28.25276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{0.0pt}{202.01476pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(63.0,10.0){
\leavevmode\hbox to0.4pt{\vbox to173.96pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower 28.25276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{0.0pt}{202.01476pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(33.0,12.0){
\leavevmode\hbox to60.15pt{\vbox to0.4pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{59.75096pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}
\put(63.0,68.0){$f_{+}(\cdot)$} \put(53.0,25.0){$f_{-}(\cdot)$}
\put(47.0,15.0){${\mathcal{I}_{k}}$} \put(31.0,8.0){\tiny${kN^{2/3}}$}
\put(50.0,8.0){\tiny${(M_{1}+2k)N^{2/3}}$} \end{overpic} Figure 10. Events
considered in Proposition 3.4. Here $\mathcal{L}_{1}^{N}(2p+1)$ is given by
the black rough curve. The parabolic curves $f_{\pm}(x):=-(N\nu)^{-1}x^{2}\pm
M_{2}N^{1/3}$ are also depicted. Horizontal lines eminate from these parabolas
starting at $x=kN^{2/3}$. The event in (3.14) tells us that on the horizontal
interval $\mathcal{I}_{k}:=\llbracket kN^{2/3},(M_{1}+2k)N^{2/3}\rrbracket$
the black rough curve stays entirely below the black horizontal line while the
event in (3.15) tells us that there is a point in $\mathcal{I}_{k}$ at which
the black rough curve exceeds the red horizontal curve.

######  Proposition 3.5 (Low point on the first curve).

Fix any $\varepsilon\in(0,1)$. There exists $M_{0}(\varepsilon)$ such that for
all $M\geq M_{0}$,

$\displaystyle\liminf_{N\to\infty}\mathbb{P}\left(\mathcal{L}_{1}^{N}(2MN^{2/3}+1)\leq-\tfrac{1}{8}M^{2}N^{1/3}\nu\right)>1-\varepsilon,$
(3.16)

where $\nu$ is defined in (3.13).

######  Definition 3.6 ($n$-step random walk and bridge measures).

Recall the spaces $(\Omega^{p}_{n},\mathcal{F}^{p}_{n})$ for $p\in\\{1,2\\}$
from Definition 1.7. For $p=2$ that definition provided coordinate function
notation $S_{i}(k):=\omega_{i}(k)$ for $i\in\\{1,2\\}$ and $k\in\llbracket
1,n\rrbracket$. For $p=1$ similarly define coordinate functions
$S(k):=\omega(k)$ for $k\in\llbracket 1,n\rrbracket$. Recall
$\mathsf{f}_{\theta}$ from (2.3).

For $a\in\mathbb{R}$ define the probability measure $\mathbb{P}^{n;a}$ on
$(\Omega^{1}_{n},\mathcal{F}^{1}_{n})$ for a single $n$-step random walk
started at $a$ to be proportional to the product of the Dirac delta function
$\delta_{\omega(1)=a}$ and a density (against Lebesgue on $\mathbb{R}^{n-1}$)
given by

$\displaystyle\prod_{k=2}^{n}\mathsf{f}_{\theta}\big{(}\omega(k)-\omega(k-1)\big{)}\,d\omega(k).$

Similarly, for $(a_{1},a_{2})\in\mathbb{R}^{2}$ define the probability measure
$\mathbb{P}^{n;(a_{1},a_{2})}$ on $(\Omega^{2}_{n},\mathcal{F}^{2}_{n})$ for a
pair of independent $n$-step random walk started at $a_{1}$ and $a_{2}$ by
taking the product of $\mathbb{P}`^{n;a_{1}}$ and $\mathbb{P}^{n;a_{2}}$.

For $a,b\in\mathbb{R}$ define the probability measure $\mathbb{P}^{n;a;b}$
$(\Omega^{1}_{n},\mathcal{F}^{1}_{n})$ for a single $n$-step random bridge
started at $a$ and ended at $b$ to be proportional to the product of two Dirac
delta function $\delta_{\omega(1)=a}\delta_{\omega(n)=b}$ and a density
(against Lebesgue on $\mathbb{R}^{n-2}$) given by

$\displaystyle\prod_{k=2}^{n}\mathsf{f}_{\theta}\big{(}\omega(k)-\omega(k-1)\big{)}\prod_{k=2}^{n-1}d\omega(k).$

Similarly, for $(a_{1},a_{2}),(b_{1},b_{2})\in\mathbb{R}^{2}$ define the
probability measure $\mathbb{P}^{n;(a_{1},a_{2});(b_{1},b_{2})}$ on
$(\Omega^{2}_{n},\mathcal{F}^{2}_{n})$ for a pair of independent $n$-step
random briges started at $a_{1}$ and $a_{2}$ and ended (respectively) at
$b_{1}$ and $b_{2}$ by taking the product of $\mathbb{P}^{n;a_{1};b_{1}}$ and
$\mathbb{P}^{n;a_{2};b_{2}}$.

The proofs of Propositions 3.4 and 3.5 rely on the fluctuation results from
[BW22], as restated earlier in Theorem 1.5, and are postponed to the next
subsection. Assuming their validity, we complete the proof of Theorem 3.3.

###### Proof of Theorem 3.3.

For clarity we divide the proof into two steps.

Step 1. In this step we define notation and events used in the proof. Fix
$\varepsilon\in(0,1)$ and $k>0$. Take $M_{0}$ from Proposition 3.4. We set $R$
large enough so that

$\displaystyle 2^{-5}R\geq 2k+1,\qquad
M_{0}-2^{-5}(\tfrac{1}{8}R^{2}\nu+M_{0})+R^{3/2}\leq-
M_{0}-2^{-10}R^{2}\nu,\qquad R\geq 2M_{0}$ (3.17)

and $Q:=2^{-5}R$. We will assume some additional conditions on $R$ later,
which will depend on certain probability bounds that will be specified in the
next step. For convenience, we will also assume $kN^{2/3}$ and $RN^{2/3}$ are
integers (instead of using floor functions below). We set

$\displaystyle a:=M_{0}N^{1/3},\ \ b:=-\tfrac{1}{8}R^{2}N^{1/3}\nu,\ \
n:=RN^{2/3}-kN^{2/3}+1,\ \
v:=-\big{(}\tfrac{1}{8}R^{2}\nu+2\sqrt{R}\big{)}N^{1/3}.$

Let us define the sets $\mathcal{I}:=\llbracket
QN^{2/3},(M_{0}+2Q)N^{2/3}\rrbracket$ and $\mathcal{J}:=\llbracket
kN^{2/3},RN^{2/3}\rrbracket$. Due to (3.17), we have
$\mathcal{I}\subset\mathcal{J}$. Next we define the following events:

$\displaystyle\mathsf{A}$
$\displaystyle:=\bigg{\\{}\sup_{p\in\mathcal{J}}{\mathcal{L}_{2}(2p)}\leq
v\bigg{\\}},\quad\mathsf{B}:=\bigg{\\{}\mathcal{L}_{1}(2kN^{2/3}+1)\leq
a,\mathcal{L}_{1}(2RN^{2/3}+1)\leq b\bigg{\\}}.$

The $\mathsf{A}$ event demands that the second curve $\mathcal{L}^{N}_{2}(2p)$
does not rise above $v$ for any $p\in\mathcal{J}$. The $\mathsf{B}$ event
requires both $\mathcal{L}^{N}_{1}(2kN^{2/3}+1)$ and
$\mathcal{L}^{N}_{1}(2RN^{2/3}+1)$ to be less than $a$ and $b$ respectively.
Finally we set

$\displaystyle\mathsf{C}:=\bigg{\\{}\sup_{p\in\mathcal{I}}{\mathcal{L}^{N}_{1}(2p+1)}+Q^{2}\nu
N^{1/3}\geq-a\bigg{\\}}$

In words, $\mathsf{C}$ ensures there exists some $p\in\mathcal{I}$ such that
$\mathcal{L}^{N}_{1}(2p+1)$ is greater than $-a-Q^{2}\nu N^{1/3}$.

Note that by Proposition 3.4 we have $\mathbb{P}(\mathsf{C})\geq
1-\varepsilon$ . Furthermore, by Propositions 3.4 and 3.5 for large enough $R$
we also have $\mathbb{P}(\neg\mathsf{B})\leq 2\varepsilon$. We claim that for
all large enough $R$ we have

$\displaystyle\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{C})\leq\varepsilon.$
(3.18)

We prove (3.18) in the next step. Assuming this, note that by union bound we
have

$\displaystyle\mathbb{P}(\neg\mathsf{A})\geq\mathbb{P}(\mathsf{C})-\mathbb{P}(\neg\mathsf{B})-\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{C})\geq
1-4\varepsilon.$

Changing $\varepsilon\mapsto\varepsilon/4$ we arrive at (3.12). This completes
the proof modulo (3.18).

\begin{overpic}[width=199.16928pt]{bm2.pdf} \put(2.7,-5.0){ \leavevmode\hbox
to0.4pt{\vbox to213.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{213.39569pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(25.7,40.0){
\leavevmode\hbox to0.4pt{\vbox to99.98pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower 85.15828pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{85.35828pt}\pgfsys@lineto{0.0pt}{184.94293pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(49.0,40.0){
\leavevmode\hbox to0.4pt{\vbox to99.98pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower 85.15828pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{85.35828pt}\pgfsys@lineto{0.0pt}{184.94293pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(93.3,-5.0){
\leavevmode\hbox to0.4pt{\vbox to213.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{213.39569pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(2.3,97.0){
\leavevmode\hbox to2.4pt{\vbox to2.4pt{\pgfpicture\makeatletter\hbox{\hskip
1.2pt\lower-1.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(92.8,50.0){
\leavevmode\hbox to2.4pt{\vbox to2.4pt{\pgfpicture\makeatletter\hbox{\hskip
1.2pt\lower-1.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(25.0,82.0){
\leavevmode\hbox to4.4pt{\vbox to4.4pt{\pgfpicture\makeatletter\hbox{\hskip
2.2pt\lower-2.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.0pt}{0.0pt}\pgfsys@curveto{2.0pt}{1.10458pt}{1.10458pt}{2.0pt}{0.0pt}{2.0pt}\pgfsys@curveto{-1.10458pt}{2.0pt}{-2.0pt}{1.10458pt}{-2.0pt}{0.0pt}\pgfsys@curveto{-2.0pt}{-1.10458pt}{-1.10458pt}{-2.0pt}{0.0pt}{-2.0pt}\pgfsys@curveto{1.10458pt}{-2.0pt}{2.0pt}{-1.10458pt}{2.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(0.0,28.0){
\leavevmode\hbox to191.03pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip-5.49046pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@invoke{
}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{196.32385pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}
\put(43.5,30.0){$y=v$} \put(26.0,50.0){ \leavevmode\hbox to47.8pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip 2.07999pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{{ {\pgfsys@beginscope
\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{}
\pgfsys@moveto{-2.07999pt}{2.39998pt}\pgfsys@curveto{-1.69998pt}{0.95998pt}{-0.85318pt}{0.28pt}{0.0pt}{0.0pt}\pgfsys@curveto{-0.85318pt}{-0.28pt}{-1.69998pt}{-0.95998pt}{-2.07999pt}{-2.39998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.4pt}{0.0pt}\pgfsys@lineto{45.12459pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.2pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{45.32458pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(3.0,0.0){
\leavevmode\hbox to179.65pt{\vbox
to5.2pt{\pgfpicture\makeatletter\hbox{\hskip-5.49046pt\lower-2.59998pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{6.09045pt}{0.0pt}\pgfsys@lineto{184.54294pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{5.89046pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{184.74294pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(1.3,100.0){$a$}
\put(96.0,53.0){$b$} \put(63.0,68.0){$f(\cdot)$}
\put(37.0,53.0){${\mathcal{I}}$} \put(48.0,3.0){${\mathcal{J}}$} \end{overpic}
Figure 11. In this figure $\mathcal{L}^{N}_{1}(2p+1)$ (black curve) and
$\mathcal{L}^{N}_{2}(2p)$ (blue curve) are plotted for $p\in\mathcal{J}$.
$\mathsf{A}$ denotes the event that the blue curve lies below the horizontal
line $y=v$. $\mathsf{B}$ denotes the event that the black curve starts below
$a$ and ends below $b$. The curve $f$ in the figure is given by
$f(x)=-(N\nu)^{-1}x^{2}-a$. The event $\mathsf{C}$ denotes that there is a
point $p^{\prime}\in\mathcal{I}$ where the black rough curve stays above the
red horizontal line (this event does not occur in the above figure). The key
idea is that on $\mathsf{A}\cap\mathsf{B}$, the blue curve lies below $y=v$
completely, and the black curve behaves like a simple random bridge and
follows a linear trajectory with starting and ending points less than $a$ and
$b$ respectively. As a result, the event $\mathsf{C}$ (which requires the
black curve to follow parabolic trajectory) does not occur with high
probability. But we know both $\mathsf{B}$ and $\mathsf{C}$ occurs with high
probability. Thus the event $\mathsf{A}$ occurs with low probability.
\begin{overpic}[width=199.16928pt]{bm3.pdf} \put(2.7,-5.0){ \leavevmode\hbox
to0.4pt{\vbox to213.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{213.39569pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(34.0,40.0){
\leavevmode\hbox to0.4pt{\vbox to99.98pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower 85.15828pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{85.35828pt}\pgfsys@lineto{0.0pt}{184.94293pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(67.0,40.0){
\leavevmode\hbox to0.4pt{\vbox to99.98pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower 85.15828pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{85.35828pt}\pgfsys@lineto{0.0pt}{184.94293pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(93.3,-5.0){
\leavevmode\hbox to0.4pt{\vbox to213.8pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{213.39569pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(2.3,61.0){
\leavevmode\hbox to2.4pt{\vbox to2.4pt{\pgfpicture\makeatletter\hbox{\hskip
1.2pt\lower-1.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(92.8,9.0){
\leavevmode\hbox to2.4pt{\vbox to2.4pt{\pgfpicture\makeatletter\hbox{\hskip
1.2pt\lower-1.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(34.0,50.0){
\leavevmode\hbox to65.84pt{\vbox to5.2pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-2.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.4pt}{0.0pt}\pgfsys@lineto{65.04143pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.2pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{65.24142pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} \put(1.0,61.0){$a$}
\put(96.0,9.0){$b$} \put(51.0,53.0){${\mathcal{K}}$}
\put(36.0,90.0){\tiny{$y=-a-Q^{2}\nu N^{1/3}$}} \end{overpic} Figure 12. In
the above figure the random bridge $S(i)$ from $a$ to $b$ is depicted by the
black curve. The event $\mathsf{D}$ ensures the random bridge lies below the
blue line $y=a+\frac{x}{n}(b-a)+\sqrt{Rn}$. The event $\mathsf{C}$ requires
$S(i)\geq-\big{(}M_{0}+Q^{2}\nu\big{)}N^{1/3}$ for some
$i\in\mathcal{K}:=\llbracket(Q-k)N^{2/3},(M_{0}+2Q-k)N^{2/3}\rrbracket$. One
can choose $R$ large enough so that the horizontal black line $y=-a-Q^{2}\nu
N^{1/3}=-\big{(}M_{0}+Q^{2}\nu\big{)}N^{1/3}$ lies above the blue line
$y=a+\frac{x}{n}(b-a)+\sqrt{Rn}$ for all $x\geq(Q-k)N^{2/3}$. This forces
$\mathsf{D}\subset\neg\mathsf{C}$.

Step 2. In this step we will prove (3.18). The reader is encouraged to consult
with Figure 11 and its caption to get an overview of the key idea behind the
proof.

We consider the $\sigma$-algebra:

$\mathcal{F}:=\sigma\big{(}\mathcal{L}^{N}_{2}\llbracket
1,2N-2i\rrbracket,\mathcal{L}^{N}_{1}(\llbracket
1,2kN^{2/3}+1\rrbracket\cup\llbracket 2RN^{2/3}+1,2N\rrbracket)\big{)}.$

Note that $\mathsf{A}\cap\mathsf{B}$ is measurable with respect to
$\mathcal{F}$. Hence

$\displaystyle\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{C})=\mathbb{E}\left[\mathbf{1}_{\mathsf{A}\cap\mathsf{B}}\mathbb{E}\left[\mathbf{1}_{\mathsf{C}}\mid\mathcal{F}\right]\right].$

Using the Gibbs property for two-sided boundaries (see Lemma 2.5), the
conditional law is determined by the boundary data and is monotone with
respect to the boundary data (see Proposition 2.6). On the event
$\mathsf{A}\cap\mathsf{B}$, $\mathcal{L}^{N}_{2}$ (on even points) is at most
$v$, $\mathcal{L}^{N}_{1}(2kN^{2/3}+1)$ is at most $a$ and
$\mathcal{L}^{N}_{1}(2RN^{2/3}+1)$ is at most $b$. Thus by stochastic
monotonicity we have

$\displaystyle\mathbf{1}_{\mathsf{A}\cap\mathsf{B}}\cdot\mathbb{E}\big{(}\mathbf{1}_{\mathsf{C}}\mid\mathcal{F}\big{)}\leq\mathbf{1}_{\mathsf{A}\cap\mathsf{B}}\cdot\frac{\mathbb{E}^{n;a;b}\left(W(S,v)\mathbf{1}_{\mathsf{C}}\right)}{\mathbb{E}^{n;a;b}\left(W(S,v)\right)}\leq\mathbf{1}_{\mathsf{A}\cap\mathsf{B}}\cdot\frac{\mathbb{E}^{n;a;b}\left(\mathsf{C}\right)}{\mathbb{E}^{n;a;b}\left(W(S,v)\right)}.$
(3.19)

where $S=(S(1),\ldots,S(n))$ is distributed according to $\mathbb{P}^{n;a;b}$,
the $n$-step random bridge measure from $a$ to $b$, and where
$W(S,v):=\exp\left(-2\sum_{i=2}^{n-1}e^{v-S(i)}\right)$. The event
$\mathsf{C}$ should now be treated as being defined in terms of $S$ as

$\displaystyle\mathsf{C}=\bigg{\\{}\sup_{p\in\llbracket
QN^{2/3},(M_{0}+2Q)N^{2/3}\rrbracket}S(p-kN^{2/3}+1)+Q^{2}\nu
N^{1/3}\geq-a\bigg{\\}}.$

Note that

$\displaystyle\mathbb{E}^{n;a;b}\left(W(S,v)\right)$
$\displaystyle\geq\exp\left(-2ne^{-\sqrt{n}}\right)\mathbb{P}^{n;a;b}\left(S(i)\geq
v+\sqrt{n}\mbox{ for all }i\in\llbracket 1,n\rrbracket\right)$
$\displaystyle\geq\exp\left(-2ne^{-\sqrt{n}}\right)\mathbb{P}^{n;a;b}\left(S(i)-a-\tfrac{i(b-a)}{{n}}\geq-\sqrt{n}\mbox{
for all }i\in\llbracket 1,n\rrbracket\right).$ (3.20)

where the last inequality follows by noting that
$S(i)-a-\frac{i(b-a)}{n}\geq-\sqrt{n}$ implies $S(i)\geq b-\sqrt{n}\geq
v+\sqrt{n}$. Since random bridges weakly converge to Brownian bridges (see
[Lig68] and [DW21a] for a quantitative version), using estimates for Brownian
bridges, we see that the r.h.s. (3.20) is uniformly bounded below by some
absolute constant $\delta$. We now claim that for all large enough $R$

$\displaystyle\mathsf{D}\subset\neg\mathsf{{C}},\quad\mathbb{P}^{n;a;b}(\mathsf{D})\geq
1-\varepsilon\delta,\ \ \mbox{ where }\
\mathsf{D}:=\left\\{\sup_{i\in\llbracket
1,n\rrbracket}\left(S(i)-a-\tfrac{i(b-a)}{n}\right)\leq\sqrt{R}\sqrt{n}\right\\}.$
(3.21)

Note that (3.21) implies
$\mathbb{P}^{n;a;b}(\mathsf{C})\leq\varepsilon\delta$. Plugging this back in
(3.19) along with the bound $\mathbb{E}^{n;a;b}\left(W(S,z)\right)\geq\delta$,
yields that r.h.s. (3.19) is at most $\varepsilon$. This proves (3.18).

