We have the following result that is proved in Section 5.2.

######  Proposition 5.1.

Fix $r,\gamma>0$ and $p\in\\{1,2\\}$. Set $\alpha=\alpha_{p}$ according to
(3.11). We have

$\displaystyle\lim_{\delta\downarrow
0}\limsup_{N\to\infty}\mathbb{P}_{\alpha_{p}}\left(\omega_{\delta}^{N}(\mathcal{L}_{1}^{N},\llbracket
1,2\lfloor rN^{2/3}\rfloor-1\rrbracket)\geq\gamma N^{1/3}\right)=0.$ (5.2)

###### Proof of Theorem 1.1.

By a standard criterion for functional tightness [Bil13, Theorem 7.3],
Proposition 5.1 along with endpoint tightness from Theorem 3.10 implies
tightness of the probability law of $N^{-1/3}\mathcal{L}_{1}^{N}(\llbracket
1,2\lfloor rN^{2/3}\rfloor-1\rrbracket)$. In light of the matching in
distribution in Theorem 1.3 (i), this immediately translates into tightness of
the measure $\mathbb{P}^{N}_{\alpha}$ desired to prove Theorem 1.1. Note that
cases (1) and (2) of Theorem 1.1 follow by using the $p=1$ and $p=2$ cases of
Proposition 5.1 and Theorem 3.10. ∎

The rest of this section is devoted to proving Proposition 5.1. This relies on
the following technical result which deals with the modulus of continuity for
the bottom-free measure.

######  Proposition 5.2.

Fix any $M,V,k_{1},k_{2},\gamma>0$ with $k_{2}>k_{1}$. For each $N>0$, define
the sets

$\displaystyle I_{1,M}:=\\{y\in\mathbb{R},|y|\leq 2MN^{1/3}\\},\mbox{ and
}I_{2,M}:=\\{(y_{1},y_{2})\in\mathbb{R}^{2}:y_{i}\in
I_{1,M/2},y_{1}-y_{2}\geq-(\log N)^{7/6}\\}.$

For each $p\in\\{1,2\\}$, there exist
$\delta=\delta(M,V,k_{1},k_{2},\gamma,\varepsilon)>0$ and
$N_{0}=N_{0}(M,V,k_{1},k_{2},\gamma,\varepsilon)>0$ such that for all
$\vec{x}\in I_{p,M}$, $T\in\llbracket k_{1}N^{2/3},k_{2}N^{2/3}\rrbracket$,
and $N\geq N_{0}$ we have

$\displaystyle\sum_{i=1}^{p}\mathbb{P}_{\alpha_{p}}^{\vec{x},(-\infty)^{T};p,T}\left(\sum_{k=1}^{i}|L_{k}(k)|\leq
VN^{1/3},\mbox{ and }\omega_{\delta}^{N}(L_{i},\llbracket
1,T/4+i-2\rrbracket)\geq\gamma N^{1/3}\right)\leq\varepsilon.$

We postpone the proof of Proposition 5.2 to Section 5.3. Section 5.1 contains
a few lemmas that are used in the proof of Proposition 5.1 later in Section
5.2.

###  5.1. Preparatory Lemmas

We first discuss a few consequences of Proposition 3.9 that form the
preparatory tools for our modulus of continuity analysis.

######  Lemma 5.3.

Fix any $\varepsilon\in(0,\frac{1}{2})$ and $T\geq 2$. Let
$(X(i))_{i=1}^{2T-1}$ be a random vector with $X(1)=X(2T-1)\equiv 0$ and
density at $(X_{i})_{i=2}^{2T-2}=(u_{i})_{2=1}^{2T-2}$ proportional to

$\prod_{i=1}^{2T-2}G_{\theta,(-1)^{i+1}}(u_{i}-u_{i+1})$

respectively where $u_{1}=0$ and $u_{2T-1}=0$ and $G$ is defined in (2.2).
Similarly, define an independent random vector $(Y(i))_{i=1}^{2T-1}$ precisely
as with $X$ except that $G_{\theta,(-1)^{i+1}}$ is replaced by
$G_{\theta,(-1)^{i}}$. Then, there exists $M_{0}(\varepsilon)>0$ such that for
all $T\geq 2$ we have

$\displaystyle\mathbb{P}\bigg{(}\sup_{i\in\llbracket
1,2T-1\rrbracket}\big{(}|X(i)|+|Y(i)|\big{)}\geq
M_{0}\sqrt{T}\bigg{)}\leq\varepsilon.$ (5.3)

We refer to Figure 17 for graphical representation of the distributions
appearing in Lemma 5.3.

$0$$0$$0$$0$ Figure 17. Graphical representation of $X$ (left) and $Y$ (right)
distribution from Lemma 5.3.

###### Proof.

Fix $\varepsilon\in(0,1)$. Note that $(X(2i-1))_{i=1}^{T}$ forms a random
bridge from $0$ to $0$ with increment from $G_{\theta,+1}\ast G_{\theta,-1}$.
By the KMT coupling for random bridges [DW21a] along with Brownian bridge
estimates, there exists a constant $M>0$ such that (here we temporarily use
$\mathbb{P}$ and $\mathbb{E}$ for the probability and expectation for the $X$
and $Y$ vectors)

$\displaystyle\mathbb{P}(\mathsf{A})\leq\tfrac{\varepsilon}{4},\mbox{ where
}\mathsf{A}:=\left\\{\sup_{i\in\llbracket 1,T\rrbracket}|X(2i-1)|\geq
M\sqrt{T}\right\\}.$

Let us write $\mathcal{F}:=\sigma\big{(}(X(2i-1))_{i=1}^{T}\big{)}$. By a
union bound we have

$\displaystyle\mathbb{P}\Big{(}\sup_{i\in\llbracket
1,2T-1\rrbracket}|X(i)|\geq
5M\sqrt{T}\Big{)}\leq\frac{\varepsilon}{4}+\sum_{i=1}^{T-1}\mathbb{E}\left[\mathbf{1}_{\neg\mathsf{A}}\cdot\mathbb{E}\left[\mathbf{1}_{|X(2i)|\geq
5M\sqrt{T}}\mid\mathcal{F}\right]\right].$ (5.4)

Note that the distribution of even points given the odd points are given by
the $\xi$-distribution introduced in (4.5). Observe that by Lemma B.4,

$\displaystyle\mathbf{1}_{X(2i-1),X(2i+1)\in(-M\sqrt{T},M\sqrt{T})}\cdot\mathbb{E}\left[\mathbf{1}_{|X(2i)|\geq
5M\sqrt{T}}\mid\mathcal{F}\right]\leq\mathrm{C}\exp(-\tfrac{1}{\mathrm{C}}\sqrt{T}),$

for some absolute constant $\mathrm{C}>0$. Plugging the above bound back in
(5.4) and taking $T$ large enough we get that r.h.s. (5.4) is at most
$\frac{\varepsilon}{2}$. Similarly we see that
$\mathbb{P}\big{(}\sup_{i\in\llbracket 1,2T-1\rrbracket}|Y(i)|\geq
5M\sqrt{T}\big{)}\leq\frac{\varepsilon}{2}$. Adjusting $M$, we arrive at
(5.7). ∎

######  Lemma 5.4.

Recall for $p\in\\{1,2\\}$, $\alpha:=\alpha_{p}$ from (3.11). Fix any $r\geq
1$ and $\varepsilon>0$. Set $T=\lfloor rN^{2/3}\rfloor$. There exists
$M=M(\varepsilon)>0$ and $N_{0}(\varepsilon)>0$ such that for all $N\geq
N_{0}$ we have

$\displaystyle\mathbb{P}_{\alpha_{1}}^{0,(-\infty)^{T};1,T}\bigg{(}\sup_{i\in\llbracket
1,2T-1\rrbracket}|L_{1}(i)|\geq M\sqrt{T}\bigg{)}\leq\varepsilon,$ (5.5)
$\displaystyle\mathbb{P}_{\alpha_{2}}^{(0,-\sqrt{T}),(-\infty)^{T};2,T}\bigg{(}\sup_{i\in\llbracket
1,2T-1\rrbracket}|L_{1}(i)|+\sup_{j\in\llbracket 1,2T\rrbracket}|L_{2}(j)|\geq
M\sqrt{T}\bigg{)}\leq\varepsilon,$ (5.6)

where the bottom free law
$\mathbb{P}_{\alpha_{p}}^{\vec{x},(-\infty)^{2T};2,T}$ is defined in
Definition 2.4.

###### Proof.

For clarity we divide the proof into two steps.

Step 1. Fix any $\varepsilon\in(0,\frac{1}{2})$ and consider
$M_{0}(\varepsilon)$ from Lemma 5.3. In this step we prove (5.5). From
Proposition 3.9 choose $M_{1}(\varepsilon)>0$ such that for all large enough
$T$ we have

$\displaystyle\mathbb{P}_{\alpha_{1}}^{0,(-\infty)^{T};1,T}\big{(}|L_{1}(1)|\geq
M_{1}\sqrt{T}\big{)}\geq\varepsilon,\quad\mathbb{P}_{\alpha_{2}}^{(0,-\sqrt{T}),(-\infty)^{T};2,T}\big{(}|L_{1}(1)|+|L_{2}(2)|\geq
M_{1}\sqrt{T}\big{)}\geq\varepsilon.$ (5.7)

We will use the first bound immediately, and the second a bit later. Set
$M_{3}:=2M_{0}+M_{1}+1$, and

$\displaystyle\mathsf{A}_{1}:=\left\\{\sup_{i\in\llbracket
1,2T-1\rrbracket}L_{1}(i)\geq(M_{3}+M_{0})\sqrt{T}\right\\},\qquad\mathsf{A}_{2}:=\left\\{\sup_{i\in\llbracket
2,2T\rrbracket}L_{2}(i)\geq(M_{0}+M_{1})\sqrt{T}\right\\}.$

Finally, introduce shorthand notation $\mathbb{P}_{1}$ for
$\mathbb{P}_{\alpha_{1}}^{(0,-\sqrt{T}),(-\infty)^{T};1,T}$ and
$\mathbb{P}_{2}$ for
$\mathbb{P}_{\alpha_{2}}^{(0,-\sqrt{T}),(-\infty)^{T};2,T}$ (and likewise for
$\mathbb{E}$). In view of (5.7), by a union bound we have

$\displaystyle\mathbb{P}_{1}(\mathsf{A}_{1})\leq\varepsilon+\mathbb{E}_{1}\left[\mathbf{1}_{L_{1}(1)\leq
M_{1}\sqrt{T}}\mathbb{E}_{1}\left[\mathbf{1}_{\mathsf{A}_{1}}\mid\sigma(L_{1}(1))\right]\right].$

As $\mathsf{A}_{1}$ is an increasing event with respect to the boundary data,
due to stochastic monotonicity (Proposition 2.6), increasing the boundaries
will only increase the conditional probability. Thus to get an upper bound, we
may assume $L_{1}(1)=L_{1}(2T-1)=M_{1}\sqrt{T}$. But note that under this
boundary condition we have
$(L_{1}(i)-M_{1}\sqrt{T})_{i=1}^{2T-1}\stackrel{{\scriptstyle(d)}}{{=}}(X(i))_{i=1}^{2T-1}$,
where $X(\cdot)$ is defined in Lemma 5.3. Thus, owing to (5.3), almost surely
we have

$\displaystyle\mathbf{1}_{L_{1}(1)\leq
M_{1}\sqrt{T}}\mathbb{E}_{1}\left[\mathbf{1}_{\mathsf{A}_{1}}\mid\sigma(L_{1}(1))\right]\leq\mathbb{P}\bigg{(}\sup_{i\in\llbracket
1,2T-1\rrbracket}|X(i)|\geq(2M_{0}+M)\sqrt{T}\bigg{)}\leq\varepsilon.$

This implies $\mathbb{P}_{1}(\mathsf{A}_{1})\leq 2\varepsilon$. Following
similar calculations one can show

$\mathbb{P}_{1}\bigg{(}\inf_{i\in\llbracket
1,2T-1\rrbracket}L_{1}(i)\leq-(M_{3}+M_{0})\sqrt{T}\bigg{)}\leq 2\varepsilon.$

This proves (5.5) with $M=M_{3}$ for $\varepsilon\mapsto 2\varepsilon$.

Step 2. In this step we prove (5.6). At this point we encourage the readers to
look at Figure 18 and its caption for an overview of the proof idea.

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Figure 18. In the above figure, we have plotted $L_{1}\llbracket
1,2T-1\rrbracket$ (black curve) and $L_{2}\llbracket 2,2T\rrbracket$ (blue
curve). Endpoint tightness, Proposition 3.9, ensures that
$L_{1}(1),L_{2}(2)\in(-M_{0}\sqrt{T},M_{0}\sqrt{T})$. Assuming this, in order
to seek an uniform upper bound for the blue curve, by stochastic monotonicity
we may push the black curve all the way to $+\infty$. The resulting law for
the blue curve is given by $Y(\cdot)$ (upto a translation) introduced in Lemma
5.3. A uniform upper bound for the resulting law for the blue curve law can
then be estimated by Lemma 5.3. The upper bound is shown in the dashed line
above. Once we have a uniform upper bound for the blue curve, we may elevate
the endpoints of black curve much higher (from black points to red points in
the above right figure) so that the curve no longer feels the effect of the
blue curve. The red curve above denotes a sample for $L_{1}$ from this
elevated end points. Without interaction with the blue curve, its law (upto a
translation) equals to $X(\cdot)$ in Lemma 5.3. A uniform upper bound for the
red curve can then be estimated by Lemma 5.3.

Let us set $\mathcal{F}_{1}=\sigma(L_{2}(2),(L_{1}(i))_{i=1}^{2T-1})$ and
$\mathcal{F}_{2}=\sigma(L_{1}(1),(L_{2}(i))_{i=2}^{2T})$. In view of (5.7), by
a union bound

$\displaystyle\mathbb{P}_{2}(\mathsf{A}_{2})$
$\displaystyle\leq\varepsilon+\mathbb{P}_{2}\bigg{(}\\{L_{2}(2)\leq
M_{0}\sqrt{T}\\}\cap\mathsf{A}_{2}\bigg{)}\leq\varepsilon+\mathbb{E}_{2}\left[\mathbf{1}_{L_{2}(2)\leq
M_{0}\sqrt{T}}\mathbb{E}_{2}\left[\mathbf{1}_{\mathsf{A}_{2}}\mid\mathcal{F}_{1}\right]\right].$

As $\mathsf{A}_{2}$ is an increasing event with respect to the boundary data,
due to stochastic monotonicity (Proposition 2.6), increasing the boundaries
will only increase the conditional probability of $\mathsf{A}_{2}$. Thus, to
get an upper bound, we may assume $L_{2}(2T)=L_{2}(2)=M_{1}\sqrt{T}$, and
$L_{1}(i)=+\infty$ for all $i\in\llbracket 1,2T-1\rrbracket$. Under this
boundary condition we have
$(L_{2}(i+1)-M_{1}\sqrt{T})_{i=1}^{2T-1}\stackrel{{\scriptstyle(d)}}{{=}}(Y(i))_{i=1}^{2T-1}$
where $Y(\cdot)$ is defined in Lemma 5.3. Thus, almost surely we have (recall
$\mathbb{P}$ is the law of $Y$ below)

$\displaystyle\mathbf{1}_{L_{2}(2)\leq
M_{0}\sqrt{T}}\cdot\mathbb{E}_{2}\left[\mathbf{1}_{\mathsf{A}_{2}}\mid\mathcal{F}_{1}\right]\leq\mathbb{P}\bigg{(}\sup_{i\in\llbracket
1,2T-1\rrbracket}|X(i)|\geq M_{0}\sqrt{T}\bigg{)}\leq\varepsilon.$

Thus $\mathbb{P}_{2}(\mathsf{A}_{2})\leq 2\varepsilon$. In view of this bound,
applying a union bound we have

$\displaystyle\mathbb{P}_{2}(\mathsf{A}_{1})\leq
3\varepsilon+\mathbb{E}_{2}\left[\mathbf{1}_{\\{L_{1}(1)\leq
M_{1}\sqrt{T}\\}\cap\neg\mathsf{A}_{2}}\mathbb{E}_{2}\left[\mathbf{1}_{\mathsf{A}_{1}}\mid\mathcal{F}_{2}\right]\right].$

As $\mathsf{A}_{1}$ is an increasing event with respect to the boundary data,
due to stochastic monotonicity (Proposition 2.6), increasing the boundaries
will only increase the conditional probability. Thus, to get an upper bound we
may assume $L_{1}(1)=L_{1}(2T-1)=M_{3}\sqrt{T}$ and
$L_{2}(i)=(M_{0}+M_{1})\sqrt{T}$ for all $i\in\llbracket 2,2T\rrbracket$. From
the definition of the Gibbs measure, almost surely we have

$\displaystyle\mathbf{1}_{\\{L_{1}(1)\leq
M_{1}\sqrt{T}\\}\cap\neg\mathsf{A}_{2}}\mathbb{E}_{2}\left[\mathbf{1}_{\mathsf{A}_{1}}\mid\mathcal{F}_{2}\right]\leq\frac{1}{\mathbb{E}[\Delta]}\mathbb{E}\left[\Delta\cdot\mathbf{1}_{\mathsf{A}_{1}}\right],$

where on the right-hand side, $\mathsf{A}_{1}$ is defined as the event
$\\{\sup_{i\in\llbracket 1,2T-1\rrbracket}X(i)\geq M_{0}\sqrt{T}\\}$ and

$\Delta=\exp\left(-\sum_{i=1}^{T-1}\left(e^{-(M_{0}+1)\sqrt{T}-X_{1}(2i-1)}+e^{-(M_{0}+1)\sqrt{T}-X_{1}(2i+1)}\right)\right).$

As $\Delta\leq 1$, by (5.3),
$\mathbb{E}\left[\Delta\\!\cdot\\!\mathbf{1}_{\mathsf{A}_{1}}\right]\leq\mathbb{E}[\mathbf{1}_{\mathsf{A}_{1}}]\leq\varepsilon$.
By (5.3) we have $\mathbb{E}[\Delta]\geq(1-\varepsilon)\cdot
e^{-2(T-1)e^{-\sqrt{T}}}\geq\beta$ for some absolute constant $\beta>0$. Thus,
$\mathbb{P}_{2}(\mathsf{A}_{1})\leq(3+\beta^{-1})\varepsilon$. Similarly one
can show

$\displaystyle\mathbb{P}_{2}\left(\inf_{i\in\llbracket
2,2T\rrbracket}L_{2}(i)\leq-(M_{3}+M_{0})\sqrt{T}\right)\leq(3+\beta^{-1})\varepsilon,$
$\displaystyle\mathbb{P}_{2}\left(\inf_{i\in\llbracket
1,2T-1\rrbracket}L_{1}(i)\leq-(M_{0}+M_{1})\sqrt{T}\right)\leq 2\varepsilon.$

Thus adjusting the constants we can find $\widetilde{M}$ such that

$\displaystyle\mathbb{P}_{2}\bigg{(}\sup_{i\in\llbracket
1,2T-1\rrbracket}|L_{1}(i)|+\sup_{j\in\llbracket
2,2T\rrbracket}|L_{2}(j)|\geq(M-1)\sqrt{T}\bigg{)}\leq\varepsilon/3.$

Finally via Lemma 4.4 we know $L_{2}(1)-L_{2}(2)\sim G_{\alpha_{2}+\theta,1}$.
Thus, by a union bound, for all $T$ large enough we have
$\mathbb{P}_{2}(|L_{2}(1)|\geq
M\sqrt{T})\leq\varepsilon/3+\mathbb{P}_{2}(|L_{2}(1)-L_{2}(2)|\geq\sqrt{T})\leq
2\varepsilon/3$. By another union bound, we arrive at (5.6). ∎

Recall the normalizing constant $V_{p}^{T}(\vec{y},\vec{z})$ from (2.8). One
can easily obtain a lower bound for this normalizing constant as a consequence
of the Lemma 5.4.

######  Corollary 5.5.

Fix any $r>0$ and for each $N>0$ set $T=\lfloor rN^{2/3}\rfloor$. Fix any
$p\in\\{1,2\\}$ and set $\alpha=\alpha_{p}$ according to (3.11). Recall
$V_{p}^{T}(\vec{y},\vec{z})$ from (2.8). There exists
$Q_{0}=Q_{0}(r)>0,N_{0}=N_{0}(r)>0$ such that for all $Q\geq Q_{0}$ and $N\geq
N_{0}$, $V_{p}^{T}(\vec{y},\vec{z})\geq\tfrac{1}{2}$ for all
$\vec{z}\in\mathbb{R}^{T}$ with $z_{i}\leq QN^{1/3}$ for $i\in\llbracket
1,T\rrbracket$ and $\vec{y}\in\mathbb{R}^{p}$ with $y_{i}\geq(2Q-1)N^{1/3}$
for $i\in\llbracket 1,p\rrbracket$. Here we assume $L_{p}(2T+1):=\infty$.

