# Discrete geodesic local time converges under KPZ scaling

Shirshendu Ganguly Department of Statistics, UC Berkeley, Berkeley, CA, USA.
e-mail<EMAIL_ADDRESS>Lingfu Zhang Department of Statistics, UC
Berkeley, Berkeley, CA, USA. e-mail<EMAIL_ADDRESS>
###### Abstract

The directed landscape constructed in [DOV18] produces a directed, planar,
random geometry, and is believed to be the universal scaling limit of two-
dimensional first and last passage percolation models in the Kardar-Parisi-
Zhang (KPZ) universality class. Geodesics in this random geometry form an
important class of random continuous curves exhibiting fluctuation theory
quite different from that of Brownian motion. In this vein, counterpart to
Brownian local time, BLT (a self-similar measure supported on the set of zeros
of Brownian motion), a local time for geodesics, GLT, was recently constructed
and used to study fractal properties of the directed landscape in [GZ22]. It
is a classical fact and can be proven using the Markovian property of Brownian
motion that the uniform discrete measure on the set of zeros of the simple
random walk converges to BLT. In this paper, we prove the “KPZ analog” of this
by showing that the local times for discrete geodesics in pre-limiting
integrable last passage percolation models converge to GLT under suitable
scaling guided by KPZ exponents.

In absence of any Markovianity, our arguments rely on the recently proven
convergence of geodesics in pre-limiting models to that in the directed
landscape [DV21]. However, this input concerns macroscopic properties and is
too coarse to capture the microscopic information required for local time
analysis. To relate the macroscopic and microscopic behavior, a key ingredient
is an _a priori_ smoothness estimate of the local time in the discrete model,
proved relying on geometric ideas such as the coalescence of geodesics as well
as their stability under perturbations of boundary data.

Contents

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##  1 Introduction and main results

The zero set of simple random walks is an object whose study is as old as that
of the random walk itself. Estimates on the random walk hitting zero are
crucial in understanding the properties of random walk bridges and excursions.
The Brownian counterpart gives rise to a canonical fractal subset $Z$ of the
real line analogous to the Cantor set but random. The Hausdorff dimension of
$Z$ is known to be $1/2$ stemming from the fact that the probability a simple
random walk is at $0$ at time $n$ decays as $\frac{1}{\sqrt{n}}.$ There is a
natural measure supported on $Z$ known as the Brownian local time (BLT) which
can be obtained as the density of the occupation measure of Brownian motion
(This is the measure $\mu$ on $\mathbb{R}$ such that
$\mu(A)=\int_{0}^{1}\mathbf{1}(\mathbb{B}_{s}\in A)\mathrm{d}s$) at zero.

While Brownian motion is a canonical one-dimensional random curve, this
article focuses on the local time of another class of random curves arising
from models of random geometry. A paradigmatic example is what is known as
First passage percolation (FPP) where one considers a distortion of the usual
graph metric on $\mathbb{Z}^{2}$ by putting i.i.d. random non-negative weights
on the edges and considering the corresponding random weighted path metric. A
natural class of random curves arising in this situation is formed by the
weight-minimizing paths between points; namely the geodesics in the associated
metric space.

The geodesic behavior, encoding a lot of information about the underlying
geometry, has been an important object of investigation both in the rigorous
and non- rigorous literature going back to the 1960s when the model of FPP was
first proposed by Hammersley and Welsh.

However, despite the interest, the innate hardness of the model has made it
very hard to verify rigorously any of these predictions. Nonetheless, an
important related class of models for which some progress has been made is
known as Last passage percolation (LPP). Ignoring microscopic specifications,
in general terms they all consist of an environment of random noise through
which _directed_ paths travel, accruing the integral of the noise along it — a
quantity known as energy or weight. Given two endpoints, a maximization is
done over the weights of all paths with these endpoints with the optimizing
path still being called a _geodesic_ , despite its maximizing nature as
opposed to minimizing as in FPP.

Remarkably, particular choices of the underlying noise environment make LPP
“integrable”, admitting certain exact algebraic properties such as
distributional identities involving eigenvalues of random matrices, young
diagrams, and so on, which provide key inputs for probabilistic analysis.