Let us now verify (3.21). Indeed, $\mathbb{P}^{n;a;b}(\mathsf{D})$ can be made
arbitrarily close to $1$ by choosing $R$ large enough (as random bridges
weakly converge to Brownian bridges [Lig68]). We choose $R$ so large that
$\mathbb{P}^{n;a;b}(\mathsf{D})$ is at least $1-\varepsilon\delta$. Let us now
verify $\mathsf{D}\subset\neg\mathsf{C}$ (see also Figure 12 and its caption).
For $q\geq Q$ we see that

$\displaystyle a+\tfrac{(q-k)(b-a)}{R-k}+\sqrt{R}\sqrt{n}$
$\displaystyle\leq\left(M_{0}-\tfrac{Q-k}{R-k}(\tfrac{1}{8}R^{2}\nu+M_{0})+R^{3/2}\right)N^{1/3}$
$\displaystyle\leq\left(M_{0}-2^{-5}(\tfrac{1}{8}R^{2}\nu+M_{0})+R^{3/2}\right)N^{1/3}\leq-\left(M_{0}+Q^{2}\nu\right)N^{1/3}$

The penultimate inequality follows by observing that as $Q=2^{-5}R$, we have
$Q-k\geq 2^{-5}(R-k)>0$. Finally the last inequality follows from (3.17). Thus
for all $p\geq QN^{2/3}$,

$\displaystyle x+\tfrac{(p-kN^{2/3})(y-x)}{(R-k)N^{2/3}}+\sqrt{R}\sqrt{n}\leq
M_{0}N^{1/3}-Q^{2}\nu N^{1/3}$

Clearly this implies $\mathsf{D}\subset\neg\mathsf{C}$, completing the proof
(3.21). ∎

####  3.2.1. Proof of Propositions 3.4 and 3.5

The proofs of Propositions 3.4 and 3.5 uses the following.

######  Lemma 3.7 (Uniform tightness).

Recall $Z_{N}^{\operatorname{line}}(m)$, the point-to-(partial)line partition
function defined in (1.8). Fix $\varepsilon\in(0,1)$. There exists
$K_{0}=K_{0}(\varepsilon)>0$, such that for all $M>0$ and $K\geq K_{0}$ we
have

$\displaystyle\liminf_{N\to\infty}\mathbb{P}\left(-K\leq\frac{\log
Z_{N}^{\operatorname{line}}(MN^{2/3})+2\Psi(\theta)N}{N^{1/3}}+M^{2}\nu\leq
K\right)>1-\varepsilon$

where $\nu$ is defined in (3.13).

We remark that the above lemma was alluded in the introduction in the form of
(1.9).

###### Proof.

We recall the notations introduced in Section 1.3.2. Fix any $M>0$. Set
$k=MN^{2/3}$ and $p:=1+\frac{2k}{N-k}$. Let $\theta_{c}$ be the unique
solution to $\Psi^{\prime}(\theta_{c})-p\Psi^{\prime}(2\theta-\theta_{c})=0$.
Set $f_{\theta,p}=-\Psi(\theta_{c})-p\Psi(2\theta-\theta_{c})$ and
$\sigma_{\theta,p}^{3}=\frac{1}{2}(-\Psi^{\prime\prime}(\theta_{c})-\Psi^{\prime\prime}(2\theta-\theta_{c}))$
where $\Psi$ is the digamma function defined in (1.4). A straightforward
calculation (done at the end of Appendix D) shows

$(N-k)f_{\theta,p}=-2N\Psi(\theta)+M^{2}N^{1/3}(\Psi^{\prime}(\theta))^{2}/\Psi^{\prime\prime}(\theta)+O(1),\,\,\,\mbox{
and
}\,\,\,\sigma_{\theta,p}/(-\Psi^{\prime\prime}(\theta))^{1/3}\stackrel{{\scriptstyle
N\to\infty}}{{\to}}1,$ (3.22)

where $O(1)$ terms depend on $M,\theta$, but are bounded in $N$. When
$\alpha=\alpha_{2}>0$, we have that
$\lim_{N\to\infty}(N-k)^{1/3}\sigma_{\theta,p}(\alpha_{2}+\theta-\theta_{c})=\infty$
for each fixed $M>0$. Thus by Theorem 1.5 we get

$\displaystyle\frac{\log
Z_{N}^{\operatorname{line}}(MN^{2/3})+2\Psi(\theta)N}{(-N\Psi^{\prime\prime}(\theta))^{1/3}}+M^{2}\nu\stackrel{{\scriptstyle(d)}}{{\Longrightarrow}}\operatorname{TW}_{\operatorname{GUE}},$

where $\operatorname{TW}_{\operatorname{GUE}}$ is the GUE Tracy-Widom
distribution [TW94] and $\nu$ is defined in (3.13). For
$\alpha=\alpha_{1}=N^{-1/3}\mu$, we have
$\lim_{N\to\infty}(N-k)^{1/3}\sigma_{\theta,p}(\alpha_{1}+\theta-\theta_{c})=y:=\sigma_{\theta,1}(\mu-M\Psi^{\prime}(\theta)/\Psi^{\prime\prime}(\theta)).$
Another application of Theorem 1.5 yields

$\displaystyle\frac{\log
Z_{N}^{\operatorname{line}}(MN^{2/3})+2\Psi(\theta)N}{(-N\Psi^{\prime\prime}(\theta))^{1/3}}+M^{2}\nu\stackrel{{\scriptstyle(d)}}{{\Longrightarrow}}U_{-y}.$

where $U_{-y}$ is the Baik-Ben Arous-Péché distribution [BAP05] (see [BW22,
(5.2)] for definition). As $M\to\infty$, so does $y\to\infty$. Since
$U_{-y}\stackrel{{\scriptstyle(d)}}{{\Longrightarrow}}\operatorname{TW}_{\operatorname{GUE}}$
as $y\to\infty$ (see [BR01b, (2.36)]), we thus get tightness uniformly in $M$.
∎

###### Proof of Proposition 3.4.

Fix $k>0$, $\varepsilon\in(0,1)$. Since for any $M_{1}>0$

$\displaystyle\sup_{j\in\llbracket
kN^{2/3},(M_{1}+2k)N^{2/3}\rrbracket}Z(N+j,N-j)\leq
Z_{N}^{\operatorname{line}}(kN^{2/3}),$

appealing to Lemma 3.7 with $M\mapsto k$ we see that

$\displaystyle\mathbb{P}\left(\sup_{j\in\llbracket
kN^{2/3},(M_{1}+2k)N^{2/3}\rrbracket}\frac{\log
Z(N+j,N-j)+2\Psi(\theta)N}{N^{1/3}}+k^{2}\nu\leq M_{2}\right)\geq
1-\varepsilon,$

where $M_{2}$ can be chosen to be any $M\geq K_{0}$ where $K_{0}(\varepsilon)$
comes from Lemma 3.7. Recalling that $\mathcal{L}^{N}_{1}(2j+1)=\log
Z(N+j,N-j)+2\Psi(\theta)N$ from (1.7), we get (3.14).

The remainder of the proof is now devoted in proving (3.15). Towards this end,
set $K_{1}=\frac{1}{2}(M_{1}+2k)^{2}\nu$. Choose $M_{1}$ large enough so that
$K_{1}\geq K_{0}(\varepsilon/4)$ where $K_{0}$ comes from Lemma 3.7. Applying
Lemma 3.7 with $M\mapsto M_{1}+2k$, $K\mapsto K_{1}$, and
$\varepsilon\mapsto\varepsilon/4$ we have

$\displaystyle\liminf_{N\to\infty}\mathbb{P}\left(\frac{\log
Z_{N}^{\operatorname{line}}((M_{1}+2k)N^{2/3})+2\Psi(\theta)N}{N^{1/3}}\leq-\tfrac{1}{2}(M_{1}+2k)^{2}\nu\right)>1-\tfrac{1}{4}\varepsilon.$
(3.23)

Now we take $K_{2}=(\frac{(M_{1}+2k)^{2}}{4}-k^{2})\nu-\log
2\geq\frac{1}{4}M_{1}^{2}\nu$. We again choose $M_{1}$ large enough so that
$K_{2}\geq K_{0}(\varepsilon/4)$. Then applying Lemma 3.7 with $M\mapsto k$,
$K\mapsto K_{2}$, and $\varepsilon\mapsto\varepsilon/4$ we have

$\displaystyle\liminf_{N\to\infty}\mathbb{P}\left(\frac{\log
Z_{N}^{\operatorname{line}}(kN^{2/3})+2\Psi(\theta)N}{N^{1/3}}\geq-\tfrac{1}{4}(M_{1}+2k)^{2}\nu+\log
2\right)>1-\tfrac{1}{4}\varepsilon.$ (3.24)

By union bound the above two estimates implies for all large enough $M_{1}$ we
have

$\displaystyle\liminf_{N\to\infty}\mathbb{P}\left(Z_{N}^{\operatorname{line}}(kN^{2/3})>2\cdot
Z_{N}^{\operatorname{line}}((M_{1}+2k)N^{2/3})\right)>1-\tfrac{1}{2}\varepsilon.$
(3.25)

Let us temporarily set
$A=Z_{N}^{\operatorname{line}}(kN^{2/3})-Z_{N}^{\operatorname{line}}((M_{1}+2k)N^{2/3}$
and $B=Z_{N}^{\operatorname{line}}((M_{1}+2k)N^{2/3}$. Observe that $A+B>2B$
implies $2A>A+B$. Recall from (1.8) that

$A=\\!\\!\\!\\!\\!\\!\sum_{\lceil
kN^{2/3}\rceil}^{\lceil(M_{1}+2k)N^{2/3}\rceil-1}\\!\\!\\!\\!\\!\\!Z(N+j,N-j)\leq(M_{1}+k)N^{\frac{2}{3}}\\!\\!\\!\\!\\!\\!\sup_{j\in\llbracket
kN^{\frac{2}{3}},(M_{1}+2k)N^{\frac{2}{3}}\rrbracket}\\!\\!\\!\\!\\!\\!Z(N+j,N-j).$

We thus have

$\displaystyle\big{\\{}Z_{N}^{\operatorname{line}}(kN^{2/3})>2\cdot
Z_{N}^{\operatorname{line}}((M_{1}+2k)N^{2/3})\big{\\}}$ (3.26)
$\displaystyle\subset\bigg{\\{}\sup_{j\in\llbracket
kN^{\frac{2}{3}},(M_{1}+2k)N^{\frac{2}{3}}\rrbracket}\\!\\!\\!\\!\\!\\!\log
Z(N+j,N-j)>\log
Z_{N}^{\operatorname{line}}(kN^{\frac{2}{3}})-\log(2(M_{1}+k)N^{\frac{2}{3}})\bigg{\\}}.$

By Lemma 3.7, one can choose $M_{2}$ large enough (but free of $k$) so that

$\liminf_{N\to\infty}\mathbb{P}\Big{(}\log
Z_{N}^{\operatorname{line}}(kN^{2/3})+2\Psi(\theta)N+k^{2}\nu N^{1/3}\geq-
M_{2}N^{\frac{1}{3}}+\log(2(M_{1}+k)N^{\frac{2}{3}})\Big{)}>1-\tfrac{1}{2}\varepsilon.$

Using this, in view of (3.26) and (3.25), and using
$\mathcal{L}^{N}_{1}(2j+1)=\log Z(N+j,N-j)+2\Psi(\theta)N$ (see (1.7)) we
arrive at (3.15). This proves Proposition 3.4. ∎

###### Proof of Proposition 3.5.

We use the same notations as from the proof of Proposition 3.4 and utilize
(3.23) and (3.24) obtained there with $k=1$. Set $M=M_{1}+2$. Combining (3.23)
and (3.24) implies

$\displaystyle\liminf_{N\to\infty}\mathbb{P}\left(\log
Z_{N}^{\operatorname{line}}(N^{2/3})>\tfrac{1}{4}M^{2}N^{1/3}\nu+\log
Z_{N}^{\operatorname{line}}(MN^{2/3})\right)\geq 1-\tfrac{1}{2}\varepsilon.$

As $Z_{N}^{\operatorname{line}}(MN^{2/3})\geq Z(N+MN^{2/3};N-MN^{2/3})$, this
leads to

$\displaystyle\liminf_{N\to\infty}\mathbb{P}\left(\log
Z_{N}^{\operatorname{line}}(N^{2/3})>\tfrac{1}{4}M^{2}N^{1/3}\nu+\log
Z(N+MN^{2/3},N-MN^{2/3})\right)\geq 1-\tfrac{1}{2}\varepsilon.$

Again by Lemma 3.7, one can choose $M$ large enough so that

$\liminf_{N\to\infty}\mathbb{P}\bigg{(}\log
Z_{N}^{\operatorname{line}}(N^{2/3})\leq\tfrac{1}{8}M^{2}N^{1/3}\nu-2\Psi(\theta)N\bigg{)}>1-\tfrac{1}{2}\varepsilon,$

which forces

$\displaystyle\liminf_{N\to\infty}\mathbb{P}\left(\log
Z(N+MN^{2/3};N-MN^{2/3})<-2N\Psi(\theta)-\tfrac{1}{8}M^{2}N^{1/3}\nu\right)\geq
1-\varepsilon.$

By (1.7), $\mathcal{L}^{N}_{1}(2MN^{2/3}+1)=\log
Z(N+MN^{2/3};N-MN^{2/3})-2\Psi(\theta)N$ hence (3.16) follows. ∎

###  3.3. Spatial properties of the lower curves

In this subsection, we study spatial properties of the lower curves of the
$\mathpzc{HSLG}$ line ensemble. The main result of this section is the
following.

######  Theorem 3.8.

Fix any $p\in\\{1,2\\}$. Set $\alpha:=\alpha_{p}$ according to (3.11).
Consider the $\mathpzc{HSLG}$ line ensemble from Definition 2.7 with
parameters $(\alpha,\theta)$. Given any $k,\varepsilon>0$, there exist
constants $M=M(k,\varepsilon)\geq 1$ and $N_{0}(k,\varepsilon)\geq 1$ such
that for all $N\geq N_{0}(k,\varepsilon)$ and $v\in\\{2,3\\}$ we have

$\displaystyle\mathbb{P}\left(\sup_{s\in\llbracket
1,kN^{2/3}\rrbracket}\mathcal{L}_{v}^{N}(s)\geq
MN^{1/3}\right)\leq\varepsilon.$ (3.27)

In plain words, Theorem 3.8 argues that with high probability on the domain
$\llbracket 1,kN^{2/3}\rrbracket$, the entire second curve and third curve
lies below a threshold $MN^{1/3}$. The proof of Theorem 3.8 can be easily
extended to include other lower indexed curves as well. However, for the
proofs of our main results, it suffices to consider the first three curves.

Recall from Theorem 1.3 that the conditional laws of the $\mathpzc{HSLG}$ line
ensemble are given by $\mathpzc{HSLG}$ Gibbs measures introduced in Definition
1.2. The key technical ingredient in proving Theorem 3.8 is the tightness of
left boundary points of the first two curves under the bottom-free measure
defined in Definition 2.4.

######  Proposition 3.9.

Fix any $p\in\\{1,2\\}$. Set $\alpha:=\alpha_{p}$ according to (3.11). Fix any
$r\geq 1$ and $\varepsilon>0$. Set $T=\lfloor rN^{2/3}\rfloor$. Define

$\displaystyle
A:=\begin{cases}1+\sqrt{r}|\mu|\Psi^{\prime}(\frac{1}{2}\theta)&\mbox{if
}p=1,\\\ 1&\mbox{if }p=2.\end{cases}$ (3.28)

There exists $M=M(\varepsilon)>0$ and $N_{0}(\varepsilon)>0$ such that for all
$N\geq N_{0}$ we have

$\displaystyle\mathbb{P}_{\alpha_{p}}^{(0,-A\sqrt{T}),(-\infty)^{T};2,T}\big{(}|L_{1}(1)|+|L_{2}(2)|\geq
M\sqrt{T}\big{)}\leq\varepsilon.$ (3.29)

where the law $\mathbb{P}_{\alpha_{p}}^{\vec{y},(-\infty)^{2T};2,T}$ is
defined in Definition 2.4. Furthremore, there exists
$\widetilde{M}=\widetilde{M}(\varepsilon)>0$ and
$\widetilde{N}_{0}(\varepsilon)>0$ such that for all $N\geq\widetilde{N}_{0}$
we have

$\displaystyle\mathbb{P}_{\alpha_{1}}^{0,(-\infty)^{T};1,T}\big{(}|L_{1}(1)|\geq\widetilde{M}\sqrt{T}\big{)}\leq\varepsilon.$
(3.30)

As we shall see in the next section, the proof of the above lemma can be
extended to include $L_{2}(1)$ instead of $L_{2}(2)$. For technical reasons we
work with $L_{2}(2)$ here.

As mentioned in the introduction, the proof of Proposition 3.9 relies on
several ingredients related to non-intersecting random walks. We postpone its
proof to Section 4. We now complete the proof of Theorem 3.8 assuming
Proposition 3.9.

###### Proof of Theorem 3.8.

We prove the $v=2$ case and then use it to show the $v=3$ case.

Part I: $v=2$ case. For clarity we divide the proof into two steps.

Step 1. Recall that the points in the line ensemble satisfy certain high
probability ordering due to Theorem 3.1. In particular, if we know the even
points on $\mathcal{L}^{N}_{2}$ are not too high, Theorem 3.1 will force that
with high probability the odd points are not too high as well. Thus it
suffices to control the even points on $\mathcal{L}^{N}_{2}$. In this step, we
flesh out the details of the above idea. The proof of control on even points
on $\mathcal{L}^{N}_{2}$ appears in the second step of the proof.

We begin by defining a few events that will appear in the rest of the proof.
Fix $k,\varepsilon>0$. For any $r\in\llbracket 1,kN^{2/3}\rrbracket\cap
2\mathbb{Z}$, define

$\displaystyle\mathsf{A}(r,M):=\big{\\{}\mathcal{L}^{N}_{2}(r)\geq
MN^{1/3}\big{\\}},\quad\mathsf{F}(r,M):=\\{\mathcal{L}^{N}_{1}(r-1)\geq\tfrac{3M}{4}N^{1/3}\\}.$

Define

$\displaystyle\mathsf{B}(r,M):=\mathsf{A}(r,M)\cap\bigcap_{s\in\llbracket
r+2,kN^{2/3}\rrbracket\cap 2\mathbb{Z}}\neg\mathsf{A}(s,M),$

so that $(\mathsf{B}(r,M))_{r\in\llbracket 1,kN^{2/3}\rrbracket}$ forms a
disjoint collections of events. Note that

$\bigsqcup_{r\in\llbracket 1,kN^{2/3}\rrbracket\cap
2\mathbb{Z}}\mathsf{B}(r,M)=\bigcup_{r\in\llbracket 1,kN^{2/3}\rrbracket\cap
2\mathbb{Z}}\mathsf{A}(r,M)=\bigg{\\{}\sup_{r\in\llbracket
1,kN^{2/3}\rrbracket\cap 2\mathbb{Z}}\mathcal{L}^{N}_{2}(r)\geq
MN^{1/3}\bigg{\\}}.$

In the above equation, we use $\sqcup$ instead of $\cup$ to stress on the fact
that it is an union of disjoint events. Thus the above union demands at least
one of the even points in $\llbracket 1,kN^{2/3}\rrbracket$ of
$\mathcal{L}^{N}_{2}$ to exceed $MN^{1/3}$. We next define

$\displaystyle\mathsf{G}^{+}(M):=\bigsqcup_{r\in\llbracket
1,kN^{2/3}\rrbracket\cap
2\mathbb{Z}}\mathsf{B}(r,M)\cap\mathsf{F}(r,M),\quad\mathsf{G}^{-}(M):=\bigsqcup_{r\in\llbracket
1,kN^{2/3}\rrbracket\cap 2\mathbb{Z}}\mathsf{B}(r,M)\cap\neg\mathsf{F}(r,M).$

Finally set $\mathsf{G}(M):=\mathsf{G}^{+}(M)\sqcup\mathsf{G}^{-}(M).$ Observe
that the event

$\displaystyle\neg\mathsf{G}(M)=\left\\{\sup_{s\in\llbracket
1,kN^{2/3}\rrbracket\cap 2\mathbb{Z}}\mathcal{L}^{N}_{2}(s)<MN^{1/3}\right\\}$

controls the supremum of the second curve over the even points. Take
$0<k^{\prime}<k$. By the union bound we get that

$\displaystyle\mathbb{P}\left(\sup_{s\in\llbracket
1,k^{\prime}N^{2/3}\rrbracket}\mathcal{L}^{N}_{2}(s)\geq
3MN^{\frac{1}{3}}\right)\leq\mathbb{P}(\mathsf{G}(2M))+\mathbb{P}\bigg{(}\sup_{\begin{subarray}{c}s\in\llbracket
1,k^{\prime}N^{2/3}\rrbracket\\\
s\in(2\mathbb{Z}+1)\end{subarray}}\mathcal{L}^{N}_{2}(s)\geq
3MN^{\frac{1}{3}},\neg\mathsf{G}(2M)\bigg{)}.$ (3.31)

Note that on $-\mathsf{G}(2M)$ the supremum of $\mathcal{L}^{N}_{2}(s)$ over
all $s\in\llbracket 1,kN^{2/3}\rrbracket$ is at most $2MN^{1/3}$. Then by the
ordering of the line ensemble (Theorem 3.1) on $\neg\mathsf{G}(2M)$ it is
exponentially unlikely that any odd point within $\llbracket
1,k^{\prime}N^{2/3}\rrbracket$ will exceed $2MN^{1/3}+(\log N)^{7/6}$. In
particular the second term in r.h.s. (3.31) can be made smaller than
$\frac{\varepsilon}{2}$ by choosing $N$ large enough and taking $M\geq 1$. For
the first term we claim that there exists $M_{0},N_{0}$ depending on
$k,\varepsilon$ such that for all $N\geq N_{0}$ and $M\geq M_{0}$ we have

$\displaystyle\mathbb{P}(\mathsf{G}(2M))\leq\tfrac{\varepsilon}{2}.$ (3.32)

Clearly plugging this bound back in r.h.s. (3.31) proves (3.27) with $M\mapsto
3M$ and $k^{\prime}\mapsto k$. For the remainder of the proof we focus on
proving (3.32).