###### Proof.

Consider the event

$\mathsf{A}:=\left\\{\inf_{j\in\llbracket
1,T\rrbracket}L_{p}(2j-1)\geq(Q+1)N^{1/3}\right\\}.$

Observe that

$\displaystyle V_{p}^{T}(\vec{y},\vec{z})$
$\displaystyle\geq\mathbb{E}_{\alpha_{p}}^{\vec{y},(-\infty)^{T};p,T}\left[\mathbf{1}_{\mathsf{A}}\prod_{j=1}^{T}W(z_{j};L_{p}(2j+1),L_{p}(2j-1))\right]$
$\displaystyle\geq\exp(-2Te^{-N^{1/3}})\cdot\mathbb{P}_{\alpha_{p}}^{\vec{y},(-\infty)^{T};p,T}\left(\mathsf{A}\right).$

Taking $N$ large enough ensures $\exp(-2Te^{-N^{1/3}})\geq 1/\sqrt{2}$. Since
$\mathsf{A}$ is an increasing event with respect to the boundary data,
applying stochastic monotonicity (Proposition 2.6) and translation invariance
(Lemma 2.1 (a)) of $\mathpzc{HSLG}$ Gibbs measures we have

$\displaystyle\mathbb{P}_{\alpha_{p}}^{\vec{y},(-\infty)^{T};p,T}\left(\mathsf{A}\right)=\mathbb{P}_{\alpha_{p}}^{\vec{x},(-\infty)^{T};p,T}\left(\inf_{j\in\llbracket
1,T\rrbracket}L_{p}(2j-1)\geq-(Q-2)N^{1/3}\right)$

where $\vec{x}=0$ if $p=1$ and $\vec{x}=(0,-\sqrt{T})$ if $p=2$. Appealing to
Lemma 5.4 we may choose $Q$ large enough so that the above probability is at
least $1/\sqrt{2}$. ∎

###  5.2. Proof of Proposition 5.1

For clarity we divide the proof into three steps.

Step 1. In this step, we give a roadmap of the proof of (5.1) leaving the
technical details to later steps. Fix $r,\varepsilon,\delta>0$ and
$p\in\\{1,2\\}$. Fix $N\geq 3$ large enough so that $T=8\lfloor
rN^{2/3}\rfloor\geq 24$. Set $\alpha=\alpha_{p}$ according to (3.11) and
consider the $\mathpzc{HSLG}$ line ensemble $\mathcal{L}^{N}$ from Definition
2.7 with parameters $(\alpha,\theta)$. Consider the modulus of continuity
event

$\displaystyle\mathsf{MC}_{\delta}:=\big{\\{}\omega_{\delta}^{N}(\mathcal{L}_{1}^{N},\llbracket
1,T/4-1\rrbracket)\geq\gamma N^{1/3}\big{\\}}.$

By Theorem 3.10, there exists $V=V(\varepsilon)>0$ such that

$\displaystyle\mathbb{P}(\mathsf{A}_{1})\geq 1-\varepsilon,\,\,\mbox{ where
}\,\,\mathsf{A}_{1}:=\left\\{N^{-1/3}|\mathcal{L}_{1}^{N}(1)|+N^{-1/3}|\mathcal{L}_{2}^{N}(2)|\leq
V\right\\}.$ (5.8)

By Proposition 3.4, there exists $M_{1}(\varepsilon)>0$ such that for all
large enough $N$

$\displaystyle\mathbb{P}\left(\mathcal{L}_{1}^{N}(2T-1)\geq
M_{1}N^{1/3}\right)\leq\varepsilon.$ (5.9)

We claim that there exists $M_{2}(r,\varepsilon)>0$ such that for all large
enough $N$

$\displaystyle\mathbb{P}\left(\mathcal{L}_{p}^{N}(2T+p-2)\leq-
M_{2}N^{1/3}\right)\leq\varepsilon.$ (5.10)

We shall prove (5.10) in Step 2. Let us assume it for now. Set
$M=\max\\{M_{1},M_{2},4\\}$ and consider the events

$\displaystyle\mathsf{B}_{1}$
$\displaystyle:=\big{\\{}|\mathcal{L}_{1}^{N}(2T-1)|\leq 2MN^{1/3}\big{\\}},$
$\displaystyle\mathsf{B}_{2}$
$\displaystyle:=\Big{\\{}\mathcal{L}_{2}^{N}(2T)\geq-
MN^{\frac{1}{3}},\,\,\mathcal{L}_{1}^{N}(2T-1)\leq
MN^{\frac{1}{3}},\,\,\mathcal{L}_{1}^{N}(2T-1)\geq\mathcal{L}_{1}^{N}(2T)-(\log
N)^{7/6}\Big{\\}},$

For each $\beta>0$ we define

$\displaystyle\mathsf{C}(p,\beta)$
$\displaystyle:=\Big{\\{}V_{p}^{T}\left((\mathcal{L}_{j}^{N}(2T+j-2))_{j\in\llbracket
1,p\rrbracket};(\mathcal{L}_{p+1}^{N}(2k))_{k=1}^{T}\right)\geq\beta\Big{\\}},$
(5.11)

where $V_{p}^{T}(\cdot,\cdot)$ is defined in (2.8). We now claim that there
exists $\beta(r,\varepsilon)>0$ such that

$\displaystyle\mathbb{P}\left(\neg\mathsf{C}(p,\beta)\right)\leq\varepsilon.$
(5.12)

We work with this choice of $\beta$ for the rest of this step. We postpone the
proof of (5.12) to Step 3. Let us now complete the proof of Proposition 5.1
assuming it. Consider the following $\sigma$-algebra:

$\displaystyle\mathcal{F}_{p,k}:=\sigma\left((\mathcal{L}_{i}^{N}\llbracket
1,2N-2i+2\rrbracket)_{i\geq p+1},(\mathcal{L}_{i}^{N}(j))_{j\geq
2k+i-2,i\in\llbracket 1,p\rrbracket}\right).$ (5.13)

Clearly $\mathsf{B}_{p}\cap\mathsf{C}(p,\beta)$ is measurable with respect to
$\mathcal{F}_{p,T}$. By union bound and tower property of conditional
expectation we have

$\displaystyle\mathbb{P}(\mathsf{MC}_{\delta})$
$\displaystyle\leq\mathbb{P}(\neg\mathsf{A}_{1})+\mathbb{P}\left(\neg\mathsf{B}_{p}\right)+\mathbb{P}\left(\neg\mathsf{C}(p,\beta)\right)$
(5.14) $\displaystyle\hskip
56.9055pt+\mathbb{E}\left[\mathbf{1}_{\mathsf{B}_{p}\cap\mathsf{C}(p,\beta)}\mathbb{E}\left[\mathbf{1}_{\mathsf{A}_{1}\cap\mathsf{MC}_{\delta}}\mid\mathcal{F}_{p,T}\right]\right].$

We bound the four terms on the right-hand side of the above equation
separately.

$\mathsf{A}_{1}$ event: We have
$\mathbb{P}(\neg\mathsf{A}_{1})\leq\varepsilon$ due to (5.8).

$\mathsf{B}_{p}$ event: Note that for large enough $N$,
$\mathsf{B}_{2}\subset\mathsf{B}_{1}$. Combining (5.9), (5.10), and Theorem
3.1 (with $\rho\mapsto\frac{1}{2},M\mapsto M$), by a union bound we see that
for all large enough $N$,

$\displaystyle\mathbb{P}(\neg\mathsf{B}_{p})$
$\displaystyle\leq\mathbb{P}(\neg\mathsf{B}_{2})$
$\displaystyle\leq\mathbb{P}(\mathcal{L}_{2}^{N}(2T)\leq-
MN^{1/3})+\mathbb{P}(\mathcal{L}_{1}^{N}(2T-1)\geq MN^{1/3})$
$\displaystyle\hskip
28.45274pt+\mathbb{P}\left(\mathcal{L}_{1}^{N}(2T-1)\leq\mathcal{L}_{1}^{N}(2T)-(\log
N)^{7/6}\right)$ $\displaystyle\leq 2\varepsilon+2^{-N}\leq 3\varepsilon.$

$\mathsf{C}(p,\beta)$ event: We have
$\mathbb{P}(\neg\mathsf{C}(p,\beta))\leq\varepsilon$ due to (5.12).

Conditional probability: By Theorem 1.3 and (2.7) we have

$\displaystyle\mathbb{E}\left[\mathbf{1}_{\mathsf{A}_{1}\cap\mathsf{MC}_{\delta}}\mid\mathcal{F}_{p,T}\right]=\frac{\mathbb{E}_{\alpha_{p}}^{\vec{y},(-\infty)^{2T};p,T}\left[V_{p}^{T}\left(\vec{y};(\mathcal{L}_{p+1}^{N}(2i))_{i=1}^{T}\right)\cdot\mathbf{1}_{\mathsf{A}_{1}\cap\mathsf{MC}_{\delta}}\right]}{V_{p}^{T}\left(\vec{y};(\mathcal{L}_{p+1}^{N}(2i))_{i=1}^{T}\right)}$
(5.15)

where $\vec{y}:=(\mathcal{L}_{j}^{N}(2T+j-2))_{j\in\llbracket 1,p\rrbracket}$
and $V_{p}^{T}(\cdot;\cdot)$ is defined in (2.8). From definition we have
$V_{p}^{T}\left(\vec{y};(\mathcal{L}_{p+1}^{N}(2i))_{i=1}^{T}\right)\in[0,1]$.
On $\mathsf{C}(p,\beta)$ we have

$\displaystyle\mathbf{1}_{\mathsf{C}(p,\beta)}\cdot\mbox{r.h.s.~{}\eqref{e45}}\leq\mathbf{1}_{\mathsf{C}(p,\beta)}\cdot\beta^{-1}\cdot\mathbb{P}_{\alpha_{p}}^{\vec{y},(-\infty)^{2T};p,T}\left(\mathsf{A}_{1}\cap\mathsf{MC}_{\delta}\right).$

Observe that the event $\mathsf{B}_{p}$ ensures $\vec{y}\in I_{p,M}$ where the
set $I_{p,M}$ is defined in the statement of Proposition 5.2. We can thus
apply Proposition 5.2 with $\varepsilon\mapsto\beta\cdot\varepsilon$, to get a
$\delta>0$ such that

$\displaystyle\mathbf{1}_{\mathsf{B}_{p}}\cdot\mathbb{P}_{\alpha_{p}}^{\vec{y},(-\infty)^{2T};p,T}\left(\mathsf{A}_{1}\cap\mathsf{MC}_{\delta}\right)\leq\mathbf{1}_{\mathsf{B}_{p}}\cdot\varepsilon,$

for all large enough $N$. Thus overall we have

$\displaystyle\mathbb{E}\big{[}\mathbf{1}_{\mathsf{B}_{p}\cap\mathsf{C}(p,\beta)}\mathbb{E}\left[\mathbf{1}_{\mathsf{A}_{1}\cap\mathsf{MC}_{\delta}}\mid\mathcal{F}_{p,T}\right]\big{]}\leq\varepsilon.$

Plugging the above four estimates into r.h.s. (5.14) and taking limsup
$N\to\infty$, then $\delta\downarrow 0$, yields

$\displaystyle\limsup_{\delta\downarrow
0}\limsup_{N\to\infty}\mathbb{P}(\mathsf{MC}_{\delta})\leq 6\varepsilon.$

As $\varepsilon$ is arbitrary, we thus have (5.2), completing the proof.

Step 2. In this step we prove (5.10). We write $\mathbb{P}_{\alpha_{p}}$
instead of $\mathbb{P}$ to stress the fact that the $\mathpzc{HSLG}$ line
ensemble has boundary parameter $\alpha_{p}$, defined in (3.11). We claim that
there exists $M_{2}(r,\varepsilon)$ such that for all large enough $N$

$\displaystyle\mathbb{P}_{\alpha_{p}}\left(\mathsf{Fall}_{p}^{(M_{2})}\right)\leq\tfrac{\varepsilon}{4},\quad\mathsf{Fall}_{p}^{(M_{2})}:=\left\\{\inf_{j\in\llbracket
1,4T+4\rrbracket,i\in\llbracket 1,p\rrbracket}\mathcal{L}_{i}^{N}(j)\leq-
M_{2}N^{1/3}\right\\}.$ (5.16)

Note that as $\\{\mathcal{L}_{p}^{N}(2T+p-2)\leq-
M_{2}N^{1/3}\\}\subset\mathsf{Fall}_{p}^{(M_{2})}$, (5.16) implies (5.10). To
show (5.16), we first define a few more events. For each $R\geq 32r+1$ we
define

$\displaystyle\mathsf{B}_{i}^{(R,j)}$
$\displaystyle:=\left\\{\mathcal{L}_{i}^{N}(2j+i-2)\geq-R^{2}N^{1/3}\right\\},\quad\widetilde{\mathsf{B}}_{i}^{(R,j)}:=\mathsf{B}_{i}^{(R,j)}\cap\bigcup_{k\in\llbracket
j+1,RN^{2/3}\rrbracket}\neg\mathsf{B}_{i}^{(R,k)},$
$\displaystyle\mathsf{B}_{i}^{(R)}$ $\displaystyle:=\bigcup_{j\in\llbracket
4T+4,RN^{2/3}\rrbracket}\mathsf{B}_{i}^{(R,j)}=\bigsqcup_{j\in\llbracket
4T+4,RN^{2/3}\rrbracket}\widetilde{\mathsf{B}}_{i}^{(R,j)}=\bigg{\\{}\sup_{j\in\llbracket
4T+4,RN^{2/3}\rrbracket}\mathcal{L}_{i}^{N}(2j+i-2)\geq-R^{2}N^{1/3}\bigg{\\}},$
$\displaystyle\mathsf{Dif}^{(R)}$
$\displaystyle:=\left\\{\mathcal{L}_{1}^{N}(2j-1)\geq\mathcal{L}_{2}^{N}(2j)+(\log
N)^{2}\mbox{ for all }j\in\llbracket 1,RN^{2/3}\rrbracket\right\\}.$

By Theorem 3.1, Theorem 3.3, and Proposition 3.4, we can find a
$R=R(r,\varepsilon)\geq 1$ such that for all large enough $N$, and for
$v\in\\{1,2\\}$

$\displaystyle\mathbb{P}_{\alpha_{v}}\left(\neg\mathsf{B}_{1}^{(R)}\right)+\mathbb{P}_{\alpha_{v}}\left(\neg\mathsf{B}_{2}^{(R)}\right)+\mathbb{P}_{\alpha_{v}}\left(\neg\mathsf{Dif}^{(R)}\right)\leq\tfrac{\varepsilon}{8}.$
(5.17)

We fix this choice of $R$. Observe that for large enough $N$, we have

$\displaystyle\widetilde{\mathsf{B}}_{2}^{(R,i)}\cap\mathsf{Dif}^{(R)}\subset\widetilde{\mathsf{B}}_{2}^{(R,i)}\cap\mathsf{B}_{1}^{(2R,i)},$

uniformly for all $i\in\llbracket 4T+4,RN^{2/3}\rrbracket$. For $p=2$, by the
union bound and the tower property of conditional expectation, in view of
(5.17), we have

$\displaystyle\mathbb{P}_{\alpha_{2}}\left(\mathsf{Fall}_{2}^{(M_{2})}\right)$
$\displaystyle\leq\mathbb{P}_{\alpha_{2}}\left(\neg\mathsf{B}_{2}^{(R)}\right)+\mathbb{P}\left(\neg\mathsf{Dif}^{(R)}\right)+\sum_{j\in\llbracket
4T+4,RN^{2/3}\rrbracket}\mathbb{P}_{\alpha_{2}}\left(\widetilde{\mathsf{B}}_{2}^{(R,j)}\cap\mathsf{B}_{1}^{(2R,j)}\cap\mathsf{Fall}_{2}^{(M_{2})}\right)$
(5.18) $\displaystyle\leq\tfrac{\varepsilon}{8}+\sum_{j\in\llbracket
4T+4,RN^{2/3}\rrbracket}\mathbb{E}\left[\mathbf{1}_{\widetilde{\mathsf{B}}_{2}^{(R,j)}\cap\mathsf{B}_{1}^{(2R,j)}}\mathbb{E}_{\alpha_{2}}\left[\mathbf{1}_{\mathsf{Fall}_{2}^{(M_{2})}}\mid\mathcal{F}_{2,j}\right]\right],$

where $\mathcal{F}_{p,k}$ is defined in (5.13). For $p=1$, applying union
bound and using (5.17) we have

$\displaystyle\mathbb{P}_{\alpha_{1}}\left(\mathsf{Fall}_{1}^{(M_{2})}\right)$
$\displaystyle\leq\mathbb{P}_{\alpha_{1}}\left(\neg\mathsf{B}_{1}^{(R)}\right)+\sum_{j\in\llbracket
4T,RN^{2/3}\rrbracket}\mathbb{P}_{\alpha_{1}}\left(\widetilde{\mathsf{B}}_{1}^{(R,j)}\cap\mathsf{Fall}_{1}^{(M_{2})}\right)$
(5.19) $\displaystyle\leq\tfrac{\varepsilon}{8}+\sum_{j\in\llbracket
4T+4,RN^{2/3}\rrbracket}\mathbb{E}\left[\mathbf{1}_{\widetilde{\mathsf{B}}_{1}^{(R,j)}}\mathbb{E}_{\alpha_{1}}\left[\mathbf{1}_{\mathsf{Fall}_{1}^{(M_{2})}}\mid\mathcal{F}_{1,j}\right]\right].$

We now proceed to control the conditional expectation
$\mathbb{P}_{\alpha_{p}}\left(\mathsf{Fall}_{p}^{(M_{2})}\mid\mathcal{F}_{p,j}\right)$
separately for $p=1$ and $p=2$. Applying the Gibbs property (Theorem 1.3), we
have

$\displaystyle\mathbf{1}_{\widetilde{\mathsf{B}}_{2}^{(R,j)}\cap\mathsf{B}_{1}^{(2R,j)}}\cdot\mathbb{E}_{\alpha_{2}}\left[\mathsf{Fall}_{2}^{(M_{2})}\mid\mathcal{F}_{2,j}\right]$
$\displaystyle=\mathbf{1}_{\widetilde{\mathsf{B}}_{2}^{(R,j)}\cap\mathsf{B}_{1}^{(2R,j)}}\cdot\mathbb{P}_{\alpha_{2}}^{\vec{y},\vec{z};2,j}\bigg{(}\mathsf{Fall}_{2}^{(M_{2})}\bigg{)}$
$\displaystyle\leq\mathbf{1}_{\widetilde{\mathsf{B}}_{2}^{(R,j)}\cap\mathsf{B}_{1}^{(2R,j)}}\cdot\mathbb{P}_{\alpha_{2}}^{(0,-\sqrt{j}),(-\infty)^{j};2,j}\bigg{(}\mathsf{Fall}_{2}^{(M_{2}-4R^{2})}\bigg{)}.$

Here $\vec{y}=(\mathcal{L}_{1}^{N}(2j-1),\mathcal{L}_{2}^{N}(2j))$ and
$\vec{z}=(\mathcal{L}_{3}^{N}(2m))_{m=1}^{j}$. Let us briefly explain the
above inequality. Note that on
$\widetilde{\mathsf{B}}_{2}^{(R,j)}\cap\mathsf{B}_{1}^{(2R,j)}$ we have
$y_{i}\geq(-4R^{2}N^{1/3}-(i-1)\sqrt{j})$ for $i=1,2$. Furthermore
$\mathsf{Fall}_{2}^{(M_{2})}$ is an event which decreases with respect to
boundary data. Thus to obtain an upper bound, by stochastic monotonicity
(Proposition 2.6), we may take the boundary data from $(y_{1},y_{2})$ to
$(-4R^{2}N^{1/3},-4R^{2}N^{1/3}-\sqrt{j})$ and $\vec{z}$ to $(-\infty)^{j}$.
The above inequality then follows by translation invariance (see Lemma 2.1
(a)). Similar applications of the Gibbs property and stochastic monotonicity
yield that on $\widetilde{\mathsf{B}}_{1}^{(R,j)}$ we have

$\displaystyle\mathbb{E}_{\alpha_{1}}\left[\mathbf{1}_{\mathsf{Fall}_{1}^{(M_{2})}}\mid\mathcal{F}_{2,j}\right]\leq\mathbb{P}_{\alpha_{1}}^{0,(-\infty)^{j};1,j}\left(\mathsf{Fall}_{1}^{(M_{2}-4R^{2})}\right).$

We now claim that one can choose $M_{2}(r,\varepsilon)>0$ large enough such
that for all $j\in\llbracket 4T+4,RN^{2/3}\rrbracket$,

$\displaystyle\mathbb{P}_{\alpha_{p}}^{\vec{x},(-\infty)^{j};p,j}\left(\mathsf{Fall}_{p}^{(M_{2}-4R^{2})}\right)\leq\tfrac{\varepsilon}{8},$
(5.20)

where $\vec{x}:=0$ (if $p=1$) or $\vec{x}:=(0,-\sqrt{j})$ (if $p=2$). Plugging
the above bound back in (5.19) and (5.18) and using the fact that
$\\{\widetilde{\mathsf{B}}_{p}^{(R,j)}\\}_{j\in\llbracket
4T+4,RN^{2/3}\rrbracket}$ is a disjoint collection of events we arrive at the
bound in (5.16). Thus we are left to verify (5.20) in this step. Observe that

$\displaystyle\mathbb{P}_{\alpha_{p}}^{\vec{x},(-\infty)^{j};p,j}\left(\mathsf{Fall}_{p}^{(M_{2}-4R^{2})}\right)\leq\mathbb{P}_{\alpha_{p}}^{\vec{x},(-\infty)^{j};p,j}\left(\inf_{k\in\llbracket
1,2j+i-2\rrbracket,i\in\llbracket
1,p\rrbracket}L_{i}(k)\leq-(M_{2}-4R^{2})N^{1/3}\right)$

By Lemma 5.4, one can choose $M_{2}$ large enough such that the above
expression is bounded above by $\varepsilon/8$ for all $j\in\llbracket
4T,RN^{2/3}\rrbracket$. This proves (5.20) completing our work for this step.

