the proof that it holds at $i=1$ is entirely analogous. To this
end, observe that

(12.10) $\displaystyle\begin{aligned}
\big{\|}\mathfrak{y}_{0}^{(k)}-\widetilde{\mathfrak{z}}_{0}^{(k-1)}\big{\|}_{\mathcal{C}^{0}}&=\displaystyle\sup_{|x|\leq
2/3}\bigg{|}\widetilde{\gamma}_{-1/2L_{k}}^{(k-1)}(x)-\widetilde{G}_{k-1}^{+}\Big{(}-\displaystyle\frac{1}{2L_{k}},x\Big{)}\bigg{|}\\\
&\leq\displaystyle\sup_{j\in\llbracket-2n_{k}/3,2n_{k}/3\rrbracket}\bigg{|}\widetilde{\gamma}_{-1/2L_{k}}^{(k-1)}\Big{(}\displaystyle\frac{j}{n_{k}}\Big{)}-\widetilde{G}_{k-1}^{+}\Big{(}-\displaystyle\frac{1}{2L_{k}},\displaystyle\frac{j}{n_{k}}\Big{)}\bigg{|}+\displaystyle\frac{10B}{n_{k}},\end{aligned}$

where in the statement we used (11.27), and in the second we used the facts
that $\big{[}\widetilde{\gamma}_{-1/2L_{k}}^{(k-1)}\big{]}_{1}\leq 4B$ and
$\big{[}\widetilde{G}_{k-1}^{+}\big{]}_{1}\leq 5B$ (where the former holds by
(11.25) and the latter by (11.29)). We also have

(12.11) $\displaystyle\begin{aligned}
&\displaystyle\sup_{j\in\llbracket-2n_{k}/3,2n_{k}/3\rrbracket}\bigg{|}\widetilde{\gamma}_{-1/2L_{k}}^{(k-1)}\Big{(}\displaystyle\frac{j}{n_{k}}\Big{)}-\widetilde{G}_{k-1}^{+}\Big{(}-\displaystyle\frac{1}{2L_{k}},\displaystyle\frac{j}{n_{k}}\Big{)}\bigg{|}\\\
&\qquad\leq\displaystyle\sup_{j\in\llbracket-2n_{k}/3,2n_{k}/3\rrbracket}\bigg{|}x_{j+n_{k}/2}^{(k)}\Big{(}-\displaystyle\frac{1}{2L_{k}}\Big{)}-4G_{k-1}^{+}\Big{(}-\displaystyle\frac{1}{8L_{k}},\displaystyle\frac{j}{n_{k-1}}\Big{)}\bigg{|}+n^{-9/10}\\\
&\qquad=4\cdot\displaystyle\sup_{j\in\llbracket-
n_{k-1}/6,n_{k-1}/6\rrbracket}\bigg{|}x_{j+n_{k-1}/2}^{(k-1)}\Big{(}-\displaystyle\frac{1}{8L_{k}}\Big{)}-G_{k-1}^{+}\Big{(}-\displaystyle\frac{1}{8L_{k}},\displaystyle\frac{j}{n_{k-1}}\Big{)}\bigg{|}+n^{-9/10}.\end{aligned}$

where in the first statement we used (11.26) and (11.25) (with the facts that
$n_{k-1}=4n_{k}$ and that we are restricting to $\Omega_{k}^{(1)}$), and in
the second we used the facts that
$x_{j+n_{k}/2}^{(k)}(s)=4x_{j+n_{k-1}/2}^{(k-1)}(s/4)$ (which holds by (11.19)
and the facts that $\omega_{k-1}=4\omega_{k}$ and $n_{k-1}=4n_{k}$). Next,
since we are restricting to the event
$\Omega_{k-1}^{(2)}\subseteq\Omega_{k-1}\subseteq\Omega_{k}^{(1)}$, we have by
(11.32) (with the fact that
$(-1/8L_{k},jn_{k-1}^{-1})\notin\frac{1}{4}\cdot\mathfrak{R}_{k}$, by
(11.17)), we have

(12.12) $\displaystyle\displaystyle\sup_{|j|\leq
n_{k-1}/6}\bigg{|}x_{j+n_{k-1}/2}^{(k-1)}\Big{(}-\displaystyle\frac{1}{8L_{k}}\Big{)}-G_{k-1}^{+}\Big{(}-\displaystyle\frac{1}{8L_{k}},\displaystyle\frac{j}{n_{k-1}}\Big{)}\bigg{|}\leq\displaystyle\frac{\vartheta_{k-1}}{5}.$

Combining (12.10), (12.11), (12.12), and the fact that
$4\vartheta_{k-1}/5+n^{-9/10}+10Bn_{k}^{-1}\leq
4\vartheta_{k-1}/5+2n^{-9/10}\leq\vartheta_{k-1}$ for sufficiently large $n$
(due to (11.20) and the fact that $\vartheta_{k-1}\geq n^{-13/15}$ by (11.16))
we deduce that (11.30) holds; this establishes the lemma. ∎

####  12.3. Proof of 12.2

In this section we establish 12.2, following the ideas outlined in the
beginning of Section 11.2 and Section 11.3.

###### Proof of 12.2.

Both statements of the lemma will follow from a suitable application of 10.27.
We begin with the first statement, indicating that $\Omega_{0}$ is likely.

To this end, apply 10.27 (translated by $(-\xi,0)$) to $\bm{x}$ from (3.9)
(which are non-intersecting Brownian bridges with variances $n^{-1}$, by 4.4),
with the $(L,B,\varkappa,m,\delta;G)$ there given by
$(1/2\xi,2B,\delta,50,1/25000;G)$ here; 10.26 is then verified by (3.8) and
the hypotheses of 11.2. This yields a constant $c_{1}=c_{1}(\varepsilon,B)>0$
such that

(12.13)
$\displaystyle\mathbb{P}\big{[}\mathscr{E}_{0}^{\complement}\big{]}\leq
c_{1}^{-1}e^{-c_{1}(\log
n)^{2}},\quad\text{where}\quad\mathscr{E}_{0}=\Bigg{\\{}\displaystyle\sup_{|s|\leq\xi}\bigg{(}\displaystyle\max_{j\in\llbracket
1,n\rrbracket}\big{|}x_{j}(s)-G(s,jn^{-1})\big{|}\bigg{)}\leq\delta+n^{-11/12}\Bigg{\\}}.$

Then, observe that

$\displaystyle\mathscr{E}_{0}$
$\displaystyle\subseteq\Bigg{\\{}\displaystyle\sup_{|s-s_{0}n^{-1/3}|\leq\omega_{0}/2L_{0}}\bigg{(}\displaystyle\max_{-n_{0}/2<|j|\leq
n_{0}/2}\big{|}x_{j_{0}+j}(s)-G(s,jn^{-1}+j_{0}n^{-1})\big{|}\bigg{)}\leq\delta+n^{-11/12}\Bigg{\\}}$
$\displaystyle=\Bigg{\\{}\displaystyle\sup_{(j/n_{0},s)\in\overline{\mathfrak{R}}_{0}}\big{|}x_{j+n_{0}/2}^{(0)}(s)-G_{0}(s,jn_{0}^{-1})\big{|}\leq\displaystyle\frac{1}{4}\cdot(\delta+n^{-11/12})\Bigg{\\}}\subseteq\Omega_{0},$

where in the first statement we restricted the range of $(j,s)$ in the
definition (12.13) of $\mathscr{E}_{0}$ (using the facts that $n/3\leq
j_{0}\leq 2n/3$; that $n_{0}=n/4$ by (11.16); that $|n^{-1/3}s_{0}|\leq\xi/2$;
and that $\omega_{0}/2L_{0}<\delta_{0}^{1/2}<(2B)^{-1}\leq\xi/2$ for
sufficiently small $\delta_{0}$, again by (11.16)); in the second we used the
fact that $G_{0}(s,jn_{0}^{-1})=4G(n^{-1/3}s+\omega_{0}x,jn^{-1}+j_{0}n^{-1})$
(by (11.18) and the fact that $\omega_{0}=1/4$), the fact that
$x_{j+n_{0}/2}^{(0)}(s)=4x_{j+j_{0}}(n^{-1/3}s_{0}+\omega_{0}s)$ (by (11.19),
(3.9), and the fact that $\omega_{0}=1/4$) and the definition (11.17) of
$\mathfrak{R}_{k}$; and in the third we used (11.21), the definition (11.22)
of $\Omega_{0}$, and the fact that $\delta_{0}^{3/4}>\delta+n^{-11/2}$ for
sufficiently large $n$ and small $\delta$ (by (11.16)). Together with (12.13),
this yields the first statement of the lemma.