Step 2. In this step we prove (3.32). Observe that from the definition of
$\mathsf{G}^{-}(2M)$ we have

$\mathbb{P}(\mathsf{G}^{-}(2M))\leq\mathbb{P}\Big{(}\mathcal{L}^{N}_{1}(r-1)-\mathcal{L}^{N}_{2}(r)\geq-\tfrac{M}{2}N^{1/3}\mbox{
for some }r\in\llbracket 1,kN^{1/3}\rrbracket\cap 2\mathbb{Z}\Big{)}.$

However by Theorem 3.1 the right-hand side of the above equation can be made
smaller that $\frac{\varepsilon}{4}$ for all $N\geq N_{0}$ and $M\geq 1$, by
choosing $N_{0}:=N_{0}(k,\varepsilon)>0$ appropriately. We next claim that

$\displaystyle\mathbb{P}(\mathsf{G}^{+}(2M))\leq
2\mathbb{P}(\mathsf{A}(2,M))\leq\tfrac{\varepsilon}{4}.$ (3.33)

As $\mathsf{G}(2M)=\mathsf{G}^{-}(2M)\cup\mathsf{G}^{+}(2M)$, in view of the
above claim, (3.32) follows via a union bound.

Let us now prove (3.33). Observe that by definition of $G^{+}(2M)$ we have

$\displaystyle\mathbb{P}\big{(}\mathsf{A}(2,M)\big{)}\geq\mathbb{P}\big{(}\mathsf{G}^{+}(2M)\cap\mathsf{A}(2,M)\big{)}$
$\displaystyle=\sum_{r\in\llbracket 1,kN^{2/3}\rrbracket\cap
2\mathbb{Z}}\mathbb{P}\big{(}\mathsf{B}(r,2M)\cap\mathsf{F}(r,2M)\cap\mathsf{A}(2,M)\big{)}.$
(3.34)

We focus on each of the terms in the above sum. Using the tower property we
have

$\displaystyle\mathbb{P}\big{(}\mathsf{B}(r,2M)\cap\mathsf{F}(r,2M)\cap\mathsf{A}(2,M)\big{)}$
(3.35)
$\displaystyle=\mathbb{E}\left[\mathbf{1}_{\mathsf{B}(r,2M)\cap\mathsf{F}(r,2M)}\mathbb{E}\left(\mathbf{1}_{\mathsf{A}(2,M)}\mid\sigma\big{(}\mathcal{L}^{N}_{3},\mathcal{L}^{N}_{1}\llbracket
r-1,kN^{2/3}\rrbracket,\mathcal{L}^{N}_{2}\llbracket
r,kN^{2/3}\rrbracket\big{)}\right)\right].$

Using the Gibbs property (see Theorem 1.3 and Lemma 2.2 (a)) we have almost
surely that

$\displaystyle\mathbf{1}_{\mathsf{B}(r,2M)\cap\mathsf{F}(r,2M)}\mathbb{E}\left(\mathbf{1}_{\mathsf{A}(2,M)}\mid\sigma\big{(}\mathcal{L}^{N}_{3},\mathcal{L}^{N}_{1}\llbracket
r-1,kN^{2/3}\rrbracket,\mathcal{L}^{N}_{2}\llbracket
r,kN^{2/3}\rrbracket\big{)}\right)$ (3.36)
$\displaystyle=\mathbf{1}_{\mathsf{B}(r,2M)\cap\mathsf{F}(r,2M)}\mathbb{P}_{\alpha_{p}}^{\vec{y},\vec{z};2,r/2}(L_{2}(2)>MN^{1/3})$
$\displaystyle\geq\mathbf{1}_{\mathsf{B}(r,2M)\cap\mathsf{F}(r,2M)}\mathbb{P}_{\alpha_{p}}^{\vec{w},(-\infty)^{r};2,r/2}(L_{2}(2)>MN^{1/3}),$

where $\vec{y}=(\mathcal{L}^{N}_{1}(r-1),\mathcal{L}^{N}_{2}(r))$,
$\vec{z}=(\mathcal{L}^{N}_{3}(2v))_{v=1}^{r/2}$ and
$\vec{w}:=(\frac{3M}{2}N^{1/3},\frac{3M}{2}N^{1/3}-A\sqrt{r/2})$ ($A\geq 1$ is
defined in (3.28)). The last inequality above follows by stochastic
monotonicity (Proposition 2.6). We now briefly explain how stochastic
monotonicity works here. Note that the event $\\{L_{2}(2)>MN^{1/3}\\}$ is
decreasing thus by stochastic monotonicity to achieve a lower bound, we can
reduce the boundary $\vec{z}$ to $(-\infty)^{r}$. Furthermore, on
$\mathsf{B}(r,2M)\cap\mathsf{F}(r,2M)$, we may reduce $\vec{y}$ to $\vec{w}$
as $y_{i}\geq w_{i}$ on $\mathsf{B}(r,2M)\cap\mathsf{F}(r,2M)$.

Note that $MN^{1/3}\geq Mk^{-\frac{1}{2}}\sqrt{r/2}$. By translation
invariance (Lemma 2.1 (a)) and Proposition 3.9, we may choose
$M_{0}(k,\varepsilon)$ large enough so that for all $M\geq M_{0}$ and
$r\in\llbracket 1,kN^{2/3}\rrbracket\cap 2\mathbb{Z}$ we have

$\displaystyle\mathbb{P}_{\alpha_{p}}^{\vec{w},(-\infty)^{r};2,r/2}(L_{2}(2)>MN^{1/3})=\mathbb{P}_{\alpha_{p}}^{(0,-A\sqrt{r/2}),(-\infty)^{r};2,r/2}\big{(}L_{2}(2)>-\tfrac{1}{2}MN^{1/3}\big{)}\geq\tfrac{1}{2}.$

Inserting the above bound in (3.36) and then going back to (3.35) we get

$\displaystyle\mbox{r.h.s.~{}\eqref{e:gmp1}}\geq\tfrac{1}{2}\mathbb{P}\big{(}\mathsf{B}(r,2M)\cap\mathsf{F}(r,2M)\big{)}.$

Recall that $\mathsf{B}(r,2M)\cap\mathsf{F}(r,2M)$ are all disjoint events
whose union over $r\in\llbracket 1,kN^{2/3}\rrbracket\cap 2\mathbb{Z}$ is
given by $\mathsf{G}^{+}(2M)$. Summing the above inequality over
$r\in\llbracket 1,kN^{2/3}\rrbracket\cap 2\mathbb{Z}$, in view of (3.34), we
thus arrive at
$\mathbb{P}(\mathsf{A}(2,M))\geq\frac{1}{2}\mathbb{P}(\mathsf{G}^{+}(2M))$.
This proves the first inequality in (3.33). For the second one observe that by
union bound

$\displaystyle\mathbb{P}(\mathsf{A}(2,M))\leq\mathbb{P}\big{(}\mathcal{L}^{N}_{1}(3)-\mathcal{L}^{N}_{2}(2)\leq-N^{1/3}\big{)}+\mathbb{P}\big{(}\mathcal{L}^{N}_{1}(3)\geq(M-1)N^{1/3}\big{)}.$

By Theorem 3.1 the first term on the right-hand side of the above equation can
be made arbitrarily small by choosing $N$ large enough. As for the second
term, recall the point-to-line partition function
$Z_{N}^{\operatorname{line}}(\cdot)$ from (1.8). From Theorem 1.5 we know
$N^{-1/3}[\log Z_{N}^{\operatorname{line}}(1)+2\Psi(\theta)N]$ is tight. Since
$\mathcal{L}^{N}_{1}(3)\leq\log Z_{N}^{\operatorname{line}}(1)+2\Psi(\theta)N$
(see (1.7)), one can make the second term arbitrarily small enough by choosing
$M,N$ large enough. This completes the proof of (3.33).

Part II: $v=3$ case. Fix $k>0$. Let us define

$\mathsf{E}:=\bigg{\\{}\sup_{s\in\llbracket
1,kN^{2/3}\rrbracket}\mathcal{L}^{N}_{3}(s)\geq
MN^{1/3}\bigg{\\}},\quad\mathsf{F}:=\bigg{\\{}\sup_{s\in\llbracket
1,kN^{2/3}\rrbracket}\mathcal{L}^{N}_{2}(s)\geq\tfrac{1}{2}MN^{1/3}\bigg{\\}}.$

By repeated application of the union bound we have

$\displaystyle\mathbb{P}(\mathsf{E})$
$\displaystyle\leq\mathbb{P}(\mathsf{F})+\mathbb{P}(\mathsf{E}\cap\neg\mathsf{F})$
$\displaystyle\leq\mathbb{P}(\mathsf{F})+\mathbb{P}\bigg{(}\mathcal{L}^{N}_{2}(s)-\mathcal{L}^{N}_{3}(s)\leq-\tfrac{1}{2}MN^{1/3},\mbox{
for some }s\in\llbracket 1,kN^{2/3}\rrbracket\bigg{)}$
$\displaystyle\leq\mathbb{P}(\mathsf{F})+\sum_{\llbracket
1,kN^{2/3}\rrbracket}\mathbb{P}\big{(}\mathcal{L}^{N}_{2}(s)-\mathcal{L}^{N}_{3}(s)\leq-\tfrac{1}{2}MN^{1/3}\big{)}.$
(3.37)

By Theorem 3.1, there exists an absolute constant $N_{0}$ such that for all
$s\geq 1$, and $M\geq 1$, we have
$\mathbb{P}\left(\mathcal{L}^{N}_{2}(s)-\mathcal{L}^{N}_{3}(s)<-\tfrac{1}{2}MN^{1/3}\right)\leq
2^{-N}$. Since we have established $v=2$ case of Theorem 3.8, we may directed
use (3.27) with $v\mapsto 2$, $M\mapsto\frac{1}{2}M$ and
$\varepsilon\mapsto\frac{1}{2}\varepsilon$, to get that
$\mathbb{P}(\mathsf{F})\leq\frac{1}{2}\varepsilon$ for all large enough $N,M$.
Thus for all $N,M$ large enough we have
$\eqref{lastwo}\leq\frac{1}{2}\varepsilon+kN^{2/3}2^{-N}\leq\varepsilon$. ∎

Theorem 3.8 and Proposition 3.9 can be used to deduce left boundary tightness
for the $\mathpzc{HSLG}$ line ensemble. We shall refer to this property as
endpoint tightness.

######  Theorem 3.10 (Endpoint Tightness).

Fix any $p\in\\{1,2\\}$. Set $\alpha:=\alpha_{p}$ according to (3.11). Recall
the $\mathpzc{HSLG}$ line ensemble from Definition 2.7 with parameters
$(\alpha,\theta)$. The sequences
$\\{{N^{-1/3}}\mathcal{L}_{1}^{N}(1)\\}_{N\geq 1}$ and
$\\{{N^{-1/3}}\mathcal{L}_{2}^{N}(2)\\}_{N\geq 1}$ are tight.

Again the proof can be extended to include tightness of
$N^{-1/3}\mathcal{L}_{2}^{N}(1)$ as well, once we have the corresponding
version in Proposition 3.9. We again refrain from doing so, as it is
inconsequential to the proofs of our main theorem. We refer to the discussion
in the introduction (Remark 1.6) about how Theorem 3.10 relates to the work of
[IMS22].

###### Proof of Theorem 3.10.

Fix an $\varepsilon>0$. We shall show that for all large enough $N,M$ we have

$\displaystyle\mathbb{P}(\mathcal{L}_{1}^{N}(1)\leq MN^{1/3})\geq
1-3\varepsilon,\quad\mathbb{P}(\mathcal{L}_{2}^{N}(2)\leq-MN^{1/3})\leq
3\varepsilon.$ (3.38)

In view of the ordering of points in the line ensemble (Theorem 3.1), we know
$\mathcal{L}_{1}^{N}(1)\geq\mathcal{L}_{2}^{N}(2)-(\log N)^{7/6}$ with
probability at least $1-2^{-N}$. This along with the above equation ensures
endpoint tightness. We thus focus on proving (3.38).

Proof of the first inequality in (3.38). Recall the point-to-line partition
function $Z_{N}^{\operatorname{line}}(\cdot)$ from (1.8). From Theorem 1.5, we
know $N^{-1/3}\big{(}\log
Z_{N}^{\operatorname{line}}(1)+2\Psi(\theta)N\big{)}$ is tight. Since
$\mathcal{L}_{1}^{N}(3)\leq\log
Z_{N}^{\operatorname{line}}(1)+2\Psi(\theta)N$, there exists
$M_{1}(\varepsilon)>0$ such that for all $N\geq 3$ we have
$\mathbb{P}(\mathcal{L}_{1}^{N}(3)\leq M_{1}N^{1/3})\geq 1-\varepsilon.$
Thanks to Theorem 3.1, there exists $M_{2}(\varepsilon)>M_{1}(\varepsilon)$
such that for all $N\geq 3$

$\displaystyle\mathbb{P}(\mathsf{A})\geq
1-2\varepsilon,\quad\mathsf{A}:=\bigg{\\{}\mathcal{L}_{1}^{N}(3)\leq
M_{1}N^{1/3},\sup_{j\in\llbracket 1,4\rrbracket}\mathcal{L}_{2}^{N}(j)\leq
M_{2}N^{1/3}\bigg{\\}}.$

Define $\mathcal{F}:=\sigma\big{(}(\mathcal{L}_{1}^{N}(j))_{j\geq
3},(\mathcal{L}_{i}^{N}\llbracket 1,2N-2i+2\rrbracket)_{i\geq 2}\big{)}$. By
the union bound and tower property of the conditional expectation, for any
$M_{3}>0$ we have

$\displaystyle\mathbb{P}(\mathcal{L}_{1}^{N}(1)\geq M_{2}N^{1/3}+M_{3})\leq
2\varepsilon+\mathbb{E}\left[\mathbf{1}_{\mathsf{A}}\mathbb{E}\big{[}\mathbf{1}_{\mathcal{L}_{1}^{N}(1)\geq
M_{2}N^{1/3}+M_{3}}\mid\mathcal{F}\big{]}\right]$

Using Theorem 1.3 we have

$\displaystyle\mathbb{E}\big{[}\mathbf{1}_{\mathcal{L}_{1}^{N}(1)\geq
M_{2}N^{1/3}+M_{3}}\mid\mathcal{F}\big{]}=\mathbb{P}_{\alpha_{p}}^{\mathcal{L}_{1}^{N}(3),(\mathcal{L}_{2}^{N}(2),\mathcal{L}_{2}^{N}(4));1,2}(L_{1}(1)\geq
M_{2}N^{1/3}+M_{3})$

On event $\mathsf{A}$, the boundary data are at most $M_{2}N^{1/3}$. By
stochastic monotonicity (Proposition 2.6) and translation invariance of the
Gibbs measure (Lemma 2.1 (a)) we have

$\displaystyle\mathbf{1}_{\mathsf{A}}\cdot\mathbb{P}_{\alpha_{p}}^{\mathcal{L}_{1}^{N}(3),(\mathcal{L}_{2}^{N}(2),\mathcal{L}_{2}^{N}(4));1,2}(L_{1}(1)\geq
M_{2}N^{1/3}+M_{3})\leq\mathbf{1}_{\mathsf{A}}\cdot\mathbb{P}_{\alpha_{p}}^{0,(0,0);1,2}(L_{1}(1)\geq
M_{3}).$

The last probability can be made less than $\varepsilon$ by taking $M_{3}$
large enough. Thus setting $M_{4}=M_{4}(\varepsilon):=M_{3}+M_{2}$, we see
that for all $N\geq 3$, the first inequality in (3.38) holds with $M=M_{4}$.

Proof of the second inequality in (3.38). We start by defining two high
probability events $\mathsf{B}_{1}$ and $\mathsf{B}_{2}$. The idea is to then
show $\mathbb{P}\big{(}\\{\mathcal{L}_{2}^{N}(2)\leq-
MN^{1/3}\\}\cap\mathsf{B}_{1}\cap\mathsf{B}_{2}\big{)}$ can be made
arbitrarily small by choosing $N,M$ large enough.