Step 3. In this step we prove (5.12). For each $Q>0$ consider the event

$\displaystyle\mathsf{D}_{Q}$ $\displaystyle:=\left\\{\sup_{i\in\llbracket
1,4T+4\rrbracket}\mathcal{L}_{p+1}^{N}(i)\leq
QN^{1/3},\,\,\inf_{j\in\llbracket
1,p\rrbracket}\mathcal{L}_{j}^{N}(4T+j+2)\geq-QN^{1/3}+\sqrt{2T+1}\right\\}.$
(5.21)

By Theorem 3.1, Proposition 3.8, and (5.16) there exists $Q(r,\varepsilon)>0$
large enough such that
$\mathbb{P}(\neg\mathsf{D}_{p,Q})\leq\frac{\varepsilon}{2}$. Consider
$\mathcal{F}_{p,2T+2}$ from (5.13). Recall the event ${\mathsf{C}}(p,\beta)$
from (5.11). By union bound and the tower-property of the expectation, we have

$\displaystyle\mathbb{P}(\neg\mathsf{C}(p,\beta))$
$\displaystyle\leq\mathbb{P}(\neg\mathsf{C}(p,\beta)\cap\mathsf{D}_{Q})+\tfrac{\varepsilon}{2}=\mathbb{E}\Big{[}\mathbf{1}_{\mathsf{D}_{Q}}\mathbb{E}\big{[}\mathbf{1}_{\mathsf{C}(p,\beta)}\mid\mathcal{F}_{p,2T+2}\big{]}\Big{]}+\tfrac{\varepsilon}{2}.$
(5.22)

Applying the Gibbs property and (2.7) we have

$\displaystyle\mathbb{E}[\mathbf{1}_{\neg\mathsf{C}(p,\beta)}\mid\mathcal{F}_{p,2T+2}]$
$\displaystyle=\mathbb{P}_{\alpha_{p}}^{\vec{y};\vec{z};p,2T+2}\big{(}\neg\mathsf{C}(p,\beta)\big{)}$

with $\vec{y}=(y_{1},\ldots,y_{p})$ and $y_{j}=\mathcal{L}_{j}^{N}(4T+j+2)$
for $j\in\llbracket 1,p\rrbracket$, and
$\vec{z}=(\mathcal{L}_{p+1}^{N}(2k))_{k=1}^{2T+2}$. Set
$\vec{x}=(-QN^{1/3}+\sqrt{2T+1})^{p}$. We claim that there exists
$Q_{0}(r,\varepsilon)>0$, $N_{0}(r,\varepsilon)>0$ and
$\beta(r,\varepsilon)>0$, such that for all $N\geq N_{0}$, $Q\geq Q_{0}$,
$y_{i}\geq x_{i}$ and $\vec{z}\in\mathbb{R}^{2T+2}$ with $\sup_{i\in\llbracket
1,2T+2\rrbracket}z_{i}\leq QN^{1/3}$ we have

$\displaystyle\mathbb{P}_{\alpha_{p}}^{\vec{y};\vec{z};p,2T+2}\left(\neg\widetilde{\mathsf{C}}(p,\beta)\right)\leq\tfrac{\varepsilon}{2},\mbox{
where
}\widetilde{\mathsf{C}}(p,\beta):=\left\\{\mathcal{V}_{p}\geq\beta\right\\},$
(5.23)

where we set (see (2.8))

$\mathcal{V}_{p}:=V_{p}^{T}\big{(}(L_{i}(2T+i-2))_{i\in\llbracket
1,p\rrbracket},(z_{1},\ldots,z_{T})\big{)}.$ (5.24)

Clearly in view of the definition of $\mathsf{D}_{Q}$ from (5.21), the above
claim shows that r.h.s. (5.22) is at most $\varepsilon$ (5.12). Thus, to
complete our proof it suffices to check (5.23). Towards this end, we first
claim that for all $\vec{y}\in\mathbb{R}^{p},\vec{z}\in\mathbb{R}^{2T+2}$

$\displaystyle\mathbb{P}_{\alpha_{p}}^{\vec{y};\vec{z};p,2T+2}\left(\neg\widetilde{\mathsf{C}}(p,\beta)\right)=\frac{\mathbb{E}_{\alpha_{p}}^{\vec{y};\vec{w};p,2T+2}\left[\mathbf{1}_{\widetilde{\mathsf{C}}(p,\beta)}\cdot\mathcal{R}_{p}\cdot\mathcal{V}_{p}\right]}{\mathbb{E}_{\alpha_{p}}^{\vec{y};\vec{w};p,2T+2}\left[\mathcal{R}_{p}\cdot\mathcal{V}_{p}\right]},\quad\mbox{
where }\,\,\mathcal{R}_{p}:=e^{-e^{z_{T}-L_{2}(2T+1)}\mathbf{1}_{p=2}},$
(5.25)

and where $\vec{w}\in[-\infty,\infty)^{2T+2}$ is defined by setting
$w_{i}=-\infty$ for $i\leq T$ and $w_{i}=z_{i}$ for $i>T$. We postpone the
proof of (5.25) to the next step.

Assuming (5.25), to prove (5.23), we provide upper and lower bounds for the
numerator and denominator of r.h.s. (5.25) respectively. Consider the events

$\displaystyle\mathsf{R}_{1}$ $\displaystyle:=\left\\{L_{1}(2T-1)\geq
2QN^{1/3}\right\\},$ $\displaystyle\mathsf{R}_{2}$
$\displaystyle:=\left\\{L_{2}(2T)\geq 2QN^{1/3},L_{2}(2T+1)\geq
2QN^{1/3},L_{1}(2T-1)\geq(2Q-1)N^{1/3}\right\\}$

Note that

$\displaystyle\mathbb{E}_{\alpha_{p}}^{\vec{y},\vec{w};p,2T+2}\left[\mathcal{R}_{p}\cdot\mathcal{V}_{p}\right]$
$\displaystyle\geq\mathbb{E}_{\alpha_{p}}^{\vec{y},\vec{w};p,2T+2}\left[\mathbf{1}_{\mathsf{R}_{p}}\cdot\mathcal{R}_{p}\cdot\mathcal{V}_{p}\right]$
$\displaystyle\geq\tfrac{1}{2}\exp(-e^{-QN^{1/3}})\cdot\mathbb{P}_{\alpha_{p}}^{\vec{y},\vec{w};p,2T+2}\big{(}\mathsf{R}_{p}\big{)}$
$\displaystyle\geq\tfrac{1}{2}\exp(-e^{-QN^{1/3}})\cdot\mathbb{P}_{\alpha_{p}}^{\vec{x},(-\infty)^{2T+2};p,2T+2}\big{(}\mathsf{R}_{p}\big{)}.$
(5.26)

where the penultimate inequality follows from the definition of
$\mathcal{R}_{p}$ and Corollary 5.5 and the final inequality follows via
stochastic monotonicity (Proposition 2.6) as $\mathsf{R}_{p}$ is an increasing
event with respect to the boundary data (recall $y_{i}\geq x_{i}$). To lower
bound the above expression, we proceed into two cases depending on the value
of $p$.

Case 1. $p=1$. Note that $\mathsf{R}_{1}\supset\mathsf{RP}_{1,Q}$ event
defined in (4.2). By Proposition 4.1, we have
$\mathbb{P}_{\alpha_{1}}^{\vec{x},(-\infty)^{2T+2};1,2T+2}(\mathsf{R}_{1})\geq\mathbb{P}_{\alpha_{1}}^{-QN^{1/3},(-\infty)^{2T+2};1,2T+2}(\mathsf{RP}_{1,Q})\geq\phi_{1}>0$
for some $\phi_{1}$ free of $N$.

Case 2. $p=2$. Let $\vec{u}:=(-QN^{1/3}+\sqrt{2T+2},-QN^{1/3})$. Let us use
the shorthand notation $\mathbb{P}_{2}^{\gamma_{1},\gamma_{2}}$ for
$\mathbb{P}_{\alpha_{2}}^{(\gamma_{1},\gamma_{2}),(-\infty)^{2T+2};2,2T+2}$.
Note that by stochastic monotonicity and union bound we have

$\displaystyle\mathbb{P}_{2}^{\vec{x}}(\mathsf{R}_{2})\geq\mathbb{P}_{2}^{\vec{u}}(\mathsf{R}_{2})$
$\displaystyle\geq\mathbb{P}_{2}^{\vec{u}}\Big{(}\\{L_{2}(2T)\geq
2QN^{1/3}\\}\cap\\{L_{2}(2T+1)\geq 2QN^{1/3}\\}\Big{)}$ (5.27)
$\displaystyle\hskip 56.9055pt-\mathbb{P}_{2}^{\vec{u}}\big{(}L_{1}(2T-1)\leq
L_{2}(2T)-N^{1/3}\big{)}.$

Note that $\mathsf{RP}_{2,Q}\subset\\{L_{2}(2T)\geq
2QN^{1/3}\\}\cap\\{L_{2}(2T+1)\geq 2QN^{1/3}\\}$ (with $T$ replaced by $T+1$
in (4.2)). Applying stochastic monotonicity (Proposition 2.6) and Proposition
4.1 with $p\mapsto 2$ and $T\mapsto T+1$, we see that the first term in the
above equation can be bounded as

$\displaystyle\mathbb{P}_{2}^{\vec{u}}\Big{(}\\{L_{2}(2T)\geq
2QN^{1/3}\\}\cap\\{L_{2}(2T+1)\geq
2QN^{1/3}\\}\Big{)}\geq\mathbb{P}_{2}^{(-QN^{1/3},-(Q+1)N^{1/3})}\big{(}\mathsf{RP}_{2,Q}\big{)}\geq\phi_{2},$
(5.28)

for some $\phi_{2}>0$ free of $N$. As for the second term in r.h.s. (5.27), by
translation invariance (Lemma 2.1 (a)) we have

$\displaystyle\mathbb{P}_{2}^{\vec{u}}\big{(}L_{1}(2T-1)\leq
L_{2}(2T)-N^{1/3}\big{)}$
$\displaystyle=\mathbb{P}_{2}^{(0,-\sqrt{2T+1})}\big{(}L_{1}(2T-1)\leq
L_{2}(2T)-N^{1/3}\big{)}$
$\displaystyle=\frac{{\mathbb{E}}_{\operatorname{PRW}}^{2T+2;(0,\sqrt{2T+1})}\left[W_{\operatorname{sc}}\mathbf{1}_{S_{1}(T-1)\leq
S_{2}(T-1)-N^{1/3}}\right]}{{{\mathbb{E}}_{\operatorname{PRW}}^{2T+2;(0,\sqrt{2T+1})}[W_{\operatorname{sc}}]}},$

where the last equality follows from Lemma 4.4 (recall the PRW law from
Definition 1.7 and $W_{\operatorname{sc}}$ from (1.15)). Now by Corollary
(4.12),
${\mathbb{E}}_{\operatorname{PRW}}^{2T+2;(0,\sqrt{2T+1})}[W_{\operatorname{sc}}]\geq\mathrm{C}/\sqrt{2T+2}$
for some absolute constant $\mathrm{C}>0$. However on the event
$\\{S_{1}(T-1)\leq S_{2}(T-1)-N^{1/3}\\}$,
$W_{\operatorname{sc}}\leq\exp(-e^{N^{1/3}})$. Thus,

$\mathbb{P}_{2}^{\vec{u}}\big{(}L_{1}(2T-1)\leq L_{2}(2T)-N^{1/3}\big{)}\to 0$

as $N\to\infty$. Hence, inserting (5.28) back in r.h.s. (5.27), we see that
for all large enough $N$,

$\mathbb{P}_{2}^{\vec{x}}(\mathsf{R}_{2})\geq\phi_{2}-\mathbb{P}_{2}^{\vec{u}}\big{(}L_{1}(2T-1)\leq
L_{2}(2T)-N^{1/3}\big{)}\geq\frac{1}{2}\phi_{2}.$

Summarizing the above two cases, for all large enough $N$, (5.26) is lower
bounded by some $\phi>0$ free of $N$. For the numerator in r.h.s. (5.25)
observe that as $\mathcal{R}_{p}\leq 1$, by definition of the event
$\widetilde{\mathsf{C}}(p,\beta)$, we have
$\mathbf{1}_{\neg\widetilde{\mathsf{C}}(p,\beta)}\cdot\mathcal{R}_{p}\cdot\mathcal{V}_{p}\leq\beta$.
Let us now choose $\beta=\phi\varepsilon$. Plugging these bounds back in
r.h.s. (5.25) yields (5.23).

Step 5. All that remains is to prove (5.25). We will do this for the $p=2$
case. The $p=1$ case is done analogously. Fix any $\vec{y}\in\mathbb{R}^{2}$,
$\vec{z}\in\mathbb{R}^{2T+2}$ and define $\vec{w}\in[-\infty,\infty)^{2T+2}$
such that $w_{i}=-\infty$ for $i\leq T$ and $w_{i}=z_{i}$ for $i>T$. Assume
$(L_{1}\llbracket 1,4T+3\rrbracket,L_{2}\llbracket
1,4T+4\rrbracket)\sim\mathbb{P}_{\alpha_{2}}^{\vec{y},\vec{z};2,2T+2}$. Let
$\mathcal{G}:=\sigma(L_{i}\llbracket 2T+i-2,4T+i+2\rrbracket)_{i\in\llbracket
1,2\rrbracket}$. Fix any event $\mathsf{F}$ measurable with respect to
$\mathcal{G}$. Set $L_{2}(4T+1)=\infty$. We claim that (recall $W(a;b,c)$ from
(1.5), $\mathcal{V}_{p}$ from (5.24), and $\mathcal{R}_{2}$ from (5.25))

$\displaystyle\mathbb{E}_{\alpha_{2}}^{\vec{y},(-\infty)^{2T+2};2,2T+2}\left[\mathbf{1}_{\mathsf{F}}\cdot\prod_{j=1}^{2T+2}{W(z_{j};L_{2}(2j+1),L_{2}(2j-1))}\right]$
(5.29)
$\displaystyle=\mathbb{E}_{\alpha_{2}}^{\vec{y},(-\infty)^{2T+2};2,2T+2}\left[\mathbf{1}_{\mathsf{F}}\cdot\mathcal{R}_{2}\cdot\mathcal{V}_{2}\prod_{j=1}^{2T+2}{W(w_{j};L_{2}(2j+1),L_{2}(2j-1))}\right].$

Assuming (5.29) we can finish the proof of (5.25) (for $p=2$) via the
following string of equalities:

$\displaystyle\mathbb{P}_{\alpha_{2}}^{\vec{y};\vec{z};2,2T+2}\bigg{(}\neg\widetilde{\mathsf{C}}(2,\beta)\bigg{)}$
$\displaystyle=\frac{\mathbb{E}_{\alpha_{2}}^{\vec{y};(-\infty)^{2T+2};2,2T+2}\bigg{[}\mathbf{1}_{\neg\widetilde{\mathsf{C}}(2,\beta)}\cdot\prod\limits_{j=1}^{2T+2}W(z_{j};L_{2}(2j+1),L_{2}(2j-1))\bigg{]}}{\mathbb{E}_{\alpha_{2}}^{\vec{y};(-\infty)^{2T+2};2,2T+2}\bigg{[}\prod\limits_{j=1}^{2T+2}{W(z_{j};L_{2}(2j+1),L_{2}(2j-1))}\bigg{]}}$
$\displaystyle=\frac{\mathbb{E}_{\alpha_{2}}^{\vec{y};(-\infty)^{2T+2};2,2T+2}\bigg{[}\mathbf{1}_{\neg\widetilde{\mathsf{C}}(2,\beta)}\cdot\mathcal{R}_{2}\cdot\mathcal{V}_{2}\prod\limits_{j=1}^{2T+2}W(w_{j};L_{2}(2j+1),L_{2}(2j-1))\bigg{]}}{\mathbb{E}_{\alpha_{2}}^{\vec{y};(-\infty)^{2T+2};2,2T+2}\bigg{[}\mathcal{R}_{2}\cdot\mathcal{V}_{2}\prod\limits_{j=1}^{2T+2}{W(w_{j};L_{2}(2j+1),L_{2}(2j-1))}\bigg{]}}$
$\displaystyle=\frac{\mathbb{E}_{\alpha_{2}}^{\vec{y};\vec{w};2,2T+2}\bigg{[}\mathbf{1}_{\neg\widetilde{\mathsf{C}}(2,\beta)}\cdot\mathcal{R}_{2}\cdot\mathcal{V}_{2}\bigg{]}}{\mathbb{E}_{\alpha_{2}}^{\vec{y};\vec{w};2,2T+2}\bigg{[}\mathcal{R}_{2}\cdot\mathcal{V}_{2}\bigg{]}}.$

Let us briefly explain the above equalities. The first equality is due to
(2.7) and (2.8). In the second equality we have applied (5.29) to the
numerator and denominator by taking $\mathsf{F}=\neg\mathsf{C}(2,\beta)$ and
$\mathsf{F}=\Omega$ (the full set, i.e., $\mathbf{1}_{\mathsf{F}}=1$)
respectively. The last equality follows by applying (2.7) and (2.8) again.
This proves (5.25) modulo (5.29).

To see why (5.29) holds, observe that

$\displaystyle\mathbb{E}_{\alpha_{2}}^{\vec{y},(-\infty)^{2T+2};2,2T+2}\left[\mathbf{1}_{\mathsf{F}}\cdot\prod_{j=1}^{2T+2}{W(z_{j};L_{2}(2j+1),L_{2}(2j-1))}\right]$
$\displaystyle=\mathbb{E}_{\alpha_{2}}^{\vec{y},(-\infty)^{2T+2};2,2T+2}\left[\mathbf{1}_{\mathsf{F}}\cdot\mathcal{R}_{2}\prod_{j=T+1}^{2T+2}{W(z_{j};L_{2}(2j+1),L_{2}(2j-1))}\right.$
$\displaystyle\hskip
71.13188pt\left.\cdot\mathbb{E}_{\alpha_{2}}^{\vec{y},(-\infty)^{2T+2};2,2T+2}\left[\mathcal{R}_{2}\prod_{j=1}^{T-1}{W(z_{j};L_{2}(2j+1),L_{2}(2j-1))}\mid\mathcal{G}\right]\right].$

By the Gibbs property, the inner expectation, when viewed as a random
variable, is almost surely equal to $\mathcal{V}_{2}$ defined in (2.8). On the
other hand, we have

$\prod_{j=T+1}^{2T+2}{W(z_{j};L_{2}(2j+1),L_{2}(2j-1))}=\prod_{j=1}^{2T+2}W(w_{j};L_{2}(2j+1),L_{2}(2j-1)).$

Combining the above two observations, leads to (5.29).

###  5.3. Proof of Proposition 5.2

As with the proof of Propositions 3.9 and 4.1, we divide the proof of
Proposition 5.2 into two parts depending on $p=1$ (critical) or $p=2$
(supercritical).

###### Proof of Proposition 5.2 in the $p=1$ case (critical phase).

Fix any $T\in\llbracket k_{1}N^{\frac{2}{3}},k_{2}N^{\frac{2}{3}}\rrbracket$.
Fix any $\delta\leq\gamma/{6\kappa}$. We recall the representation of bottom-
free law in $p=1$ case from Lemma 4.3. Consider the Brownian motion $B_{1}$
obtained via the KMT coupling that satisfies (4.9). Define

$\displaystyle\mathsf{A}_{\delta}$
$\displaystyle:=\bigg{\\{}\sup_{\begin{subarray}{c}i_{1},i_{2}\in\llbracket
1,T\rrbracket\\\
|i_{1}-i_{2}|\leq\frac{\delta}{2}N^{2/3}\end{subarray}}|L_{1}(2i_{1}-1)-L_{1}(2i_{2}-1)|\geq\tfrac{1}{6}\gamma
N^{\frac{1}{3}}\bigg{\\}},$ $\displaystyle\mathsf{B}(k)$
$\displaystyle:=\left\\{|L_{1}(2k-1)-L_{1}(2k)|,|L_{1}(2k+1)-L_{1}(2k)|\geq\tfrac{1}{3}\gamma
N^{\frac{1}{3}}\right\\}.$

Fix any $x\in\mathbb{R}$ and write
$\mathbb{P}_{1}:=\mathbb{P}_{\alpha_{1}}^{x,(-\infty)^{T};1,T}$. Observe that
by union bound we have

$\displaystyle\mathbb{P}_{1}\left(\omega_{\delta}^{N}(L_{1},\llbracket
1,2T-1\rrbracket)\geq\gamma
N^{1/3}\right)\leq\mathbb{P}_{1}(\mathsf{A}_{\delta})+\sum_{k=1}^{T-1}\mathbb{P}_{1}(\neg\mathsf{A}_{\delta}\cap\mathsf{B}(k)).$
(5.30)

We now proceed to bound each of the above term separately. For the first term,
by (4.7) and (4.9), in view of the estimate in (4.10) we have for all large
enough $N$ that

$\displaystyle\mathbb{P}_{1}(\mathsf{A}_{\delta})$
$\displaystyle\leq\mathbb{P}_{1}\bigg{(}\sup_{\begin{subarray}{c}i_{1},i_{2}\in\llbracket
1,T\rrbracket\\\
|i_{1}-i_{2}|\leq\frac{\delta}{2}N^{2/3}\end{subarray}}\sigma|B_{1}{(T-i_{1}-1)}-B_{1}{(T-i_{2}-1)}|\geq\tfrac{\gamma}{12}N^{1/3}-2\mathrm{C}\log
T\bigg{)}$
$\displaystyle\leq\mathbb{P}_{1}\bigg{(}\sup_{\begin{subarray}{c}i_{1},i_{2}\in\llbracket
1,T\rrbracket\\\
|i_{1}-i_{2}|\leq\frac{\delta}{2}N^{2/3}\end{subarray}}\sigma|B_{1}{(i_{1})}-B_{1}{(i_{2})}|\geq\tfrac{\gamma}{24}N^{1/3}\bigg{)}.$

By modulus of continuity of Brownian motion, the right-hand side of the above
equation can be made smaller than $\frac{1}{2}\varepsilon$ by choosing
$\delta$ small enough depending on $\mu,\theta,\gamma,k_{1},k_{2}$. For the
second term in r.h.s. (5.30) we use Lemma B.4 to get

$\displaystyle\mathbb{P}_{1}(\neg\mathsf{A}_{\delta}\cap\mathsf{B}(k))\leq\mathrm{C}e^{-\tfrac{1}{\mathrm{C}}\gamma
N^{\frac{1}{3}}}.$

Plugging the bounds back in (5.30) and taking $N$ large enough we get the
desired result. ∎

###### Proof of Proposition 5.2 in the $p=2$ case (supercritical phase).

Fix any $(x_{1},x_{2})\in I_{2,M}$, and $T\in\llbracket
k_{1}N^{\frac{2}{3}},k_{2}N^{\frac{2}{3}}\rrbracket$. Set $n:=T$. Recall the
law paired random walk and weighted paired random walk defined in Definition
1.7. We recall from Lemma 4.4 that the bottom-free law
$\mathbb{P}_{\alpha_{2}}^{(x_{1},x_{2}),(-\infty)^{T};2,T}$ is equal to
${\mathbb{P}}_{\operatorname{WPRW}}^{n;(x_{1},x_{2})}$ for the supercritical
case. At this point is it also good to recall the random walk measures from
Definition 3.6.