Both FPP and LPP models are expected to belong to the Kardar-Parisi-Zhang
(KPZ) universality class which encompasses a broad family of models of one-
dimensional random growth which are believed to exhibit certain common
features, such as local growth driven by white noise and a smoothing effect
from some notion of surface tension and a slope-dependent non-linear growth.
It is predicted that the fluctuation theories of such models are dictated by
universal scaling exponents and limiting distributions, exhibited by solutions
to a stochastic PDE known as the KPZ equation [Cor12, Cor16, HQ18]. For these
$(1+1)$-dimensional models, the resulting evolution of the growth interface
$h(t,x)$ is described by the characteristic triple
$(1,\frac{1}{3},\frac{2}{3})$ of exponents: at time $t$, the value of $h(t,x)$
is of order $t^{1}$, with order $t^{1/3}$ deviations from its mean, and
further, non-trivial correlations are observed when $x$ is varied on the scale
of $t^{2/3}$. Furthermore, $h(t,x)$ on proper centering and scaling according
to these exponents, is expected to converge to a universal limit $t\to\infty$
[QR14, MQR16]. In terms of geodesics in, say, LPP, this corresponds to having
a weight fluctuation exponent $1/3$ and a transversal fluctuation exponent
(deviation from the straight line joining its endpoints) $2/3$.

Another key feature in such random settings is the phenomenon of geodesic
coalescence where geodesics between different points tend to merge together
and share a significant fraction of their journey, passing through the
atypically high values of the underlying noise variables. Note that this is
quite different from standard Euclidean geometry where geodesics are merely
straight lines and hence have trivial intersection patterns.

Owing to such coalescence properties, even though the geodesic is a global
object, with high probability, a local segment of it is likely to depend only
on the nearby noise. In other words, it exhibits a form of weak decay of
correlation property.

In [DOV18] the directed landscape was constructed. This is a random continuum
metric space expected to be the universal scaling limit of a class of FPP and
LPP models. Moreover, in a follow-up work [DV21], a class of exactly solvable
LPP models was indeed shown to converge to the landscape. The latter also
admits geodesics (a non-trivial fact) and it was further shown that the pre-
limiting geodesics converge to the limiting one in a certain topology.

However, the topology of convergence is not refined enough to conclude the
convergence of several observables of interest associated with the geodesics
to their limiting counterparts.

In this article, we focus on the convergence of one such observable, namely
that of the discrete local time, i.e., the amount of time a pre-limiting
discrete geodesic intersects a line. While defining the discrete local time is
straightforward (the formal definition appears later in (1.2)), its analog for
the directed landscape is rather non-trivial and was one of the key
accomplishments in [GZ22] by the same authors. The notion of local time turned
out to be a crucial ingredient in the analysis of the exceptional set of
points witnessing geodesic non-coalescence in the above-mentioned paper.

Thus this paper along with [GZ22] initiates the study of local times in the
context of geodesics, akin to the study of the same for random walks and
Brownian motion which can be considered classical, and hence forms a part of a
bigger program of investigating the similarities and dissimilarities between
the geodesic and Brownian motion. The arguments in this paper can be seen as
one exploiting the weak decay of correlation the geodesic exhibits as alluded
to above — a style of argument that we expect would be useful in several other
applications going forward. We now move on to the statement of our results
first by introducing the key objects in play.

###  1.1 Directed landscape

The directed landscape $\mathcal{L}$ is a random continuous function from the
parameter space

$\mathbb{R}^{4}_{\uparrow}=\\{u=(p;q)=(x,s;y,t)\in\mathbb{R}^{4}:s<t\\}$

to $\mathbb{R}$ constructed in the breakthrough work [DOV18]. It satisfies the
following composition law

$\mathcal{L}(x,r;z,t)=\max_{y\in\mathbb{R}}\mathcal{L}(x,r;y,s)+\mathcal{L}(y,s;z,t)$
(1.1)

for any $x,z\in\mathbb{R}$ and $r<s<t$. For a continuous path
$\pi:[s,t]\to\mathbb{R}$, its weight under the metric $\mathcal{L}$ is given
by

$\|\pi\|_{\mathcal{L}}=\inf_{k\in\mathbb{N}}\inf_{s=t_{0}<t_{1}<\cdots<t_{k}=t}\sum_{i=1}^{k}\mathcal{L}(\pi(t_{i-1}),t_{i-1};\pi(t_{i}),t_{i}).$