We shall use Theorem 3.3 (high point on the second curve) with $k\mapsto 1$.
Consider $R_{0}=R_{0}(1,\varepsilon)>0$ from Theorem 3.3. Set
$R=\max\\{R_{0},1\\}$. By Theorem 3.3 with $k\mapsto 1$, there exists
$M_{5}(\varepsilon)>0$ such that for all large enough $N$

$\displaystyle\mathbb{P}(\mathsf{B}_{1})\geq
1-\varepsilon,\quad\mathsf{B}_{1}$ $\displaystyle:=\bigcup_{q\in\llbracket
N^{2/3},RN^{2/3}\rrbracket}\mathsf{B}_{1}(p),\quad\mathsf{B}_{1}(q):=\big{\\{}\mathcal{L}_{2}^{N}(2q)\geq-
M_{5}N^{1/3}\big{\\}}.$

We write the set $\mathsf{B}_{1}$ as a union of disjoint sets as follows:

$\displaystyle\mathsf{C}_{1}(q):=\mathsf{B}_{1}(q)\cap\bigcap_{s\in\llbracket
q+1,RN^{2/3}\rrbracket}\neg\mathsf{B}_{1}(s),\quad\mathsf{C}_{1}:=\bigsqcup_{q\in\llbracket
N^{2/3},RN^{2/3}\rrbracket}\mathsf{C}_{1}(q)=\mathsf{B}_{1}.$

By Theorem 3.1, for large enough $N$ we have

$\displaystyle\mathbb{P}(\mathsf{B}_{2})\geq
1-\varepsilon,\quad\mathsf{B}_{2}$ $\displaystyle:=\bigcap_{q\in\llbracket
N^{2/3},RN^{2/3}\rrbracket}\mathsf{B}_{2}(q),\quad\mathsf{B}_{2}(q):=\big{\\{}\mathcal{L}_{2}^{N}(2q)-\mathcal{L}_{1}^{N}(2q-1)\leq
N^{1/3}\big{\\}}.$

Set
$\mathcal{F}_{q}:=\sigma\big{(}(\mathcal{L}_{1}^{N}(j-1),\mathcal{L}_{2}^{N}(j))_{j\geq
2q},(\mathcal{L}_{i}^{N}\llbracket 1,2N-2i+2\rrbracket)_{i\geq 3}\big{)}$.
Observe that $\mathsf{B}_{2}(q)\cap\mathsf{C}_{1}(q)$ is measurable with
respect to $\mathcal{F}_{q}$. Note that for any $M_{6}>0$ we have

$\displaystyle\mathbb{P}\bigg{(}\left\\{\mathcal{L}_{2}^{N}(2)\leq-{M}_{6}N^{1/3}\right\\}\cap\mathsf{B}_{1}\cap\mathsf{B}_{2}\bigg{)}$
$\displaystyle\leq\sum_{q\in\llbracket
N^{2/3},RN^{2/3}\rrbracket}\mathbb{P}\bigg{(}\left\\{\mathcal{L}_{2}^{N}(2)\leq-{M}_{6}N^{1/3}\right\\}\cap\mathsf{C}_{1}(p)\cap\mathsf{B}_{2}(p)\bigg{)}$
$\displaystyle=\sum_{q\in\llbracket
N^{2/3},RN^{2/3}\rrbracket}\mathbb{E}\left[\mathbf{1}_{\mathsf{B}_{2}(q)\cap\mathsf{C}_{1}(q)}\mathbb{E}\left[\mathbf{1}_{\mathcal{L}_{2}^{N}(2)\leq-{M}_{6}N^{1/3}}\mid\mathcal{F}_{q}\right]\right].$
(3.39)

By the Gibbs property (Theorem 1.3) we have

$\displaystyle\mathbf{1}_{\mathsf{B}_{2}(q)\cap\mathsf{C}_{1}(q)}\cdot\mathbb{E}\left[\mathbf{1}_{\mathcal{L}_{2}^{N}(2)\leq-{M}_{6}N^{1/3}}\mid\mathcal{F}_{q}\right]$
$\displaystyle=\mathbf{1}_{\mathsf{B}_{2}(q)\cap\mathsf{C}_{1}(q)}\cdot\mathbb{P}_{\alpha_{p}}^{(\mathcal{L}_{1}^{N}(2q-1),\mathcal{L}_{2}^{N}(2q)),(\mathcal{L}_{3}^{N}(2i))_{i=1}^{q};2,q}(L_{2}(2)\leq-{M}_{6}N^{\frac{1}{3}})$
$\displaystyle\leq\mathbf{1}_{\mathsf{B}_{2}(q)\cap\mathsf{C}_{1}(q)}\cdot\mathbb{P}_{\alpha_{p}}^{(y_{1},y_{2}),(-\infty)^{q};2,q}(L_{2}(2)\leq-{M}_{6}N^{1/3}),$

where $y_{1}=-(M_{5}+1)N^{1/3}$, $y_{2}=-M_{5}N^{1/3}$. The last inequality
follows due to stochastic monotonicity (Proposition 2.6) as on the event
$\mathsf{B}_{2}(q)\cap\mathsf{C}_{1}(q)$ we have $\mathcal{L}_{2}^{N}(2q)\geq-
M_{5}N^{1/3}$ and $\mathcal{L}_{1}^{N}(2q-1)\geq-(M_{5}+1)N^{1/3}$. By
translation invariance (Lemma 2.1 (a)) and stochastic monotonicity
(Proposition 2.6) we have

$\displaystyle\mathbb{P}_{\alpha_{p}}^{(y_{1},y_{2}),(-\infty)^{q};2,q}(L_{2}(2)\leq-{M}_{6}N^{1/3})\leq\mathbb{P}_{\alpha_{p}}^{(0,-A\sqrt{q}),(-\infty)^{q};2,q}(L_{2}(2)\leq(M_{5}+1-{M}_{6})N^{1/3})\leq\varepsilon,$

where the last inequality is uniform over $q\in\llbracket
N^{2/3},RN^{2/3}\rrbracket$ and follows from Proposition 3.9 by taking $M_{6}$
large enough ($A\geq 1$ is defined in (3.28)). Plugging the above bound back
in (3.39), and noting that $(\mathsf{B}_{2}(q))_{q\in\llbracket
N^{2/3},RN^{2/3}\rrbracket}$ forms a disjoint collection of events we have
that $\eqref{abd}\leq\varepsilon$. Using the fact that
$\mathbb{P}(\neg\mathsf{B}_{i})\leq\varepsilon$ for $i=1,2$, an application of
the union bound yields the second inequality in (3.38) with $M={M}_{6}$. ∎

##  4\. Properties of the first two curves of Gibbs measures with no bottom
curve

In this section, we prove Proposition 3.9 that asserts endpoint tightness of
bottom-free measures defined in Definition 2.4. Along with Proposition 3.9, we
also study probabilities of a certain event which we call region pass event
under the bottom-free measure.

######  Proposition 4.1.

Fix any $r,M>0$ and $p\in\\{1,2\\}$. Set $T=\lfloor rN^{2/3}\rfloor$. We set
$\alpha=\alpha_{p}$ according to (3.11). Recall the bottom-free measure from
Definition 2.4. Let $\vec{y}\in\mathbb{R}^{p}$ with $y_{i}=-(M+i-1)N^{1/3}$.
There exists $\phi=\phi(r,M)>0$ and $N_{0}(r,M)>0$ such that for all $N\geq
N_{0}$ we have

$\displaystyle\mathbb{P}_{\alpha_{p}}^{\vec{y},(-\infty)^{2T};p,2T}(\mathsf{RP}_{p,M})\geq\phi,$
(4.1)

where the region pass event is defined as

$\displaystyle\mathsf{RP}_{p,M}:=\left\\{\inf_{j\in\llbracket
1,2T+p-2\rrbracket}L_{p}(j)\geq 2MN^{1/3}\right\\}.$ (4.2)

$2MN^{1/3}$$-MN^{1/3}$$-(M+1)N^{1/3}$$2T$$4T$ Figure 13. The above figure
depicts the event $\mathsf{RP}_{2,M}$ under the law
$\mathbb{P}_{\alpha_{2}}^{\vec{y},(-\infty)^{2T};2,2T}$.

The event $\mathsf{RP}_{p,M}$ requires the first $2T+p-2$ points of the $p$-th
curve to lie above $2MN^{1/3}$. Although this is a low probability event,
Proposition 4.1 says that this event always has positive probability
(independent of $N$) under the bottom-free measure. (see Figure 13 for $p=2$
case).

Recall from (3.11) that $\alpha_{1}$ and $\alpha_{2}$ are the boundary
parameters corresponding to critical and supercritical phases respectively.
Depending on the phase being critical or supercritical, the arguments for
proving Proposition 3.9 and Proposition 4.1 are markedly different. We first
give interpretation of the bottom-free laws under the two phases in Section
4.1. In Section 4.2 and 4.3, we provide proofs of the aforementioned lemmas
for critical and supercritical phases respectively.

In the critical phase, the left boundary attraction between the first two
curves is weak as $\alpha_{1}=O(N^{-1/3})$ – it is the soft-intersection that
only comes into the effect. The analysis of the Gibbs measures in this case is
similar to the one done in studying full-space line ensembles and relies on
KMT coupling type results. In the supercritical phase, we have
$\alpha_{2}=O(1)$ and the soft-intersection and attraction acts as two
opposite forces: one tries to repel the curves and another tries to attract.
This situation has asymptotically zero probability. The KMT coupling is no
longer suitable to analyze events under this setting. This makes the argument
for the supercritical phase more involved.

Let us first introduce a piece of notation that we will use frequently for the
remainder of the paper. Consider any probability measure
$\mathbb{P}^{\bullet}$ on $\mathbb{R}^{|\mathcal{K}_{k,T}|}$ equipped with
Borel $\sigma$-algebra on $\mathbb{R}^{|\mathcal{K}_{k,T}|}$ where
$\mathcal{K}_{k,T}$ is defined in (2.4). For
$\omega\in\mathbb{R}^{|\mathcal{K}_{k,T}|}$, we denote the coordinate
functions as $L_{i}(j)(\omega):=\omega_{i}(j)$ for
$(i,j)\in\mathcal{K}_{k,T}$. We will simply write
$(L_{i}(j))_{(i,j)\in\mathcal{K}_{k,T}}\sim\mathbb{P}^{\bullet}$ for the
random variables $(L_{i}(j))_{(i,j)\in\mathcal{K}_{k,T}}$ under the measure
$\mathbb{P}^{\bullet}$.

###  4.1. The bottom-free laws under critical and supercritical phase

In this section, we provide two alternative (and ultimately equivalent)
representations of the bottom-free laws defined in Definition 2.4. The first
representation, provided in Lemma 4.3, is most suitable for studying the
critical phase while the second representation, provided in Lemma 4.4 (see
also Definition 1.7 and the discussion in the introduction), is most suitable
for studying the supercritical phase.

We begin with the following lemma where we mention how the bottom-free measure
on the domain $\mathcal{K}_{k,T}$ (see (2.4)) with boundary condition
$\vec{y}$ is well-defined under certain cases.

######  Lemma 4.2 (Well-definedness of bottom-free measures).

Take $\vec{y}\in\mathbb{R}^{k}$. For $k\geq 1$ and
$\alpha\in(-\theta,\theta)$, $f_{k,T}^{\vec{y},(-\infty)^{T}}(\mathbf{u})$
(see (2.5)) is proportional to

$\displaystyle\prod_{i=1}^{k-1}\prod_{j=1}^{T-\mathbf{1}_{i=1}}W(u_{i+1,2j};u_{i,2j+1},u_{i,2j-1})\prod_{i=1}^{k}\prod_{j=1}^{2T-1-\mathbf{1}_{i=1}}G_{\theta+(-1)^{i+j-1}\alpha,(-1)^{j+1}}(u_{i,j}-u_{i,j+1}).$
(4.3)

where $W$ and $G$ are defined in (2.1) and (2.2). Furthermore,
$f_{2,T}^{\vec{y},(-\infty)^{T}}(\mathbf{u})$ is proportional to

$\displaystyle\exp\left(-e^{u_{2,2}-u_{1,3}}\right)G_{\alpha,1}(u_{2,2}-u_{1,1})G_{\theta,1}(u_{1,1}-u_{1,2})G_{\alpha+\theta,1}(u_{2,1}-u_{2,2})$
(4.4) $\displaystyle\hskip
56.9055pt\prod_{j=2}^{T-1}W(u_{2,2j};u_{1,2j+1},u_{1,2j-1})\prod_{i=1}^{2}\prod_{j=2}^{2T-1-\mathbf{1}_{i=1}}G_{\theta,(-1)^{j+1}}(u_{i,j}-u_{i,j+1}).$

Moreover, the above two densities are integrable.

###### Proof.

Recall the definition of the $\mathpzc{HSLG}$ Gibbs measure from (2.5) and the
corresponding graphical representation from Figure 3. Note that the red
colored edges comes with a weight of the form
$\exp(-\alpha(u_{2i-1,1}-u_{2i,1}))$ which can be written as a product
$e^{-\alpha u_{2i-1,1}}\cdot e^{\alpha u_{2i,1}}$. Thus we will think of each
red edge as two red rings on the endpoints of the edges that comes with the
(vertex) weight $e^{(-1)^{i}\alpha u_{i,1}}$ (see Figure 14 (C)). Upon doing
this vertex weight identification, the case $k=1$ and $T=3$ corresponds to the
top graph of Figure 14 (C). One can check that the weights corresponding two
graphs in Figure 14 (C) are equal. The red vertex weight on the right boundary
of Figure 14 (C) can be absorbed in the constant of proportionality of the
Gibbs measure. Thus ignoring this weight, the remaining weight is precisely
given by the expression in (4.3). The general $k$ odd case follows by
redistributing the weights according to Figure 14 (A). For the case when $k$
is even and $\alpha>0$, we redistribute according to Figure 14 (B). This leads
to the $k=2$ density given in (4.4). One can compute also the explicit density
for the general even case from Figure 14 (B). Since $0\leq W\leq 1$ and $G$s
are densities, it is clear that the expressions in (4.3) and (4.4) are
integrable. ∎

$\cdots\cdots$$\cdots\cdots$$\cdots\cdots$$\cdots\cdots$ (a)

$\cdots\cdots$$\cdots\cdots$$\cdots\cdots$$\cdots\cdots$ (b)

$=$ (c)

$=$ (d)

Figure 14. Redistribution of edge weights for $\alpha\in(-\theta,\theta)$ (A)
and for $\alpha>0$ and $k$ even (B). The weights of green, gray (dashed), and
yellow edges are $e^{(\theta-\alpha)x-e^{x}}$, $e^{\alpha x-e^{x}}$, and
$e^{(\theta+\alpha)x-e^{x}}$ respectively. The weights of black, blue edges
are defined in (1.5). (A) can be derived from Figure 3 by observing the
equality (as weight functions) in (C). The red circle around vertex $v$
signifies a vertex weight of the form $e^{\alpha x}$. The red vertex weight on
the right boundary can be absorbed in the constant of proportionality of the
Gibbs measure. (B) can be derived from Figure 3 by observing the equality (as
weight functions) in (D).

This redistribution described in Figure 14 will allows us to view the bottom
free laws as laws which are absolutely continuous with respect to random
walks. We give two such representations which will be useful in our critical
and supercritical phase analysis. Towards this end, we introduce
$\xi$-distributions. Given $\theta_{1},\theta_{2}>0$ and $a,b\in\mathbb{R}$,
we consider the following two probability density functions

$\displaystyle\xi_{\theta_{1},\theta_{2};\pm 1}^{(a,b)}(x)\propto
G_{\theta_{1},\pm 1}(a-x)G_{\theta_{2},\pm 1}(b-x).$ (4.5)

The graphical representation of the above two distributions are given in
Figure 15 (B).

$y_{1}$ (a) $\mathbb{P}_{\alpha_{1}}^{y_{1},(-\infty)^{4};1,4}$

$b$$a$$\theta_{1}$$\theta_{2}$$b$$a$$\theta_{1}$$\theta_{2}$ (b)
$\xi_{\theta_{1},\theta_{2};+1}^{(a,b)}$ and
$\xi_{\theta_{1},\theta_{2};-1}^{(a,b)}$

$y_{1}$$y_{2}$$y_{1}$$y_{2}$$y_{1}$$y_{2}$$\cdot$$=$ (c)
$\mathbb{P}_{\alpha_{1}}^{(y_{1},y_{2}),(-\infty)^{3};2,3}$

Figure 15. Figures (A) and (B) are graphical representations of the
probability distributions $\mathbb{P}_{\alpha_{1}}^{y_{1},(-\infty)^{4};1,4}$
and $\xi_{\theta_{1},\theta_{2};\pm 1}^{(a,b)}$ respectively. We use the
representation from Figure 14 (A) here. The black edge labeled as $\theta_{i}$
in (B) represents that the edge carries a weight of $e^{\theta_{i}x-e^{x}}$.
(C) shows the decomposition of
$\mathbb{P}_{\alpha_{1}}^{(y_{1},y_{2}),(-\infty)^{4};2,4}$ into
$\widehat{\mathbb{P}}^{(y_{1},y_{2})}$ (middle figure) and
$W_{\operatorname{cr}}$ (right figure). The marginal law of the gray (blue
resp.) shaded region is a random walk started at $y_{1}$ ($y_{2}$ resp.) with
increment $G_{\theta+\alpha_{1},-1}\ast G_{\theta-\alpha_{1},+1}$
($G_{\theta+\alpha_{1},+1}\ast G_{\theta-\alpha_{1},-1}$ resp.).

######  Lemma 4.3 (Critical phase representation).

Consider an independent collection of random variables
$Y_{i}(j)\stackrel{{\scriptstyle i.i.d.}}{{\sim}}G_{\theta+\alpha_{1},1}$ and
$U_{i}(j)\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\operatorname{Beta}(\theta-\alpha_{1},2\alpha_{1})$ for
$i=1,2$ and $j\in\mathbb{Z}_{\geq 1}$. Define

$\displaystyle V_{i}(j):=Y_{i}(2j)+\log U_{i}(2j)-\mathbb{E}[\log
U_{i}(2j)]-Y_{i}(2j-1).$ (4.6)

so that $V_{i}(j)$ form an i.i.d. collection of mean zero random variables.
Set $L_{i}(2T+i-2)=y_{i}$ and $\mbox{for }k\in\llbracket 1,T-1\rrbracket$
define

$\displaystyle L_{i}(2T+i-2k-2):=\left(y_{i}+(-1)^{i+1}(k-1)\mathbb{E}[\log
U_{1}(1)]\right)+(-1)^{i+1}\sum_{j=1}^{k-1}V_{i}(j),$ (4.7)

and set

$\displaystyle
W_{\operatorname{cr}}:=\exp\left(-\sum_{k=1}^{T-1}\left(e^{L_{2}(2k)-L_{1}(2k-1)}+e^{L_{2}(2k)-L_{1}(2k+1)}\right)\right),$
(4.8)

where ‘cr’ stands for critical. Conditioned on
$(L_{i}(2j+i-2))_{i\in\\{1,2\\},j\in\llbracket 1,T\rrbracket}$, we set

$L_{i}(2k+i-1)\sim\xi_{\theta-\alpha_{1},\theta+\alpha_{1},(-1)^{i+1}}^{(L_{i}(2k+i-2),L_{i}(2k+i))}\mbox{
for }i\in\\{1,2\\},k\in\llbracket 1,T-1\rrbracket$

and $L_{2}(1)=X+L_{2}(2)$ where $X\sim G_{\theta+\alpha_{1},1}$. We have

  1. (a)

$(L_{1}(j))_{(1,j)\in\mathcal{K}_{1,T}}$ is distributed as
$\mathbb{P}_{\alpha_{1}}^{y_{1},(-\infty)^{T};1,T}$.

  2. (b)

Let $\overline{\mathbb{P}}^{(y_{1},y_{2})}$ denotes the joint law of
$\\{(L_{i}(j))_{(i,j)\in\mathcal{K}_{2,T}}\\}$. This law has graphical
representation given by the middle figure in Figure 15 (C). The law
$\mathbb{P}_{\alpha_{1}}^{(y_{1},y_{2}),(-\infty)^{T};2,T}$ is absolutely
continuous with respect to $\overline{\mathbb{P}}^{(y_{1},y_{2})}$ with

$\displaystyle\frac{d\mathbb{P}_{\alpha_{1}}^{(y_{1},y_{2}),(-\infty)^{S};2,T}}{d\overline{\mathbb{P}}^{(y_{1},y_{2})}}\propto
W_{\operatorname{cr}}.$

###### Proof.

Let us consider
$(L_{1}(j))_{(1,j)\in\mathcal{K}_{1,T}}\sim\mathbb{P}_{\alpha_{1}}^{y_{1},(-\infty)^{T};1,T}$.
See Figure 15 (A) for the graphical representation of the law. We focus on the
odd points (shaded inside the gray box in the figure). Note that
$(L_{1}(2T-1-2k))_{k=0}^{T-1}$ is a random walk starting at
$L_{1}(2T-1)=y_{1}$ with increments distributed as
$G_{\theta+\alpha_{1},1}\ast G_{\theta-\alpha_{1},-1}$. Conditioned on the odd
points, we have
$L_{1}(2k)\sim\xi_{\theta-\alpha_{1},\theta+\alpha_{1};1}^{(L(2k-1),L(2k+1))}$.
Since $V_{1,j}+\mathbb{E}[\log U_{1,1}]\sim G_{\theta+\alpha,1}\ast
G_{\theta-\alpha,-1}$, Part (a) of the lemma follows.

Let us now consider the
$\mathbb{P}_{\alpha_{1}}^{(y_{1},y_{2}),(-\infty)^{T};2,T}$ law whose
graphical representation is given in Figure 15 (C). We view the graph as
superimposition of two graphs where in one graph we collect all the non-black
edges and the other graph we include only the black edges (see Figure 15 (C)).
We denote the law of the Gibbs measure formed by deleting the black edges as
$\widehat{\mathbb{P}}^{(y_{1},y_{2})}$ (middle figure in Figure 15 (C)). The
law $\mathbb{P}_{\alpha_{1}}^{(y_{1},y_{2}),(-\infty)^{T};2,T}$ can be
recovered from $\widehat{\mathbb{P}}^{(y_{1},y_{2})}$ by viewing the black
edges as a Radon-Nikodym derivative. Note that $W_{\operatorname{cr}}$,
defined in (4.8), precisely contains all the effect of the black edges in the
Gibbs measure.

