where in the last line we used (6.12). Plugging this bound back in (6.8) we
get

$\displaystyle\mathbb{P}_{t}(A)$
$\displaystyle\geq(1-\delta)\mathbb{E}\left[\mathbf{1}\\{\mathsf{Fav}_{t}(\delta)\\}\mathbb{P}_{\operatorname{free},t}(A)\right]-(1-\delta)\delta$
$\displaystyle\geq(1-\delta)\mathbb{E}\left[\mathbf{1}\\{\mathsf{Fav}_{t}(\delta)\\}\gamma(A)-\delta\right]-(1-\delta)\delta$
$\displaystyle=\gamma(A)(1-\delta)\mathbb{P}(\mathsf{Fav}_{t}(\delta))-2\delta(1-\delta).$

where the inequality in the penultimate line follows from (6.13). Taking
$\liminf$ both sides as $t\to\infty$, in view of (6.7) we see that

$\displaystyle\liminf_{t\to\infty}\mathbb{P}_{t}(A)\geq(1-\delta)(1-\varepsilon)\gamma(A)-2\delta(1-\delta).$

Taking $\liminf_{\delta\downarrow 0}$ and using the fact that $\varepsilon$ is
arbitrary, we get that $\liminf_{t\to\infty}\mathbb{P}_{t}(A)\geq\gamma(A)$.
Similarly for the upper bound, on the event $\mathsf{Fav}_{t}(\delta)$ we have

$\displaystyle\frac{\mathbb{E}_{\operatorname{free},t}\left[W\mathbf{1}_{A}\right]}{\mathbb{E}_{\operatorname{free},t}\left[W\right]}$
$\displaystyle\leq\frac{\mathbb{P}_{\operatorname{free},t}(A)}{(1-\delta)\mathbb{P}_{\operatorname{free},t}(W\geq
1-\delta)}\leq\frac{1}{(1-\delta)^{2}}\mathbb{P}_{\operatorname{free},t}(A),$

where we again use (6.12) for the last inequality. Inserting the above bound
in (6.9) we get

$\displaystyle\mathbb{P}_{t}(A)$
$\displaystyle\leq\frac{1}{(1-\delta)^{2}}\mathbb{E}\left[\mathbf{1}\\{\mathsf{Fav}_{t}(\delta)\\}\mathbb{P}_{\operatorname{free},t}(A)\right]+\mathbb{P}\left(\neg\mathsf{Fav}_{t}(\delta)\right)$
$\displaystyle\leq\frac{\delta}{(1-\delta)^{2}}+\frac{1}{(1-\delta)^{2}}\mathbb{E}\left[\mathbf{1}\\{\mathsf{Fav}_{t}(\delta)\\}\gamma(A)\right]+\mathbb{P}\left(\neg\mathsf{Fav}_{t}(\delta)\right)$
$\displaystyle\leq\frac{\delta}{(1-\delta)^{2}}+\frac{1}{(1-\delta)^{2}}\gamma(A)+\mathbb{P}\left(\neg\mathsf{Fav}_{t}(\delta)\right).$

The inequality in the penultimate line above follows from (6.13). Taking
$\limsup$ both sides as $t\to\infty$, in view of (6.7) we see that

$\displaystyle\limsup_{t\to\infty}\mathbb{P}_{t}(A)\leq\frac{\delta}{(1-\delta)^{2}}+\frac{1}{(1-\delta)^{2}}\gamma(A)+\varepsilon.$

As before taking $\limsup_{\delta\downarrow 0}$ and using the fact that
$\varepsilon$ is arbitrary, we get that
$\limsup_{t\to\infty}\mathbb{P}_{t}(A)\leq\gamma(A)$. With the matching upper
bound for $\liminf$ derived above, we thus arrive at (6.1), completing the
proof of Theorem 1.11.

Step 4. In this step we prove (6.7). Fix any $\delta>0$. Recall the
distributional convergence of KPZ line ensemble to Airy line ensemble from
Proposition 2.7. By the Skorokhod representation theorem, we may assume that
our probability space are equipped with $\mathcal{A}_{1}(x)$
$\mathcal{A}_{2}(x)$, such that as $t\to\infty$, almost surely we have

$\displaystyle\max_{i=1,2}\sup_{x\in[-1,1]}|2^{1/3}\mathfrak{h}_{t}^{(i)}(2^{1/3}x)-\mathcal{A}_{i}(x)|\to
0.$ (6.14)

Figure 6. In the above figure $\mathsf{Gap}_{t}(\delta)$ defined in (6.3)
denotes the event that the value of the blue point is smaller than the value
of each of the red points at least by $\delta$, The
$\mathsf{Rise}_{t}(\delta)$ event defined in (6.4) requires no point on the
whole blue curve (restricted to ${I}_{t}=(-t^{-\alpha},t^{-\alpha})$) exceed
the value of the blue point by a factor $\frac{1}{4}\delta$ (i.e., there is no
significant rise). The $\mathsf{Tight}_{t}(\delta)$ defined in (6.5) event
ensures the value of the red points are within $[-\delta^{-1},\delta^{-1}]$.
The $\mathsf{Fluc}_{t}^{(i)}(\delta)$ event defined in (6.15) signifies every
value of every point on the $i$-th curve (restricted to ${I}_{t}$) is within
$\frac{1}{4}\delta$ distance away from its value on the left boundary:
$\mathfrak{h}_{t}^{(1)}(-t^{-\alpha})$. Finally, $\mathsf{Sink}_{t}(\delta)$
event defined in (6.20) denotes the event that no point on the black curve
(restricted to ${I}_{t}$) drops below the value of the red points by a factor
larger than $\frac{1}{4}\delta$, (i.e., there is no significant sink).

For $i=1,2$, consider the event

$\displaystyle\mathsf{Fluc}_{t}^{(i)}(\delta):=\left\\{\sup_{x\in
I_{t}}|\mathfrak{h}_{t}^{(i)}(x)-\mathfrak{h}_{t}^{(i)}(-t^{-\alpha})|\leq\tfrac{1}{4}\delta\right\\}.$
(6.15)

See Figure 6 and its caption for an interpretation of this event. We claim
that for every $\delta>0$,

$\displaystyle\liminf_{t\to\infty}\mathbb{P}\left(\mathsf{Fluc}_{t}^{(i)}(\delta)\right)=1.$
(6.16)

Let us complete the proof of (6.7) assuming (6.16). Fix any $\varepsilon>0$.
Note that
$\\{|\mathfrak{h}_{t}^{(1)}(-t^{-\alpha})-\mathfrak{h}_{t}^{(1)}(t^{-\alpha})|\leq\tfrac{1}{4}\delta\\}\supset\mathsf{Fluc}_{t}^{(1)}(\delta)$.
Recall $\mathsf{Gap}_{t}(\delta)$ from (6.3). Observe that

$\displaystyle\neg\mathsf{Gap}_{t}(\delta)\cap\left\\{|\mathfrak{h}_{t}^{(1)}(-t^{-\alpha})-\mathfrak{h}_{t}^{(1)}(t^{-\alpha})|\leq\tfrac{1}{4}\delta\right\\}$
$\displaystyle\subset\left\\{\mathfrak{h}_{t}^{(2)}(-t^{\alpha})-\mathfrak{h}_{t}^{(1)}(-t^{-\alpha})\geq-\tfrac{5}{4}\delta\right\\}$
$\displaystyle\subset\left\\{\inf_{x\in[-1,0]}[\mathfrak{h}_{t}^{(2)}(x)-\mathfrak{h}_{t}^{(1)}(x)]\geq-\tfrac{5}{4}\delta\right\\}.$

Using these two preceding set relations, by union bound we have

$\displaystyle\mathbb{P}\left(\neg\mathsf{Gap}_{t}(\delta)\right)$
$\displaystyle\leq\mathbb{P}\left(|\mathfrak{h}_{t}^{(1)}(-t^{-\alpha})-\mathfrak{h}_{t}^{(1)}(t^{-\alpha})|\geq\tfrac{1}{4}\delta\right)+\mathbb{P}\left(\neg\mathsf{Gap}_{t}(\delta)\cap|\mathfrak{h}_{t}^{(1)}(-t^{-\alpha})-\mathfrak{h}_{t}^{(1)}(t^{-\alpha})|\leq\tfrac{1}{4}\delta\right)$
$\displaystyle\leq\mathbb{P}\left(\neg\mathsf{Rise}_{t}^{(1)}(\delta)\right)+\mathbb{P}\left(\inf_{x\in[-1,0]}[\mathfrak{h}_{t}^{(2)}(x)-\mathfrak{h}_{t}^{(1)}(x)]\geq-\tfrac{5}{4}\delta\right).$

As $t\to\infty$, the first term goes to zero due (6.16) and by Proposition
2.7, the second term goes to

$\mathbb{P}\left(\inf_{x\in[-1,0]}[\mathcal{A}_{2}(2^{-1/3}x)-\mathcal{A}_{1}(2^{-1/3}x)]\geq-\tfrac{5}{4\cdot
2^{1/3}}\delta\right).$

But by (2.1) we know Airy line ensembles are strictly ordered. Thus the above
probability can be made arbitrarily small be choose $\delta$ small enough. In
particular, there exists a $\delta_{1}\in(0,1)$ such that for all
$\delta\in(0,\delta_{1})$ the above probability is always less than
$\frac{\varepsilon}{2}$. This forces

$\displaystyle\liminf_{t\to\infty}\mathbb{P}\left(\mathsf{Gap}_{t}(\delta)\right)\geq
1-\tfrac{\varepsilon}{2}.$ (6.17)

Recall $\mathsf{Rise}_{t}(\delta)$ from (6.4). Clearly
$\mathsf{Rise}_{t}(\delta)\subset\mathsf{Fluc}_{t}^{(2)}(\delta)$. Thus for
every $\delta>0$,

$\displaystyle\liminf_{t\to\infty}\mathbb{P}(\mathsf{Rise}_{t}(\delta))=1.$
(6.18)

Finally using Proposition 2.8 (a) and (b) we see that
$\mathfrak{h}_{t}^{(1)}(t^{-\alpha}),\mathfrak{h}_{t}^{(1)}(t^{-\alpha})$ are
tight. Thus there exists $\delta_{2}\in(0,1)$ such that for all
$\delta\in(0,\delta_{2})$, we have

$\displaystyle\liminf_{t\to\infty}\mathbb{P}\left(\mathsf{Tight}_{t}(\delta)\right)\geq
1-\tfrac{\varepsilon}{2}.$ (6.19)

Combining (6.17), (6.18), (6.19), and recalling the definition of
$\mathsf{Fav}_{t}(\delta)$ from (6.6), by union bound we get (6.7) for all
$\delta\in(0,\min\\{\delta_{1},\delta_{2}\\})$.

Let us now prove (6.16). Recall $\mathsf{Fluc}_{t}^{(i)}(\delta)$ from (6.15).
Define the event:

$\displaystyle\mathsf{Conv}_{t}(\delta):=\left\\{\sup_{x\in[-1,1]}|\mathfrak{h}_{t}^{(i)}(x)-2^{-1/3}\mathcal{A}_{i}(2^{-1/3}x)|\leq\tfrac{1}{16}\delta\right\\}.$

Observe that

$\displaystyle\left\\{\neg\mathsf{Fluc}_{t}^{(i)}(\delta),\mathsf{Conv}_{t}(\delta)\right\\}\subset\left\\{\sup_{|x|\leq
2^{-1/3}t^{-\alpha}}\left[\mathcal{A}_{i}(x)-\mathcal{A}_{i}(-2^{-1/3}t^{-\alpha})\right]\geq\tfrac{2^{1/3}}{8}\delta\right\\}.$

Thus by union bound

$\displaystyle\mathbb{P}\left(\neg\mathsf{Fluc}_{t}^{(i)}(\delta)\right)$
$\displaystyle\leq\mathbb{P}\left(\neg\mathsf{Conv}_{t}(\delta)\right)+\mathbb{P}\left(\neg\mathsf{Fluc}_{t}^{(i)}(\delta),\mathsf{Conv}_{t}(\delta)\right)$
$\displaystyle\leq\mathbb{P}\left(\neg\mathsf{Conv}_{t}(\delta)\right)+\mathbb{P}\left(\sup_{|x|\leq
2^{-1/3}t^{-\alpha}}\left[\mathcal{A}_{i}(x)-\mathcal{A}_{i}(-2^{-1/3}t^{-\alpha})\right]\geq\tfrac{2^{1/3}}{8}\delta\right).$

By (6.14), the first term above goes to zero as $t\to\infty$, whereas the
second term goes to zero as $t\to\infty$, via modulus of continuity of Airy
line ensembles from Proposition 2.4. Note that in Proposition 2.4 the modulus
of continuity is stated for $\mathcal{A}_{i}(x)+x^{2}$. However, in the above
scenario since we deal with a vanishing interval
$[-2^{-1/3}t^{-\alpha},2^{-1/3}t^{-\alpha}]$, the parabolic term does not play
any role. This establishes (6.16).

Step 5. In this step we prove (6.12). Let us consider the event

$\displaystyle\mathsf{Sink}_{t}(\delta):=\left\\{\inf_{x\in
I_{t}}\mathfrak{h}_{t}^{(1)}(x)\geq-\tfrac{1}{4}\delta+\min\\{\mathfrak{h}_{t}(-t^{-\alpha}),\mathfrak{h}_{t}(t^{-\alpha})\\}\right\\}.$
(6.20)

See Figure 6 and its caption for an interpretation of this event. Recall
$\mathsf{Gap}_{t}(\delta)$ and $\mathsf{Rise}_{t}(\delta)$ from (6.3) and
(6.4). Observe that on the event
$\mathsf{Gap}_{t}(\delta)\cap\mathsf{Rise}_{t}(\delta)$, we have $\sup_{x\in
I_{t}}\mathfrak{h}_{t}^{(2)}(x)\leq\min\\{\mathfrak{h}_{t}(-t^{-\alpha}),\mathfrak{h}_{t}(t^{-\alpha})\\}-\frac{3}{4}\delta$.
Thus on
$\mathsf{Gap}_{t}(\delta)\cap\mathsf{Rise}_{t}(\delta)\cap\mathsf{Sink}_{t}(\delta)$,
we have

$\inf_{x\in{I}_{t}}\left[\mathfrak{h}_{t}^{(1)}(x)-\mathfrak{h}_{t}^{(2)}(x)\right]\geq\tfrac{1}{2}\delta.$

Recall $W$ from (6.11). On the event $\\{\inf_{x\in
I_{t}}\left[\mathfrak{h}_{t}^{(1)}(x)-\mathfrak{h}_{t}^{(2)}(x)\right]\geq\tfrac{1}{2}\delta\\}$
we have the pointwise inequality

$W>\exp(-2t^{2/3-\alpha}e^{-\frac{1}{2}t^{1/3}\delta})\geq 1-\delta,$

where we choose a $t_{1}(\delta)>0$ so that the last inequality is true for
all $t\geq t_{1}$. Thus for all $t\geq t_{1}$,

$\displaystyle\mathbf{1}\\{\mathsf{Fav}_{t}(\delta)\\}\mathbb{P}_{\operatorname{free},t}(W\geq
1-\delta)\geq\mathbf{1}\\{\mathsf{Fav}_{t}(\delta)\\}\mathbb{P}_{\operatorname{free},t}(\mathsf{Sink}_{t}(\delta)).$
(6.21)

Recall that $\mathbb{P}_{\operatorname{free},t}$ denotes the law of a Brownian
bridge $B_{1}(\cdot)$ on $I_{t}$ starting at
$B_{1}(-t^{-\alpha})=\mathfrak{h}_{t}(-t^{-\alpha})$ and ending at
$B_{1}(t^{-\alpha})=\mathfrak{h}_{t}(t^{-\alpha})$. Let us consider another
Brownian bridge $\widetilde{B}_{1}(\cdot)$ on $I_{t}$ starting and ending at
$\min\\{\mathfrak{h}_{t}(-t^{-\alpha}),\mathfrak{h}_{t}(t^{-\alpha})\\}$. By
standard estimates for Brownian bridge (see Lemma 2.11 in [CH16] for example)

$\displaystyle\mathbb{P}\left(\inf_{x\in
I_{t}}\widetilde{B}_{1}(x)\geq-\tfrac{1}{4}\delta+\min\\{\mathfrak{h}_{t}(-t^{-\alpha}),\mathfrak{h}_{t}(t^{-\alpha})\\}\right)=1-\exp\left(-\tfrac{\delta^{2}}{8|I_{t}|}\right)=1-\exp\left(-\tfrac{\delta^{2}}{16}t^{\alpha}\right).$

Note that $B_{1}(\cdot)$ is stochastically larger than
$\widetilde{B}_{1}(\cdot)$. Since the above event is increasing, we thus have
$\mathbb{P}_{\operatorname{free},t}\left(\mathsf{Sink}_{t}(\delta)\right)$ is
at least $1-\exp\left(-\tfrac{\delta^{2}}{16}t^{\alpha}\right)$. We now choose
$t_{2}(\delta)>0$, such that
$1-\exp\left(-\tfrac{\delta^{2}}{16}t^{\alpha}\right)\geq 1-\delta$. Taking
$t_{0}=\max\\{t_{1},t_{2}\\}$, we thus get (6.12) from (6.21).

Step 6. In this step we prove (6.13). As before consider the Brownian bridge
$B_{1}(\cdot)$ on $I_{t}$ starting at
$B_{1}(-t^{-\alpha})=\mathfrak{h}_{t}(-t^{-\alpha})$ and ending at
$B_{1}(t^{-\alpha})=\mathfrak{h}_{t}(t^{-\alpha})$. We may write $B_{1}$ as

$B_{1}(x)=\mathfrak{h}_{t}^{(1)}(-t^{-\alpha})+\frac{x+t^{-\alpha}}{2t^{-\alpha}}(\mathfrak{h}_{t}^{(1)}(t^{-\alpha})-\mathfrak{h}_{t}^{(1)}(-t^{-\alpha}))+\overline{B}(x).$

where $\overline{B}$ is a Brownian bridge on $I_{t}$ starting and ending at
zero. Thus,

$\displaystyle
t^{1/3}(B_{1}(t^{-2/3}x)-B_{1}(0))=t^{1/3}\left[\overline{B}(t^{-2/3}x)-\overline{B}(0)\right]+\tfrac{1}{2}{t^{\alpha-1/3}x}(\mathfrak{h}_{t}^{(1)}(t^{-\alpha})-\mathfrak{h}_{t}^{(1)}(-t^{-\alpha})).$
(6.22)

Recall that $\alpha=\frac{1}{6}$. By Brownian scaling,
$B_{*}(x):=t^{1/3}\overline{B}(t^{-2/3}x)$ is a Brownian bridge on the large
interval $[-\sqrt{t},\sqrt{t}]$ starting and ending at zero. By computing the
covariances, it is easy to check that as $t\to\infty$, $B_{*}(x)-B_{*}(0)$
converges weakly to a two-sided Brownian motion $B(\cdot)$ on $[-a,a]$. This
gives us the weak limit for the first term on the r.h.s. of (6.22). For the
second term, recall the event $\mathsf{Tight}_{t}(\delta)$ from (6.5). As
$|x|\leq a$, on $\mathsf{Tight}_{t}(\delta)$, we have

$\tfrac{1}{2}{t^{\alpha-1/3}x}(\mathfrak{h}_{t}^{(1)}(t^{-\alpha})-\mathfrak{h}_{t}^{(1)}(-t^{-\alpha}))\leq{t^{-1/6}a}\delta^{-1}.$

This gives an uniform bound (uniform over the event
$\mathsf{Fav}_{t}(\delta))$ on the second term in (6.22). Thus as long as the
boundary data is in $\mathsf{Tight}_{t}(\delta)$,
$\mathbb{P}_{\operatorname{free},t}(A)\to\gamma(A)$ where
$\gamma(A)=\mathbb{P}(B(\cdot)\in A)$. This proves (6.13).