We next establish the second. Throughout the remainder of this proof, we
condition on $\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{k})$ and restrict
to the event $\Omega_{k}^{(1)}$ (which, as stated below (11.23), is measurable
with respect to $\mathcal{F}_{\operatorname{ext}}(\mathsf{R}_{k})$); since
$\Omega_{k}^{(3)}\subseteq\Omega_{k}^{(2)}$, it then suffices to show that
$\Omega_{k}^{(3)}$ holds with probability at least $1-n^{-20}$. To this end,
for each index $\pm\in\\{+,-\\}$, define the sequences
$\bm{u}^{(k;\pm)},\bm{v}^{(k;\pm)},\bm{u}^{(k)},\bm{v}^{(k)}\in\overline{\mathbb{W}}_{n_{k}}$
and functions
$f_{k}^{\pm},g_{k}^{\pm},f_{k},g_{k}:[-1/2L_{k},1/2L_{k}]\rightarrow\mathbb{R}$
by for each $j\in\llbracket 1-n_{k}/2,n_{k}/2\rrbracket$ and
$s\in[-1/2L_{k},1/2L_{k}]$ setting

(12.14) $\displaystyle\begin{aligned}
&u_{j+n_{k}/2}^{(k;\pm)}=G_{k}^{\pm}\Big{(}-\displaystyle\frac{1}{2L_{k}};\displaystyle\frac{j}{n_{k}}\Big{)}\pm
n^{-9/10};\qquad\bm{u}^{(k)}=\bm{x}^{(k)}\Big{(}-\displaystyle\frac{1}{2L_{k}}\Big{)};\\\
&v_{j+n_{k}/2}^{(k;\pm)}=G_{k}^{\pm}\Big{(}-\displaystyle\frac{1}{2L_{k}};\displaystyle\frac{j}{n_{k}}\Big{)}\pm
n^{-9/10};\qquad\bm{v}^{(k)}=\bm{x}^{(k)}\Big{(}\displaystyle\frac{1}{2L_{k}}\Big{)};\\\
&f_{k}^{\pm}(s)=G_{k}^{\pm}\pm
n^{-9/10}\Big{(}s,\displaystyle\frac{1}{2}\Big{)};\qquad
g_{k}^{\pm}(s)=G_{k}^{\pm}\Big{(}s,-\displaystyle\frac{1}{2}\Big{)}\pm
n^{-9/10};\\\ &f_{k}(s)=x_{n_{k}+1}^{(k)}(s);\qquad
g_{k}(s)=x_{0}^{(k)}(s).\end{aligned}$

For each index $\pm\in\\{+,-\\}$, denote the line ensemble
$\bm{x}^{(k;\pm)}=\big{(}x_{1}^{(k;\pm)},x_{2}^{(k;\pm)},\ldots,x_{n_{k}}^{(k;\pm)}\big{)}\in\llbracket
1,n_{k}\rrbracket\times\mathcal{C}\big{(}[-1/2L_{k},1/2L_{k}]\big{)}$ sampled
from the measure
$\mathsf{Q}_{f_{k}^{\pm};g_{k}^{\pm}}^{\bm{u}^{(k;\pm)};\bm{v}^{(k;\pm)}}(n_{k}^{-1})$.
Further denote the events

$\displaystyle\mathscr{E}_{k}^{\pm}=\Bigg{\\{}\displaystyle\max_{j\in\llbracket
1-n_{k}/2,n_{k}/2\rrbracket}\Big{|}x_{j+n_{k}/2}^{(k;\pm)}(s)-\big{(}G_{k}^{\pm}(s,jn_{k}^{-1})\pm
n^{-9/10}\big{)}\Big{|}\leq
n^{-9/10}\Bigg{\\}};\qquad\mathscr{E}_{k}=\mathscr{E}_{k}^{-}\cap\mathscr{E}_{k}^{+}.$

We next show $\mathscr{E}_{k}$ is likely, by applying 10.27 (translated by
$(-1/2L_{k},-1/2)$) with the
$(n;\varepsilon,\delta,B_{0},m;\varkappa,L;G;\bm{u},\bm{v})$ there equal to
$(n_{k};\varepsilon/4,1/25000,2C_{0},45;0,L_{k};G_{k}^{\pm}\pm
n^{-9/10};\bm{u}^{(k;\pm)};\bm{v}^{(k;\pm)})$ here. To verify 10.26, the first
statement in (10.15) there follows from (11.20), the fact that $L_{k}\geq 1$
is increasing in $k$, and the estimate $e^{\sqrt{\log n}}\leq n^{1/25000}$;
the second and third statements in (10.15) are verified by Definition 11.8;
and the bound (10.16) there is verified by (12.14). This yields a constant
$c_{2}=c_{2}(\varepsilon,B)>0$ such that

(12.15)
$\displaystyle\mathbb{P}\big{[}\mathscr{E}_{k}^{\complement}\big{]}\leq
c_{2}^{-1}e^{-c_{2}(\log n_{k})^{2}}\leq n^{-20},$

where here we have implicitly used the fact that $\log n_{k}\geq(\log n)/2$
(as $n_{k}\geq n_{K_{0}}\geq n^{1/2}$ by (11.20) and the fact that
$n_{k}=\omega_{k}n$ is decreasing in $k$).

Now, observe from (11.19), 4.4, and the Brownian Gibbs property that the
family $\bm{x}^{(k)}$ of non-intersecting Brownian bridges has law
$\mathsf{Q}_{f_{k};g_{k}}^{\bm{u}^{(k)};\bm{v}^{(k)}}(n_{k}^{-1})$. We claim
that it is possible to couple the three famlies of non-intersecting Brownian
bridges $\big{(}\bm{x}^{(k;-)},\bm{x}^{(k)},\bm{x}^{(k;+)}\big{)}$ so that

(12.16) $\displaystyle x_{j}^{(k;-)}(s)\leq x_{j}^{(k)}(s)\leq
x_{j}^{(k;+)}(s),\qquad\text{almost surely, for each $(j,s)\in\llbracket
1,n_{k}\rrbracket\times\Big{[}-\displaystyle\frac{1}{2L_{k}},\displaystyle\frac{1}{2L_{k}}\Big{]}$}.$

See the right side of Figure 3.10 for a depiction. To this end, it suffices by
height monotonicity (4.6) to show that

(12.17) $\displaystyle\begin{aligned}
\bm{u}^{(k;-)}\leq\bm{x}^{(k)}&\Big{(}-\displaystyle\frac{1}{2L_{k}}\Big{)}\leq\bm{u}^{(k;+)};\qquad\bm{v}^{(k;-)}\leq\bm{x}^{(k)}\Big{(}\displaystyle\frac{1}{2L_{k}}\Big{)}\leq\bm{v}^{(k;+)};\\\
&f_{k}^{-}\leq x_{n_{k}+1}^{(k)}\leq f_{k}^{+};\qquad g_{k}^{-}\leq
x_{0}^{(k)}\leq g_{k}^{+}.\end{aligned}$

To do this, observe for any $j\in\llbracket 1-n_{k}/2,n_{k}/2\rrbracket$ that

$\displaystyle
u_{j+n_{k}/2}^{(k;-)}(0)\leq\mathfrak{y}_{0}^{(k)}(jn_{k}^{-1})-n^{-9/10}$
$\displaystyle=\widetilde{\gamma}_{-1/2L_{k}}^{(k-1)}(jn_{k}^{-1})-n^{-9/10}\leq
x_{j+n_{k}/2}^{(k)}\Big{(}-\displaystyle\frac{1}{2L_{k}}\Big{)},$

where in the first statement we applied (12.14) and the second statement of
Definition 11.8; in the second we applied (11.27); and in the third we applied
(11.25). This shows that $\bm{u}^{(k;-)}\leq\bm{x}^{(k)}(-1/2L_{k})$. By
similar reasoning we also have $\bm{x}^{(k)}(-1/2L_{k})\leq\bm{u}^{(k;+)}$,
establishing the first statement of (12.17); the proof of the second is
entirely analogous and is thus omitted. To establish the third, observe that