Such a path $\pi$ is said to be a geodesic from $(\pi(s),s)$ to $(\pi(t),t)$
if $\|\pi\|_{\mathcal{L}}=\mathcal{L}(\pi(s),s;\pi(t),t)$. It is shown in
[DOV18] that almost surely, there exists at least one geodesic from $p$ to $q$
for any $u=(p;q)=(x,s;y,t)\in\mathbb{R}^{4}_{\uparrow}$. We let
$\pi_{u}=\pi_{(p;q)}=\pi_{(x,s;y,t)}:[s,t]\to\mathbb{R}$ denote any such
geodesic from $p$ to $q$. Further, for any fixed
$u\in\mathbb{R}^{4}_{\uparrow}$, almost surely the geodesic $\pi_{u}$ is
unique. Towards establishing its universal nature, in [DOV18, DV21], the
directed landscape $\mathcal{L}$ and the associated geodesics are proved to be
the scaling limits of their counterparts for various exactly solvable LPP
models, including Brownian LPP on $\mathbb{R}\times\mathbb{Z}$, Poissonian LPP
on $\mathbb{R}^{2}$, and LPP on $\mathbb{Z}^{2}$ with i.i.d. Exponential or
Geometric weights (Exponential/Geometric LPP).

Semi-infinite geodesics: Going beyond point-to-point geodesics, for each
$p=(x,s)\in\mathbb{R}^{2}$ and $r\in\mathbb{R}$, we denote by
$\pi_{p}^{r}:[s,\infty)\to\mathbb{R}$ a semi-infinite geodesic started from
$p$ in the $r$ direction; i.e. we have $t^{-1}\pi_{p}^{r}(t)\to r$ as
$t\to\infty$, and for any $s\leq s_{1}<s_{2}<\infty$, the restriction of
$\pi_{p}^{r}$ on $[s_{1},s_{2}]$ is a geodesic (from
$(\pi_{p}^{r}(s_{1}),s_{1})$ to $(\pi_{p}^{r}(s_{2}),s_{2})$). When $r=0$, we
will drop the $r$ dependence and simply use $\pi_{p}=\pi_{p}^{0}$. While the
issues regarding the existence and uniqueness of such semi-infinite geodesics
are not straightforward, they were treated in a systematic way in [GZ22,
Section 4] and [RV21, BSS22], with the following established: almost surely,
such semi-infinite geodesic $\pi_{p}$ exists for any $p\in\mathbb{R}^{2}$; and
for fixed $p$, almost surely there is a unique semi-infinite geodesic.

The landscape and thereby the semi-infinite geodesic $\pi_{(0,0)}$ admits
invariance under appropriate scalings guided by the exponent triple
$(1,1/3,2/3)$, characteristic of the KPZ universality class, much like
Brownian motion which is invariant under diffusive scaling. Hence, as in the
case of the latter, both $\mathcal{L}$ and $\pi_{(0,0)}$ are expected to
exhibit various fractal or self-similar properties.

###  1.2 Pre-limiting model: Exponential LPP

On the pre-limiting side, our choice of the model is Exponential LPP111Our
arguments would work equally well for Geometric LPP, the other known
integrable LPP model on $\mathbb{Z}^{2}$. Two other known integrable models
are Poissonian LPP on $\mathbb{R}^{2}$ and Brownian LPP on
$\mathbb{R}\times\mathbb{Z}$, and the arguments in this paper also imply
similar conclusions for the natural notions of local times in these settings.
A longer discussion appears following Theorem 1.2.. Consider directed last
passage percolation (LPP) on $\mathbb{Z}^{2}$ with i.i.d. Exponential weights
on the vertices, i.e., we have a random field
$\\{\omega_{p}:p\in\mathbb{Z}^{2}\\}$ where $\omega_{p}$ are i.i.d.
$\operatorname{Exp}(1)$ random variables. For any up/right path $\gamma$ from
$p$ to $q$ where $p\preceq q$ (i.e., $p$ is co-ordinate wise smaller than or
equal to $q$) the weight of $\gamma$ is defined by

$T(\gamma):=\sum_{w\in\gamma}\omega_{w}.$

(Note that owing to the continuity of the Exponential random variables, almost
surely, $T(\gamma_{1})\neq T(\gamma_{2})$ for $\gamma_{1}\neq\gamma_{2}$). For
any two points $p$ and $q$ with $p\preceq q$, we shall denote by $T_{p,q}$ the
last passage time from $p$ to $q$; i.e.,

$T_{p,q}=\max_{\gamma}T(\gamma),$

where the maximum is over all directed paths from $p$ to $q$. The (almost
surely unique) directed path with the maximum weight is the geodesic from $p$
to $q$, denoted as $\Gamma_{p,q}$. When $p\not\preceq q$, we let
$T_{p,q}=-\infty$.