###  6.2. Dyson Behavior around joint maximum

In this subsection we state and prove Proposition 6.1.

######  Proposition 6.1 (Dyson behavior around joint maximum).

Fix $p\in(0,1)$. Set $q=1-p$. Consider $2$ independent copies of the KPZ
equation $\mathcal{H}_{\uparrow}(x,t)$, and $\mathcal{H}_{\downarrow}(x,t)$,
both started from the narrow wedge initial data. Let $\mathcal{M}_{p,t}$ be
the almost sure unique maximizer of the process
$x\mapsto(\mathcal{H}_{\uparrow}(x,pt)+\mathcal{H}_{\downarrow}(x,qt))$ which
exists via Lemma 3.1. Set

$\displaystyle D_{1}(x,t)$
$\displaystyle:=\mathcal{H}_{\uparrow}(\mathcal{M}_{p,t},pt)-\mathcal{H}_{\uparrow}(x+\mathcal{M}_{p,t},pt),$
(6.23) $\displaystyle D_{2}(x,t)$
$\displaystyle:=\mathcal{H}_{\downarrow}(x+\mathcal{M}_{p,t},qt)-\mathcal{H}_{\downarrow}(\mathcal{M}_{p,t},qt).$

As $t\to\infty$, we have the following convergence in law

$\displaystyle(D_{1}(x,t),D_{2}(x,t))\stackrel{{\scriptstyle
d}}{{\to}}(\mathcal{D}_{1}(x),\mathcal{D}_{2}(x))$ (6.24)

in the uniform-on-compact topology. Here
$\mathcal{D}=(\mathcal{D}_{1},\mathcal{D}_{2}):\mathbb{R}\to\mathbb{R}^{2}$ is
a two-sided $\mathsf{DBM}$, that is,
$\mathcal{D}_{+}(\cdot):=\mathcal{D}(\cdot)\mid_{[0,\infty)}$ and
$\mathcal{D}_{-}(\cdot):=\mathcal{D}(-\cdot)\mid_{(-\infty,0]}$ are
independent copies of $\mathsf{DBM}$ defined in Definition 5.1.

For clarity, the proof is completed over several subsections (Sections
6.2.1-6.2.9) below and we refer to Figure 7 for the structure of the proof.

Recasting Proposition 6.1 in the KPZ line ensemble framework (Section 6.2.1)
Heuristics and outline of proof of Proposition 6.1 (Section 6.2.2) Reducing
the global maximizer to the local maximizer (Section 6.2.3) Defining “Nice”
events that happen with high probability (Lemma 6.2, Section 6.2.4)
Conditioning w.r.t. large boundaries to obtain Brownian bridge law (Section
6.2.5) Conditioning w.r.t. the max data and small boundaries to obtain
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ law (Section 6.2.6) Obtaining
matching upper and lower bounds for (6.31) and the desired convergence
(Section 6.2.7) Proof of Lemma 6.2 (Section 6.2.8) Proofs of Lemmas 6.6 and
6.7 (Section 6.2.9) Figure 7. Structure of Section 6.2.

####  6.2.1. KPZ line ensemble framework

In this subsection, we convert Proposition 6.1 into the language of scaled KPZ
line ensemble defined in Proposition 2.5. We view
$\mathcal{H}_{\uparrow}(x,t)=\mathcal{H}_{t,\uparrow}^{(1)}(x),\mathcal{H}_{\downarrow}(x,t)=\mathcal{H}_{t,\downarrow}^{(1)}(x)$
as the top curves of two (unscaled) KPZ line ensembles:
$\\{\mathcal{H}_{t,\uparrow}^{(n)}(x),\mathcal{H}_{t,\downarrow}^{(n)}(x)\\}_{n\in\mathbb{N},x\in\mathbb{R}}$.
Following (2.5) we define their scaled versions:

$\displaystyle\mathfrak{h}_{t,\uparrow}^{(n)}(x)$
$\displaystyle:=t^{-1/3}\left(\mathcal{H}_{t,\uparrow}^{(n)}(t^{2/3}x)+\tfrac{t}{24}\right),\qquad\mathfrak{h}_{t,\downarrow}^{(n)}(x):=t^{-1/3}\left(\mathcal{H}_{t,\downarrow}^{(n)}(t^{2/3}x)+\tfrac{t}{24}\right).$

Along with the full maximizer $\mathcal{M}_{p,t}$, we will also consider local
maximizer defined by

$\displaystyle\mathcal{M}_{p,t}^{M}:=\mathop{\mathrm{argmax}}_{x\in[-Mt^{2/3},Mt^{2/3}]}(\mathcal{H}_{pt,\uparrow}^{(1)}(x)+\mathcal{H}_{qt,\downarrow}^{(1)}(x)),\qquad
M\in[0,\infty].$ (6.25)

For each $M>0$, $\mathcal{M}_{p,t}^{M}$ is unique almost surely by
$\mathbf{H}_{t}$-Brownian Gibbs property. We now set

$\displaystyle Y_{M,t,\uparrow}^{(n)}(x)$
$\displaystyle:=p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(n)}\big{(}(pt)^{-2/3}\mathcal{M}_{p,t}^{M}\big{)}-p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(n)}\big{(}p^{-2/3}x\big{)},$
(6.26) $\displaystyle Y_{M,t,\downarrow}^{(n)}(x)$
$\displaystyle:=q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(n)}\big{(}q^{-2/3}x\big{)}-q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(n)}\big{(}(qt)^{-2/3}\mathcal{M}_{p,t}^{M}\big{)}.$

Taking into account of (6.23) and all the above new notations, it can now be
checked that for each $t>0$,

$\displaystyle D_{1}(x,t)\stackrel{{\scriptstyle
d}}{{=}}t^{1/3}Y_{\infty,t,\uparrow}^{(1)}\big{(}t^{-2/3}(\mathcal{M}_{p,t}^{\infty}+x)\big{)},\quad
D_{2}(x,t)\stackrel{{\scriptstyle
d}}{{=}}t^{1/3}Y_{\infty,t,\downarrow}^{(1)}\big{(}t^{-2/3}(\mathcal{M}_{p,t}^{\infty}+x)\big{)},$
(6.27)

both as functions in $x$. Thus it is equivalent to verify Proposition 6.1 for
the above $Y_{\infty,t,\uparrow}^{(1)},Y_{\infty,t,\downarrow}^{(1)}$
expressions. In our proof we will mostly deal with local maximizer version,
and so for convenience we define:

$\displaystyle
D_{M,t,\uparrow}(x):{=}t^{1/3}Y_{M,t,\uparrow}^{(1)}\big{(}t^{-2/3}(\mathcal{M}_{p,t}^{M}+x)\big{)},\quad
D_{M,t,\downarrow}(x)=t^{1/3}Y_{M,t,\downarrow}^{(1)}\big{(}t^{-2/3}(\mathcal{M}_{p,t}^{M}+x)\big{)}.$
(6.28)

where $Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)}$ are defined in (6.26).
We will introduce several other notations and parameters later in the proof.
For the moment, the minimal set of notations introduced here facilitate our
discussion of ideas and outline of the proof of Proposition 6.1 in the next
subsection.

####  6.2.2. Ideas and Outline of Proof of Proposition 6.1

Before embarking on a rather lengthy proof, in this subsection we explain the
core ideas behind the proof and provide an outline for the remaining
subsections.

First we contrast the proof idea with that of Theorem 1.11. Indeed, similar to
the proof of Theorem 1.11, from (6.27) we see that at the level
$Y_{\infty,t,\uparrow}^{(1)},Y_{\infty,t,\downarrow}^{(1)}$ we are interested
in understanding their laws restricted to a very small symmetric interval of
order $O(t^{-2/3})$ around the point $t^{-2/3}\mathcal{M}_{p,t}^{\infty}$.
However, the key difference from the conceptual argument presented at the
beginning of the proof if Theorem 1.11 is that the centered point
$t^{-2/3}\mathcal{M}_{p,t}^{\infty}$ is random and it does not go to zero.
Rather by Theorem 1.8 it converges in distribution to a nontrivial random
quantity (namely $\Gamma(p\sqrt{2})$). Hence one must take additional care of
this random point. This makes the argument significantly more challenging
compared to that of Theorem 1.11.

Figure 8. An overview of the proof for Proposition 6.1. The top and bottom
black curves are $Y_{M,t,\uparrow}^{(1)}$ and $Y_{M,t,\downarrow}^{(1)}$
respectively. Note that the way they are defined in (6.26),
$Y_{M,t,\uparrow}^{(1)}(x)\geq Y_{M,t,\downarrow}^{(1)}(x)$ with equality at
$x=\Phi=t^{-2/3}\mathcal{M}_{p,t}^{M}$ labelled as the red dot in the above
figure. The blue curves are $Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(2)}$.
There is no such ordering within blue curves. They may intersect among
themselves as well as with the black curves. With $\alpha=\frac{1}{6}$, we
consider the interval $K_{t}=(\Phi-t^{-\alpha},\Phi+t^{-\alpha})$. In this
vanishing interval around $\Phi$, the curves will be ordered with high
probability. In fact, with high probability, there will be a uniform
separation. For instance, for small enough $\delta$, we will have
$Y_{M,t,\uparrow}^{(2)}(x)-Y_{M,t,\uparrow}^{(1)}(x)\geq\frac{1}{4}\delta$,
and
$Y_{M,t,\downarrow}^{(1)}(x)-Y_{M,t,\downarrow}^{(2)}(x)\geq\frac{1}{4}\delta$,
for all $x\in K_{t}$ wth high probability. This will allow us to conclude
black curves are behave approximately like two-sided
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$s on that narrow window. Then upon
going into a even smaller window of $O(t^{-2/3})$, the two-sided
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$s turn into a two-sided
$\mathsf{DBM}$.

We now give a road-map of our proof. At this point, readers are also invited
to look into Figure 8 alongside the explanation offered in its caption.

  * •

As noted in Lemma 3.1, the random centering
$t^{-2/3}\mathcal{M}_{p,t}^{\infty}$ has decaying properties and can be
approximated by $t^{-2/3}\mathcal{M}_{p,t}^{M}$ by taking large enough $M$.
Hence on a heuristic level it suffices to work with the local maximizers
instead. In Subsection 6.2.3, this heuristics will be justified rigorously. We
will show there how to pass from
$Y_{\infty,t,\uparrow}^{(1)},Y_{\infty,t,\downarrow}^{(1)}$ defined in (6.27)
to their finite centering analogs:
$Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)}$. The rest of the proof then
boils down to analyzing the laws of the latter.

  * •

We now fix a $M>0$ for the rest of the proof. Our analysis will now operate
with $\mathcal{M}_{p,t}^{M}$. For simplicity, let us also use the notation

$\displaystyle\Phi:=t^{-2/3}\mathcal{M}_{p,t}^{M}$ (6.29)

for the rest of the proof. We now perform several conditioning on the laws of
the curves. Recall that by Proposition 2.5,
$\\{\mathfrak{h}_{pt,\uparrow}^{(n)}(\cdot)\\}_{n\in\mathbb{N}}$
$\\{\mathfrak{h}_{qt,\downarrow}^{(n)}(\cdot)\\}_{n\in\mathbb{N}}$ satisfy the
$\mathbf{H}_{pt}$-Brownian Gibbs property and the $\mathbf{H}_{qt}$-Brownian
Gibbs property respectively with $\mathbf{H}_{t}$ given by (2.4). Conditioned
on the end points of $\mathfrak{h}_{pt,\uparrow}^{(1)}(\pm Mp^{-2/3})$ and
$\mathfrak{h}_{qt,\downarrow}^{(1)}(\pm Mq^{-2/3})$ and the second curves
$\mathfrak{h}_{pt,\uparrow}^{(2)}(\cdot)$ and
$\mathfrak{h}_{qt,\downarrow}^{(2)}(\cdot)$, the laws of
$\mathfrak{h}_{pt,\uparrow}^{(1)}(\cdot)$, and
$\mathfrak{h}_{pt,\uparrow}^{(1)}(\cdot)$ are absolutely continuous w.r.t.
Brownian bridges with appropriate end points. This conditioning is done in
Subsection 6.2.5.

  * •

We then condition further on Max data :
$\mathcal{M}_{p,t}^{M},\mathfrak{h}_{pt,\uparrow}^{(1)}((pt)^{-2/3}\mathcal{M}_{p,t}^{M}),\mathfrak{h}_{qt,\downarrow}^{(1)}((qt)^{-2/3}\mathcal{M}_{p,t}^{M})$.
Under this conditioning, via the decomposition result in Proposition 4.10, the
underlying Brownian bridges mentioned in the previous point, when viewed from
the joint maximizer, becomes two-sided
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$s defined in Definition 4.4. This
viewpoint from the joint maximizer is given by
$Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)}$. See Figure 8 for more
details.

  * •

We emphasize the fact that the deduction of
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$s done above is only for the
underlying Brownian law. One still needs to analyze the Radon-Nikodym (RN)
derivative. As we are interested in the laws of
$Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)}$ on an interval of order
$t^{-2/3}$ around $\Phi$, we analyze the RN derivative only on a small
interval around $\Phi$. To be precise, we consider a slightly larger yet
vanishing interval of length $2t^{-\alpha}$ for $\alpha=\frac{1}{6}$ around
the random point $\Phi$. We show that the RN derivative on this small random
patch is close to $1$. Thus upon further conditioning on the boundary data of
this random small interval, the trajectories of $Y_{M,t,\uparrow}^{(1)}$ and
$Y_{M,t,\downarrow}^{(1)}$ defined in (6.26) around $\Phi$ turns out to be
close to two-sided $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ with appropriate
(random) endpoints.

  * •

Finally, we zoom further into a tiny interval of order $O(t^{-2/3})$ symmetric
around the random point $\Phi$. Utilizing Lemma 5.3, we convert the two-sided
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$s to two-sided $\mathsf{DBM}$s.

We now provide an outline of the rest of the subsections. In Subsection 6.2.3
we reduce our proof from understanding laws around global maximizers to that
of local maximizers. As explained in the above road-map, the proof follows by
performing several successive conditioning. On a technical level, this
requires defining several high probability events on which we can carry out
our conditional analysis. These events are all defined in Subsection 6.2.4 and
are claimed to happen with high probability in Lemma 6.2. We then execute the
first layer of conditioning in Subsection 6.2.5. The two other layers of
conditioning are done in Subsection 6.2.6. Lemma 6.6 and Lemma 6.7 are the
precise technical expressions for the heuristic claims in the last two bullet
points of the road-map. Assuming them, we complete the final steps of the
proof in Subsection 6.2.7. Proof of Lemma 6.2 is then presented in Subsection
6.2.8. Finally, in Subsection 6.2.9, we prove the remaining lemmas: Lemma 6.6
and 6.7.

####  6.2.3. Global to Local maximizer

We now fill out the technical details of the road-map presented in the
previous subsection. Fix any $a>0$. Consider any Borel set $A$ of
$C([-a,a]\to\mathbb{R}^{2})$ which is a continuity set of a two-sided
$\mathsf{DBM}$ $\mathcal{D}(\cdot)$ restricted to $[-a,a].$ By Portmanteau
theorem, it is enough to show that

$\displaystyle\mathbb{P}((D_{1}(\cdot,t),D_{2}(\cdot,t))\in
A)\rightarrow\mathbb{P}(\mathcal{D}(\cdot)\in A),$ (6.30)

where $D_{1},D_{2}$ are defined in (6.23). In this subsection, we describe how
it suffices to check (6.30) with $\mathcal{M}_{p,t}^{M}$. Recall
$D_{M,t,\uparrow}(\cdot),D_{M,t,\downarrow}(\cdot)$ from (6.28). We claim that
for all $M>0$:

$\displaystyle\lim_{t\to\infty}\mathbb{P}((D_{M,t,\uparrow}(\cdot),D_{M,t,\downarrow}(\cdot))\in
A)\rightarrow\mathbb{P}(\mathcal{D}(\cdot)\in A).$ (6.31)

Note that when $\mathcal{M}_{p,t}^{\infty}=\mathcal{M}_{p,t}^{M}$,
$(D_{M,t,\uparrow}(\cdot),D_{M,t,\downarrow}(\cdot))$ is exactly equal to

$t^{1/3}Y_{\infty,t,\uparrow}^{(1)}\big{(}t^{-2/3}(\mathcal{M}_{p,t}^{\infty}+\cdot)\big{)}\
,\
t^{1/3}Y_{\infty,t,\downarrow}^{(1)}\big{(}t^{-2/3}(\mathcal{M}_{p,t}^{\infty}+\cdot)\big{)}$

which via (6.27) is same in distribution as $D_{1}(\cdot,t),D_{2}(\cdot,t)$.
Thus,

$\displaystyle{\left|\mathbb{P}((D_{1}(\cdot,t),D_{2}(\cdot,t))\in
A)-\mathbb{P}((D_{M,t,\uparrow}(\cdot),D_{M,t,\downarrow}(\cdot))\in
A)\right|\leq 2\mathbb{P}(\mathcal{M}_{p,t}\neq\mathcal{M}_{p,t}^{M}).}$

Now given any $\varepsilon>0$, by Lemma 3.1, we can take $M=M(\varepsilon)>0$
large enough so that
$2\mathbb{P}(\mathcal{M}_{p,t}\neq\mathcal{M}_{p,t}^{M})\leq\varepsilon$. Then
upon taking $t\to\infty$ in the above equation, in view of (6.31), we see that

$\limsup_{t\to\infty}\left||\mathbb{P}((D_{1}(\cdot,t),D_{2}(\cdot,t))\in
A)-\mathbb{P}((\mathcal{D}(\cdot)\in A)\right|\leq\varepsilon.$

As $\varepsilon$ is arbitrary, this proves (6.30). The rest of the proof is
now devoted in proving (6.31).

