# The half-space log-gamma polymer in the bound phase

Sayan Das  S. Das, Department of Mathematics, University of Chicago,
S. Das 5734 S. University Avenue, Chicago, Illinois 60637 USA
<EMAIL_ADDRESS>and  Weitao Zhu  W. Zhu, Department of Mathematics,
Columbia University,
S. Das 2990 Broadway, New York, NY 10027 USA<EMAIL_ADDRESS>
###### Abstract.

We consider the log-gamma polymer in the half-space with bulk weights
distributed as $\operatorname{Gamma}^{-1}(2\theta)$ and diagonal weights as
$\operatorname{Gamma}^{-1}(\alpha+\theta)$ for $\theta>0$ and
$\alpha>-\theta$. We show that in the bound phase, i.e., when
$\alpha\in(-\theta,0)$, the endpoint of the polymer lies within an $O(1)$
stochastic window of the diagonal. This result gives the first rigorous proof
of the pinned phenomena for the half-space polymers in the bound phase
conjectured by Kardar [Kar85]. We also show that the limiting quenched
endpoint distribution of the polymer around the diagonal is given by a random
probability mass function proportional to the exponential of a random walk
with log-gamma type increments.

###### Key words and phrases:

Directed Polymer, Kardar–Parisi–Zhang equation, stochastic heat equation,
Brownian bridge.

###### 2020 Mathematics Subject Classification:

Primary 60K37, 82B21, Secondary 82D60.

Figure 1. The bound and the unbound phase.

##  1\. Introduction

Directed polymers in random environments, first appeared in [HHF85, IS88,
Bol89], are a rich class of mathematical physics models that have been
extensively studied over the last several decades (see books [Szn98, Gia07,
DH09, Com17] and the references therein). More recently, a particular variant
of the polymer models, the half-space polymers, has garnered considerable
attention. The structure of the half-space polymers resembles the behavior of
an interface in the presence of an attractive wall and their understanding
renders importance to the studies of the wetting phenomena ([Abr80, PSW82,
BHL83]). Depending on the attraction force of the wall, it was conjectured in
[Kar85] that these models exhibit a “depinning” phase transition. When the
attraction force exceeds a certain critical threshold (colloquially known as
the bound phase), [Kar85] conjectured that the endpoint of the polymer stays
within a $O(1)$ window around the wall, i.e., it gets pinned to the wall. In
this paper, we focus on the half-space polymers with log-gamma weights which
make the model integrable and demonstrate a $O(1)$ pinning phenomena of the
endpoint in the bound phase. Our work is the first rigorous proof of such
$O(1)$ endpoint pinning phenomena.

Presently, we begin with an introduction to the model and the statements of
our main results.

###  1.1. The model and the main results

Fix any $\theta>0$ and $\alpha>-\theta$ and define the half-space index set:
$\mathcal{I}^{-}=\\{(i,j)\in\mathbb{Z}_{>0}^{2}\mid j\leq i\\}$. We consider a
family of independent variables $(W_{i,j})_{(i,j)\in\mathcal{I}^{-}}$:

$\displaystyle W_{i,i}\sim\operatorname{Gamma}^{-1}(\alpha+\theta)\qquad
W_{i,j}\sim\operatorname{Gamma}^{-1}(2\theta)\ \mbox{ for }i<j,$ (1.1)

where $\operatorname{Gamma}(\beta)$ denotes a random variable with density
$\mathbf{1}\\{x>0\\}[\Gamma(\beta)]^{-1}x^{\beta-1}e^{-x}$. Let
$\Pi_{N}^{\operatorname{half}}$ be the set of all upright lattice paths of
length $2N-2$ starting from $(1,1)$ that are confined to the half-space
$\mathcal{I}^{-}$ (see Figure 2). Given the weights in (1.1), the half-space
log-gamma ($\mathpzc{HSLG}$) polymer is a random measure on
$\Pi_{N}^{\operatorname{half}}$ defined as

$\displaystyle\mathbb{P}^{W}(\pi)=\frac{1}{Z(N)}\prod_{(i,j)\in\pi}W_{i,j}\cdot\mathbf{1}_{\pi\in\Pi_{N}^{\operatorname{half}}},$
(1.2)

where $Z(N)$ is the normalizing constant.

$(1,1)$$(11,5)$$(9,7)$$\operatorname{Gamma}^{-1}(\alpha+\theta)$$\operatorname{Gamma}^{-1}(2\theta)$
Figure 2. Two possible paths of length $14$ in $\Pi_{8}^{\operatorname{half}}$
are shown in the figure.

The parameter $\alpha$ controls the strength of the boundary weights, i.e. the
attractiveness of the wall, and a “depinning” phase transition occurs when
$\alpha=0$ (see [Kar85, PS02, BC20]). When $\alpha\geq 0$, [BW22, BCD23a]
showed that the polymer measure is unpinned and the endpoint lies in a
$O(N^{2/3})$ window. For $\alpha<0,$ the conjecture is that the attraction is
strong enough so that the polymer measure is pinned to the diagonal (see
Figure 1). Indeed, our first main result below confirms that in the bound
phase, i.e., when $\alpha\in(-\theta,0)$, the endpoint of the $\mathpzc{HSLG}$
polymer is within $O(1)$ window of the diagonal and is the first such result
to capture the “pinning” phenomenon of the half-space polymer measure to the
diagonal.

######  Theorem 1.1 (Bounded endpoint).

Fix $\theta>0$ and $\alpha\in(-\theta,0)$ and consider the random measure
$\mathbb{P}^{W}$ from (1.2). For a path $\pi\in\Pi_{N}^{\operatorname{half}}$,
we denote $\pi(2N-2)$ as the height (i.e., $y$-coordinate) of the endpoint of
the polymer. We have

$\displaystyle\lim_{k\to\infty}\limsup_{N\to\infty}\mathbb{P}^{W}(\pi(2N-2)\leq
N-k)=0,\quad\mbox{ in probability}.$ (1.3)

Theorem 1.1 is a quenched result and naturally implies its annealed version.
Following the above theorem, our next point of inquiry is the limiting
behavior of the quenched distribution of the endpoints around the diagonal. We
introduce and clarify a few more notations below before stating our results in
this direction. Let $\Pi_{m,n}^{\operatorname{half}}$ is the set of all
upright lattice paths starting from $(1,1)$ and ending at $(m,n)$ that reside
solely in the half-space $\mathcal{I}^{-}$. We define the point-to-point
partition function as

$\displaystyle
Z(m,n):=\sum_{\pi\in\Pi_{m,n}^{\operatorname{half}}}\prod_{(i,j)\in\pi}W_{i,j}.$
(1.4)

Under the above definition, the normalizing constant $Z(N)$ in (1.2), can also
be viewed as the point-to-line partition function, i.e.

$\displaystyle Z(N)=\sum_{p=0}^{N-1}Z(N+p,N-p).$

The natural logarithm of the partition function is termed as the free energy
of the polymer. The aforementioned depinning phase transition can be observed
by studying the fluctuations of the free energy of the polymer. In this
context, [BW22] obtained precise one-point fluctuations for the point-to-line
free energy $\log Z(N)$ in both the bound and unbound phases and observed the
BBP phase transition. A similar fluctuation result and Baik-Rains phase
transition were later shown in [IMS22] for the point-to-point free energy
$\log Z(N,N)$ on the diagonal. For $\alpha\geq 0$, it was recently proven in
[BCD23a] that the point-to-point free energy process

$\big{(}\log Z(N+pN^{2/3},N-pN^{2/3})\big{)}_{p\in[0,r]}$

after appropriate centering and scaling by $N^{1/3}$ is functionally tight.
This result captures the characteristic KPZ $1/3$ fluctuation and $2/3$
transversal scaling exponents. In our present work, we study the point-to-
point free energy process under $\alpha<0$ case. Our second main result below
obtains precise fluctuations for the increments of the point-to-point free
energy process when $\alpha<0$. To state the result, we introduce the
definition of the log-gamma random walk.

######  Definition 1.2.

Fix $\theta>0$ and $\alpha\in(-\theta,0]$. Let
$Y_{1}\sim\operatorname{Gamma}(\theta+\alpha)$ and
$Y_{2}\sim\operatorname{Gamma}(\theta-\alpha)$ be independent random
variables. We refer to $X:=\log Y_{2}-\log Y_{1}$ as a log-gamma random
variable. It has a density given by

$\displaystyle
p(x):=\frac{1}{\Gamma(\theta+\alpha)\Gamma(\theta-\alpha)}\int_{\mathbb{R}}\exp\left((\theta-\alpha)y-e^{y}+(\theta+\alpha)(y-x)-e^{y-x}\right)dy.$
(1.5)

Let $(X_{i})_{i\geq 0}$ be a sequence of such iid log-gamma random variables.
Set $S_{0}=0$ and $S_{k}=\sum_{i=1}^{k}X_{i}$. We refer to $(S_{k})_{k\geq 0}$
as a log-gamma random walk.

Our next result states that in the bound phase, the above random walk is an
attractor for the increments of the half-space log-partition function.

######  Theorem 1.3.

Fix $\theta>0$ and $\alpha\in(-\theta,0)$. For each $k\geq 1$, as
$N\to\infty$, we have the following multi-point convergence in distribution

$\displaystyle\left(\frac{Z(N+r,N-r)}{Z(N,N)}\right)_{r\in\llbracket
0,k\rrbracket}\stackrel{{\scriptstyle
d}}{{\to}}\left(e^{-S_{r}}\right)_{r\in\llbracket 0,k\rrbracket},$ (1.6)

where $(S_{r})_{r\geq 0}$ is a log-gamma random walk from Definition 1.2.

From the above result, we deduce the following limiting quenched distribution
of the endpoint when viewed around the diagonal.

######  Theorem 1.4.

Fix $\theta>0$ and $\alpha\in(-\theta,0)$ and consider the random measure
$\mathbb{P}^{W}$ from (1.2). Let $(S_{k})_{k\geq 0}$ be a log-gamma random
walk from Definition 1.2. Set $Q:=\sum_{p\geq 0}e^{-S_{p}}$. For a path
$\pi\in\Pi_{N}^{\operatorname{half}}$, we denote $\pi(2N-2)$ as the height
(i.e., $y$-coordinate) of the endpoint of the polymer. Then for each $k\geq
1$, as $N\to\infty$, we have the following multi-point convergence in
distribution

$\displaystyle\left(\mathbb{P}^{W}(\pi(2N-2)=N-r)\right)_{r\in\llbracket
0,k\rrbracket}\stackrel{{\scriptstyle d}}{{\to}}\left(Q^{-1}\cdot
e^{-S_{r}}\right)_{r\in\llbracket 0,k\rrbracket}.$ (1.7)

######  Remark 1.5.

Lemma A.2 ensures that $\mathbb{P}(Q\in[1,\infty))=1$ and makes the right-hand
side of (1.7) a valid (random) probability distribution. In fact, by Theorem
1.2 in [ABO21], once we multiply both the denominator and numerator of the
r.h.s. of (1.7) by $R_{0}\sim\operatorname{Gamma}^{-1}(\theta-\alpha)$, then
the new denominator $QR_{0}\sim\operatorname{Gamma}^{-1}(-2\alpha).$

Beyond proving the $O(1)$ transversal fluctuation around the point $(N,N)$ and
pinning down the exact density within this region, our main theorems above
also shed light on the attractive properties of half-space log-gamma
stationary measures. In [BKLD20] a stationary version of the half-space log-
gamma polymer was considered for $\alpha\in(-\theta,\theta)$, where the
horizontal weights along the first row are assumed to be distributed as
$\operatorname{Gamma}^{-1}(\theta-\alpha)$ (i.e.,
$W_{i,1}\sim\operatorname{Gamma}^{-1}(\theta-\alpha)$). Let us denote
$Z^{\operatorname{stat}}(n,m)$ to be the point-to-point $\mathpzc{HSLG}$
partition function computed using these weights. It was shown in [BKLD20,
Proposition 4.5], that this model is stationary in the sense that for all
$k\geq 1$, and $N\geq k+1$

$(\log Z^{\operatorname{stat}}(N,N)-\log
Z^{\operatorname{stat}}(N+r,N-r))_{r\in\llbracket
0,k\rrbracket}\stackrel{{\scriptstyle d}}{{=}}(S_{r})_{r\in\llbracket
0,k\rrbracket}.$

where $(S_{r})_{r\geq 0}$ is a log-gamma random walk defined in Definition
1.2.

######  Remark 1.6.

Using the above stationary weights, one can define an associated polymer
measure $\mathbb{P}_{\operatorname{stat}}^{W}$ in the spirit of (1.2). We
remark that both Theorem 1.1 and Theorem 1.4 continue to hold under
$\mathbb{P}_{\operatorname{stat}}^{W}$. This is not hard to check from our
log-gamma random walk results presented in Appendix A.

Theorem 1.3 shows that for $\alpha<0$ the above log-gamma random walk measure
is an attractor for the original polymer model in the sense that the increment
of the log-partition function of the original model converges to the same log-
gamma random walk measure. We believe that our broad techniques should also
lead to a similar convergence result for $\alpha\geq 0$. We leave this for
future consideration.

We end this section by mentioning a recent work [BC23] on the stationary
measures for the $\mathpzc{HSLG}$ polymer. The point-to-point log-gamma
polymer partition function $Z(n,m)$ satisfies a recurrence relation

$\displaystyle Z(n,m)$ $\displaystyle=W_{n,m}\cdot(Z(n-1,m)+Z(n,m-1))\mbox{
for }n>m\geq 1,$ $\displaystyle Z(n,n)$ $\displaystyle=W_{n,n}\cdot
Z(n,n-1)\mbox{ for }n\geq 1,$

We refer to a process $(h(k))_{k\geq 0}$ as horizontal-stationary for the
$\mathpzc{HSLG}$ polymer if the solution to the above recurrence relation with
initial data $z(\cdot,0)=e^{h(\cdot)}$ has stationary horizontal increments.
For instance, the distribution of horizontal increments $(\log Z(N+k,N)-\log
Z(N,N))_{k\geq 0}$ is same for all $N\geq 0$ (and equal to that of the initial
data). Recently, [BC23] posited a one-parameter family of horizontal-
stationary measures for the $\mathpzc{HSLG}$ polymer model and conjectured
that these stationary measures are attractors for a large class of initial
data $(Z(n,0))_{n\geq 0}$ subject to the condition $\lim_{k\to\infty}\log
Z(k,0)/k=d\in\mathbb{R}$. However, the initial data relevant to our polymer
model corresponds to $Z(k,0)=\mathbf{1}_{k=1}$ and is not covered in [BC23].
Nevertheless, our result is consistent with [BC23], in the sense that it
corresponds to the $v=\infty$ limit of their Conjecture 1.9. Our limiting
distributon also matches with stationary measures for log-gamma polymer models
on the strip obtained in [BCY23] after taking the right side of the strip to
infinity.

####  1.1.1. Implications of Gaussian fluctuations on the diagonal

In [IMS22], the authors studied one point fluctuations of the $\mathpzc{HSLG}$
log-partition function on the diagonal, $\log Z(N,N)$, in both phases. In
bound phase, they showed that

$\displaystyle\frac{\log Z(N,N)-RN}{\sigma\sqrt{N}}\to G,$ (1.8)

where $G\sim\mathcal{N}(0,1)$ and

$\displaystyle R(\theta,\alpha)$
$\displaystyle:=-\Psi(\theta+\alpha)-\Psi(\theta-\alpha),\quad\sigma^{2}(\theta,\alpha)$
$\displaystyle:=\Psi^{\prime}(\theta+\alpha)-\Psi^{\prime}(\theta-\alpha).$

Here $\Psi(\cdot)$ denote the digamma function defined on $\mathbb{R}_{>0}$ by

$\displaystyle\Psi(z)=\partial_{z}\log\Gamma(z)=-\gamma+\sum_{n=0}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+z}\right),$
(1.9)

where $\gamma$ is the Euler-Mascheroni constant. Combining the above result
from [IMS22] with our results, we prove gaussianity away from the diagonal.

######  Theorem 1.7.

Fix any $k\in\mathbb{Z}_{\geq 1}$. For each $N>0$, fix
$(a_{N,1},\ldots,a_{N,k})\in\mathbb{Z}_{\geq 0}^{k}$. Suppose that as
$N\to\infty$, $a_{N,i}/\sqrt{N}\to 0$ for each $i\in\\{1,\ldots,k\\}$. We have

$\displaystyle\bigg{(}\frac{\log
Z(N+a_{N,i},N-a_{N,i})-RN}{\sigma\sqrt{N}}\bigg{)}_{i=1}^{k}\to(G,G,\ldots,G).$

where $G\sim\mathcal{N}(0,1)$.

The above theorem establishes gaussianity in the $o(\sqrt{N})$ window around
the diagonal with trivial correlations. In fact, we expect the above theorem
to hold even if $a_{N,i}/N\to 0$. When $a_{N,i}$ are precisely of the order
$N$, we still expect to see gaussianity but with nontrivial correlations. The
above result is proved using a strong coupling result (Proposition 5.3) that
we prove in Section 5. The gaussianity in the above theorem essentially comes
from the [IMS22] input. However, we believe that it is possible to establish
(1.8) using the machinery developed in this paper. We leave this for future
work.

###  1.2. Proof Ideas

In this section we sketch the key ideas behind the proofs of our main results.
Our proof relies on inputs from the recently developed $\mathpzc{HSLG}$
Gibbsian line ensemble in [BCD23a], one-point fluctuation results for point-
to-(partial)line half-space log-partition functions from [BW22] and the
localization techniques from [DZ22a]. At the heart of our argument lies an
innovative combinatorial argument that bridges the aforementioned inputs and
enables our proof.

$H_{N}^{(1)}(\cdot)$$H_{N}^{(2)}(\cdot)$$H_{N}^{(3)}(\cdot)$$M_{1}\sqrt{N}$$\sqrt{N}$
Figure 3. First three curves of the $\mathpzc{HSLG}$ line ensemble. There is a
high probability uniform separation of length $\sqrt{N}$ between the first two
curves in the above $M_{1}\sqrt{N}$ window.

The starting point of our analysis is the $\mathpzc{HSLG}$ Gibbsian line
ensemble in [BCD23a], which allows us to embed the free energy $\log
Z(N+r,N-r)$ of the $\mathpzc{HSLG}$ polymer as the top curve of a Gibbsian
line ensemble $(H_{N}^{(k)}(\cdot))_{k\in\llbracket 1,N\rrbracket}$ of log-
gamma increment random walks interacting through a soft version of non-
intersection (Theorem 2.4) conditioning and subject to an energetic
interaction at the left boundary (where $r=0$) depending on the value of
$\alpha$. This fact is due to the geometric RSK correspondence ([COSZ14,
OSZ14, NZ17, BZ19]) and the half-space Whittaker process ([BBC20]). The key
idea of our proof is to show that with high probability, the first and the
second curves in our line ensemble ensemble (see Figure 3) are sufficiently
uniformly separated. Then the separation allows us to conclude that the first
curve indeed behaves similarly to a log-gamma random walk by a localization
analysis.

The existing literature contains some information about the locations of the
top two curves. When $\alpha<0$, one can deduce from the line ensemble
description in [BCD23a] that the first and the second curves are repulsed from
each other at the left boundary. Results in [BW22] also supply information
about the location for the first curve. However, one cannot deduce that the
entire second curve lies uniformly much lower than the first curve from the
above two inputs and line ensemble techniques alone.