$\displaystyle
x_{n_{k}+1}^{(k)}(s)=4x_{5n_{k-1}/8+1}^{(k-1)}\Big{(}\displaystyle\frac{s}{4}\Big{)}$
$\displaystyle\leq
4G_{k-1}^{+}\Big{(}\displaystyle\frac{s}{4},\displaystyle\frac{1}{8}+\displaystyle\frac{1}{n_{k-1}}\Big{)}+\displaystyle\frac{4\vartheta_{k-1}}{5}$
$\displaystyle\leq
4G_{k-1}^{+}\Big{(}\displaystyle\frac{s}{4},\displaystyle\frac{1}{8}\Big{)}+\displaystyle\frac{4\vartheta_{k-1}}{5}+\displaystyle\frac{2C_{0}}{n_{k}}$
$\displaystyle\leq
4G_{k-1}^{+}\Big{(}\displaystyle\frac{s}{4},\displaystyle\frac{1}{8}\Big{)}+\vartheta_{k-1}\leq\widetilde{G}_{k-1}^{+}\Big{(}s,\displaystyle\frac{1}{2}\Big{)}+\vartheta_{k-1}\leq
G_{k}^{+}\Big{(}s,\displaystyle\frac{1}{2}\Big{)}\leq f_{k}^{+}(s).$

Here, in the first statement we used that
$x_{n_{k}+1}^{(k)}(s)=4x_{n_{k-1}/2+n_{k}/2+1}^{(k-1)}(s/4)=4x_{5n_{k-1}/8+1}^{(k-1)}(s/4)$
(which follows from (11.19), with the equalities $\omega_{k-1}=4\omega_{k}$
and $n_{k-1}=4n_{k}$); in the second we used (11.32), the fact that
$(s/4,1/8+n_{k-1}^{-1})\in\mathfrak{R}_{k-1}\setminus\frac{1}{4}\cdot\mathfrak{R}_{k}$
for $s\in[-1/2L_{k},1/2L_{k}]$ (by (11.17)), and our restriction to the event
$\Omega_{k}^{(1)}\subseteq\Omega_{k-1}\subseteq\Omega_{k-1}^{(2)}$; in the
third we used the third statement of Definition 11.8; in the fourth we used
the bound $\vartheta_{k-1}\geq n^{-13/15}\geq 5C_{0}n_{k}^{-1}$ (which holds
for sufficiently large $n$ by (11.16) and (11.20)); in the fifth we used
(11.26); in the sixth we used the sixth statement of Definition 11.8; and in
the seventh we used (12.14). Similar reasoning indicates that
$x_{n_{k}+1}^{(k)}\geq f_{k}^{-}$, which yields the third statement of
(12.17); the proof of the fourth is entirely analogous. This verifies (12.17)
and thus (12.16).

In view of (12.16) and (12.15), we have that

$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=1-n_{k}/2}^{n_{k}/2}\bigcap_{|s|\leq
1/2L_{k}}\big{\\{}G_{k}^{-}(s,jn_{k}^{-1})-2n^{-9/10}\leq
x_{j+n_{k}/2}^{(k)}(s)\leq G_{k}^{+}(s,jn_{k}^{-1}$
$\displaystyle)+2n^{-9/10}\big{\\}}\Bigg{]}$
$\displaystyle\geq\mathbb{P}[\mathscr{E}_{k}]\geq 1-n^{-20}.$

Since by (11.16) and the fifth statement of Definition 11.8 we have
$\big{|}G_{k}^{-}(s,x)-G_{k}^{+}(s,x)\big{|}\leq
C_{0}e^{-c_{0}L_{k}^{1/8}}\leq\varsigma_{k}/5$ whenever $|x|\leq 1/4$, it
follows that

(12.18)
$\displaystyle\mathbb{P}\Bigg{[}\bigcap_{j=-n_{k}/4}^{n_{k}/4}\bigcap_{|s|\leq
1/2L_{k}}\Big{\\{}\big{|}x_{j+n_{k}/2}^{(k)}(s)-G_{k}^{+}(s,jn_{k}^{-1})\big{|}\leq\displaystyle\frac{\varsigma_{k}}{5}+4n^{-9/10}\Big{\\}}\Bigg{]}\geq
1-n^{-20}.$

Due to the bound $\vartheta_{k}/5=(n^{-13/15}+\varsigma_{k})/5\geq
4n^{-9/10}+\varsigma_{k}/5$, the event on the left side of (12.18) is
contained in $\Omega_{k}^{(3)}$; this verifies 12.2. ∎

##  Chapter 4 Limit Shapes Near the Edge

In this chapter we analyze how bridge-limiting measure processes
$\bm{\mu}^{\star}=(\mu_{t})$ behave near the edges of their supports; see 14.1
below. In what follows, we recall the notation on bridge-limiting measure
processes from Definition 10.2 (and, more broadly, from Section 10).

###  13\. Density Estimates for Limit Shapes

####  13.1. Free Convolution Estimates

In this section we collect some estimates on free convolution measures, which
are subject to the following assumption that bounds their integrals. In what
follows, we recall notation on free convolutions from Section 4.3.

######  Assumption 13.1.

Let $B,L\geq 1$ be real numbers, and let $\tau\in[B^{-1},B]$ be a real number;
let $\nu\in\mathscr{P}_{\operatorname{fin}}$ be a measure with total mass
$\nu(\mathbb{R})=L^{3/2}$ that is supported on $[-BL,0]$. We denote the
measure $\nu_{\tau}=\nu\boxplus\mu_{\operatorname{sc}}^{(\tau)}$, which is the
free convolution of $\nu$ with the rescaled semicircle distribution. As
mentioned in Section 4.3, $\nu_{\tau}$ admits a density with respect to
Lebesgue measure, which we denote by $\varrho_{\tau}\in L^{1}(\mathbb{R})$.

######  Assumption 13.2.

Adopting 13.1, further assume that $\varrho_{\tau}$ satisfies

(13.1) $\displaystyle\int_{x}^{\infty}\varrho_{\tau}(y)dy\leq
B|x|^{3/2},\qquad\text{for each $x\in[-BL,-1]$}.$

We then have the following two propositions. The former, established in
Section 24.2 below, bounds the support of $\varrho_{\tau}$ under 13.1 and
bounds its magnitude under 13.2. The latter, established in Section 24.4
below, bounds the derivatives of $\varrho_{\tau}$ under 13.2, assuming a lower
bound on $\varrho_{\tau}$ (made precise through the function $\gamma_{\tau}$
in (13.3) below).

######  Proposition 13.3.

For any real number $B\geq 1$, there exists a constant $C=C(B)>1$ such that
the following holds.

  1. (1)

Adopting 13.1, we have
$\operatorname{supp}\varrho_{\tau}\subseteq[-CL,CL^{3/4}]$.

  2. (2)

If we further adopt 13.2 then

(13.2) $\displaystyle\varrho_{\tau}(x)\leq
C\max\\{1,-x\\}^{3/4},\qquad\text{for each $x\in{\mathbb{R}}$}.$

######  Proposition 13.4.

For any integer $\ell\geq 1$ and real numbers $A\geq 1$ and $B\geq 3$, there
exist constants $\varepsilon=\varepsilon(A,B)>0$ and $C=C(\ell,A,B)>1$ such
that the following holds. Adopt 13.1 and 13.2. Defining the function
$\gamma_{\tau}:[0,L^{3/2}]\mapsto{\mathbb{R}}$ by for each $y\in[0,L^{3/2}]$
setting

(13.3)
$\displaystyle\gamma_{\tau}(y)=\sup\left\\{x\in\mathbb{R}:\int_{x}^{\infty}\varrho_{\tau}(u)du\geq
y\right\\},$

we further assume the following two bounds.

  1. (1)

We have $\gamma_{\tau}(B)\geq-A$.