####  6.2.4. Nice events

In this subsection, we focus on defining several events that are collectively
‘nice’ in the sense that they happen with high probability. We fix an $M>0$
for the rest of the proof and work with the local maximizer
$\mathcal{M}_{p,t}^{M}$ defined in (6.25). We will also make use of the
notation $\Phi$ defined in (6.29) heavily in this and subsequent subsections.
We now proceed to define a few events based on the location and value of the
maximizer and values at the endpoints of an appropriate interval. Fix any
arbitrary $\delta>0$. Let us consider the event:

$\displaystyle\mathsf{ArMx}(\delta):=\left\\{\Phi\in[-M+\delta,M-\delta]\right\\}.$
(6.32)

The $\mathsf{ArMx}(\delta)$ controls the location of the local maximizer
$\Phi$. Set $\alpha=\frac{1}{6}$. We define tightness event that corresponds
to the boundary of the interval of length $2t^{-\alpha}$ around $\Phi:$

$\displaystyle\mathsf{Bd}_{\uparrow}(\delta)$
$\displaystyle:=\mathsf{Bd}_{+,\uparrow}(\delta)\cap\mathsf{Bd}_{-,\uparrow}(\delta),\quad\mathsf{Bd}_{\downarrow}(\delta):=\mathsf{Bd}_{+,\downarrow}(\delta)\cap\mathsf{Bd}_{-,\downarrow}(\delta),$
(6.33)

where

$\displaystyle\mathsf{Bd}_{\pm,\uparrow}(\delta)$
$\displaystyle:=\left\\{\left|\mathfrak{h}_{pt,\uparrow}^{(1)}\big{(}p^{-2/3}(\Phi\pm
t^{-\alpha})\big{)}-\mathfrak{h}_{pt,\uparrow}^{(1)}\big{(}\Phi
p^{-2/3})\right|\leq\tfrac{1}{\delta}t^{-\alpha/2}\right\\}$ (6.34)
$\displaystyle\mathsf{Bd}_{\pm,\downarrow}(\delta)$
$\displaystyle:=\left\\{\left|\mathfrak{h}_{qt,\downarrow}^{(1)}\big{(}q^{-2/3}(\Phi\pm
t^{-\alpha})\big{)}-\mathfrak{h}_{qt,\downarrow}^{(1)}\big{(}\Phi
q^{-2/3})\right|\leq\tfrac{1}{\delta}t^{-\alpha/2}\right\\},$

Finally we consider the gap events that provide a gap between the first curve
and the second curve for each of the line ensemble:

$\displaystyle\mathsf{Gap}_{M,\uparrow}(\delta)$
$\displaystyle:=\left\\{p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}\big{(}\Phi
p^{-2/3}\big{)}\geq p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}\big{(}\Phi
p^{-2/3}\big{)}+\delta\right\\},$ (6.35)
$\displaystyle\mathsf{Gap}_{M,\downarrow}(\delta)$
$\displaystyle:=\left\\{q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(1)}\big{(}\Phi
q^{-2/3}\big{)}\geq q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(2)}\big{(}\Phi
q^{-2/3}\big{)}+\delta\right\\}.$ (6.36)

We next define the ‘rise’ events which roughly says the second curves
$\mathfrak{h}_{pt,\uparrow}^{(1)}$ and $\mathfrak{h}_{qt,\downarrow}^{(2)}$ of
the line ensembles does not rise too much on a small interval of length
$2t^{-\alpha}$ around $\Phi p^{-2/3}$ and $\Phi q^{-2/3}$ respectively.

$\displaystyle\mathsf{Rise}_{M,\uparrow}(\delta)$
$\displaystyle:=\left\\{\sup_{x\in[-t^{-\alpha},t^{-\alpha}]}p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}\big{(}\Phi
p^{-2/3}+x\big{)}\leq p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}\big{(}\Phi
p^{-2/3}\big{)}+\tfrac{\delta}{4}\right\\},$ (6.37)
$\displaystyle\mathsf{Rise}_{M,\downarrow}(\delta)$
$\displaystyle:=\left\\{\sup_{x\in[-t^{-\alpha},t^{-\alpha}]}q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(2)}\big{(}\Phi
q^{-2/3}+x\big{)}\leq q^{1/3}\mathfrak{h}_{pt,\downarrow}^{(2)}\big{(}\Phi
q^{-2/3}\big{)}+\tfrac{\delta}{4}\right\\}.$ (6.38)

$\mathsf{Bd}$, $\mathsf{Gap}$, $\mathsf{Rise}$ type events and their
significance are discussed later in Subsection 6.2.8 in greater details. See
also Figure 9 and its caption for explanation of some of these events. We put
all the above events into one final event:

$\displaystyle\mathsf{Nice}_{M}(\delta):=\left\\{\mathsf{ArMx}(\delta)\cap\bigcap_{x\in\\{\uparrow,\downarrow\\}}\mathsf{Bd}_{x}(\delta)\cap\mathsf{Gap}_{M,x}(\delta)\cap\mathsf{Rise}_{M,x}(\delta)\right\\}.$
(6.39)

All the above events are dependent on $t$. But we have suppressed this
dependence from the notations. The $\mathsf{Nice}_{M}(\delta)$ turns out to be
a favorable event. We isolate this fact as a lemma below.

######  Lemma 6.2.

For any $M>0$, under the above setup we have

$\displaystyle\liminf_{\delta\downarrow
0}\liminf_{t\to\infty}\mathbb{P}_{t}\left(\mathsf{Nice}_{M}(\delta)\right)=1.$
(6.40)

We postpone the proof of this technical lemma to Section 6.2.8 and for the
moment we continue with the current proof of Proposition 6.1 assuming its
validity.

####  6.2.5. Conditioning with respect to large boundaries

As alluded in Subsection 6.2.2, the proof involves conditioning on different
$\sigma$-fields successively. We now specify all the different $\sigma$-fields
that we will use throughout the proof. Set $\alpha=\frac{1}{6}$. We consider
the random interval

$\displaystyle K_{t}:=(\Phi-t^{-\alpha},\Phi+t^{-\alpha}).$ (6.41)

Let us define:

$\displaystyle\mathcal{F}_{1}$
$\displaystyle:=\sigma\left(\left\\{\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}x),\mathfrak{h}_{qt,\downarrow}^{(1)}(q^{-2/3}x)\right\\}_{x\in(-M,M)^{c}},\left\\{\mathfrak{h}_{pt,\uparrow}^{(2)}(x),\mathfrak{h}_{qt,\downarrow}^{(2)}(x)\right\\}_{x\in\mathbb{R}}\right)$
(6.42) $\displaystyle\mathcal{F}_{2}$
$\displaystyle:=\sigma\left(\Phi,\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3}),\mathfrak{h}_{qt,\downarrow}^{(1)}(\Phi q^{-2/3})\right),$ (6.43)
$\displaystyle\mathcal{F}_{3}$
$\displaystyle:=\sigma\left(\left\\{\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}x),\mathfrak{h}_{qt,\downarrow}^{(1)}(q^{-2/3}x)\right\\}_{x\in
K_{t}^{c}}\right).$ (6.44)

In this step we perform conditioning w.r.t. $\mathcal{F}_{1}$ for the
expression on the l.h.s. of (6.31). We denote
$\mathbb{P}_{t}(A):=\mathbb{P}\big{(}(D_{M,t,\uparrow}(\cdot),D_{M,t,\downarrow}(\cdot))\in
A\big{)}$. Taking the $\mathsf{Nice}_{M}(\delta)$ event defined in (6.39)
under consideration, upon conditioning with $\mathcal{F}_{1}$ we have the
following upper and lower bounds:

$\displaystyle\mathbb{P}_{t}(A)$
$\displaystyle\geq\mathbb{P}_{t}(\mathsf{Nice}_{M}(\delta),A)=\mathbb{E}_{t}\left[\mathbb{P}_{t}(\mathsf{Nice}_{M}(\delta),A\mid\mathcal{F}_{1})\right],$
(6.45) $\displaystyle\mathbb{P}_{t}(A)$
$\displaystyle\leq\mathbb{P}_{t}(\mathsf{Nice}_{M}(\delta),A)+\mathbb{P}_{t}(\neg\mathsf{Nice}_{M}(\delta))=\mathbb{E}_{t}\left[\mathbb{P}_{t}(\mathsf{Nice}_{M}(\delta),A\mid\mathcal{F}_{1})\right]+\mathbb{P}_{t}(\neg\mathsf{Nice}_{M}(\delta)).$
(6.46)

Note that the underlying measure consists of the mutually independent
$\mathfrak{h}_{pt,\uparrow}^{(1)}(\cdot)$ and
$\mathfrak{h}_{qt,\downarrow}^{(1)}(\cdot)$ which by Proposition 2.5 satisfy
$\textbf{H}_{pt}$ and $\textbf{H}_{qt}$ Brownian Gibbs property respectively.
Applying the respectively Brownian Gibbs properties and following (2.3) we
have

$\displaystyle\mathbb{P}_{t}(\mathsf{Nice}_{M}(\delta),A\mid\mathcal{F}_{1})=\frac{\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta),A}W_{\uparrow}W_{\downarrow}]}{\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]}.$
(6.47)

Here

$\displaystyle
W_{\uparrow}:=\exp\left(-t^{2/3}\int_{-M}^{M}\exp\left(t^{1/3}\big{[}p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}(p^{-2/3}x)-p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}x)\big{]}\right)\mathrm{d}x\right)$
(6.48)

and

$\displaystyle
W_{\downarrow}:=\exp\left(-t^{2/3}\int_{-M}^{M}\exp\left(t^{1/3}\big{[}q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(2)}(q^{-2/3}x)-q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(1)}(q^{-2/3}x)\big{]}\right)\mathrm{d}x\right).$
(6.49)

In (6.47), $\mathbb{P}_{\operatorname{free},t}$ and
$\mathbb{E}_{\operatorname{free},t}$ are the probability and the expectation
operator respectively corresponding to the joint ‘free’ law for
$(p^{1/3}\mathfrak{h}_{pt,\uparrow}(p^{-2/3}x),q^{1/3}\mathfrak{h}_{qt,\downarrow}(q^{-2/3}x))_{x\in[-M,M]}$
which by Brownian scaling is given by a pair of independent Brownian bridges
$(B_{1}(\cdot),B_{2}(\cdot))$ on $[-M,M]$ with starting points
$(p^{1/3}\mathfrak{h}_{pt,\uparrow}(-Mp^{-2/3}),q^{1/3}\mathfrak{h}_{qt,\downarrow}(-Mq^{-2/3}))$
and endpoints
$(q^{1/3}\mathfrak{h}_{pt,\uparrow}(Mp^{-2/3}),q^{1/3}\mathfrak{h}_{qt,\downarrow}(Mq^{-2/3})).$

####  6.2.6. Conditioning with respect to maximum data and small boundaries

In this subsection we perform conditioning on the numerator of r.h.s. of
(6.47) w.r.t. $\mathcal{F}_{2}$ and $\mathcal{F}_{3}$ defined in (6.43) and
(6.44). Recall that by Proposition 4.10, upon conditioning Brownian bridges on
$\mathcal{F}_{2}$, the conditional laws around the joint local maximizer
$\Phi$ over $[-M,M]$ is now given by two
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$s (defined in Definition 4.4) with
appropriate lengths and endpoints. Indeed, based on Proposition 4.10, given
$\mathcal{F}_{1},\mathcal{F}_{2}$, we may construct the conditional laws for
the two functions on $[-M,M]$:

######  Definition 6.3 ($\mathsf{Nlarge}$ Law).

Consider two independent $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$
$V_{\ell}^{\mathsf{large}}$ and $V_{r}^{\mathsf{large}}$ with following
description:

  1. (1)

$V_{\ell}^{\mathsf{large}}$ is a $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ on
$[0,\Phi+M]$ ending at

$\left(p^{1/3}\left[\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3})-\mathfrak{h}_{pt,\uparrow}^{(1)}(-Mp^{-2/3})\right],q^{1/3}\left[\mathfrak{h}_{qt,\downarrow}^{(1)}(-Mq^{-2/3})-\mathfrak{h}_{qt,\downarrow}^{(1)}(\Phi
q^{-2/3})\right]\right),$

  2. (2)

$V_{r}^{\mathsf{large}}$ is a $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ on
$[0,M-\Phi]$ ending at

$\left(p^{1/3}\left[\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3})-\mathfrak{h}_{pt,\uparrow}^{(1)}(Mp^{-2/3})\right],q^{1/3}\left[\mathfrak{h}_{qt,\downarrow}^{(1)}(Mq^{-2/3})-\mathfrak{h}_{qt,\downarrow}^{(1)}(\Phi
q^{-2/3})\right]\right).$

We then define ${B}^{\mathsf{large}}:[-M,M]\to\mathbb{R}^{2}$ as follows:

$\displaystyle{B}^{\mathsf{large}}(x)=\begin{cases}V_{\ell}(\Phi-x)&x\in[-M,\Phi]\\\
V_{r}(x-\Phi)&x\in[\Phi,M]\end{cases}.$

We denote the expectation and probability operator under above law for
${B}^{\mathsf{large}}$ (which depends on $\mathcal{F}_{1},\mathcal{F}_{2}$) as
$\mathbb{E}_{\mathsf{Nlarge|2,1}}$ and $\mathbb{P}_{\mathsf{Nlarge|2,1}}$.

Thus we may write

$\displaystyle\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta),A}W_{\uparrow}W_{\downarrow}]$
$\displaystyle=\mathbb{E}_{\operatorname{free},t}[\mathbb{E}_{\mathsf{Nlarge|2,1}}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta),A}W_{\uparrow}W_{\downarrow}]].$
(6.50)

Since $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$s are Markovian, we may
condition further upon $\mathcal{F}_{3}$ to get
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$s again but on a smaller interval.
To precisely define the law, we now give the following definitions:

######  Definition 6.4 ($\mathsf{Nsmall}$ law).

Consider two independent $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$
$V_{\ell}^{\mathsf{small}}$ and $V_{r}^{\mathsf{small}}$ with the following
descriptions:

  1. (1)

$V_{\ell}^{\mathsf{small}}$ is a $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ on
$[0,t^{-\alpha}]$ ending at

$\left(p^{1/3}\left[\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3})-\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}(\Phi-t^{-\alpha}))\right],q^{1/3}\left[\mathfrak{h}_{qt,\downarrow}^{(1)}(q^{-2/3}(\Phi-t^{-\alpha}))-\mathfrak{h}_{qt,\downarrow}^{(1)}(\Phi
q^{-2/3})\right]\right),$

  2. (2)

$V_{r}^{\mathsf{small}}$ is a $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ on
$[0,t^{-\alpha}]$ ending at

$\left(p^{1/3}\left[\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3})-\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}(\Phi+t^{-\alpha}))\right],q^{1/3}\left[\mathfrak{h}_{qt,\downarrow}^{(1)}(q^{-2/3}(\Phi+t^{-\alpha}))-\mathfrak{h}_{qt,\downarrow}^{(1)}(\Phi
q^{-2/3})\right]\right).$

We then define
${B}^{\mathsf{small}}:[\Phi+t^{-\alpha},\Phi-t^{-\alpha}]\to\mathbb{R}^{2}$ as
follows:

$\displaystyle{B}^{\mathsf{small}}(x)=\begin{cases}V_{\ell}(\Phi-x)&x\in[\Phi-t^{-\alpha},\Phi]\\\
V_{r}(x-\Phi)&x\in[\Phi,\Phi+t^{-\alpha}]\end{cases}.$

We denote the the expectation and probability operators under the above law
for ${B}^{\mathsf{small}}$ (which depends on
$\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3}$) as
$\mathbb{E}_{\mathsf{Nsmall|3,2,1}}$ and $\mathbb{P}_{\mathsf{Nsmall|3,2,1}}$
respectively.

We thus have

r.h.s. of (6.50)
$\displaystyle=\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}\mathbb{E}_{\mathsf{Nsmall|3,2,1}}[\mathbf{1}_{A}W_{\uparrow}W_{\downarrow}]].$
(6.51)

The $\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}$ comes of the interior expectation
above as $\mathsf{Nice}_{M}(\delta)$ is measurable w.r.t.
$\mathcal{F}_{1}\cup\mathcal{F}_{2}\cup\mathcal{F}_{3}$ (see its definition in
(6.39)).

Next note that due to the definition of $W_{\uparrow},W_{\downarrow}$ from
(6.48) and (6.49), we may extract certain parts of it which are measurable
w.r.t. $\mathcal{F}_{1}\cup\mathcal{F}_{2}\cup\mathcal{F}_{3}$. Indeed, we can
write $W_{\uparrow}=W_{\uparrow,1}W_{\uparrow,2}$ and
$W_{\downarrow}=W_{\downarrow,1}W_{\downarrow,2}$ where

$\displaystyle
W_{\uparrow,1}:=\exp\left(-t^{2/3}\int_{K_{t}}\exp\left(t^{1/3}\big{[}p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}(p^{-2/3}x)-p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}x)\big{]}\right)\mathrm{d}x\right)$
(6.52) $\displaystyle W_{\uparrow,2}:=\exp\left(-t^{2/3}\int_{[-M,M]\cap
K_{t}^{c}}\exp\left(t^{1/3}\big{[}p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}(p^{-2/3}x)-p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}x)\big{]}\right)\mathrm{d}x\right),$

and

$\displaystyle
W_{\downarrow,1}:=\exp\left(-t^{2/3}\int_{K_{t}}\exp\left(t^{1/3}\big{[}q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(2)}(q^{-2/3}x)-q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(1)}(q^{-2/3}x)\big{]}\right)\mathrm{d}x\right).$
(6.53) $\displaystyle W_{\downarrow,2}:=\exp\left(-t^{2/3}\int_{[-M,M]\cap
K_{t}^{c}}\exp\left(t^{1/3}\big{[}q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(2)}(q^{-2/3}x)-q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(1)}(q^{-2/3}x)\big{]}\right)\mathrm{d}x\right),$

where recall $K_{t}$ from (6.41). The key observation is that
$W_{\uparrow,2},W_{\downarrow,2}$ are measurable w.r.t.
$\mathcal{F}_{1}\cup\mathcal{F}_{2}\cup\mathcal{F}_{3}$. Thus we have

r.h.s. of (6.51)
$\displaystyle=\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow,2}W_{\downarrow,2}\cdot\mathbb{E}_{\mathsf{Nsmall|3,2,1}}[\mathbf{1}_{A}W_{\uparrow,1}W_{\downarrow,1}]].$
(6.54)

######  Remark 6.5.

It is crucial to note that in (6.51) the event $\mathsf{Nice}_{M}(\delta)$
includes the event $\mathsf{ArMx}(\delta)$ defined in (6.32). Indeed, the
$\mathsf{ArMx}(\delta)$ event is measurable w.r.t.
$\mathcal{F}_{1}\cup\mathcal{F}_{2}$ and ensures that
$[\Phi-t^{-\alpha},\Phi+t^{-\alpha}]\subset[-M,M]$ for all large enough $t$,
which is essential for going from $\mathsf{Nlarge}$ law to $\mathsf{Nsmall}$
law. Thus such a decomposition is not possible for
$\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]$ which appears
in the denominator of r.h.s. of (6.47). Nonetheless, we may still provide a
lower bound for
$\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]$ as follows:

$\displaystyle\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]\geq\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow}W_{\downarrow}]=\mathbb{E}_{\operatorname{free},t}[W_{\uparrow,2}W_{\downarrow,2}\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}\cdot\mathbb{E}_{\mathsf{Nsmall|3,2,1}}[W_{\uparrow,1}W_{\downarrow,1}]].$
(6.55)

With the deductions in (6.54) and (6.55), we now come to the task of analyzing
$W_{\uparrow,1}W_{\downarrow,1}$ under $\mathsf{Nsmall}$ law. The following
lemma ensures that on $\mathsf{Nice}_{M}(\delta)$,
$W_{\uparrow,1}W_{\downarrow,1}$ is close to $1$ under $\mathsf{Nsmall}$ law.