A key to this proof is the following estimate for
${\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}]$
(recall $W_{\operatorname{sc}}$ from (1.15)).

######  Lemma 5.6.

There exist constants $\mathrm{C}_{1},\mathrm{C}_{2}>0$, depending on $M$,
such that for all $(x_{1},x_{2})\in I_{2,M}$ we have

$\displaystyle{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}]\geq\tfrac{1}{\sqrt{n}}{\mathrm{C}_{1}^{-1}}\cdot\mathbb{P}^{\lfloor
n/4\rfloor;(x_{1},x_{2})}(\widetilde{\mathsf{NI}})\geq{\mathrm{C}_{2}^{-1}}e^{-\mathrm{C}_{2}(\log
n)^{5/4}},$ (5.31)

where $\widetilde{\mathsf{NI}}:=\\{S_{1}(k)\geq S_{2}(k)\mbox{ for all
}k\in\llbracket 1,n/4\rrbracket\\}$ and $S_{1},S_{2}$ are random walks under
the law $\mathbb{P}^{\lfloor n/4\rfloor;(x_{1},x_{2})}$.

Before proving Lemma 5.6 we complete the proof of Proposition 5.2 in the
following two steps.

Step 1. Fix any $V,\gamma>0$. Set $v=\gamma/\sqrt{k_{2}},u=V/\sqrt{k_{1}}$,
and $t=\lfloor 2\log\log n\rfloor$. Let
$\mathcal{F}:=\sigma(S_{1}(1),S_{2}(1))$. Consider the events

$\displaystyle\mathsf{MC}_{\delta}$
$\displaystyle:=\Big{\\{}|S_{1}(1)|+|S_{2}(1)|\leq
u\sqrt{n},\omega_{\delta}^{N}(S_{i}(\cdot),\llbracket
0,\tfrac{n}{8}\rrbracket)\geq\tfrac{1}{6}v\sqrt{n},\mbox{ for
}i=1,2\Big{\\}},\mbox{ for }\delta>0.$

We claim that given $\varepsilon>0$, there exists $\delta$ small enough and
$N$ large enough such that

$\displaystyle{\mathbb{P}}_{\operatorname{WPRW}}^{n;(x_{1},x_{2})}(\mathsf{MC}_{\delta})\leq\varepsilon.$
(5.32)

We finish the proof of the lemma assuming (5.32). Lemma 4.4 implies that
$\big{(}L_{1}(2j+1),L_{2}(2j+2)\big{)}_{j=0}^{n}$ is distributed as
$\mathsf{WPRW}$. Write $L_{2}^{\mathsf{even}}(k):=L_{2}(2k)$ and
$\mathbb{P}_{2}:=\mathbb{P}_{\alpha_{2}}^{\vec{x},(-\infty)^{T};2,T}$. Then
(5.32) implies

$\displaystyle\mathbb{P}_{2}\big{(}\\{|L_{1}(1)|+|L_{2}(2)|\leq
VN^{1/3}\\}\cap\mathsf{A}\big{)}\leq\varepsilon,\quad\mathsf{A}:=\Big{\\{}\omega_{\delta}^{N}(L_{2}^{\mathsf{even}},\llbracket
1,T/8\rrbracket)\geq\tfrac{1}{6}\gamma N^{1/3}\Big{\\}}.$ (5.33)

On the event $\neg\mathsf{A}$ the increments of $L_{2}^{\mathsf{even}}$ are
well controlled. By Lemma 4.4, conditioned on the even points of $L_{2}$, the
distribution of the odd points of $L_{2}$ are given by $\xi$-distributions
defined in (4.5). Once we have a bound on the increments of
$L_{2}^{\mathsf{even}}$, we may invoke the tails estimates of
$\xi$-distributions from Lemma B.4 to control increments of $L_{2}$. In
particular, due to Lemma B.4,

$\mathbf{1}_{\neg\mathsf{A}}\cdot\mathbb{E}_{2}\left[\mathbf{1}_{|L_{2}(2k+1)-L_{2}(2k)|,|L_{2}(2k+1)-L_{2}(2k+2)|\geq\frac{1}{3}\gamma
N^{1/3}}\mid\sigma\big{(}L_{2}^{\mathsf{even}}\llbracket
1,T/8\rrbracket\big{)}\right]\leq\mathrm{C}\exp(-\tfrac{1}{\mathrm{C}}\gamma
N^{1/3})$

for all $k\geq 1$. For the first point in $L_{2}$, i.e., $L_{2}(1)$, we recall
from Lemma 4.4 that $L_{2}(1)\sim X+L_{2}(2)$ where $X\sim
G_{\alpha_{2}+\theta,1}$. The explicit form of $G_{\alpha_{2}+\theta,1}$ from
(2.2) allow us to derive that

$\mathbb{P}_{2}\bigg{(}|L_{2}(1)-L_{2}(2)|\geq\tfrac{1}{6}\gamma
N^{1/3}\mid\sigma\big{(}L_{2}^{\mathsf{even}}\llbracket
1,T/8\rrbracket\big{)}\bigg{)}\leq\mathrm{C}\exp(-\tfrac{1}{\mathrm{C}}\gamma
N^{1/3}).$

Thus, in view of (5.33), by the union bound

$\displaystyle\mathbb{P}_{2}\left(|L_{1}(1)|+|L_{2}(2)|\leq
VN^{\frac{1}{3}},\omega_{\delta}^{N}(L_{2},\llbracket
1,T/4\rrbracket)\geq\gamma
N^{\frac{1}{3}}\right)\leq\varepsilon+\mathrm{C}\cdot
k_{2}N^{\frac{2}{3}}\exp(-\tfrac{1}{\mathrm{C}}\gamma N^{\frac{1}{3}})$

which can be made arbitrarily small taking $N$ large enough. A similar
argument shows that

$\displaystyle\mathbb{P}_{2}\left(|L_{1}(1)|+|L_{2}(2)|\leq
VN^{\frac{1}{3}},\omega_{\delta}^{N}(L_{1},\llbracket
1,T/4-1\rrbracket)\geq\gamma N^{\frac{1}{3}}\right)$

can be made arbitrarily small as well taking $N$ large enough. This proves
Proposition 5.2.

Step 2. In this step we prove (5.32). First, recall that due to (1.14),

${\mathbb{P}}_{\operatorname{WPRW}}^{n;(x_{1},x_{2})}(\mathsf{MC}_{\delta})=\frac{{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{MC}_{\delta}}]}{{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}]}$

where $W_{\operatorname{sc}}$ is defined in (1.15). We first define a few more
necessary events.

$\displaystyle\mathsf{G}_{1}$ $\displaystyle:=\\{|S_{1}(1)|+|S_{2}(1)|\leq
u\sqrt{n},|S_{1}(1)-S_{2}(1)|\leq(\log n)^{3/2}\\},$
$\displaystyle\mathsf{G}_{2}$ $\displaystyle:=\\{|S_{1}(1)|+|S_{2}(1)|\leq
u\sqrt{n},1\leq S_{1}(1)-S_{2}(1)\leq 2\\}.$

Recall the non-intersection event $\mathsf{NI}_{p}$ from (4.30). Let us
temporarily set $t=\lceil 2\log\log n\rceil$. As $W_{\operatorname{sc}}\leq
1$, we write

$\displaystyle{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{MC}_{\delta}}]\leq\underbrace{{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{MC}_{\delta}\cap\mathsf{G}_{1}\cap\mathsf{NI}_{t}}]}_{(\mathbf{I})}+\underbrace{{{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}\mathbf{1}_{\neg\mathsf{NI}_{t}}]+{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[\mathbf{1}_{\neg\mathsf{G}_{1}}]}}_{(\mathbf{II})}.$

For $(\mathbf{II})$, note that on $\neg\mathsf{NI}_{t}$, we have
$W_{\operatorname{sc}}\leq e^{-e^{t}}\leq e^{-(\log n)^{2}}$ and by Lemma 4.7,
${\mathbb{P}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}(\neg\mathsf{G}_{1})\leq\mathrm{C}e^{-\mathrm{C}^{-1}(\log
n)^{3/2}}$. Thus, $(\mathbf{II})\leq\mathrm{C}e^{-\mathrm{C}^{-1}(\log
n)^{3/2}}$. In view of Lemma 5.6,
$({\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}])^{-1}\cdot(\mathbf{II})\to
0$. For $(\mathbf{I})$, note that

$\displaystyle(\mathbf{I})$
$\displaystyle={\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{MC}_{\delta}\cap\mathsf{G}_{1}\cap\mathsf{NI}_{0}}]+\sum_{p=1}^{t}{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{MC}_{\delta}\cap\mathsf{G}_{1}\cap\mathsf{NI}_{p}\cap\neg\mathsf{NI}_{p-1}}]$
(5.34)
$\displaystyle\leq\sum_{p=0}^{t}\mathrm{C}e^{-e^{p}}{{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}}\big{[}\mathbf{1}_{\mathsf{G}_{1}}{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[\mathbf{1}_{\mathsf{MC}_{\delta}\cap\mathsf{NI}_{p}}\mid\mathcal{F}]\big{]}.$

Upon conditioning on $\mathcal{F}$, the conditional law is the law of two
independent $n$-step random bridges from $(S_{1}(1),S_{2}(1))$ to
$(x_{1},x_{2})$. We may lift the bridges by $p$ units. The modulus of
continuity event remains unchanged and $\mathsf{NI}_{p}$ event turns into
$\mathsf{NI}$. Now we apply the comparison trick between random bridges and
modified random bridges via Lemma 4.10. By Lemma 4.10, there exists a constant
$\mathrm{C}$ depending only on $u$ such that

$\displaystyle\mathbf{1}_{\mathsf{G}_{1}}{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[\mathbf{1}_{\mathsf{MC}_{\delta}\cap\mathsf{NI}_{p}}\mid\mathcal{F}]$
$\displaystyle=\mathbf{1}_{\mathsf{G}_{1}}\mathbb{P}^{n;(S_{1}(1),S_{2}(1)),(x_{1},x_{2})}(\mathsf{MC}_{\delta}\cap\mathsf{NI}_{p})$
(5.35)
$\displaystyle=\mathbf{1}_{\mathsf{G}_{1}}\mathbb{P}^{n;(S_{1}(1)+p,S_{2}(1)),(x_{1}+p,x_{2})}(\mathsf{MC}_{\delta}\cap\mathsf{NI})$
$\displaystyle\leq\mathrm{C}\cdot\mathbf{1}_{\mathsf{G}_{1}}\widetilde{\mathbb{P}}_{p}(\mathsf{MC}_{\delta}\cap\mathsf{NI})=\mathrm{C}\cdot\mathbf{1}_{\mathsf{G}_{1}}\widetilde{\mathbb{P}}_{p}(\mathsf{MC}_{\delta}\mid\mathsf{NI})\cdot\widetilde{\mathbb{P}}_{p}(\mathsf{NI})$

where $\widetilde{\mathbb{P}}_{p}$ denote the law of a $(n;\lfloor
n/4\rfloor,\lfloor n/4\rfloor)$-modified random bridge defined in Definition
4.9 starting from $(S_{1}(1)+p,S_{2}(1))$ to $(x_{1}+p,x_{2})$. Observe that
$\widetilde{\mathbb{P}}_{p}(\mathsf{NI})$ is $\mathcal{F}$-measurable. By
Lemmas C.3 and C.8 (recall Corollary 4.6),

$\displaystyle\mathbf{1}_{\mathsf{G}_{1}}\cdot\widetilde{\mathbb{P}}_{p}(\mathsf{NI})\leq\mathbf{1}_{\mathsf{G}_{1}}\cdot\tfrac{\mathrm{C}}{\sqrt{n}}\cdot
e^{\mathrm{C}p}\cdot\max\\{S_{1}(1)-S_{2}(1),1\\}\cdot\mathbb{P}^{\lfloor
n/4\rfloor,(x_{1},x_{2})}\big{(}\widetilde{\mathsf{NI}}\big{)}.$ (5.36)

We plug the estimates from (5.35) and (5.36) back in (5.34). Thus setting
$\mathrm{C}_{3}:=\sum_{r=1}^{\infty}2\mathrm{C}_{1}\mathrm{C}^{3}e^{\mathrm{C}r}e^{-e^{r}}$
(with $\mathrm{C}_{1}$ coming from Lemma 5.6) and utilizing the lower bound
for
${\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}]$
from Lemma 5.6, we have

$\displaystyle({\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}])^{-1}\cdot(\mathbf{I})\leq\mathrm{C}_{3}\cdot{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}\left[\mathbf{1}_{\mathsf{G}_{1}}\cdot\max\\{S_{1}(1)-S_{2}(1),1\\}\cdot\sup_{p\in\llbracket
0,t\rrbracket}\widetilde{\mathbb{P}}_{p}(\mathsf{MC}_{\delta}\mid\mathsf{NI})\right]$

Now we claim that one can choose $\delta$ sufficiently small such that

$\displaystyle{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}\left[\mathbf{1}_{\mathsf{G}_{1}}\cdot\max\\{S_{1}(1)-S_{2}(1),1\\}\cdot\sup_{p\in\llbracket
0,t\rrbracket}\widetilde{\mathbb{P}}_{p}(\mathsf{MC}_{\delta}\mid\mathsf{NI})\right]\leq\tfrac{1}{2}\mathrm{C}_{1}^{-1}\varepsilon.$
(5.37)

We write
$\mathsf{G}_{1}=\mathsf{G}_{1,M_{2}}\cup\widetilde{\mathsf{G}}_{1,M_{2}}$,
where

$\displaystyle\mathsf{G}_{1,M_{2}}:=\\{|S_{1}(1)|+|S_{2}(1)|\leq
u\sqrt{n},|S_{1}(1)-S_{2}(1)|\leq
M_{2}\\},\quad\widetilde{\mathsf{G}}_{1,M_{2}}:=\mathsf{G}_{1}\cap\neg\mathsf{G}_{1,M_{2}}.$

Given the tail estimates, one can choose $M_{2}$ large enough such that

$\displaystyle{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}\left[\mathbf{1}_{\widetilde{\mathsf{G}}_{1,M_{2}}}\cdot\max\\{S_{1}(1)-S_{2}(1),1\\}\right]\leq\tfrac{1}{4}\mathrm{C}_{1}^{-1}\varepsilon.$

This fixes our choice for $M_{2}$. Now note that the event
$\mathsf{MC}_{\delta}$ depends only on the first $\lfloor n/8\rfloor$ points
of the two $(n;\lfloor n/4\rfloor,\lfloor n/4\rfloor)$-modified random
bridges. By definition, the first $\lfloor n/4\rfloor$ points of a $(n;\lfloor
n/4\rfloor,\lfloor n/4\rfloor)$-modified random bridge is just a random walk.
Thus, in view of Lemma C.12 (recall Corollary 4.6), one can then choose
$\delta$ small enough and $N$ large enough such that on uniformly on
$\mathsf{G}_{1,M_{2}}$ we have

$\sup_{p\in\llbracket
0,t\rrbracket}\widetilde{\mathbb{P}}_{p}(\mathsf{MC}_{\delta}\mid\mathsf{NI})\leq\tfrac{1}{4}\mathrm{C}_{1}^{-1}M_{2}^{-1}\varepsilon.$

Thus, we have

l.h.s. (5.37)
$\displaystyle\leq{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}\Big{[}\mathbf{1}_{\widetilde{\mathsf{G}}_{1,M_{2}}}\\!\\!\\!\cdot\max\\{S_{1}(1)-S_{2}(1),1\\}\Big{]}+M_{2}\\!\cdot\\!{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}\Big{[}\mathbf{1}_{\mathsf{G}_{1,M_{2}}}\\!\cdot\\!\sup_{p\in\llbracket
0,t\rrbracket}\widetilde{\mathbb{P}}_{p}(\mathsf{MC}_{\delta}\mid\mathsf{NI})\Big{]}$
$\displaystyle\leq\tfrac{1}{4}\mathrm{C}_{1}^{-1}\varepsilon+M_{2}\\!\cdot\\!\tfrac{1}{4}\mathrm{C}_{1}^{-1}M_{2}^{-1}\varepsilon=\tfrac{1}{2}\mathrm{C}_{1}^{-1}\varepsilon,$

verifying the inequality in (5.37). ∎

###### Proof of Lemma 5.6.

Recall the definition of $(n,p,q)$-modified random bridge from Definition 4.9,
in particular that $\widetilde{\mathbb{P}}^{(n;\lfloor n/4\rfloor,\lfloor
n/4\rfloor);(a_{1},a_{2}),(x_{1},x_{2})}$ denotes the law of two independent
$(n,\lfloor n/4\rfloor,\lfloor n/4\rfloor)$-modified random bridge started at
$(a_{1},a_{2})$ and ended at $(x_{1},x_{2})$. We shall use the shorthand
$\widetilde{\mathbb{P}}^{(a_{1},a_{2})}$ for
$\widetilde{\mathbb{P}}^{(n;\lfloor n/4\rfloor,\lfloor
n/4\rfloor);(a_{1},a_{2}),(x_{1},x_{2})}$. Also recall the notation
$\mathbb{P}^{m;(b_{1},b_{2})}$ from Definition 3.6 to denote the law of two
independent random walks of length $m$ started at $(b_{1},b_{2})$ with same
increment law as the modified bridges.

Recall the events $\mathsf{NI}$ and $\mathsf{Gap}_{\beta}$ from (4.30) and
(4.22) respectively. Invoking Lemma C.11 we first fix a
$\beta=\beta(M)\leq\frac{1}{2}$ small enough so that it satisfies

$\widetilde{\mathbb{P}}^{(a_{1},a_{2})}(\mathsf{Gap}_{\beta}\mid\mathsf{NI})\geq\tfrac{3}{4},$

for all $|a_{i}|\leq\sqrt{n}$ with $1\leq a_{1}-a_{2}\leq 2$. Next by Lemma
C.5, we fix $\xi=\xi(M)>0$ so that

$\mathbb{P}^{\lfloor n/4\rfloor;(b_{1},b_{2})}\Big{(}|S_{1}(\lfloor
n/4\rfloor)|,|S_{2}(\lfloor
n/4\rfloor)|\leq\xi\sqrt{n}\mid\widetilde{\mathsf{NI}}\Big{)}\geq\sqrt{\tfrac{3}{4}}$

for all $|b_{i}|\leq(M+1)\sqrt{n}$. Here
$\widetilde{\mathsf{NI}}:=\\{S_{1}(k)\geq S_{2}(k)\mbox{ for all
}k\in\llbracket 2,n/4\rrbracket\\}$ is the non-intersection event over
$\lfloor n/4\rfloor$ points.