If
$(L_{i}(j))_{(i,j)\in\mathcal{K}_{2,T}}\sim\widehat{\mathbb{P}}^{(y_{1},y_{2})}$,
we have $L_{1}(\cdot)$ independent of $L_{2}(\cdot)$ and $L_{1}$ is
distributed as $\mathbb{P}_{\alpha_{1}}^{y_{1},(-\infty)^{T};1,T}$. $L_{2}$
has a similar representation with even points $(L_{2}(2T-2k))_{k=0}^{T-1}$
forming a random walk starting at $y_{2}$ with increments distributed as
$G_{\theta+\alpha,-1}\ast G_{\theta-\alpha,1}$. Conditioned on the even
points, we have
$L_{2}(2k+1)\sim\xi_{\theta-\alpha_{1},\theta+\alpha_{1};-1}^{(L(2k),L(2k+2))}$
and $L_{2}(1)\sim G_{\theta+\alpha_{1},1}+L_{2}(2)$. Since
$-V_{2,j}-\mathbb{E}[\log U_{1,1}]\sim G_{\theta+\alpha,-1}\ast
G_{\theta-\alpha,1}$, we see that $\widehat{\mathbb{P}}^{(y_{1},y_{2})}$ is
equal to the law $\overline{\mathbb{P}}^{(y_{1},y_{2})}$ defined in Part (b)
of the lemma. This completes the proof of Part (b). ∎

In the supercritical phase, the weighted paired random walk ($\mathsf{WPRW}$)
measure (recall this and the $\mathsf{PRW}$ measure from Definition 1.7)
provides a useful way to describe the measure
$\mathbb{P}_{\alpha_{2}}^{\vec{y},(-\infty)^{T};2,T}$.

######  Lemma 4.4 (Supercritical phase representation).

Fix any $\vec{y}\in\mathbb{R}^{2}$ and $T\in\mathbb{Z}_{\geq 2}$. Suppose
$(L_{1}(2j-1),L_{2}(2j))_{j\in\llbracket
1,T-1\rrbracket}\sim\mathbb{P}_{\operatorname{WPRW}}^{T;\vec{y}}$. Conditioned
on $(L_{1}(2i-1),L_{2}(2i))_{i\in\llbracket 1,T-1\rrbracket}$, set
$L_{2}(1)\sim X+L_{2}(2)$ where $X\sim G_{\alpha_{2}+\theta,1}$ and

$L_{1}(2k)\sim\xi_{\theta,\theta;1}^{(L_{1}(2k-1),L_{1}(2k+1))},\ \
L_{2}(2k+1)\sim\xi_{\theta,\theta;-1}^{(L_{2}(2k),L_{2}(2k+2))}\mbox{ for
}k\in\llbracket 1,T-1\rrbracket.$

Then $(L_{i}(j))_{(i,j)\in\mathcal{K}_{2,T}}$ is distributed as
$\mathbb{P}_{\alpha_{2}}^{\vec{y},(-\infty)^{T};2,T}$.

###### Proof.

We use the alternative graph representation of
$\mathbb{P}_{\alpha_{2}}^{\vec{y},(-\infty)^{T};2,T}$ law from Figure 14 (B).
We decompose this graph into two graphs: one without the black edges (middle
figure of Figure 16), and one with the black edges, (right figure of Figure
16). However, unlike the critical phase, the Gibbs measure corresponding to
the middle figure does not split into two independent parts because of the
gray (dashed) edge. For this measure, the marginal law of the odd points of
the first curve and even points of the second curve together form the paired
random walk. Upon taking the black edges into consideration (which corresponds
to the $W_{\operatorname{sc}}$ weight), the odd points of the first curve and
even points of the second curve jointly follow the $\mathsf{WPRW}$ law. ∎

$y_{1}$$y_{2}$$y_{1}$$y_{2}$$y_{1}$$y_{2}$$\cdot$$=$ Figure 16.
$\mathbb{P}_{\alpha_{2}}^{(y_{1},y_{2}),(-\infty)^{3};2,3}$ law is decomposed
into two parts. We use the representation from Figure 14 (B) here. The first
part (middle figure) shaded region corresponds to a paired random walk. The
second part (right figure) corresponds to $W_{\operatorname{sc}}$.

###  4.2. Proof of Proposition 3.9 and Proposition 4.1 in the critical phase

We continue with the notations from Lemma 4.3. By the KMT coupling for random
walks [KMT75], we may find independent Brownian motions $B_{1},B_{2}$ defined
on the same probability space such that the following holds. There exists a
constant $\mathrm{C}>0$ depending only on $\theta$ and $\mu$ such that for all
$T$ large enough,

$\displaystyle{\mathbb{P}}\left(\max_{k\leq
T-2}\big{|}\sum_{j=1}^{k}V_{i}(j)-\sigma B_{i}(k)\big{|}\geq\mathrm{C}\log
T\right)\leq 1/T.$ (4.9)

$V_{1}(1)$ defined in (4.6) and $\sigma^{2}:=\operatorname{Var}(V_{1}(1))$.
Recall that in the critical phase we have $\alpha_{1}=N^{-1/3}\mu$. Set
$\kappa:=\tfrac{1}{4}|\mu|\Psi^{\prime}(\tfrac{1}{2}\theta)\geq 0$. As
$\Psi^{\prime}$ is a decreasing nonnegative function on $[0,\infty)$, for
large enough $N$

$\displaystyle\big{|}\mathbb{E}[\log
U_{1}(1)]\big{|}=\big{|}\Psi(\theta-\alpha_{1})-\Psi(\theta+\alpha_{1})\big{|}\leq
2|\alpha_{1}|\Psi^{\prime}(\tfrac{1}{2}\theta)=\tfrac{1}{2}\kappa N^{-1/3}.$
(4.10)

Propositions 3.9 and 4.1 can now be proven using the above coupling and
estimate for $|\mathbb{E}[\log U_{1}(1)]|$.

###### Proof of Proposition 3.9 in the case $p=1$ (critical phase).

Fix $\varepsilon\in(0,1)$. Set

$\beta_{1}:=\mathbb{P}\Big{(}\sup_{x\in[0,T]}B_{1}(x)\leq\tfrac{\sqrt{T}}{8}\Big{)}=\mathbb{P}\Big{(}\sup_{x\in[0,1]}B_{1}(x)\leq\tfrac{1}{8}\Big{)}>0,\quad\beta_{2}:=\inf_{n\in\mathbb{N}}\exp\big{(}-2(n-1)e^{-\frac{1}{2}\sqrt{n}}\big{)}>0.$

Set $T:=\lfloor rN^{2/3}\rfloor$. Continuing with the notations from Lemma
4.3, let us assume

$(L_{i}(j))_{(i,j)\in\mathcal{K}_{2,T}}\sim\overline{\mathbb{P}}^{(0,-A\sqrt{T})}.$

Observe that $|(T-1)\mathbb{E}[\log U_{1}(1)]|\leq\sqrt{r}\kappa\sqrt{T}$.
Following the relation in (4.7) and the estimate in (4.9) we get that

$\displaystyle\overline{\mathbb{P}}^{(0,-A\sqrt{T})}\bigg{(}L_{1}(2k-1)$
$\displaystyle\geq-\tfrac{1}{8}\sqrt{T}-\sqrt{r}\kappa\sqrt{T}-\mathrm{C}\log
T\mbox{ for all }k\in\llbracket 1,T\rrbracket,\mbox{ and}$ $\displaystyle
L_{2}(2k)$
$\displaystyle\leq-A\sqrt{T}+\tfrac{1}{8}\sqrt{T}+\sqrt{r}\kappa\sqrt{T}+\mathrm{C}\log
T\mbox{ for all }k\in\llbracket
1,T\rrbracket\bigg{)}\geq\beta_{1}^{2}-\tfrac{2}{T}.$

Recall that $A=1+2\sqrt{r}\kappa$ from (3.28). Thus for large enough $T$ we
can ensure that

$\big{(}-\tfrac{1}{8}\sqrt{T}-\sqrt{r}\kappa\sqrt{T}-\mathrm{C}\log
T\big{)}-\big{(}-A\sqrt{T}+\tfrac{1}{8}\sqrt{T}+\sqrt{r}\kappa\sqrt{T}+\mathrm{C}\log
T\big{)}\geq\tfrac{1}{2}\sqrt{T},$

and $\beta_{1}^{2}-\frac{2}{T}\geq\frac{1}{2}\beta_{1}^{2}$. Thus for large
enough $T$ we have

$\displaystyle\overline{\mathbb{P}}^{(0,-A\sqrt{T})}\Big{(}L_{1}(2k-1)\wedge
L_{1}(2k+1)\geq L_{2}(2k)+\tfrac{1}{2}\sqrt{T},\mbox{ for all }k\in\llbracket
1,T-1\rrbracket\Big{)}\geq\tfrac{1}{2}\beta_{1}^{2}.$

Following the definition of $W_{\operatorname{cr}}$ from (4.8) we thus get

$\displaystyle\overline{\mathbb{E}}^{(0,-A\sqrt{T})}[W_{\operatorname{cr}}]\geq\tfrac{1}{2}\beta_{1}^{2}\cdot\exp\big{(}-2(T-1)e^{-\frac{1}{2}\sqrt{T}}\big{)}\geq\tfrac{1}{2}\beta_{1}^{2}\beta_{2}.$

By Lemma 4.3, this forces

$\displaystyle\mathbb{P}_{\alpha_{1}}^{(0,-A\sqrt{T}),(-\infty)^{T};2,T}\big{(}|L_{2}(2)|\geq
M\sqrt{T}\big{)}$
$\displaystyle=\frac{\overline{\mathbb{E}}^{(0,-A\sqrt{T})}\big{[}W_{\operatorname{cr}}\mathbf{1}_{|L_{2}(2)|\geq
M\sqrt{T}}\big{]}}{\overline{\mathbb{E}}^{(0,-A\sqrt{T})}[W_{\operatorname{cr}}]}$
$\displaystyle\leq
2\beta_{1}^{-2}\beta_{2}^{-1}\cdot\overline{\mathbb{P}}^{(0,-A\sqrt{T})}\big{(}|L_{2}(2)|\geq
M\sqrt{T}\big{)}.$

Under $\overline{\mathbb{P}}^{(0,-A\sqrt{T})}$, $L_{2}(2)$ has variance
$T\cdot\operatorname{Var}(V_{1}(1))$ and mean $-A\sqrt{T}+(T-1)\mathbb{E}[\log
U_{1}(1)]$. One can thus choose $M$ large enough so that the last term in the
above equation is at most $\varepsilon$. Similarly one can show
$\mathbb{P}_{\alpha_{2}}^{(0,-A\sqrt{T}),(-\infty)^{T};2,T}\big{(}|L_{1}(1)|\geq
M\sqrt{T}\big{)}\leq\varepsilon$ for all large enough $M$. This proves (3.29)
for $p=1$. For (3.30), observe that by Lemma 4.3 (a) and the Markov inequality
one can take $\widetilde{M}$ large enough so that have

$\displaystyle\mathbb{P}_{\alpha_{1}}^{0,(-\infty)^{T};1,T}\big{(}|L_{1}(1)|\geq\widetilde{M}\sqrt{T}\big{)}$
$\displaystyle\leq\tfrac{1}{\widetilde{M}^{2}T}\big{(}T\cdot\operatorname{Var}(V_{1}(1))+(|(T-1)\mathbb{E}[\log
U_{1}(1)])^{2}\big{)}\leq\varepsilon.$

∎

###### Proof of Proposition 4.1 in the case $p=1$ (critical case).

We continue with the same notations as in Lemma 4.3 with $T\mapsto 2T$. Set
$T:=\lfloor rN^{2/3}\rfloor$. Consider the collection of random variables
$(L_{i}(j))_{(i,j)\in\mathcal{K}_{2,2T}}$ defined in (4.7) with
$y_{1}=-MN^{\frac{1}{3}}$. By Lemma 4.3 (a), we get that

$(L_{1}(j))_{(i,j)\in\mathcal{K}_{1,2T}}\sim\mathbb{P}_{\alpha_{1}}^{-MN^{1/3};(-\infty)^{2T};1,2T}.$

We may assume $V_{1}(j)$ are defined in a probability space that includes a
Brownian motion $B_{1}(\cdot)$ such that (4.9) holds. Recall that given a
standard Brownian motion $B$ and an open set $\mathcal{U}\subset C([0,1])$
with $\\{f:f(0)=0\\}\subset\mathcal{U}$, we have
$\mathbb{P}(B|_{[0,1]}\in\mathcal{U})>0$. Thus by the scale invariance of
Brownian motion, there exists $\phi(\theta,\mu,r,M)>0$ such that

$\displaystyle\mathbb{P}\Big{(}0\leq\sigma B_{1}(x)-(16M+5\kappa
r)N^{1/3}\leq{MN^{1/3}}\mbox{ for all }x\in[\tfrac{T}{2},2T]\Big{)}\geq
2\phi.$

Here $\kappa=\frac{1}{4}|\mu|\Psi^{\prime}(\frac{1}{2}\theta)\geq 0$. Now for
$y=-MN^{\frac{1}{3}}$ we have $|y+(k-1)\mathbb{E}[\log U_{1}(1)]|\leq(M+\kappa
r)N^{\frac{1}{3}}$ for all $k\leq 2T$. For large enough $N$ we also have
$\mathrm{C}\log 2T\leq MN^{1/3}$ where $\mathrm{C}$ comes from (4.9). Thus in
view of (4.7) and (4.9) we have

$\displaystyle\mathbb{P}_{1}\bigg{(}(14M+4\kappa r)N^{\frac{1}{3}}\leq
L_{1}(4T-1-2k)$ (4.11) $\displaystyle\hskip 99.58464pt\leq(19M+6\kappa
r)N^{\frac{1}{3}}\mbox{ for all
}k\in\llbracket\tfrac{T}{2},2T-1\rrbracket\bigg{)}\geq 2\phi-\tfrac{1}{2T},$

where for simplicity we write
$\mathbb{P}_{1}:=\mathbb{P}_{\alpha_{1}}^{-MN^{\frac{1}{3}},(-\infty)^{2T};1,2T}$.
Let us set

$\displaystyle\mathsf{A}$ $\displaystyle:=\left\\{(14M+4\kappa
r)N^{\frac{1}{3}}\leq L_{1}(4T-1-2k)\leq(19M+6\kappa r)N^{\frac{1}{3}}\mbox{
for all }k\in\llbracket\tfrac{T}{2},2T-1\rrbracket\right\\},$
$\displaystyle\mathsf{B}(k)$
$\displaystyle:=\left\\{|L_{1}(2k-1)-L_{1}(2k)|,|L_{1}(2k+1)-L_{1}(2k)|\geq
2(5M+2\kappa r)N^{\frac{1}{3}}\right\\}.$

Recall the event $\mathsf{RP}_{1,M}$ from (4.2). Observe that

$\displaystyle\mathsf{RP}_{1,M}\supset\mathsf{A}\cap\bigcap_{k\in\llbracket
1,3T/2-1\rrbracket}\mathsf{B}(k)$

Thus by applying the union bound we get

$\displaystyle\mathbb{P}_{1}(\mathsf{RP}_{1,M})\geq\mathbb{P}_{1}\bigg{(}\mathsf{A}\cap\bigcap_{k\in\llbracket
1,3T/2-1\rrbracket}\mathsf{B}(k)\bigg{)}\geq\mathbb{P}_{1}(\mathsf{A})-\sum_{k\in\llbracket
1,3T/2-1\rrbracket}\mathbb{P}_{1}\bigg{(}\mathsf{A}\cap\neg\mathsf{B}(k)\bigg{)}$
(4.12)

Let us denote
$\mathcal{F}_{\operatorname{odd}}:=\sigma\big{(}(L_{1}(2k-1))_{k=1}^{2T}\big{)}$.
Note that the event $\mathsf{A}$ is measurable with respect to
$\mathcal{F}_{\operatorname{odd}}$. On the event $\mathsf{A}$,
$|L_{1}(2k+1)-L_{1}(2k-1)|\leq(5M+2\kappa r)N^{\frac{1}{3}}$ for all
$k\in\llbracket 1,3T/2-1\rrbracket$. Recall that the distribution of even
points of $L_{1}$ conditioned on $\mathcal{F}_{\operatorname{odd}}$ are given
by the $\xi$-distributions (see (4.5) and Lemma 4.3). Applying the tail bound
for the $\xi$-distribution from Lemma B.4 we have

$\displaystyle\mathbf{1}_{\mathsf{A}}\mathbb{E}_{1}\big{(}\mathbf{1}_{\neg\mathsf{B}(k)}\mid\mathcal{F}_{\operatorname{odd}}\big{)}\leq\mathbf{1}_{\mathsf{A}}\cdot\exp\left(-\mathrm{C}(5M+2\kappa
r)N^{\frac{1}{3}}\right),$

for all $k\in\llbracket 1,3T/2-1\rrbracket$. Taking another expectation with
respect to $\mathcal{F}_{\operatorname{odd}}$ above and then plugging the
bound back in (4.12), along with the lower bound of
$\mathbb{P}_{1}(\mathsf{A})$ from (4.11) we get that

$\displaystyle\mathbb{P}_{1}(\mathsf{RP}_{1,M})\geq
2\phi-\tfrac{1}{2T}-3rN^{\frac{2}{3}}\exp\left(-\mathrm{C}(5M+2\kappa
r)N^{\frac{1}{3}}\right).$

For large enough $N$, the right side of above equation is always larger than
$\phi$. ∎

###  4.3. Proof of Propositions 3.9 and 4.1 in the supercritical phase

Recall that Lemma 4.4 establishes that the law of
$\mathbb{P}_{\alpha_{2}}^{\vec{y},(-\infty)^{T};2,T}$ is related to the law of
weighted paired random walk ($\mathsf{WPRW}$) defined in Definition 1.7. We
will start by developing a few important properties of the paired random walk
and weighted paired random walk measures before going into the proof of
Propositions 3.9 and 4.1 in the $p=2$ case (supercritical phase).

######  Lemma 4.5 (Properties of the increments).

The densities $\mathsf{f}_{\theta}$ and $\mathsf{g}_{\zeta}$, defined in
(2.3), enjoy the following properties.

  1. (1)

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

  2. (2)

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

  3. (3)

For any $a<b$, $\inf_{x\in[a,b]}\mathsf{f}_{\theta}(x)>0$ and
$\inf_{x\in[a,b]}\mathsf{g}_{\zeta}(x)>0$.

  4. (4)

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

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

In other words $X$ and $Y$ are subexponential random variables.

###### Proof.

Recall that $\mathsf{f}_{\theta}=G_{\theta,+1}\ast G_{\theta,-1}$. Thus the
random variable corresponding to $\mathsf{f}_{\theta}$ can be viewed as
difference of two independent random variables drawn from $G_{\theta,+1}$.
Hence symmetricity claim of part (1) follows. Concavity of
$\log\mathsf{f}_{\theta}$ can be checked by computing the second derivative
explicitly. For the characteristic function from [DLMF, Formula 5.8.3] one has

$\displaystyle\psi(t)=\left|\frac{\Gamma(\theta+it)}{\Gamma(\theta)}\right|^{2}=\prod_{n=0}^{\infty}\left(1+\frac{t^{2}}{(\theta+n)^{2}}\right)^{-1}\leq\left(1+\frac{t^{2}}{\theta^{2}}\right)^{-1}.$
(4.13)

From here, one can verify part (2) of the above lemma. Part (3) and (4)
follows from the explicit form of the $G$ function. ∎

The following corollary allows us to use estimates developed in Appendix C
regarding non-intersection probabilities for random walks and bridges with
general jump distributions.

######  Corollary 4.6.

The density $\mathsf{f}_{\theta}$ defined in (2.3) satisfies Assumption C.1.

###### Proof.

This follows immediately from Lemma 4.5. ∎

Recall the $\mathsf{PRW}$ law from Definition 1.7. Let
$\overline{\mathsf{f}}_{\theta}^{(n)}$ be the density of
$\frac{X(1)+\cdots+X(n)}{\sqrt{n}}$ where $X(i)$ are i.i.d. drawn from
$\mathsf{f}_{\theta}$. Assume $U(n),V(n)\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}\overline{\mathsf{f}}_{\theta}^{(n-1)}$. Note that for any
Borel set $A\subset\mathbb{R}^{2}$ and $x,y\in\mathbb{R}$ we have

$\displaystyle\mathbb{P}_{\operatorname{PRW}}^{n;(x,y)}\big{(}(S_{1}(1),S_{2}(1))\in
A\big{)}=\frac{\mathbb{E}\Big{[}\mathsf{g}_{\zeta}(\sqrt{n}(V(n)-U(n)-\frac{x-y}{\sqrt{n}}))\mathbf{1}_{(U(n)+x,V(n)+y)\in{n}^{-1/2}A}\Big{]}}{\mathbb{E}[\mathsf{g}_{\zeta}(\sqrt{n}(V(n)-U(n)-\frac{x-y}{\sqrt{n}}))]}.$
(4.14)

The above formula is the guiding principle for extracting tail estimates of
various kinds of functions of $(S_{1}(1),S_{2}(1))$. We list few of them that
are indispensable for our later analysis.