######  Lemma 6.6.

There exist $t_{0}(\delta)>0$ such that for all $t\geq t_{0}$ we have

$\displaystyle\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(W_{\uparrow,1}W_{\downarrow,1}>1-\delta)\geq\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}(1-\delta).$
(6.56)

This allow us to ignore $W_{\uparrow,1}W_{\downarrow,1}$, in
$\mathbb{E}_{\mathsf{Nsmall|3,2,1}}[\mathbf{1}_{A}W_{\uparrow,1}W_{\downarrow,1}]$.
Hence it suffices to study $\mathbb{P}_{\mathsf{Nsmall}|3,2,1}(A)$. The
following lemma then compares this conditional probability with that of
$\mathsf{DBM}$.

######  Lemma 6.7.

There exist $t_{0}(\delta)>0$ such that for all $t\geq t_{0}$ we have

$\displaystyle\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}|\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(A)-\tau(A)|\leq\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}\cdot\delta,$
(6.57)

where $\tau(A):=\mathbb{P}(\mathcal{D}(\cdot)\in A)$, $\mathcal{D}$ being a
two-sided $\mathsf{DBM}$ defined in the statement of Proposition 6.1.

We prove these two lemmas in Section 6.2.9. For now, we proceed with the
current proof of (6.31) in the next section.

####  6.2.7. Matching Lower and Upper Bounds

In this subsection, we complete the proof of (6.31) by providing matching
lower and upper bounds in the two steps below. We assume throughout this
subsection that $t$ is large enough, so that (6.56) and (6.57) holds.

Step 1: Lower Bound. We start with (6.45). Following the expression in (6.47),
and our deductions in (6.50), (6.51), (6.54) we see that

$\displaystyle\mathbb{P}_{t}(A)$
$\displaystyle\geq\mathbb{E}_{t}\left[\mathbb{P}_{t}(\mathsf{Nice}_{M}(\delta),A\mid\mathcal{F}_{1})\right]$
$\displaystyle=\mathbb{E}\left[\frac{\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow,2}W_{\downarrow,2}\cdot\mathbb{E}_{\mathsf{Nsmall|3,2,1}}[\mathbf{1}_{A}W_{\uparrow,1}W_{\downarrow,1}]]}{\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]}\right]$
(6.58)
$\displaystyle\geq(1-\delta)\mathbb{E}_{t}\left[\frac{\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow,2}W_{\downarrow,2}\cdot\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(A,W_{\uparrow,1}W_{\downarrow,1}>1-\delta)]}{\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]}\right]$
(6.59)

where in the last inequality we used the fact
$W_{\uparrow,1}W_{\downarrow,1}\leq 1$. Now applying Lemma 6.6 and Lemma 6.7
successively we get

$\displaystyle\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(A,W_{\uparrow,1}W_{\downarrow,1}>1-\delta)$
$\displaystyle\geq\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}[\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(A)-\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(W_{\uparrow,1}W_{\downarrow,1}\leq
1-\delta)]$
$\displaystyle\geq\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}[\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(A)-\delta]$
$\displaystyle\geq\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}[\tau(A)-2\delta]$

where recall $\tau(A)=\mathbb{P}(\mathcal{D}(\cdot)\in A)$. As
$W_{\uparrow,1}W_{\downarrow,1}\leq 1$ and probabilities are nonnegative,
following the above inequalities we have

$\displaystyle\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(A,W_{\uparrow,1}W_{\downarrow,1}>1-\delta)\geq\max\\{0,\tau(A)-2\delta\\}\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow,1}W_{\downarrow,1}.$

Substituting the above bound back to (6.59) and using the fact that
$W_{\uparrow,2}W_{\downarrow,2}W_{\uparrow,1}W_{\downarrow,1}=W_{\uparrow}W_{\downarrow}$,
we get

$\displaystyle\mathbb{P}_{t}(A)$
$\displaystyle\geq(1-\delta)\max\\{0,\tau(A)-2\delta\\}\mathbb{E}_{t}\left[\frac{\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow}W_{\downarrow}]}{\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]}\right]$
$\displaystyle=(1-\delta)\max\\{0,\tau(A)-2\delta\\}\mathbb{P}_{t}(\mathsf{Nice}_{M}(\delta)).$

In view of Lemma 6.2, taking $\liminf_{t\to\infty}$ followed by
$\liminf_{\delta\downarrow 0}$ we get that
$\liminf_{t\to\infty}\mathbb{P}_{t}(A)\geq\tau(A)$. This proves the lower
bound.

Step 2: Upper Bound. We start with (6.46). Using the equality in (6.58) we get

$\displaystyle\mathbb{P}_{t}(A)$
$\displaystyle\leq\mathbb{E}\left[\frac{\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow,2}W_{\downarrow,2}\cdot\mathbb{E}_{\mathsf{Nsmall|3,2,1}}[\mathbf{1}_{A}W_{\uparrow,1}W_{\downarrow,1}]]}{\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]}\right]+\mathbb{P}_{t}(\neg\mathsf{Nice}_{M}(\delta))$
$\displaystyle\leq\mathbb{E}\left[\frac{\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow,2}W_{\downarrow,2}\cdot\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(A)]}{\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]}\right]+\mathbb{P}_{t}(\neg\mathsf{Nice}_{M}(\delta))$
$\displaystyle\leq(\tau(A)+\delta)\mathbb{E}\left[\frac{\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow,2}W_{\downarrow,2}]}{\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]}\right]+\mathbb{P}_{t}(\neg\mathsf{Nice}_{M}(\delta)).$
(6.60)

Let us briefly justify the inequalities presented above. Going from first line
to second line we used the fact $W_{\uparrow,1}W_{\downarrow,1}\leq 1$. The
last inequality follows from Lemma 6.7 where recall that
$\tau(A)=\mathbb{P}(\mathcal{D}(\cdot)\in A).$ Now note that by Lemma 6.6, on
$\mathsf{Nice}_{M}(\delta)$,

$\displaystyle\mathbb{E}_{\mathsf{Nsmall|3,2,1}}[W_{\uparrow,1}W_{\downarrow,1}]$
$\displaystyle\geq\mathbb{E}_{\mathsf{Nsmall|3,2,1}}[\mathbf{1}_{W_{\uparrow,1}W_{\downarrow,1}\geq
1-\delta}\cdot W_{\uparrow,1}W_{\downarrow,1}]$
$\displaystyle\geq(1-\delta)\mathbb{P}_{\mathsf{Nsmall|3,2,1}}({W_{\uparrow,1}W_{\downarrow,1}\geq
1-\delta})\geq(1-\delta)^{2}.$

Using the expression from (6.55) we thus have

$\displaystyle\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]$
$\displaystyle\geq\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow,2}W_{\downarrow,2}\cdot\mathbb{E}_{\mathsf{Nsmall|3,2,1}}[W_{\uparrow,1}W_{\downarrow,1}]]$
$\displaystyle\geq(1-\delta)^{2}\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}W_{\uparrow,2}W_{\downarrow,2}].$

Going back to (6.60), this forces

$\displaystyle\mbox{r.h.s.~{}of
\eqref{627.5}}\leq\frac{\tau(A)+\delta}{(1-\delta)^{2}}+\mathbb{P}_{t}(\neg\mathsf{Nice}_{M}(\delta)).$

In view of Lemma 6.2, taking $\limsup_{t\to\infty}$, followed by
$\limsup_{\delta\downarrow 0}$ in above inequality we get that
$\limsup_{t\to\infty}\mathbb{P}_{t}(A)\leq\tau(A).$ Along with the matching
lower bound obtained in Step 1 above, this establishes (6.31).

####  6.2.8. Proof of Lemma 6.2

Recall from (6.39) that $\mathsf{Nice}_{M}(\delta)$ event is an intersection
of several kinds of events. To show (6.40), it suffices to prove the same for
each of the events. That is, given an event $\mathsf{E}$ which is part of
$\mathsf{Nice}_{M}(\delta)$ we will show

$\displaystyle\limsup_{\delta\to\infty}\limsup_{t\to\infty}\mathbb{P}(\mathsf{E})=1.$
(6.61)

Below we analyze each such possible choices for $\mathsf{E}$ separately.

$\mathsf{ArMx}(\delta)$ event. Recall $\mathsf{ArMx}(\delta)$ event from
(6.32). As noted in (3.9),

$\mathcal{M}_{p,t}^{M}\stackrel{{\scriptstyle
d}}{{\to}}\mathop{\mathrm{argmax}}_{x\in[-M,M]}\mathcal{A}(x),$

where $\mathcal{A}$ is defined in (3.8). Since $\mathcal{A}$ restricted to
$[-M,M]$ is absolutely continuous with Brownian motion with appropriate
diffusion coefficients,
$\mathop{\mathrm{argmax}}_{x\in[-M,M]}\mathcal{A}(x)\in(-M,M)$ almost surely.
In other words, maximum is not attained on the boundaries almost surely. But
then

$\displaystyle\liminf_{\delta\downarrow
0}\liminf_{t\to\infty}\mathbb{P}(\mathsf{ArMx}(\delta))$
$\displaystyle=\liminf_{\delta\downarrow
0}\mathbb{P}(\mathop{\mathrm{argmax}}_{x\in[-M,M]}\mathcal{A}(x)\in[-M+\delta,M-\delta])$
$\displaystyle=\mathbb{P}(\mathop{\mathrm{argmax}}_{x\in[-M,M]}\mathcal{A}(x)\in(-M,M))=1.$

This proves (6.61) with $\mathsf{E}\mapsto\mathsf{ArMx}(\delta).$

$\mathsf{Bd}_{\uparrow}(\delta),\mathsf{Bd}_{\downarrow}(\delta)$ events. We
first define

$\displaystyle\mathsf{Tight}_{\pm,\uparrow}(\lambda):=\left\\{p^{1/3}\left|\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3})-\mathfrak{h}_{pt,\uparrow}^{(1)}(\pm
Mp^{-2/3})\right|\leq\tfrac{1}{\lambda}\right\\},$
$\displaystyle\mathsf{Tight}_{\pm,\downarrow}(\lambda):=\left\\{q^{1/3}\left|\mathfrak{h}_{qt,\downarrow}^{(1)}(\Phi
q^{-2/3})-\mathfrak{h}_{qt,\downarrow}^{(1)}(\pm
Mq^{-2/3})\right|\leq\tfrac{1}{\lambda}\right\\},$

and set

$\displaystyle\mathsf{Sp}(\lambda):=\mathsf{ArMx}(\lambda)\cap\mathsf{Tight}_{+,\uparrow}(\lambda)\cap\mathsf{Tight}_{-,\uparrow}(\lambda)\cap\mathsf{Tight}_{+,\downarrow}(\lambda)\cap\mathsf{Tight}_{-,\downarrow}(\lambda)$
(6.62)

where $\mathsf{ArMx}(\lambda)$ is defined in (6.32). We claim that

$\displaystyle\limsup_{\lambda\downarrow
0}\limsup_{t\to\infty}\mathbb{P}(\neg\mathsf{Sp}(\lambda)))=0.$ (6.63)

Let us assume (6.63) for the time being and consider the main task of
analyzing the probability of the events
$\mathsf{Bd}_{\uparrow}(\delta),\mathsf{Bd}_{\downarrow}(\delta)$ defined in
(6.33). We have
$\mathsf{Bd}_{\uparrow}(\delta)=\mathsf{Bd}_{+\uparrow}(\delta)\cap\mathsf{Bd}_{-,\uparrow}(\delta)$
where $\mathsf{Bd}_{\pm,\uparrow}(\delta)$ is defined in (6.34). Let us focus
on $\mathsf{Bd}_{+,\uparrow}(\delta)$. Recall the $\sigma$-fields
$\mathcal{F}_{1},\mathcal{F}_{2}$ from (6.42) and (6.43). As described in
Subsection 6.2.6, upon conditioning on $\mathcal{F}_{1}\cup\mathcal{F}_{2}$,
the conditional law on $[-M,M]$ are given by $\mathsf{Nlarge}$ defined in
Definition 6.3, which are made up of
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$s
$V_{\ell}^{\mathsf{large}},V_{r}^{\mathsf{large}}$ defined in Definition 6.3.

Note that applying Markov inequality conditionally we have

$\displaystyle\mathbf{1}_{\mathsf{Sp}(\lambda)}\mathbb{P}\left(\mathsf{Bd}_{+,\uparrow}(\delta)\mid\mathcal{F}_{1},\mathcal{F}_{2}\right)$
$\displaystyle=\mathbf{1}_{\mathsf{Sp}(\lambda)}\cdot\mathbb{P}\left(|\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}(\Phi+t^{-\alpha}))-\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3})|>\tfrac{1}{\delta}t^{-\alpha/2}\mid\mathcal{F}_{1},\mathcal{F}_{2}\right)$
$\displaystyle\leq\mathbf{1}_{\mathsf{Sp}(\lambda)}\cdot\delta^{2}t^{2\alpha}\cdot\mathbb{E}_{\mathsf{Nlarge}|2,1}\left[[V_{r,1}^{\mathsf{large}}(p^{-2/3}t^{-\alpha})]^{4}\right]$

However, on $\mathbf{1}_{\mathsf{Sp}(\lambda)}$, the
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ has length bounded away from zero
and the endpoints are tight. Applying (5.20) with $K\mapsto 2,t\mapsto
1,s\mapsto 0,n\mapsto p^{2/3}t^{\alpha},M\mapsto 1/\lambda$, for all large
enough $t$ we get
$\mathbb{E}_{\mathsf{Nlarge}|2,1}\left[[V_{r,1}^{\mathsf{large}}(p^{-2/3}t^{-\alpha})]^{4}\right]\leq\mathrm{C}_{p,\lambda}t^{-2\alpha}$.
Thus,

$\displaystyle\limsup_{t\to\infty}\mathbb{P}\left(\neg\mathsf{Bd}_{+,\uparrow}(\delta)\right)\leq\limsup_{t\to\infty}\mathbb{P}(\neg\mathsf{Sp}(\lambda))+\delta^{2}{\mathrm{C}_{p,\lambda}.}$

Taking $\delta\downarrow 0$, followed by $\lambda\downarrow 0$, in view of
(6.63) we get $\limsup_{\delta\downarrow
0}\limsup_{t\to\infty}\mathbb{P}(\neg\mathsf{Bd}_{+,\uparrow}(\delta))=0$.
Similarly one can conclude $\limsup_{\delta\downarrow
0}\limsup_{t\to\infty}\mathbb{P}(\neg\mathsf{Bd}_{-,\uparrow}(\delta))=0$
Thus, this two together yields $\liminf_{\delta\downarrow
0}\liminf_{t\to\infty}\mathbb{P}(\mathsf{Bd}_{\uparrow}(\delta))=1$. By
exactly the same approach one can derive that
$\mathbb{P}(\mathsf{Bd}_{\downarrow}(\delta))$ goes to $1$ under the same
iterated limit. Thus it remains to show (6.63).

Let us recall from (6.62) that $\mathsf{Sp}(\lambda)$ event is composed of
four tightness events and one event about the $\mathop{\mathrm{argmax}}$. We
first claim that $\limsup_{\lambda\downarrow
0}\limsup_{t\to\infty}\mathbb{P}(\mathsf{Tight}_{x,y}(\lambda))=1$ for each
$x\in\\{+,-\\}$ and $y\in\\{\uparrow,\downarrow\\}$. The earlier analysis of
$\mathsf{ArMx}(\lambda)$ event in (6.2.8) then enforces (6.63). Since all the
tightness events are similar, it suffices to prove any one of them say
$\mathsf{Tight}_{+,\uparrow}$. By Proposition 2.5 we have the distributional
convergence of $2^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(2^{1/3}x)$ to
$\mathcal{A}_{1}(x)$ in the uniform-on-compact topology, where
$\mathcal{A}_{1}(\cdot)$ is the parabolic $\operatorname{Airy}_{2}$ process.
As $\Phi\in[-M,M]$, we thus have

$\displaystyle\limsup_{t\to\infty}\mathbb{P}(\mathsf{Tight}_{+,\uparrow}(\lambda))$
$\displaystyle\leq\limsup_{t\to\infty}\mathbb{P}\left(p^{1/3}\sup_{x\in[-M,M]}\left|\mathfrak{h}_{pt,\uparrow}^{(1)}(xp^{-2/3})-\mathfrak{h}_{pt,\uparrow}^{(1)}(Mp^{-2/3})\right|\leq\tfrac{1}{\lambda}\right)$
$\displaystyle=\mathbb{P}\left(p^{1/3}\sup_{|x|\leq
2^{-1/3}M}\left|\mathcal{A}_{1}(xp^{-2/3})-\mathcal{A}_{1}(2^{-1/3}Mp^{-2/3})\right|\leq\tfrac{2^{1/3}}{\lambda}\right).$

For fixed $p,M$, by tightness of parabolic $\operatorname{Airy}_{2}$ process
on a compact interval, the last expression goes to one as $\lambda\downarrow
0$, which is precisely what we wanted to show.