  2. (2)

For any $B^{-1}\leq y\leq y^{\prime}\leq B$ with
$y^{\prime}-y\geq\varepsilon$, we have
$\big{|}\gamma_{\tau}(y)-\gamma_{\tau}(y^{\prime})\big{|}\leq
A|y-y^{\prime}|$.

Then, we have

(13.4)
$\displaystyle\big{\|}\gamma_{\tau}-\gamma_{\tau}(1)\big{\|}_{\mathcal{C}^{\ell}([2/B,B/2])}\leq
C.$

####  13.2. Density Upper and Lower Bound Estimates

In this section we obtain upper and lower bounds for the density associated
with a bridge-limiting measure process. We begin by stating three assumptions,
which will be used at various points below (though not necessarily all at
once). The first sets notation for the types of boundary measures; bridge-
limited measure processes; and associated inverted height functions, inverted
liquid regions, and density processes that we will consider in this chapter.
In what follows, we recall the inverted height function and density process
associated with a bridge-limiting measure process from Definition 10.4 (the
latter of which exists by the first part of 10.5), and the associated inverted
liquid region from Definition 10.6.

######  Assumption 13.5.

Let $L\geq B\geq 10$ be real numbers and
$\mu_{0},\mu_{1}\in\mathscr{P}_{\operatorname{fin}}$ be two measures with
total masses $\mu_{0}(\mathbb{R})=L^{3/2}=\mu_{1}(\mathbb{R})$, satisfying
$\operatorname{supp}\mu_{0}\subseteq[-BL,0]$ and
$\operatorname{supp}\mu_{1}\subseteq[-BL,0]$. Let $\bm{\mu}=(\mu_{t})$ denote
the bridge-limiting measure process on $[0,1]$ with boundary data
$(\mu_{0};\mu_{1})$. Further denote the associated density process by
$(\varrho_{t})$; height function by
$H:[0,1]\times\mathbb{R}\rightarrow[0,L^{3/2}]$ as in (10.3); inverted height
function by $G:[0,1]\times[0,L^{3/2}]\rightarrow{\mathbb{R}}$ as in (10.4);
liquid region by $\Omega\subseteq(0,1)\times\mathbb{R}$; inverted liquid
region by $\Omega^{\operatorname{inv}}\subseteq(0,1)\times(0,L^{3/2})$; the
function $u:\Omega\rightarrow\mathbb{R}$ as in (10.6); and the complex slope
$f:\Omega\rightarrow\overline{\mathbb{H}}$ as in (10.8). Further define the
function $\gamma:[0,1]\rightarrow\mathbb{R}$ by setting $\gamma(t)=G(t,0)$ for
each $t\in[0,1]$.

Observe by (10.4) that the curve $\gamma(t)$ traces the upper edge for the
support of $\bm{\mu}$. We sometimes refer to it as the _arctic boundary_
associated with $\bm{\mu}$; see the left side of Figure 3.5. We first show the
following result indicating that $G$, $u$, and $\varrho$ are real analytic on
$\Omega$.

######  Lemma 13.6.

Adopting 13.5, the functions $G$, $u$, and $\varrho$ are real anaytic on
$\Omega$.

###### Proof.

Fix a point $(t_{0},x_{0})\in\Omega$. By Definition 10.6,
$\varrho_{t_{0}}(y_{0})>0$ and so $\partial_{y}G(t_{0},y_{0})<0$. Since $G$ is
smooth on $\Omega$ by 10.8, there exist a real number
$\varepsilon=\varepsilon(t_{0},x_{0})>0$ and a neighborhood
$U=U(t_{0},x_{0})\subset\Omega$ containing $(t_{0},x_{0})$ such that
$-\varepsilon^{-1}<\partial_{y}G(t,x)<-\varepsilon$, for each $(t,x)\in U$.
Thus, $G\in\operatorname{Adm}_{\varepsilon}(U)$ (recall from Definition
10.20). Furthermore, by 10.17, $G$ solves (10.14) on $U$, implying by 10.23
that $G$ is real analytic on $U$. By (10.6) and (10.7), this implies that $u$
and $\varrho$ are also real analytic on $\Omega$ (where for $\varrho$ we used
the fact that $\partial_{y}G$ is bounded away from $0$ on $U$). Since
$(t_{0},x_{0})\in\Omega$ was arbitrary, this confirms the lemma. ∎

The next two assumptions impose estimates on the boundary measures $\mu_{0}$
and $\mu_{1}$; the first states that its integrals are bounded above, and the
second states that their densities are bounded below (which we formally
express through an upper bound on the gaps of the associated inverted height
function).

######  Assumption 13.7.

Adopt 13.5; assume that

(13.5) $\displaystyle\int_{x}^{0}\mu_{0}(dy)\leq
B|x|^{3/2},\quad\text{and}\quad\int_{x}^{0}\mu_{1}(dy)\leq
B|x|^{3/2},\qquad\text{for each $x\in[-BL,-1]$}.$

######  Assumption 13.8.

Adopt 13.5; assume that $G(0,0)=G(1,0)=0$ and for any real numbers $0\leq
y\leq y^{\prime}\leq L^{3/2}$ that

(13.6) $\displaystyle
G(t,y)-G(t,y^{\prime})\leq\frac{3B}{2}\big{(}(y^{\prime})^{2/3}-y^{2/3}\big{)},\quad
t\in\\{0,1\\}.$

The below result states that, under the integral bound 13.7, $\varrho_{t}$ is
bounded above at intermediate times $t\in(0,1)$. Its proof, which appears in
Section 13.3 below, uses 13.3 and the continuum height comparison 10.14.

######  Proposition 13.9.

Adopting 13.7, the following two statements hold.

  1. (1)

For each real number $t\in[0,1]$, we have
$\operatorname{supp}\varrho_{t}\subseteq[-2BL,4B^{2}]$ and

(13.7) $\displaystyle
G(t,r)\leq(2B)^{2}-\Big{(}\displaystyle\frac{r}{B}\Big{)}^{2/3},\qquad\text{for
each $r\in[0,L^{3/2}]$}.$

  2. (2)

There exists a constant $C=C(B)>1$ such that

(13.8) $\displaystyle\varrho_{t}(x)\leq C\max\\{1,-x\\}^{3/4},\qquad\text{for
each $(t,x)\in[B^{-1},1-B^{-1}]\times\mathbb{R}$}.$

The next result states that, under the gap bound 13.8, $\varrho_{t}$ is
bounded below at intermediate times $t\in(0,1)$. Its proof appears in Section
13.4 and uses the continuum gap comparison 10.15.

######  Proposition 13.10.

Adopting 13.8, the following two statements hold.

  1. (1)

For any $t\in[0,1]$, we have $\gamma(t)\geq 0$, and

(13.9) $\displaystyle G(t,r)\geq-3Br^{2/3},\qquad\text{for each
$r\in[0,L^{3/2}]$}.$

  2. (2)

We have $\Omega^{\operatorname{inv}}=(0,1)\times(0,L^{3/2})$. Moreover, we
have

(13.10)
$\displaystyle\varrho_{t}\big{(}G(t,r)\big{)}\geq\frac{r^{1/3}}{4B}\geq\bigg{(}\frac{\gamma(t)-G(t,r)}{96B^{3}}\bigg{)}^{1/2},\qquad\text{for
any $(t,r)\in(0,1)\times\Big{[}0,\displaystyle\frac{L^{3/2}}{2}\Big{]}$}.$

####  13.3. Proof of Density Upper Bound

In this section we establish 13.9. We first require the following lemma
bounding the inverted height function $G$ at intermediate times by its values
on the boundary.