We consider the following events

$\displaystyle\mathsf{G}_{3}$
$\displaystyle:=\big{\\{}|S_{i}(1)|\leq\sqrt{n}\mbox{ for }i=1,2,\ 1\leq
S_{1}(1)-S_{2}(1)\leq 2\big{\\}},$ $\displaystyle\mathsf{T}_{\xi}$
$\displaystyle:=\big{\\{}|S_{i}(\lfloor n/4\rfloor)|,|S_{i}(n-\lfloor
n/4\rfloor)|\leq\xi\sqrt{n}\mbox{ for }i=1,2\big{\\}},$

where $\mathsf{T}$ stands for tightness. Observe that by Lemma 4.8 we have

$\displaystyle{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}]\geq{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{Gap}_{\beta}\cap\mathsf{G}_{3}\cap\mathsf{T}_{\xi}}]$
$\displaystyle\geq\tfrac{1}{\mathrm{C}}{\mathbb{P}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}(\mathsf{Gap}_{\beta}\cap\mathsf{G}_{3}\cap\mathsf{T}_{\xi})$
(5.38)
$\displaystyle=\tfrac{1}{\mathrm{C}}{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}\left[\mathbf{1}_{\mathsf{G}_{3}}{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[\mathbf{1}_{\mathsf{Gap}_{\beta},\mathsf{T}_{\xi}}\mid\mathcal{F}]\right]$

where $\mathcal{F}:=\sigma(S_{1}(1),S_{2}(1))$. Under the event
$\mathsf{G}_{3}$ and $\mathsf{T}_{\xi}$ we may invoke Lemma 4.10 to get

$\displaystyle\mathbf{1}_{\mathsf{G}_{3}}\cdot{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[\mathbf{1}_{\mathsf{Gap}_{\beta}\cap\mathsf{T}_{\xi}}\mid\mathcal{F}]\geq\mathrm{C}^{-1}\cdot\mathbf{1}_{\mathsf{G}_{3}}\cdot\widetilde{\mathbb{P}}^{(S_{1}(1),S_{1}(2))}(\mathsf{Gap}_{\beta}\cap\mathsf{T}_{\xi})$
(5.39)

almost surely. By Corollary C.10 (recall Corollary 4.6),

$\displaystyle\mathbf{1}_{\mathsf{G}_{3}}\cdot\widetilde{\mathbb{P}}^{(S_{1}(1),S_{1}(2))}({\mathsf{Gap}_{\beta}\cap\mathsf{T}_{\xi}})$
(5.40)
$\displaystyle=\mathbf{1}_{\mathsf{G}_{3}}\cdot\widetilde{\mathbb{P}}^{(S_{1}(1),S_{1}(2))}({\mathsf{Gap}_{\beta}\cap\mathsf{T}_{\xi}}\mid\mathsf{NI})\widetilde{\mathbb{P}}_{(a_{1},a_{2})}(\mathsf{NI})$
$\displaystyle\geq\mathrm{C}^{-1}\mathbf{1}_{\mathsf{G}_{3}}\cdot\widetilde{\mathbb{P}}^{(S_{1}(1),S_{1}(2))}({\mathsf{Gap}_{\beta}\cap\mathsf{T}_{\xi}}\mid\mathsf{NI})\cdot\mathbb{P}^{\lfloor
n/4\rfloor;(S_{1}(1),S_{1}(2))}(\widetilde{\mathsf{NI}})\mathbb{P}^{\lfloor
n/4\rfloor;(x_{1},x_{2})}(\widetilde{\mathsf{NI}}).$

By our choice of $\beta$ and $\xi$, we have
$\mathbf{1}_{\mathsf{G}_{3}}\cdot\widetilde{\mathbb{P}}^{(S_{1}(1),S_{1}(2))}({\mathsf{Gap}_{\beta},\mathsf{T}_{\xi}}\mid\mathsf{NI})\geq\frac{1}{2}\mathbf{1}_{\mathsf{G}_{3}}$
almost surely. By Lemma C.3 (recall Corollary 4.6), we have
$\mathbf{1}_{\mathsf{G}_{3}}\cdot\mathbb{P}^{\lfloor
n/4\rfloor;(S_{1}(1),S_{1}(2))}(\widetilde{\mathsf{NI}})\geq\frac{\mathrm{C}^{-1}}{\sqrt{n}}$
almost surely. Thus combining (5.38), (5.39), and (5.40) we have

$\displaystyle{\mathbb{E}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}[W_{\operatorname{sc}}]\geq\tfrac{1}{\sqrt{n}}\mathrm{C}^{-1}\cdot\mathbb{P}^{\lfloor
n/4\rfloor;(x_{1},x_{2})}(\widetilde{\mathsf{NI}})\cdot{\mathbb{P}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}(\mathsf{G}_{3}).$

By Lemma 4.7 ((4.17) in particular),
${\mathbb{P}}_{\operatorname{PRW}}^{n;(x_{1},x_{2})}(\mathsf{G}_{3})\geq\mathrm{C}^{-1}$.
Plugging this back in the above equation we get the first inequality in
(5.31). For the second inequality, we consider the event:

$\displaystyle\mathsf{G}_{4}$ $\displaystyle:=\\{|S_{1}(2)-x_{1}|\leq
1,\,\,|S_{2}(2)-\min\\{x_{1}-3,x_{2}\\}|\leq 1\\}.$

Observe that

$\displaystyle\mathbb{P}^{\lfloor
n/4\rfloor;(x_{1},x_{2})}(\widetilde{\mathsf{NI}})\geq\mathbb{P}^{\lfloor
n/4\rfloor;(x_{1},x_{2})}\bigg{(}\mathsf{G}_{4}\cap\\{S_{1}(j)\geq
S_{2}(j)\mbox{ for all }j\in\llbracket 3,n/4\rrbracket\\}\bigg{)}.$

By the tail bounds of the increments from Lemma B.3, and given the condition
$x_{1}-x_{2}\geq-(\log N)^{7/6}$, we have $\mathbb{P}^{\lfloor
n/4\rfloor;(x_{1},x_{2})}(\mathsf{G}_{4})\geq\mathrm{C}^{-1}\exp(-\mathrm{C}(\log
n)^{7/6})$ (recall $n\geq k_{1}N^{2/3}-1$). Furthermore, on $\mathsf{G}_{4}$
we must have $S_{1}(2)\geq S_{2}(2)$. By Lemma C.3 (recall Corollary 4.6), we
have

$\mathbb{P}^{\lfloor n/4\rfloor-1;(a_{1},a_{2})}\big{(}S_{1}(j)\geq
S_{2}(j)\mbox{ for all }j\in\llbracket
2,n/4-1\rrbracket\big{)}\geq\mathrm{C}^{-1}/\sqrt{n}$

for all $a_{1}\geq a_{2}$. Thus we have

$\mathbb{P}^{\lfloor
n/4\rfloor;(x_{1},x_{2})}\bigg{(}\mathsf{G}_{4}\cap\\{S_{1}(j)\geq
S_{2}(j)\mbox{ for all }j\in\llbracket
3,n/4\rrbracket\\}\bigg{)}\geq\mathrm{C}^{-1}\exp(-\mathrm{C}(\log
n)^{7/6})\cdot\tfrac{1}{\sqrt{n}}.$

Adjusting the constant we get the second inequality in (5.31). ∎

##  Appendix A Stochastic monotonicity

The goal of this section is to prove the stochastic monotonicity of
$\mathpzc{HSLG}$ Gibbs measure (Proposition 2.6). Let
$\Lambda=\\{(i,j):k_{1}\leq i\leq k_{2},a_{i}\leq j\leq b_{i}\\}$. Let
$w_{1},\ldots,w_{|\Lambda|}$ be the enumeration of points in $\Lambda$ in the
lexicographic order. Set $\Lambda_{r}=\\{w_{1},w_{2},\ldots,w_{r}\\}$, so that
$\Lambda_{|\Lambda|}=\Lambda$. Let
$E_{r}:=E(\Lambda_{r}\cup\partial\Lambda_{r})$ (the edges in $\mathbb{Z}^{2}$
connecting points in $\Lambda_{r}\cup\partial\Lambda_{r}$), and, recalling the
weights $W_{e}$ from (1.5), let

$H_{r}(x;(u_{v})_{v\in\partial\Lambda_{r}}):=\int_{\mathbb{R}^{|\Lambda_{r-1}|}}\prod_{e=\\{v_{1}\to
v_{2}\\}\in E_{r}}W_{e}(u_{v_{1}}-u_{v_{2}})\prod_{v\in\Lambda_{r-1}}du_{v},$
(A.1)

where $u_{w_{r}}=x$. The proof of Proposition 2.6 relies on the following
technical lemma.

######  Lemma A.1.

Fix $r\in\llbracket 1,|\Lambda|\rrbracket$. For each
$v\in\partial\Lambda_{r}$, fix any $u_{v},u_{v}^{\prime}\in\mathbb{R}$ with
$u_{v}\leq u_{v}^{\prime}$. For all $s\geq t$

$\displaystyle
H_{r}\big{(}s;(u_{v})_{v\in\partial\Lambda_{r}}\big{)}H_{r}\big{(}t;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}}\big{)}\leq
H_{r}\big{(}s;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}}\big{)}H_{r}\big{(}t;(u_{v})_{v\in\partial\Lambda_{r}}\big{)}$
(A.2)

We prove Lemma A.1 at the end of this section and now complete the proof of
Proposition 2.6.

###### Proof of Proposition 2.6.

Fix $r\in\llbracket 1,|\Lambda|\rrbracket$. We first claim that for all
boundary conditions $(u_{v})_{v\in\partial\Lambda_{r}}$ and
$(u_{v}^{\prime})_{v\in\partial\Lambda_{r}}$ with $u_{v}\leq u_{v}^{\prime}$
for all $v\in\partial\Lambda_{r}$, and $s\in\mathbb{R}$,

$\mathbb{P}\big{(}L(w_{r})\leq s\mid L(v)=u_{v}\textrm{ for all
}v\in\partial\Lambda_{r}\big{)}\geq\mathbb{P}\big{(}L(w_{r})\leq s\mid
L(v)=u_{v}^{\prime}\textrm{ for all }v\in\partial\Lambda_{r}\big{)}.$ (A.3)

To show this, observe that $H_{r}(x;(u_{v})_{v\in\partial\Lambda_{r}})$ in
(A.1) is proportional to the conditional density at $x$ of $L(w_{r})$ given
$\big{(}L(v)\big{)}_{v\in\partial\Lambda_{r}}=(u_{v})_{v\in\partial\Lambda_{r}}$.
Thus,

$\mathbb{P}\big{(}L(w_{r})\leq s\mid L(v)=u_{v}\textrm{ for all
}v\in\partial\Lambda_{r}\big{)}=F_{r}\big{(}s;(u_{v})_{v\in\partial\Lambda_{r}}\big{)}:=\frac{\int_{-\infty}^{s}H_{r}(x;(u_{v})_{v\in\partial\Lambda_{r}})dx}{\int_{-\infty}^{\infty}H_{r}(x;(u_{v})_{v\in\partial\Lambda_{r}})dx}$
(A.4)

To prove (A.3) observe that owing to Lemma A.1, the derivative of

$\log\int_{-\infty}^{s}H_{r}(x;(u_{v})_{v\in\partial\Lambda_{r}})dx-\log\int_{-\infty}^{s}H_{r}(x;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}})dx.$

is non-positive for all $s$. This implies for $s^{\prime}\geq s$ we have

$\displaystyle\frac{\int_{-\infty}^{s}H_{r}(x;(u_{v})_{v\in\partial\Lambda_{r}})dx}{\int_{-\infty}^{s}H_{r}(x;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}})dx}\geq\frac{\int_{-\infty}^{s^{\prime}}H_{r}(x;(u_{v})_{v\in\partial\Lambda_{r}})dx}{\int_{-\infty}^{s^{\prime}}H_{r}(x;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}})dx}$

Taking $s^{\prime}\to\infty$ and cross-multiplying yields the desired
inequality (A.3), in light of (A.4).

Given $(u_{v})_{v\in\partial\Lambda}\in\mathbb{R}^{|\partial\Lambda|}$, we now
define a sequence of random variables according to the following algorithm.
Note that below, $x\leftarrow y$ means to assign the value $y$ to the variable
$x$.

Algorithm 1 Defining the random vectors

Generate $U_{1},\ldots,U_{|\Lambda|}$ i.i.d. random variables from $U[0,1]$

$Y_{|\Lambda|}\leftarrow(u_{v})_{v\in\partial\Lambda}$

$r\leftarrow|\Lambda|$

while $r\geq 1$ do

$L(w_{r};(u_{w})_{v\in\partial\Lambda})\leftarrow F_{r}^{-1}(U_{r};Y_{r})$

$\widetilde{u}_{v}\leftarrow u_{v}$ for all
$v\in\partial\Lambda_{r-1}\cap\partial\Lambda_{r}$

$\widetilde{u}_{w_{r}}\leftarrow L(w_{r};(u_{w})_{v\in\partial\Lambda})$

$Y_{r-1}\leftarrow(\widetilde{u}_{v})_{v\in\partial\Lambda_{r-1}}$

$r\leftarrow r-1$

end while

This defines a collection of random variables
$L(w_{i};(u_{v})_{v\in\partial\Lambda})$ indexed by $i\in\llbracket
1,|\Lambda|\rrbracket$ and
$(u_{v})_{v\in\partial\Lambda}\in\mathbb{R}^{|\partial\Lambda|}$, all on the
common probability space on which $U_{1},\ldots,U_{|\Lambda|}$ are defined. It
is clear from the definition that for each
$(u_{v})_{v\in\partial\Lambda}\in\mathbb{R}^{|\partial\Lambda|}$, the law of
$\big{(}L(w_{i};(u_{v})_{v\in\partial\Lambda})\big{)}_{i\in\llbracket
1,|\Lambda|\rrbracket}$ is given by the $\mathpzc{HSLG}$ Gibbs measure on the
domain $\Lambda$ with boundary condition $(u_{v})_{v\in\partial\Lambda}$. Take
two boundary conditions $(u_{v})_{v\in\partial\Lambda}$ and
$(u_{v}^{\prime})_{v\in\partial\Lambda}$ with $u_{v}\leq u_{v}^{\prime}$ for
all $v\in\partial\Lambda$. As each $F_{r}$ is stochastically increasing with
respect to the boundary condition, i.e., (A.3), sequentially we obtain that
with probability 1 on our probability space
$L(w_{r};(u_{v})_{v\in\partial\Lambda})\leq
L(w_{r};(u_{v}^{\prime})_{v\in\partial\Lambda})$ for all $r$, thus completing
the proof. ∎

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

Let us begin with a few pieces of notations. Fix any $1\leq r\leq|\Lambda|$.
Set $e_{r}:=\\{w_{r}\to(w_{r}+(0,1)),(w_{r}+(0,1))\to w_{r}\\}\cap E_{r}$. In
words, this is the directed blue edge (see Figure 19 A) with $w_{r}$ as the
left point of $e_{r}$.

$w_{1}$$w_{2}$$w_{3}$$w_{4}$$w_{5}$$w_{6}$$w_{7}$$w_{8}$$w_{9}$$w_{10}$$w_{11}$$w_{12}$$e_{5}$$e_{8}$
(a)

$e_{5}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$w_{5}$$w_{6}$ (b)

Figure 19. (A) A possible domain $\Lambda$ includes all the vertices in the
shaded region. $w_{i}$’s are the vertices of $\Lambda$ enumerated in
lexicographic order. Directed edges $e_{r}$ going are shown above for $r=5$
and $r=8$. These are the blue edges with $w_{r}$ as the left point of $e_{r}$.
(B) The domain $\Lambda_{5}$ includes the vertices in the shaded region.
$Q_{5}$ is the set of all red and black edges that have one vertex as $w_{6}$
and one vertex in $\partial\Lambda_{6}$. In the above figure, $Q_{5}$ is
composed of two black edges that points toward $w_{6}$.

Define

$\displaystyle
h_{r}\big{(}x;(u_{v})_{v\in\partial\Lambda_{r}}\big{)}:=\int_{\mathbb{R}^{|\Lambda_{r-1}|}}\prod_{e=\\{v_{1}\to
v_{2}\\}\in
E_{r}\setminus\\{e_{r}\\}}W_{e}(u_{v_{1}}-u_{v_{2}})\prod_{v\in\Lambda_{r-1}}du_{v}$

with the convention $u_{w_{r}}=x$. Observe that the difference between $H_{r}$
from (A.1) and $h_{r}$ above is that the weight of the directed blue edge
$e_{r}$ is included in the former but not in the latter. Note that the
vertices of $e_{r}$ are not in $\Lambda_{r-1}$. Thus in the definition of
$H_{r}$, the edge weight function corresponding to $e_{r}$ can be pulled out
of the integrand leading to

$\displaystyle
H_{r}\big{(}x;(u_{v})_{v\in\partial\Lambda_{r}}\big{)}=h_{r}\big{(}x;(u_{v})_{v\in\partial\Lambda_{r}}\big{)}\cdot
F_{r}(u_{w_{r}+(0,1)}-x)$ (A.5)

where $F_{r}(y)$ is the directed blue edge weight corresponding to $e_{r}$,
i.e., $F_{r}(y):=e^{\vartheta_{r}y-e^{y}}$ or
$F_{r}(y)=e^{-\vartheta_{r}y-e^{-y}}$ depending on the direction of the
$e_{r}$ edge between $w_{r}$ and $w_{r}+(0,1)$. Here $\vartheta_{r}$ is the
parameter linked to the blue edge $e_{r}$.

With the above introduced notation, we now turn towards the proof of (A.2).
Note that given a function $P(x)=e^{-R(x)}$ with $R$ being convex, we have

$\displaystyle P(\delta-\beta)P(\gamma-\alpha)\geq
P(\delta-\alpha)P(\gamma-\beta)$ (A.6)

for all $\alpha,\beta,\gamma,\delta\in\mathbb{R}$ with $\alpha\leq\beta$ and
$\gamma\leq\delta$. All our weight functions in (1.5) are of this type. In
particular, this implies that (A.6) holds for $P=F_{r}$. In view of this and
the relation (A.5), to show (A.2) it suffices to show the same holds for
$h_{r}$ replacing $H_{r}$, i.e.,

$\displaystyle
h_{r}\big{(}s;(u_{v})_{v\in\partial\Lambda_{r}}\big{)}h_{r}\big{(}t;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}}\big{)}\leq
h_{r}\big{(}s;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}}\big{)}h_{r}\big{(}t;(u_{v})_{v\in\partial\Lambda_{r}}\big{)}.$
(A.7)

We shall prove (A.7) via induction. Note that

$\displaystyle h_{1}(x;(u_{v})_{v\in\partial\\{w_{1}\\}})=\prod_{e=\\{v_{1}\to
v_{2}\\}\in E_{1}\setminus\\{e_{1}\\}}W_{e}(u_{v_{1}}-u_{v_{2}})$

is the product of edge weights without any integration and with the convention
$u_{w_{1}}=x$. Applying (A.6) to each such weight function yields (A.7) for
$r=1$. Observe the recursion relation for $h_{r}$:

$\displaystyle
h_{r+1}\big{(}x;(u_{v})_{v\in\partial\Lambda_{r+1}}\big{)}=d_{r}\big{(}x;(u_{v})_{v\in\partial\Lambda_{r+1}}\big{)}\cdot\int_{\mathbb{R}}h_{r}\big{(}y;(u_{v})_{v\in\partial\Lambda_{r}}\big{)}F_{r}(x-y)dy$

where by convention we set $u_{w_{r+1}}=x$ and where we define

$\displaystyle
d_{r}(x;(u_{v})_{v\in\partial\Lambda_{r+1}})=\prod_{e=\\{v_{1}\to v_{2}\\}\in
Q_{r}}W_{e}(u_{v_{1}}-u_{v_{2}})$

with $Q_{r}$ being the set of all red and black edges that have one vertex as
$w_{r+1}$ and another vertex in $\partial\Lambda_{r+1}$, see Figure 19 (B).
Note that the blue edge $e_{r+1}$ between $w_{r+1}$ and $w_{r+1}+(0,1)$ is
excluded from $Q_{r}$. Appealing to (A.6) again, we have

$\displaystyle
d_{r}(s;(u_{v})_{v\in\partial\Lambda_{r+1}})d_{r}(t;(u_{v}^{\prime})_{v\in\partial\Lambda_{r+1}})\leq
d_{r}(s;(u_{v}^{\prime})_{v\in\partial\Lambda_{r+1}})d_{r}(t;(u_{v})_{v\in\partial\Lambda_{r+1}})$
(A.8)

for all $s\geq t$ and for all $u_{v}^{\prime}\geq u_{v}$ with
$v\in\partial\Lambda_{r+1}$. Under same conditions we claim that

$\displaystyle\int_{\mathbb{R}^{2}}h_{r}(y;(u_{v})_{v\in\partial\Lambda_{r}})F_{r}(s-y)h_{r}(x;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}})F_{r}(t-x)dxdy$
(A.9) $\displaystyle\hskip
85.35826pt\leq\int_{\mathbb{R}^{2}}h_{r}(y;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}})F_{r}(s-y)h_{r}(x;(u_{v})_{v\in\partial\Lambda_{r}})F_{r}(t-x)dxdy.$

Combining the above inequality with (A.8) we have (A.7) completing the proof.
To see why (A.9) holds, we split the integrals in (A.9) over $\\{x<y\\}$ and
$\\{y<x\\}$ and swap the $x,y$ labels in the region $\\{y<x\\}$ to get that
(A.9) is equivalent to

$\int_{x<y}A(y)Y(y)B(x)X(x)+C(x)Z(x)D(y)W(y)\leq\int_{x<y}D(y)Y(y)C(x)X(x)+B(x)Z(x)A(y)Z(y)$

where we let

$A(y)=h_{r}(y;(u_{v})_{v\in\partial\Lambda_{r}}),B(x)=h_{r}(x;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}}),C(x)=h_{r}(x;(u_{v})_{v\in\partial\Lambda_{r}}),D(y)=h_{r}(y;(u_{v}^{\prime})_{v\in\partial\Lambda_{r}}),$
$X(x)=F_{r}(t-x),Y(y)=F_{r}(s-y),W(y)=F_{r}(t-y),Z(x)=F_{r}(s-x).$

The integral above can be rewritten as
$\int_{x<y}\big{(}A(y)B(x)-C(x)D(y)\big{)}\big{(}X(x)Y(y)-W(y)Z(x)\big{)}$ and
thus it suffices to show for each $x\leq y$ the integrand is non-positive. By
induction hypothesis, $A(y)B(x)\leq C(x)D(y)$ for all $x\leq y$ and since the
weight function $F_{r}$ satisfies (A.6) (with $P=F_{r}$), we also have
$X(x)Y(y)\geq W(y)Z(x)$. This proves (A.9), completing the proof of the lemma.
∎

##  Appendix B Basic properties of log-gamma type random variables

In this section we collect some basic facts about log-gamma type random
variables. Towards this end, for each $\theta,\kappa>0$, and
$m\in\mathbb{Z}_{\geq 1}$ we consider the following function:

$\displaystyle
H_{\theta,(-1)^{m},\kappa}(y):=\frac{\kappa^{\theta}}{\Gamma(\theta)}\exp(\theta(-1)^{m}y-\kappa
e^{(-1)^{m}y}).$

It is plain to check $H$ is a valid probability density function. Observe that
$H_{\theta,(-1)^{m},1}\equiv G_{\theta,(-1)^{m}}$ where $G$ is defined in
(2.2). The following lemma collects some useful properties of $H$. Its proof
follows via straightforward computations and is hence omitted.

######  Lemma B.1.

Suppose $X\sim H_{\theta,1,\kappa}$. Then $-X\sim H_{\theta,-1,\kappa}$. For
all $\alpha>-\theta$ we have $\mathbb{E}[e^{\alpha
X}]=\frac{\Gamma(\alpha+\theta)}{\kappa^{\alpha}\Gamma(\theta)}$.