$\mathsf{Gap}_{M,\uparrow}(\delta),\mathsf{Gap}_{M,\downarrow}(\delta)$
events. Recall the definitions of $\mathsf{Gap}_{M,\uparrow}(\delta)$ and
$\mathsf{Gap}_{M,\downarrow}(\delta)$ from(6.35) and (6.36).We begin with the
proof of $\mathsf{Gap}_{M,\uparrow}(\delta)$. Let

$\mathsf{Diff}_{M,\uparrow}(\delta):=\left\\{\inf_{|x|\leq
M}p^{1/3}\left(\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}x)-\mathfrak{h}_{pt,\uparrow}^{(2)}(p^{-2/3}x)\right)\geq\delta\right\\}.$

Note that $\Phi\in[-M,M]$. Thus
$\mathsf{Gap}_{M,\uparrow}(\delta)\supset\mathsf{Diff}_{M,\uparrow}(\delta).$
Thus to show (6.61) with $\mathsf{E}\mapsto\mathsf{Gap}_{M,\uparrow}(\delta)$
it suffices to prove

$\displaystyle\liminf_{\delta\downarrow
0}\liminf_{t\rightarrow\infty}\mathbb{P}(\mathsf{Diff}_{M,\uparrow}(\delta))=1,$
(6.64)

We recall from Proposition 2.7 the distributional convergence of the KPZ line
ensemble to the Airy line ensemble in the uniform-on-compact topology. By
Skorokhod representation theorem, we may assume that our probability space is
equipped with $\mathcal{A}_{1}(\cdot)$ and $\mathcal{A}_{2}(\cdot)$ such that
almost surely as $t\to\infty$

$\displaystyle\max_{i=1,2}\sup_{|x|\leq
Mp^{-2/3}}|2^{1/3}\mathfrak{h}_{t,\uparrow}^{(i)}(2^{1/3}x)-\mathcal{A}_{i}(x)|\rightarrow
0.$ (6.65)

We thus have

$\displaystyle\liminf_{t\rightarrow\infty}\mathbb{P}(\mathsf{Diff}_{M,\uparrow}(\delta))=\mathbb{P}\left(\inf_{|x|\leq
M2^{-1/3}p^{-2/3}}p^{1/3}\left(\mathcal{A}_{1}(x)-\mathcal{A}_{2}(x)\right)\geq
2^{1/3}\delta\right).$ (6.66)

As the Airy line ensemble is absolutely continuous w.r.t. non-intersecting
Brownian motions, it is strictly ordered with touching probability zero (see
(2.1)). Hence r.h.s. of (6.66) goes to zero as $\delta\downarrow 0$. This
proves (6.64). The proof is similar for $\mathsf{Gap}_{M,\downarrow}(\delta).$

$\mathsf{Rise}_{M,\uparrow}(\delta),\mathsf{Rise}_{M,\uparrow}(\delta)$
events. Recall
$\mathsf{Rise}_{M,\uparrow}(\delta),\mathsf{Rise}_{M,\uparrow}(\delta)$ events
from (6.37) and (6.38). Due to their similarities, we only analyze the
$\mathsf{Rise}_{M,\uparrow}(\delta)$ event. As with the previous case, we
assume that our probability space is equipped with $\mathcal{A}_{1}(\cdot)$
and $\mathcal{A}_{2}(\cdot)$ (first two lines of the Airy line ensemble) such
that almost surely as $t\to\infty$ (6.65) holds. Applying union bound we have

$\displaystyle\mathbb{P}\left(\neg\mathsf{Rise}_{M}(\delta)\right)$
$\displaystyle\leq\mathbb{P}\left(\sup_{{|x|\leq
Mp^{-2/3}}}p^{1/3}|2^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}(2^{1/3}x)-\mathcal{A}_{2}(x)|\geq\tfrac{\delta}{16}\right)$
$\displaystyle\hskip
56.9055pt+\mathbb{P}\left(\neg\mathsf{Rise}_{M}(\delta),\sup_{|x|\leq
Mp^{-2/3}}p^{1/3}|2^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}(2^{1/3}x)-\mathcal{A}_{2}(x)|\leq\tfrac{\delta}{16}\right)$
$\displaystyle\leq\mathbb{P}\left(\sup_{|x|\leq
Mp^{-2/3}}p^{1/3}|2^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}(2^{1/3}x)-\mathcal{A}_{2}(x)|\geq\tfrac{\delta}{16}\right)$
$\displaystyle\hskip
56.9055pt+\mathbb{P}\bigg{(}\sup_{\begin{subarray}{c}x,y\in[-M,M]\\\ |x-y|\leq
t^{-\alpha}\end{subarray}}p^{1/3}|\mathcal{A}_{2}(x)-\mathcal{A}_{2}(y)|\geq\tfrac{\delta}{8}\bigg{)}.$

In the r.h.s. of above equation, the first term goes to zero as $t\to\infty$
by (6.65). The second term on the other hand goes to zero as $t\to\infty$ by
modulus of continuity estimates for Airy line ensemble from Proposition 2.4.
This shows,
$\lim_{t\to\infty}\mathbb{P}(\mathsf{Rise}_{M,\uparrow}(\delta))=1$. Similarly
one has $\lim_{t\to\infty}\mathbb{P}(\mathsf{Rise}_{M,\downarrow}(\delta))=1$
as well. This proves (6.61) for
$\mathsf{E}\mapsto\mathsf{Rise}_{M,\uparrow}(\delta),\mathsf{Rise}_{M,\downarrow}(\delta)$.

We have thus shown (6.61) for all the events listed in (6.39). This
establishes (6.40) concluding the proof of Lemma 6.2.

####  6.2.9. Proof of Lemma 6.6 and 6.7

In this subsection we prove Lemma 6.6 and 6.7.

Proof of Lemma 6.6. Recall $W_{\uparrow,1}$ and $W_{\downarrow,1}$ from (6.52)
and (6.53) respectively. We claim that for all large enough $t$, on
$\mathsf{Nice}_{M}(\delta)$ we have

$\displaystyle\mathbb{P}_{\mathsf{Nsmall}|3,2,1}(W_{\uparrow,1}>\sqrt{1-\delta})\geq
1-\tfrac{1}{2}\delta,\quad\mathbb{P}_{\mathsf{Nsmall}|3,2,1}(W_{\downarrow,1}>\sqrt{1-\delta})\geq
1-\tfrac{1}{2}\delta$ (6.67)

simultaneously. (6.56) then follows via union bound. Hence we focus on proving
(6.67). In the proof below we only focus on first part of (6.67) and the
second one follows analogously. We now define the ‘sink’ event:

$\displaystyle\mathsf{Sink}_{\uparrow}(\delta)$
$\displaystyle:=\left\\{\inf_{x\in[-t^{-\alpha},t^{-\alpha}]}p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3}+x)\geq p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3})-\tfrac{\delta}{4}\right\\}.$ (6.68)

Figure 9. In the above figure we have plotted the curves
$f(x):=p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}x)$ (black) and
$g(x):=p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}(p^{-2/3}x)$ (blue) restricted
to the interval $K_{t}:=(\Phi-t^{-\alpha},\Phi+t^{-\alpha})$. For convenience,
we have marked two blue points along with their values as $(A,f(A))$,
$(B,g(B))$. $\mathsf{Gap}_{M,\uparrow}(\delta)$ defined in (6.35) denote the
event that the blue points are separated by $\delta$, i.e,
$f(A)-g(B)\geq\delta$. The $\mathsf{Rise}_{M,\uparrow}(\delta)$ defined in
(6.37) ensures no point on the blue curve (restricted to $K_{t}$) has value
larger than $g(B)+\frac{1}{4}\delta$ (that is no significant rise). The
$\mathsf{Bd}_{\uparrow}(\delta)$ event defined in (6.33) indicates the red
points on the black curve are within
$[f(A)-\frac{1}{\delta}t^{-\alpha/2},f(A)+\frac{1}{\delta}t^{-\alpha/2}]$. The
$\mathsf{Sink}_{\uparrow}(\delta)$ event defined in (6.68) ensures that all
points on the black curve (restricted to $K_{t}$) have values larger than
$f(A)-\frac{1}{4}\delta$ (that is no significant sink). Clearly then on
$\mathsf{Sink}_{\uparrow}(\delta)\cap\mathsf{Rise}_{M,\uparrow}(\delta)\cap\mathsf{Gap}_{M,\uparrow}(\delta)$
for all $x\in K_{t}$, we have $f(x)-g(x)\geq
f(A)-\frac{1}{4}\delta-g(B)-\frac{1}{4}\delta\geq\frac{1}{2}\delta$.

Recall $\mathsf{Rise}_{M,\uparrow}(\delta)$ and
$\mathsf{Gap}_{M,\uparrow}(\delta)$ from (6.37) and (6.35). Note that on
$\mathsf{Sink}_{\uparrow}(\delta)\cap\mathsf{Rise}_{M,\uparrow}(\delta)\cap\mathsf{Gap}_{M,\uparrow}(\delta)$
we have uniform separation between $\mathfrak{h}_{pt,\uparrow}^{(1)}$ and
$\mathfrak{h}_{pt,\downarrow}^{(2)}$ on the interval $p^{-2/3}{K}_{t}$, that
is

$\displaystyle\inf_{x\in[\Phi-t^{-\alpha},\Phi+t^{-\alpha}]}\left[p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}x)-p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(2)}(p^{-2/3}x)\right]\geq\tfrac{\delta}{2}.$
(6.69)

See Figure 9 alongside its caption for further explanation of the above fact.
But then (6.69) forces
$W_{\uparrow,1}\geq\exp(-t^{2/3}2t^{-\alpha}e^{-\frac{1}{4}t^{1/3}\delta})$
which can be made strictly larger than $\sqrt{1-\delta}$ for all large enough
$t$. Thus,

$\displaystyle\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(W_{\uparrow,1}>\sqrt{1-\delta})\geq\mathbf{1}_{\mathsf{Nice}_{M}(\delta)}\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(\mathsf{Sink}_{\uparrow}(\delta)).$
(6.70)

Now we divide the sink event into two parts:
$\mathsf{Sink}_{\uparrow}(\delta)=\mathsf{Sink}_{+\uparrow}(\delta)\cap\mathsf{Sink}_{-,\uparrow}(\delta)$
where

$\displaystyle\mathsf{Sink}_{\pm,\uparrow}(\delta)$
$\displaystyle:=\left\\{\inf_{x\in[0,t^{-\alpha}]}p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3}\pm x)\geq p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3})-\tfrac{\delta}{4}\right\\},$

In view of (6.70), to prove first part of (6.67), it suffices to show for all
large enough $t$, on $\mathsf{Nice}_{M}(\delta)$ we have

$\displaystyle\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(\mathsf{Sink}_{+,\uparrow}(\delta))\geq
1-\tfrac{\delta}{4},\quad\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(\mathsf{Sink}_{-,\uparrow}(\delta))\geq
1-\tfrac{\delta}{4}.$ (6.71)

We only prove first part of (6.71) below. Towards this end, recall
$Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)}$ from (6.26). Observe that

$\displaystyle
Y_{M,t,\uparrow}^{(1)}(\Phi+x)=p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi
p^{-2/3})-p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(\Phi p^{-2/3}+x).$

Recall $\mathsf{Nsmall}$ law from Definition 6.4. Our discussion in Subsection
6.2.6 implies that under $\mathbb{P}_{\mathsf{Nsmall}|3.2,1}$,

$(Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)})(\Phi+\cdot)|_{[0,t^{-\alpha}]}\stackrel{{\scriptstyle
d}}{{=}}V_{r}^{\mathsf{small}}(\cdot),\quad(Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)})(\Phi+\cdot)|_{[-t^{-\alpha},0]}\stackrel{{\scriptstyle
d}}{{=}}V_{\ell}^{\mathsf{small}}(-\cdot),$

where recall that $V_{\ell}^{\mathsf{small}}$ and $V_{r}^{\mathsf{small}}$ are
conditionally independent $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ on
$[0,t^{-\alpha}]$ with appropriate end points, defined in Definition 6.4. In
particular we have,

$\displaystyle\mathbb{P}_{\mathsf{Nsmall|3,2,1}}(\mathsf{Sink}_{+,\uparrow}(\delta))=\mathbb{P}_{\mathsf{Nsmall|3,2,1}}\left(\sup_{x\in[0,t^{-\alpha}]}V_{r,1}^{\mathsf{small}}(x)\leq\tfrac{1}{4}\delta\right)$
(6.72)

where
$V_{r}^{\mathsf{small}}=(V_{r,1}^{\mathsf{small}},V_{r,2}^{\mathsf{small}})$.
Recall $\mathsf{Nice}_{M}(\delta)$ event from (6.39). It contains
$\mathsf{Bd}_{\uparrow}(\delta)$ event defined in (6.33). On this event,
$-\frac{1}{\delta}\leq
V_{r,1}^{\mathsf{Small}}(t^{-\alpha}),V_{r,2}^{\mathsf{Small}}(t^{-\alpha})\leq\frac{1}{\delta}t^{-\alpha/2}$.
We consider another $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$
$U=(U_{1},U_{2})$ on $[0,t^{-\alpha}]$ with non-random endpoints
$U_{1}(t^{-\alpha})=U_{2}(t^{-\alpha})=\frac{1}{\delta}t^{-\alpha/2}$. On
$\mathsf{Bd}_{\uparrow}(\delta)$ event, by monotonicity of non-intersecting
Brownian bridges (Lemma 2.6 in [CH14]), one may couple $U=(U_{1},U_{2})$ and
$V_{r}^{\mathsf{small}}$ so that $U_{i}$ always lies above
$V_{r,i}^{\mathsf{small}}$ for $i=1,2$. Thus on
$\mathsf{Bd}_{\uparrow}(\delta)$ event,

$\displaystyle\mathbb{P}_{\mathsf{Nsmall|3,2,1}}\left(\sup_{x\in[0,t^{-\alpha}]}V_{r,1}^{\mathsf{small}}(x)\leq\lambda
t^{-\alpha/2}\right)\geq\mathbb{P}\left(\sup_{x\in[0,1]}t^{\alpha/2}U_{1}(xt^{-\alpha})\leq\lambda\right)\geq
1-\tfrac{\delta}{4},$

where the last inequality is true by taking $\lambda$ large enough. This
choice of $\lambda$ is possible as by Brownian scaling,
$t^{\alpha/2}U_{1}(xt^{-\alpha}),t^{\alpha/2}U_{2}(xt^{-\alpha})$ is
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ on $[0,1]$ ending at
$(\frac{1}{\delta},\frac{1}{\delta})$. Taking $t$ large enough one can ensure
$\lambda t^{-\alpha/2}\leq\frac{\delta}{4}$. Using the equality in (6.72) we
thus establish the first part of (6.71). The second part is analogous. This
proves the first part of (6.67). The second part of (6.67) follows similarly.
This completes the proof of Lemma 6.6.

Proof of Lemma 6.7. The idea behind this proof is Proposition 5.8, which
states that a $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ after Brownian
rescaling converges in distribution to a $\mathsf{DBM}$. The following fills
out the details. Recall that

$\mathbb{P}_{\mathsf{Nsmall}|3.2,1}(A)=\mathbb{P}_{\mathsf{Nsmall}|3.2,1}(D_{M,t,\uparrow},D_{M,t,\downarrow}(\cdot)\in
A).$

Recall from (6.28) that $D_{M,t,\uparrow},D_{M,t,\downarrow}$ is a diffusive
scaling of $Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)}$ when centering at
$\Phi$, where $Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)}$ are defined in
(6.26). Recall $\mathsf{Nsmall}$ law from Definition 6.4. Our discussion in
Subsection 6.2.6 implies that under $\mathbb{P}_{\mathsf{Nsmall}|3.2,1}$,

$(Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)})(\Phi+\cdot)|_{[0,t^{-\alpha}]}\stackrel{{\scriptstyle
d}}{{=}}V_{r}^{\mathsf{small}}(\cdot),\quad(Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)})(\Phi+\cdot)|_{[-t^{-\alpha},0]}\stackrel{{\scriptstyle
d}}{{=}}V_{\ell}^{\mathsf{small}}(-\cdot),$

where $V_{\ell}^{\mathsf{small}}$ and $V_{r}^{\mathsf{small}}$ are
conditionally independent $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ on
$[0,t^{-\alpha}]$ with appropriate end points defined in Definition 6.4. Using
Brownian scaling, we consider

$V_{\ell}^{0}(x):=t^{\alpha/2}V_{\ell}^{\mathsf{small}}(xt^{-\alpha}),\quad
V_{r}^{0}(x):=t^{\alpha/2}V_{r}^{\mathsf{small}}(xt^{-\alpha}),$

which are now $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ on $[0,1]$. Note that
on $\mathsf{Bd}_{\uparrow}(\delta),\mathsf{Bd}_{\downarrow}(\delta)$ (defined
in (6.33)), we see that endpoints of $V_{\ell}^{0},V_{r}^{0}$ are in
$[-\frac{1}{\delta},\frac{1}{\delta}]$. Thus as $\alpha=\frac{1}{6}$,
performing another diffusive scaling by Proposition 5.8 we see that as
$t\to\infty$

$t^{1/4}V_{\ell}^{0}(xt^{-1/2})\ ,\ t^{1/4}V_{r}(xt^{-1/2})$

converges to two independent copies of $\mathsf{DBM}$s (defined in Definition
5.1) in the uniform-on-compact topology. Hence we get two-sided $\mathsf{DBM}$
convergence for the pair $(D_{M,t,\uparrow},D_{M,t,\downarrow})$ under
$\mathbb{P}_{\mathsf{Nsmall}|3.2,1}$ as long as
$\mathbf{1}\\{\mathsf{Nice}_{M}(\delta)\\}$ holds. This proves (6.57).

###  6.3. Proof of Theorem 1.10

We take $p\mapsto\frac{1}{2}$ and $t\mapsto 2t$ in Proposition 6.1. Then by
Lemma 3.2, $\mathcal{P}_{2,t}$ defined in the statement of Theorem 1.10 is
same as $\mathcal{M}_{\frac{1}{2},2t}$ considered in Proposition 6.1. Its
uniqueness is already justified in Lemma 3.1. Furthermore,

$R_{2}(x,t)\stackrel{{\scriptstyle d}}{{=}}D_{1}(x,t)-D_{2}(x,t),$

as functions in $x$, where $R_{2}(x,t)$ is defined in (1.11) and $D_{1},D_{2}$
are defined in (6.24). By Proposition 6.1 and Lemma 5.3 we get that
$D_{1}(x,t)-D_{2}(x,t)\stackrel{{\scriptstyle d}}{{\to}}\mathcal{R}_{2}(x)$ in
the uniform-on-compact topology. This proves Theorem 1.10 for $k=2$ case.

For $k=1$ case, by Lemma 3.2, $\mathcal{P}_{1,t}$ is same as
$\mathcal{M}_{*,t}$ which is unique almost surely by Lemma 3.1. This
guarantees $\mathcal{P}_{1,t}$ is unique almost surely as well. Thus we are
left to show

$\displaystyle\mathcal{H}(\mathcal{P}_{1,t},t)-\mathcal{H}(x+\mathcal{P}_{1,t},t)\stackrel{{\scriptstyle
d}}{{\to}}\mathcal{R}_{1}(x).$ (6.73)

where $\mathcal{R}_{1}(x)$ is a two-sided Bessel process with diffusion
coefficient $1$ defined in Definition 5.2. The proof of (6.73) is exactly
similar to that of Proposition 6.1 with few minor alterations listed below.

  1. (1)

Just as in Subsection 6.2.1, one may put the problem in (6.73) under the
framework of KPZ line ensemble. Compared to Subsection 6.2.1, in this case,
clearly there will be just one set of line ensemble.

  2. (2)

Given the decay estimates for $\mathcal{M}_{*,t}$ from Lemma 3.1, it boils
down to show Bessel behavior around local maximizers. The rigorous
justification follows from a soft argument analogous to what is done in
Subsection 6.2.3.