####  1.2.1. Intuition behind the separation

Before we proceed to further break down our argument about the separation, it
is worth dwelling on the mathematical intuition behind the separation between
the first and second curves, which originates from the definition of the line
ensemble defined in Section 2.1. For simplicity, let us focus only on the left
boundary. By Definition 2.1, we have $H_{N}^{(1)}(1)=\log Z(N,N)$, and

$\displaystyle
H_{N}^{(1)}(1)+H_{N}^{(2)}(1):=\log\left[2\sum_{\pi_{1},\pi_{2}}\prod_{(i,j)\in\pi_{1}\cup\pi_{2}}\widetilde{W}_{i,j}\right],$
(1.10)

where the above sum is over all pair of non-intersecting upright paths
$\pi_{1},\pi_{2}$ from $(1,1)$ to $(N,N-1)$ and from $(1,2)$ to $(N,N)$
confined in the entire quadrant $\mathbb{Z}_{\geq 1}^{2}$ (instead of octant).
Here $\widetilde{W}_{i,j}$ is the symmetrized version of the weights defined
in (1.1) on the entire quadrant as:

$\displaystyle\widetilde{W}_{i,i}=W_{i,i}/2\ \mbox{ for }\ i\geq
1,\qquad\widetilde{W}_{i,j}=\widetilde{W}_{j,i}=W_{i,j}\ \mbox{ for }\ i>j.$
(1.11)

Using point-to-(partial)line log-partition function fluctuation results from
[BW22] and line ensemble techniques, it is not hard to deduce that
$\tfrac{1}{N}H_{N}^{(1)}(1)\to R:=-\Psi(\theta+\alpha)-\Psi(\theta-\alpha),$
where $\Psi$ is the digamma function defined in (1.9). However,
$H_{N}^{(2)}(1)$ should follow a different law of large numbers. This can be
understood intuitively from (1.10) as follows. For $\alpha$ close to
$-\theta$, the weights on the diagonal are huge and stochastically dominate
all the other weights. The sum in (1.10) then concentrates on the pair of
paths $\pi_{1}^{*},\pi_{2}^{*}$ which jointly have the maximal numbers of
diagonal points. This occurs when one of the paths carries all the diagonal
weights and the other path has no diagonal weights. Thus we expect,

$\displaystyle\sum_{\pi_{1},\pi_{2}}\prod_{(i,j)\in\pi_{1}\cup\pi_{2}}\widetilde{W}_{i,j}\asymp\left[\sum_{\pi_{1}}\prod_{(i,j)\in\pi_{1}}\widetilde{W}_{i,j}\right]\cdot\left[\sum_{\pi_{2}\mid\operatorname{diag}(\pi_{2})=\varnothing}\prod_{(i,j)\in\pi_{2}}\widetilde{W}_{i,j}\right]$
(1.12)

Upon taking logarithms and dividing by $N$, the first term goes to $R$.
However, the second term does not feel the effect of the diagonal and hence
should follow the law of large numbers corresponding to the unbound phase,
i.e., $\alpha>0$. The unbound phase law of large numbers is given by
$-\Psi(\theta)$ noted in [BW22, BCD23a]. Thus overall, we expect
$\frac{1}{N}(H_{N}^{(1)}(1)+H_{N}^{(2)}(1))\to R-\Psi(\theta)$. As $\Psi$ is
concave, the above heuristics suggests $H_{N}^{(2)}(1)$ follow a lower law of
large numbers. While our technical arguments to be presented later do not
yield exactly (1.12), we utilize the above idea to obtain a large enough
separation between the two curves, which turns out to be sufficient for
proving our main theorems.

####  1.2.2. The $U$ map and its consequences

We now describe the key idea that makes the above intuition work. All the
statements mentioned in this subsection should be understood as high
probability statements. The above idea of having one path having all diagonal
weights is made precise in Section 3, where we develop a combinatorial map in
Lemma 3.1, referred to as the $U$ map.

$\pi_{1}$$\pi_{2}$ (a)

$\pi_{1}^{\prime}$$\pi_{2}^{\prime}$ (b)

Figure 4. The $U$ map takes $\pi_{1},\pi_{2}$ from (A) and returns
$\pi_{1}^{\prime},\pi_{2}^{\prime}$ in (B). The precise description of the map
is given in the proof of Lemma 3.1

The $U$ map takes every pair of paths $\pi_{1},\pi_{2}$ in the sum in (1.10)
and returns a pair of non-intersecting paths
$\pi_{1}^{\prime},\pi_{2}^{\prime}$ while preserving their shared weights up
to reflections (see Figure 4). Moreover, the diagonal weights collectively
carried by the pair will only rest on one of the paths among
$\pi_{1}^{\prime},\pi_{2}^{\prime}$. The $U$ map is not injective but has at
most $2^{N}$ many inverses for each pair in its image. When we apply the $U$
map to a single pair of adjacent paths, we get that

$\tfrac{1}{N}(H_{N}^{(1)}(1)+H_{N}^{(2)}(1))\leq\log 2+R-\Psi(\theta).$

The $\log 2$ is an entropy factor that comes from overcounting the number of
inverses of our $U$ map. To remove its influence, we rely on the definition of
the lower curves of the line ensemble. Indeed, similar to (1.10),
$\sum_{i=1}^{2k}H_{N}^{(i)}(1)$ admits a representation in terms of $2k$-many
non-intersecting paths. When we apply the $U$ map to $k$ pairs of adjacent
paths simultaneously, it leads to the following average law of large numbers
of the top $2k$ curves:

$\frac{1}{2kN}\sum_{i=1}^{2k}H_{N}^{(i)}(1)\leq\tfrac{1}{2k}\log
2-\tfrac{1}{2}\Psi(\theta)-\tfrac{1}{2}\Psi(\theta+\alpha)-\tfrac{1}{2}\Psi(\theta-\alpha).$

Taking $k$ large enough, one can ensure the right-hand side constant is
strictly less than $R$. In fact, the above argument can be strengthened to
conclude that for large enough $k$

$\displaystyle\sup_{p\in\llbracket
1,2N-4k+2\rrbracket}\frac{1}{2kN}\sum_{i=1}^{2k}H_{N}^{(i)}(p)\leq R-\delta,$

for some $\delta>0$. This is obtained in Proposition 3.4.

As a consequence of this result, using soft non-intersection property of the
line ensemble (Theorem 2.4), we derive that with high probability, the
$(2k+2)$-th curve $H_{N}^{2k+2}(\cdot)$ is uniformly $\mbox{Const}\cdot N$
below $RN$ over $\llbracket 1,N\rrbracket$ in Section 4. Employing one-point
results from [BW22], one can ensures the point $H_{N}^{(1)}(M_{1}\sqrt{N})$ on
the top curve is $(M_{2}+1)\sqrt{N}$ below $RN.$ Combining the last two
results and line ensemble techniques we are able to benchmark the second curve
from above:

$\displaystyle\sup_{p\in\llbracket
1,M_{1}\sqrt{N}\rrbracket}H_{N}^{(2)}(p)\leq RN-M_{2}\sqrt{N}$ (1.13)

in Proposition 4.2. The details of the argument are presented in Section 4.
While we are unable to obtain a mismatch in the laws of large numbers for the
first two curves following the above procedures, the fact that the second
curve is below the diffusive regime of the first curve (since $M_{2}$ can be
chosen as large as possible) over an interval of length $M_{1}\sqrt{N}$ is
sufficient for our next step of the analysis.

####  1.2.3. Localization analysis

The remaining piece of our proof of main theorems boils down to a localization
analysis of the first curve in Section 5. Our proof roughly follows the
techniques developed in our paper [DZ22a]. First, to prove Theorem 1.1 we
divide the tail into a deep and a shallow tail depending on the distance away
from $(N,N)$, see Figure 5.

$y=N-M\sqrt{N}$$y=N-k$$(N,N)$$(1,1)$$(2N-1,1)$deep tail starting pointdeep
tailshallow tail Figure 5. If the height of the endpoint of the polymer is
less than $N-k$, it either lies in the shallow tail or in the deep tail
(illustrated above). Lemma 5.1 shows it is exponentially unlikely to lie in
the deep tail.

Our argument in Lemma 5.1 uses one-point fluctuations results of point-
to-(partial)line log-partition function from [BW22] as input and shows that
the probability of the endpoint living in the deep tail region is
exponentially small. To show that the shallow tail contribution is also small
and to prove our remaining theorems, we establish the following strong
convergence result in Proposition 5.3:

  1. (a)

the law of the top curve within the $[1,M\sqrt{N}]$ window is arbitrarily
close to that of a log-gamma random walk for large enough $N$ (Proposition
5.3).

In light of (a), the conclusion that the shallow tail contribution is small
follows from estimating the probability of the same event under the log-gamma
random walk law. Theorem 1.3 is immediate from (a) and Theorem 1.4 also
follows from (a) after some calculations. The details are presented in Section
5.2.

Finally, we briefly explain how we establish (a). A detailed discussion
appears in the Step 1 of the proof of Proposition 5.3. As $H_{N}^{(1)}(\cdot)$
is a log-gamma random walk subject to soft non-intersecting condition with
$H_{N}^{(2)}(\cdot)$, it suffices to show that there’s sufficient distance
between the first and the second curves. Indeed, this will imply $H_{N}^{(1)}$
behaves like a true log-gamma random walk. As we have already benchmarked the
second curve in (1.13), it remains to determine a suitable lower bound for the
first curve. The key idea here is to find a point $p=O(\sqrt{N})$ on the first
curve in the deep tail region such that with high probability

$H_{N}^{(1)}(p)\geq RN-M^{\prime}\sqrt{N}$

for some $M^{\prime}$. This is achieved in Lemma 5.2 using fluctuation results
from [BW22]. Then using standard random walk tools such as Kolmogorov’s
maximal inequality, we derive that with high probability $H_{N}^{(1)}(q)\geq
RN-(M^{\prime}+1)\sqrt{N}$ for all $q\in\llbracket 1,p\rrbracket$. Choosing
$M_{2}=M^{\prime}+2$ in (1.13) implies that with high probability the first
curve is at least $\sqrt{N}$ above the second curve, This completes our
deduction and consequently establishes (a).

###  1.3. Related works and future directions

Our study of half-space polymers succeeds an extensive history of endeavors
that attempt to unravel their full-space variant. These full-space polymer
models have rich connections with symmetric functions, random matrices,
stochastic PDEs and integrable systems and are believed to belong to the KPZ
universality class (see [Com17, Gia07, BC20]). Yet in spite of intense efforts
in the past decade, rigorous results proving either the $1/3$ fluctuation
exponent or the $2/3$ transversal exponent for general polymers have been
scarce outside a few integrable cases (see [Com17, Sep12, BC20, BCD23b, DZ22a,
DZ22b] and the references therein).

In the half-space geometry, a wealth of literature has focused on the phase
diagram for limiting distributions based on the diagonal strength. One of the
first mathematical works goes to the series of joint works [BR01a, BR01c,
BR01b] on the geometric last passage percolation (LPP), i.e. polymers with
zero temperature. Their multi-point fluctuations were studied in [SI04] and
similar results were later proved for exponential LPP in [BBCS18a, BBCS18b]
using Pfaffian Schur processes. For further recent works on half-space LPP, we
refer to [BBNV18, BFO20, BFO22, FO22] and the references therein.

For positive temperature models, i.e., polymers, as they are no longer
directly related to the Pfaffian point processes, the first rigorous proof of
the depinning transition appeared much later in [BW22]. Here the authors also
included precise fluctuation results such as the BBP phase transition [BBAP05]
for the point-to-line log-gamma free energy. For the point-to-point log-gamma
free energy, the limit theorem as well as the Baik-Rains phase transition were
conjectured in [BBC20] based on steepest descent analysis of half-space
Macdonald processes. This result has been recently proved in [IMS22] by
relating the half- space model to a free boundary version of the Schur
process.

Similar to their full-space counterparts, in addition to fluctuations, another
dimension of interest to half-space polymers is their localization behaviors,
which refer to the concentration of polymers in a very small region given the
environment. Figure 1 is a simulation of $30$ samples of $\mathpzc{HSLG}$
polymers of length $120$ sampled from the same environment with $\theta=1$,
$\alpha=-0.2$ and $\alpha=+0.2$. The simulation suggests that even in the
unbound phase, we expect a localization phenomenon around a favorite site
given by the environment. Localization is a unique behavior of the polymer
path in the strong disorder regime. In the full space, various levels of
localization results have been established for discrete and continuous
polymers. The mathematical work began with the strong localization result of
[CH02] that confirmed the existence of the favorite sites for the endpoint
distributions of point-to-line polymers and has been upgraded to the notion of
atomic and geometric localization for general reference random walks in a
series of joint works [Bat18, BC20, BS20, BD22]. An even stronger notion, the
“favorite region conjecture”, which conjectures the favorite corridor of a
polymer to be stochastically bounded, has been proved for two integrable
models: the stationary log-gamma polymer in the discrete case ([CN16]) and the
continuous directed random polymer (CDRP) in the continuous case ([DZ22a]). In
this direction, building up on [DZ22a] work, recently [DGL23] have studied the
localization distance of the CDRP.

Investigating the geometry of the half-space CDRP is an interesting question
to consider next. Recently, a number of new results have appeared on the half-
space KPZ equation, which arises as the free energy of the half-space CDRP
[Wu20, BC23], in both the mathematics [CS18, BBCW18, BBC20, Par19, Par22,
BC23, IMS22] and the physics literature [GLD12, BBC16, IT18, DNKLDT20, KLD18,
BKLD20, BLD21, BKLD22]. These results on the free energy render the half-space
continuous polymers amenable to analysis. However, the challenge with further
studying the geometry of the half-space CDRP remains, due to the lack of an
analogous half-space KPZ line ensemble.

### Outline

The rest of the paper is organized as follows. Section 2 reviews some of the
existing results related to $\mathpzc{HSLG}$ line ensemble and one-point
fluctuations of point-to-(partial)line free energy of $\mathpzc{HSLG}$
polymer. In Section 3 we prove our key combinatorial lemma and use it to
control the average law of large numbers for the top curves of the line
ensemble. In Section 4, we establish control over the second curve of the line
ensemble. Finally, in Section 5, we complete the proofs of our main theorems.
Appendix A contains basic properties of log-gamma random walks.

### Notation

Throughout this paper, we will assume $\theta>0$ and $\alpha\in(-\theta,0)$
are fixed parameters. We write $\llbracket a,b\rrbracket:=[a,b]\cap\mathbb{Z}$
to denote the set of integers between $a$ and $b$. We will use serif fonts
such as $\mathsf{A},\mathsf{B},\ldots$ to denote events. The complement of an
event $\mathsf{A}$ will be denoted as $\neg\mathsf{A}$.

### Acknowledgements

We thank Guillaume Barraquand and Ivan Corwin for useful discussions,
comments, and for their encouragement during the completion of this
manuscript. SD’s research was partially supported by Ivan Corwin’s W.M. Keck
Foundation Science and Engineering Grant.

##  2\. Basic framework and tools

In this section, we present the necessary background on the half-space log-
gamma ($\mathpzc{HSLG}$) line ensemble and point-to-(partial) line partition
function. From [BCD23a] and [BW22] we gather a few of the known results on
these objects that are crucial in our proofs.

###  2.1. The $\mathpzc{HSLG}$ line ensemble and its Gibbs property

We begin with the description of the $\mathpzc{HSLG}$ line ensemble and its
Gibbs property. The definition of the $\mathpzc{HSLG}$ line ensemble is based
on the point-to-point symmetrized partition function for multiple paths
defined in (2.1). These are sum over multiple non-intersecting upright paths
on the entire quadrant $\mathbb{Z}_{>0}^{2}$ of products of the symmetrized
version defined in (1.11) of the weights defined in (1.1). Fix
$m,n,r\in\mathbb{Z}_{>0}$ with $n\geq r$, let $\Pi_{m,n}^{(r)}$ be the set of
all $r$-tuples of non-intersecting upright paths in $\mathbb{Z}_{>0}^{2}$
starting from $(1,r),(1,r-1),\ldots,(1,1)$ and going to
$(m,n),(m,n-1),\ldots,(m,n-r+1)$ respectively. We define the point-to-point
symmetrized partition function for $r$ paths as

$\displaystyle
Z_{\operatorname{sym}}^{(r)}(m,n):=\sum_{(\pi_{1},\ldots,\pi_{r})\in\Pi_{m,n}^{(r)}}\prod_{(i,j)\in\pi_{1}\cup\cdots\cup\pi_{r}}\widetilde{W}_{i,j}.$
(2.1)

where $\widetilde{W}_{i,j}$ are defined in (1.11). We write
$Z_{\operatorname{sym}}(m,n):=Z_{\operatorname{sym}}^{(1)}(m,n)$ and use the
convention that $Z_{\operatorname{sym}}^{(0)}(m,n)\equiv 1$. One can recover
$\mathpzc{HSLG}$ partition function from symmetrized partition function via
the following identity. For each $(m,n)\in\mathcal{I}^{-}$ we have

$\displaystyle 2Z_{\operatorname{sym}}(m,n)=Z(m,n).$ (2.2)

The above identity appears in Section 2.1 of [BW22] and follows easily due to
the symmetry of the weights. We stress that the above relation is an exact
equality not just in distribution.

######  Definition 2.1 ($\mathpzc{HSLG}$ line ensemble).

Fix $N>1$. For each $k\in\llbracket 1,N-1\rrbracket$ and $p\in\llbracket
1,2N-2k+2\rrbracket$ set

$\displaystyle
H_{N}^{(k)}(p):=\log\left(\frac{2Z_{\operatorname{sym}}^{(k)}(N+\lfloor
p/2\rfloor,N-\lceil p/2\rceil+1)}{Z_{\operatorname{sym}}^{(k-1)}(N+\lfloor
p/2\rfloor,N-\lceil p/2\rceil+1)}\right)$ (2.3)

We view the $k$-th curve $H_{N}^{(k)}$ as a random continuous function
$H_{N}^{(k)}:[1,2(N-k+1)]\to\mathbb{R}$ by linearly interpolating its values
on integer points. We call the collection of curves
$H_{N}:=(H_{N}^{(1)},H_{N}^{(2)},\ldots,H_{N}^{(N)})$ the $\mathpzc{HSLG}$
line ensemble.

We remark that in Definition 2.7 in [BCD23a], the authors defined the
$\mathpzc{HSLG}$ line ensemble by defining
$\mathcal{L}_{i}^{N}(j)=H_{N}^{(i)}(j)+\mbox{Const}\cdot N$ where the ‘Const’
is explicit and encodes the law of large numbers for the $\mathpzc{HSLG}$ free
energy process (as well as the entire line ensemble) in the unbound phase.
Since the laws of large numbers for the first curve and the second curve in
the bound phase are possibly different (recall our discussion of the proof
idea in the introduction), we choose to not add this constant in our
definition of line ensemble. All the results from [BCD23a] can be easily
translated to results in our setting by adding this appropriate constant.

In view of (2.2), for all $p\leq 2N$ we have

$\displaystyle H_{N}^{(1)}(p)=\log Z(N+\lfloor p/2\rfloor,N-\lceil
p/2\rceil+1).$ (2.4)

The $\mathpzc{HSLG}$ line ensemble enjoys a property known as the
$\mathpzc{HSLG}$ Gibbs property. To state the $\mathpzc{HSLG}$ Gibbs property,
we introduce the $\mathpzc{HSLG}$ Gibbs measures via graphical representation.

Consider a diamond lattice on the lower-right quadrant with vertices
$\\{(m,-n),(m+\frac{1}{2},-n+\frac{1}{2})\mid m,n\in\mathbb{Z}_{>0}^{2}\\}$
and nearest neighbor edges as shown in Figure 6. We label the vertices by
setting $\phi((m,n))=(-\lfloor n\rfloor,2m-1)$. In the rest of the paper, we
identify a vertex in this lattice by this labeling and will not mention its
actual coordinates.

On the diamond lattice domain, we add potential directed colored edges. A
directed colored edge $\vec{e}=\\{v_{1}\to v_{2}\\}$ on this lattice is a
directed edge from $v_{1}$ to $v_{2}$ that has three choices of colors:
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\textbf{blue}},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\textbf{red}},\mbox{
and
}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\textbf{black}}$.
Given a directed colored edge, we associate a weight function based on the
color of the edge defined as follows:

$\displaystyle W_{\vec{e}}(x)=\begin{cases}\exp((\theta-\alpha)x-e^{x})&\mbox{
if $\vec{e}$ is
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{blue}}}\\\
\exp((\theta+\alpha)x-e^{x})&\mbox{ if $\vec{e}$ is
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{red}}}\\\
\exp(-e^{x})&\mbox{ if $\vec{e}$ is
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(2.5)

We consider a graph $G_{N}$ on the diamond lattice with vertex set

$\displaystyle K_{N}$ $\displaystyle:=\\{(i,j)\mid i\in\llbracket
1,N\rrbracket,j\in\llbracket 1,2N-2i+2\rrbracket\\}.$

with directed colored edges described below. For each $(p,q)\in K_{N}$,

  * •

If $q$ is odd and $p$ is odd (even resp.), we put a blue (red resp.) edge:
$(p,q)\to(p,q+1)$.

  * •

If $q\geq 3$ is odd and $p$ is odd (even resp.), we put a red (blue resp.)
edge: $(p,q)\to(p,q-1)$.

  * •

If $q$ is even, we put two black edges: $(p,q)\to(p-1,q)\mbox{ and
}(p,q)\to(p+1,q).$

The corresponding graph is shown in Figure 6. We write $E(F)$ for the set of
edges of any graph $F\subset G_{N}$.

$\vdots$$\cdots$$\vdots$$\cdots$$\vdots$$\cdots$$\vdots$$\cdots$$\vdots$$\cdots$${(1,1)}$${(2,1)}$${(3,1)}$${(1,2)}$${(1,4)}$${(3,4)}$${(3,5)}$${(3,6)}$${(3,7)}$
(a)

(b)

Figure 6. (A) Diamond lattice with a few of the labeling of the vertices shown
in the figure. The $m$-th gray-shaded region have vertices with labels of the
form $\\{(m,n)\mid n\in\mathbb{Z}_{>0}^{2}\\}$. Thus each such region consists
of vertices with the same first coordinate labeling. Potential directed
colored edges on the lattice are also drawn above. (B) $K_{N}$ with $N=4$.
$\Lambda_{N}^{*}$ consists of all vertices in the shaded region.

The following result from [BCD23a] shows how the conditional distribution of
the $\mathpzc{HSLG}$ line ensemble is given by certain measures called
$\mathpzc{HSLG}$ Gibbs measures.