######  Lemma 13.11.

Adopting 13.5, we have for each $(t,r)\in[0,1]\times[0,L^{3/2}]$ that

(13.11) $\displaystyle-L^{3/4}\leq
G(t,r)-\big{(}(1-t)G(0,r)+tG(1,r)\big{)}\leq L^{3/4}.$

###### Proof.

We only establish the lower bound in (13.11), as the proof of the upper bound
is entirely analogous. Fixing $r\in[0,L^{3/2}]$, we will compare $G$ with the
limiting Brownian watermelon of Example 10.11, with the $(a,b;A;u,v)$ there
equal to $\big{(}0,1;r,G(0,r),G(1,r)\big{)}$ here, so define (recalling
$\gamma_{\operatorname{sc}}$ from (4.23))

(13.12) $\displaystyle
G^{-}(t,y)=\big{(}r(1-t)t\big{)}^{1/2}\cdot\gamma_{\operatorname{sc}}(r^{-1}y)+(1-t)\cdot
G(0,r)+t\cdot G(1,r).$

Then, for each $(t,y)\in\\{0,1\\}\times[0,r]$, we have $G(t,y)\geq
G(t,r)=G^{-}(t,y)$, where the first bound follows from the fact that $G(t,y)$
is non-increasing in $y$ and the second follows from the definition (13.12) of
$G^{-}$. Thus, by the first statement in 10.14 we have for each
$(t,y)\in[0,1]\in[0,r]$ that $G(t,y)\geq G^{-}(t,y)$. At $y=r$, this implies

(13.13) $\displaystyle\begin{split}G(t,r)\geq G^{-}(t,r)\geq(1-t)\cdot
G(0,r)+t\cdot G(1,r)-r^{1/2}\geq(1-t)\cdot G(0,r)+t\cdot
G(1,r)-L^{3/4},\end{split}$

where in the second inequality we used (13.12) with the bound
$\big{(}r(1-t)t\big{)}^{1/2}\gamma_{\operatorname{sc}}(1)\geq-r^{1/2}$ (as
$\gamma_{\operatorname{sc}}(1)=-2$ by (4.23) and $t(1-t)\leq 1/4$), and in the
third we used the bound $r\leq L^{3/2}$. This confirms (13.11). ∎

Now we can establish 13.9].

###### Proof of Item 1 in 13.9.

Since $\operatorname{supp}\mu_{0}\subseteq[-BL,0]$ and
$\operatorname{supp}\mu_{1}\subseteq[-BL,0]$, we have by (10.4) that
$G(0,y)\leq 0$ and $G(1,y)\leq 0$ for each $y\in[0,L^{3/2}]$. For
$y\in[B,L^{3/2}]$ we have by (10.4) and (13.5) that $G(0,y)\leq-(y/B)^{2/3}$
and $G(1,y)\leq-(y/B)^{2/3}$. Combining these yields

(13.14) $\displaystyle G(0,y)\leq
1-\Big{(}\displaystyle\frac{y}{B}\Big{)}^{2/3},\quad\text{and}\quad G(1,y)\leq
1-\Big{(}\displaystyle\frac{y}{B}\Big{)}^{2/3},\qquad\text{for each
$y\in[0,L^{3/2}]$}.$

By taking $r=L^{3/2}$ in (13.11) (and using the fact that $G(0,y)\geq-BL$ and
$G(1,y)\geq-BL$ for each $y\in[0,L^{3/2}]$, which holds by (10.4) with the
facts that $\operatorname{supp}\mu_{0}\subseteq[-BL,0]$ and
$\operatorname{supp}\mu_{1}\subseteq[-BL,0]$), the lower bound in (13.11)
implies that $G(t,L^{3/2})\geq-BL-L^{3/4}\geq-2BL$ (where in the last bound we
used the fact that $L\geq B>1$). By (10.4), this implies for each $t\in[0,1]$
that

(13.15) $\displaystyle\operatorname{supp}\mu_{t}\subseteq[-2BL,\infty].$

Next we prove $G(t,0)\leq 1+2B^{2}\leq(2B)^{2}$. To this end, we will compare
$G$ to the limiting Airy profile of 10.13 with the
$(a,b;\mathfrak{a},\mathfrak{b},\mathfrak{c})$ there equal to
$(0,1;1,\mathfrak{c},\mathfrak{c})$ here, where ${\mathfrak{c}}=(3\pi
B/4)^{2}$. So, for $(t,y)\in[0,1]\times\mathbb{R}_{\geq 0}$, define

(13.16)
$\displaystyle\begin{split}G^{+}(t,y):=1+{\mathfrak{c}}(1-t)t-\left(\frac{3\pi}{4{\mathfrak{c}}^{1/2}}\right)^{2/3}y^{2/3}=1+{\mathfrak{c}}(1-t)t-\left(\frac{y}{B}\right)^{2/3}.\end{split}$

By (13.14), we then have the lower bound

(13.17) $\displaystyle G^{+}(t,y)=1-\left(\frac{y}{B}\right)^{2/3}\geq
G(t,y),\qquad\text{for $t\in\\{0,1\\}$}.$

Thus the first statement in continuum height comparison 10.16 gives
$G(t,y)\leq G^{+}(t,y)$ for $(t,y)\in[0,1]\times[0,L^{3/2}]$. Using the
explicit formula (13.16), we get

(13.18) $\displaystyle\begin{split}G(t,y)\leq G^{+}(t,y)\leq
1+\frac{{\mathfrak{c}}}{4}-\left(\frac{y}{B}\right)^{2/3}\leq
1+2B^{2}-\left(\frac{y}{B}\right)^{2/3}\leq
4B^{2}-\left(\frac{y}{B}\right)^{2/3},\end{split}$

where in the first inequality we used
${\mathfrak{c}}t(1-t)\leq{\mathfrak{c}}/4$; in the second inequality, we used
${\mathfrak{c}}=(3\pi B/4)^{2}\leq 8B^{2}$; and in the last inequality we used
$1\leq 2B^{2}$. This finishes the proof of (13.7). By taking $y=0$ in (13.18)
we get $G(t,0)\leq 4B^{2}$, which with (10.4) implies that
$\operatorname{supp}\varrho_{t}\subseteq(-\infty,4B^{2}]$. Together with
(13.15), this yields $\operatorname{supp}\varrho_{t}\subseteq[-2BL,4B^{2}]$,
verifying the first part of the proposition. ∎

###### Proof of Item 2 in 13.9.

For $y\in\big{[}(2B)^{4},L^{3/2}]$, (13.7) implies

(13.19) $\displaystyle
G(t,y)\leq(2B)^{2}-\left(\frac{y}{B}\right)^{2/3}\leq-\left(\frac{y}{(2B)^{4}}\right)^{2/3},$

Let $C_{1}=(2B)^{4}$. From Item 1 in 13.9, we have
$\operatorname{supp}\varrho_{t}\subseteq[-2BL,4B^{2}]$ for each $t\in[0,1]$.
So, by (10.4) (with the fact that $\mu_{t}(\mathbb{R})=L^{3/2}$), (13.19)
implies

(13.20) $\displaystyle\int_{x}^{\infty}\varrho_{t}(y)dy\leq
C_{1}|x|^{3/2},\qquad\text{for $x\in[-C_{1}^{-2/3}L,-1]$}.$

By the second part of 10.5, for any $t\in[B^{-1},1-B^{-1}]$, there exists a
measure $\nu_{t}$ with $\nu_{t}(\mathbb{R})=L^{3/2}$ and
$\operatorname{supp}\nu_{t}\subseteq\operatorname{supp}\mu_{0}+\operatorname{supp}\mu_{1}\subseteq[-2BL,0]$,
such that $\mu_{t}=\nu_{t}\boxplus\mu_{\operatorname{sc}}^{(\tau)}$ for
$\tau=t(1-t)$. Since $B^{-1}(1-B^{-1})\leq\tau\leq 1/2$, (13.20) verifies 13.2
(with the $B$ there equal to $(2B)^{4}$ here, and using the fact that the left
side of (13.20) is at most $L^{3/2}\leq C_{1}|x|^{3/2}$ for $x\leq-
C_{1}^{-2/3}L$), and so the second part of 13.3 yields (13.8). ∎

####  13.4. Proof of Density Lower Bound

In this section we establish 13.10.