######  Lemma 4.7 (Tail estimates for the entrance law).

Fix any $M>0$, $n\geq 1$, and consider $x_{n},y_{n}\in\mathbb{R}$ with
$|x_{n}|,|y_{n}|\leq M\sqrt{n}$. Fix two open intervals $I_{1},I_{2}>0$. Under
the above setup, there exists a constant
$\mathrm{C}=\mathrm{C}(M,I_{1},I_{2})>1$ such that for all $n\geq 1$ and
$\tau\geq 1$, we have

$\displaystyle\mathbb{P}_{\operatorname{PRW}}^{n;(x_{n},y_{n})}\big{(}|S_{1}(1)|\geq\tau\sqrt{n}\big{)}\leq\mathrm{C}e^{-\frac{1}{\mathrm{C}}\tau},$
(4.15)
$\displaystyle\mathbb{P}_{\operatorname{PRW}}^{n;(x_{n},y_{n})}\big{(}|S_{1}(1)-S_{2}(1)|\geq\tau\big{)}\leq\mathrm{C}e^{-\frac{1}{\mathrm{C}}\tau},$
(4.16)
$\displaystyle\mathbb{P}_{\operatorname{PRW}}^{n;(x_{n},y_{n})}\big{(}S_{1}(1)-S_{2}(1)\in
I_{1},S_{1}(1)\in\sqrt{n}I_{2}\big{)}\geq\tfrac{1}{\mathrm{C}}.$ (4.17)

###### Proof of Lemma 4.7.

For simplicity let us write $z_{n}:=\frac{x_{n}-y_{n}}{\sqrt{n}}$. It is
enough to prove the Lemma 4.7 for large enough $n$. So, throughout the proof
we will assume $n$ is large enough. We first claim that the denominator of the
right-hand side of (4.14) is of the order $n^{-1/2}$, i.e. there exists a
$\mathrm{C}>1$ such that for all large enough $n$ we have

$\displaystyle\frac{1}{\mathrm{C}\sqrt{n}}\leq\mathbb{E}[\mathsf{g}_{\zeta}(\sqrt{n}(V(n)-U(n)-z_{n}))]\leq\frac{\mathrm{C}}{\sqrt{n}}$
(4.18)

Fix any $\tau\geq 0$. Using the exponential tails of $\mathsf{g}_{\zeta}$
(part (4) from Lemma 4.5) we have

$\displaystyle\mathbb{E}\bigg{[}\mathsf{g}_{\zeta}(\sqrt{n}(V(n)-U(n)-z_{n}))\mathbf{1}_{|\sqrt{n}(V(n)-U(n)-z_{n})|\geq\tau}\bigg{]}$
(4.19)
$\displaystyle\leq\mathbb{E}\bigg{[}\mathsf{g}_{\zeta}(\sqrt{n}(V(n)-U(n)-z_{n}))\mathbf{1}_{|\sqrt{n}(V(n)-U(n)-z_{n})|\geq\tau+\sqrt{n}}\bigg{]}$
$\displaystyle\hskip
85.35826pt+\sum_{p\in\llbracket\tau,\tau+\sqrt{n}\rrbracket}\mathbb{E}\bigg{[}\mathsf{g}_{\zeta}(\sqrt{n}(V(n)-U(n)-z_{n}))\mathbf{1}_{|\sqrt{n}(V(n)-U(n)-z_{n})|\in[p,p+1]}\bigg{]}$
$\displaystyle\leq\mathrm{C}e^{-\frac{1}{\mathrm{C}}(\sqrt{n}+\tau)}+\sum_{p\in\llbracket\tau,\tau+\sqrt{n}\rrbracket}\mathrm{C}e^{-\frac{1}{\mathrm{C}}p}\cdot\mathbb{P}\big{(}\tfrac{p}{\sqrt{n}}\leq|V(n)-U(n)-z_{n}|\leq\tfrac{p+1}{\sqrt{n}}\big{)}.$

Note that
$\overline{\mathsf{f}}_{\theta}^{(n)}(x)=\sqrt{n}\mathsf{f}_{\theta}^{\ast
n}(x\sqrt{n})$ where $\mathsf{f}_{\theta}^{\ast n}$ denotes the $n$-fold
convolution of $\mathsf{f}_{\theta}$. As $n\to\infty$, we know by central
limit theorem that this should converge to a Gaussian density with appropriate
variance. Lemma C.2 (recall Corollary 4.6) records a sharp quantitative
version of this convergence. Indeed, the estimate from Lemma C.2 (recall
Corollary 4.6) ensures that given any interval
$B:=[\frac{p}{\sqrt{n}},\frac{p+1}{\sqrt{n}}]\subset[-2,2]$, for all large
enough $n$, we have

$\mathbb{P}\big{(}(V(n)-U(n)-z_{n})\in
B\big{)}=(1+o(1))\mathbb{P}\big{(}Z_{1}-Z_{2}-z_{n}\in B\big{)}$

where $Z_{1},Z_{2}$ are independent Gaussian random variables with same
variance as of $\mathsf{f}_{\theta}$. By Gaussian computations we can ensure
that for all large enough $n$ we have

$\mathbb{P}\big{(}(V(n)-U(n)-z_{n})\in
B\big{)}\in[R^{-1}/\sqrt{n},R/\sqrt{n}]$

for some $R>1$ depending only on $M$. This ensures
$\mathbb{P}\big{(}\tfrac{p}{\sqrt{n}}\leq|V(n)-U(n)-z_{n}|\leq\tfrac{p+1}{\sqrt{n}}\big{)}\leq\frac{R}{\sqrt{n}}$
for all $p\in\llbracket 0,\sqrt{n}\rrbracket$. Plugging this bound back in
r.h.s. (4.19) leads to

$\displaystyle\mathbb{E}\bigg{[}\mathsf{g}_{\zeta}(\sqrt{n}(V(n)-U(n)-z_{n}))\mathbf{1}_{|\sqrt{n}(V(n)-U(n)-z_{n})|\geq\tau}\bigg{]}\leq\frac{\mathrm{C}}{\sqrt{n}}e^{-\tau/\mathrm{C}}.$
(4.20)

Taking $\tau=0$ leads to the upper bound in (4.18). For the lower bound we
note

$\displaystyle\mathbb{E}[g(\sqrt{n}(V(n)-U(n)-z_{n}))]$
$\displaystyle\geq\mathbb{E}\bigg{[}g(\sqrt{n}(V(n)-U(n)-z_{n}))\mathbf{1}_{V_{n}-U_{n}-x_{n}+y_{n}\in(\frac{1}{\sqrt{n}},\frac{2}{\sqrt{n}})}\bigg{]}$
(4.21)
$\displaystyle\geq\mathbb{P}\bigg{(}V(n)-U(n)-z_{n}\in(\tfrac{1}{\sqrt{n}},\tfrac{2}{\sqrt{n}})\bigg{)}\cdot\inf_{x\in[1,2]}\mathsf{g}_{\zeta}(x)$
$\displaystyle\geq\tfrac{R^{-1}}{\sqrt{n}}\cdot\inf_{x\in[1,2]}\mathsf{g}_{\zeta}(x),$

which is bounded below by $1/[\mathrm{C}\sqrt{n}]$, by the property of
$\mathsf{g}_{\zeta}$ from part (3) of Lemma 4.5. This proves the lower bound
in (4.18).

Let us now prove the inequalities in Lemma 4.7 one by one. Inserting the upper
bound in (4.20) and lower bound in (4.18) in the formula (4.14) leads to
(4.16). For (4.17) notice that due to (4.14) and (4.21) we have

$\displaystyle\mathbb{P}\big{(}S_{1}(1)-S_{2}(1)\in
I_{1},S_{1}(1)\in\sqrt{n}I_{2}\big{)}$
$\displaystyle\geq\mathrm{C}_{1}^{-1}\sqrt{n}\cdot\inf_{x\in
I_{1}}g(-x)\cdot\mathbb{P}\Big{(}U(n)+\tfrac{x_{n}}{\sqrt{n}}\in\sqrt{n}I_{2},U(n)+z_{n}-V(n)\in{n}^{-1/2}I_{1}\Big{)}.$

Using Lemma C.2 (recall Corollary 4.6) again, the probability above can be
shown lower bounded by $\frac{\mathrm{C}_{2}^{-1}}{\sqrt{n}}$ for some
$\mathrm{C}_{2}$ depending on $M,I_{1},I_{2}$ but free of $n$. This proves
(4.17). For (4.15) we observe

$\displaystyle\mathbb{E}\big{[}g(\sqrt{n}(V(n)-U(n)-z_{n}))\mathbf{1}_{|U(n)|\geq\tau}\big{]}$
$\displaystyle\leq\mathrm{C}e^{-\frac{\sqrt{n}}{\mathrm{C}}}+\sum_{p\in\llbracket
0,\sqrt{n}\rrbracket}\mathrm{C}e^{-\frac{p}{\mathrm{C}}}\mathbb{P}\Big{(}\tfrac{p}{\sqrt{n}}\leq|V(n)-U(n)-z_{n}|\leq\tfrac{p+1}{\sqrt{n}},|U(n)|\geq\tau\Big{)}.$

By a union bound followed by tower property of conditional expectation, we
have

$\displaystyle\mathbb{P}\Big{(}\tfrac{p}{\sqrt{n}}\leq|V(n)-U(n)-z_{n}|\leq\tfrac{p+1}{\sqrt{n}},|U(n)|\geq\tau\Big{)}$
$\displaystyle\leq\mathbb{E}\left[\mathbf{1}_{\tau\leq|U(n)|\leq(\log
n)^{3/2}}\mathbb{P}\left(\tfrac{p}{\sqrt{n}}\leq|V(n)-U(n)-z_{n}|\leq\tfrac{p+1}{\sqrt{n}}\mid
U(n)\right)\right]+\mathbb{P}\big{(}|U(n)|\geq(\log n)^{3/2}\big{)}.$

By Lemma C.2 (recall Corollary 4.6), the conditional probability above can be
uniformly bounded above by $\frac{\mathrm{C}_{3}}{\sqrt{n}}$ for some
$\mathrm{C}_{3}$ independent of $p$ and $n$. Exponential tail estimates of
$U(n)$, which follows from sub-exponential property (part (4) of Lemma 4.5) of
$\mathsf{f}_{\theta}$ (see Theorem 2.8.1 from [Ver18]), show that the right-
hand side of the above equation is at most
$\frac{\mathrm{C}}{\sqrt{n}}e^{-\frac{1}{\mathrm{C}}\tau}$. Combining these
estimates yields

$\mathbb{E}\left[g(\sqrt{n}(V(n)-U(n)-z_{n}))\mathbf{1}_{|U(n)|\geq\tau}\right]\leq\tfrac{\mathrm{C}}{\sqrt{n}}e^{-\frac{1}{\mathrm{C}}\tau}.$

Using the lower bound for the denominator from (4.21), in view of (4.14), we
get (4.15). ∎

In order to deal with the $\mathsf{WPRW}$ law, the weighted version of the
$\mathsf{PRW}$ law (see Definition 1.7), we next analyze the
$W_{\operatorname{sc}}$ weight defined in (1.15). We record a convenient lower
bound for $W_{\operatorname{sc}}$ that will be useful in our later analysis.
Fix any $p,q\in\mathbb{Z}_{\geq 1}$ with $p+q\leq n-1$. Given any $\beta>0$,
we consider several ‘Gap’ events:

$\displaystyle\mathsf{Gap}_{1,\beta}$
$\displaystyle:=\\{S_{1}(k)-S_{2}(k)\geq\beta k^{1/4}\mbox{ for all
}k\in\llbracket 2,p\rrbracket\\},$ $\displaystyle\mathsf{Gap}_{2,\beta}$
$\displaystyle:=\\{S_{1}(k)-S_{2}(k)\geq\beta(n-k)^{1/4}\mbox{ for all
}k\in\llbracket n-q,n-1\rrbracket\\},$ $\displaystyle\mathsf{Gap}_{3,\beta}$
$\displaystyle:=\\{S_{1}(k)-S_{2}(k)\geq n^{1/4}\mbox{ for all }k\in\llbracket
p+1,n-q\rrbracket\\},$ $\displaystyle\mathsf{Gap}_{4,\beta}$
$\displaystyle:=\\{S_{1}(k-1)-S_{1}(k)\leq\beta^{-1}k^{1/8}\mbox{ for all
}k\in\llbracket 2,p\rrbracket\\},$ $\displaystyle\mathsf{Gap}_{5,\beta}$
$\displaystyle:=\\{S_{1}(k-1)-S_{1}(k)\leq\beta^{-1}(n-k+1)^{1/8}\mbox{ for
all }k\in\llbracket n-q+1,n\rrbracket\\},$
$\displaystyle\mathsf{Gap}_{6,\beta}$
$\displaystyle:=\\{|S_{1}(k)-S_{1}(k-1)|\leq\beta^{-1}(\log n)\mbox{ for all
}k\in\llbracket p+1,n-q\rrbracket\\}.$

The events depend on $p$ and $q$ as well, but we have suppressed it from the
notation. $\mathsf{Gap}_{1,\beta},\mathsf{Gap}_{2,\beta},$ and
$\mathsf{Gap}_{3,\beta}$ requires $S_{1}(k)-S_{2}(k)$ to be bigger than a
threshold pointwise in the left ($\llbracket 2,p\rrbracket$), right
($\llbracket n-q,n-1\rrbracket$), and middle ($\llbracket p+1,n-q\rrbracket$)
region respectively. The type of threshold depends on the region.
$\mathsf{Gap}_{4,\beta},\mathsf{Gap}_{5,\beta},$ and $\mathsf{Gap}_{6,\beta}$
controls the increments of $S_{1}(k)$. Set

$\displaystyle\mathsf{Gap}_{\beta}:=\bigcap_{i=1}^{6}\mathsf{Gap}_{i,\beta}.$
(4.22)

We have the following deterministic inequality for $W_{\operatorname{sc}}$.

######  Lemma 4.8.

Recall $W_{\operatorname{sc}}$ from (1.15). Given any $\beta>0$, there exists
$a_{\beta}>0$ such that for all $n\geq 1$,

$W_{\operatorname{sc}}\geq
a_{\beta}\cdot\mathbf{1}_{\mathsf{Gap}_{\beta}\cap\\{|S_{1}(1)-S_{2}(1)|\leq\beta^{-1}\\}}.$

where $W_{\operatorname{sc}}$ is defined in (1.15).

###### Proof.

Assume $\mathsf{Gap}_{\beta}$ holds. For $k\in\llbracket 2,n-1\rrbracket$ we
have

$\displaystyle S_{2}(k)-S_{1}(k)$ $\displaystyle\leq-\min(\beta
k^{1/4},\beta(n-k)^{1/4},n^{1/4})=:\tau^{(n)}(k).$

Clearly
$\sum_{k=2}^{n-1}e^{S_{2}(k)-S_{1}(k)}\leq\sum_{k=1}^{n-1}e^{\tau^{(n)}(k)}$
is uniformly bounded in $n$ and hence can be bounded by some constant
${C}_{\beta}\in(0,\infty)$. Similarly for $k\in\llbracket 2,n-1\rrbracket$ we
have

$\displaystyle S_{2}(k)-S_{1}(k+1)$
$\displaystyle=S_{2}(k)-S_{1}(k)+S_{1}(k)-S_{1}(k+1)$
$\displaystyle\leq\tau_{n}^{(1)}(k)+\beta^{-1}\max((k+1)^{1/8},(n-k)^{1/8},(\log
n))=:\widetilde{\tau}^{(n)}(k).$

Clearly
$\sum_{k=2}^{n-1}e^{S_{2}(k)-S_{1}(k+1)}\leq\sum_{k=1}^{n-1}e^{\widetilde{\tau}^{(n)}(k)}$
is uniformly bounded in $n$ and hence can be bounded by some constant
$\widetilde{C}_{\beta}\in(0,\infty)$. Finally,

$S_{2}(1)-S_{1}(1)=S_{1}(1)-S_{2}(2)+S_{2}(2)-S_{1}(2)\leq 3\beta^{-1}$

on the event
$\\{|S_{1}(1)-S_{2}(1)|\leq\beta^{-1}\\}\cap\mathsf{Gap}_{4,\beta}$. Thus from
the definition of $W_{\operatorname{sc}}$ in (1.15) we have

$\displaystyle
W_{\operatorname{sc}}\geq\mathbf{1}_{\mathsf{Gap}_{\beta}\cap\\{|S_{1}(1)-S_{2}(1)|\leq\beta^{-1}\\}}\cdot\exp(-e^{3\beta^{-1}}-C_{\beta}-\widetilde{C}_{\beta}).$

Taking $a_{\beta}:=\exp(-e^{3\beta^{-1}}-C_{\beta}-\widetilde{C}_{\beta})$
completes the proof. ∎

Note that upon conditioning on the values of $S_{1}(1)$ and $S_{2}(1)$, a
paired random walk (recall Definition 3.6) has the law of two independent
$n$-step random bridges (recall Definition 3.6) starting from
$(S_{1}(1),S_{2}(1))$ and ending at $(x,y)$.

We now introduce modified random bridge above as they are easier to work with
than random bridges. Indeed, as described in the proof idea section of the
introduction, the main advantage of working with modified random bridges is
that they have a (true) random walk portion and one can appeal to classical
non-intersection probability estimates available for the random walks. On the
other hand, the laws of random bridge and modified random bridge can be
compared with the help of Lemma 4.10 below. Figure 5 contains an illustration
of such a bridge for $p=q=\lfloor n/4\rfloor$.

######  Definition 4.9 ($(n;p,q)$-modified random bridge).

Fix $n\geq 1$, and $p,q\in\llbracket 0,n\rrbracket$ with $p+q\leq n$ and
$p\neq 0$. Take any $a,b\in\mathbb{R}$. Let $(X(i),Y(i))_{i\in\mathbb{Z}_{\geq
1}}\stackrel{{\scriptstyle i.i.d.}}{{\sim}}\mathsf{f}_{\theta}$ where
$\mathsf{f}_{\theta}$ is defined in (2.3). Set $S(1):=a$ and $S(n):=b$. Set
For $k\in\llbracket 2,p\rrbracket$, set

$S(k):=a+\sum_{j=1}^{k-1}X_{j}\mbox{ for }k\in\llbracket 2,p\rrbracket,\quad
S(n-k):=b-\sum_{j=1}^{k}Y_{j}\mbox{ for }k\in\llbracket 1,q\rrbracket.$

Conditioned on $(S(k))_{k\in\llbracket 1,p\rrbracket\cup\llbracket
n-q,n\rrbracket}$, set
$(S(k))_{k=p}^{n-q}\sim\mathbb{P}^{n-p-q+1;\widetilde{a},\widetilde{b}}$ where
$\widetilde{a}:=S(p)$, $\widetilde{b}:=S(n-q)$, and $\mathbb{P}^{m;a,b}$ is a
$m$-step random walk from $a$ to $b$. We call $(S(k))_{k\in\llbracket
1,n\rrbracket}$ a $(n;p,q)$-modified random bridge of length $n$ starting at
$a$ and ending at $b$ and denote its law as
$\widetilde{\mathbb{P}}^{(n;p,q);a,b}$. Again, we shall often consider two
independent $(n;p,q)$-modified random bridges starting from $(a_{1},a_{2})$
and ending at $(b_{1},b_{2})$. Such bridges can be viewed as a measure on
$(\Omega^{2}_{n},\mathcal{F}^{2}_{n})$ space introduced in Definition 1.7. We
write $\widetilde{\mathbb{P}}^{(n;p,q);(a_{1},a_{2}),(b_{1},b_{2})}$ to denote
its law.