######  Theorem 2.2 (Gibbs property).

Consider the directed colored graph $G_{N}$ described above. Set $\Lambda_{N}$
to be a connected subset of the graph $G_{N}$ on the diamond lattice $K_{N}$

$\Lambda_{N}^{*}=\\{(i,j)\mid i\in\llbracket 1,N-1\rrbracket,j\in\llbracket
1,2N-2i+1\rrbracket\\}.$

Let $\Lambda$ be a connected subset of $\Lambda_{N}$. Recall the
$\mathpzc{HSLG}$ line ensemble $H^{N}$ from Theorem 2.1. The law of
$\\{H_{i}^{N}(j)\mid(i,j)\in\Lambda\\}$ conditioned on
$\\{H_{i}^{N}(j)\mid(i,j)\in\Lambda^{c}\\}$ is a measure on
$\mathbb{R}^{|\Lambda|}$ with density at $(u_{i,j})_{(i,j)\in\Lambda}$
proportional to

$\displaystyle\prod_{\vec{e}=\\{v_{1}\to v_{2}\\}\in
E(\Lambda\cup\partial\Lambda)}W_{\vec{e}}(u_{v_{1}}-u_{v_{2}}),$ (2.6)

where $u_{i,j}=H_{i}^{N}(j)$ for $(i,j)\in\partial\Lambda$.

We refer to the above conditional law as the $\mathpzc{HSLG}$ Gibbs measure
with boundary condition $\vec{u}=(u_{i,j})_{(i,j)\in\partial\Lambda}$ and
denote this measure as $\mathbb{P}_{\operatorname{gibbs}}^{\vec{u}}(\cdot)$.
The above theorem follows directly from the results in [BCD23a]). Theorem 1.3
in [BCD23a] specifies the Gibbs property for the centered line ensemble
$L_{i}^{N}(j)$. The same Gibbs property holds for $H_{N}^{(i)}(j)$ as
$\mathpzc{HSLG}$ Gibbs measures are translation-invariant (Observation 2.1 (b)
in [BCD23a]. The Gibbs property stated in Theorem 1.3 is different and valid
for all $\alpha>-\theta$. When $\alpha\in(-\theta,\theta)$, one can
redistribute the edge-weights (see Observation 4.2 in [BCD23a]) to obtain the
above stated Gibbs property.

The $\mathpzc{HSLG}$ Gibbs measures satisfy stochastic monotonicity w.r.t. the
boundary data.

######  Proposition 2.3 (Stochastic monotonicity, Proposition 2.6 in
[BCD23a]).

Fix $k_{1}\leq k_{2}$, $a_{i}\leq b_{i}$ for $k_{1}\leq i\leq k_{2}$ and
$\alpha>-\theta$. Let

$\displaystyle\Lambda:=\\{(i,j)\mid k_{1}\leq i\leq k_{2},a_{i}\leq j\leq
b_{i}\\}.$

There exists a probability space consisting of a collection of random
variables

$\displaystyle\\{L(v;(u_{w})_{w\in\partial\Lambda})\mid
v\in\Lambda,(u_{w})_{w\in\partial\Lambda}\in\mathbb{R}^{|\partial\Lambda|}\\}$

such that

  1. (1)

for each $(u_{w})_{w\in\partial\Lambda}\in\mathbb{R}^{|\partial\Lambda|}$, the
law of $\\{L(v;(u_{w})_{w\in\partial\Lambda})\mid v\in\Lambda\\}$ is given by
the $\mathpzc{HSLG}$ Gibbs measure for the domain $\Lambda$ with boundary
condition $(u_{w})_{w\in\partial\Lambda}\in\mathbb{R}^{|\partial\Lambda|}$.

  2. (2)

with probability $1$, for all $v\in\Lambda$ we have

$L(v;(u_{w})_{w\in\partial\Lambda})\leq
L(v;(u_{w}^{\prime})_{w\in\partial\Lambda})\mbox{ whenever }u_{w}\leq
u_{w}^{\prime}\mbox{ for all }w\in\partial\Lambda.$

As mentioned in the introduction, the $\mathpzc{HSLG}$ line ensemble enjoys a
certain soft non-intersection property. This property is captured in our next
theorem.

######  Theorem 2.4 (Ordering of points, Theorem 3.1 in [BCD23a]).

Fix any $k\in\mathbb{Z}_{>0}$ and $\rho\in(0,1)$. There exists
$N_{0}=N_{0}(\rho,k)>0$ such that for all $N\geq N_{0}$, $i\in\llbracket
1,k\rrbracket$ and $p\in\llbracket 1,N-i\rrbracket$ the following inequalities
holds:

$\displaystyle\mathbb{P}(H_{N}^{(i)}(2p+1)\leq H_{N}^{(i)}(2p)+\log^{2}N)$
$\displaystyle\geq 1-\rho^{N},$ $\displaystyle\mathbb{P}(H_{N}^{(i)}(2p-1)\leq
H_{N}^{(i)}(2p)+\log^{2}N)$ $\displaystyle\geq 1-\rho^{N},$
$\displaystyle\mathbb{P}(H_{N}^{(i+1)}(2p)\leq H_{N}^{(i)}(2p+1)+\log^{2}N)$
$\displaystyle\geq 1-\rho^{N},$ $\displaystyle\mathbb{P}(H_{N}^{(i+1)}(2p)\leq
H_{N}^{(i)}(2p-1)+\log^{2}N)$ $\displaystyle\geq 1-\rho^{N}.$

We remark that the above theorem is true in the unbound phase as well (i.e.,
for all $\alpha>-\theta$).

We now introduce the interacting random walks which are a specialized version
of $\mathpzc{HSLG}$ Gibbs measures (see Figure 7).

######  Definition 2.5 (Interacting random walk).

We say $(L_{1},L_{2})=(L_{1}\llbracket 1,2T-2\rrbracket,L_{2}\llbracket
1,2T-1\rrbracket)$ is an interacting random walk ($\mathsf{IRW}$) of length
$T$ with boundary condition $(a,b)$ if its law is a measure on
$\mathbb{R}^{4T-3}$ with density at
$(u_{1,j})_{j=1}^{2T-2},(u_{2,j})_{j=1}^{2T-1}$ proportional to

$\displaystyle\prod_{j=1}^{T-1}\exp(-e^{u_{2,2j}-u_{1,2j-1}}-e^{u_{2,2j}-u_{1,2j+1}})\prod_{i=1}^{2}\prod_{j=1}^{2T-1}G_{\theta+(-1)^{i+j}\alpha}\big{(}(-1)^{j+1}(u_{i,j}-u_{i,j+1})\big{)}$

where $G_{\beta}(x)=[\Gamma(\beta)]^{-1}\exp(\beta x-e^{x})$, $u_{1,2T-1}=a$,
$u_{2,2T}=b$, and $u_{1,2T}=0$ (which forces
$G_{\theta+\alpha}(u_{1,2T-1}-u_{1,2T})$ to be a constant). Figure 7 provides
the graphical representation of $\mathsf{IRW}$.

$b$$a$ Figure 7. $\mathsf{IRW}$ of length $6$ with boundary condition $a$ and
$b$.

Note that the directed colored graph associated with $\mathsf{IRW}$ can be
viewed as a subset of $G_{N}$ (introduced above). Specifically, for each
$i\geq 1$, if we consider the vertex set

$V_{i,T}:=\\{(2i,j),(2i+1,j)\mid j\in\llbracket
1,2T-1\rrbracket\\}\cup\\{(2i+1,2T)\\},$

the subgraph induced by $V_{i,T},E(V_{i,T})$ corresponds to the graph
associated with $\mathsf{IRW}$. Note that the graph associated with
$\mathsf{IRW}$ can also be viewed as the subgraph induced by
$\hat{V}_{T},E(\hat{V}_{T})$ where

$\hat{V}_{T}:=\\{(1,j),(2,j)\mid j\in\llbracket
1,2T-1\rrbracket\\}\cup\\{(2,2T)\\},$

provided we switch $\alpha$ to $-\alpha$ in (2.5) (i.e., switching red and
blue edges). Since we have restricted $\alpha\in(-\theta,0)$ (bound phase),
under this switching $\mathsf{IRW}$ can be viewed as certain $\mathpzc{HSLG}$
Gibbs measures in the unbound phase. Indeed, after switching $\alpha$ to
$-\alpha$, in the language of [BCD23a], $\mathsf{IRW}$ precisely corresponds
to bottom-free measure on the domain $\mathcal{K}_{2,T}$ with boundary
condition $(a,b)$ (see Definition 2.3 in [BCD23a]). This allows us to use the
unbound phase estimates developed in [BCD23a]. We end this section by
recording one such estimate.

######  Proposition 2.6 (Lemma 5.3 in [BCD23a]).

Fix any $T\geq 2$. Let $(L_{1},L_{2})$ be an $\mathsf{IRW}$ of length $T$ with
boundary condition $(0,-\sqrt{T})$. Fix $\varepsilon\in(0,1)$. There exists
$M_{0}=M_{0}(\varepsilon)>0$ such that

$\displaystyle\mathbb{P}\left(\sup_{p\in\llbracket
1,2T-1\rrbracket}|L_{1}(p)|+\sup_{q\in\llbracket 1,2T\rrbracket}|L_{2}(q)|\geq
M_{0}\sqrt{T}\right)\leq\varepsilon.$

###  2.2. One-point fluctuations of point-to-(partial)line free energy

In this section, we gather the point-to-(partial)line free energy fluctuation
results from [BW22]. To state the theorem, we introduce a few necessary
objects first.

Recall the point-to-point half-space partition function $Z(m,n)$ from (1.4).
For $k\in\llbracket 0,N-1\rrbracket$, we define the point-to-(partial)line
half-space partition function as

$\displaystyle\mathpzc{Z}_{N}^{\mathsf{PL}}(m)=\sum_{p=m}^{N-1}Z(N+p,N-p)=\sum_{p=m}^{N-1}e^{H_{N}^{(1)}(2p+1)}.$
(2.7)

For the second equality, note that by (2.4) we have $H_{N}^{(1)}(2p+1)=\log
Z(N+p,N-p)$ and thus we can translate the point-to-(partial)line partition
function in Definition 1.8 (or equivalently in Definition 1.3) of [BW22] into
sums of $e^{H_{N}^{(1)}(2p+1)}$ by way of the full-space point-to-point
partition function $Z(n+p,n-p)$.

Recall the digamma function $\Psi(\cdot)$ from (1.9). For any
$k\in\mathbb{Z}_{>0}$, we set

$\displaystyle R(\theta,\alpha)$
$\displaystyle:=-\Psi(\theta+\alpha)-\Psi(\theta-\alpha),$ (2.8)
$\displaystyle\tau(\theta,\alpha)$
$\displaystyle:=\Psi(\theta-\alpha)-\Psi(\theta+\alpha),$
$\displaystyle\sigma^{2}(\theta,\alpha)$
$\displaystyle:=\Psi^{\prime}(\theta+\alpha)-\Psi^{\prime}(\theta-\alpha),$
$\displaystyle\Delta_{k}(\theta,\alpha)$
$\displaystyle:=\Psi(\theta)-\tfrac{1}{2}[\Psi(\theta+\alpha)+\Psi(\theta-\alpha)]-\tfrac{1}{2k}\log
2.$

For the remainder of the paper, we will make use of the above notation
repeatedly. As $\Psi$ is a strictly concave function, for all large enough $k$
(depending on $\alpha,\theta$) we have $\Delta_{k}>0$. For the results and
proofs in the remainder of the paper, we always choose $k$ large enough such
that $\Delta_{k}>0.$

We now state the necessary results from [BW22] about the point-
to-(partial)line partition function $\mathpzc{Z}_{N}^{\mathsf{PL}}(m)$ that we
need in our subsequent analysis.

######  Theorem 2.7.

Suppose $g:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$. Suppose further that $N$ is an
integer that tends to infinity in such a way that $\frac{g(N)}{N}\to 0$. We
have

$\displaystyle\frac{1}{N^{1/2}\sigma}\left[\log\mathpzc{Z}_{N}^{\mathsf{PL}}(g(N))-RN+g(N)\tau\right]\stackrel{{\scriptstyle
d}}{{\to}}\mathcal{N}(0,1).$ (2.9)

where $R,\tau,\sigma$ are defined in (2.8). We have the following law of large
numbers

$\displaystyle\frac{1}{N}\log\left[\sum_{p=1}^{N-1}Z(N+p,N-p)\right]\stackrel{{\scriptstyle
p}}{{\to}}R\quad\frac{1}{N}\log\left[\sum_{p=1}^{N}Z(N+p,N-p+1)\right]\stackrel{{\scriptstyle
p}}{{\to}}R.$ (2.10)

Furthermore, the above law of large numbers continues to hold when $\alpha=0$,
i.e., the diagonal weights are assumed to be distributed as
$\operatorname{Gamma}^{-1}(\theta)$. In that case $R(\theta,\alpha)$ is
interpreted as $R(\theta,0)=-2\Psi(\theta)$.

###### Proof.

Theorem 1.10 in [BW22] discusses several fluctuation results for point-
to-(partial)line partition function for the $\mathpzc{HSLG}$ polymer,
including Theorem 1.10(3) which applies to the bound phase in this paper.
Letting $n=N-g(N)$ and $m=N+g(N)$ in (1.12) of [BW22] yields

$\displaystyle\frac{1}{(N-g(N))^{1/2}\sigma_{p}}\left[\log\mathpzc{Z}_{N}^{\mathsf{PL}}(g(N))+(N-g(N))\mu_{p}\right]\stackrel{{\scriptstyle
d}}{{\to}}\mathcal{N}(0,1).$

where $\mu_{p}:=\Psi(\theta+\alpha)+p\Psi(\theta-\alpha)$ and
$\sigma_{p}^{2}:=\Psi^{\prime}(\theta+\alpha)-p\Psi^{\prime}(\theta-\alpha)$
with $p=\frac{N+g(N)}{N-g(N)}$. Observe that $(N-g(N))\mu_{p}=-RN+g(N)\tau$.
As $g(N)/N\to 0$, we have that

$\frac{(N-g(N))^{1/2}\sigma_{p}}{N^{1/2}\sigma}\to 1.$

Therefore the above fluctuation result implies (2.9). For the law of large
numbers, the first convergence in (2.10) follows by taking $g(N)\equiv 1$ and
appealing to (2.9). The second law of large numbers also follows from Theorem
1.10(3) in [BW22] as their result also gives fluctuation results for point-
to-(partial)line free energy of the same form with the same law of large
numbers. Finally, the last point of Theorem 2.7 follows from Theorem 1.10(2)
in [BW22] which deals with the $\alpha=0$ case. ∎

##  3\. Controlling the average law of large numbers of the top curves

In this section, we control the average law of large numbers of the top $2k$
curves for large enough $k$ (Proposition 3.4). As explained in the
introduction, the key idea behind this proposition is to show that the
contribution of diagonal weights in the $2k$ many non-intersecting paths of
$Z_{\operatorname{sym}}^{(2k)}(m,n)$ (defined in (2.1)) essentially comes from
$k$ many of them. The starting point of this idea is Lemma 3.1. Given a pair
of non-intersecting paths $(\pi_{1},\pi_{2})$ starting and ending at adjacent
locations with the same $x$-coordinate, Lemma 3.1 constructs two new non-
intersecting paths $(\pi_{1}^{\prime},\pi_{2}^{\prime})$ from
$(\pi_{1},\pi_{2})$ such that the new paths collectively carry the same weight
variables but the diagonal weights only rest on the lower path. This
combinatorial result proceeds to help us decompose the symmetrized multilayer
partition function $Z_{\operatorname{sym}}^{(2k)}(m,n)$ into pairs of single-
layer ones in Lemmas 3.2 and 3.3 before culminating into the final result in
Proposition 3.4.

Let $\Pi(v_{1}\to v_{2},u_{1}\to u_{2})$ denote the set of pairs of non-
intersecting upright paths in $\mathbb{Z}_{>0}^{2}$ starting from
$u_{1},v_{1}$ and ending at $u_{2},v_{2}$ respectively. Recall that
$\mathcal{I}^{-}=\left\\{(i,j)\in\mathbb{Z}_{>0}^{2}|j\leq i\right\\}$. Define
$\mathcal{I}^{+}:=\left\\{(i,j)\in\mathbb{Z}_{>0}^{2}|j\geq i\right\\}$ which
represents the half-space index set that includes points on and above the
diagonal. The first lemma constructs the $U$ map.

######  Lemma 3.1 (Construction of $U$ map).

Fix $x\in\mathbb{Z}_{>0}$ and any $(m,n)\in\mathcal{I}^{-}$ with $n\geq 2$.
Then there exists a map $U:\Pi_{1}\to\Pi_{2}$ where

$\displaystyle\Pi_{1}$ $\displaystyle:=\Pi((1,x+1)\to(m,n),(1,x)\to(m,n-1))$
$\displaystyle\Pi_{2}$ $\displaystyle:=\Pi((1,x+1)\to(n-1,m),(1,x)\to(n,m)),$

such that the following properties hold (let
$(\pi_{1}^{\prime},\pi_{2}^{\prime}):=U(\pi_{1},\pi_{2})$):

  1. (a)

$\pi_{1}^{\prime}$ has no diagonal points, i.e.,
$\\{(i,i)\in\mathbb{Z}_{>0}^{2}\\}\cap\pi_{1}^{\prime}$ is empty and

$\\{(i,i)\in\mathbb{Z}_{>0}^{2}\\}\cap\pi_{2}^{\prime}=\\{(i,i)\in\mathbb{Z}_{>0}^{2}\\}\cap\\{\pi_{1}\cup\pi_{2}\\}.$

  2. (b)

Recall the symmetrized weights
$(\widetilde{W}_{i,j})_{(i,j)\in\mathbb{Z}_{>0}^{2}}$ from (1.11). We have

$\prod_{(i,j)\in\pi_{1}\cup\pi_{2}}\widetilde{W}_{i,j}\overset{a.s.}{=}\prod_{(i,j)\in\pi_{1}^{\prime}\cup\pi_{2}^{\prime}}\widetilde{W}_{i,j}.$

  3. (c)

For each $(\pi_{1}^{\prime},\pi_{2}^{\prime})\in\Pi_{2}$ we have

$\left|U^{-1}(\\{(\pi_{1}^{\prime},\pi_{2}^{\prime})\\})\right|\leq
2^{|\\{(i,i)\in\pi_{1}\cup\pi_{2}\\}|}.$

(a)

(b)

Figure 8. The $U$ map takes (A) to (B).

We remark that Lemma 3.1 is entirely combinatorial and does not use any
results about the integrability of the model. Lemma 3.1 continues to hold for
any collection of symmetrized weights that are not necessarily distributed as
inverse-Gamma random variables.

###### Proof.

We define a partial order $<$ on the points $\mathbb{Z}_{>0}^{2}$ by requiring
$P_{1}=(a_{1},b_{1})<P_{2}=(a_{2},b_{2})$ whenever $a_{1}+b_{1}<a_{2}+b_{2}$.
Let $\pi_{1}$ denote the path from $(1,x+1)$ to $(m,n)$ and $\pi_{2}$ the path
from $(1,x)$ to $(m,n-1)$. We denote $\operatorname{diag}(\pi_{i})$ as the set
of points on $\pi_{i}$ that lie on the diagonal set
$D:=\\{(i,i)\in\mathbb{Z}_{>0}^{2}\\}$. Recall that
$\mathcal{I}^{+}=\\{(i,j)\in\mathbb{Z}_{>0}^{2}\mid i\leq j\\}$ and
$\mathcal{I}^{-}=\\{(i,j)\in\mathbb{Z}_{>0}^{2}\mid j\leq i\\}$.

We first define a special collection of points, $\mathsf{SPDiag}$ from
$\operatorname{diag}(\pi_{2})$. Let $D_{1}<D_{2}<D_{3}<\cdots<D_{s}$ be all
the points in $\operatorname{diag}(\pi_{1}\cup\pi_{2})$ arranged in the
increasing order. We put the point $D_{j}\in\operatorname{diag}(\pi_{2})$ in
the set $\mathsf{SPDiag}$ if $D_{j-1}\in\operatorname{diag}(\pi_{1})$ or
$D_{j+1}\in\operatorname{diag}(\pi_{1})$. In other words, $\mathsf{SPDiag}$
consists of the diagonal points in $\pi_{2}$ that bookend contiguous clusters
of $\operatorname{diag}(\pi_{1})$ in
$\operatorname{diag}(\pi_{1}\cup\pi_{2}).$ We enumerate the points in
$\mathsf{SPDiag}$ as $A_{1}<A_{2}<\cdots<A_{r}$. Let $B_{j}$ be the first
point on $\pi_{1}$ that has the same $x$-coordinate as $A_{j}$. Note that by
construction, either only $\pi_{1}$ intersects the diagonal or only $\pi_{2}$
intersects the diagonal between $A_{j}$ and $A_{j+1}$, $j=1,\ldots,r$. Let us
denote $A_{r+1}:=(m,n-1)$ and $B_{r+1}:=(m,n)$.