We next define generalized $\mathpzc{HSLG}$ $\Theta$-Gibbs measures in the
same vein as $\mathpzc{HSLG}$ $\Theta$-Gibbs measures (see Definition 1.2) but
by considering the weight function

$\displaystyle\widetilde{W}_{e}(x)=\begin{cases}\exp(\vartheta x-\kappa
e^{x})&\mbox{ if $e$ is
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{blue}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}($\vartheta$)}}\\\
\exp(-\gamma e^{x})&\mbox{ if $e$ is
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{black}}}\\\
\exp(-\alpha x)&\mbox{ if $e$ is
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{red}}.}\end{cases}$

instead of $W$ defined in (1.5). $\kappa=\gamma=1$ in above weights lead to
the usual Gibbs measures. The following result ensures that generalized
$\mathpzc{HSLG}$ $\Theta$-Gibbs measures (and hence the usual ones from
Definition 1.2) are well-defined.

######  Lemma B.2.

Fix any $\gamma,\kappa>0,$
$\Theta:=\\{\vartheta_{m,n}>0:(m,n)\in\mathbb{Z}_{\geq 1}^{2}\\}$ and
$\alpha>-\min\\{\vartheta_{m,n}:(m,n)\in\mathbb{Z}_{\geq 1}^{2}\\}$. Recall
the graph $G$ from Section 1.3.1 used in defining $\mathpzc{HSLG}$
$\Theta$-Gibbs measures. Given a domain $\Lambda$ and a boundary condition
$\\{u_{i,j}:(i,j)\in\partial\Lambda\\}$, we have

$\displaystyle\int_{\mathbb{R}^{|\Lambda|}}\prod_{e=\\{v_{1}\to v_{2}\\}\in
E(\Lambda\cup\partial\Lambda)}\widetilde{W}_{e}(u_{v_{1}}-u_{v_{2}})\prod_{v\in\Lambda}du_{v}<\infty.$

Let us suppose $|u_{i,j}|\leq R$ for all $(i,j)\in\partial\Lambda$. Let us
assume $\Lambda=\mathcal{K}_{k,T}$ or $\mathcal{K}^{\prime}_{k,T}$ defined in
(2.4). There exists a constant $\mathrm{C}$ that depends only on
$\gamma,\kappa,\theta,$ and $\alpha$ such that

$\displaystyle\int_{\mathbb{R}^{|\Lambda|}}\prod_{e=\\{v_{1}\to v_{2}\\}\in
E(\Lambda\cup\partial\Lambda)}\widetilde{W}_{e}(u_{v_{1}}-u_{v_{2}})\prod_{v\in\Lambda}du_{v}\leq\mathrm{C}^{kT+R}.$

###### Proof.

We shall prove this lemma only for the homogeneous case, i.e.
$\vartheta_{m,n}\equiv\theta>0$. The general case is notationally more
cumbersome but follows in an exact same manner as the homogeneous case. First
note that for red edges $\\{v_{1}\to v_{2}\\}$ the corresponding weight
function $W_{e}(u_{v_{1}}-u_{v_{2}})$ factors out as $e^{-\alpha
u_{v_{1}}}\cdot e^{\alpha u_{v_{2}}}$. Hence they can be viewed as vertex
weight functions. More specifically, at each vertex $(k,1)$ we can associate
the vertex weight function $V_{k}(u):=e^{(-1)^{k}\alpha u}$. They replace the
role of red edge weights. We denote this vertex weights as red circles in
Figure 20. We now divide our analysis into two cases based on the value of
$\alpha$.

(a)

(b)

(c)

$u_{0}$$u_{k}$$u_{0}$$u_{k}$ (d)

$u_{k}$ (e)

$c$$u$ (f)

Figure 20. (A) A possible domain $\Lambda$. (B) Reduction in the case of
$\alpha\in(-\theta,\theta)$. (C) Reduction in the case of $\alpha>0$. (D) Type
I Gibbs measures. The figure shows two of them of even length. It may also
have odd length with one edge at either of the end removed. (E) Type II Gibbs
measures. It may also have odd length with one edge at right end removed. (F)
Few examples of Type III Gibbs measures.

Suppose $\alpha\in(-\theta,\theta)$. As black edge weights are less than $1$,
we may drop all of them to get a Gibbs measure based on the blue and red edge
weights only, see Figure 20 (B). The integral of the reduced Gibbs measure can
be viewed as a product of integrals of several smaller Gibbs measures that are
two types: Type I and Type II as in Figure 20 (D) and (E) respectively. Type I
Gibbs measures are those for which red vertex weights do not appear. The
integral corresponding to Type I takes the following form:

$\displaystyle\left(\kappa^{\theta}(\Gamma(\theta))^{-1}\right)^{k}\int_{\mathbb{R}^{k-1}}\prod_{i=1}^{k}H_{\theta,\kappa,(-1)^{i+m}}(u_{i-1}-u_{i})\prod_{i=1}^{k-1}du_{i}$

where $u_{0}$ and $u_{k}$ are in $\partial\Lambda$. In this case, we may use
$H_{\theta,\kappa,(-1)^{i+m}}(u_{k-1}-u_{k})\leq\mathrm{C}$ and the fact that
$H$ is a probability density function to get that the integral is bounded by
$\mathrm{C}\cdot\left(\kappa^{\theta}(\Gamma(\theta))^{-1}\right)^{k}$. Type
II Gibbs measures are the ones where red vertex weights are present. The
integral corresponding to the Type II Gibbs measures takes the form

$\displaystyle\int_{\mathbb{R}^{k}}\prod_{i=1}^{k}e^{(-1)^{m}\alpha
u_{0}}\cdot e^{(-1)^{i+m}\theta(u_{i-1}-u_{i})-\kappa
e^{(-1)^{i+m}(u_{i-1}-u_{i})}}\prod_{i=1}^{k}du_{i}.$

The integrand can be manipulated to show that the above integral is equal to

$\displaystyle e^{(-1)^{m+k-1}\alpha
u_{k}}\prod_{i=1}^{k-1}(\Gamma(\theta+(-1)^{m+i+1}\alpha))\kappa^{-\theta+(-1)^{m+i}\alpha}\int_{\mathbb{R}^{k}}\prod_{i=1}^{k}H_{\theta+(-1)^{m+i+1}\alpha,\kappa,(-1)^{i-1}}(x_{i})\prod_{i=1}^{k}dx_{i}$
$\displaystyle=e^{(-1)^{m+k-1}\alpha
u_{k}}\prod_{i=1}^{k-1}(\Gamma(\theta+(-1)^{m+i+1}\alpha))\kappa^{-\theta+(-1)^{m+i}\alpha}.$

Since we factor out the measure into these independent pieces and all
integrals are finite, the claim follows for $\alpha\in(-\theta,\theta)$.

For $\alpha>0$, we remove all the black edges except the ones connecting
$(2i-1,1)$ to $(2i,1)$ (again since weights of black edges are atmost $1$,
removal of them only increases the integrand). This leads to a reduced Gibbs
measures shown in Figure 20 (C). The reduced Gibbs measure decomposes into
several Type I Gibbs measures and Type III Gibbs measures. Type III Gibbs
measures are those for which red vertex weights do appear. Because of the
presence of black edge in this case, Type III Gibbs measures are different
from Type II. A few of the possible Type III Gibbs measures are shown in
Figure 20 (F).

  * •

If a Type III Gibbs measure has two red vertices in its domain or boundary, we
may use the fact that the weight of the figure

$a$$b$$c$

is $e^{(\theta+\alpha)(b-c)-\kappa e^{b-c}}\cdot e^{\alpha(c-a)-\gamma
e^{c-a}}\leq e^{(\theta+\alpha)(b-c)-\kappa
e^{b-c}}\cdot\left(\sup_{x\in\mathbb{R}}e^{\alpha x-\gamma
e^{x}}\right)\leq\mathrm{C}\cdot e^{(\theta+\alpha)(b-c)-\kappa e^{b-c}}$.

  * •

If a Type III Gibbs measure has only one red vertex in its domain or boundary,
then it must contain either of the two following figures

$b$$c$$a$$c$

with $c\in\partial\Lambda$. The corresponding weights are $e^{\alpha c}\cdot
e^{(\theta+\alpha)(b-c)-\kappa e^{b-c}}\leq\mathrm{C}^{\prime}e^{\alpha c}$
and $e^{-\alpha c}\cdot e^{\alpha(c-a)-\gamma
e^{c-a}}\leq\mathrm{C}^{\prime}e^{-\alpha c}$ respectively where
$\mathrm{C}^{\prime}:=\sup_{x\in\mathbb{R}}e^{\alpha x-\gamma e^{x}}$.

Based on the kind of Type III Gibbs measures, we may insert the above bounds
on the Gibbs weights in the integrand of this type of Gibbs measures. The
resulting integral can then be computed explicitly to yield a bound of the
form $\mathrm{C}^{V}e^{|\alpha c|}$ where $V$ is the number of vertices in the
Gibbs measures. For example, for the middle figure in Figure 20 (F) we have
(with $u_{4}:=u$)

$\displaystyle\left(\kappa^{-\theta}\Gamma(\theta)\right)^{4}\int_{\mathbb{R}^{4}}e^{-\alpha
u_{0}}e^{-\gamma
e^{c-u_{0}}}\prod_{i=0}^{3}H_{\theta,\kappa,(-1)^{i}}(u_{i}-u_{i+1})du_{i}$
$\displaystyle\leq\left(\kappa^{-\theta}\Gamma(\theta)\right)^{4}\cdot\mathrm{C}e^{-\alpha
c}\int_{\mathbb{R}^{4}}\prod_{i=0}^{3}H_{\theta,\kappa,(-1)^{i}}(u_{i}-u_{i+1})du_{i}\leq\left(\kappa^{-\theta}\Gamma(\theta)\right)^{4}\cdot\mathrm{C}e^{|\alpha
c|}.$

This establishes the lemma for $\alpha>0$. ∎

We end this section we two lemmas concerning with the tail properties of
$\mathsf{f}_{\theta}$ and $\xi$-distributions defined in (2.3) and (4.5)
respectively.

######  Lemma B.3.

For all $x\in\mathbb{R}$ we have

$e^{-2e}\leq(\Gamma(\theta))^{2}\mathsf{f}_{\theta}(x)e^{\theta|x|}\leq\Gamma(2\theta),$

where $\mathsf{f}_{\theta}$ is defined in (2.3).

###### Proof.

Since $\mathsf{f}_{\theta}$ is symmetric, it suffices to show the lemma for
$x>0$. We have

$\displaystyle\big{(}\Gamma(\theta)\big{)}^{2}\mathsf{f}_{\theta}(x)=\int_{\mathbb{R}}e^{\theta
y-e^{y}+\theta(y-x)-e^{y-x}}dy=e^{-\theta x}\int_{\mathbb{R}}e^{2\theta
y-e^{y}-e^{y-x}}dy.$

Now for the lower bound we observe

$\displaystyle\int_{\mathbb{R}}e^{2\theta
y-e^{y}-e^{y-x}}dy\geq\int_{0}^{1}e^{2\theta y-e^{y}-e^{y-x}}dy\geq e^{-2e},$

whereas for the upper bound we have

$\displaystyle\int_{\mathbb{R}}e^{2\theta
y-e^{y}-e^{y-x}}dy\leq\int_{\mathbb{R}}e^{2\theta y-e^{y}}dy=\Gamma(2\theta).$

∎

######  Lemma B.4.

Fix any $\theta_{0}>1$. For any
$\theta_{1},\theta_{2}\in[\theta_{0}^{-1},\theta_{0}]$ and $a,b\in\mathbb{R}$,
define the random variable $X_{\theta_{1},\theta_{2};\pm
1}^{(a,b)}\sim\xi_{\theta_{1},\theta_{2};\pm 1}^{(a,b)}$ where
$\xi_{\theta_{1},\theta_{2};\pm 1}^{(a,b)}$ is defined in (4.5). There exists
a constant $\mathrm{C}>0$ depending only $\theta_{0}$ such that for all
$\theta_{1},\theta_{2}\in[\theta_{0}^{-1},\theta_{0}]$, for all
$a,b\in\mathbb{R}$ and for all $r\geq|a-b|$ we have

$\displaystyle\mathbb{P}\Big{(}X_{\theta_{1},\theta_{2};\pm
1}^{(a,b)}\notin\big{[}\min\\{a,b\\}-2r,\max\\{a,b\\}+2r]\Big{)}\leq\mathrm{C}e^{-\frac{1}{\mathrm{C}}r}.$

###### Proof.

Fix any $\theta_{1},\theta_{2}\in[\theta_{0}^{-1},\theta_{0}]$. We shall prove
the bound for $X_{\theta_{1},\theta_{2};1}^{(a,b)}$. The proof for the case
$X_{\theta_{1},\theta_{2};-1}^{(a,b)}$ is analogous. Without loss of
generality assume $b\leq a$. Observe that

$\mathbb{P}\Big{(}X_{\theta_{1},\theta_{2};\pm
1}^{(a,b)}\notin\big{[}\min\\{a,b\\}-2r,\max\\{a,b\\}+2r\big{]}\Big{)}\leq\frac{\int_{(-\infty,b-2r]\cup[a+2r,\infty)}G_{\theta_{1},1}(a-x)G_{\theta_{2},1}(b-x)}{\int_{a}^{a+1}G_{\theta_{1},1}(a-x)G_{\theta_{2},1}(b-x)}.$
(B.1)

Note that

$\displaystyle\int_{a}^{a+1}G_{\theta_{1},1}(a-x)G_{\theta_{2},1}(b-x)\geq\tfrac{1}{\mathrm{C}}\cdot
e^{-\max\\{\theta_{1},\theta_{2}\\}\cdot(a-b)},$

where we have used the fact that $G_{\beta,1}(-y)\geq\mathrm{C}^{-1}e^{-\beta
y}$ (recall $G$ from (2.2)). Similarly

$\displaystyle\int_{x\leq
b-2r}G_{\theta_{1},1}(a-x)G_{\theta_{2},1}(b-x)dx+\int_{x\geq
a+2r}G_{\theta_{1},1}(a-x)G_{\theta_{2},1}(b-x)dx\leq\mathrm{C}\cdot
e^{-2(\theta_{1}+\theta_{2})r}.$

Thus as long as $r\geq a-b$, inserting the above two bounds back in (B.1) and
adjusting the constant $\mathrm{C}$ we get the desired result. ∎

##  Appendix C Estimates for non-intersection probability

In this section, we study non-intersection probability of random walks and
random bridges (defined in Definition 3.6), and modified random bridges
(defined in Definition 4.9). Throughout this section we shall assume the
increments are drawn from a density $\mathsf{f}$ that satisfies the following
assumptions. It is worth recalling that due to Corollary 4.6,
$\mathsf{f}_{\theta}$ defined in (2.3) satisfies the conditions of Assumption
C.1 and hence all results of this section can be applied to random walks with
that increment law.

######  Assumption C.1 (Assumption on the increments).

The density $\mathsf{f}$ satisfies the following properties.

  1. (1)

The density $\mathsf{f}$ is symmetric and $\log\mathsf{f}$ is concave.

  2. (2)

Let $\psi$ denote the characteristic function corresponding to $\mathsf{f}$.
$|\psi|$ is integrable. Given any $\delta>0$, there exists $\eta$ such that
$\sup_{t\geq\delta}|\psi(t)|=\eta<1$.

  3. (3)

There exists a constant $\mathrm{C}>0$ such that
$\mathsf{f}(x)\leq\mathrm{C}e^{-|x|/\mathrm{C}}$. In particular, this implies
that if $X\sim\mathsf{f}$, there exists $v>0$ such that and

$\displaystyle\sup_{|t|\leq v}\big{[}\mathbb{E}[e^{tX}]\big{]}<\infty.$

In other words $X$ is an subexponential random variable.

The following lemma concerns with sharp rate of convergence of the probability
density function of $(X(1)+X(2)+\cdots+X(n))/\sqrt{n}$, where
$X(i)\stackrel{{\scriptstyle i.i.d.}}{{\sim}}\mathsf{f}$, to the Gaussian
density with appropriate variance.

######  Lemma C.2.

Let $\mathsf{f}^{*n}$ be the $n$-fold convolution of $\mathsf{f}$. There
exists a constant $\mathrm{C}>0$ such that

$\displaystyle\sup_{|x|\leq(\log
n)^{2}}\left|\frac{\sqrt{n}\mathsf{f}^{*n}(x\sqrt{n})}{\phi_{\sigma}(x)}-1\right|\leq\mathrm{C}\cdot
n^{-3/4}.$

where
$\phi_{\sigma}(x):=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{x^{2}}{2\sigma^{2}}}$
and $\sigma^{2}:=\int x^{2}\mathsf{f}(x)dx$.

###### Proof.

This proof is adapted from Theorem 5 in Chapter XV in [Fel08]. In what follows
we shall use the big $O$-notation and write $a_{n}=O(b_{n})$ if $a_{n}/b_{n}$
is uniformly bounded above by some universal constant. Let $\psi$ denote the
characteristic function of $\mathsf{f}_{\theta}$. The explicit form of $\psi$
was given in (4.13). In particular, $|\psi|$ is integrable. In what follows,
for simplicity we will assume $\sigma^{2}=1$. Set
$f_{n}(x):=\int_{\mathbb{R}}e^{itx}\psi^{n}(t/\sqrt{n})dt$. By the Fourier
inversion formula, $f_{n}(x)$ is the density of
$(X(1)+X(2)+\cdots+X(n))/\sqrt{n}$ where $X(i)\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\mathsf{f}$. Hence we have
$\sqrt{n}\mathsf{f}^{*n}(x/\sqrt{n})=f_{n}(x)$. Since $\mathsf{f}$ is
symmetric and has all finite moments, by Taylor expansion we have

$\displaystyle\psi(t/\sqrt{n})=1-\tfrac{t^{2}}{2n}+O(\tfrac{t^{4}}{n^{2}}).$

Set $\alpha=1/16$. Thus for $|t|\leq n^{\alpha}$, we have
$\psi(t/\sqrt{n})=1-\frac{t^{2}}{2n}+O(n^{4\alpha-2})=e^{-t^{2}/2n+O(n^{4\alpha-2})}$.
Thus $\psi^{n}(t/\sqrt{n})=e^{-t^{2}/2}(1+O(n^{-3/4}))$, where the $O$ term is
free of $t$ in that specified range. Thus,

$\displaystyle f_{n}(x)$ $\displaystyle=(1+O(n^{-3/4}))\int_{|t|\leq
n^{\alpha}}e^{itx}e^{-t^{2}/2}dt+\int_{|t|\geq
n^{\alpha}}e^{itx}\psi^{n}(t/\sqrt{n})dt$
$\displaystyle=(1+O(n^{-3/4}))\int_{\mathbb{R}}e^{itx}e^{-t^{2}/2}dt+\int_{|t|\geq
n^{\alpha}}e^{itx}\psi^{n}(t/\sqrt{n})dt-(1+O(n^{-3/4}))\int_{|t|\geq
n^{\alpha}}e^{itx}e^{-t^{2}/2}dt.$

We next compute the order of the last two integrals above. Clearly
$\int_{|t|\geq n^{\alpha}}e^{-t^{2}/2}dt\leq\mathrm{C}e^{-cn^{2\alpha}}$. For
the second one, we choose $\delta>0$ small enough such that $|\psi(t)|\leq
e^{-t^{2}/4}$ for all $|t|\leq\delta$. This implies

$\displaystyle\int_{n^{\alpha}\leq
t\leq\sqrt{n}\delta}|\psi^{n}(t/\sqrt{n})|dt\leq\mathrm{C}e^{-cn^{2\alpha}}.$

For $|t|\geq\sqrt{n}\delta$, we know $\sup_{t\geq\delta}|\psi(t)|=\eta<1$ by
part (2) of Lemma 4.7. This forces

$\displaystyle\int_{|t|\geq\sqrt{n}\delta}|\psi^{n}(t/\sqrt{n})|dt\leq\eta^{n-1}\sqrt{n}\int_{\mathbb{R}}\psi(t)dt.$

Thus the error integrals are at most
$\mathrm{C}e^{-\frac{1}{\mathrm{C}}n^{1/8}}$ in absolute value uniform in $x$.
Furthermore if we assume $|x|\leq(\log n)^{2}$,
$\phi_{1}(x)\geq\frac{1}{\sqrt{2\pi}}e^{-(\log n)^{2}/2}$, which dominates the
error coming from the integrals. Hence we may divide $\phi_{1}(x)$ and still
obtain that the errors are going to zero. ∎

For any $p\in[0,\infty)$, $s,t\in\llbracket 1,n\rrbracket$, we set

$\displaystyle\mathsf{NI}_{p}\llbracket
s,t\rrbracket:=\\{S_{1}(k)-S_{2}(k)\geq-p,\mbox{ for all }k\in\llbracket
s,t\rrbracket\\}.$

When $s=2,t=n-1$ we write $\mathsf{NI}_{p}:=\mathsf{NI}_{p}\llbracket
2,n-1\rrbracket$ so that it coincides with the $\mathsf{NI}_{p}$ event defined
in (1.18). When $a_{1}-a_{2}=O(1)$, it is well known that
$\mathbb{P}^{n;(a_{1},a_{2})}(\mathsf{NI})=O(n^{-1/2})$. We record this
classical fact in the following lemma.

######  Lemma C.3.

For all $(a_{1},a_{2})\in\mathbb{R}^{2}$ we have
$\mathbb{P}^{n;(a_{1},a_{2})}(\mathsf{NI})\leq\mathrm{C}\frac{\max\\{a_{1}-a_{2},1\\}}{\sqrt{n}}$
for some absolute constant $\mathrm{C}>0$. If in addition $a_{1}\geq a_{2}$,
we have
$\mathbb{P}^{n;(a_{1},a_{2})}(\mathsf{NI})\geq\frac{\mathrm{C}^{-1}}{\sqrt{n}}$.