######  Lemma 4.10 (Comparison Lemma).

Fix any $M,\widetilde{M}>0$ and $\delta_{1},\delta_{2}\in[0,1/2)$, and $n\geq
1$. Set $p=\lfloor n\delta_{1}\rfloor$ and $q=\lfloor n\delta_{2}\rfloor$. For
$\vec{x}\in\mathbb{R}^{n-2}$, let $V_{a,b}(\vec{x})$ and
$\widetilde{V}_{a,b}(\vec{x})$ be the joint density of a $n$-step random
bridge and $(n;p,q)$-modified random bridge starting at $a$ and ending at $b$.
Suppose $a,b\in\mathbb{R}$ with $|a-b|\leq M\sqrt{n}$. Then, there exists two
constants $\mathrm{C}_{1}=\mathrm{C}_{1}(M,\delta_{1},\delta_{2})>0$ and
$\mathrm{C}_{2}=\mathrm{C}_{2}(M,\widetilde{M},\delta_{1},\delta_{2})>0$ such
that for all $\vec{x}\in\mathbb{R}^{n-2}$ and all $a,b\in\mathbb{R}$ with
$|a-b|\leq M\sqrt{n}$ we have

$\displaystyle
V_{a,b}(\vec{x})\leq\mathrm{C}_{1}\\!\cdot\\!\widetilde{V}_{a,b}(\vec{x}),$
(4.23) $\displaystyle
V_{a,b}(\vec{x})\\!\cdot\\!\mathbf{1}_{|x_{p}-x_{n-q}|\leq\widetilde{M}\sqrt{n}}\geq\mathrm{C}_{2}^{-1}\\!\cdot\\!\widetilde{V}_{a,b}(\vec{x})\\!\cdot\\!\mathbf{1}_{|x_{p}-x_{n-q}|\leq\widetilde{M}\sqrt{n}}.$
(4.24)

###### Proof.

We have

$\displaystyle
V_{a,b}(\vec{x}):=\frac{\prod_{j=0}^{n-2}\mathsf{f}_{\theta}(x_{j+1}-x_{j})}{\mathsf{f}_{\theta}^{\ast(n-1)}(b-a)},\quad\widetilde{V}_{a,b}(\vec{x}):=\frac{\prod_{j=0}^{n-2}\mathsf{f}_{\theta}(x_{j+1}-x_{j})}{\mathsf{f}_{\theta}^{\ast(n-p-q)}(x_{p}-x_{n-q})}.$

where $x_{0}:=a$ and $x_{n-1}:=b$. We thus have

$\displaystyle\frac{V_{a,b}(\vec{x})}{\widetilde{V}_{a,b}(\vec{x})}=\frac{\mathsf{f}_{\theta}^{\ast(n-p-q)}(x_{p}-x_{n-q})}{\mathsf{f}_{\theta}^{\ast(n-1)}(b-a)}.$
(4.25)

By [Fel08, Theorem 2, Chapter XV.5] we know

$\displaystyle\sup_{z\in\mathbb{R}}\bigg{|}{\sqrt{k}}\mathsf{f}_{\theta}^{\ast
k}(z)-\tfrac{1}{\sqrt{2\pi\sigma}}e^{-\frac{z^{2}}{2k\sigma^{2}}}\bigg{|}\xrightarrow{k\to\infty}0,$
(4.26)

where $\sigma=\int_{\mathbb{R}}x^{2}\mathsf{f}_{\theta}(x)dx$. Thus, there
exist a constant $\widetilde{\mathrm{C}}_{1}>1$ depending on
$M,\delta_{1},\delta_{2}$ such that

$\displaystyle\sqrt{n}\mathsf{f}_{\theta}^{\ast(n-p-q)}(z)\leq\widetilde{\mathrm{C}}_{1}\mbox{
for all
}z\in\mathbb{R},\quad\tfrac{1}{\widetilde{\mathrm{C}}_{1}}\leq\sqrt{n}\mathsf{f}_{\theta}^{\ast(n-1)}(b-a)\leq{\widetilde{\mathrm{C}}_{1}}.$

for all large enough $n$. Inserting these bounds in the numerator and
denominator of r.h.s. (4.25) we get the (4.23) by setting
$\mathrm{C}_{1}:=\widetilde{\mathrm{C}}_{1}^{2}$. When
$|x_{p}-x_{n-q}|\leq\widetilde{M}\sqrt{n}$, we may utilize the limit result in
(4.26) to obtain a new constant $\widetilde{\mathrm{C}}_{2}>0$ depending on
$\widetilde{M},\delta_{1},\delta_{2}$ such that

$\displaystyle\sqrt{n}\mathsf{f}_{\theta}^{\ast(n-p-q)}(x_{p}-x_{n-q})\geq\tfrac{1}{\widetilde{\mathrm{C}}_{2}}\mbox{
whenever }|x_{p}-x_{n-q}|\leq\widetilde{M}\sqrt{n},$

for all large enough $n$. Using this bound and the upper bound for
$\sqrt{n}\mathsf{f}_{\theta}^{\ast(n-1)}(b-a)$ we get the desired result. We
arrive at (4.24) by setting
$\mathrm{C}_{2}:=\widetilde{\mathrm{C}}_{2}\cdot\widetilde{\mathrm{C}}_{1}$. ∎

With all the preparatory results in hand, we are now ready to prove
Propositions 3.9 and 4.1. Recall that in the introduction we gave a proof
sketch for Proposition 3.9 (that does not appeal to Proposition 4.1). In what
follows, we shall use the techniques outlined in that sketch to establish more
sophisticated intermediate results (such as Lemma 4.11). These results will
allow us to prove Proposition 4.1 first. Then using those intermediate results
we shall then establish Proposition 3.9.

###### Proof of Proposition 4.1 for the $p=2$ case (supercritical phase).

We split the proof into several steps.

Step 1. In this step, we reduce the proof of Proposition 4.1 to the claim
around (4.27). Fix $r>0$. Set $T:=\lfloor rN^{2/3}\rfloor$ and $n=2T$. Recall
$y_{i}$’s and the event $\mathsf{RP}_{2,M}$ from the statement of the lemma.
Since $\mathsf{RP}_{2,M}$ is a monotone event, by Proposition 2.6 we have

$\displaystyle\mathbb{P}_{\alpha_{2}}^{\vec{y},(-\infty)^{2T};2,2T}(\mathsf{RP}_{2,M})\geq\mathbb{P}_{\alpha_{2}}^{\vec{x},(-\infty)^{2T};2,2T}(\mathsf{RP}_{2,M}).$

where $x_{1}=-2MN^{1/3}$, $x_{2}=-2MN^{1/3}-\sqrt{n}$. By translation
invariance (Lemma 2.1 (a)), we may lift the Gibbs measure by $2MN^{1/3}$ units
so that the boundary conditions changes from $(x_{1},x_{2})$ to
$(0,-\sqrt{n})$. The $\mathsf{RP}_{2,M}$ event now requires the second curve
to be above the (lifted) barrier $4MN^{1/3}$ under this new boundary
condition. Since $4MN^{1/3}\leq 8Mr^{-1/2}\sqrt{n}$. it thus suffices to show
that there exists $\phi=\phi(r,M)>0$ such that

$\displaystyle\mathbb{P}_{\alpha_{2}}^{(0,-\sqrt{n}),(-\infty)^{n};2,n}\bigg{(}\inf_{i\in\llbracket
1,n\rrbracket}L_{2}(i)\geq 8Mr^{-1/2}\sqrt{n}\bigg{)}\geq\phi.$

for all large enough $n$. Towards this end we claim that there exists
$\phi=\phi(r,M)>0$ such that

$\displaystyle\liminf_{n\to\infty}\mathbb{P}_{\alpha_{2}}^{(0,-\sqrt{n}),(-\infty)^{2n};2,n}(\mathsf{D}_{m})\geq
2\phi,$ (4.27)

where

$\displaystyle\mathsf{D}_{m}:=\bigg{\\{}(L_{1}(2i-1),L_{2}(2i))\in(10m\sqrt{n},11m\sqrt{n})^{2}\mbox{
for all }i\in\llbracket 1,n/2\rrbracket\bigg{\\}},$

and $m:=Mr^{-1/2}$. Let us complete the proof assuming (4.27). Note that
(4.27) controls the even points of the second curve. By Lemma 4.4, we know
conditioned on the even points,
$L_{2}(2k+1)\sim\xi_{\theta,\theta;-1}^{(L_{2}(2k),L_{2}(2k+2))}$ for
$k=1,2,\ldots,2n-1$. In view of Lemma B.4, on the event $\mathsf{D}_{m}$ we
have

$\mathbb{E}\left[\mathbf{1}_{L_{2}(2k+1)\leq
8m\sqrt{n}}\mid\sigma\big{(}L_{2}(2k),L_{2}(2k+2)\big{)}\right]\leq\mathrm{C}e^{-\frac{1}{\mathrm{C}}m\sqrt{n}}.$

By Lemma 4.4, $L_{2}(1)\sim G_{\alpha_{2}+\theta,1}+L_{2}(2)$. Since the
density $G_{\alpha_{2}+\theta,1}$ have exponential tails, we see that on the
event $\mathsf{D}_{m}$ we have $\mathbb{P}(L_{2}(1)\leq 8m\sqrt{n}\mid
L_{2}(2))\leq\mathrm{C}e^{-\frac{1}{\mathrm{C}}m\sqrt{n}}.$ Thus by a union
bound,

$\displaystyle\mathbb{P}_{\alpha_{2}}^{(0,-\sqrt{n}),(-\infty)^{2n};2,n}\bigg{(}\inf_{i\in\llbracket
1,n\rrbracket}L_{2}(i)\geq
8m\sqrt{n}\bigg{)}\geq\mathbb{P}_{\alpha_{2}}^{(0,-\sqrt{n}),(-\infty)^{2n};2,n}(\mathsf{D}_{m})-\mathrm{C}\cdot
ne^{-\frac{1}{\mathrm{C}}m\sqrt{n}}\geq\phi,$

for large enough $n$. This establishes Proposition 4.1 for $p=2$, modulo
(4.27).

Step 2. In this and subsequent steps we prove (4.27). Recall the
$\mathsf{PRW}$ and $\mathsf{WPRW}$ laws from Definition 1.7. Recall from Lemma
4.4 that $(L_{1}(2i-1),L_{2}(2i))_{i\in\llbracket
1,n\rrbracket}\sim{\mathbb{P}}_{\operatorname{WPRW}}^{n;(0,-\sqrt{n})}$. We
the terminology from Definition 1.7 to write

$\displaystyle{\mathbb{P}}_{\operatorname{WPRW}}^{n;(0,-\sqrt{n})}(\mathsf{D}_{m})=\frac{{\mathbb{E}}_{\operatorname{PRW}}^{n;(0,-\sqrt{n})}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{D}_{m}}]}{{\mathbb{E}}_{\operatorname{PRW}}^{n;(0,-\sqrt{n})}[W_{\operatorname{sc}}]}$
(4.28)

where $W_{\operatorname{sc}}$ is defined in (1.15) and $\mathsf{D}_{m}$ is now
defined as

$\displaystyle\mathsf{D}_{m}:=\big{\\{}(S_{1}(i),S_{2}(i))\in(10m\sqrt{n},11m\sqrt{n})^{2}\mbox{
for all }i\in\llbracket 1,n/2\rrbracket\big{\\}}.$

For the remainder of the proof we write $\mathbb{P}$ and $\mathbb{E}$ for
${\mathbb{P}}_{\operatorname{PRW}}^{n;(0,-\sqrt{n})}$ and
${\mathbb{E}}_{\operatorname{PRW}}^{n;(0,-\sqrt{n})}$ respectively. We claim
that there exists constants $\mathrm{C}>0$ and
$\widetilde{\mathrm{C}}=\widetilde{\mathrm{C}}(m)>0$ such that

$\displaystyle\mathbb{E}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{D}_{m}}]\leq\tfrac{\widetilde{\mathrm{C}}^{-1}}{\sqrt{n}},\qquad\mathbb{E}[W_{\operatorname{sc}}]\geq\tfrac{\mathrm{C}}{\sqrt{n}}.$
(4.29)

Clearly plugging these bounds back in (4.28) verifies (4.27). Let us thus
focus on proving (4.29). For the upper bound we use the following lemma.

######  Lemma 4.11.

There exists a constant $\mathrm{C}>0$ such that for all Borel sets
$A\subset\mathbb{R}^{2}$ we have

$\displaystyle\mathbb{E}\left[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{A}}\right]\leq\tfrac{\mathrm{C}}{n}+\tfrac{\mathrm{C}}{\sqrt{n}}\mathbb{E}\bigg{[}\mathbf{1}_{\mathsf{A}}\Big{(}(S_{1}(1)-S_{2}(1)+1)\vee
1\Big{)}\Big{(}\tfrac{|S_{1}(1)|}{\sqrt{n}}\vee 2\Big{)}^{3/2}\bigg{]},$

where $\mathsf{A}:=\\{(S_{1}(1),S_{2}(1))\in A\\}$.

Note that taking $A=\mathbb{R}^{2}$ in Lemma 4.11 and utilizing the
exponential tail estimates of $|S_{1}(1)-S_{2}(1)|$ and $|S_{1}(1)|/\sqrt{n}$
from Lemma 4.7, the upper bound in (4.29) follows.

###### Proof of Lemma 4.11.

As in (1.18), define

$\displaystyle\mathsf{NI}_{p}:=\left\\{S_{1}(k)-S_{2}(k)\geq-p\mbox{ for all
}k\in\llbracket 2,n-1\rrbracket\right\\}.$ (4.30)

We set $\mathsf{NI}:=\mathsf{NI}_{0}$. Here $\mathsf{NI}$ stands for non-
intersection. Observe that for any $q\in\mathbb{Z}_{\geq 1}$ we have

$\displaystyle\mathbf{1}_{\mathsf{NI}}+\sum_{p=0}^{q}\mathbf{1}_{\mathsf{NI}_{p+1}\cap\mathsf{NI}_{p}^{c}}+\mathbf{1}_{\mathsf{NI}_{q}^{c}}=1.$
(4.31)

Thus, taking $q=\lfloor\log\log n\rfloor$ we have

$\displaystyle\mathbb{E}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{A}}]$
$\displaystyle=\mathbb{E}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{A}\cap\mathsf{NI}}]+\sum_{p=0}^{\lfloor\log\log
n\rfloor-1}\mathbb{E}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{A}\cap\mathsf{NI}_{p+1}\cap\mathsf{NI}_{p}^{c}}]+\mathbb{E}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{A}\cap\mathsf{NI}_{\lfloor\log\log
n\rfloor}^{c}}]$ $\displaystyle\leq\sum_{p=0}^{\lfloor\log\log
n\rfloor}e^{-e^{p-1}}\mathbb{P}\left(\mathsf{A}\cap\mathsf{NI}_{p}\right)+\tfrac{1}{n},$
(4.32)

where the above inequality follows by noting that on $\mathsf{NI}_{p}^{c}$ we
have $W\leq e^{-e^{p-1}}$. For the probability term above we condition on
$\mathcal{F}:=\sigma(S_{1}(1),S_{2}(1))$ and write
$\mathbb{P}(\mathsf{A}\cap\mathsf{NI}_{p})=\mathbb{E}\left[\mathbf{1}_{\mathsf{A}}\mathbb{E}[\mathbf{1}_{\mathsf{NI}_{p}}\mid\mathcal{F}]\right]$.
Upon conditioning on $S_{1}(1),S_{2}(1)$, the paired random walk
$\mathbb{P}_{\operatorname{PRW}}^{n;(0,-\sqrt{n})}$ law is equal to the law of
two independent $n$-step random walks starting from $(S_{1}(1),S_{2}(1))$ to
$(0,-\sqrt{n})$. Thanks to this observation, we may now appeal to Lemma C.8 to
conclude that $\mathbb{E}[\mathbf{1}_{\mathsf{NI}_{p}}\mid\mathcal{F}]\leq
e^{\mathrm{C}p}\mathbb{E}[\mathbf{1}_{\mathsf{NI}}\mid\mathcal{F}]$ holds for
some deterministic constant $\mathrm{C}>0$. This allows us to conclude

$\mathbb{P}(\mathsf{A}\cap\mathsf{NI}_{p})\leq
e^{\mathrm{C}p}\cdot\mathbb{P}(\mathsf{A}\cap\mathsf{NI}).$

Plugging this into (4.32) and observing that the series $\sum_{p\geq
0}e^{-e^{p-1}}\cdot e^{\mathrm{C}p}$ is summable shows

$\displaystyle\mathbb{E}[W_{\operatorname{sc}}\mathbf{1}_{A}]\leq\tfrac{1}{n}+\mathrm{C}\cdot\mathbb{P}(\mathsf{A}\cap\mathsf{NI}).$
(4.33)

Thus to suffices to bound $\mathbb{P}(\mathsf{A}\cap\mathsf{NI})$. Towards
this end, we first define the event

$\displaystyle\mathsf{B}:=\left\\{|S_{i}(1)|\leq(\log n)^{3/2}\sqrt{n}\mbox{
for }i=1,2,\mbox{ and }|S_{1}(1)-S_{2}(1)|\leq(\log n)^{3/2}\right\\}.$

By the union bound we have
$\mathbb{P}(\mathsf{A}\cap\mathsf{NI})\leq\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{NI})+\mathbb{P}(\mathsf{B}^{c}).$
For the second term note that by tail estimates from Lemma 4.7 we have

$\displaystyle\mathbb{P}(\mathsf{B}^{c})\leq\sum_{i=1}^{2}\mathbb{P}(|S_{i}(1)|\geq(\log
n)^{3/2})+\mathbb{P}\left(|S_{1}(1)-S_{2}(1)|\geq(\log
n)^{3/2}\right)\leq\tfrac{\mathrm{C}}{n}.$ (4.34)

For the first term we write
$\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{NI})=\mathbb{E}\left[\mathbf{1}_{\mathsf{A}\cap\mathsf{B}}\mathbb{E}[\mathbf{1}_{\mathsf{NI}}\mid\mathcal{F}]\right]$.
Again, since upon conditioning on $S_{1}(1),S_{2}(1)$, the paired random walk
law is equal to the law of of two independent $n$-step random walks starting
from $(S_{1}(1),S_{2}(1))$ to $(0,-\sqrt{n})$, we may use random bridge
estimates from Appendix C. In particular, Lemma C.9 (recall Corollary 4.6)
shows there exists a constant $\mathrm{C}>0$ such that

$\displaystyle\mathbf{1}_{\mathsf{B}}\mathbb{E}[\mathbf{1}_{\mathsf{NI}}\mid\mathcal{F}]\leq\mathbf{1}_{\mathsf{B}}\cdot\tfrac{\mathrm{C}}{\sqrt{n}}\Big{(}(S_{1}(1)-S_{2}(1)+1)\vee
1\Big{)}\Big{(}\tfrac{|S_{1}(1)|}{\sqrt{n}}\vee 2\Big{)}^{3/2}.$

for all $n$. Taking expectation with respect to $\mathcal{F}$ on both sides of
the above equation and then using the fact that
$\mathbf{1}_{\mathsf{A}\cap\mathsf{B}}\leq\mathbf{1}_{\mathsf{A}}$ leads to

$\displaystyle\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{NI})\leq\tfrac{\mathrm{C}}{\sqrt{n}}\cdot\mathbb{E}\left[\mathbf{1}_{\mathsf{A}}\Big{(}(S_{1}(1)-S_{2}(1)+1)\vee
1\Big{)}\Big{(}\tfrac{|S_{1}(1)|}{\sqrt{n}}\vee 2\Big{)}^{3/2}\right].$ (4.35)

Inserting this bound along with the bound in (4.34) back in (4.33), we arrive
at the desired bound stated in the statement of the lemma. ∎

Step 3. In this step we prove the lower bound in (4.29). Towards this end,
consider the event

$\displaystyle\mathsf{E}_{m}:=\big{\\{}1\leq S_{1}(1)-S_{2}(1)\leq 2,\mbox{
and
}S_{1}(1),S_{2}(1)\in(\tfrac{41}{4}m\sqrt{n},\tfrac{43}{4}m\sqrt{n})\big{\\}},$

and $\sigma$-algebra $\mathcal{F}:=\sigma(S_{1}(1),S_{2}(1))$. Fix any
$\beta>0$ and recall $\mathsf{Gap}_{\beta}$ from (4.22). We have