$A_{j}$$B_{j}$$A_{j+1}$$B_{j+1}$$Q$$Q^{\prime}$$P_{1}$$P_{2}$$\pi_{3}$$\pi_{1}$$\pi_{2}$
(a)

$A_{j}$$B_{j}$$A_{j+1}$$B_{j+1}$$P_{1}$$P_{2}$$P_{1}^{\prime}$$P_{2}^{\prime}$$\pi_{2}^{\prime}$$\pi_{1}^{\prime}$
(b)

Figure 9. The second case when $j\leq r-1$ and only $\pi_{1}$ intersects with
the diagonal. $\pi_{1}$ and $\pi_{2}$ are the black and blue paths in Figure
(A) respectively. $\pi_{3}$ is the black dashed path in Figure (A).
$\pi_{1}^{\prime}$ is the path in Figure (B) which is formed by the
concatenation of solid blue paths and the black dashed path.
$\pi_{2}^{\prime}$ is the path in Figure (B) which is formed by the
concatenation of solid black paths and the blue dashed path. The $U$ map takes
$\pi_{1},\pi_{2}$ and returns $\pi_{1}^{\prime},\pi_{2}^{\prime}$.

We now construct new paths $\pi_{2}^{\prime}$ and $\pi_{1}^{\prime}$ from
$\pi_{2}$ and $\pi_{1}$ by reconstructing each segment between $A_{j}$ and
$A_{j+1}$ (and $B_{j}$ and $B_{j+1}$ for $\pi_{1}$ respectively),
$j=1,\ldots,r$. We separate the reconstruction procedures for each segment
into the following cases: if only $\pi_{2}$ intersects the diagonal and $j\leq
r-1$, if only $\pi_{1}$ intersects the diagonal and $j\leq r-1$, or if $j=r.$

  1. (1)

When $1\leq j\leq r-1$ and only $\pi_{2}$ intersects the diagonal, we keep the
original paths. We set $\pi_{1}^{\prime}$ and $\pi_{2}^{\prime}$ on these
segments to be the same as those on $\pi_{1}$ and $\pi_{2}$ respectively.

  2. (2)

When $1\leq j\leq r-1$ and only $\pi_{1}$ intersects the diagonal between
$A_{j}$ and $A_{j+1}$ (see Figure 9), the portion of the path $\pi_{2}$ from
$A_{j}$ to $A_{j+1}$ (excluding $A_{j}$ and $A_{j+1}$) lies in
$\mathcal{I}^{-}\setminus D$. Reflecting the portion of the path $\pi_{2}$
from $A_{j}$ to $A_{j+1}$ (black path in Figure 9) across the diagonal yields
a path $\pi_{3}$ (black dashed path in Figure 9). Let $Q$ be the first point
on $\operatorname{diag}(\pi_{1})$ that lies between $A_{j}$ and $A_{j+1}$ and
$Q^{\prime}$ be the last, which exist by construction of the $\mathsf{SPdiag}$
set ($Q$ and $Q^{\prime}$ may overlap). As the $y$-coordinate of $A_{i}$ is
strictly smaller than that of $B_{i}$ and $Q,Q^{\prime}$ are on the
$\operatorname{diag}(\pi_{1})$, $\pi_{1}$ and $\pi_{3}$ must intersect on the
segments between $A_{j}$ and $Q$ and $Q^{\prime}$ and $A_{j+1}$. Let $P_{1}$
be the first point of intersection and $P_{2}$ the last point of intersection.
Clearly $P_{1}\neq P_{2}$ as the former is between $A_{j}$ and $Q$ and the
latter lies between $Q^{\prime}$ and $A_{j+1}$. Replacing the portion of
$\pi_{1}$ between $P_{1}$ and $P_{2}$ with that of $\pi_{3}$ yields a path
$\pi_{1}^{\prime}$ from $B_{j}$ to $B_{j+1}$. As the part of $\pi_{3}$ between
$P_{1}$ and $P_{2}$ lies in $\mathcal{I}^{+}\setminus D$, $\pi_{1}^{\prime}$
lies entirely in $\mathcal{I}^{+}\setminus D$. We denote the reflections of
$P_{1}$ and $P_{2}$ across the diagonal as $P_{1}^{\prime}$ and
$P_{2}^{\prime}$, which must lie on the original $\pi_{2}$ by construction.
Similarly replacing the portion of $\pi_{2}$ between $P_{1}^{\prime}$ and
$P_{2}^{\prime}$ with the reflection of $\pi_{1}$ between $P_{1}$ and $P_{2}$
across the diagonal yields a path $\pi_{2}^{\prime}$ from $A_{j}$ to
$A_{j+1}$. As $\pi_{1}$ and $\pi_{2}$ are non-intersecting, the reflected
paths are also non-intersecting. Thus the new paths $\pi_{1}^{\prime}$ and
$\pi_{2}^{\prime}$ are non-intersecting.

  3. (3)

When $j=r$, consider the portion of the path $\pi_{2}$ from $A_{r}$ to
$A_{r+1}$ (see Figure 10). Note that in this segment, all the diagonal points
belong to $\pi_{1}$. Reflecting this portion of $\pi_{2}$ across the diagonal
gives us $\pi_{3}$ (black dashed path in Figure 10). Let $Q$ be the first
point on $\operatorname{diag}(\pi_{1})$ that lies between $A_{j}$ and
$A_{j+1}$ and $Q$ exists as $\pi_{1}$ ends at
$B_{r+1}:=(m,n)\in\mathcal{I}^{-}$. Note that $\pi_{3}$ lies entirely in
$\mathcal{I}^{+}\setminus D$, excluding $A_{r}$. Thus $\pi_{1}$ and $\pi_{3}$
necessarily intersect in $\mathcal{I}^{+}\setminus D$. Again, we locate the
first point intersection $P$ and replace the portion of $\pi_{1}$ from $P$ to
$B_{r+1}$ with the portion of $\pi_{3}$ from $P$ to
$A_{r+1}^{\prime}:=(n-1,m)$. Similarly, reflecting the portion of $\pi_{1}$
from $P$ and $B_{r+1}$ across the diagonal and replacing the portions of
$\pi_{2}$ between $P^{\prime}$ and $A_{r+1}$ with the portion of reflection
between $P$ and $B_{r+1}^{\prime}:=(n,m)$ yields a path $\pi_{2}^{\prime}$
from $A_{r}$ to $B^{\prime}_{r+1}$. Clearly, the new path $\pi_{1}^{\prime}$
lies in $\mathcal{I}^{+}\setminus D$ and the paths $\pi_{1}^{\prime}$ and
$\pi_{2}^{\prime}$ are non-intersecting as the reflected portions are non-
intersecting.

$A_{r}$$B_{r}$$A_{r+1}$$B_{r+1}$$Q$$P$$\pi_{1}$$\pi_{2}$$\pi_{3}$ (a)

$A_{r}$$B_{r}$$A_{r+1}^{\prime}$$B_{r+1}^{\prime}$$P$$P^{\prime}$$\pi_{2}^{\prime}$$\pi_{1}^{\prime}$
(b)

Figure 10. The $j=r$ case. $\pi_{1}$ and $\pi_{2}$ are the black and blue
paths in Figure (A) respectively. $\pi_{3}$ is the black dashed path in Figure
(A). $\pi_{1}^{\prime}$ is the path in Figure (B) which is formed by the
concatenation of the solid blue path and the black dashed path.
$\pi_{2}^{\prime}$ is the path in Figure (B) which is formed by the
concatenation of the solid black path and the blue dashed path. The $U$ map
takes $\pi_{1},\pi_{2}$ and spits out $\pi_{1}^{\prime},\pi_{2}^{\prime}$.

As $A_{j}$ and $B_{j}$ remain unchanged, connecting all the segments between
$A_{j}$’s (and $B_{j}$’s respectively) for $j\leq r$ and $A_{r}$ and
$B^{\prime}_{r+1}$ (and $B_{r}$ and $A^{\prime}_{r+1}$) yields the new path
$\pi_{2}^{\prime}$ from $(1,x)$ to $(n,m)$ and the new path $\pi_{1}^{\prime}$
from $(1,x+1)$ to $(n-1,m)$ (see Figure 8). At each step of the above
construction, the paths remain non-intersecting. Thus
$(\pi_{1}^{\prime},\pi_{2}^{\prime})$ form a non-intersecting pair. We call
this explicitly constructed map $U$. By construction, $\pi_{1}^{\prime}$ lies
entirely in $\mathcal{I}^{+}\setminus D$ and has no diagonal points. This
proves (a). Since the construction involves only exchanges of reflected
portions, due to the symmetry of the weights $\widetilde{W}_{i,j}$ across the
diagonal, we have (b). Finally to verify (c), note that there are at most
$2^{\operatorname{diag}(\pi_{1}^{\prime}\cup\pi_{2}^{\prime})}$ possible
choices of $\operatorname{diag}(\pi_{1})$ and $\operatorname{diag}(\pi_{2})$
for a given pair of two paths $(\pi_{1}^{\prime},\pi_{2}^{\prime})$ in the
pre-image of $U$. As $\operatorname{diag}(\pi_{1})$ and
$\operatorname{diag}(\pi_{2})$ uniquely determine $\mathsf{SPDiag}$ where
reflections are performed, reverting the same operations on $\pi_{1}^{\prime}$
and $\pi_{2}^{\prime}$ between consecutive points in $\mathsf{SPDiag}$ leads
to original $\pi_{1}$ and $\pi_{2}$. Thus the map has at most
$2^{\operatorname{diag}(\pi_{1}^{\prime}\cup\pi_{2}^{\prime})}$ inverses for
$(\pi_{1}^{\prime},\pi_{2}^{\prime})$, which completes the proof. ∎

Note that the $U$ map in Lemma 3.1 gives us a path that does not contain any
diagonal vertex. To capture the contribution of this path, we now introduce
the diagonal-avoiding symmetrized partition function. Let
$\widetilde{\Pi}_{m,n}^{(1)}$ be the collection of all upright paths from
$(1,1)$ to $(m,n)$ that do not touch the diagonal after $(1,1)$. Set

$\displaystyle\widetilde{Z}_{\operatorname{sym}}(m,n):=\sum_{\pi\in\widetilde{\Pi}_{m,n}^{(1)}}\prod_{(i,j)\in\pi}\widetilde{W}_{i,j},\quad\widetilde{V}_{q}:=\sum_{(i,j)\mid
i+j=q}\widetilde{Z}_{\operatorname{sym}}(i,j)$ (3.1)

where $\widetilde{W}_{i,j}$ is defined in (1.11). We call
$\widetilde{Z}_{\operatorname{sym}}(m,n)$ the diagonal-avoiding symmetrized
partition function. Let us recall $Z_{\operatorname{sym}}(m,n)$ from (2.1) and
we similarly define

$\displaystyle V_{q}:=\sum_{(i,j)\mid i+j=q}Z_{\operatorname{sym}}(i,j)$ (3.2)

The next lemma establishes a relation between
$Z_{\operatorname{sym}}^{(2k)}(m,n)$, $V_{m+n}$ and $\widetilde{V}_{m+n}$.

######  Lemma 3.2.

For all $(m,n)\in\mathcal{I}^{-}$, almost surely we have

$\displaystyle Z_{\operatorname{sym}}^{(2k)}(m,n)\leq
2^{n}\cdot\prod_{i=2}^{2k}\prod_{j=1}^{i-1}(\widetilde{W}_{1,j})^{-1}\cdot\prod_{i=1}^{k}\left[V_{m+n+2-2i}\widetilde{V}_{m+n+1-2i}\right]$
(3.3)

where $Z_{\operatorname{sym}}^{(i)}(m,n)$, $V_{m+n+2-2i}$ and
$\widetilde{V}_{m+n+1-2i}$ are defined in (2.1), (3.2) and (3.1) respectively.

###### Proof.

We extend our definition of $U$ map from Lemma 3.1 to the domain
$\Pi_{m,n}^{(2k)}$ by defining
$U(\pi_{1},\ldots,\pi_{2k}):=(U(\pi_{1},\pi_{2}),\ldots,U(\pi_{2k-1},\pi_{2k})).$
Let $R_{m,n}^{i,k}$ be the collection of all upright paths from $(1,2k-2i+1)$
to $(n-2i+2,m)$. Let $\widetilde{R}_{m,n}^{i,k}$ be the collection of all
upright paths from $(1,2k-2i+2)$ to $(n-2i+1,m)$ that avoid the diagonal.
Given any $(\pi_{1}^{\prime},\ldots,\pi_{2k}^{\prime})\in
U(\Pi_{m,n}^{(2k)})$, by (c), there are at most

$\displaystyle\prod_{i=1}^{k}2^{|(j,j)\in\pi_{2i-1}^{\prime}\cup\pi_{2i}^{\prime}|}$

many inverses in the pre-image of the $U$ map. The $U$ map preserves the
number of diagonal vertices by (a). Furthermore by non-intersection, a
$2k$-tuple of paths in $\Pi_{m,n}^{(2k)}$ has at most $n$ many diagonal
vertices. Thus there are at most $2^{n}$ many inverses. Hence by (a) we have

$\displaystyle Z_{\operatorname{sym}}^{(2k)}(m,n)\leq
2^{n}\cdot\prod_{i=1}^{k}\left[\sum_{\pi_{1}\in\widetilde{R}_{m,n}^{i,k}}\prod_{(i,j)\in\pi_{1}}\widetilde{W}_{i,j}\right]\cdot\prod_{i=1}^{k}\left[\sum_{\pi_{2}\in{R}_{m,n}^{i,k}}\prod_{(i,j)\in\pi_{2}}\widetilde{W}_{i,j}\right].$
(3.4)

We may elongate each of the path in $R_{m,n}^{i,k}$ and
$\widetilde{R}_{m,n}^{i,k}$ by appending an up-path from $(1,1)$ to
$(1,2k-2i+2)$ and from $(1,1)$ to $(1,2k-2i+1)$ respectively. This produces
elongated paths in $\widetilde{\Pi}_{n-2i+2,m}^{(1)}$ and $\Pi_{n-2i+1,m}$
respectively. In terms of weights, we need to multiply the existing weights in
(3.4) by $\prod_{j=1}^{2k-2i+1}\widetilde{W}_{1,j}$ and
$\prod_{j=1}^{2k-2i}\widetilde{W}_{1,j}$ respectively to get the corresponding
weights of elongated paths. After doing precisely the above, we have

$\displaystyle Z_{\operatorname{sym}}^{(2k)}(m,n)\leq
2^{n}\cdot\prod_{i=2}^{2k}\prod_{j=1}^{i-1}(\widetilde{W}_{1,j})^{-1}\cdot\prod_{i=1}^{k}\left[Z_{\operatorname{sym}}(n-2i+2,m)\widetilde{Z}_{\operatorname{sym}}(n-2i+1,m)\right].$
(3.5)

We get (3.3) from the above inequality in (3.5) by observing the definition of
$V_{q}$ and $\widetilde{V}_{q}$ from (3.2). This completes the proof. ∎

The next lemma bounds $\log V_{q}$ and $\log\widetilde{V}_{q}$ from above with
high probability.

######  Lemma 3.3.

Recall $R$ from (2.8). For every $\delta>0$ and $1\leq p<n$, we have

$\displaystyle\lim_{q\to\infty}\mathbb{P}\left(\log
V_{q}\leq(R+\delta)\tfrac{q}{2}\right)=1,\quad\lim_{q\to\infty}\mathbb{P}\left(\log\widetilde{V}_{q}\leq(-2\Psi(\theta)+\delta)\tfrac{q}{2}\right)=1.$
(3.6)

###### Proof.

Fix any $\delta>0$. By Lemma 2.2 we have

$\displaystyle V_{2N}=\sum_{p=0}^{N-1}Z(N+p,N-p),\quad
V_{2N+1}=\sum_{p=1}^{N}Z(N+p,N-p+1).$

From Theorem 2.7 ((2.10) in particular) we have that

$\displaystyle\frac{1}{N}\log\left[\sum_{p=1}^{N}Z(N+p,N-p+1)\right]\stackrel{{\scriptstyle
p}}{{\to}}R,\quad\frac{1}{N}\log\left[\sum_{p=1}^{N-1}Z(N+p,N-p)\right]\stackrel{{\scriptstyle
p}}{{\to}}R,$ (3.7)

Note that in the above equation, we have excluded $Z(N,N)$ as their result
does not contain $Z(N,N)$ in the sum. However, in our case, we may include
$Z(N,N)$ by appealing to Theorem 2.4. First, in view of the above law of large
numbers in (3.7), we have

$\displaystyle\mathbb{P}(\log V_{2N+1}\leq(R+\tfrac{1}{2}\delta)N)\to 1.$
(3.8)

On the other hand, by (2.4) we have
$\sum_{p=1}^{N}e^{H_{N}^{(1)}(2p)}=\sum_{p=1}^{N}Z(N+p,N-p+1)$. Since
$H_{N}^{(1)}(2)\leq\log\sum_{p=1}^{N}e^{H_{N}^{(1)}(2p)}=\log V_{2N+1}$, (3.8)
implies

$\displaystyle\mathbb{P}(H_{N}^{(1)}(2)\leq(R+\tfrac{1}{2}\delta)N)\to 1,$

as $N\to\infty$. In addition, by ordering of points in the line ensemble
(Theorem 2.4) we know that with probability at least $1-2^{-N}$,
$H_{N}^{(1)}(1)\leq H_{N}^{(2)}(2)+\log^{2}N$. Thus we have

$\displaystyle\mathbb{P}(H_{N}^{(1)}(1)\leq(R+\delta)N)\to 1,$ (3.9)

as $N\to\infty$. Given that $H_{N}^{(1)}(1)=Z(N,N)$, combining (3.9) and the
second convergence in (3.7) yields $\mathbb{P}(\log V_{2N}\leq(R+\delta)N)\to
1$ and together with (3.8) this concludes the proof of the first convergence
in (3.6).

Next, for the diagonal-avoiding case, let $(W_{i,i}^{\alpha=0})_{i\geq 1}$ be
a family of weights distributed as $\operatorname{Gamma}(\theta)$ independent
of $(W_{i,j})$. We set $W_{i,j}^{\alpha=0}:=\widetilde{W}_{i,j}$ for $i\neq
j$. This gives us a new collection of symmetrized weights. We denote the
corresponding symmetrized partition function and the diagonal-avoiding
symmetrized partition function as $Z_{\operatorname{sym}}^{\alpha=0}$ and
$\widetilde{Z}_{\operatorname{sym}}^{\alpha=0}$ respectively. Observe that

$\displaystyle\widetilde{Z}_{\operatorname{sym}}(i,j)\leq\frac{\widetilde{W}_{1,1}}{W_{1,1}^{\alpha=0}}\cdot\widetilde{Z}_{\operatorname{sym}}^{\alpha=0}(i,j)\leq\frac{\widetilde{W}_{1,1}}{W_{1,1}^{\alpha=0}}\cdot
Z_{\operatorname{sym}}^{\alpha=0}(i,j).$ (3.10)

The first equality in (3.10) is due to the fact that the weight corresponding
$(1,1)$ is common in all paths and that is the only diagonal weight that
appears in the diagonal avoiding symmetrized partition functions. The next
inequality is obvious as we have just removed the diagonal avoiding
restriction. This leads to

$\displaystyle\log\widetilde{V}_{q}\leq\log\widetilde{W}_{1,1}-\log
W_{1,1}^{\alpha=0}+\log\left[\sum_{(i,j)\mid
i+j=q}Z_{\operatorname{sym}}^{\alpha=0}(i,j)\right].$

The first two terms on the right-hand side of the above display are tight. An
upper bound on the third term can be computed by the exact same analysis as
$V_{q}$. Indeed, the law of large numbers and Theorem 2.4 continue to hold for
$\alpha=0$ when $R$ becomes $-2\Psi(\theta)$ (see the last point in Theorem
2.7). This concludes the proof of (3.6). ∎

Finally, with Lemmas 3.2 and 3.3 in place, we are ready to control the average
law of large numbers of the top curves of the $\mathpzc{HSLG}$ line ensemble.

######  Proposition 3.4.

Recall $\Delta_{k},R$ in (2.8). Fix any $\varepsilon>0$ and
$k\in\mathbb{Z}_{>0}$ large such that $\Delta_{k}>0$. Then there exists
$N_{0}(k,\varepsilon)>2k+1$ such that for all $N\geq N_{0}$ we have

$\displaystyle\mathbb{P}\left(\sup_{p\in\llbracket
1,2N-4k+2\rrbracket}\frac{1}{2k}\sum_{i=1}^{2k}H_{N}^{(i)}(p)\leq(R-\tfrac{1}{2}\Delta_{k})N\right)\geq
1-\varepsilon.$

In other words, Proposition 3.4 claims that when $k$ is large enough so that
$\Delta_{k}>0$, the average law of large numbers of top $2k$ curves is
strictly less than $R$, which is the law of large numbers for point-
to-(partial)line free energy process (see Theorem 2.7).