###### Proof.

The first part is [Koz77, Theorem A] and the second part is [Spi60, Theorem
3.5]. ∎

We again remind the readers that the above result, as well as all the results
stated below within this section, the random walks/bridges or the modified
random bridges are assumed to have increments drawn from a density
$\mathsf{f}$ satisfying Assumption C.1. In many of our arguments below, we
shall often appeal to stochastic monotonicity of non-intersecting random walks
or bridges with respect to boundary data. We record this result below.

######  Proposition C.4 (Stochastic monotonicity of random bridges and random
walks).

Fix $n\in\mathbb{Z}_{\geq 1}$. et $a_{i}^{(j)},b_{i}^{(j)}\in[-\infty,\infty]$
for $i,j\in\\{1,2\\}$. Suppose $a_{i}^{(1)}\geq a_{i}^{(2)}$ and
$b_{i}^{(1)}\geq b_{i}^{(2)}$ for $i\in\\{1,2\\}$.

  1. (a)

There exists a probability space that supports a collection of random
variables

$\displaystyle\big{(}S_{1}^{(j)}(k),S_{2}^{(j)}(k):j\in\\{1,2\\},k\in\llbracket
1,n\rrbracket\big{)}$

such that $S_{i}^{(1)}(k)\geq S_{i}^{(2)}(k)$ for all $i\in\\{1,2\\}$ and
$k\in\llbracket 1,n\rrbracket$, and marginally
$(S_{1}^{(j)}(\cdot),S_{2}^{(j)}(\cdot))\sim\mathbb{P}^{n;(a_{1}^{(j)},a_{2}^{(j)}),(b_{1}^{(j)},b_{2}^{(j)})}(\cdot\mid\mathsf{NI})$
for each $j\in\\{1,2\\}$.

  2. (b)

There exists a probability space that supports a collection of random
variables

$\displaystyle\big{(}S_{1}^{(j)}(k),S_{2}^{(j)}(k):j\in\\{1,2\\},k\in\llbracket
1,n\rrbracket\big{)}$

such that $S_{i}^{(1)}(k)\geq S_{i}^{(2)}(k)$ for all $i\in\\{1,2\\}$ and
$k\in\llbracket 1,n\rrbracket$, and marginally
$(S_{1}^{(j)}(\cdot),S_{2}^{(j)}(\cdot))\sim\mathbb{P}^{n;(a_{1}^{(j)},a_{2}^{(j)})}(\cdot\mid\mathsf{NI})$
for each $j\in\\{1,2\\}$.

###### Proof.

This proposition of as a discrete analogue of Lemma 2.6 in [CH14] and is true
under log-concavity assumption on $\mathsf{f}$. Instead of giving the full
details, we explain the two possible ways in proving this proposition. One is
via Markov Chain arguments as done, for example, in [CH14, Wu20, Ser23]
previously. In [Ser23] for example, the stochastic monotonicity was proved for
non-intersecting random bridges under discrete bounded increment assumption.
One can take a discrete to continuous limit of the increments to obtain the
above result. The second route is via direct construction argument is the
style of [BCD23, DW21b]. We have, in fact, already adapted that technique in
proving stochastic monotonicity for $\mathpzc{HSLG}$ Gibbs measures in
Appendix A and a similar argument can be carried out to prove Proposition C.4.
∎

Next we study diffusive properties of the random walks under the non-
intersecting event.

######  Lemma C.5.

Given any $\varepsilon>0$ there exists a constant $\delta(\varepsilon)>0$ such
that for all $n\in\mathbb{Z}_{\geq 1}$ and $(a_{1},a_{2})\in\mathbb{R}^{2}$ we
have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}S_{1}(n)-S_{2}(n)\geq\delta\sqrt{n}\mid\mathsf{NI}\bigg{)}\geq
1-\varepsilon,$ (C.1)
$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}\sup_{k\in\llbracket
1,n\rrbracket}(S_{1}(k)-S_{2}(k))\leq\delta^{-1}\sqrt{n}+\max\\{a_{1}-a_{2},0\\}\mid\mathsf{NI}\bigg{)}\geq
1-\varepsilon,$ (C.2)
$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}\inf_{k\in\llbracket
1,n\rrbracket}S_{1}(k)-a_{1}\geq-\delta^{-1}\sqrt{n}\mid\mathsf{NI}\bigg{)}\geq
1-\varepsilon,$ (C.3)
$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}\sup_{k\in\llbracket
1,n\rrbracket}S_{2}(k)-a_{2}\leq\delta^{-1}\sqrt{n}\mid\mathsf{NI}\bigg{)}\geq
1-\varepsilon.$ (C.4)

We remark that Lemma C.5 holds if $\mathsf{NI}=\mathsf{NI}_{0}\llbracket
2,n-1\rrbracket$ is replaced by $\mathsf{NI}_{0}\llbracket 2,n\rrbracket$. The
same argument presented below essentially works when the conditional event is
the latter one instead.

###### Proof.

Proof of Eq. (C.1). Set $U(k):=S_{1}(k)-S_{2}(k)$. Under
$\mathbb{P}^{n;(a_{1},a_{2})}$, $(U(k))_{k=1}^{n}$ is a random walk starting
from $a_{1}-a_{2}$ with increments drawn from $\mathsf{f}\ast\mathsf{f}$. The
non-intersection condition for $(S_{1}(k),S_{2}(k))_{k=1}^{n}$ translates to
$(U(k))_{k=1}^{n}$ staying non-negative. If $a_{1}\geq a_{2}$, since
$\\{U(n)\geq\delta\sqrt{n}\\}$ is an increasing event with respect to the
boundary conditions, we have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}S_{1}(n)-S_{2}(n)\geq\delta\sqrt{n}\mid\mathsf{NI}\bigg{)}\geq\mathbb{P}^{n;(0,0)}\bigg{(}S_{1}(n)-S_{2}(n)\geq\delta\sqrt{n}\mid\mathsf{NI}\bigg{)}$

But under $\mathbb{P}^{n;(0,0)}$, it is known from [Igl74] that the random
walk $(U(k))_{k=1}^{n}$, conditioned to stay non-negative converges weakly to
a Brownian meander under diffusive scaling. Since the endpoint of a Brownian
meander is a strictly positive continuous random variable, we thus have (C.1).
If $a_{1}\leq a_{2}$, the argument is a bit more involved. We first write the
(complement of the) conditional probability as a ratio:

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\big{(}S_{1}(n)-S_{2}(n)\leq\delta\sqrt{n}\mid\mathsf{NI}\big{)}=\frac{\mathbb{P}^{n;(a_{1},a_{2})}\big{(}\big{\\{}S_{1}(n)-S_{2}(n)\leq\delta\sqrt{n}\big{\\}}\cap\mathsf{NI}\big{)}}{\mathbb{P}^{n;(a_{1},a_{2})}\big{(}\mathsf{NI}\big{)}}.$

For the denominator we condition on $(S_{1}(2),S_{2}(2))$ and use the lower
bound from Lemma C.3:

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}(\mathsf{NI})$
$\displaystyle=\mathbb{E}^{n;(a_{1},a_{2})}\left[\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\mathbb{E}^{n;(a_{1},a_{2})}\left[\mathbf{1}_{\mathsf{NI}_{0}\llbracket
3,n-1\rrbracket}\mid\sigma(S_{1}(2),S_{2}(2))\right]\right]$ (C.5)
$\displaystyle\geq\frac{\mathrm{C}^{-1}}{\sqrt{n}}\mathbb{P}^{n;(a_{1},a_{2})}({S_{1}(2)\geq
S_{2}(2)}).$ (C.6)

For the numerator, we again condition on $(S_{1}(2),S_{2}(2))$ to get

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\big{(}\big{\\{}S_{1}(n)-S_{2}(n)\leq\delta\sqrt{n}\big{\\}}\cap\mathsf{NI}\big{)}$
$\displaystyle=\mathbb{E}\left[\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\mathbb{E}\left[\mathbf{1}_{\\{S_{1}(n)-S_{2}(n)\leq\delta\sqrt{n}\\}\cap\mathsf{NI}_{0}\llbracket
3,n-1\rrbracket}\mid\sigma(S_{1}(2),S_{2}(2))\right]\right].$ (C.7)

Upon conditioning on $(S_{1}(2),S_{2}(2))$, the random walks starts at
$(S_{1}(2),S_{2}(2))$. For any $b_{1}\geq b_{2}$, utilizing the upper bound
from Lemma C.3 and stochastic monotonicity we have

$\displaystyle\mathbb{P}^{n-1;(b_{1},b_{2})}\left(\\{S_{1}(n-1)-S_{2}(n-1)\leq\delta\sqrt{n}\\}\cap\mathsf{NI}_{0}\llbracket
2,n-2\rrbracket\right)$
$\displaystyle=\mathbb{P}^{n-1;(b_{1},b_{2})}(\mathsf{NI}_{0}\llbracket
2,n-2\rrbracket)\mathbb{P}^{n-1;(b_{1},b_{2})}\left(\\{S_{1}(n-1)-S_{2}(n-1)\leq\delta\sqrt{n}\\}\mid\mathsf{NI}_{0}\llbracket
2,n-2\rrbracket\right)$
$\displaystyle\leq\frac{\mathrm{C}}{\sqrt{n}}\max\\{b_{1}-b_{2},1\\}\cdot\mathbb{P}^{n-1;(0,0)}\left({\\{S_{1}(n-1)-S_{2}(n-1)\leq\delta\sqrt{n}\\}\mid\mathsf{NI}_{0}\llbracket
2,n-2\rrbracket}\right)$

Taking $b_{1}=S_{1}(2)$ and $b_{2}=S_{2}(2)$, we insert the above bound back
in (C.7) to get

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\big{(}\big{\\{}S_{1}(n)-S_{2}(n)\leq\delta\sqrt{n}\big{\\}}\cap\mathsf{NI}\big{)}$
$\displaystyle\hskip
28.45274pt\leq\frac{\mathrm{C}}{\sqrt{n}}\mathbb{E}^{n;(a_{1},a_{2})}\left[\mathbf{1}_{S_{1}(2)\geq
S_{2}(2)}\max\\{S_{1}(2)-S_{2}(2),1\\}\right]$ (C.8) $\displaystyle\hskip
113.81102pt\cdot\mathbb{P}^{n-1;(0,0)}\left({\\{S_{1}(n-1)-S_{2}(n-1)\leq\delta\sqrt{n}\\}\mid\mathsf{NI}_{0}\llbracket
2,n-2\rrbracket}\right)$

Note that under $\mathbb{P}^{n;(a_{1},a_{2})}$,
$S_{1}(2)-S_{2}(2)\stackrel{{\scriptstyle(d)}}{{=}}Z-b$ where
$Z\sim\mathsf{f}\ast\mathsf{f}$ and $b=a_{2}-a_{1}\geq 0$. We claim that there
exists a constant $\mathrm{C}>0$ such that for all $b\geq 0$ we have

$\displaystyle{\mathbb{E}\left[\max\\{Z-b,1\\}\mathbf{1}_{Z\geq
b}\right]}\leq\mathrm{C}\cdot\mathbb{P}(Z\geq b).$ (C.9)

Plugging this bound in the expectation in (C.8) we have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\big{(}\big{\\{}S_{1}(n)-S_{2}(n)\leq\delta\sqrt{n}\big{\\}}\cap\mathsf{NI}\big{)}$
$\displaystyle\leq\frac{\mathrm{C}}{\sqrt{n}}\mathbb{P}^{n;(a_{1},a_{2})}({S_{1}(2)\geq
S_{2}(2)})\\!\cdot\\!\mathbb{P}^{n-1;(0,0)}\left({\\{S_{1}(n-1)-S_{2}(n-1)\leq\delta\sqrt{n}\\}\mid\mathsf{NI}_{0}\llbracket
2,n-2\rrbracket}\right)$

Combining this with the lower bound on the denominator from (C.6) we get that

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\big{(}S_{1}(n)-S_{2}(n)\leq\delta\sqrt{n}\mid\mathsf{NI}\big{)}\leq\mathrm{C}\\!\cdot\\!\mathbb{P}^{n-1;(0,0)}\left({\\{S_{1}(n-1)-S_{2}(n-1)\leq\delta\sqrt{n}\\}\mid\mathsf{NI}_{0}\llbracket
2,n-2\rrbracket}\right)\\!.\\!$

From here, we can again appeal to [Igl74] and Brownian meander properties to
show that the above bound can be made arbitrarily small by choosing $\delta$
small enough. Thus we are left to (C.9).

Suppose $Z\sim\mathsf{f}\ast\mathsf{f}$. Observe that for any $b\in\mathbb{R}$
we have

$\displaystyle\mathbb{E}\left[\max\\{Z-b,1\\}\mathbf{1}_{Z\geq b}\right]$
$\displaystyle=\sum_{k=1}^{\infty}\mathbb{E}\left[\max\\{Z-b,1\\}\mathbf{1}_{Z\in[b+k-1,b+k]}\right]$
$\displaystyle\leq\mathbb{P}(Z\geq
b)\left[1+\sum_{k=2}^{\infty}k\cdot\frac{\mathbb{P}(Z\geq
b+k-1)}{\mathbb{P}(Z\geq b)}\right]$

If we assume $b\geq 0$ additionally, using exponential tail bounds for
$\mathsf{f}$ (and hence $\mathsf{f}\ast\mathsf{f}$), we may get a constant
$\mathrm{C}>0$ free of $b$, such that $\frac{\mathbb{P}(Z\geq
b+k-1)}{\mathbb{P}(Z\geq b)}\leq\mathrm{C}e^{-k/\mathrm{C}}$ for all $k\geq
2$. This ensure the infinite sum above can be bounded uniformly over
$b\in[0,\infty)$. This proves (C.9).

Proof of Eq. (C.2). Set $U(k):=S_{1}(k)-S_{2}(k)$. To obtain (C.2), observe
the following inequalities

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}\sup_{k\in\llbracket
1,n\rrbracket}U(k)\leq\delta^{-1}\sqrt{n}+\max\\{a_{1}-a_{2},0\\}\mid\bigcap_{k=2}^{n}\\{U(k)\geq
0\\}\bigg{)}$
$\displaystyle\geq\mathbb{P}^{n;(\max\\{a_{1},a_{2}\\},a_{2})}\bigg{(}\sup_{k\in\llbracket
1,n\rrbracket}U(k)\leq\delta^{-1}\sqrt{n}+\max\\{a_{1}-a_{2},0\\}\mid\bigcap_{k=2}^{n}\\{U(k)\geq
0\\}\bigg{)}$
$\displaystyle\geq\mathbb{P}^{n;(\max\\{a_{1},a_{2}\\},a_{2})}\bigg{(}\sup_{k\in\llbracket
1,n\rrbracket}U(k)\leq\delta^{-1}\sqrt{n}+\max\\{a_{1}-a_{2},0\\}\mid\bigcap_{k=2}^{n}\\{U(k)\geq\max\\{a_{1}-a_{2},0\\}\\}\bigg{)}$
$\displaystyle=\mathbb{P}^{n;(a_{2},a_{2})}\bigg{(}\sup_{k\in\llbracket
1,n\rrbracket}U(k)\leq\delta^{-1}\sqrt{n}\mid\bigcap_{k=2}^{n}\\{U(k)\geq
0\\}\bigg{)}\geq 1-\varepsilon.$

Let us briefly explain the above inequalities that imply (C.2). The first
inequality follows from stochastic monotonicity applied to the boundary point.
We are conditioning on the event that requires the random walk
$(U(k))_{k=1}^{n}$ to stay above the barrier zero. By stochastic monotonicity,
increasing this barrier will only decrease the conditional probability. This
implies the second inequality. The equality in the last line follows by
translating the random walk. The final inequality follows by taking $\delta$
small enough due to the tightness of the random walk paths conditioned to stay
positive (when scaled by diffusively) [Igl74].

Proof of Eq. (C.3) and Eq. (C.4) Note that due to stochastic monotonicity
(Proposition C.4), taking $a_{2}\downarrow-\infty$ we get

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}\inf_{k\in\llbracket
1,n\rrbracket}S_{1}(k)-a_{1}\geq-\delta^{-1}\sqrt{n}\mid\mathsf{NI}\bigg{)}$
$\displaystyle\geq\mathbb{P}^{n;(a_{1},-\infty)}\bigg{(}\inf_{k\in\llbracket
1,n\rrbracket}S_{1}(k)-a_{1}\geq-\delta^{-1}\sqrt{n}\mid\mathsf{NI}\bigg{)}$
$\displaystyle=\mathbb{P}^{n;(a_{1},-\infty)}\bigg{(}\inf_{k\in\llbracket
1,n\rrbracket}S_{1}(k)-a_{1}\geq-\delta^{-1}\sqrt{n}\bigg{)}$
$\displaystyle=\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}\inf_{k\in\llbracket
1,n\rrbracket}S_{1}(k)-a_{1}\geq-\delta^{-1}\sqrt{n}\bigg{)}.$

The first equality above is due to the fact that $\mathsf{NI}$ happens almost
surely when the second walk starts at $-\infty$. The second equality follows
from noting that $S_{1}(\cdot)$ and $S_{2}(\cdot)$ are independent and hence
the probability is independent of the starting point of the second walk. Thus
the non-intersecting condition makes $S_{1}(\cdot)$ stochastically larger than
a usual random walk. By diffusive behavior of random walks one can choose
$\delta$ small enough so that the above quantity is at least $1-\varepsilon$.
Similarly the non-intersecting condition makes $S_{2}(\cdot)$ stochastically
smaller than a usual random walk. Combining this with the diffusive behavior
of random walks leads to (C.4). ∎

######  Corollary C.6.

Fix any $n\in\mathbb{Z}_{\geq 2}$. Suppose $a_{1},a_{2}\in\mathbb{R}$ with
$|a_{1}-a_{2}|\leq n/\log n$. Given any $\varepsilon,\gamma>0$ there exists a
constant $\rho(\varepsilon,\gamma)\in(0,\frac{1}{4}]$ such that for all large
enough $n$ we have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}\sup_{k\in\llbracket
1,n\rho\rrbracket,i=1,2}\big{|}S_{i}(k)-a_{i}\big{|}\geq\gamma\sqrt{n}\mid\mathsf{NI}\bigg{)}\geq
1-\varepsilon.$

###### Proof.

Let us focus only on $S_{1}(\cdot)$. We may control lower drift of
$S_{1}(\cdot)$ around $a_{1}$, i.e., $\inf_{k\in\llbracket
1,n\rho\rrbracket}[S_{1}(k)-a_{1}]$ by an argument similar to the proof of
(C.3). For upper drift we use

$\displaystyle\sup_{k\in\llbracket 1,n\rho\rrbracket}[S_{1}(k)-a_{1}]\leq
a_{2}-a_{1}+\sup_{k\in\llbracket
1,n\rho\rrbracket}[S_{1}(k)-S_{2}(k)]+\sup_{k\in\llbracket
1,n\rho\rrbracket}[S_{2}(k)-a_{2}]$

The second and third term can be controlled by an argument similar to the
proof of (C.2) and (C.4) respectively. Note that by diffusive properties all
the fluctuations are of the order $\sqrt{n\rho}$. Hence one can choose $\rho$
small enough so that

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2})}\bigg{(}\sup_{k\in\llbracket
1,n\rho\rrbracket}\big{|}S_{1}(k)-a_{1}\big{|}\geq\gamma\sqrt{n}\mid\mathsf{NI}\bigg{)}\geq
1-\varepsilon.$

∎

We now study non-intersecting probabilities for random bridges
$\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}$ defined in Definition 3.6
(increments drawn from $\mathsf{f}$). The following lemma shows that when the
starting points and endpoints are far apart in the diffusive scale, non-
intersection probability is bounded from zero.

######  Lemma C.7.

Fix $\delta>0$. For each $n\in\mathbb{Z}_{\geq 4}$, consider the set

$\displaystyle R_{n,\delta}:=\\{(x_{1},x_{2}):|x_{i}|\leq 2\sqrt{n}(\log
n)^{3/2},x_{1}-x_{2}\geq\delta\sqrt{n}\\}$ (C.10)

There exists $\phi=\phi(\delta)>0$ such that for all $n$ large enough and all
$(a_{1},a_{2}),(b_{1},b_{2})\in R_{n,\delta}$ we have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}\left(\inf_{k\in\llbracket
1,n\rrbracket}\big{[}S_{1}(k)-S_{2}(k)\big{]}\geq\tfrac{1}{4}\delta\sqrt{n}\right)\geq\phi.$

###### Proof.

Fix any $(a_{1},a_{2}),(b_{1},b_{2})\in R_{n,\delta}$. For simplicity let us
write $\mathbb{P}$ for $\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}$. Note that
$|b_{i}-a_{i}|\leq 4\sqrt{n}(\log n)^{3/2}$. By the KMT coupling for Brownian
bridges (Theorem 2.3 in [DW21a] with $z=b_{i}-a_{i}$ and $p=0$), there exists
a constant $\mathrm{C}>0$ such that for all $n\in\mathbb{Z}_{\geq 1}$ we have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}\bigg{(}\neg\mathsf{SC}_{(a_{1},a_{2})}^{(b_{1},b_{2})}\bigg{)}\leq\tfrac{1}{n},$
(C.11) $\displaystyle\hskip 14.22636pt\mbox{where
}\mathsf{SC}_{(a_{1},a_{2})}^{(b_{1},b_{2})}:=\bigg{\\{}\sup_{k\in\llbracket
1,n\rrbracket,i=1,2}\left|S_{i}(k)-\sqrt{n}B_{i}(k/n)-a_{i}-\tfrac{k}{n}(b_{i}-a_{i})\right|\leq\mathrm{C}\log^{3}n\bigg{\\}},$

and where $B_{1},B_{2}$ are Brownian bridges on the same probability space
with variance $\int x^{2}\mathsf{f}(x)dx$ ($\mathsf{SC}$ stands for ‘strong
coupling’). By Brownian bridge properties, there exists $\phi=\phi(\delta)>0$
so that

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}\bigg{(}\sup_{x\in[0,1]}(|B_{1}(x)|+|B_{2}(x)|)\leq\tfrac{1}{8}\delta\bigg{)}\geq
2\phi.$

Combining the previous two math displays we see that with probability
$2\phi-\frac{1}{n}$ we have

$\displaystyle S_{1}(k)-S_{2}(k)$ $\displaystyle\geq
a_{1}-a_{2}+\tfrac{k}{n}(b_{1}-a_{1}-b_{2}+a_{2})-2\mathrm{C}(\log
n)^{3}-\tfrac{1}{4}\delta\sqrt{n}$
$\displaystyle=\tfrac{n-k}{n}a_{1}-a_{2}+\tfrac{k}{n}(b_{1}-b_{2})-2\mathrm{C}(\log
n)^{3}-\tfrac{1}{4}\delta\sqrt{n}$ $\displaystyle\geq-2\mathrm{C}(\log
n)^{3}+\tfrac{1}{2}\delta\sqrt{n}>\tfrac{1}{4}\delta\sqrt{n}$

for all large enough $n$. Taking $n$ large enough ensures
$2\phi-\frac{1}{n}\geq\phi$ completing the proof. ∎

Our next lemma gives a crude bound for the weak non-intersection probability
in terms of true non-intersection probability.