  3. (3)

In the spirit of Subsection 6.2.4, one can define a similar
$\mathsf{Nice}^{\prime}_{M}(\delta)$ event but now for a single line ensemble.
$\mathsf{Nice}^{\prime}_{M}(\delta)$ will contain similar events, such as:

     * •

control on the location of local maximizer (analog of $\mathsf{ArMx}(\delta)$
event (6.32)),

     * •

control on the gap between first curve and second curve at the maximizer
(analog of $\mathsf{Gap}_{M,\uparrow}(\delta)$ event (6.35)),

     * •

fluctuations of the first curve on a small interval say $I$ around maximizer
(analog of $\mathsf{Rise}_{M,\uparrow}(\delta)$ event (6.37),

     * •

and control on the value of the endpoints of $I$ (analog of
$\mathsf{Bd}_{\uparrow}(\delta)$ event (6.33)).

On $\mathsf{Nice}^{\prime}_{M}(\delta)$ event, the conditional analysis can be
performed in the same manner.

  4. (4)

Next, as in proof of Proposition 6.1, we proceed by three layers of
conditioning. For first layer, we use the $\mathbf{H}_{t}$ Brownian Gibbs
property of the single line ensemble under consideration. Next, conditioning
on the location and values of the maximizer, we similarly apply the same
Bessel bridge decomposition result from Proposition 4.8 to convert the
conditional law to that of the Bessel bridges over a large interval (see
Subsection 6.2.5). Finally, analogous to Subsection 6.2.6, the third layer of
conditioning reduces large Bessel bridges to smaller ones following the
Markovian property of Bessel bridges, see Lemma 4.2.

  5. (5)

Since a Bessel bridge say on $[0,1]$ is a Brownian bridge conditioned to stay
positive on $[0,1]$, it has the Brownian scaling property and it admits
monotonicity w.r.t. endpoints. These are two crucial tools that went into the
Proof of Lemma 6.6 in Subsection 6.2.9. Thus the Bessel analogue of Lemma 6.6
can be derived using the scaling property and monotonicity stated above in the
exact same way. Finally, the Bessel analogue of Lemma 6.7 can be obtained from
Corollary 5.9. Indeed Corollary 5.9 ensures that small Bessel bridges
converges to Bessel process under appropriate diffusive limits on the
$\mathsf{Nice}^{\prime}_{M}(\delta)$ event.

Executing all the above steps in an exact same manner as proof of Proposition
6.1, (6.73) is established. This completes the proof of Theorem 1.10.

##  7\. Proof of localization theorems

In this section we prove our main results: Theorem 1.4 and Theorem 1.5. In
Section 7.1 we study certain tail properties (Lemma 7.1 and Proposition 7.2)
of the quantities that we are interested in and prove Theorem 1.4. Proof of
Proposition 7.2 is then completed in Section 7.2 along with proof of Theorem
1.5.

###  7.1. Tail Properties and proof of Theorem 1.4

We first settle the question of finiteness of the Bessel integral appearing in
the statements of Theorems 1.4 and 1.5 in the following Lemma.

######  Lemma 7.1.

Let $R_{\sigma}(\cdot)$ be a Bessel process with diffusion coefficient
$\sigma>0$, defined in Definition 5.2. Then

$\displaystyle\mathbb{P}\left(\int_{\mathbb{R}}e^{-R_{\sigma}(x)}\mathrm{d}x\
{\in(0,\infty)}\right)=1.$

###### Proof.

Since $R_{\sigma}(\cdot)$ has continuous paths,
$\sup_{x\in[0,1]}R_{\sigma}(x)$ is finite almost surely. Thus almost surely we
have

$\int_{\mathbb{R}}e^{-R_{\sigma}(x)}\mathrm{d}x\geq\int_{0}^{1}e^{-R_{\sigma}(x)}\mathrm{d}x>0.$

On the other hand, by the classical result from [Mot59] it is known that

$\mathbb{P}(R_{\sigma}(x)<x^{1/4}\mbox{ infinitely often})=0.$

Thus, there exists $\Omega$ such that $\mathbb{P}(\Omega)=1$ and for all
$\omega\in\Omega$, there exists $x_{0}(\omega)\in(0,\infty)$ such that

$\displaystyle R_{\sigma}(x)(\omega)\geq x^{1/4}\mbox{ for all }x\geq
x_{0}(\omega).$

Hence for this $\omega$,

$\displaystyle{\int_{0}^{\infty}e^{-R_{\sigma}(x)(\omega)}\mathrm{d}x=\int_{0}^{x_{0}(\omega)}e^{-R_{\sigma}(x)(\omega)}\mathrm{d}x+\int_{x_{0}(\omega)}^{\infty}e^{-R_{\sigma}(x)(\omega)}\mathrm{d}x<x_{0}(\omega)+\int_{0}^{\infty}e^{-x^{1/4}}\mathrm{d}x<\infty.}$

This establishes that $\int_{\mathbb{R}}e^{-R_{\sigma}(x)}\mathrm{d}x$ is
finite almost surely. ∎

Our next result studies the tail of the integral of the pre-limiting process.

######  Proposition 7.2.

Fix $p\in(0,1)$. Set $q=1-p$. Consider $2$ independent copies of the KPZ
equation $\mathcal{H}_{\uparrow}(x,t)$, and $\mathcal{H}_{\downarrow}(x,t)$,
both started from the narrow wedge initial data. Let $\mathcal{M}_{p,t}$ be
the almost sure unique maximizer of the process
$x\mapsto(\mathcal{H}_{\uparrow}(x,pt)+\mathcal{H}_{\downarrow}(x,qt))$ which
exists via Lemma 3.1. Set

$\displaystyle D_{1}(x,t)$
$\displaystyle:=\mathcal{H}_{\uparrow}(\mathcal{M}_{p,t},pt)-\mathcal{H}_{\uparrow}(x+\mathcal{M}_{p,t},pt),$
(7.1) $\displaystyle D_{2}(x,t)$
$\displaystyle:=\mathcal{H}_{\downarrow}(x+\mathcal{M}_{p,t},qt)-\mathcal{H}_{\downarrow}(\mathcal{M}_{p,t},qt).$

For all $\rho>0$ we have

$\displaystyle\limsup_{K\to\infty}\limsup_{t\to\infty}\mathbb{P}\left(\int_{[-K,K]^{c}}e^{D_{2}(x,t)-D_{1}(x,t)}\,\mathrm{d}x\geq\rho\right)=0.$
(7.2)

As a corollary, we derive that for any $p\in(0,1)$ the $pt$-point density of
point-to-point $\mathsf{CDRP}$ of length $t$ indeed concentrates in a
microscopic region of size $O(1)$ around the favorite point.

######  Corollary 7.3.

Recall the definition of $\mathsf{CDRP}$ and the notation $\mathbb{P}^{\xi}$
from Definition 1.1. Fix $p\in(0,1)$. Suppose $X\sim\mathsf{CDRP}(0,0;0,t)$.
Consider $\mathcal{M}_{p,t}$ the almost sure unique mode of $f_{p,t}$, the
quenched density of $X(pt)$. We have

$\displaystyle\limsup_{K\to\infty}\limsup_{t\to\infty}\mathbb{P}^{\xi}\left(|X(pt)-\mathcal{M}_{p,t}|\geq
K\right)=0,\mbox{ in probability}.$

One also has the analogous version of Proposition 7.2 involving one single
copy of the KPZ equation viewed around its maximum. This leads to a similar
corollary about tightness of the quenched endpoint distribution for point-to-
line $\mathsf{CDRP}$ (see Definition 1.2) when re-centered around its mode.
The details are skipped for brevity.

The proof of Proposition 7.2 is heavily technical and relies on the tools as
well as notations from Proposition 6.1. For clarity, we first prove Corollary
7.3 and Theorem 1.4 assuming the validity of Proposition 7.2. The proof of
Proposition 7.2 is then presented in Section 7.2.

###### Proof of Corollary 7.3.

We have $\mathcal{Z}(0,0;x,pt)\stackrel{{\scriptstyle
d}}{{=}}e^{\mathcal{H}_{\uparrow}(x,pt)}$ and by time reversal property
$\mathcal{Z}(x,pt;0,t)\stackrel{{\scriptstyle
d}}{{=}}e^{\mathcal{H}_{\downarrow}(x,qt)}$ as functions in $x$, where
$\mathcal{H}_{\uparrow},\mathcal{H}_{\downarrow}$ are independent copies of
KPZ equation started from narrow wedge initial data. The uniqueness of the
mode $\mathcal{M}_{p,t}$ for $f_{p,t}$ is already settled in Lemma 3.1. Thus,
the quenched density of $X(pt)-\mathcal{M}_{p,t}$ is given by

$\displaystyle
f_{p,t}(x+\mathcal{M}_{p,t})=\frac{\exp(D_{2}(x,t)-D_{1}(x,t))}{\int\limits_{\mathbb{R}}\exp(D_{2}(y,t)-D_{1}(y,t))\mathrm{d}y},$
(7.3)

where $D_{i}(x,t),i=1,2$ are defined in (6.23). Thus,

$\displaystyle\mathbb{P}^{\xi}\left(|X(pt)-\mathcal{M}_{p,t}|\geq
K\right)=\frac{\int\limits_{[-K,K]^{c}}e^{D_{2}(x,t)-D_{1}(x,t)}\,\mathrm{d}x}{\int\limits_{\mathbb{R}}e^{D_{2}(x,t)-D_{1}(x,t)}\,\mathrm{d}x}\leq\frac{\int\limits_{[-K,K]^{c}}e^{D_{2}(x,t)-D_{1}(x,t)}\,\mathrm{d}x}{\int\limits_{[-K,K]}e^{D_{2}(x,t)-D_{1}(x,t)}\,\mathrm{d}x}.$
(7.4)

Notice that by by (7.2) the numerator of r.h.s. of (7.4) goes to zero in
probability under the iterated limit $\limsup_{t\to\infty}$ followed by
$\limsup_{K\to\infty}$. Whereas due to Proposition 6.1, under the iterated
limit, the denominator converges in distribution to
$\int_{\mathbb{R}}e^{-R_{2}(x)}\mathrm{d}x$ which is strictly positive by
Lemma 7.1. Thus overall the r.h.s. of (7.4) goes to zero in probability under
the iterated limit. This completes the proof. ∎

###### Proof of Theorem 1.4.

Fix any $p\in(0,1).$ Set $q=1-p$. Recall from (7.3) that

$\displaystyle
f_{p,t}(x+\mathcal{M}_{p,t})=\frac{\exp(D_{2}(x,t)-D_{1}(x,t))}{\int\limits_{\mathbb{R}}\exp(D_{2}(y,t)-D_{1}(y,t))\mathrm{d}y}$
(7.5)

where $D_{i}(x,t),i=1,2$ are defined in (6.23). Note that by Proposition 6.1,
a continuous mapping theorem immediately implies that for any $K<\infty$

$\displaystyle\frac{\exp(D_{2}(x,t)-D_{1}(x,t))}{\int_{-K}^{K}\exp(D_{2}(y,t)-D_{1}(y,t))\mathrm{d}y}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\frac{e^{-\mathcal{R}_{2}(x)}}{\int_{-K}^{K}e^{-\mathcal{R}_{2}(y)}\mathrm{d}y}$
(7.6)

in the uniform-on-compact topology. Here $\mathcal{R}_{2}$ is a 3D Bessel
process with diffusion coefficient $2$. For simplicity, we denote

$\displaystyle\Lambda_{t}(x):=\exp(D_{2}(x,t)-D_{1}(x,t))\text{ and
}\Lambda(x)=\exp(-\mathcal{R}_{2}(x)).$

We can then rewrite (7.5) as product of four factors:

$\displaystyle
f_{p,t}(x+\mathcal{M}_{p,t})=\frac{\Lambda_{t}(x)}{\int_{\mathbb{R}}\Lambda_{t}(y)\mathrm{d}y}=\frac{\int_{-K}^{K}\Lambda_{t}(y)\mathrm{d}y}{\int_{\mathbb{R}}\Lambda_{t}(y)\mathrm{d}y}\cdot\frac{\int_{\mathbb{R}}\Lambda(y)\mathrm{d}y}{\int_{-K}^{K}\Lambda(y)\mathrm{d}y}\cdot\frac{\int_{-K}^{K}\Lambda(y)\mathrm{d}y}{\int_{\mathbb{R}}\Lambda(y)\mathrm{d}y}\cdot\frac{\Lambda_{t}(x)}{\int_{-K}^{K}\Lambda_{t}(y)\mathrm{d}y}.$

Corollary 7.3 ensures

$\frac{\int_{-K}^{K}\Lambda_{t}(y)\mathrm{d}y}{\int_{\mathbb{R}}\Lambda_{t}(y)\mathrm{d}y}=\mathbb{P}^{\xi}(|X(pt)-\mathcal{M}_{p,t}|\leq
K)\stackrel{{\scriptstyle p}}{{\to}}1$

as $t\to\infty$ followed by $K\to\infty$. Lemma 7.1 with $\sigma=2$ yields
that
$\int_{[-K,K]^{c}}\Lambda(y)\mathrm{d}y=\int_{[-K,K]^{c}}e^{-\mathcal{R}_{2}(y)}\mathrm{d}y\stackrel{{\scriptstyle
p}}{{\rightarrow}}0$ as $K\rightarrow\infty.$ Thus as $K\rightarrow\infty$

$\displaystyle\frac{\int_{\mathbb{R}}\Lambda(y)\mathrm{d}y}{\int_{-K}^{K}\Lambda(y)\mathrm{d}y}\stackrel{{\scriptstyle
p}}{{\rightarrow}}1.$

Meanwhile, (7.6) yields that as $t\rightarrow\infty,$

$\displaystyle\frac{\int_{-K}^{K}\Lambda(y)\mathrm{d}y}{\int_{\mathbb{R}}\Lambda(y)\mathrm{d}y}\cdot\frac{\Lambda_{t}(x)}{\int_{-K}^{K}\Lambda_{t}(y)\mathrm{d}y}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\frac{\int_{-K}^{K}\Lambda(y)\mathrm{d}y}{\int_{\mathbb{R}}\Lambda(y)\mathrm{d}y}\cdot\frac{\Lambda(x)}{\int_{-K}^{K}\Lambda(y)\mathrm{d}y}=\frac{\Lambda(x)}{\int_{\mathbb{R}}\Lambda(y)\mathrm{d}y}.$

in the uniform-on-compact topology. Thus, overall we get that
$f_{p,t}(x+\mathcal{M}_{p,t})\stackrel{{\scriptstyle
d}}{{\to}}\frac{\Lambda(x)}{\int_{\mathbb{R}}\Lambda(y)\mathrm{d}y},$ in the
uniform-on-compact topology. This establishes (1.7), completing the proof of
Theorem 1.4. ∎

###  7.2. Proof of Proposition 7.2 and Theorem 1.5

Coming to the proof of Proposition 7.2, we note that the setup of Proposition
7.2 is same as that of Proposition 6.1. Hence all the discussions pertaining
to Proposition 6.1 are applicable here. In particular, to prove Proposition
7.2, we will be using few notations and certain results from the proof of
Proposition 6.1.

###### Proof of Proposition 7.2.

Fix any $M>0$. The proof of (6.24) proceeds by dividing the integral into two
parts depending on the range:

$\displaystyle U_{1}$
$\displaystyle:=[-t^{2/3}M-\mathcal{M}_{p,t},t^{2/3}M-\mathcal{M}_{p,t}]^{c},$
(Deep Tail) $\displaystyle U_{2}$
$\displaystyle:=[K,K]^{c}\cap[-t^{2/3}M-\mathcal{M}_{p,t},t^{2/3}M-\mathcal{M}_{p,t}],$
(Shallow Tail)

and controlling each of them individually. See Figure 10 for details. In the
following two steps, we control these two kind of tails respectively.

Figure 10. Illustration for the proof of Proposition 7.2. In Deep Tail region
we use parabolic decay of KPZ line ensemble, and in Shallow Tail we use non-
intersecting Brownian bridge separation estimates from Proposition 5.6.

Step 1. In this step, we control the Deep Tail region: $U_{1}$. The goal of
this step is to show

$\displaystyle\limsup_{t\to\infty}\mathbb{P}\left(\int_{U_{1}}e^{D_{2}(x,t)-D_{1}(x,t)}\,\mathrm{d}x\geq\tfrac{\rho}{2}\right)\leq\mathrm{C}\exp(-\tfrac{1}{\mathrm{C}}M^{3}),$
(7.7)

for some constant $\mathrm{C}=\mathrm{C}(p)>0$. We now recall the framework of
KPZ line ensemble discussed in Subsection 6.2.1. We define

$\displaystyle\mathcal{S}_{p,t}(x):=p^{1/3}\mathfrak{h}^{(1)}_{pt,\uparrow}(p^{-2/3}x)+q^{1/3}\mathfrak{h}^{(1)}_{qt,\downarrow}(q^{-2/3}x)$
(7.8)

where $\mathfrak{h}_{t,\uparrow},\mathfrak{h}_{t,\downarrow}$ are scaled KPZ
line ensembles corresponding to
$\mathcal{H}_{\uparrow},\mathcal{H}_{\downarrow}$, see (2.6). Observe that

$\displaystyle D_{2}(x,t)-D_{1}(x,t)\stackrel{{\scriptstyle
d}}{{=}}t^{1/3}\left[\mathcal{S}_{p,t}(t^{-2/3}(x+\mathcal{M}_{p,t}))-\sup_{z\in\mathbb{R}}\mathcal{S}_{p,t}(z)\right],$

where $D_{1},D_{2}$ are defined in (7.1). Thus we have

$\displaystyle\int_{U_{1}}\exp(D_{2}(x,t)-D_{1}(x,t))\mathrm{d}x\stackrel{{\scriptstyle
d}}{{=}}\int_{|x|\geq
M}\exp\left(t^{1/3}\left[\mathcal{S}_{p,t}(x)-\sup_{z\in\mathbb{R}}\mathcal{S}_{p,t}(z)\right]\right)\mathrm{d}x$

where $U_{1}$ is defined in (Deep Tail). Towards this end, we define two
events

$\displaystyle\mathsf{A}:=\left\\{\sup_{z\in\mathbb{R}}\mathcal{S}_{p,t}(z)\leq-\tfrac{M^{2}}{4}\right\\},\quad\mathsf{B}:=\left\\{\sup_{x\in\mathbb{R}}\left(\mathcal{S}_{p,t}(x)+x^{2}\right)>\tfrac{M^{2}}{4}\right\\},$

Note that on $\neg A\cap\neg B$, for all $|x|\geq M$, we have

$\displaystyle\mathcal{S}_{p,t}(x)-\sup_{z\in\mathbb{R}}\mathcal{S}_{p,t}(z)\leq\tfrac{M^{2}}{4}+\tfrac{M^{2}}{4}-x^{2}\leq\tfrac{M^{2}}{2}-\tfrac{3M^{2}}{4}-\tfrac{x^{2}}{4}\leq-\tfrac{M^{2}}{4}-\tfrac{x^{2}}{4}.$