###### Proof.

Fix any $\varepsilon>0$. The definition of the $\mathpzc{HSLG}$ line ensemble
in (2.3) and (3.3) collectively yield that, for all $p\in\llbracket
1,N-2k+1\rrbracket$,

$\displaystyle\sum_{i=1}^{2k}H_{N}^{(i)}(2p)$ $\displaystyle=2k\log 2+\log
Z_{\operatorname{sym}}^{(2k)}(N+p,N-p+1)$ $\displaystyle\leq 2k\log 2+N\log
2-\log\left[\prod_{i=2}^{2k}\prod_{j=1}^{i-1}\widetilde{W}_{1,j}\right]+\log\prod_{i=1}^{k}\left[V_{2N+3-2i}\widetilde{V}_{2N+2-2i}\right],$

where the r.h.s. is free of $p$. Hence we may take supremum over
$p\in\llbracket 1,N-2k+1\rrbracket$ over both sides of the above display to
get

$\displaystyle\sup_{p\in\llbracket
1,N-2k+1\rrbracket}\sum_{i=1}^{2k}H_{N}^{(i)}(2p)$
$\displaystyle\leq(2k+N)\log
2-\log\left[\prod_{i=2}^{2k}\prod_{j=1}^{i-1}\widetilde{W}_{1,j}\right]$
(3.11) $\displaystyle\hskip
85.35826pt+\log\prod_{i=1}^{k}\left[V_{2N+3-2i}\widetilde{V}_{2N+2-2i}\right].$

We now provide high probability upper bounds for each of the terms on the
r.h.s. of (3.11). Let us take $\delta:=\frac{\Delta_{k}}{4}$. By Lemma 3.3, we
may choose $N_{0}(k,\varepsilon)>2k+1$ large enough such that for all $N\geq
N_{0}$

$\mathbb{P}(\log V_{N}\leq(R+\delta)\tfrac{N}{2})\geq
1-\tfrac{\varepsilon}{8k},\quad\mathbb{P}(\log\widetilde{V}_{N}\leq(-2\Psi(\theta)+\delta)\tfrac{N}{2})\geq
1-\tfrac{\varepsilon}{8k}.$

Thus applying a union bound we see that for all large enough $N$, with
probability $1-\frac{\varepsilon}{4}$,

$\displaystyle\log\prod_{i=1}^{k}\left[V_{N+3-2i}\widetilde{V}_{N+2-2i}\right]\leq
RkN-2\Psi(\theta)kN+2k\delta N.$ (3.12)

Note that the random variable
$\log\left[\prod_{i=2}^{2k}\prod_{j=1}^{i-1}\widetilde{W}_{1,j}\right]$ is
tight and free of $N$. Hence with probability $1-\frac{\varepsilon}{4}$ one
can ensure that

$\displaystyle(2k+N)\log
2-\log\left[\prod_{i=2}^{2k}\prod_{j=1}^{i-1}\widetilde{W}_{1,j}\right]\leq
N\log 2+2k\delta N.$ (3.13)

holds for all large enough $N$. Inserting the above two bounds in (3.12) and
(3.13) back in (3.11), we have that with probability at least
$1-\frac{\varepsilon}{2},$

$\displaystyle\sup_{p\in\llbracket
1,N-2k+1\rrbracket}\frac{1}{2k}\sum_{i=1}^{2k}H_{N}^{(i)}(2p)\leq\left[\tfrac{1}{2k}\log
2+2\delta+\tfrac{R}{2}-\Psi(\theta)\right]N,$ (3.14)

for all large enough $N$. As $\delta=\frac{\Delta_{k}}{4}$, the r.h.s. of
(3.14) is precisely $(R-\tfrac{1}{2}\Delta_{k})N$. By the exact same argument,
one can check that with probability at least $1-\frac{\varepsilon}{2}$ we have

$\displaystyle\sup_{p\in\llbracket
0,N-2k\rrbracket}\frac{1}{2k}\sum_{i=1}^{2k}H_{N}^{(i)}(2p+1)\leq(R-\tfrac{1}{2}\Delta_{k})N,$
(3.15)

for all large enough $N$. Taking another union bound of (3.14) and (3.15), we
get the desired result. ∎

##  4\. Controlling the second curve

In this section, we establish the separation between the first and the second
curve of our $\mathpzc{HSLG}$ line ensemble. On the basis of Proposition 3.4,
Lemma 4.1 first establishes that for large enough $k$ with high probability
the $(2k+2)$-th curve $H_{N}^{(2k+2)}(\cdot)$ is uniformly $\mbox{const}\cdot
N$ below than $RN$ over an interval of $\llbracket 1,N\rrbracket$, where $R$
defined in (2.8) is the law of large numbers for point-to-(partial)line free
energy process (Theorem 2.7). This helps us show that with high probability
the second curve $H_{N}^{(2)}(\cdot)$ over an interval of length $O(\sqrt{N})$
is $M\sqrt{N}$ below $RN$ next in Proposition 4.2 for any $M>0$.

######  Lemma 4.1.

Recall $R$ in (2.8). Fix any $\varepsilon>0$ and $k\in\mathbb{Z}_{>0}$ large
enough such that $\Delta_{k}>0$. Then there exists $N_{0}(k,\varepsilon)$ such
that for all $N\geq N_{0}$ we have

$\displaystyle\mathbb{P}\left(\sup_{p\in\llbracket
1,N\rrbracket}H_{N}^{(2k+2)}(p)\leq(R-\tfrac{1}{4}\Delta_{k})N\right)\geq
1-\varepsilon.$ (4.1)

###### Proof.

Let us consider the following events

$\displaystyle\mathsf{A}$ $\displaystyle:=\left\\{\sup_{p\in\llbracket
1,N\rrbracket}H_{N}^{(2k+2)}(p)\leq(R-\tfrac{1}{4}\Delta_{k})N\right\\},$
$\displaystyle\mathsf{B}$ $\displaystyle:=\left\\{H_{N}^{(i+1)}(p)\leq
H_{N}^{(i)}(p)+2\log^{2}N,\mbox{ for all }i\in\llbracket
1,2k+1\rrbracket,p\in\llbracket 1,N\rrbracket\right\\},$
$\displaystyle\mathsf{C}$ $\displaystyle:=\left\\{\sup_{p\in\llbracket
1,N\rrbracket}\frac{1}{2k}\sum_{i=1}^{2k}H_{N}^{(i)}(p)\leq(R-\tfrac{1}{2}\Delta_{k})N\right\\}.$

We claim that for all large enough $N$, we have
$(\mathsf{B}\cap\neg\mathsf{A})\subset\mathsf{\neg C}$. To see this, note that
on $\mathsf{B}\cap\neg\mathsf{A}$, there exists a point $p^{*}\in\llbracket
1,N\rrbracket$ such that $H_{N}^{(2k+2)}(p^{*})>(R-\tfrac{1}{4}\Delta_{k})N)$
and hence (as $\mathsf{B}$ holds)

$H_{N}^{(i)}(p^{*})>(R-\tfrac{1}{4}\Delta_{k})N-(4k+4)\log^{2}N,$

for all $i\in\llbracket 1,2k+1\rrbracket$. However, the above display also
implies that

$\sup_{p\in\llbracket
1,N\rrbracket}\frac{1}{2k}\sum_{i=1}^{2k}H_{N}^{(i)}(p)>(R-\tfrac{1}{4}\Delta_{k})N-(4k+4)\log^{2}N$

which is strictly bigger than $(R-\tfrac{1}{2}\tau)N$ and implies
$\neg\mathsf{C}$. Thus by a union bound, we have

$\displaystyle\mathbb{P}(\neg\mathsf{A})\leq\mathbb{P}(\neg\mathsf{B})+\mathbb{P}(\mathsf{B}\cap\neg\mathsf{A})\leq\mathbb{P}(\neg\mathsf{B})+\mathbb{P}(\neg\mathsf{C}).$
(4.2)

Note that for fixed $k$, by Theorem 2.4 with $\rho=\frac{1}{2}$ and a union
bound, we have $\mathbb{P}(\neg\mathsf{B})\leq N\cdot(2k+1)\cdot
2^{-N}\leq\frac{\varepsilon}{2}$ for all $N\geq N_{1}(k,\varepsilon)$. On the
other hand, Proposition 3.4 yields that for fixed $k$ and $\varepsilon$,
$\mathbb{P}(\neg\mathsf{C})\leq\frac{\varepsilon}{2}$ for all $N$ greater than
some $N_{2}(k,\frac{\varepsilon}{2})$. Letting
$N_{0}(k,\varepsilon)=\max\\{N_{1},N_{2}\\}$ and inserting these two bounds in
(4.2) leads to (4.1). ∎

Building on Lemma 4.1, the next result demonstrates that on a given interval
of length $O(\sqrt{N})$ starting from $1$ and any $M_{2}>0$, the second curve
$H_{N}^{(2)}(\cdot)$ is uniformly lower than $RN-M_{2}\sqrt{N}$ with high
probability (see Figure 11).

######  Proposition 4.2.

Recall $\Delta_{k},R$ in (2.8). Fix $\varepsilon\in(0,1)$, $M_{1},M_{2}\geq 1$
and $k\in\mathbb{Z}_{>0}$ such that $\Delta_{k}>0$. Then there exists a
constant $N_{2}(\varepsilon,M_{1},M_{2})>0$ such that for all $N\geq N_{2}$ we
have

$\displaystyle\mathbb{P}\left(\sup_{p\in[1,2\lfloor
M_{1}\sqrt{N}\rfloor+1]}H_{N}^{(2)}(p)\leq RN-M_{2}\sqrt{N}\right)\geq
1-\tfrac{1}{2}\epsilon.$ (4.3)

$y=RN-M_{2}\sqrt{N}$$H_{N}^{(2)}(\cdot)$$x=1$$x=2\lfloor
M_{1}\sqrt{N}\rfloor+1$ Figure 11. The high probability event in Proposition
4.2.

###### Proof.

The proof of Proposition 4.2 is conducted in the following stages:

  * •

Using Theorem 2.7 and Lemma 4.1, we determine the high probability locations
of $H_{N}^{(1)}(2M\sqrt{N}+1)$ and $H_{N}^{(2k+2)}(\cdot)$. Using the ordering
of points in Theorem 2.4, we then bound the endpoints
$H_{N}^{(i)}(2M\sqrt{N}+1)$, $i\in\llbracket 1,2k+1\rrbracket$ from above
based on the high probability locations of $H_{N}^{(1)}(2M\sqrt{N}+1)$ and the
$(2k+2)$-th curve.

  * •

We next consider the conditional law of $(H_{N}^{(i)}\llbracket
1,2M\sqrt{N}\rrbracket)_{i\in\llbracket 1,2k+1\rrbracket}$ given the above
boundary conditions. By Theorem 2.2, this law is given by an appropriate
$\mathpzc{HSLG}$ Gibbs measure. Applying stochastic monotonicity, we may also
assume that the $H_{N}^{(2i-1)}(2{M}\sqrt{N}+1)$ and
$H_{N}^{(2i)}(2{M}\sqrt{N}+1)$ are sufficiently far apart. This will allow us
to approximate the Gibbs measure as a product of interacting random walks
defined in Definition 2.5.

  * •

Lastly, we use the associated estimates of interacting random walks from
Proposition 2.6 to dissect the Gibbs measure and yield a quantitative bound in
our favor.

We now flesh out the technical details of the above stages. In the following
proof, we assume all the multiples of $\sqrt{N}$ appearing below are integers
for convenience in notation. The general case follows verbatim by considering
the floor function. For clarity, we split our proof into several steps.

Step 1. In this step, we reduce our proof of (4.3) to (4.7). Let us consider
the $\mathpzc{HSLG}$ line ensemble $H_{N}=(H_{N}^{(1)},\ldots,H_{N}^{(N)})$.
Fix any $\varepsilon\in(0,1)$, $M_{1},M_{2}\geq 1$ and any
$k\in\mathbb{Z}_{>0}$ such that $\Delta_{k}>0$. Let $\Phi(x)$ be the
cumulative distribution function of a standard Gaussian random variable. Set
$\tau:=|\Psi(\theta-\alpha)-\Psi(\theta+\alpha)|$. Let $M\in\mathbb{Z}_{>0}$
whose precise value is to be determined. Taking $g(N)=M\sqrt{N}$ in Theorem
2.7 yields

$\displaystyle\frac{1}{\sigma\sqrt{N}}\left[\log\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})-RN+M\tau\sqrt{N}\right]\stackrel{{\scriptstyle
d}}{{\to}}\mathcal{N}\big{(}0,1\big{)}.$ (4.4)

Note that (4.4) implies

$\displaystyle\mathbb{P}\left(\frac{1}{\sigma\sqrt{N}}\left[\log\left[\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})\right]-RN+M\tau\sqrt{N}\right]\leq\Phi(1-\frac{\varepsilon}{2})\right)\to
1-\frac{\varepsilon}{2}.$

Thus for $N$ large enough, we have that with probability greater than
$1-\varepsilon$,

$\displaystyle\log\left[\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})\right]\leq
RN-\big{(}M\tau-\Phi(1-\tfrac{\varepsilon}{2})\sigma\big{)}\sqrt{N}.$ (4.5)

Set
$M\geq\max\\{M_{1},\frac{1}{\tau}(M_{2}+k+1+\Phi(1-\tfrac{\varepsilon}{2})\sigma)\\}.$
Note that by definition,
$H_{N}^{(1)}(2M\sqrt{N}+1)\leq\log\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})$
and as $M\tau-\Phi(1-\frac{\varepsilon}{2})\sigma>M_{2}+k+1$, (4.5) yields
that

$\displaystyle\mathbb{P}(\mathsf{A})\geq 1-\varepsilon,\mbox{ where
}\mathsf{A}:=\left\\{H_{N}^{(1)}(2M\sqrt{N}+1)\leq
RN-(M_{2}+k+1)\sqrt{N}\right\\}$ (4.6)

for all large enough $N$. Set $T=M\sqrt{N}+1$. We claim that

$\displaystyle\mathbb{P}(\neg\mathsf{E})\leq
3\varepsilon+\frac{k\varepsilon}{(1-\varepsilon)^{k+1}},\mbox{ where
}\mathsf{E}$ $\displaystyle:=\left\\{\sup_{p\in\llbracket
1,2T-1\rrbracket}H_{N}^{(2)}(p)\leq RN-M_{2}\sqrt{N}\right\\}.$ (4.7)

Since $2T-1\geq 2M_{1}\sqrt{N}+1$, assuming (4.7) and adjusting $\varepsilon$
yield (4.3).

Step 2. In this step we prove (4.7). To begin with, we consider several
events:

$\displaystyle\mathsf{B}$
$\displaystyle:=\bigcap_{i=1}^{2k}\left\\{H_{N}^{(i+1)}(2T)\leq
H_{N}^{(i)}(2T-1)+\log^{2}N,\right.$ $\displaystyle\hskip
113.81102pt\left.H_{N}^{(i+1)}(2T-1)\leq
H_{N}^{(i+1)}(2T)+\log^{2}N\right\\},$ $\displaystyle\mathsf{C}$
$\displaystyle:=\left\\{\sup_{p\in\llbracket
1,N\rrbracket}H_{N}^{(2k+2)}(p)\leq(R-\tfrac{1}{4}\Delta_{k})N\right\\},$
$\displaystyle\mathsf{D}$
$\displaystyle:=\bigcap_{i=1}^{k}\bigg{\\{}\max\big{\\{}H_{N}^{(2i)}(2T-1),H_{N}^{(2i)}(2T),H_{N}^{(2i+1)}(2T)\big{\\}}$
$\displaystyle\hskip 142.26378pt\leq
RN-(M_{2}+k+1)\sqrt{N}+2k\log^{2}N\bigg{\\}}.$

Let us consider the $\sigma$-field

$\displaystyle\mathcal{F}$
$\displaystyle:=\sigma\left\\{H_{N}^{(2i)}\llbracket
2T-1,2N-4i+2\rrbracket,H_{N}^{(2i+1)}\llbracket 2T,2N-4i\rrbracket,\
i\in\llbracket 1,k\rrbracket,\right.$ $\displaystyle\hskip
113.81102pt\left.H_{N}^{(1)}\llbracket 1,2N\rrbracket,H_{N}^{(j)}\llbracket
1,2N-2j+2\rrbracket,\ j\in\llbracket 2k+2,N\rrbracket\right\\}.$

By Theorem 2.4 with $\rho=\frac{1}{2}$, we have
$\mathbb{P}(\neg\mathsf{B})\leq 4k2^{-N}\leq\varepsilon$ for all large enough
$N$. Observe that $\mathsf{A}\cap\mathsf{B}\subset\mathsf{D}$ and recall that
$\mathbb{P}(\neg\mathsf{A})<\varepsilon$ in (4.6). Thus via the union bound,
we have
$\mathbb{P}(\neg\mathsf{D})\leq\mathbb{P}(\neg\mathsf{A})+\mathbb{P}(\neg\mathsf{B})\leq
2\varepsilon$. Note that $\mathsf{C}\cap\mathsf{D}$ is measurable w.r.t.
$\mathcal{F}$. Applying the union bound and tower property of conditional
expectation we get

$\displaystyle\mathbb{P}(\neg\mathsf{E})$
$\displaystyle\leq\mathbb{P}(\neg\mathsf{C})+\mathbb{P}(\neg\mathsf{D})+\mathbb{P}(\mathsf{C}\cap\mathsf{D}\cap\neg\mathsf{E})\leq
3\varepsilon+\mathbb{E}\left[\mathbf{1}_{\mathsf{C}\cap\mathsf{D}}\cdot\mathbb{E}\left[\mathbf{1}_{\neg\mathsf{E}}\mid\mathcal{F}\right]\right].$
(4.8)

where in the last inequality we have used Lemma 4.1 to get that
$\mathbb{P}(\neg\mathsf{C})\leq\varepsilon$ for all large enough $N$. We claim
that

$\displaystyle\mathbb{E}\left[\mathbf{1}_{\mathsf{C}\cap\mathsf{D}}\cdot\mathbb{E}\left[\mathbf{1}_{\neg\mathsf{E}}\mid\mathcal{F}\right]\right]\leq\frac{k\varepsilon}{(1-\varepsilon)^{k+1}}.$
(4.9)

We will demonstrate (4.9) in the Steps 3-4. Currently, assuming the validity
of (4.9) and appealing to (4.8) prove (4.7).

Step 3. In this step we study
$\mathbf{1}_{\mathsf{C}\cap\mathsf{D}}\mathbb{E}\left[\mathbf{1}_{\neg\mathsf{E}}\mid\mathcal{F}\right]$
by invoking the Gibbs property (Theorem 2.2). Let us consider the domain

$\displaystyle\Theta_{k,T}:=\\{(i,j)\mid i\in\llbracket
2,2k+1\rrbracket,j\in\llbracket
1,2T-1-\mathbf{1}_{i=\operatorname{even}}\rrbracket\\}.$

Figure 12. $\Theta_{k,T}$ for $k=3$, $T=4$ shown in the shaded region. The
$\mathpzc{HSLG}$ Gibbs measure on $\Theta_{3,4}$ with boundary condition
$(u_{i,j})_{(i,j)\in\partial\Theta_{3,4}}.$

By Theorem 2.2, the distribution of the line ensemble conditioned on
$\mathcal{F}$ is given by $\mathbb{P}_{\operatorname{gibbs}}^{\vec{u}}$, i.e.
the $\mathpzc{HSLG}$ Gibbs measure on the domain $\Theta_{k,T}$ with boundary
condition $\vec{u}:=(H_{N}^{(i)}(j))_{(i,j)\in\partial\Theta_{k,T}}$ and the
boundary set of $\Theta_{k,T}$ is given by

$\displaystyle\partial\Theta_{k,T}:=\left\\{(1,2j-1),(2,2T-1),(3,2T),(2i,2T-1),(2i,2T),(2i+1,2T)\mid
i\in\llbracket 2,k\rrbracket,j\in\llbracket 1,T\rrbracket\right\\}.$

Note that for large enough $N$, on the event $\mathsf{C}\cap\mathsf{D}$ we
have

$\displaystyle H_{N}^{(1)}(2j-1)\leq x_{1,2j-1}:=\infty,\quad j\in\llbracket
1,T\rrbracket,$ (4.10) $\displaystyle H_{N}^{(2i)}(2T-1)\leq
x_{2i,2T-1}=RN-(M_{2}+i)\sqrt{N},\quad i\in\llbracket 1,k\rrbracket,$
$\displaystyle H_{N}^{(2i)}(2T)\leq x_{2i,2T}:=RN-(M_{2}+i)\sqrt{N},\quad
i\in\llbracket 2,k\rrbracket,$ $\displaystyle H_{N}^{(2i+1)}(2T)\leq
x_{2i+1,2T}:=RN-(M_{2}+i)\sqrt{N}-\sqrt{T},\quad i\in\llbracket
1,k\rrbracket,$ $\displaystyle H_{N}^{(2k+2)}(2j)\leq
x_{2k+2,2j}:=RN-(M_{2}+k+1)\sqrt{N},\quad j\in\llbracket 1,T\rrbracket.$

where $\mathsf{C}$ holds only in the last inequality. Since $\neg\mathsf{E}$
event is increasing with respect to the boundary data, by stochastic
monotonicity we have

$\displaystyle\mathbf{1}_{\mathsf{C}\cap\mathsf{D}}\cdot\mathbb{E}\left[\mathbf{1}_{\neg\mathsf{E}}\mid\mathcal{F}\right]\leq\mathbf{1}_{\mathsf{C}\cap\mathsf{D}}\cdot\mathbb{P}_{\operatorname{gibbs}}^{\vec{u}}(\neg\mathsf{E})\leq\mathbb{P}_{\operatorname{gibbs}}^{\vec{x}}(\neg\mathsf{E}).$
(4.11)

To bound $\mathbb{P}_{\operatorname{gibbs}}^{\vec{x}}(\neg\mathsf{E})$ we need
a convenient alternative representation for the
$\mathbb{P}_{\operatorname{gibbs}}^{\vec{x}}$ measure. Towards this end,
through the Gibbs property, we dissect the
$\mathbb{P}_{\operatorname{gibbs}}^{\vec{x}}$ measure into blocks of
independent interacting random walks (Definition 2.5) and the Radon-Nikodym
derivatives interleaved between adjacent blocks (see Figure 13). Let us now
describe this decomposition.