###### Proof of Item 1 in 13.10.

Since $G(0,0)=0=G(1,0)$, taking $y=0$ and $y^{\prime}=r$ in the assumption
(13.6) gives

(13.21) $\displaystyle-G(t,r)=G(t,0)-G(t,r)\leq\frac{3B}{2}\cdot
r^{2/3},\qquad\text{for $(t,r)\in\\{0,1\\}\times[1,L^{3/2}]$}.$

Next we prove $\gamma(t)\geq 0$ by comparing $G$ to the limiting Airy profile
as in 10.13 with the $(a,b;\mathfrak{a},\mathfrak{b},\mathfrak{c})$ there
equal to $(0,1;0;\mathfrak{c},\mathfrak{c})$ here, where
${\mathfrak{c}}=\pi^{2}/(12B^{3})$. So, for
$(t,y)\in[0,1]\times\mathbb{R}_{\geq 0}$, define

(13.22)
$\displaystyle\begin{split}G^{-}(t,y):={\mathfrak{c}}(1-t)t-\left(\frac{3\pi}{4{\mathfrak{c}}^{1/2}}\right)^{2/3}y^{2/3}={\mathfrak{c}}(1-t)t-3By^{2/3}.\end{split}$

When $t\in\\{0,1\\}$, using (13.21) and (13.22) we deduce the upper bound

(13.23) $\displaystyle G^{-}(t,y)=-3By^{2/3}\leq-\frac{3B}{2}\cdot y^{2/3}\leq
G(t,y),\qquad\text{for $(t,y)\in\\{0,1\\}\times[0,L^{3/2}]$}.$

For $t\in[0,1]$, we have from (13.11) that

$\displaystyle G(t,L^{3/2})$ $\displaystyle\geq(1-t)\cdot G(0,L^{3/2})+t\cdot
G(1,L^{3/2})-L^{3/4}$
$\displaystyle\geq-\frac{3BL}{2}-L^{3/4}\geq\frac{{\mathfrak{c}}}{4}-3BL\geq{\mathfrak{c}}t(1-t)-3BL=G^{-}(t,L^{3/2}),$

where the second statement is from (13.21) with $r=L^{3/2}$; the third holds
since $B\geq 1$ and $L\geq 1$; the fourth uses $1/4\geq t(1-t)$; and the fifth
uses the definition (13.22) of $G^{-}$. Thus, the second statement in
continuum height comparison 10.16 gives $G(t,y)\geq G^{-}(t,y)$ for each
$(t,y)\in[0,1]\times[0,L^{3/2}]$. Using the explicit formula (13.22) for
$G^{-}$, it follows for each $(t,y)\in[0,1]\times[0,L^{3/2}]$ that

(13.24) $\displaystyle\begin{split}G(t,y)\geq
G^{-}(t,y)={\mathfrak{c}}(1-t)t-3By^{2/3}\geq-3By^{2/3},\end{split}$

verifying (13.9). Consequently, $\gamma(t)=G(t,0)\geq 0$ by setting $y=0$ in
(13.24), verifying the first statement of the first part of the proposition. ∎

###### Proof of Item 2 in 13.10.

To prove (13.10), we will compare $G$ to the inverted height function from
Example 10.12 with $(a,b)=(0,1)$, $A=L^{3/2}$, and $d=8B^{2}L^{2}/\pi^{2}$.
So, for $(t,y)\in[0,1]\times[0,L^{3/2}]$, define

(13.25)
$\displaystyle\widetilde{G}(t,y)=\bigg{(}d+\displaystyle\frac{At(1-t)}{1+2\kappa}\bigg{)}^{1/2}\cdot\gamma_{\operatorname{sc}}(A^{-1}y),$

where we recall $\kappa$ from (10.11) and the classical location
$\gamma_{\operatorname{sc}}(y)$ from (4.23). Observe that

(13.26) $\displaystyle d+\displaystyle\frac{At(1-t)}{1+2\kappa}\leq
d+\displaystyle\frac{A}{8\kappa}\leq
d+\displaystyle\frac{A^{2}}{16d}=d+\displaystyle\frac{\pi^{2}L}{128B^{2}}\leq
2d,$

where in the first statement we used the bound $t(1-t)\leq 1/4$; in the second
we used the fact that $\kappa\geq 2A^{-1}d$ (which follows from (10.11)); in
the third we used the definitions of $A$ and $d$; and in the fourth we used
the definition of $d$ with the facts that $B\geq 1$ and $L\geq 1$. Thus, for
$y\in[0,A/2]$, we obtain

(13.27)
$\displaystyle-\partial_{y}\widetilde{G}(t,y)=-A^{-1}\bigg{(}d+\displaystyle\frac{A(1-t)t}{1+2\kappa}\bigg{)}^{1/2}\gamma^{\prime}_{\operatorname{sc}}\Big{(}\displaystyle\frac{y}{A}\Big{)}\leq\bigg{(}d+\displaystyle\frac{A(1-t)t}{1+2\kappa}\bigg{)}^{1/2}\frac{\pi}{A^{2/3}y^{1/3}}\leq\frac{4B}{y^{1/3}},$

where in the first statement we differentiated (13.25) with respect to $y$; in
the second we applied the second part of 4.31; and in the third we used
(13.26) and the definitions of $A$ and $d$. Moreover, for $y\in[0,A]$, we have

(13.28)
$\displaystyle\frac{B}{y^{1/3}}=\frac{d^{1/2}\pi}{2^{3/2}A^{2/3}y^{1/3}}\leq-A^{-1}\bigg{(}d+\displaystyle\frac{At(1-t)}{1+2\kappa}\bigg{)}^{1/2}\cdot\gamma_{\operatorname{sc}}^{\prime}\Big{(}\displaystyle\frac{y}{A}\Big{)}=-\partial_{y}\widetilde{G}(t,y),$

where in the first statement we used the definitions of $A$ and $d$, in the
second we used the second part of 4.31, and in the third we used the
definition (13.25) of $\widetilde{G}$. Thus, for any $0\leq y\leq
y^{\prime}\leq A=L^{3/2}$, we have

(13.29)
$\displaystyle\widetilde{G}(t,y)-\widetilde{G}(t,y^{\prime})\geq\int_{y}^{y^{\prime}}\frac{Bdr}{r^{1/3}}=\frac{3B}{2}\big{(}(y^{\prime})^{2/3}-y^{2/3}\big{)}\geq
G(t,y)-G(t,y^{\prime}),$

where the first statement follows from integrating (13.28); the second from
performing the integral; and the third from (13.6). This verifies the
assumption in the first statement of continuum gap comparison 10.15; and we
conclude for $t\in[0,1]$ and $0\leq y\leq y^{\prime}\leq L^{3/2}$ that

(13.30) $\displaystyle\widetilde{G}(t,y)-\widetilde{G}(t,y^{\prime})\geq
G(t,y)-G(t,y^{\prime}).$

Since $\widetilde{G}(t,y)$ is differentiable (and has negative derivative) in
$y\in(0,L^{3/2})$ for any $t\in(0,1)$, this implies (by Definition 10.4) that
$\varrho_{t}\big{(}G(t,y)\big{)}>0$ for any $(t,y)\in(0,1)\times(0,L^{3/2})$,
meaning by (LABEL:omega12) that
$\Omega^{\operatorname{inv}}=(0,1)\times(0,L^{3/2})$. This establishes the
first statement of the second part of the proposition.

Next, by integrating the upper bound (13.27), together with (13.30), we deduce

(13.31) $\displaystyle
G(t,y)-G(t,y^{\prime})\leq\widetilde{G}(t,y)-\widetilde{G}(t,y^{\prime})\leq\int_{y}^{y^{\prime}}\frac{4Bdr}{r^{1/3}}=6B\big{(}(y^{\prime})^{2/3}-y^{2/3}\big{)},$

for any $0<t<1$ and $0\leq y<y^{\prime}\leq L^{3/2}/2$. From Definition 10.4
(and recalling from 10.5 that for $t\in(0,1)$ that the density $\varrho_{t}$
exists), (13.31) implies for $(t,y)\in(0,1)\times(0,L^{3/2}/2]$ that

(13.32)
$\displaystyle\frac{1}{\varrho_{t}\big{(}G(t,y)\big{)}}=-\partial_{y}G(t,y)=\lim_{y^{\prime}\rightarrow
y^{+}}\frac{G(t,y)-G(t,y^{\prime})}{y^{\prime}-y}\leq\frac{4B}{y^{1/3}}.$

By rearranging, this gives the first inequality in (13.10).