######  Lemma C.8.

There exists $\mathrm{C}>0$ such that for all $p\in[0,\infty)$,
$(a_{1},a_{2}),(b_{1},b_{2})\in\mathbb{R}^{2}$, $n\in\mathbb{Z}_{\geq 1}$

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{NI}_{p})\leq
e^{\mathrm{C}p}\cdot\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{NI}).$

###### Proof.

By lifting the first random bridge by $p$ units we see that

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{NI}_{p})=\mathbb{P}^{n;(a_{1}+p,a_{2}),(b_{1}+p,b_{2})}(\mathsf{NI}).$

Conditioning on the second point and the penultimate point of both the random
bridges we get

$\displaystyle\mathbb{P}^{n;(a_{1}+p,a_{2}),(b_{1}+p,b_{2})}(\mathsf{NI})=\frac{\int_{x_{1}\geq
x_{2},y_{1}\geq
y_{2}}\Lambda_{x_{1},x_{2}}^{n;(y_{1},y_{2})}(\mathsf{NI})\Upsilon_{p}(x_{1},x_{2};y_{1},y_{2})dx_{1}dx_{2}dy_{1}dy_{2}}{\mathsf{f}^{\ast(n-1)}(a_{1}-b_{1})\mathsf{f}^{\ast(n-1)}(a_{2}-b_{2})}.$
(C.12)

where

$\displaystyle\Upsilon_{p}(x_{1},x_{2};y_{1},y_{2}):=\mathsf{f}(a_{1}+p-x_{1})\mathsf{f}(a_{2}-x_{2})\mathsf{f}(y_{1}-b_{1}-p)\mathsf{f}(y_{2}-b_{2}),$
$\displaystyle\Lambda_{x_{1},x_{2}}^{n;(y_{1},y_{2})}(\mathsf{NI}):=\int_{x_{j,1}\geq
x_{j,2},j\in\llbracket
3,n-2\rrbracket}\prod_{j=2}^{n-2}\mathsf{f}(x_{j,1}-x_{j+1,1})\mathsf{f}(x_{j,2}-x_{j+1,2})\prod_{j=3}^{n-2}dx_{j,1}dx_{j,2}.$

Here in the above integration we set $x_{2,1}:=x_{1}$, $x_{2,2}:=x_{2}$,
$x_{n-1,1}:=y_{1}$, $x_{n-1,2}:=y_{2}$. From Lemma B.3, we have that
$\Upsilon_{p}(x_{1},x_{2};y_{1},y_{2})\leq
e^{\mathrm{C}p}\Upsilon_{0}(x_{1},x_{2};y_{1},y_{2})$, where the
$\mathrm{C}>0$ depends only on $\theta$. Plugging this bound back in (C.12) we
get the desired result. ∎

The following technical lemma, which can be thought of as the bridge analog of
Lemma C.3, studies the non-intersection probability for random bridges when
the starting points are close.

######  Lemma C.9.

Fix $M>0$ and $n\in\mathbb{Z}_{\geq 2}$. There exist a constant
$\mathrm{C}=\mathrm{C}(M)>0$ such that for all $|a_{i}|\leq\sqrt{n}(\log
n)^{3/2}$ with $|a_{1}-a_{2}|\leq(\log n)^{3/2}$, and $|b_{i}|\leq M\sqrt{n}$
with $b_{1}\geq b_{2}$ we have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{NI})\leq\mathrm{C}\tfrac{1}{\sqrt{n}}{\max\\{a_{1}-a_{2},1\\}\cdot\max\left\\{\tfrac{1}{\sqrt{n}}|a_{1}-b_{1}|,2\right\\}^{3/2}}.$

###### Proof.

It suffices to prove the lemma only for large enough $n$ (since we can always
choose the $\mathrm{C}$ large enough). Set
$r=\max\\{\frac{1}{\sqrt{n}}|a_{1}-b_{1}|,2\\}$ and $p=\lfloor
nr^{-3}\rfloor$. We first claim that there exists $m(M)>0$ such that

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{NI})\leq
2\cdot\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}\left(\\{|S_{i}(p)-a_{i}|\leq
m\sqrt{n}r^{-1}\mbox{ for }i=1,2\\}\cap\mathsf{NI}\right).$ (C.13)

Let us first complete the proof of the lemma assuming (C.13). Note that the
density of $S_{i}(p)$ at $x$ is given by
$\frac{\mathsf{f}^{*(p-1)}(x-a_{i})\mathsf{f}^{*(n-p)}(b_{i}-x)}{\mathsf{f}^{*(n-1)}(b_{i}-a_{i})}$.
By Lemma C.2, we may replace the $n$-fold convolution with Gaussian densities
at the expensive of a multiplicative factor close to $1$. In particular for
large enough $n$ we have

$\displaystyle\sup_{|x-a_{i}|\leq
m\sqrt{n}r^{-1}}\frac{\mathsf{f}^{*(n-p)}(b_{i}-x)}{\mathsf{f}^{*(n-1)}(b_{i}-a_{i})}$
$\displaystyle=2\exp\left(\tfrac{1}{2\sigma^{2}}\left(r^{2}-\tfrac{(r-mr^{-1})^{2}}{1-r^{-3}}\right)\right)$
$\displaystyle=2\exp\left(\tfrac{1}{2\sigma^{2}(1-r^{-3})}\left(-r^{-1}-m^{2}r^{-2}+2m\right)\right)\leq
2e^{2m/\sigma^{2}}.$

Thus
$\frac{\mathsf{f}^{*(p-1)}(x-a_{i})f^{*(n-p)}(b_{i}-x)}{\mathsf{f}^{*(n-1)}(b_{i}-a_{i})}\leq
2e^{2m/\sigma^{2}}\cdot\mathsf{f}^{*(p-1)}(x-a_{i})$ whenever $|x-a_{i}|\leq
m\sqrt{n}r^{-1}$. This allows us to go from random bridge laws to random walk
laws. We thus have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}\left(\\{|S_{i}(p)-a_{i}|\leq
m\sqrt{n}r^{-1}\mbox{ for }i=1,2\\}\cap\mathsf{NI}\right)$
$\displaystyle\leq\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}\left(\\{|S_{i}(p)-a_{i}|\leq
m\sqrt{n}r^{-1}\mbox{ for }i=1,2\\}\cap\bigcap_{k=1}^{p}\\{S_{1}(k)\geq
S_{2}(k)\\}\right)$ $\displaystyle\leq
2e^{2m/\sigma^{2}}\cdot\mathbb{P}^{n;(a_{1},a_{2})}\left(\\{|S_{i}(p)-a_{i}|\leq
m\sqrt{n}r^{-1}\mbox{ for }i=1,2\\}\cap\bigcap_{k=1}^{p}\\{S_{1}(k)\geq
S_{2}(k)\\}\right)$ $\displaystyle\leq
2e^{2m/\sigma^{2}}\cdot\mathbb{P}^{p;(a_{1},a_{2})}\left(\bigcap_{k=1}^{p}\\{S_{1}(k)\geq
S_{2}(k)\\}\right)\leq\tfrac{\mathrm{C}}{\sqrt{n}}r^{3/2}\cdot\max\\{a_{1}-a_{2},1\\}.$

where the last inequality uses Lemma C.3. This completes the proof modulo
(C.13). The rest of the proof is devoted to showing (C.13).

We claim that

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(S_{1}(p)-a_{1}\leq-m\sqrt{n}r^{-1}\mid\mathsf{NI})\leq\tfrac{1}{8},$
(C.14)
$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(S_{1}(p)-a_{1}\geq
m\sqrt{n}r^{-1}\mid\mathsf{NI})\leq\tfrac{1}{8},$ (C.15)
$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(S_{2}(p)-a_{2}\leq-m\sqrt{n}r^{-1}\mid\mathsf{NI})\leq\tfrac{1}{8},$
$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(S_{2}(p)-a_{2}\geq
m\sqrt{n}r^{-1}\mid\mathsf{NI})\leq\tfrac{1}{8},$

for all large enough $n$. Applying an union bound, leads to (C.13). We shall
prove the first two inequalities: (C.14) and (C.15), the remaining two follows
in a similar fashion.

Proof of Eq. (C.15). Similar to the proof of (C.3) and (C.4), by stochastic
monotonicity for random bridges (Proposition C.4) we have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(S_{1}(p)-a_{1}\leq-m\sqrt{n}r^{-1}\mid\mathsf{NI})\leq\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(S_{1}(p)-a_{1}\leq-m\sqrt{n}r^{-1})$
(C.16)

We invoke the KMT coupling for random bridges [DW21a] to define Brownian
bridge $B_{1},B_{2}$ on $[0,1]$ on a common probability space such that (C.11)
holds. By (C.11), with probability $1-\frac{1}{n}$,

$\displaystyle S_{1}(p)-a_{1}$
$\displaystyle\geq\sqrt{n}B_{1}(p/n)+\tfrac{p}{n}(b_{1}-a_{1})-\mathrm{C}\log^{3}n$
$\displaystyle=\sqrt{n}B_{1}(p/n)-\sqrt{n}r^{-2}-\mathrm{C}\log^{3}n\geq\sqrt{n}B_{1}(p/n)-2\sqrt{n}r^{-1}.$

for large enough $n$. Since $p/n$ is of the order $r^{-3}$, $B_{1}(p/n)$
fluctuates of the order $r^{-3/2}$. By Brownian bridge one point tail
estimates, there exists a constant $c>0$ such that for all $m\geq 3$

$\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(B_{1}(p/n)>-(m-2)r^{-1})\geq
1-e^{-cm^{2}r}.$

Thus by an union bound we have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(S_{1}(p)-a_{1}>-m\sqrt{n}r^{-1})\geq
1-\tfrac{1}{n}-e^{-cm^{2}r}.$ (C.17)

Taking $m,n$ are large enough, ensure that
$1-\frac{1}{n}-e^{-cm^{2}r}\geq\frac{7}{8}$. This verifies (C.14).

Proof of Eq. (C.15). By stochastic monotonicity (Proposition C.4) at the
starting points,

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(S_{1}(p)-a_{1}\geq
m\sqrt{n}r^{-1}\mid\mathsf{NI})$
$\displaystyle\leq\mathbb{P}^{n;(a_{1}+\sqrt{n}r^{-1},a_{2}),(b_{1},b_{2})}(S_{1}(p)-a_{1}\geq
m\sqrt{n}r^{-1}\mid\mathsf{NI})$
$\displaystyle\leq\frac{\mathbb{P}^{n;(a_{1}+\sqrt{n}r^{-1},a_{2}),(b_{1},b_{2})}(S_{1}(p)-a_{1}\geq
m\sqrt{n}r^{-1})}{\mathbb{P}^{n;(a_{1}+\sqrt{n}r^{-1},a_{2}),(b_{1},b_{2})}(\mathsf{NI})}.$
(C.18)

Using an argument similar to the derivation of (C.17), we find that

$\displaystyle\mathbb{P}^{n;(a_{1}+\sqrt{n}r^{-1},a_{2}),(b_{1},b_{2})}(S_{1}(p)-a_{1}\geq
m\sqrt{n}r^{-1})\leq\tfrac{1}{n}+e^{-cm^{2}r}.$ (C.19)

This gives an upper bound for the numerator of (C.18). For the denominator,
recall the event $\mathsf{SC}_{(a_{1},a_{2})}^{(b_{1},b_{2})}$ and the
Brownian bridges $B_{1},B_{2}$ from (C.11). Note that on the event

$\mathsf{SC}_{(a_{1}+\sqrt{n}r^{-1},a_{2})}^{(b_{1},b_{2})}\cap\\{\inf_{x\in[0,1]}(B_{1}(x)-B_{2}(x))\geq-\frac{1}{2}r^{-1}\\},$

for large enough $n$ we have

$\displaystyle S_{1}(k)$
$\displaystyle\geq\sqrt{n}B_{1}(k/n)+a_{1}+\sqrt{n}r^{-1}+\tfrac{k}{n}(b_{1}-a_{1})-\mathrm{C}(\log
n)^{3}$
$\displaystyle\geq\sqrt{n}B_{2}(k/n)+\tfrac{1}{2}\sqrt{n}r^{-1}+a_{2}+\tfrac{k}{n}(b_{2}-a_{2})-2\mathrm{C}(\log
n)^{3}$ $\displaystyle\geq
S_{2}(k)+\tfrac{1}{2}\sqrt{n}r^{-1}-3\mathrm{C}(\log n)^{3}\geq S_{2}(k).$

where we used that $|a_{1}-a_{2}|\leq(\log n)^{3/2}$, $b_{1}\geq b_{2}$, and
$r\leq(\log n)^{3/2}$. Thus for large enough $n$,

$\displaystyle\mathbb{P}^{n;(a_{1}+\sqrt{n}r^{-1},a_{2}),(b_{1},b_{2})}(\mathsf{NI})$
$\displaystyle\geq\mathbb{P}^{n;(a_{1}+\sqrt{n}r^{-1},a_{2}),(b_{1},b_{2})}\bigg{(}\inf_{x\in[0,1]}(B_{1}(x)-B_{2}(x))\geq-\tfrac{1}{2}r^{-1}\bigg{)}$
$\displaystyle\quad-\mathbb{P}^{n;(a_{1}+\sqrt{n}r^{-1},a_{2}),(b_{1},b_{2})}\bigg{(}\neg\mathsf{SC}_{(a_{1}+\sqrt{n}r^{-1},a_{2})}^{(b_{1},b_{2})}\bigg{)}$
$\displaystyle\geq\mathrm{C}r^{-2}-\tfrac{1}{n}\geq\tfrac{1}{2}\mathrm{C}r^{-2},$

where the penultimate inequality follows from (C.11) and Brownian bridge
calculations (see Lemma 2.11 in [CH16] for example). Combining (C.19) and the
above lower bound we have

$\displaystyle\mbox{r.h.s.~{}\eqref{eeb0}}\leq\tfrac{2}{\mathrm{C}}\left(\tfrac{r^{2}}{n}+r^{2}e^{-cm^{2}r}\right)\leq\tfrac{1}{8},$

for all large enough $n$ and $m$ (as $r\leq(\log n)^{3/2}$). ∎

######  Corollary C.10.

Fix any $M>0$ and $n\geq 1$. Suppose $|a_{i}|,|b_{i}|\leq M\sqrt{n}$ for
$i=1,2$. There exists a constant $\mathrm{C}=\mathrm{C}(M)>0$ such that

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{NI})\leq\mathrm{C}\\!\cdot\\!\mathbb{P}^{\lfloor
n/4\rfloor;(a_{1},a_{2})}(\widetilde{\mathsf{NI}})\mathbb{P}^{\lfloor
n/4\rfloor;(b_{1},b_{2})}(\widetilde{\mathsf{NI}})$
$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{NI})\geq\tfrac{1}{\mathrm{C}}\\!\cdot\\!\mathbb{P}^{\lfloor
n/4\rfloor;(a_{1},a_{2})}(\widetilde{\mathsf{NI}})\mathbb{P}^{\lfloor
n/4\rfloor;(b_{1},b_{2})}(\widetilde{\mathsf{NI}}).$

where $\widetilde{\mathsf{NI}}:=\\{S_{1}(k)\geq S_{1}(k)\mbox{ for all
}k\in\llbracket 2,n/4\rrbracket\\}.$

###### Proof.

The upper bound follows by applying (4.23) with $\delta_{i}=\frac{1}{4}$ and
integrating over the non-intersection event. Let us focus on the lower bound.
For simplicity we will drop the floor functions from $\lfloor n/4\rfloor$. By
Lemma C.5, we can choose a constant $\widetilde{M}$ depending only on $M$ such
that for all $|c_{i}|\leq M\sqrt{n}$ with $|x_{i}|\leq M\sqrt{n}$, we have

$\displaystyle\mathbb{P}^{n/4;(c_{1},c_{2})}\bigg{(}\bigcap_{i=1}^{2}\\{|S_{i}(n/4)|\leq\widetilde{M}\sqrt{n}\\}\mid\widetilde{\mathsf{NI}}\bigg{)}\geq\tfrac{3}{4}.$
(C.20)

By Lemma C.5 we can next choose a $\delta=\delta(M)>0$ small enough such that

$\displaystyle\mathbb{P}^{n/4;(c_{1},c_{2})}\bigg{(}(S_{1}(n/4),S_{2}(n/4))\in
R_{n,\delta}\mid\widetilde{\mathsf{NI}}\bigg{)}\geq\tfrac{3}{4}.$ (C.21)

where $R_{n,\delta}$ is from (C.10). By Lemma C.7, there exists
$\phi(\delta)>0$ so that for all $(x_{1},x_{2}),(y_{1},y_{2})\in R_{n,\delta}$

$\displaystyle\mathbb{P}^{n/2;(x_{1},x_{2}),(y_{1},y_{2})}\bigg{(}\bigcap_{k=1}^{n/2}\\{S_{1}(k)\geq
S_{2}(k)\\}\bigg{)}\geq\phi.$ (C.22)

We next consider the events

$\displaystyle\mathsf{E}_{1}:=\left\\{|S_{i}(n/4)|\leq\widetilde{M}\sqrt{n}\mbox{
for
}i=1,2\right\\},\quad\mathsf{E}_{2}:=\left\\{|S_{i}(3n/4)|\leq\widetilde{M}\sqrt{n}\mbox{
for }i=1,2\right\\}.$

Using (4.24) with $\delta=\frac{1}{4}$ we have

$\displaystyle\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{NI})\geq\mathbb{P}^{n;(a_{1},a_{2}),(b_{1},b_{2})}(\mathsf{E}_{1}\cap\mathsf{E}_{2}\cap\mathsf{NI})$
$\displaystyle\geq\mathrm{C}^{-1}\\!\cdot\\!\widetilde{\mathbb{P}}(\mathsf{E}_{1}\cap\mathsf{E}_{2}\cap\mathsf{NI})$
$\displaystyle=\mathrm{C}^{-1}\\!\cdot\\!\widetilde{\mathbb{P}}(\mathsf{NI})\widetilde{\mathbb{P}}(\mathsf{E}_{1}\cap\mathsf{E}_{2}\mid{\mathsf{NI}}).$

for some $\mathrm{C}>0$ depending on $\widetilde{M},M$. Here
$\widetilde{\mathbb{P}}:=\widetilde{\mathbb{P}}^{(n;n/4,n/4);(a_{1},a_{2}),(b_{1},b_{2})}$
denotes the joint law of two independent $(n;n/4,n/4)$-modified random bridges
of length $n$ starting at $(a_{1},a_{2})$ and ending at $(b_{1},b_{2})$ (see
Definition 4.9). In view of our $\widetilde{M}$ choice and by the definition
of modified random bridges, we have

$\widetilde{\mathbb{P}}(\mathsf{E}_{1}\cap\mathsf{E}_{2}\mid{\mathsf{NI}})=\mathbb{P}^{n/4;(a_{1},a_{2})}\bigg{(}\bigcap_{i=1}^{2}\\{|S_{i}(n/4)|\leq\widetilde{M}\sqrt{n}\\}\mid\widetilde{\mathsf{NI}}\bigg{)}\mathbb{P}^{n/4;(b_{1},b_{2})}\bigg{(}\bigcap_{i=1}^{2}\\{|S_{i}(n/4)|\leq\widetilde{M}\sqrt{n}\\}\mid\widetilde{\mathsf{NI}}\bigg{)}$

which is lower bounded by $(3/4)^{2}$ from (C.20) and (C.21). Furthermore, in
view of (C.22), we have

$\widetilde{\mathbb{P}}(\mathsf{NI})\geq\phi\cdot\mathbb{P}^{n/4;(a_{1},a_{2})}(\widetilde{\mathsf{NI}})\mathbb{P}^{n/4;(b_{1},b_{2})}(\widetilde{\mathsf{NI}}).$

We thus have the desired lower bound. ∎

We now analyze the $\mathsf{Gap}_{\beta}$ event defined in (4.22) under
modified random bridge law. Fix any $M>0$, $n\geq 1$, and
$(a_{1},a_{2}),(b_{1},b_{2})\in\mathbb{R}^{2}$. Suppose $|a_{i}|,|b_{i}|\leq
M\sqrt{n}$ and $a_{1}\geq a_{2}$. Take $p,q\in\llbracket 0,n\rrbracket$ with
$p+q\leq n/2$ and $p\neq 0$. Suppose further that there exists $\rho\in(0,1)$
such that either $q\geq n\rho$ or $b_{1}-b_{2}\geq\rho\sqrt{n}$. Consider two
independent $(n;p,q)$\- modified random bridges $(S_{i}(k))_{k\in\llbracket