Recall the interacting random walk ($\mathsf{IRW}$) from Definition 2.5. Let
$(L_{2i}\llbracket 1,2T-2\rrbracket,L_{2i+1}\llbracket
1,2T-1\rrbracket)_{i=1}^{k}$ be $k$ independent $\mathsf{IRW}$s of length $T$
with boundary condition $(x_{2i,2T-1},x_{2i+1,2T})$. Let us denote the joint
law and expectation of $L$ as $\mathbb{P}_{\operatorname{block}}^{\vec{x}}$
and $\mathbb{E}_{\operatorname{block}}^{\vec{x}}$ respectively. Set

$\displaystyle
W_{\operatorname{br}}:=\exp\left(-\sum_{i=1}^{k}\sum_{j=1}^{T}\left[e^{L_{2i+2}(2j)-L_{2i+1}(2j+1)}+e^{L_{2i+2}(2j)-L_{2i+1}(2j-1)}\right]\right)$
(4.12)

with the convention $L_{2i+1}(2T+1)=\infty$ for $i\in\llbracket 1,k\rrbracket$
and $L_{i}(j)=x_{i,j}$ for all $(i,j)\in\partial\Theta_{k,T}$. Note that here
only $H_{N}^{(1)}(2j+1),j\in\llbracket 1,T\rrbracket$ are in the boundary and
are set to $\infty$ in (4.10). Thus, their contributions to the Radon-Nikodym
derivative $W_{\operatorname{br}}$ would be
$\prod_{j=1}^{2T-2}\exp(-e^{H_{N}^{(2)}(j)-\infty})=1.$ From the description
of the $\mathpzc{HSLG}$ Gibbs measure, we have

$\displaystyle\mathbb{P}_{\operatorname{gibbs}}^{\vec{x}}(\neg\mathsf{E})=\frac{\mathbb{E}_{\operatorname{block}}^{\vec{x}}[W_{\operatorname{br}}\mathbf{1}_{\neg\mathsf{E}}]}{\mathbb{E}_{\operatorname{block}}^{\vec{x}}[W_{\operatorname{br}}]},$
(4.13)

=$\times$ Figure 13. Proof Scheme: The Gibbs measure on $\Theta_{2,4}$ domain
(left figure) can be decomposed into two parts: One is the combination of the
top colored row and 2 $\mathsf{IRW}$s (middle figure) and two are the
remaining black weights (right figure) which will be viewed as a Radon-Nikodym
derivative. Here note that in the middle figure, the only contribution from
the top row comes from the odd points, $H_{N}^{(1)}(2j-1)$ for $j\in\llbracket
1,T\rrbracket$, which are set to $\infty$. Thus, their contribution to (4.12)
from (2.6) would be $\exp(-e^{-\infty})=1.$

Step 4. Finally in this step, we provide an upper bound for the right-hand
side of (4.13) by bounding its numerator and denominator separately. Let us
consider the event:

$\displaystyle\mathsf{G}$
$\displaystyle:=\bigcap_{i=1}^{k}\left\\{\sup_{p\in\llbracket
1,2T-1\rrbracket}|L_{2i}(p)-x_{2i,2T-1}|+\sup_{q\in\llbracket
1,2T\rrbracket}|L_{2i+1}(q)-x_{2i,2T-1}|\leq M_{0}\sqrt{T}\right\\}.$

where $M_{0}$ comes from Proposition 2.6. From the description of the Gibbs
measure, it is clear that if $(L_{2i}(\cdot),L_{2i+1}(\cdot))$ is an
$\mathsf{IRW}$ with boundary condition $(x_{2i,2T-1},x_{2i,2T-1}-\sqrt{T})$,
then $(L_{2i}(\cdot)-x_{2i,2T-1},L_{2i+1}(\cdot)-x_{2i,2T-1})$ is an
$\mathsf{IRW}$ with boundary condition $(0,-\sqrt{T})$. Thus, appealing to
Proposition 2.6, we see that

$\mathbb{P}_{\operatorname{block}}^{\vec{x}}(\mathsf{G})\geq(1-\varepsilon)^{k}\geq
1-k\varepsilon.$

Let us assume $N$ is large enough so that
$\sqrt{N}-2M_{0}\sqrt{T}\geq\frac{1}{2}\sqrt{N}$ (recall $T=O(\sqrt{N})$).
Observe that under the event $\mathsf{G}$, we have for all $p\leq 2T-1$

$L_{2i}(p)\leq x_{2,2T-1}+M_{0}\sqrt{T}=RN-(M_{2}+1)\sqrt{N}+M_{0}\sqrt{T}\leq
RN-M_{2}\sqrt{N}.$

Thus, $\mathsf{E}$ defined in (4.7) holds. This implies
$\neg\mathsf{E}\subset\neg\mathsf{G}$. Hence

$\displaystyle\mathbb{E}_{\operatorname{block}}^{\vec{x}}[W_{\operatorname{br}}\mathbf{1}_{\neg\mathsf{E}}]\leq\mathbb{P}_{\operatorname{block}}^{\vec{x}}(\neg\mathsf{E})\leq\mathbb{P}_{\operatorname{block}}^{\vec{x}}(\neg\mathsf{G})\leq
k\varepsilon.$ (4.14) $\displaystyle\mathbf{1}_{\mathsf{G}}\cdot
W_{\operatorname{br}}\geq\mathbf{1}_{\mathsf{G}}\cdot\exp\left(-k(2T-1)e^{\sqrt{N}-2M_{0}\sqrt{T}}\right)\geq(1-\varepsilon).$

where the last one follows by taking $N$ large enough (recall
$T=O(\sqrt{N})$). Thus,
$\mathbb{E}_{\operatorname{block}}[W_{\operatorname{br}}]\geq(1-\varepsilon)\mathbb{P}_{\operatorname{block}}(\mathsf{G})\geq(1-\varepsilon)^{k+1}$.
Inserting this bound and the bound in (4.14) back in (4.13) we get that
$\mathbb{P}_{\operatorname{gibbs}}^{\vec{x}}(\neg\mathsf{E})\leq\frac{k\varepsilon}{(1-\varepsilon)^{k+1}}.$
Combining this bound with (4.11) yields (4.9). This completes the proof. ∎

##  5\. Proof of main theorems

In this section, we prove our main theorems, Theorems 1.1, 1.3, 1.4, and 1.7.
In Section 5.1, we first present a few supporting technical results. Next in
Section 5.2 we complete the proof of our main theorems by assuming a technical
proposition (Proposition 5.3) which is proved in Section 5.3.

###  5.1. Preparatory lemmas

The first lemma settles a weaker version of Theorem 1.1. Recall the polymer
measure $\mathbb{P}^{W}$ from (1.2), the partition function $Z(m,n)$ from
(1.4), and the $\mathpzc{HSLG}$ line ensemble $H_{N}$ from Definition 2.1.
Note that the quenched distribution of the endpoint of the polymer is related
via

$\displaystyle\mathbb{P}^{W}(\pi(2N-2)=N-r)=\frac{Z(N+r,N-r)}{\sum_{p=0}^{N-1}Z(N+p,N-p)}=\frac{e^{H_{N}^{(1)}(2r+1)}}{\sum_{p=0}^{N-1}e^{H_{N}^{(1)}(2p+1)}}.$
(5.1)

where the second equality follows from the relation (2.4). Recalling
$\mathpzc{Z}_{N}^{\mathsf{PL}}(k)=\sum_{p=k}^{N-1}e^{H_{N}^{(1)}(2p+1)}$ from
(2.7), we obtain

$\mathbb{P}^{W}(\pi(2N-2)\leq
N-k)=\frac{\mathpzc{Z}_{N}^{\mathsf{PL}}(k)}{\mathpzc{Z}_{N}^{\mathsf{PL}}(0)}.$

Theorem 1.1 claims that this quenched probability decays as $N\to\infty$
followed by $k\to\infty$. Lemma 5.1 takes $k=\lfloor M\sqrt{N}\rfloor$. For
notational convenience, we assume all the multiples of $\sqrt{N}$ appearing in
the proofs in this section are integers. The general case follows verbatim by
considering the floor function.

######  Lemma 5.1.

Fix $\varepsilon>0$ and recall $\mathpzc{Z}_{N}^{\mathsf{PL}}(\cdot)$ from
Theorem 2.7. There exist constants $M(\varepsilon)>0,N_{1}(\varepsilon)>0$
such that for all $N\geq N_{1}$,

$\displaystyle\mathbb{P}\left(\frac{\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})}{\mathpzc{Z}_{N}^{\mathsf{PL}}(1)}\leq
e^{-\sqrt{N}}\right)\geq 1-\tfrac{1}{2}\varepsilon.$ (5.2)

###### Proof.

Fix $\varepsilon\in(0,1)$. Recall $\sigma$ from (2.8) Taking $g=1$ and
$g=M\sqrt{N}$ in Theorem 2.7 yields

$\displaystyle\frac{1}{\sigma\sqrt{N}}\left[\log\mathpzc{Z}_{N}^{\mathsf{PL}}(1)-RN\right]\stackrel{{\scriptstyle
d}}{{\to}}\mathcal{N}\big{(}0,1\big{)},$
$\displaystyle\frac{1}{\sigma\sqrt{N}}\left[\log\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})-RN+M\tau\sqrt{N}\right]\stackrel{{\scriptstyle
d}}{{\to}}\mathcal{N}\big{(}0,1\big{)}$ (5.3)

respectively, where $R,\sigma,\tau$ are defined in (2.8). Let us set
$P:=P(\varepsilon)=\Phi^{-1}(1-\tfrac{\varepsilon}{8})+1$, where $\Phi(\cdot)$
is the cumulative distribution function of $\mathcal{N}(0,1)$. For all large
enough $N$ we have

$\displaystyle\mathbb{P}\left(\log\mathpzc{Z}_{N}^{\mathsf{PL}}(1)\geq
RN-P\sigma\sqrt{N}\right)\geq 1-\tfrac{\varepsilon}{4},$
$\displaystyle\mathbb{P}\left(\log\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})\leq
RN-M\tau\sqrt{N}+P\sigma\sqrt{N}\right)\geq 1-\tfrac{\varepsilon}{4}.$

Applying a union bound gives us

$\displaystyle\mathbb{P}\left(\log\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})+(M\tau-2P\sigma)\sqrt{N}\leq\log\mathpzc{Z}_{N}^{\mathsf{PL}}(1)\right)\geq
1-\tfrac{\varepsilon}{2},$

for all large enough $N$. Taking $M:=\frac{1}{\tau}(2P\sigma+1)$ in above
equation leads to (5.2). This completes the proof. ∎

Recall our discussion in Section 1.2.3 and Figure 5. We refer to the region
$\llbracket N-M\sqrt{N},N-k\rrbracket$ and the region $\llbracket
1,N-M\sqrt{N}\rrbracket$ as the shallow tail and deep tail regions
respectively (see Figure 5). Lemma 5.1 implies that with high probability the
quenched probability of $\pi(2N-2)$ living in the deep tail region is
exponentially small. Thus the mass accumulates in the window of $M\sqrt{N}$
below the point $(N,N)$. To establish Theorem 1.1, we thus have to show the
mass in the shallow tail also goes to zero. For convenience, in our proofs
below we shall often refer to the point $(N+M\sqrt{N},N-M\sqrt{N})$ as the
deep tail starting point. Given the connection in (2.4), the deep tail
starting point corresponds to $(2M\sqrt{N}+1)$-th point for the top curve
$H_{N}^{(1)}(\cdot)$ of the $\mathpzc{HSLG}$ line ensemble. So, in the
coordinates of the $\mathpzc{HSLG}$ line ensemble, we shall refer to
$2M\sqrt{N}+1$ as the deep tail starting point.

Below, we record another important preparatory lemma which claims the
existence of a “high point” in $H_{N}^{(1)}(\cdot)$ not far after the deep
tail starting point (see Figure 14).

######  Lemma 5.2.

Fix any $\varepsilon>0$ and recall $R,\tau$ from (2.8). There exists a
constant $M_{0}(\varepsilon)>0$ such that for all $M\geq M_{0}$, there exists
$N_{0}(\varepsilon,M)$ such that for all $N\geq N_{0},$

$\displaystyle\mathbb{P}\left(\sup_{p\in\llbracket
M\sqrt{N},2M\sqrt{N}\rrbracket}H_{N}^{(1)}(2p+1)\geq
RN-\tfrac{5}{2}M\tau\sqrt{N}\right)\geq 1-\tfrac{1}{2}\varepsilon,$ (5.4)

where $\tau:=\Psi(\theta-\alpha)-\Psi(\theta+\alpha).$

###### Proof.

Let us set $P:=P(\varepsilon)=\Phi^{-1}(1-\tfrac{\varepsilon}{6})+1$, where
$\Phi(\cdot)$ is the cumulative distribution function of $\mathcal{N}(0,1)$.
In view of (5.3), for all large enough $N$ we have

$\displaystyle\mathbb{P}\left(\log\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})\geq
RN-M\tau\sqrt{N}-P\sigma\sqrt{N}\right)\geq 1-\tfrac{\varepsilon}{6},$ (5.5)
$\displaystyle\mathbb{P}\left(\log\mathpzc{Z}_{N}^{\mathsf{PL}}(2M\sqrt{N})\leq
RN-2M\tau\sqrt{N}+P\sigma\sqrt{N}\right)\geq 1-\tfrac{\varepsilon}{6}.$

Applying a union bound gives us

$\displaystyle\mathbb{P}\left(\log\mathpzc{Z}_{N}^{\mathsf{PL}}(2M\sqrt{N})+(M\tau-2P\sigma)\sqrt{N}\leq\log\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})\right)\geq
1-\tfrac{\varepsilon}{3}.$

Thus for any $M\geq\frac{2P\sigma+1}{\tau}$, we have that with probability at
least $1-\tfrac{\varepsilon}{3}$,
$\log\mathpzc{Z}_{N}^{\mathsf{PL}}(2M\sqrt{N})\leq\log\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})-\sqrt{N}$,
which implies

$2\mathpzc{Z}_{N}^{\mathsf{PL}}(2M\sqrt{N})\leq\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N}).$

However, by definition of $\mathpzc{Z}_{N}^{\mathsf{PL}}(\cdot)$, the above
display implies that with probability at least $1-\tfrac{\varepsilon}{3}$,

$\displaystyle\sup_{p\in\llbracket
M\sqrt{N},2M\sqrt{N}\rrbracket}H_{N}^{(1)}(2p+1)$
$\displaystyle\geq\log\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})-\log\mathpzc{Z}_{N}^{\mathsf{PL}}(2M\sqrt{N})-\log(2M\sqrt{N})$
$\displaystyle\geq\log\mathpzc{Z}_{N}^{\mathsf{PL}}(2M\sqrt{N})-\log(2M\sqrt{N}).$

Note that by the first entry in (5.5) with $M$ substituted by $2M$, with
probability at least $1-\frac{\varepsilon}{6}$, we have
$\log\mathpzc{Z}_{N}^{\mathsf{PL}}(2M\sqrt{N})\geq
RN-2M\tau\sqrt{N}-P\sigma\sqrt{N}$. Since for all large enough $N$, we have
$RN-(2M\tau+P\sigma)\sqrt{N}-\log(2M\sqrt{N})\geq
RN-\tfrac{5}{2}M\tau\sqrt{N}$. Thus applying another union bound helps us
arrive at (5.4) and complete the proof. ∎

###  5.2. Proof of Theorems 1.1, 1.3, 1.4, and 1.7

In this section, we prove our main theorems assuming a technical proposition,
described as follows. Fix any $M,N\geq 1$ and assume
$M\sqrt{N}\in\mathbb{Z}_{>0}$. For any Borel set $A$ of
$\mathbb{R}^{M\sqrt{N}}$ we consider the event

$\displaystyle\mathsf{A}=\left\\{(H_{N}^{(1)}(1)-H_{N}^{(1)}(2r+1))_{r=1}^{M\sqrt{N}}\in
A\right\\}.$ (5.6)

for $N>M^{2}+1$. Let $(S_{r})_{r=0}^{M\sqrt{N}}$ be the log-gamma random walk
defined in Definition 1.2. We write

$\displaystyle\mathbb{P}_{RW}(\mathsf{A}):=\mathbb{P}\left((S_{r})_{r=1}^{M\sqrt{N}}\in
A\right)$ (5.7)

The technical proposition below is the main crux of our proofs and it claims
that $\mathbb{P}$ and $\mathbb{P}_{RW}$ are close to each other when $N$ is
large. We postpone its proof to Section 5.3.

######  Proposition 5.3.

Fix any $\varepsilon\in(0,\frac{1}{2})$. Set
$M(\varepsilon)>0,N_{1}(\varepsilon)>0$ such that Lemma 5.1 and Lemma 5.2 hold
simultaneously for all $N\geq N_{1}$ for this fixed choice of $M$. Then there
exists $N_{0}(\varepsilon)>0$ such that for all $N\geq N_{0}$,

$\displaystyle|\mathbb{P}(\mathsf{A})-\mathbb{P}_{RW}(\mathsf{A})|\leq
9\varepsilon,$ (5.8)

where $\mathsf{A}$ and $\mathbb{P}_{RW}(\mathsf{A})$ are defined in (5.6) and
(5.7).

In lieu of these results, we are ready to prove our main theorems. Theorems
1.3 and 1.1 are direct applications of the supporting lemmas. For convenience,
we shall assume in the proofs below $M\sqrt{N}$ is an integer. The general
case follows verbatim by considering floor functions.