To establish the second, recall that $\gamma(t)=G(t,0)$ and take $y=0$ and
$y^{\prime}=r\leq L^{3/2}/2$ in (13.31); this yields for $t\in[0,1]$ the bound

(13.33) $\displaystyle\gamma(t)-G(t,r)=G(t,0)-G(t,r)\leq 6Br^{2/3}.$

By plugging (13.33) into (13.32), we obtain for $r\in(0,L^{3/2}/2]$ that

(13.34)
$\displaystyle\varrho_{t}\big{(}G(t,r)\big{)}\geq\frac{r^{1/3}}{4B}\geq\frac{1}{4B}\bigg{(}\frac{\gamma(t)-G(t,r)}{6B}\bigg{)}^{1/2}=\bigg{(}\frac{\gamma(t)-G(t,r)}{96B^{3}}\bigg{)}^{1/2},$

which finishes the proof of the second inequality in (13.10) for
$r\in(0,L^{3/2}/2]$. At the endpoint $r=0$ of this interval, the second bound
in (13.10) continues to hold by the nonnegativity of $\varrho_{t}$. This
verifies the second part of the proposition. ∎

####  13.5. Regularity Estimates for Limit Shapes

In this section we state the following two propositions providing estimates on
the inverted height functions $G$ subject to the integral bound 13.7 and gap
bound 13.8. The first provides approximately matching bounds on the
$y$-derivatives of $G$, and also shows that the arctic boundary $\gamma$ is
uniformly concave; we establish it later in this section. The second shows
that the functions $u$ and $\varrho$ (recall 13.5 for their definitions)
extend continuously to its arctic boundary; we establish it in Section 13.6
below.

######  Proposition 13.12.

Adopt 13.7 and 13.8. There exist constants $c=c(B)>0$ and $C=C(B)>1$ such that
the following two statements hold if $L\geq C$, for any real number
$t\in[3B^{-1},1-3B^{-1}]$.

  1. (1)

For any $y\in(0,B^{5}]$, we have $cy^{-1/3}\leq-\partial_{y}G(t,y)\leq
4By^{-1/3}$.

  2. (2)

For any real number $t^{\prime}\in[3B^{-1},1-3B^{-1}]$, we have
$\big{|}\gamma(t)-\gamma(t^{\prime})\big{|}\leq C|t-t^{\prime}|$. Moreover,
for any real numbers $t_{0},s,\tau_{0}\in\mathbb{R}$ with $3B^{-1}\leq
t_{0}-\tau_{0}<t_{0}+\tau_{0}\leq 1-3B^{-1}$ and $s\in[-\tau_{0},\tau_{0}]$,
we have

(13.35) $\displaystyle
C^{-1}(\tau_{0}^{2}-s^{2})\leq\gamma(t_{0}+s)-\left(\frac{\tau_{0}-s}{2\tau_{0}}\cdot\gamma(t_{0}-\tau_{0})+\frac{\tau_{0}+s}{2\tau_{0}}\cdot\gamma(t_{0}+\tau_{0})\right)\leq
C(\tau_{0}^{2}-s^{2}).$

######  Proposition 13.13.

Adopt 13.7 and 13.8. There exists a constant $C=C(B)>1$ such that the
following two statements hold if $L\geq C$.

  1. (1)

For any $x_{0}=G(t_{0},y_{0})$ with $t_{0}\in[4B^{-1},1-4B^{-1}]$ and
$y_{0}\in(0,B]$, we have

(13.36)
$\displaystyle\big{|}\partial_{x}u(t_{0},x_{0})\big{|}+\big{|}\partial_{x}\varrho(t_{0},x_{0})\big{|}\leq
C\big{(}\gamma(t_{0})-x_{0}\big{)}^{-1/2}.$

  2. (2)

Both $\varrho(t,x)$ and $u(t,x)$ extend continuously to the set
$\big{\\{}(t,\gamma(t)):4B^{-1}\leq t\leq 1-4B^{-1}\big{\\}}$, with
$\varrho\big{(}t,\gamma(t)\big{)}=0$ and
$u\big{(}t,\gamma(t)\big{)}=\gamma^{\prime}(t)$. In particular,

(13.37) $\displaystyle\gamma^{\prime}(t)\text{ is continuous in
}t\in[4B^{-1},1-4B^{-1}].$

We now establish 13.12.

###### Proof of Item 1 in 13.12.

First observe that for any $(t,y)\in[0,1]\times[12B^{4},B^{6}]$ we have

(13.38) $\displaystyle-3B^{5}\leq G(t,y)\leq
4B^{2}-(12B^{3})^{2/3}\leq-B^{2},$

where the lower bound is from (13.9) and the upper bound is from (13.7).
Moreover, for any $(t,y)\in[B^{-1},1-B^{-1}]\times[12B^{4},B^{6}]$, we have
for some constant $C_{1}=C_{1}(B)>1$ that

(13.39) $\displaystyle
C_{1}^{-1}\leq\frac{1}{\varrho_{t}\big{(}G(t,y)\big{)}}=-\partial_{y}G(t,y)\leq
B^{-1}\cdot(96B^{3})^{1/2}=(96B)^{1/2},$

where the first statement is from (13.8) and (13.38); the second is from
(10.7) (and the fact from Item 2 of 13.10 that
$\Omega^{\operatorname{inv}}=(0,1)\times(0,L^{3/2})$); and the third is from
(13.10), as $\gamma(t)-G(t,y)\geq-G(t,y)\geq B^{2}$ (where in the first
inequality we used the fact that $\gamma(t)\geq 0$, from Item 1 of 13.10, and
in the second we used (13.38)).

Now define the open rectangle
${\mathfrak{R}}=(B^{-1},1-B^{-1})\times(12B^{4},B^{6})$; see the left side of
Figure 4.1. By 10.17, $G$ solves (10.14) on
$\Omega^{\operatorname{inv}}=(0,1)\times(0,L^{3/2})$ (where the latter follows
from Item 2 of 13.10). Moreover, $G$ and $-\partial_{y}G(t,y)$ are bounded
above and below on $\mathfrak{R}$, by (13.38) and (13.39). Hence, recalling
Definition 10.20, there exists some constant $\varepsilon=\varepsilon(B)>0$
such that $G\in\operatorname{Adm}_{\varepsilon}(\mathfrak{R})$ (where we
observe that $G$ is Lipschitz on $\mathfrak{R}$ since it is real analytic by
13.6); this will enable us to apply the regularity results of Section 10.5 to
$G$. In particular, denoting the open rectangle
$\mathfrak{R}^{\prime}=(2B^{-1},1-2B^{-1})\times(B^{5}/2,2B^{5})\subset\mathfrak{R}$,
10.22 yields a constant $M(B)=M>1$ with

(13.40)
$\displaystyle\displaystyle\sup_{(t,y)\in\overline{\mathfrak{R}}^{\prime}}\big{|}\partial_{t}^{2}G(t,y)\big{|}\leq
M.$

We may assume in what follows that $M\geq 10B^{5}$.

We next use (13.40) to establish the first statement of the proposition,
bounding $-\partial_{y}G(t,y)$ from above and below. The upper bound is given
by the first estimate in (13.10), together with the fact (recall (10.7)) that
$-\partial_{y}G(t,y)=\varrho_{t}\big{(}G(t,y)\big{)}^{-1}$. To prove the lower
bound, we use 10.15 to compare $G$ with the limiting Brownian watermelon of
Example 10.11 with the $(a,b;A;u,v)$ there equal to
$\big{(}2B^{-1},1-2B^{-1};M^{2}/4;G(2B^{-1},A),G(1-2B^{-1},A)\big{)}$ here.
So, define $\widetilde{G}:[a,b]\times[0,A]\rightarrow\mathbb{R}$ by

(13.41)
$\displaystyle\widetilde{G}(t,y)=\bigg{(}\displaystyle\frac{A(b-t)(t-a)}{b-a}\bigg{)}^{1/2}\gamma_{\operatorname{sc}}\Big{(}\displaystyle\frac{y}{A}\Big{)}+\displaystyle\frac{b-t}{b-a}\cdot
G(2B^{-1},A)+\displaystyle\frac{t-a}{b-a}\cdot G(1-2B^{-1},A),$

where we recall the classical location $\gamma_{\operatorname{sc}}$ of the
semicircle law from (4.23). Then, for $t\in[3B^{-1},1-3B^{-1}]$, we have