$\displaystyle\mathbb{E}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{D}_{m}}]\geq\mathbb{E}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{E}_{m}}\mathbf{1}_{\mathsf{D}_{m}}\mathbf{1}_{\mathsf{Gap}_{\beta}}]\geq
a_{\beta}\mathbb{E}[\mathbf{1}_{\mathsf{E}_{m}}\mathbb{E}\big{[}\mathbf{1}_{\mathsf{D}_{m}\cap\mathsf{Gap}_{\beta}}\mid{\mathcal{F}}]\big{]}$
(4.36)

where the second inequality above follows by noting that $W\geq a_{\beta}$ on
$\mathsf{Gap}_{\beta}\cap\mathsf{E}$ (Lemma 4.8). As mentioned in the proof of
Lemma 4.11, upon conditioning on $\mathcal{F}$, the paired random walk
$\mathbb{P}_{\operatorname{PRW}}^{n;(0,-\sqrt{n})}$ law is equal to the law of
two independent $n$-step random walks starting from $(S_{1}(1),S_{2}(1))$ to
$(0,-\sqrt{n})$. For simplicity set $b_{1}=0$, $b_{2}=-\sqrt{n}$. We shall use
the comparison between random bridge to $(n;n\rho,0)$-modified random bridge
from Lemma 4.10, for a special $\rho\in(0,1)$ coming from Corollary C.6
(recall Corollary 4.6). Using the lower bound in (4.24) we get

$\displaystyle\mathbf{1}_{\mathsf{E}_{m}}\cdot\mathbb{E}[\mathbf{1}_{\mathsf{D}_{m}\cap\mathsf{Gap}_{\beta}}\mid\mathcal{F}]$
$\displaystyle\geq\mathbf{1}_{\mathsf{E}_{m}}\mathrm{C}^{-1}\cdot\widetilde{\mathbb{P}}_{\rho}^{(S_{1}(1),S_{2}(1))}(\mathsf{D}_{m}\cap\mathsf{Gap}_{\beta})$
(4.37)
$\displaystyle=\mathbf{1}_{\mathsf{E}_{m}}\mathrm{C}^{-1}\cdot\widetilde{\mathbb{P}}_{\rho}^{(S_{1}(1),S_{2}(1))}(\mathsf{D}_{m}\cap\mathsf{Gap}_{\beta}\mid\mathsf{NI})\widetilde{\mathbb{P}}_{\rho}^{(S_{1}(1),S_{2}(1))}(\mathsf{NI})$

for some $\mathrm{C}>0$ depending on $m$ and $\rho$. Here
$\widetilde{\mathbb{P}}_{\rho}^{(S_{1}(1),S_{2}(1))}:=\widetilde{\mathbb{P}}^{(n;n\rho,0);(S_{1}(1),S_{2}(1)),(0,\sqrt{n})}$
is the joint law of two independent $(n;n\rho,0)$-modified random bridge from
$S_{i}(1)$ to $b_{i}$ defined in Definition 4.9. We now claim that there
exists $\widetilde{\phi}=\widetilde{\phi}(m)>0$ such that

$\displaystyle\widetilde{\mathbb{P}}_{\rho}^{(a_{1},a_{2})}(\mathsf{D}_{m}\mid\mathsf{NI})\geq
2\widetilde{\phi}.$ (4.38)

uniformly over all $(a_{1},a_{2})\in\mathcal{P}_{1}$ where we define

$\mathcal{P}_{1}:=\big{\\{}(z_{1},z_{2})\in(\tfrac{41}{4}m\sqrt{n},\tfrac{43}{4}\sqrt{n}):1\leq
z_{1}-z_{2}\leq 2\big{\\}}.$

We postpone the proof of this claim to the next step. Let us complete the
proof of the lemma assuming it. Since, under
$\widetilde{\mathbb{P}}_{\rho}^{(a_{1},a_{2})}$, $S_{1}(\llbracket
1,n\rho\rrbracket),S_{2}(\llbracket 1,n\rho\rrbracket)$ are two independent
random walks, we may use non-intersection type estimates for random walks from
Appendix C. In particular, using Lemmas C.3 and C.5 (recall Corollary 4.6) we
can get constants $\delta>0,M_{2}>0$ and $\mathrm{C}_{1}>0$ all depending on
$m$ and $\rho$ such that uniformly over $(a_{1},a_{2})\in\mathcal{P}_{1}$ we
have

$\widetilde{\mathbb{P}}_{\rho}^{(a_{1},a_{2})}\big{(}S_{1}(k)\geq
S_{2}(k)\mbox{ for all }k\in\llbracket
2,n\rho\rrbracket,S_{1}(n\rho)-S_{2}(n\rho)\geq\delta\sqrt{n},|S_{i}(n\rho)|\leq
M_{2}\sqrt{n}\big{)}\geq\tfrac{\mathrm{C}_{1}^{-1}}{\sqrt{n}}.$

Set
$\mathsf{G}:=\\{S_{1}(n\rho)-S_{2}(n\rho)\geq\delta\sqrt{n},|S_{i}(n\rho)|\leq
M_{2}\sqrt{n}\\}$. Recall from the definition of $(n;n\rho,0)$-modified random
bridge that on $\llbracket n\rho,n\rrbracket$ the modified random bridge is
just a random bridge from $S_{i}(n\rho)$ to $b_{i}$. Applying Lemma C.7
(recall Corollary 4.6) it follows that

$\mathbf{1}_{\mathsf{G}}\cdot\widetilde{\mathbb{P}}_{\rho}^{(a_{1},a_{2})}\big{(}S_{1}(k)\geq
S_{2}(k)\mbox{ for all }k\in\llbracket
n\rho,n-1\rrbracket\big{)}\geq\mathbf{1}_{\mathsf{G}}\cdot{\mathrm{C}_{2}^{-1}},$

for some constant $\mathrm{C}_{2}>0$ depending on $m$ and $\rho$ only. Thus we
get
$\widetilde{\mathbb{P}}_{\rho}^{(a_{1},a_{2})}(\mathsf{NI})\geq\frac{\mathrm{C}_{4}^{-1}}{\sqrt{n}}$
uniformly on $\mathsf{E}_{m}$ for some deterministic constant $\mathrm{C}_{4}$
depending on $m$ and $\rho$ only. By Lemma C.11 (recall Corollary 4.6), we may
choose $\beta$ small enough depending on $m$ and $\rho$ such that
$\widetilde{\mathbb{P}}_{\rho}^{(a_{1},a_{2})}(\mathsf{Gap}_{\beta}\mid\mathsf{NI})\geq
1-\widetilde{\phi}$ uniformly over $(a_{1},a_{2})\in\mathcal{P}_{1}$. Plugging
this estimates back in (4.37), we see that

$\mathbf{1}_{\mathsf{E}_{m}}\cdot\mathbb{E}[\mathbf{1}_{\mathsf{D}_{m}\cap\mathsf{Gap}_{\beta}}\mid\mathcal{F}]\geq\mathbf{1}_{\mathsf{E}_{m}}\cdot\widetilde{\phi}\cdot\frac{\mathrm{C}_{4}^{-1}}{\sqrt{n}}.$

Now, by Lemma 4.7 (equation (4.17) in particular) we know that
$\mathbb{P}(\mathsf{E}_{m})\geq\mathrm{C}_{5}^{-1}>0$ for some
$\mathrm{C}_{5}$ depending on $m$. Plugging this back in (4.36) we see that

$\displaystyle\mathbb{E}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{D}_{m}}]\geq
a_{\beta}\cdot\mathbb{P}(\mathsf{E}_{m})\cdot\widetilde{\phi}\cdot\tfrac{\mathrm{C}_{4}^{-1}}{\sqrt{n}}=:\tfrac{\widetilde{\mathrm{C}}^{-1}}{\sqrt{n}}.$
(4.39)

where $\widetilde{\mathrm{C}}>0$ is a constant depending only on $m$ and
$\rho$.

Step 4. In this step we prove (4.38). By equation (C.1) in Lemma C.5 (recall
Corollary 4.6), we know there exists $\delta\in(0,\frac{1}{8}(m\wedge 1))$
small enough depending only on $\rho$ such that

$\displaystyle\widetilde{\mathbb{P}}_{\rho}^{(a_{1},a_{2})}\big{(}S_{1}(n\rho)-S_{2}(n\rho)\geq\delta\sqrt{n}\mid\mathsf{NI}\big{)}\geq\tfrac{15}{16}$
(4.40)

uniformly over $(a_{1},a_{2})\in\mathcal{P}_{1}$. We shall now choose $\rho$
as $\rho(\frac{1}{16},\frac{m\wedge 1}{8})$ where the latter is a constant
depending on $m$ and comes from Corollary C.6 (recall Corollary 4.6). In view
of this choice of $\rho$, applying Corollary C.6 (recall Corollary 4.6), we
see that uniformly over $(a_{1},a_{2})\in\mathcal{P}_{1}$ we have

$\displaystyle\widetilde{\mathbb{P}}_{\rho}^{(a_{1},a_{2})}\left(\sup_{k\in\llbracket
1,n\rho\rrbracket,i=1,2}|S_{i}(k)-S_{i}(1)|\leq\tfrac{m\wedge
1}{8}\sqrt{n}\mid\mathsf{NI}\right)\geq\tfrac{15}{16}.$ (4.41)

Since on $\mathcal{P}_{1}$ we also have
$(a_{1},a_{2})\in(\tfrac{41}{4}m\sqrt{n},\tfrac{43}{4}m\sqrt{n})$, combining
(4.40) and (4.41) we get

$\displaystyle\widetilde{\mathbb{P}}_{\rho}^{(a_{1},a_{2})}\bigg{(}\big{\\{}S_{1}(n\rho)-S_{2}(n\rho)\geq\delta\sqrt{n}\big{\\}}\cap\mathsf{K}_{1}\mid\mathsf{NI}\bigg{)}\geq\tfrac{7}{8},$

where

$\displaystyle\mathsf{K}_{1}:=\left\\{S_{1}(k),S_{2}(k)\in(\tfrac{81}{8}m\sqrt{n},\tfrac{87}{8}m\sqrt{n})\mbox{
for all }k\in\llbracket 1,n\rho\rrbracket,\mbox{ and
}|S_{1}(n\rho)-S_{2}(n\rho)|\leq\tfrac{\sqrt{n}}{2}\right\\}.$

Following the definition of $(n;n\rho,0)$-modified random bridge, to prove
(4.38) it suffices to show

$\displaystyle\mathbb{P}^{n-n\rho+1;(c_{1},c_{2}),(b_{1},b_{2})}\bigg{(}\big{\\{}S_{1}(k),S_{2}(k)\in(10m\sqrt{n},11m\sqrt{n})\mbox{
for all }k\in\llbracket
1,n/2\rrbracket\big{\\}}\cap\mathsf{NI}\bigg{)}\geq\tfrac{16}{7}\widetilde{\phi},$
(4.42)

uniformly over $(c_{1},c_{2})\in\mathcal{P}_{2}$ where

$\mathcal{P}_{2}:=\\{(c_{1},c_{2})\in\mathbb{R}^{2}:c_{i}\in(\tfrac{81}{8}m\sqrt{n},\tfrac{87}{8}m\sqrt{n}),\mbox{
and }\tfrac{1}{2}\sqrt{n}\geq c_{1}-c_{2}\geq\delta\sqrt{n}\\}.$

and $\mathbb{P}^{n-n\rho+1;(c_{1},c_{2}),(b_{1},b_{2})}$ is the law of two
independent random bridges of length $n-n\rho+1$ starting at $(c_{1},c_{2})$
and ending at $(b_{1},b_{2})$. For simplicity set
$u:=n-n\rho+1\geq\frac{3}{4}n$. By the KMT coupling of random bridges [DW21a]
we may assume there are two independent Brownian bridges $B_{1},B_{2}$ (with
variance $\int x^{2}\mathsf{f}_{\theta}(x)dx$) on the same probability space
such that

$\displaystyle\mathbb{P}^{u;(c_{1},c_{2}),(b_{1},b_{2})}\left(\sup_{k\in\llbracket
1,u\rrbracket,i=1,2}|S_{i}(k)-\sqrt{u}B_{i}(k/u)-c_{i}-\tfrac{k}{u}(b_{i}-c_{i})|\geq\mathrm{C}\log
n\right)\leq\tfrac{1}{n}.$ (4.43)

Let $r_{n,i}(x)$ be the piece-wise linear function interpolated by three
points: $r_{n,i}(0)=r_{n,i}(1)=0$ and
$r_{n,i}(3/4)=\frac{3}{4\sqrt{u}}(b_{i}-c_{i})$. Let $\mathcal{U}_{i}$ be the
$L^{\infty}$ open ball of $r_{n,i}(x)$ of radius $\frac{1}{4}\delta$ (this is
the same $\delta$ that was chosen at the beginning of Step 4). By properties
of Brownian bridge, there exists a $\widetilde{\phi}=\widetilde{\phi}(m)>0$
such that for all $(c_{1},c_{2})\in\mathcal{P}_{2}$, we have

$\displaystyle\mathbb{P}^{u;(c_{1},c_{2}),(b_{1},b_{2})}(B^{(i)}\in\mathcal{U}_{i}\mbox{
for }i=1,2)\geq\tfrac{32}{7}\widetilde{\phi}.$

Note that the above equation along with (4.43) implies that with probability
$\tfrac{32}{7}\widetilde{\phi}-\frac{1}{n}$, for all $n$ large enough (and
hence $u$ large enough) under the law
$\mathbb{P}^{u;(c_{1},c_{2}),(b_{1},b_{2})}$ we have the following items
simultaneously:

  * •

For all $k\in\llbracket 1,3/4u\rrbracket$

$|S_{i}(k)-c_{i}|\leq\mathrm{C}\log
n+\tfrac{1}{4}\sqrt{u}\delta\leq\tfrac{m}{8}\sqrt{u}<\tfrac{m}{8}\sqrt{n}.$

  * •

For all $k\in\llbracket 1,u\rrbracket$ we have

$\displaystyle S_{1}(k)$
$\displaystyle\geq\sqrt{u}r_{n,1}(\tfrac{k}{u})+c_{1}+\tfrac{k}{u}(b_{1}-c_{1})-\tfrac{1}{4}\sqrt{u}\delta-\mathrm{C}\log
n$ $\displaystyle\geq
S_{2}(k)+\sqrt{u}(r_{n,1}(\tfrac{k}{u})-r_{n,2}(\tfrac{k}{u}))-\tfrac{1}{2}\sqrt{u}\delta+c_{1}-c_{2}+\tfrac{k}{u}(b_{1}-b_{2}-c_{1}+c_{2})-2\mathrm{C}\log
n$
$\displaystyle\geq\sqrt{u}(r_{n,1}(k/u)-r_{n,2}(k/u))+\tfrac{1}{2}\sqrt{u}\delta-2\mathrm{C}\log
n+S_{2}(k).$

We have $r_{n,1}(x)\geq r_{n,2}(x)$ by construction, and
$c_{1}-c_{2}+\tfrac{k}{u}(b_{1}-b_{2}-c_{1}+c_{2})\geq\sqrt{u}\delta$ for all
$(c_{1},c_{2})\in\mathcal{P}_{2}$. Thus for all large enough $n$,
$S_{1}(k)>S_{2}(k)$ for all $k\in\llbracket 0,u\rrbracket$.

For $n$ large enough
$\tfrac{32}{7}\widetilde{\phi}-\frac{1}{n}\geq\tfrac{16}{7}\widetilde{\phi}$.
This establishes (4.42) and hence also Proposition 4.1. ∎

######  Corollary 4.12.

There exists an absolute constant $\mathrm{C}>0$ such that for all $n\geq 1$.

$\displaystyle{\mathbb{E}}_{\operatorname{PRW}}^{n;(0,-\sqrt{n})}[W_{\operatorname{sc}}]\geq\tfrac{\mathrm{C}^{-1}}{\sqrt{n}}.$

The above corollary follows from (4.39) as
${\mathbb{E}}_{\operatorname{PRW}}^{n;(0,-\sqrt{n})}[W_{\operatorname{sc}}]\geq{\mathbb{E}}_{\operatorname{PRW}}^{n;(0,-\sqrt{n})}[W_{\operatorname{sc}}\mathbf{1}_{\mathsf{D}_{1}}]$.
We remark that here it is important that the endpoints are $O(\sqrt{n})$ apart
to get the precise order of $\mathbb{E}[W_{\operatorname{sc}}]$. We expect a
different order if the endpoints are closer or lie in a reversed order. Later,
in Lemma 5.6, we shall prove a different lower bound for
${\mathbb{E}}_{\operatorname{PRW}}^{n;(x,y)}[W_{\operatorname{sc}}]$ that is
uniform over all possible endpoints $(x,y)$ in a specific window.

###### Proof of Proposition 3.9 in the $p=2$ case (supercritical phase).

Given the machinery developed in the above proof, proof of Proposition 3.9
follows easily. By Lemma 4.4 we have

$\displaystyle\mathbb{P}_{\alpha_{2}}^{(0,-\sqrt{T}),(-\infty)^{T};2,T}(|L_{i}(i)|\geq
M\sqrt{T})$
$\displaystyle={\mathbb{P}}_{\operatorname{WPRW}}^{T;(0,-\sqrt{T})}(|S_{i}(1)|\geq
M\sqrt{T})$
$\displaystyle=\frac{{\mathbb{E}}_{\operatorname{PRW}}^{T;(0,-\sqrt{T})}[W_{\operatorname{sc}}\mathbf{1}_{{|S_{i}(1)|\geq
M\sqrt{T}}}]}{{\mathbb{E}}_{\operatorname{PRW}}^{T;(0,-\sqrt{T})}[W_{\operatorname{sc}}]}.$
(4.44)

Now by Corollary 4.12 we have
${\mathbb{E}}_{\operatorname{PRW}}^{T;(0,-\sqrt{T})}[W_{\operatorname{sc}}]\geq\frac{\mathrm{C}}{\sqrt{T}}$
and by Lemma 4.11 we have

$\displaystyle{\mathbb{E}}_{\operatorname{PRW}}^{T;(0,-\sqrt{T})}[W_{\operatorname{sc}}\mathbf{1}_{|S_{i}(1)|\geq
M\sqrt{T}}]$
$\displaystyle\leq\tfrac{1}{T}+\tfrac{\mathrm{C}}{\sqrt{T}}{\mathbb{E}}_{\operatorname{PRW}}^{T;(0,-\sqrt{T})}\bigg{[}\mathbf{1}_{|S_{i}(1)|\geq
M\sqrt{T}}\Big{(}(S_{1}(1)-S_{2}(1)+1)\vee
1\Big{)}\Big{(}\tfrac{|S_{i}(1)|}{\sqrt{T}}\vee 2\Big{)}^{3}\bigg{]}$
$\displaystyle\leq\tfrac{1}{T}+\tfrac{\mathrm{C}}{\sqrt{T}}\sqrt{{\mathbb{E}}_{\operatorname{PRW}}^{T;(0,-\sqrt{T})}\left[\Big{(}(S_{1}(1)-S_{2}(1)+1)\vee
1\Big{)}^{2}\right]}\sqrt{{\mathbb{E}}_{\operatorname{PRW}}^{T;(0,-\sqrt{T})}\left[\mathbf{1}_{|S_{i}(1)|\geq
M\sqrt{T}}\Big{(}\tfrac{|S_{1}(1)|}{\sqrt{T}}\vee 2\Big{)}^{6}\right]},$

where the last inequality follows from Cauchy-Schwarz. Taking $T$ and $M$
large enough, in view of the tail estimates from Lemma 4.7, it follows that
(4.44) can be made arbitrarily small. By a union bound, we can thus make
$\mathbb{P}_{\alpha_{2}}^{(0,-\sqrt{T}),(-\infty)^{T};2,T}(|L_{1}(1)|+|L_{2}(2)|\geq
M\sqrt{T})$ arbitrarily small by taking $T$ and $M$ large enough. ∎

##  5\. Modulus of continuity: proof of Theorem 1.1

In this section we prove our main theorem, Theorem 1.1, about spatial
tightness of $\mathpzc{HSLG}$ polymers. Due to the relation in (1.7), Theorem
1.1 essentially follows by controlling modulus of continuity of the first
curve of log-gamma line ensemble. Towards this end, we recall the definition
of modulus of continuity function. Given a continuous function
$f:\mathbb{Z}_{\geq 1}\to\mathbb{R}$ and $U>1$, we define the modulus of
continuity function as

$\displaystyle\omega_{\delta}^{N}(f;\llbracket
1,U\rrbracket):=\sup_{\begin{subarray}{c}i_{1},i_{2}\in\llbracket
1,U\rrbracket\\\ |i_{1}-i_{2}|\leq\delta
N^{2/3}\end{subarray}}|f(i_{1})-f(i_{2})|.$ (5.1)