###### Proof of Theorem 1.7.

Given (1.8), it suffices to check that

$\displaystyle\tfrac{1}{\sqrt{N}}\left(\log Z(N,N)-\log
Z(N+a_{N},N-a_{N})\right)\stackrel{{\scriptstyle p}}{{\to}}0,$

where $\\{a_{N}\\}_{N\geq 1}$ is a sequence of nonnegative integers less than
$N$, with $a_{N}/\sqrt{N}\to 0$. In light of (2.4), it boils down to checking

$\displaystyle\tfrac{1}{\sqrt{N}}\left(H_{N}^{(1)}(1)-H_{N}^{(1)}(2a_{N}+1)\right)\stackrel{{\scriptstyle
p}}{{\to}}0.$

But thanks to Proposition 5.3, it is equivalent to argue that
$S_{a_{N}}/{\sqrt{N}}\stackrel{{\scriptstyle p}}{{\to}}0$ where
$(S_{r})_{r\geq 0}$ is the log-gamma random walk defined in Definition 1.2.
Since the increment of the walk has the finite first moment and
$a_{N}/\sqrt{N}\to 0$, by Markov inequality we deduce that
$S_{a_{N}}/{\sqrt{N}}\stackrel{{\scriptstyle p}}{{\to}}0$. This establishes
Theorem 1.7. ∎

###### Proof of Theorem 1.3.

Take the set $A$ as
$(-\infty,x_{1}]\times(-\infty,x_{2}]\times\cdots\times(-\infty,x_{k}]\times\mathbb{R}^{M\sqrt{N}-k}$
in (5.6). By Proposition 5.3,

$\displaystyle\limsup_{N\to\infty}\left|\mathbb{P}\left(\bigcap_{r=1}^{k}\\{H_{N}^{(1)}(1)-H_{N}^{(1)}(2r+1)\in(-\infty,x_{r}]\\}\right)-\mathbb{P}_{RW}\left(\bigcap_{r=1}^{k}\\{S_{r}\in(-\infty,x_{r}]\\}\right)\right|\leq
9\varepsilon,$

where $(S_{r})_{r=0}^{k}$ is defined in Definition (1.2). As $\varepsilon$ is
arbitrary, this implies

$\displaystyle\left(H_{N}^{(1)}(1)-H_{N}^{(1)}(2r+1)\right)_{r=0}^{k}\stackrel{{\scriptstyle
d}}{{\to}}(S_{r})_{r=0}^{k}.$

In conjunction with the relation (2.4), we get the desired convergence in
Theorem 1.3. ∎

###### Proof of Theorem 1.1.

Fix any $\varepsilon>0$. Get $M(\varepsilon),N_{1}(\varepsilon)>0$ such that
Lemma 5.1 and Lemma 5.2 hold simultaneously for all $N\geq N_{1}$ for this
fixed choice of $M$. Using this $M$ we split the probability as follows

$\displaystyle\mathbb{P}^{W}(\pi(2N-2)\leq N-k)$ $\displaystyle\hskip
28.45274pt=\mathbb{P}^{W}\big{(}\pi(2N-2)\in(N-M\sqrt{N},N-k]\big{)}+\mathbb{P}^{W}\big{(}\pi(2N-2)\leq
N-M\sqrt{N}\big{)}.$

For the first term observe that by (5.1)

$\displaystyle\mathbb{P}^{W}\big{(}\pi(2N-2)\in(N-M\sqrt{N},N-k]\big{)}$
$\displaystyle=\frac{\sum_{p=k}^{M\sqrt{N}-1}e^{H_{N}^{(1)}(2p+1)}}{\sum_{p=0}^{N-1}e^{H_{N}^{(1)}(2p+1)}}$
$\displaystyle\leq\frac{\sum_{p=k}^{M\sqrt{N}}e^{H_{N}^{(1)}(2p+1)}}{\sum_{p=1}^{M\sqrt{N}}e^{H_{N}^{(1)}(2p+1)}}=\frac{\sum_{p=k}^{M\sqrt{N}}e^{H_{N}^{(1)}(2p+1)-H_{N}^{(1)}(1)}}{\sum_{p=1}^{M\sqrt{N}}e^{H_{N}^{(1)}(2p+1)-H_{N}^{(1)}(1)}}.$

Fix any $\delta>0$ and consider the set

$\displaystyle\mathsf{A}_{\delta}:=\left\\{\frac{\sum_{p=K}^{M\sqrt{N}}e^{H_{N}^{(1)}(2p+1)-H_{N}^{(1)}(1)}}{\sum_{p=1}^{M\sqrt{N}}e^{H_{N}^{(1)}(2p+1)-H_{N}^{(1)}(1)}}\geq\delta\right\\}.$

By Proposition 5.3,
$\mathbb{P}(\mathsf{A}_{\delta})\leq\mathbb{P}_{RW}(\mathsf{A}_{\delta})+9\varepsilon$
for all large enough $N$. On the other hand, by Corollary A.3 we see that
$\lim_{k\to\infty}\lim_{N\to\infty}\mathbb{P}_{RW}(\mathsf{A}_{\delta})=0$.
Thus, as $\varepsilon$ is arbitrary,

$\displaystyle\lim_{k\to\infty}\lim_{N\to\infty}\mathbb{P}^{W}\big{(}\pi(2N-2)\in(N-M\sqrt{N},N-k]\big{)}=0,\mbox{
in probability}.$ (5.9)

For the second term by Lemma 5.1, we see that with probability
$1-\frac{\varepsilon}{2}$

$\displaystyle\mathbb{P}^{W}\big{(}\pi(2N-2)\leq
N-M\sqrt{N}\big{)}\leq\frac{\sum_{p=M\sqrt{N}}^{N-1}e^{H_{N}^{(1)}(2p+1)}}{\sum_{p=1}^{N-1}e^{H_{N}^{(1)}(2p+1)}}=\frac{\mathpzc{Z}_{N}^{\mathsf{PL}}(M\sqrt{N})}{\mathpzc{Z}_{N}^{\mathsf{PL}}(1)}\leq
e^{-\sqrt{N}}.$

Again, as $\varepsilon$ is arbitrary, we have that as $N\to\infty$,
$\mathbb{P}^{W}(\pi(2N-2)\leq N-M\sqrt{N})\to 0$ in probability. This
completes the proof together with (5.9). ∎

Lastly, with Theorems 1.1 and 1.3 established, we present the proof of the
limiting quenched distribution of the endpoint viewed from around the
diagonal.

###### Proof of Theorem 1.4.

Fixed $\theta>0$ and $\alpha\in(-\theta,0).$ Recall from (5.1) that

$\displaystyle\mathbb{P}^{W}(\pi(2N-2)=N-r)$
$\displaystyle=\frac{e^{H_{N}^{(1)}(2r+1)}}{\sum_{p=0}^{N-1}e^{H_{N}^{(1)}(2p+1)}}=\frac{e^{H_{N}^{(1)}(2r+1)-H_{N}^{(1)}(1)}}{\sum_{p=0}^{N-1}e^{H_{N}^{(1)}(2p+1)-H_{N}^{(1)}(1)}}$
(5.10)

where the second equality is derived through (2.4). Note that by Theorem 1.3,
a continuous mapping theorem immediately implies that for a positive integer
$k<\infty$,

$\displaystyle\left(\frac{\exp(H_{N}^{(1)}(2r+1)-H_{N}^{(1)}(1))}{\sum_{p=0}^{k}\exp(H_{N}^{(1)}(2p+1)-H_{N}^{(1)}(1))}\right)_{r\in\llbracket
0,k\rrbracket}\stackrel{{\scriptstyle
d}}{{\to}}\left(\frac{e^{-S_{r}}}{\sum_{p=0}^{k}e^{-S_{p}}}\right)_{r\in\llbracket
0,k\rrbracket}$ (5.11)

Here $(S_{i})_{i\geq 0}$ denotes a log-gamma random walk. For simplicity, we
denote

$\Lambda_{N}(p):=\exp(H_{N}^{(1)}(2p+1)-H_{N}^{(1)}(1)).$

We can then rewrite (5.10) as

$\displaystyle\mathbb{P}^{W}(\pi(2N-2)=N-r)=\frac{\Lambda_{N}(r)}{\sum_{p=0}^{N-1}\Lambda_{N}(p)}=\frac{\sum_{p=0}^{k}\Lambda_{N}(p)}{\sum_{p=0}^{N-1}\Lambda_{N}(p)}\cdot\frac{\sum_{p=0}^{\infty}e^{-S_{p}}}{\sum_{p=0}^{k}e^{-S_{p}}}\cdot\frac{\sum_{p=0}^{k}e^{-S_{p}}}{\sum_{p=0}^{\infty}e^{-S_{p}}}\cdot\frac{\Lambda_{N}(r)}{\sum_{p=0}^{k}\Lambda_{N}(p)}.$

Theorem 1.1 ensures that

$\frac{\sum_{p=0}^{k}\Lambda_{N}(p)}{\sum_{p=0}^{N-1}\Lambda_{N}(p)}=\mathbb{P}^{W}(\pi(2N-2)\geq
N-k)=1-\mathbb{P}^{W}(\pi(2N-2)<N-k)\stackrel{{\scriptstyle p}}{{\to}}1$

as $N\rightarrow\infty$ followed by $k\rightarrow\infty.$ By Lemma A.2 we have

$\lim_{k\to\infty}\frac{\sum_{p=0}^{\infty}e^{-S_{p}}}{\sum_{p=0}^{k}e^{-S_{p}}}\stackrel{{\scriptstyle
p}}{{\to}}1.$

Meanwhile, (5.11) yields that as $N\to\infty,$

$\left(\frac{\sum_{p=0}^{k}e^{-S_{p}}}{\sum_{p=0}^{\infty}e^{-S_{p}}}\cdot\frac{\Lambda_{N}(r)}{\sum_{p=0}^{k}\Lambda_{N}(p)}\right)_{r\in\llbracket
0,k\rrbracket}\stackrel{{\scriptstyle
d}}{{\to}}\left(\frac{\sum_{p=0}^{k}e^{-S_{p}}}{\sum_{p=0}^{\infty}e^{-S_{p}}}\cdot\frac{\Lambda(r)}{\sum_{p=0}^{k}e^{-S_{p}}}\right)_{r\in\llbracket
0,k\rrbracket}=\left(\frac{e^{-S_{r}}}{\sum_{p=o}^{\infty}e^{-S_{p}}}\right)_{r\in\llbracket
0,k\rrbracket}.$

Thus we establish (1.7) and complete the proof of Theorem 1.4. ∎

###  5.3. Proof of Proposition 5.3

For clarity, we divide the proof into several steps.

Step 1. In this step we sketch out the main ideas behind the proof. At this
point, we encourage the readers to consult with Figure 14. Recall the event
$\mathsf{A}$ defined in (5.6).

$y=RN-(3M\tau+1)\sqrt{N}$$H_{N}^{(2)}(\cdot)$$H_{N}^{(1)}(\cdot)$$y=RN-3M\tau\sqrt{N}$$2M\sqrt{N}+1$$4M\sqrt{N}+1$Deep
tail starting pointHigh point Figure 14. Illustration of the proof of
Proposition 5.3. As claimed by Lemma 5.2, there exists a high point $2p^{*}+1$
in $\llbracket 2M\sqrt{N}+1,4M\sqrt{N}+1\rrbracket$ such that
$H_{N}^{(1)}(2p^{*}+1)$ lies above $RN-\frac{5}{2}M\tau\sqrt{N}$ with high
probability. This high point is illustrated in blue in the figure and helps us
show that $H_{N}^{(1)}(\cdot)\geq RN-3M\tau\sqrt{N}$ between $\llbracket
1,2p^{*}+1\rrbracket.$ Meanwhile, invoking Proposition 4.2, we can ensure the
second curve stays below the benchmark of $RN-(3M\tau+1)\sqrt{N}$ on the
interval $\llbracket 1,4M\sqrt{N}+1\rrbracket$ with high probability. Thus
there is a $\sqrt{N}$ separation (with high probability) between the two
curves. By the Gibbs property, this separation ensures that the top curve is
close to a log-gamma random walk.

  * •

We take $M$ and $N_{1}$ as described in the statement of the Proposition 5.3.
In the language introduced in Figure 5 and the text before Lemma 5.2,
$2M\sqrt{N}+1$ is the deep tail starting point. Assuming Lemma 5.2, we have a
point $2p^{*}+1\in\llbracket 2M\sqrt{N}+1,4M\sqrt{N}+1\rrbracket$ where
$H_{N}^{(1)}(2p^{*}+1)$ is ‘high’ enough (see Figure 14). This high point
event is denoted as event $\mathsf{B}$ in Step 2 and has a probability of at
least $1-\tfrac{1}{2}\varepsilon$.

  * •

Invoking Proposition 4.2 with $M_{1}=2M$ and $M_{2}=3M\tau+1$, with high
probability the second curve of the line ensemble is lower than a certain
benchmark, i.e.

$\sup_{p\in\llbracket 1,4M\sqrt{N}+1\rrbracket}H_{N}^{(2)}(p)\leq
RN-(3M\tau+1)\sqrt{N}$

with probability at least $1-\frac{\varepsilon}{2}$. We denote this phenomenon
as the $\mathsf{Fluc}$ event. As $\mathsf{B}$ and $\mathsf{Fluc}$ are high
probability events, it suffices to show that
$|\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{Fluc})-\mathbb{P}_{RW}(\mathsf{A})|$
is small to prove (5.8). This result is achieved by considering the measure
conditioned on the entire second curve and the first curve beyond $2p^{*}+1$.
We remark that this is only a formal description of the proof and refer to the
last bullet point for details.

  * •

To elaborate on the above idea, by the Gibbs property in Theorem 2.2, we
introduce an explicity Radon-Nikodym derivative $W_{p^{*}}$ for the
conditional measure in Step 3. Informally, the conditional measure is
absolutely continuous w.r.t. a log-gamma random walk $(S_{k})_{k\geq 0}$ from
Definition 1.2 starting at $H_{N}^{(1)}(2p^{*}+1)$. As the free law is
precisely the limiting law we are interested in, it suffices to prove
$W_{p^{*}}$ on $[1,2p^{*}+1]$ is approximately $1$.

  * •

$W_{p^{*}}$ is close to $1$ whenever there is a wide enough separation between
the two curves. The diffusive nature of the random walk (with positive drift)
prevents the walk from dipping too low under the free law. Thus under
$\mathsf{B}\cap\mathsf{Fluc}$ we have a uniform separation of $\sqrt{N}$
between the top two curves between $\llbracket 1,2p^{*}+1\llbracket$. and
deduce that $W_{p^{*}}\approx 1$ for large $N$. This is the idea of Step 5 and
concludes that the law of the $H_{N}^{(1)}(\cdot)$ is close to the free law of
a log-gamma random walk starting at $H_{N}^{(1)}(2p^{*}+1)$.

  * •

One issue in carrying out the arguments in the last two bullet points is that
$p^{*}$ is random. Thus the Gibbs property formulated for fixed boundary
points cannot be directly applied at $p^{*}$. This issue is circumvented by a
graining argument where we denote $\mathsf{B}$ as
$\mathsf{B}=\bigsqcup\mathsf{B_{i}}$ for a disjoint collection of events
$\mathsf{B}_{i}\subset\\{H_{N}^{(1)}(2i+1)\geq RN-\tfrac{5}{2}M\sqrt{N}\\}$
defined in Step 2 and then apply the usual Gibbs property for each $i$.

Step 2. Take $M_{1}=2M$ and $M_{2}=3M\tau+1$ in Proposition 4.2. Taking
$N_{2}(\varepsilon,M_{1},M_{2})>0$ (which depends only on $\varepsilon$ as
$M_{1},M_{2}$ depends only on $\varepsilon$) from Proposition 4.2, we see that

$\displaystyle\mathbb{P}(\mathsf{Fluc})\geq 1-\tfrac{\varepsilon}{2},\mbox{
where }\mathsf{Fluc}:=\left\\{\sup_{p\in\llbracket
1,4M\sqrt{N}+1\rrbracket}H_{N}^{(2)}(p)\leq RN-(3M\tau+1)\sqrt{N}\right\\}$
(5.12)

for all $N\geq N_{2}$. Next we consider the events

$\displaystyle\mathsf{G}_{i}:=\big{\\{}H_{N}^{(1)}(2i+1)\geq
RN-\frac{5}{2}M\tau\sqrt{N}\big{\\}}\mbox{ and
}\mathsf{B}_{i}:=\bigcap_{j=i+1}^{2M\sqrt{N}}\mathsf{G}_{j}^{c}\cap\mathsf{G}_{i}.$

Note that $(\mathsf{B}_{i})_{i\in\llbracket M\sqrt{N},2M\sqrt{N}\rrbracket}$
forms a disjoint collection of events. Define

$\displaystyle\mathsf{B}$ $\displaystyle:=\bigsqcup_{i\in\llbracket
M\sqrt{N},2M\sqrt{N}\rrbracket}\mathsf{B}_{i}$
$\displaystyle=\bigcup_{i\in\llbracket
M\sqrt{N},2M\sqrt{N}\rrbracket}\mathsf{G}_{i}=\left\\{\sup_{p\in\llbracket
M\sqrt{N},2M\sqrt{N}\rrbracket}H_{N}^{(1)}(2p+1)\geq
RN-\tfrac{5}{2}M\tau\sqrt{N}\right\\},$

where we write $\sqcup$ to stress that the events are disjoint in the union.
In particular, as Lemma 5.2 holds, we have $\mathbb{P}(\mathsf{B})\geq
1-\tfrac{1}{2}\varepsilon.$ Thus for all $N\geq N_{1}+N_{2}$, by a union bound
we have

$\displaystyle|\mathbb{P}(\mathsf{A})-\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{Fluc})|\leq\mathbb{P}(\neg\mathsf{B})+\mathbb{P}(\neg\mathsf{Fluc})\leq\varepsilon.$

Hence to prove (5.8) it suffices to show

$\displaystyle|\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{Fluc})-\mathbb{P}_{RW}(\mathsf{A})|\leq
8\varepsilon.$ (5.13)

Define $\mathcal{F}_{i}$ as the $\sigma$-field
$\sigma\big{(}{H_{N}^{(1)}(x)}_{x\geq 2i+1},{H_{N}^{(j)}}(x)_{j\geq 2,x\geq
1}\big{)}.$ Note that $\mathsf{B}_{i},\mathsf{Fluc}$ are both measurable
w.r.t. $\mathcal{F}_{i}.$ Exploiting the fact that $B_{i}$’s are disjoint
yields

$\displaystyle\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{Fluc})=\sum_{i=M\sqrt{N}}^{2M\sqrt{N}}\mathbb{E}\left[\mathbf{1}_{\mathsf{B}_{i}\cap\mathsf{Fluc}}\mathbb{E}\left[\mathbf{1}_{\mathsf{A}}\mid\mathcal{F}_{i}\right]\right]$
(5.14)

where the last equality is due to the tower property of the conditional
expectation. Thus we are left to estimate
$\mathbb{E}\left[\mathbf{1}_{\mathsf{A}}\mid\mathcal{F}_{i}\right]$ for each
$i$.

Step 3. Gibbs law. To analyze
$\mathbb{E}\left[\mathbf{1}_{\mathsf{A}}\mid\mathcal{F}_{i}\right]$, we invoke
the Gibbs property (Theorem 2.2) for the $\mathpzc{HSLG}$ line ensemble. By
Theorem 2.2, the distribution of $(H_{N}^{(1)}(j))_{j=1}^{2i}$ conditioned on
$\mathcal{F}_{i}$ has a density at $(u_{j})_{j=1}^{2i}$

$\displaystyle\exp\left(-\sum_{j=1}^{i}\left[e^{H_{N}^{(2)}(2j)-u_{2j+1}}+e^{H_{N}^{(2)}(2j)-u_{2j-1}}\right]\right)$
(5.15) $\displaystyle\hskip
56.9055pt\cdot\prod_{j=1}^{i}\exp\left((\theta+\alpha)(u_{2j+1}-u_{2j})-e^{u_{2j+1}-u_{2j}}\right)$
(5.16) $\displaystyle\hskip
113.81102pt\cdot\prod_{j=1}^{i}\exp\left((\theta-\alpha)(u_{2j-1}-u_{2j})-e^{u_{2j-1}-u_{2j}}\right)$
(5.17)

with $u_{2i+1}=H_{N}^{(1)}(2i+1)$. The above explicit expression is obtained
from (2.6) and (2.5). Note that the terms in (5.15), (5.16), and (5.17)
correspond to weights of black, red, and blue edges in the graphical
representation (see left figure of Figure 15) respectively.

Based on the above decomposition, we define a free law
$\mathbb{P}_{\operatorname{free},i}$ that depends only on
$H_{N}^{(1)}{(2i+1)}$. We define that under the law
$\mathbb{P}_{\operatorname{free},i}$, the distribution of
$(H_{N}^{(1)}(j))_{j=1}^{2i}$ has a density at $(u_{j})_{j=1}^{2i}$
proportional to

$\displaystyle\prod_{j=1}^{i}\exp\left((\theta+\alpha)(u_{2j+1}-u_{2j})-e^{u_{2j+1}-u_{2j}}\right)\cdot\prod_{j=1}^{i}\exp\left((\theta-\alpha)(u_{2j-1}-u_{2j})-e^{u_{2j-1}-u_{2j}}\right)$

with $u_{2i+1}=H_{N}^{(1)}(2i+1)$. Note that free law collects all the blue
and red edge weights only. A quick comparison of the above formula with (1.5)
shows that under the free law, $(H_{N}^{(1)}(1)-H_{N}^{(1)}(2r+1))_{r=0}^{i}$
is precisely distributed as log-gamma random walk defined in Definition 1.2.

In order to obtain the original conditional distribution from the free law, we
may introduce the black weights as a Radon-Nikodym derivative (see the
decomposition in Figure 15). Indeed, we have

$\displaystyle\mathbb{E}\left[\mathbf{1}_{\mathsf{A}}\mid\mathcal{F}_{i}\right]=\frac{\mathbb{E}_{\operatorname{free},i}[W_{i}\mathbf{1}_{\mathsf{A}}]}{\mathbb{E}_{\operatorname{free},i}[W_{i}]}$
(5.18)

where

$\displaystyle
W_{i}:=\exp\left(-\sum_{j=1}^{i}\left[e^{H_{N}^{(2)}(2j)-H_{N}^{(1)}(2j+1)}+e^{H_{N}^{(2)}(2j)-H_{N}^{(1)}(2j-1)}\right]\right)$
(5.19)

$=$$\times$$a$$a$$a$$z_{1}$$z_{2}$$z_{3}$$z_{1}$$z_{2}$$z_{3}$$z_{1}$$z_{2}$$z_{3}$
Figure 15. Gibbs decomposition. The left figure shows the gibbs measure
corresponding to conditioned on $\mathcal{F}_{i}$ with $i=3$. Here
$a=H_{N}^{(1)}(2i+1)$, and $z_{j}:=H_{N}^{(2)}(2j)$ for $j\in\llbracket
1,i\rrbracket$. The measure has been decomposed into two parts. The free law
(middle) and a Radon-Nikodym derivative (right).