(13.42)
$\displaystyle-\partial_{y}\widetilde{G}(t,y)=-\left(\frac{(b-t)(t-a)}{A(b-a)}\right)^{1/2}\gamma^{\prime}_{\operatorname{sc}}\Big{(}\frac{y}{A}\Big{)}\geq\left(\frac{B^{-1}(1-5B^{-1})}{A(1-4B^{-1})}\right)^{1/2}\frac{\pi
A^{1/3}}{2^{3/2}y^{1/3}},$

where the first statement is from the definition (13.41) of $\widetilde{G}$,
and the second is from bounding $(b-t)(t-a)\geq B^{-1}(1-5B^{-1})$ (by the
fact that $t\in[3B^{-1},1-3B^{-1}]$) and
$\gamma^{\prime}_{\operatorname{sc}}(y/A)\geq(\pi A^{1/3})/(2^{3/2}y^{1/3})$
(by the second part of 4.31). Moreover, for $2B^{-1}\leq t\leq 1-2B^{-1}$, we
have

(13.43)
$\displaystyle\partial^{2}_{t}\widetilde{G}(t,B^{5})=-\frac{A^{1/2}(b-a)^{3/2}}{4(b-t)^{3/2}(t-a)^{3/2}}\cdot\gamma_{\operatorname{sc}}\Big{(}\displaystyle\frac{B^{5}}{A}\Big{)}\leq-2A^{1/2}\cdot\gamma_{\operatorname{sc}}\Big{(}\displaystyle\frac{B^{5}}{A}\Big{)}\leq-2A^{1/2},$

where the first statement follows from (13.41); the second follows from the
fact that $(b-t)(t-a)\leq(b-a)^{2}/4\leq(b-a)/4$ (as $b-a=1-4B^{-1}<1$); and
the last follows from the fact that $A=M^{2}/4\geq 25B^{5}$ (as $M\geq
10B^{5}$ and $B>1$) and $\gamma_{\operatorname{sc}}(1/25)\geq 1$ (by the first
part of 4.31).

Together with (13.40) and the fact that $4A=M^{2}$, (13.43) yields
$\partial^{2}_{t}\widetilde{G}(t,B^{5})\leq-2A^{1/2}=-M\leq\partial^{2}_{t}G(t,B^{5})$
for each $t\in[2B^{-1},1-2B^{-1}]$. Since $\widetilde{G}(2B^{-1},y)=u$ and
$\widetilde{G}(1-2B^{-1},y)=v$ are constant in $y$, we also have
$\big{|}\widetilde{G}(t,y)-\widetilde{G}(t,y^{\prime})\big{|}=0\leq\big{|}G(t,y)-G(t,y^{\prime})\big{|}$
for each $t\in\\{2B^{-1},1-2B^{-1}\\}$ and $0\leq y\leq y^{\prime}\leq A$.
This verifies the assumptions in the second statement of 10.15 (with the
$(G^{\star},\widetilde{G}^{\star})$ there equal to $(\widetilde{G},G)$ here),
which gives for each $(t,y)\in[3B^{-1},1-3B^{-1}]\times(0,B^{5}]$ that

$\displaystyle-\partial_{y}G(t,y)$
$\displaystyle=\displaystyle\lim_{y^{\prime}\rightarrow
y^{-}}\displaystyle\frac{G(t,y^{\prime})-G(t,y)}{y-y^{\prime}}$
$\displaystyle\geq\displaystyle\lim_{y^{\prime}\rightarrow
y^{-}}\displaystyle\frac{\widetilde{G}(t,y^{\prime})-\widetilde{G}(t,y)}{y-y^{\prime}}=-\partial_{y}\widetilde{G}(t,y)\geq\left(\frac{B-5}{AB(B-4)}\right)^{1/2}\frac{\pi
A^{1/3}}{2^{3/2}y^{1/3}},$

where the last inequality is from (13.42). This provides the lower bound on
$-\partial_{y}G(t,y)$ and thus finishes the proof of the first statement in
13.12. ∎

###### Proof of Item 2 in 13.12.

Fix an interval $[a,b]\subseteq[3B^{-1},1-3B^{-1}]$, and denote
$\tau=(b-a)/2$. Define the functions
$\breve{G}:[0,1]\times[0,L^{3/2}]\rightarrow\mathbb{R}$ and
$\breve{\gamma}:[0,1]\rightarrow\mathbb{R}$ by performing an affine shift on
$G$ and $\gamma$ respectively, by setting

(13.44)
$\displaystyle\breve{G}(t,y)=G(t,y)-\left(\frac{b-t}{2\tau}\cdot\gamma(a)+\frac{t-a}{2\tau}\cdot\gamma(b)\right),\qquad\text{and}\qquad\breve{\gamma}(t)=\breve{G}(t,0),$

for each $(t,y)\in[0,1]\times[0,L^{3/2}]$. Then, for each $t\in[0,1]$, we have

(13.45)
$\displaystyle\breve{\gamma}^{\prime}(t)=\gamma^{\prime}(t)+\displaystyle\frac{\gamma(b)-\gamma(a)}{2\tau},\qquad\text{and}\qquad\breve{\gamma}(a)=\breve{\gamma}(b)=0.$

From the first statements of 13.9 and 13.10, we have
$0\leq\gamma(t)=G(t,0)\leq(2B)^{2}$, which upon insertion into (13.44) gives
for each $t\in[a,b]$ that

(13.46) $\displaystyle\breve{G}(t,0)=\breve{\gamma}(t)\leq(2B)^{2}.$

Next we show that there exists a constant $C_{1}=C_{1}(B)>1$ such that, for
each $t\in[a,b]$,

(13.47) $\displaystyle C_{1}^{-1}(b-t)(t-a)\leq\breve{\gamma}(t)\leq
C_{1}(b-t)(t-a)$

We only prove the upper bound in (13.47), as the proof of the lower bound is
entirely analogous. To this end, recall from the first statement in 13.12 that
there exists some constant $c=c(B)>0$ such that
$-\partial_{y}\breve{G}(t,y)=-\partial_{y}G(t,y)\geq cy^{-1/3}$ for each
$(t,y)\in[a,b]\times(0,B^{5}]$. Integrating this estimate from $0$ to $r$ then
gives for each $(t,r)\in[a,b]\times[0,B^{5}]$ that

(13.48)
$\displaystyle\breve{G}(t,r)\leq\breve{G}(t,0)-\displaystyle\frac{3c}{2}\cdot
r^{2/3}=\breve{\gamma}(t)-\frac{3c}{2}\cdot
r^{2/3}\leq(2B)^{2}-\frac{3c}{2}\cdot r^{2/3},$

where we used (13.46) in the last inequality. By (13.44) and (13.7) (with the
fact that $\gamma(t)\geq 0$, by 13.10), we also have $\breve{G}(t,r)\leq
G(t,r)\leq(2B)^{2}-(r/B)^{2/3}$. Together with (13.48) and (13.46), this
implies for $(t,r)\in[a,b]\times(0,B^{5}]$ that

(13.49)
$\displaystyle\breve{G}(t,r)\leq(2B)^{2}-r^{2/3}\cdot\max\left\\{\frac{3c}{2},B^{-2/3}\right\\}.$

Using this, we compare $\breve{G}$ to the limiting Airy profile
$\widetilde{G}(t,y)$ from (10.12) with ${\mathfrak{c}}=243\pi^{2}/(2c^{3})$,
${\mathfrak{a}}=-ab{\mathfrak{c}}$, and ${\mathfrak{b}}=(a+b){\mathfrak{c}}$,
so define the function $\widetilde{G}:[a,b]\times\mathbb{R}_{\geq
0}\rightarrow\mathbb{R}$ by setting

(13.50)
$\displaystyle\begin{split}\widetilde{G}(t,y)={\mathfrak{c}}(b-t)(t-a)-\left(\frac{3\pi}{4{\mathfrak{c}}^{1/2}}\right)^{2/3}y^{2/3}={\mathfrak{c}}(b-t)(t-a)-\frac{c}{6}\cdot