This forces

$\displaystyle\int_{|x|\geq
M}\exp\left(t^{1/3}\left[\mathcal{S}_{p,t}(x)-\sup_{z\in\mathbb{R}}\mathcal{S}_{p,t}(z)\right]\right)\mathrm{d}x\leq\int_{[-M,M]^{c}}\exp\left(-t^{1/3}(\tfrac{M^{2}}{2}+\tfrac{y^{2}}{4})\right)\mathrm{d}y,$

which goes to zero as $t\to\infty.$ Hence $\mbox{l.h.s.~{}of
\eqref{7wts}}\leq\mathbb{P}(\neg\mathsf{A})+\mathbb{P}(\neg\mathsf{B})$. Hence
it suffices to show

$\displaystyle\mathbb{P}(\neg\mathsf{A})\leq\mathrm{C}\exp\left(-\tfrac{1}{\mathrm{C}}M^{3}\right),\quad\mathbb{P}(\neg\mathsf{B})\leq\mathrm{C}\exp\left(-\tfrac{1}{\mathrm{C}}M^{3}\right).$
(7.9)

To prove the first part of (7.9), note that

$\displaystyle\mathbb{P}\left(\neg A\right)$
$\displaystyle\leq\mathbb{P}\left(\mathcal{S}_{p,t}(0)\leq-\tfrac{M^{2}}{4}\right)$
$\displaystyle\leq\mathbb{P}\left(p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(0)\leq-\tfrac{M^{2}}{8}\right)+\mathbb{P}\left(q^{1/3}\mathfrak{h}_{qt,\downarrow}^{(1)}(0)\leq-\tfrac{M^{2}}{8}\right)\leq\mathrm{C}\exp(-\tfrac{1}{\mathrm{C}}{M^{3}}).$

where the last inequality follows by Proposition 2.8 (b), for some constant
$\mathrm{C}=\mathrm{C}(p)>0$. This proves the first part of (7.9). For the
second part of (7.9), following the definition of $\mathcal{S}_{p,t}(x)$ from
(7.8), and using the elementary inequality $\frac{1}{4p}+\frac{1}{4q}\geq 1$
by a union bound we have

$\displaystyle\mathbb{P}\left(\sup_{x\in\mathbb{R}}\left(\mathcal{S}_{p,t}(x)+x^{2}\right)>\tfrac{M^{2}}{4}\right)$
$\displaystyle\leq\mathbb{P}\left(\sup_{x\in\mathbb{R}}\left(p^{1/3}\mathfrak{h}_{pt,\uparrow}^{(1)}(p^{-2/3}x)+\tfrac{x^{2}}{4p}\right)>\tfrac{M^{2}}{8}\right)$
(7.10) $\displaystyle\hskip
28.45274pt+\mathbb{P}\left(\sup_{x\in\mathbb{R}}\left(q^{1/3}\mathfrak{h}_{qt,\uparrow}^{(1)}(q^{-2/3}x)+\tfrac{x^{2}}{4q}\right)>\tfrac{M^{2}}{8}\right).$

Applying Proposition (2.8) (c) with $\beta=\frac{1}{2}$, we get that each of
the terms on r.h.s. of (7.10) are at most
$\mathrm{C}\exp(-\frac{1}{\mathrm{C}}M^{3})$ where
$\mathrm{C}=\mathrm{C}(p)>0$. This establishes the second part of (7.9)
completing the proof of (7.7).

Step 2. In this step, we control the Shallow Tail region: $U_{2}$. We first
lay out the heuristic idea behind the Shallow Tail region controls. We recall
the nice event $\mathsf{Sp}(\lambda)$ from (6.62) which occurs with high
probability. Assuming $\mathsf{Sp}(\lambda)$ holds, we apply the the
$\mathbf{H}_{t}$ Brownian Gibbs property of the KPZ line ensembles, and
analyze the desired integral

$\int_{U_{2}}e^{D_{2}(x,t)-D_{1}(x,t)}\mathrm{d}x$

under the ‘free’ Brownian bridge law. Further conditioning on the information
of the maximizer converts the free law into the law of the
$\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ (defined in Definition 4.4). On
$\mathsf{Sp}(\lambda)$, we may apply Proposition 5.6 to obtain the desired
estimates for the ‘free’ law. One then obtain the desired estimates for KPZ
law using the lower bound for the normalizing constant from Proposition 2.5
(b).

We now expand upon the technical details. In what follows we will only work
with the right tail:

$\displaystyle
U_{+,2}:=[-t^{2/3}M-\mathcal{M}_{p,t},t^{2/3}M-\mathcal{M}_{p,t}]\cap[K,\infty)=[K,t^{2/3}M-\mathcal{M}_{p,t}]$

and the argument for the left part of the shallow tail is analogous. Note that
we also implicitly assumed $t^{2/3}M-\mathcal{M}_{p,t}\geq K$ above. Otherwise
there is nothing to prove. As before we utilize the the notations defined in
Subsection 6.2.1. Recall the local maximizer $\mathcal{M}_{p,t}^{M}$ defined
in (6.25). Recall $Y_{M,t,\uparrow}^{(1)},Y_{M,t,\downarrow}^{(1)}$ from
(6.26). Set

$\displaystyle\Gamma_{t,M,K}$
$\displaystyle:=\int_{K}^{Mt^{2/3}-\mathcal{M}_{p,t}}e^{-t^{1/3}\left[Y_{M,t,\uparrow}^{(1)}(t^{-2/3}(\mathcal{M}_{p,t}^{M}+x))-Y_{M,t,\downarrow}^{(1)}(t^{-2/3}(\mathcal{M}_{p,t}^{M}+x))\right]}\mathrm{d}x$
(7.11)
$\displaystyle=\int_{K}^{Mt^{2/3}-\mathcal{M}_{p,t}}\exp(-D_{M,t,\uparrow}(x)+D_{M,t,\downarrow}(x))\mathrm{d}x,$

where the last equality follows from the definition of
$D_{M,t,\uparrow},D_{M,t,\downarrow}$ from (6.28). Recall that the only
difference between $D_{1},D_{2}$ (defined in (6.27)) and
$D_{M,t,\uparrow},D_{M,t,\downarrow}$ is that former is defined using the
global maximizer $\mathcal{M}_{p,t}$ and the latter by local maximizer
$\mathcal{M}_{p,t}^{M}$. However, Lemma 3.1 implies that with probability at
least $1-\mathrm{C}\exp(-\frac{1}{\mathrm{C}}M^{3}),$ we have
$\mathcal{M}_{p,t}=\mathcal{M}_{p,t}^{M}$. Next, fix $\lambda>0$. Consider
$\mathsf{Sp}(\lambda)$ event defined in (6.62). We thus have

$\displaystyle\mathbb{P}\left(\int_{U_{+,2}}e^{D_{2}(x,t)-D_{1}(x,t)}\mathrm{d}x\geq\frac{\rho}{4}\right)\leq\mathrm{C}\exp(-\tfrac{1}{\mathrm{C}}M^{3})+\mathbb{P}(\neg\mathsf{Sp}(\lambda))+\mathbb{P}\left(\Gamma_{t,M,K}\geq\tfrac{\rho}{4},\mathsf{Sp}(\lambda)\right).$
(7.12)

We recall the $\sigma$-fields $\mathcal{F}_{1},\mathcal{F}_{2}$ defined in
(6.42) and (6.43). We first condition on $\mathcal{F}_{1}$. As noted in
Subsection 6.2.5, since $\mathfrak{h}_{pt,\uparrow}^{(1)}$ and
$\mathfrak{h}_{qt,\downarrow}^{(1)}$ are independent, applying
$\mathbf{H}_{pt}$ and $\mathbf{H}_{qt}$ Brownian Gibbs property from
Proposition 2.5 for $\mathfrak{h}_{pt,\uparrow}^{(1)}$,
$\mathfrak{h}_{qt,\downarrow}^{(1)}$ respectively we have

$\displaystyle\mathbb{P}\left(\Gamma_{t,M,K}\geq\tfrac{\rho}{2},\mathsf{Sp}(\lambda)\right)=\mathbb{E}\left[\frac{\mathbb{E}_{\operatorname{free},t}[\mathbf{1}_{\Gamma_{t,M,K}\geq\tfrac{\rho}{4},\mathsf{Sp}(\lambda)}W_{\uparrow}W_{\downarrow}]}{\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]}\right],$
(7.13)

where $W_{\uparrow}$, $W_{\downarrow}$ are defined in (6.48) and (6.49). Here
$\mathbb{P}_{\operatorname{free},t}$ and $\mathbb{E}_{\operatorname{free},t}$
are the probability and the expectation operator respectively corresponding to
the joint ‘free’ law for $(p^{1/3}\mathfrak{h}_{pt,\uparrow}(p^{-2/3}x)$, and
$q^{1/3}\mathfrak{h}_{qt,\downarrow}(q^{-2/3}x))_{x\in[-M,M]}$ which by
Brownian scaling is given by a pair of independent Brownian bridges
$(B_{1}(\cdot),B_{2}(\cdot))$ on $[-M,M]$ with starting points
$(p^{1/3}\mathfrak{h}_{pt,\uparrow}(-Mp^{-2/3}),q^{1/3}\mathfrak{h}_{qt,\downarrow}(-Mq^{-2/3}))$
and endpoints
$(q^{1/3}\mathfrak{h}_{pt,\uparrow}(Mp^{-2/3}),q^{1/3}\mathfrak{h}_{qt,\downarrow}(Mq^{-2/3})).$

In addition, from the last part of Proposition 2.5 we know that for any given
$\lambda>0$, there exists $\delta(M,p,\lambda)>0$ such that

$\displaystyle\mathbb{P}(\mathbb{E}_{\operatorname{free},t}[W_{\uparrow}W_{\downarrow}]>\delta)\geq
1-\lambda.$ (7.14)

Since the weight $W_{\uparrow}W_{\downarrow}\in[0,1],$ (7.13) and (7.14) give
us

$\displaystyle\mbox{r.h.s.~{}of
\eqref{c16}}\leq\mathrm{C}\exp(-\tfrac{1}{\mathrm{C}}M^{3})+\mathbb{P}(\neg\mathsf{Sp}(\lambda))+\lambda+{\frac{1}{\delta}}\mathbb{E}\left[\mathbb{P}_{\operatorname{free},t}\left(\Gamma_{t,M,K}\geq\tfrac{\rho}{4},\mathsf{Sp}(\lambda)\right)\right].$
(7.15)

Next we condition on $\mathcal{F}_{2}$ defined in (6.43). By Proposition 4.10,
upon conditioning the free measure of two Brownian bridges when viewed around
the maximizer are given by two $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$
(defined in Definition 4.4). The precise law is given by ${\mathsf{Nlarge}}$
law defined in Definition 6.3. Note that $\mathsf{Sp}(\lambda)$ is measurable
w.r.t. $\mathcal{F}_{1}\cup\mathcal{F}_{2}$. By Reverse Fatou’s Lemma and the
tower property of conditional expectations, we obtain that

$\displaystyle\limsup_{K\rightarrow\infty}\limsup_{t\rightarrow\infty}\mathbb{E}\left[\mathbb{P}_{\operatorname{free},t}\left(\Gamma_{t,M,K}\geq\tfrac{\rho}{4},\mathsf{Sp}(\lambda)\right)\right]$
$\displaystyle\leq\mathbb{E}\left[\limsup_{K\rightarrow\infty}\limsup_{t\to\infty}\mathbf{1}_{\mathsf{Sp}(\lambda)}\mathbb{P}_{\mathsf{Nlarge}|2,1}\left(\Gamma_{t,M,K}\geq\tfrac{\rho}{4}\right)\right].$
(7.16)

Following the Definition 6.3 and (7.11) we see that under $\mathsf{Nlarge}$
law,

$\displaystyle\Gamma_{t,M,K}\stackrel{{\scriptstyle
d}}{{=}}\int_{K}^{Mt^{2/3}-\mathcal{M}_{p,t}}e^{-t^{1/3}\left[V_{r,1}^{\mathsf{large}}(t^{-2/3}x)-V_{r,2}^{\mathsf{large}}(t^{-2/3}x)\right]}\mathrm{d}x.$
(7.17)

where
$V_{r}^{\mathsf{large}}=(V_{r,1}^{\mathsf{large}},V_{r,2}^{\mathsf{large}})$
is a $\mathsf{NonInt}\mbox{-}\mathsf{BrBridge}$ defined in Definition 6.3. Now
notice that by the definition in (6.62), on the $\mathsf{Sp}(\lambda)$ event,
the length of the Brownian bridges considered are bounded from below and above
and the end points are tight. Following the equality in distribution in
(7.17), the technical result of Proposition 5.6 precisely tells us that the
term inside the expectation of r.h.s. of (7.16) is zero. Thus, going back to
(7.15) we get that

$\displaystyle\limsup_{K\to\infty}\limsup_{t\to\infty}\mathbb{P}\left(\int_{U_{+,2}}e^{D_{2}(x,t)-D_{1}(x,t)}\mathrm{d}x\geq\frac{\rho}{4}\right)\leq\mathrm{C}\exp(-\tfrac{1}{\mathrm{C}}M^{3})+\limsup_{t\to\infty}\mathbb{P}(\neg\mathsf{Sp}(\lambda))+\lambda.$

Taking $\limsup_{\lambda\downarrow 0}$, in view of (6.63), we get that last
two terms in r.h.s. of the above equation are zero. Similarly one can show the
same bound for the integral under
$U_{-,2}:=[-t^{2/3}M-\mathcal{M}_{p,t},t^{2/3}M-\mathcal{M}_{p,t}]\cap(-\infty,-K]$.
Together with (7.7), we thus have

$\displaystyle\limsup_{K\to\infty}\limsup_{t\to\infty}\mathbb{P}\left(\int_{[-K,K]^{c}}e^{D_{2}(x,t)-D_{1}(x,t)}\mathrm{d}x\geq\rho\right)\leq\mathrm{C}\exp(-\tfrac{1}{\mathrm{C}}M^{3}).$

Taking $M\to\infty$ we get (7.2) completing the proof. ∎

###### Proof of Theorem 1.5.

Recall from (1.6) that

$\displaystyle
f_{*,t}(x)=\frac{\mathcal{Z}(0,0;x,t)}{\mathcal{Z}(0,0;*,t)}=\frac{e^{\mathcal{H}(x,t)}}{\int_{\mathbb{R}}e^{\mathcal{H}(y,t)}\mathrm{d}y}.$

The uniqueness of the mode $\mathcal{M}_{*,t}$ for $f_{*,t}$ is already proved
in Lemma 3.1. Thus, we have

$\displaystyle
f_{*,t}(x+\mathcal{M}_{*,t})=\frac{\exp\left(\mathcal{H}(\mathcal{M}_{*,t}+x,t)-\mathcal{H}(\mathcal{M}_{*,t},t)\right)}{\int\limits_{\mathbb{R}}\exp\left(\mathcal{H}(\mathcal{M}_{*,t}+y,t)-\mathcal{H}(\mathcal{M}_{*,t},t)\right)\mathrm{d}y}.$

Just like in Proposition 7.2, we claim that

$\displaystyle\limsup_{K\to\infty}\limsup_{t\to\infty}\mathbb{P}\left(\int_{[-K,K]^{c}}e^{\mathcal{H}(\mathcal{M}_{*,t}+y,t)-\mathcal{H}(\mathcal{M}_{*,t},t)}\mathrm{d}y\geq\rho\right)=0.$
(7.18)

The proof of (7.18) is exactly same as that of (7.2), where we divide the
integral in (7.18) into a deep tail and a shallow tail and bound them
individually. To avoid repetition, we just add few pointers for the readers.
Indeed the two key steps of proof of Proposition 7.2 that bound the deep and
shallow tails can be carried out for the (7.18) case. The deep tail regime
follows an exact similar strategy as Step 1 of the proof of Proposition 7.2
and utilizes the same parabolic decay of the KPZ equation from Proposition
2.8. The analogous shallow tail regime also follows in a similar manner by
using the uniform separation estimate for Bessel bridges from Corollary 5.7.

Now note that by Theorem 1.10 with $k=1$, we have

$\displaystyle\mathcal{H}(\mathcal{M}_{*,t}+x,t)-\mathcal{H}(\mathcal{M}_{*,t},t)\stackrel{{\scriptstyle
d}}{{\to}}\mathcal{R}_{1}(x),$ (7.19)

in the uniform-on-compact topology. Here $\mathcal{R}_{1}$ is a 3D-Bessel
process with diffusion coefficient $1$. With the tail decay estimate in (7.18)
and the same for the Bessel process from Proposition 7.1, in view of (7.19)
one can show
$f_{*,t}(x+\mathcal{M}_{*,t})\to\frac{e^{-\mathcal{R}_{1}(x)}}{\int_{\mathbb{R}}e^{-\mathcal{R}_{1}(y)}\mathrm{d}y}$
in the uniform-on-compact topology by following the analogous argument from
the proof of Theorem 1.4. This completes the proof. ∎

##  Appendix A Non-intersecting random walks

In this section we prove Lemma 4.7 that investigates the convergence of non-
intersecting random walks to non-intersecting brownian motions. We remark that
similar types of Theorems are already known in the literature such as [EK08],
where the authors considered random walks to start at different locations.
Since our walks starts at the same point, additional care is required.

We now recall Lemma 4.7 for readers’ convenience.

######  Lemma A.1.

Let $X_{j}^{i}$ be i.i.d. $\operatorname{N}(0,1)$ random variables. Let
$S_{0}^{(i)}=0$ and $S_{k}^{(i)}=\sum_{j=1}^{k}X_{j}^{i}.$ Consider
$Y_{n}(t)=(Y_{n,1}(t),Y_{n,2}(t)):=(\frac{S_{nt}^{(1)}}{\sqrt{n}},\frac{S_{nt}^{(2)}}{\sqrt{n}})$
an $\mathbb{R}^{2}$ valued process on $[0,1]$ where the in-between points are
defined by linear interpolation. Then conditioned on the non-intersecting
event $\Lambda_{n}:=\cap_{j=1}^{n}\\{S_{j}^{(1)}>S_{j}^{(2)}\\},$
$Y_{n}\stackrel{{\scriptstyle d}}{{\to}}W$, where $W(t)=(W_{1}(t),W_{2}(t))$
is distributed as $\mathsf{NonInt}\mbox{-}\mathsf{BM}$ defined in Definition
4.3.

###### Proof of Lemma A.1.

To show weak convergence, it suffices to show finite dimensional convergence
and tightness. Based on the availability of exact joint densities for non-
intersecting random walks from Karlin-McGregor formula [KM59], the
verification of weak convergence is straightforward. So, we only highlight
major steps of the computations below.