We notice that $W_{i}$ has a trivial upper bound: $W_{i}\leq 1.$ For the lower
bound, we claim that there exists $N_{0}(\varepsilon)>0$ such that for all
$N\geq N_{0}$ we have

$\displaystyle\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\mathbb{P}_{\operatorname{free},i}(W_{i}\geq
1-\varepsilon)\geq\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\cdot(1-\varepsilon).$
(5.20)

Thus, (5.20) implies that $W_{i}$ is close to 1 with high probability under
$\mathsf{Fluc}\cap\mathsf{B}_{i}$. Thus, going back to (5.18), we expect
$\mathbb{E}\left[\mathbf{1}_{\mathsf{A}}\mid\mathcal{F}_{i}\right]$ to be
close to $\mathbb{P}_{\operatorname{free},i}(\mathsf{A})$. As under the free
law
$\mathbb{P}_{\operatorname{free},i}(\mathsf{A})=\mathbb{P}_{RW}(\mathsf{A})$,
for all $i\in\llbracket M\sqrt{N},2M\sqrt{N}\rrbracket$, (5.14) eventually
leads to (5.13), which we make precise in the next step.

Step 4. Assuming (5.20), we complete the proof of (5.13) in this step. As
$W_{i}\leq 1,$ we have

$\displaystyle\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\frac{\mathbb{E}_{\operatorname{free},i}[W_{i}\mathbf{1}_{\mathsf{A}}]}{\mathbb{E}_{\operatorname{free},i}[W]}$
$\displaystyle\geq\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\mathbb{E}_{\operatorname{free},i}[W_{i}\mathbf{1}_{\mathsf{A}}]\geq(1-\varepsilon)\cdot\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\mathbb{P}_{\operatorname{free},i}(\mathsf{A}\cap\\{W\geq
1-\varepsilon\\})$
$\displaystyle\geq(1-\varepsilon)\cdot\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\left[\mathbb{P}_{\operatorname{free},i}(\mathsf{A})-\mathbb{P}_{\operatorname{free},i}(W_{i}<1-\varepsilon)\right]$
$\displaystyle\geq(1-\varepsilon)\cdot\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\left[\mathbb{P}_{\operatorname{free},i}(\mathsf{A})-\varepsilon\right]$

where we use (5.20) in the last inequality. Recall
$\mathbb{P}_{\operatorname{free},i}(\mathsf{A})=\mathbb{P}_{RW}(\mathsf{A})$.
Inserting this bound in (5.18) and then going back to (5.14) yields

$\displaystyle\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{Fluc})$
$\displaystyle\geq(1-\varepsilon)\cdot\left[\mathbb{P}_{RW}(\mathsf{A})-\varepsilon\right]\sum_{i=M\sqrt{N}}^{2M\sqrt{N}}\mathbb{P}(\mathsf{B}_{i}\cap\mathsf{Fluc})$
$\displaystyle=(1-\varepsilon)\cdot\left[\mathbb{P}_{RW}(\mathsf{A})-\varepsilon\right]\mathbb{P}(\mathsf{B}\cap\mathsf{Fluc})\geq(1-\varepsilon)^{2}\left[\mathbb{P}_{RW}(\mathsf{A})-\varepsilon\right].$

for all large enough $N$. The equality in the above equation follows by
recalling that $\mathsf{B}_{i}$’s form a disjoint collection of events and the
result implies that
$\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{Fluc})-\mathbb{P}_{RW}(\mathsf{A})\geq-3\varepsilon$.
This proves the lower bound inequality in (5.13). Similarly for the upper
bound, as $W_{i}\leq 1$, we have

$\displaystyle\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\cdot\frac{\mathbb{E}_{\operatorname{free},i}[W_{i}\mathbf{1}_{\mathsf{A}}]}{\mathbb{E}_{\operatorname{free},i}[W_{i}]}$
$\displaystyle\leq\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\cdot\frac{\mathbb{P}_{\operatorname{free},i}(\mathsf{A})}{(1-\varepsilon)\mathbb{P}_{\operatorname{free},i}(W_{i}\geq
1-\varepsilon)}\leq\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\cdot\frac{\mathbb{P}_{\operatorname{free},i}(\mathsf{A})}{(1-\varepsilon)^{2}}$

where the last inequality stems from (5.20). Again, Inserting this bound in
(5.18) and then going back to (5.14) gives us

$\displaystyle\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{Fluc})$
$\displaystyle\leq\frac{\mathbb{P}_{RW}(\mathsf{A})}{(1-\varepsilon)^{2}}\sum_{i=M\sqrt{N}}^{2M\sqrt{N}}\mathbb{P}(\mathsf{B}_{i}\cap\mathsf{Fluc})=\frac{\mathbb{P}_{RW}(\mathsf{A})}{(1-\varepsilon)^{2}}\mathbb{P}(\mathsf{B}\cap\mathsf{Fluc})\leq\frac{\mathbb{P}_{RW}(\mathsf{A})}{(1-\varepsilon)^{2}}$

where again the equality comes from the disjointness of $\mathsf{B}_{i}$’s. As
$\varepsilon\leq\frac{1}{2}$, this implies

$\displaystyle\mathbb{P}(\mathsf{A}\cap\mathsf{B}\cap\mathsf{Fluc})-\mathbb{P}_{RW}(\mathsf{A})\leq\frac{1-(1-\varepsilon)^{2}}{(1-\varepsilon)^{2}}\leq
8\varepsilon$

which proves the upper bound in (5.13). The proof of Theorem 1.3 modulo (5.20)
is thus complete.

Step 5. Finally in this step we prove (5.20). We define the event

$\displaystyle\mathsf{Sink}(i):=\left\\{\inf_{p\in\llbracket
0,i\rrbracket}H_{N}^{(1)}(2p+1)\geq RN-3M\tau\sqrt{N}\right\\}.$

We claim that there exists $N_{0}(\varepsilon)>0$ such that for all $N\geq
N_{0}$, we have

$\displaystyle\mathbf{1}_{\mathsf{B}_{i}}\mathbb{P}_{\operatorname{free},i}(\mathsf{Sink}(i))\geq\mathbf{1}_{\mathsf{B}_{i}}(1-\varepsilon),$
(5.21)

for all $i\in\llbracket M\sqrt{N},2M\sqrt{N}\rrbracket$.

Recall that the event $\mathsf{Fluc}$ in (5.12) requires the second curve
$H_{N}^{(2)}(p)$ to lie below certain threshold within the range
$p\in\llbracket 1,4M\sqrt{N}+1\rrbracket$. Recall the definition of $W_{j}$
from (5.19). Note that on $\mathsf{Sink}(j)\cap\mathsf{Fluc}$ we have

$W_{j}\geq\exp(-2je^{-\sqrt{N}})\geq\exp(-4M\sqrt{N}e^{-\sqrt{N}})$

as $j\leq 2M\sqrt{N}$. Note that $\exp(-4M\sqrt{N}e^{-\sqrt{N}})\geq
1-\varepsilon$ for all large enough $N$. Therefore, in view of (5.21) we have

$\displaystyle\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\mathbb{P}_{\operatorname{free},i}(W_{i}\geq
1-\varepsilon)\geq\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\mathbb{P}_{\operatorname{free},i}(\mathsf{Sink}(i))\geq\mathbf{1}_{\mathsf{Fluc}\cap\mathsf{B}_{i}}\cdot(1-\varepsilon)$

for all large enough $N$. This verifies (5.20). We are left to show (5.21).
Towards this end, note that on the event $\mathsf{B}_{i}$, we have
$H_{N}^{(1)}(2i+1)\geq RN-\frac{5}{2}M\tau\sqrt{N}$. Thus,

$\displaystyle\mathbf{1}_{\mathsf{B}_{i}}\mathbb{P}_{\operatorname{free},i}(\mathsf{Sink}(i))\geq\mathbf{1}_{\mathsf{B}_{i}}\mathbb{P}_{\operatorname{free},i}\left(\inf_{x\in\llbracket
0,i\rrbracket}H_{N}^{(1)}(2x+1)-H_{N}^{(1)}(2i+1)\geq-\tfrac{1}{2}M\tau\sqrt{N}\right).$
(5.22)

Recall from our discussion in Step 2 that under the law
$\mathbb{P}_{\operatorname{free},i}$,
$(H_{N}^{(1)}(1)-H_{N}^{(1)}(2r+1))_{r=0}^{i}$ is distributed as a log-gamma
random walk. Let us use $(S_{k})_{k=0}^{i}$ to denote a log-gamma random walk.
We have

$\displaystyle\mathbb{P}_{\operatorname{free},i}\left(\inf_{p\in\llbracket
0,i\rrbracket}H_{N}^{(1)}(2p+1)-H_{N}^{(1)}(2i+1)\geq-\tfrac{1}{2}M\tau\sqrt{N}\right)$
(5.23) $\displaystyle\hskip 142.26378pt=\mathbb{P}\left(\inf_{p\in\llbracket
0,i\rrbracket}(S_{i}-S_{p})\geq-\tfrac{1}{2}M\tau\sqrt{N}\right).$

Note that $(S_{i}-S_{p})_{p\geq 0}^{i}$ is again a time-reversed log-gamma
random walk. As $i\leq 2M\sqrt{N}$, appealing to Lemma A.1 yields that

$\displaystyle\mathbf{1}_{\mathsf{B}_{i}}\mathbb{P}_{\operatorname{free},i}(\mathsf{Sink}(i))\geq\mathbb{P}\left(\inf_{p\in\llbracket
0,i\rrbracket}(S_{i}-S_{p})\geq-\tfrac{1}{2}M\tau\sqrt{N}\right)\geq
1-\frac{8\operatorname{Var}(S_{1})}{M\tau^{2}\sqrt{N}}\geq 1-\varepsilon$

for all large enough $N$ (uniformly over $i\in\llbracket
M\sqrt{N},2M\sqrt{N}\rrbracket$. Inserting this bound in (5.23), in view of
the lower bound in (5.22), leads to (5.21). This completes the proof of
Proposition 5.3.

##  Appendix A Properties of random walks with positive drift

In this section, we collect some useful properties of random walks with
positive drift whose proofs follow by classical analysis. Note that the log-
gamma random walk introduced in Definition 1.2 is a random walk with positive
drift. This is because the density $p(x)$ introduced in (1.5) has mean:

$\displaystyle\int_{\mathbb{R}}xp(x)dx=\Psi(\theta-\alpha)-\Psi(\theta+\alpha),$

which is positive as the digamma function $\Psi$ is strictly increasing
(recall $\alpha<0$).

######  Lemma A.1.

Let $(X_{i})_{i\geq 0}$ be a sequence of iid random variables with
$\mathbb{E}[X_{1}]=\beta>0$ and $\operatorname{Var}[X_{1}]=\gamma<\infty$. Set
$S_{0}=0$ and $S_{k}=\sum_{i=1}^{k}X_{i}$. For all $M,N,\lambda>0$ we have

$\displaystyle\mathbb{P}\left(\inf_{k\in\llbracket
1,M\sqrt{N}\rrbracket}S_{k}\leq-\lambda\right)\leq\frac{M\sqrt{N}\gamma}{\lambda^{2}}.$

###### Proof.

As $\beta>0$, by Kolmogorov’s maximal inequality, we have

$\displaystyle\mathbb{P}\left(\inf_{k\in\llbracket
1,M\sqrt{N}\rrbracket}S_{k}\leq-\lambda\right)$
$\displaystyle=\mathbb{P}\left(\sup_{k\in\llbracket
1,M\sqrt{N}\rrbracket}|S_{k}-k\beta|\geq\lambda\right)\leq\frac{1}{\lambda^{2}}\sum_{i=1}^{M\sqrt{N}}\operatorname{Var}(X_{i})=\frac{M\sqrt{N}\gamma}{\lambda^{2}},$

which is precisely what we want to show. ∎

######  Lemma A.2.

Let $(X_{i})_{i\geq 0}$ be a sequence of iid random variables with
$\mathbb{E}[X_{1}]=\beta>0$ and $\operatorname{Var}[X_{1}]=\gamma<\infty$. Set
$S_{0}=0$ and $S_{n}=\sum_{i=1}^{n}X_{i}$. We have

$\displaystyle\mathbb{P}\left(\sum_{r=0}^{\infty}e^{-S_{r}}<\infty\right)=1$

###### Proof.

By Kolmogorov’s maximal inequality

$\displaystyle\mathbb{P}\big{(}\sup_{1\leq i\leq
n^{2}}|S_{i}-i\beta|\geq\tfrac{n^{2}}{2}\beta\big{)}\leq\frac{4}{n^{4}\beta^{2}}\sum_{i=1}^{n^{2}}\operatorname{Var}(X_{i})=\frac{4\gamma}{n^{2}\beta^{2}}.$

The last bound is summable in $n$. Thus invoking Borel-Cantelli’s lemma we
have that there exists a random $N$ with $P(7\leq N<\infty)=1$ such that

$S_{i}\geq i\beta-(N^{2}/2)\beta\geq-(N^{2}/2)\beta,\mbox{ for all }1\leq
i\leq N^{2},$

and for all $n\geq N+1$ we have

$S_{i}\geq(n-1)^{2}\beta-(n^{2}/2)\beta\geq(n^{2}/4)\beta,\mbox{ for all
}(n-1)^{2}+1\leq i\leq n^{2},$

where above we used the fact that $n\geq N+1\geq 8$. Thus with probability
$1$, we have

$\displaystyle\sum_{r=0}^{\infty}e^{-S_{r}}$
$\displaystyle=\sum_{r=0}^{N^{2}}e^{-S_{r}}+\sum_{n=N+1}^{\infty}\sum_{i=(n-1)^{2}+1}^{n^{2}}e^{-S_{i}}$
$\displaystyle\leq
N^{2}e^{(N^{2}/2)\beta}+\sum_{n=N+1}^{\infty}\sum_{i=(n-1)^{2}+1}^{n^{2}}e^{-(n^{2}/4)\beta}\leq
N^{2}e^{(N^{2}/2)\beta}+\sum_{n=N+1}^{\infty}n^{2}e^{-(n^{2}/4)\beta}<\infty.$

This completes the proof. ∎

As a corollary, we have the following double-limit result.

######  Corollary A.3.

Under the setup of Lemma A.2, almost surely we have

$\displaystyle\lim_{k\to\infty}\lim_{n\to\infty}\frac{\sum_{r=k}^{\infty}e^{-S_{r}}}{\sum_{r=0}^{n}e^{-S_{r}}}=0.$

###### Proof.

Note that $\sum_{r=0}^{n}e^{S_{r}}$ is a monotone sequence in $n$ which
converges to a random variable that is almost surely finite by Lemma A.2.
Thus,

$\displaystyle\frac{\sum_{r=k}^{\infty}e^{S_{r}}}{\sum_{r=0}^{n}e^{S_{r}}}=1-\frac{\sum_{r=0}^{k-1}e^{S_{r}}}{\sum_{r=0}^{n}e^{S_{r}}}\stackrel{{\scriptstyle
n\to\infty}}{{\to}}1-\frac{\sum_{r=0}^{k-1}e^{S_{r}}}{\sum_{r=0}^{\infty}e^{S_{r}}}.$

Taking $k\to\infty$ yields the desired result. ∎

## References

  * [ABO21] Jonas Arista, Elia Bisi, and Neil O’Connell.  Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices.  arXiv preprint arXiv:2112.12558, 2021.
  * [Abr80] Douglas B Abraham.  Solvable model with a roughening transition for a planar Ising ferromagnet.  Physical Review Letters, 44(18):1165, 1980.
  * [Bat18] Erik Bates.  Localization of directed polymers with general reference walk.  Electronic Journal of Probability, 23:1–45, 2018.
  * [BBAP05] Jinho Baik, Gérard Ben Arous, and Sandrine Péché.  Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices.  2005\.
  * [BBC16] Alexei Borodin, Alexey Bufetov, and Ivan Corwin.  Directed random polymers via nested contour integrals.  Annals of Physics, 368:191–247, 2016.
  * [BBC20] Guillaume Barraquand, Alexei Borodin, and Ivan Corwin.  Half-space Macdonald processes.  In Forum of Mathematics, Pi, volume 8. Cambridge University Press, 2020.
  * [BBCS18a] Jinho Baik, Guillaume Barraquand, Ivan Corwin, and Toufic Suidan.  Facilitated exclusion process.  In Computation and Combinatorics in Dynamics, Stochastics and Control: The Abel Symposium, Rosendal, Norway, August 2016, pages 1–35. Springer, 2018.
  * [BBCS18b] Jinho Baik, Guillaume Barraquand, Ivan Corwin, and Toufic Suidan.  Pfaffian Schur processes and last passage percolation in a half-quadrant.  2018\.
  * [BBCW18] Guillaume Barraquand, Alexei Borodin, Ivan Corwin, and Michael Wheeler.  Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process.  2018\.
  * [BBNV18] Dan Betea, Jérémie Bouttier, Peter Nejjar, and Mirjana Vuletić.  The free boundary Schur process and applications i.  In Annales Henri Poincaré, volume 19, pages 3663–3742. Springer, 2018.
  * [BC20] Erik Bates and Sourav Chatterjee.  The endpoint distribution of directed polymers.  Ann. Probab., 48(2):817–871, 2020.
  * [BC23] Guillaume Barraquand and Ivan Corwin.  Stationary measures for the log-gamma polymer and KPZ equation in half-space.  Ann. Probab., 51(5):1830–1869, 2023.
  * [BCD23a] Guillaume Barraquand, Ivan Corwin, and Sayan Das.  KPZ exponents for the half-space log-gamma polymer.  arXiv preprint arXiv:2310.10019, 2023.
  * [BCD23b] Guillaume Barraquand, Ivan Corwin, and Evgeni Dimitrov.  Spatial tightness at the edge of Gibbsian line ensembles.  Communications in Mathematical Physics, pages 1–78, 2023.
  * [BCY23] Guillaume Barraquand, Ivan Corwin, and Zongrui Yang.  Stationary measures for integrable polymers on a strip.  arXiv preprint arXiv:2306.05983, 2023.
  * [BD22] Yuri Bakhtin and Douglas Dow.  Joint localization of directed polymers.  arXiv preprint arXiv:2211.05916, 2022.
  * [BFO20] Dan Betea, Patrik L Ferrari, and Alessandra Occelli.  Stationary half-space last passage percolation.  Communications in Mathematical Physics, 377:421–467, 2020.
  * [BFO22] Dan Betea, PL Ferrari, and Alessandra Occelli.  The half-space Airy stat process.  Stochastic Processes and their Applications, 146:207–263, 2022\.
  * [BHL83] Edouard Brézin, Bertrand I. Halperin, and Stanislas Leibler.  Critical wetting in three dimensions.  Physical Review Letters, 50(18):1387, 1983.
  * [BKLD20] Guillaume Barraquand, Alexandre Krajenbrink, and Pierre Le Doussal.  Half-space stationary Kardar-Parisi-Zhang equation.  Journal of Statistical Physics, 181(4):1149–1203, 2020.
  * [BKLD22] Guillaume Barraquand, Alexandre Krajenbrink, and Pierre Le Doussal.  Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case.  Journal of Physics A: Mathematical and Theoretical, 55(27):275004, 2022.
  * [BLD21] Guillaume Barraquand and Pierre Le Doussal.  Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall.  Physical Review E, 104(2):024502, 2021.
  * [Bol89] Erwin Bolthausen.  A note on the diffusion of directed polymers in a random environment.  Communications in mathematical physics, 123(4):529–534, 1989.
  * [BR01a] Jinho Baik and Eric M. Rains.  Algebraic aspects of increasing subsequences.  Duke Mathematical Journal, 109(1):1 – 65, 2001.
  * [BR01b] Jinho Baik and Eric M Rains.  Symmetrized random permutations.  Random matrix models and their applications, Math. Sci. Res. Inst. Publ., 40:1–19, 2001.
  * [BR01c] Jinho Baik and Eric M. Rains.  The asymptotics of monotone subsequences of involutions.  Duke Mathematical Journal, 109(2):205 – 281, 2001.
  * [BS20] Yuri Bakhtin and Donghyun Seo.  Localization of directed polymers in continuous space.  Electronic Journal of Probability, 25:1–56, 2020.
  * [BW22] Guillaume Barraquand and Shouda Wang.  An Identity in Distribution Between Full-Space and Half-Space Log-Gamma Polymers.  International Mathematics Research Notices, 06 2022.  rnac132.
  * [BZ19] Elia Bisi and Nikos Zygouras.  Point-to-line polymers and orthogonal Whittaker functions.  Transactions of the American Mathematical Society, 371(12):8339–8379, 2019.
  * [CH02] Philippe Carmona and Yueyun Hu.  On the partition function of a directed polymer in a gaussian random environment.  Probability theory and related fields, 124(3):431–457, 2002.
  * [CN16] Francis Comets and Vu-Lan Nguyen.  Localization in log-gamma polymers with boundaries.  Probability Theory and Related Fields, 166:429–461, 2016.
  * [Com17] Francis Comets.  Directed polymers in random environments.  Springer, 2017.
  * [COSZ14] Ivan Corwin, Neil O’Connell, Timo Seppäläinen, and Nikolaos Zygouras.  Tropical combinatorics and Whittaker functions.  Duke Mathematical Journal, 163(3):513–563, 2014.