Step 1. One point convergence at $t=1$. Note that

$\displaystyle\mathbb{P}\left(|{\sqrt{n}Y_{n,i}(t)}-S_{\lfloor
nt\rfloor}^{(i)}|>\sqrt{n}\varepsilon\mid\Lambda_{n}\right)\leq\frac{1}{\mathbb{P}(\Lambda_{n})}\mathbb{P}\left(|X_{\lfloor
nt\rfloor+1}|>\sqrt{n}\varepsilon\right)\leq\tfrac{\mathrm{C}}{\varepsilon^{2}\sqrt{n}}$

The last inequality above follows by Markov inequality and the classical
result that $\mathbb{P}(\Lambda_{n})\geq\tfrac{\mathrm{C}}{\sqrt{n}}$ in
Spitzer [Spi60]. Thus it suffices to show finite dimensional convergence for
the cadlag process:

$\displaystyle(Z_{nt}^{(1)},Z_{nt}^{(2)}):=\frac{1}{\sqrt{n}}(S_{\lfloor
nt\rfloor}^{(1)},S_{\lfloor nt\rfloor}^{(2)}).$ (A.1)

We assume that $n$ large enough so that $\frac{n-1}{M\sqrt{n}}\geq 1$ for some
$M>0$ to be chosen later. When $t=1$, applying the Karlin-McGregor formula, we
obtain that

$\mathbb{P}(Z_{n}(1)\in\mathrm{d}y_{1},Z_{n}(1)\in\mathrm{d}y_{2}|\Lambda_{n})=\tau_{n}\cdot
f_{n,1}(y_{1},y_{2})\mathrm{d}y_{1}\mathrm{d}y_{2}$

where

$\displaystyle
f_{n,1}(y_{1},y_{2}):=\int\limits_{a_{1}>a_{2}}p_{1}(a_{1})p_{1}(a_{2})\det(p_{n-1}(a_{i}-y_{j}\sqrt{n}))_{i,j=1}^{2}\mathrm{d}a_{1}\mathrm{d}a_{2},$

and

$\displaystyle\tau_{n}^{-1}:=\int_{r_{1}>r_{2}}\int_{a_{1}>a_{2}}p_{1}(a_{1})p_{1}(a_{2})\det(p_{n-1}(a_{i}-r_{j}\sqrt{n}))_{i,j=1}^{2}\mathrm{d}a_{1}\mathrm{d}a_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}.$
(A.2)

Note that here the Karlin-McGregor formula, after we have conditioned on the
first step of the random walks with $X_{1}^{1}=a_{1}>X_{1}^{2}=a_{2}.$

We will now show that $\frac{(n-1)^{2}}{\sqrt{n}}\tau_{n}^{-1}$ and
$\frac{(n-1)^{2}}{\sqrt{n}}f_{n,1}(y_{1},y_{2})$ converges to a nontrivial
limit. Observe that

$\displaystyle\tfrac{(n-1)^{2}}{\sqrt{n}}\det(p_{n-1}(a_{i}-y_{j}\sqrt{n}))_{i,j=1}^{2}$
$\displaystyle=(n-1)p_{n-1}(a_{1}-y_{2}\sqrt{n})p_{n-1}(a_{2}-y_{1}\sqrt{n})$
(A.3) $\displaystyle\hskip
71.13188pt\cdot\tfrac{n-1}{\sqrt{n}}[e^{\frac{\sqrt{n}(a_{1}-a_{2})(y_{1}-y_{2})}{n-1}}-1].$

Thus, as $n\to\infty$, we have

$\displaystyle\tfrac{(n-1)^{2}}{\sqrt{n}}\det(p_{n-1}(a_{i}-y_{j}\sqrt{n}))_{i,j=1}^{2}$
$\displaystyle\to p_{1}(y_{1})p_{1}(y_{2})(a_{1}-a_{2})(y_{1}-y_{2}).$ (A.4)

Next we proceed to find a uniform bound for the expression in (A.3). Not that
for $x,r\geq 1$, one has the elementary inequality $x^{r}\geq x^{r}-1\geq
r(x-1)$. Now taking $r=\frac{n-1}{M\sqrt{n}}$ and
$x=\exp(\frac{\sqrt{n}}{n-1}(a_{1}-a_{2})(y_{1}-y_{2})$ we get

r.h.s. of (A.3)
$\displaystyle\leq\frac{1}{2\pi}\exp\left(-\tfrac{(a_{1}-y_{2}\sqrt{n})^{2}}{2n-2}-\tfrac{(a_{2}-y_{1}\sqrt{n})^{2}}{2n-2}+\tfrac{1}{M}(a_{1}-a_{2})(y_{1}-y_{2})\right)$
$\displaystyle\leq\frac{1}{2\pi}\exp\left(-\tfrac{y_{2}^{2}}{4}-\tfrac{y_{1}^{2}}{4}+\tfrac{1}{M}(a_{1}-a_{2})(y_{1}-y_{2})+\tfrac{1}{M}(|a_{1}y_{2}|+|a_{2}y_{1}|)\right)$
$\displaystyle\leq\frac{1}{2\pi}\exp\left(-\tfrac{y_{2}^{2}}{4}-\tfrac{y_{1}^{2}}{4}+\tfrac{2(a_{1}^{2}+y_{1}^{2}+a_{2}^{2}+y_{2}^{2})}{M})\right),$
(A.5)

where the last inequality follows by several application of the elementary
inequality $|xy|\leq\frac{1}{2}(x^{2}+y^{2})$. One can choose $M$ large enough
so that the uniform bound in (A.5) is integrable w.r.t. the measure
$p_{1}(a_{1})p_{1}(a_{2})\mathrm{d}a_{1}\mathrm{d}a_{2}$. With the pointwise
limit from (A.4), by dominated convergence theorem we have

$\displaystyle\tfrac{(n-1)^{2}}{\sqrt{n}}f_{n,1}(y_{1},y_{2})$
$\displaystyle=\tfrac{(n-1)^{2}}{\sqrt{n}}\int_{a_{1}>a_{2}}p_{1}(a_{1})p_{1}(a_{2})\det(p_{n-1}(a_{i}-y_{j}\sqrt{n}))_{i,j=1}^{2}\mathrm{d}a_{1}\mathrm{d}a_{2}$
$\displaystyle\hskip 56.9055pt\to
p_{1}(y_{1})p_{1}(y_{2})(y_{1}-y_{2})\int_{a_{1}>a_{2}}(a_{1}-a_{2})p_{1}(a_{1})p_{1}(a_{2})\mathrm{d}a_{1}\mathrm{d}a_{2}.$

Similarly one can compute the pointwise limit for the integrand in
$\tau_{n}^{-1}$ (defined in (A.2)) and the uniform bound in (A.5) works for
the denominator as well. We thus have

$\displaystyle\tfrac{(n-1)^{2}}{\sqrt{n}}\tau_{n}^{-1}\to\int_{a_{1}>a_{2}}\int_{r_{1}>r_{2}}p_{1}(r_{1})p_{1}(r_{2})(r_{1}-r_{2})(a_{1}-a_{2})p_{1}(a_{1})p_{1}(a_{2})\mathrm{d}a_{1}\mathrm{d}a_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}.$
(A.6)

Plugging these limits back in (A.1), we arrive at (4.1) (the one point density
formula for $\mathsf{NonInt}\mbox{-}\mathsf{BM}$) as the limit for (A.1).

Step 2. One point convergence at $0<t<1$. When $0<t<1$, with the Karlin-
Mcgregor formula, we similarly obtain

$\displaystyle\mathbb{P}(Z_{nt}^{(1)}\in\mathrm{d}y_{1},Z_{nt}^{(2)}\in\mathrm{d}y_{2}\mid\Lambda_{n})=\tau_{n}\cdot
f_{n,t}(y_{1},y_{2})\mathrm{d}y_{1}\mathrm{d}y_{2}$ (A.7)

where $\tau_{n}$ is defined in (A.2) and

$\displaystyle f_{n,t}(y_{1},y_{2})$
$\displaystyle=\int_{r_{1}>r_{2}}\int_{a_{1}>a_{2}}p_{1}(a_{1})p_{1}(a_{2})\left[\det(p_{\lfloor
nt\rfloor-1}(a_{i}-y_{j}\sqrt{n}))_{i,j=1}^{2}\right.$ (A.8)
$\displaystyle\hskip 85.35826pt\left.n\cdot\det(p_{n-\lfloor
nt\rfloor}(\sqrt{n}y_{i}-\sqrt{n}r_{j}))_{i,j=1}^{2}\right]\mathrm{d}a_{1}\mathrm{d}a_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}.$

One can check that as $n\to\infty$, we have

$\displaystyle n^{3/2}\det(p_{\lfloor
nt\rfloor-1}(a_{i}-y_{j}\sqrt{n}))_{i,j=1}^{2}$
$\displaystyle\to\tfrac{1}{t}p_{t}(y_{1})p_{t}(y_{2})(a_{1}-a_{2})(y_{1}-y_{2}),$
$\displaystyle n\cdot\det(p_{n-\lfloor
nt\rfloor}(\sqrt{n}y_{i}-\sqrt{n}r_{j}))_{i,j=1}^{2}$
$\displaystyle\to\det(p_{1-t}(y_{i}-r_{j}))_{i,j=1}^{2}.$

One can provide uniformly integrable bound for the integrand in
$f_{n,t}(y_{1},y_{2})$ in a similar fashion. Thus by dominated convergence
theorem,

$\displaystyle n^{3/2}f_{n,t}(y_{1},y_{2})$
$\displaystyle\to\tfrac{1}{t}p_{t}(y_{1})p_{t}(y_{2})(y_{1}-y_{2})\int_{a_{1}>a_{2}}p_{1}(a_{1})p_{1}(a_{2})(a_{1}-a_{2})\mathrm{d}a_{1}\mathrm{d}a_{2}$
$\displaystyle\hskip
85.35826pt\int_{r_{1}>r_{2}}\det(p_{1-t}(y_{i}-r_{j}))_{i,j=1}^{2}\mathrm{d}r_{1}\mathrm{d}r_{2}.$

Using (A.6) we get that $\tau_{n}\cdot f_{n,t}(y_{1},y_{2})$ converges to
(4.2), the one point density formula for $\mathsf{NonInt}\mbox{-}\mathsf{BM}$.

Step 3. Transition density convergence. For the transition densities, let
$0<t_{1}<t_{2}<1,$ and fix $x_{1}>x_{2}$. Another application of Karlin-
McGregor formula tells us

$\displaystyle\mathbb{P}(Z_{nt_{2}}^{(1)}\in\mathrm{d}y_{1},Z_{nt_{2}}^{(2)}\in\mathrm{d}y_{2}\mid
Z_{nt_{1}}^{(1)}=x_{1},Z_{nt_{1}}^{(2)}=x_{2})$ (A.9)
$\displaystyle=n\det(p_{\lfloor nt_{2}\rfloor-\lfloor
nt_{1}\rfloor}(\sqrt{n}y_{i}-\sqrt{n}x_{j}))_{i,j=1}^{2}$ $\displaystyle\hskip
56.9055pt\cdot\frac{\int\limits_{r_{1}>r_{2}}\det(p_{n-\lfloor
nt_{2}\rfloor}(\sqrt{n}y_{i}-\sqrt{n}r_{j}))_{i,j=1}^{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}y_{1}\mathrm{d}y_{2}}{\int\limits_{r_{1}>r_{2}}\det(p_{n-\lfloor
nt_{1}\rfloor}(\sqrt{n}x_{i}-\sqrt{n}r_{j}))_{i,j=1}^{2}\mathrm{d}r_{1}\mathrm{d}r_{2}}.$

One can check as $n\to\infty$

$\displaystyle\text{ r.h.s of
}\eqref{trst1}\to\frac{\det(p_{t_{2}-t_{1}}(y_{i}-x_{j}))_{i,j=1}^{2}\int_{r_{1}>r_{2}}\det(p_{1-t_{2}}(y_{i}-r_{j}))_{i,j=1}^{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}y_{1}\mathrm{d}y_{2}}{\int_{r_{1}>r_{2}}\det(p_{1-t_{1}}(x_{i}-r_{j}))_{i,j=1}^{2}\mathrm{d}r_{1}\mathrm{d}r_{2}}$

which is same as transition densities for $\mathsf{NonInt}\mbox{-}\mathsf{BM}$
as shown in (4.3). This proves finite dimensional convergence.

Step 4. Tightness. To show tightness, by Kolmogorov tightness criterion, it
suffices to show there exist $K>0$ and $n_{0}\in\mathbb{N}$ such that for all
$n\geq n_{0}$

$\displaystyle\mathbb{E}\left[|Y_{n,i}(t)-Y_{n,i}(s)|^{K}\mid\Lambda_{n}\right]\leq\mathrm{C}_{K,n_{0}}\cdot(t-s)^{2}$
(A.10)

holds for all $0\leq s<t\leq 1.$

Recall that $\mathbb{P}(\Lambda_{n})\geq\frac{\mathrm{C}}{\sqrt{n}}$. For
$t-s\leq\frac{1}{n}$ with $K\geq 5$ we have

$\displaystyle\mathbb{E}\left[|Y_{n,i}(t)-Y_{n,i}(s)|^{K}\mid\Lambda_{n}\right]$
$\displaystyle\leq\mathrm{C}\cdot\sqrt{n}\mathbb{E}\left[|Y_{n,i}(t)-Y_{n,i}(s)|^{K}\right]$
$\displaystyle\leq\mathrm{C}\cdot\sqrt{n}\frac{(nt-
ns)^{K}}{n^{K/2}}\mathbb{E}[|X_{1}^{1}|^{K}]\leq\mathrm{C}n^{\frac{1-K}{2}}(nt-
ns)^{2}\leq\mathrm{C}_{K}(t-s)^{2}.$

Thus we may assume $t-s\geq 1/n$. Then it is enough to show (A.10) for
$Z_{nt}^{(i)}$ (defined in (A.1)) instead. Note that if
$t-s\in[n^{-1},{n^{-1/4}}]$, we may take $K$ large enough so
$\frac{1}{4}(K-4)\geq 1$. Then we have

$\displaystyle\mathbb{E}\left[|Z_{nt}^{(i)}-Z_{ns}^{(i)}|^{K}\mid\Lambda_{n}\right]$
$\displaystyle\leq\mathrm{C}\cdot\sqrt{n}\mathbb{E}\left[|Z_{nt}^{(i)}-Z_{ns}^{(i)}|^{K}\right]$
$\displaystyle\leq\mathrm{C}\cdot\sqrt{n}(t-s)^{K/2}\leq\mathrm{C}\cdot
n^{1/2-(K-4)/8}(t-s)^{2}$

where in the last line we used the fact $(t-s)^{(K-4)/2}\leq n^{-(K-4)/8}$. As
$\frac{1}{4}(K-4)\geq 1$, we have
$\mathbb{E}\left[|Z_{nt}^{(i)}-Z_{ns}^{(i)}|^{K}\mid\Lambda_{n}\right]\leq\mathrm{C}(t-s)^{2}$
in this case. So, we are left with the case $t-s\geq n^{-1/4}$.

Let us assume $t=0$, $s\geq n^{-\frac{1}{4}}$. As $ns\geq n^{3/4}\to\infty$,
we will no longer make the distinction between $ns$ and $\lfloor ns\rfloor$ in
our computations. We use the pdf formula from (A.7) and (A.8) to get

$\displaystyle\mathbb{E}[|Z_{ns}^{(i)}|^{5}]$
$\displaystyle\leq\tau_{n}\int_{y_{1}>y_{2}}|y_{i}|^{5}\int_{r_{1}>r_{2}}\int_{a_{1}>a_{2}}p_{1}(a_{1})p_{1}(a_{2})\det(p_{ns-1}(a_{i}-y_{j}\sqrt{n}))_{i,j=1}^{2}$
(A.11) $\displaystyle\hskip
85.35826pt\left.n\cdot\det(p_{n-ns}(\sqrt{n}y_{i}-\sqrt{n}r_{j}))_{i,j=1}^{2}\right]\mathrm{d}a_{1}\mathrm{d}a_{2}\mathrm{d}r_{1}\mathrm{d}r_{2}\mathrm{d}y_{1}\mathrm{d}y_{2}.$

For the last determinant we may use

$\displaystyle
n\cdot\det(p_{n-ns}(\sqrt{n}y_{i}-\sqrt{n}r_{j}))_{i,j=1}^{2}\mathrm{d}r_{1}\mathrm{d}r_{2}$
$\displaystyle\leq n\cdot
p_{n-ns}(\sqrt{n}y_{1}-\sqrt{n}r_{1})p_{n-ns}(\sqrt{n}y_{2}-\sqrt{n}r_{2})\mathrm{d}r_{1}\mathrm{d}r_{2}$

which integrates to $1$ irrespective of the value of $y_{1},y_{2}$. Thus

r.h.s. of (A.11)
$\displaystyle\leq\tau_{n}\int_{y_{1}>y_{2}}|y_{i}|^{5}\int_{a_{1},a_{2}}p_{1}(a_{1})p_{1}(a_{2})\det(p_{ns-1}(a_{i}-y_{j}\sqrt{n}))_{i,j=1}^{2}\mathrm{d}a_{1}\mathrm{d}a_{2}\mathrm{d}y_{1}\mathrm{d}y_{2}.$
(A.12)

Making the change of variable $y_{i}=\sqrt{s}z_{i}$ and setting $m=ns$, we
have

$\mbox{r.h.s.~{}of \eqref{a04}}\leq\tau_{n}\cdot
s^{\frac{5}{2}+1}\mathcal{I}_{m},$

where

$\displaystyle\mathcal{I}_{m}:=\int_{z_{1}>z_{2}}|z_{i}|^{5}\int_{a_{1}>a_{2}}p_{1}(a_{1})p_{1}(a_{2})\det(p_{m-1}(a_{i}-z_{j}\sqrt{m}))_{i,j=1}^{2}\mathrm{d}a_{1}\mathrm{d}a_{2}\mathrm{d}z_{1}\mathrm{d}z_{2}.$

We claim that $\frac{(m-1)^{2}}{\sqrt{m}}{\mathcal{I}_{m}}\leq\mathrm{C}$ for
some universal constant $\mathrm{C}>0$. Clearly this integral is finite for
each $m$. And by exact same approach in Step 1, one can show as $m\to\infty$,

$\frac{(m-1)^{2}}{\sqrt{m}}\mathcal{I}_{m}:=\int_{z_{1}>z_{2}}|z_{i}|^{5}\int_{a_{1}>a_{2}}p_{1}(z_{1})p_{1}(z_{2})p_{1}(a_{1})p_{1}(a_{2})(a_{1}-a_{2})(z_{1}-z_{2})\mathrm{d}a_{1}\mathrm{d}a_{2}\mathrm{d}z_{1}\mathrm{d}z_{2}.$

Thus, $\frac{(m-1)^{2}}{\sqrt{m}}\mathcal{I}\leq\mathrm{C}$ for all $m\geq 1$.
Thus following (A.11), (A.12), in view of the above estimate we get

$\displaystyle\mathbb{E}[|Z_{ns}^{(i)}|^{5}]\leq\mathrm{C}\tau_{n}\frac{\sqrt{m}}{(m-1)^{2}}s^{\frac{5}{2}+1}.$

However, by Step 1, $n^{3/2}\tau_{n}^{-1}$ converges to a finite positive
constant. As $m=ns$, we thus get that the above term is at most
$\mathrm{C}\cdot s^{2}$. The case $t\neq 0$ can be checked similarly using the
formulas from (A.7) and (A.8) as well as transition densities formula (A.9).
This completes the proof. ∎

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