In the setting of Exponential LPP, the existence and uniqueness of semi-
infinite geodesics have been established (see [Cou11, FP05]), and we summarize
the known facts here. For any $p\in\mathbb{Z}^{2}$, there is an almost surely
unique upper-right path from it, denoted as
$\Gamma_{p}=\\{p=(x_{0},y_{0}),(x_{1},y_{1}),\cdots\\}$, such that for any
$0\leq i\leq j$, the path $(x_{i},y_{i}),\cdots,(x_{j},y_{j})$ is
$\Gamma_{(x_{i},y_{i}),(x_{j},y_{j})}$, and
$\lim_{i\to\infty}\frac{x_{i}}{y_{i}}=1$.
Discrete semi-infinite geodesic: Such a $\Gamma_{p}$ is referred to as the
$(1,1)$-direction semi-infinite geodesic from $p$.

Almost surely, for any sequence $q_{n}\in\mathbb{Z}^{2}$ that goes to infinity
in the $(1,1)$-direction, $\Gamma_{p,q_{n}}$ converges to $\Gamma_{p}$ in the
sense of local limits (i.e., any initial segment of $\Gamma_{p}$ is also the
initial segment of $\Gamma_{p,q_{n}}$ for all large $n$). Also for any
$p,q\in\mathbb{Z}^{2}$, the paths $\Gamma_{p}$ and $\Gamma_{q}$ are the same
except for finitely many points, i.e., they will eventually merge and stay
together after finitely many steps. Thus all the semi-infinite geodesics in
the $(1,1)$-direction form a tree structure.

Throughout we shall always assume (the probability $1$ event) that for any
$p\preceq q\in\mathbb{Z}^{2}$, there is a unique geodesic $\Gamma_{p,q}$, and
a unique $(1,1)$-direction semi-infinite geodesic $\Gamma_{p}$.

Further, for the rest of this paper, we let $d,ad:\mathbb{Z}^{2}\to\mathbb{Z}$
be the functions $d(x,y)=x+y$ and $ad(x,y)=x-y$. For each $n\in\mathbb{Z}$, we
let $\mathbb{L}_{n}=\\{p\in\mathbb{Z}^{2}:d(p)=2n\\}$. Given that the
orientation of the directed landscape is vertically up, for comparison
purposes it might be more natural to rotate the Exponential LPP model by
$\pi/4$ so that the vertical and horizontal directions are the temporal and
spatial directions respectively. However, we do not adopt such a change and
stick to the traditional definition so as to be consistent with the previous
literature on Exponential LPP, which will be quoted several times in this
paper.

###  1.3 Main result

We first recall the local time for the semi-infinite geodesic $\pi_{(0,0)}(t)$
in the directed landscape from [GZ22, Section 7.1] where it was introduced and
constructed. We start by considering the occupation measure by defining for
any $w>0$ and $h\geq 0$,

$L_{w}(h):=\int_{0}^{h}\mathds{1}[\pi_{(0,0)}(t)\in[-w,w]]dt.$

The following proposition defines the local time as the density of the
occupation measure at $0.$

######  Proposition 1.1 ([GZ22, Proposition 2.4, Proposition 7.4]).

Almost surely there exists a non-decreasing function
$L:[0,\infty)\to\mathbb{R}$, such that $L(h)=\lim_{w\to 0}(2w)^{-1}L_{w}(h)<0$
for any $h\geq 0$. In addition, there exist constants $c,C>0$ such that for
any $0\leq g<h$ and $M>0$, we have

$\mathbb{P}[L(h)-L(g)>M(h-g)^{1/3}]<Ce^{-cM}.$

It is clear from the definition that $L$ is supported inside the zero set of
$\pi_{(0,0)}$. Further, the above estimate indicates, and it was indeed proved
in [GZ22], that the latter has Hausdorff dimension $1/3.$ This is in contrast
to the $1/2$ dimension-ness of the zero set of Brownian motion. The disparity
stems from the fact that while Brownian motion is $\frac{1}{2}^{-}$-Hölder
regular, $\pi_{(0,0)}$ is $\frac{2}{3}^{-}$-Hölder regular [DOV18, Proposition
12.3] (see also [DSV22] where geodesics are shown to have $3/2$-variation).

Correspondingly, in Exponential LPP, for each $n\in\mathbb{N}$ define the
discrete local time function $L^{(n)}:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq
0}$ by

$L^{(n)}(h)=2^{2/3}n^{-1/3}|\\{0\leq i\leq hn:(i,i)\in\Gamma_{(0,0)}\\}|.$
(1.2)

Here the factor $2^{2/3}n^{-1/3}$ makes $L^{(n)}$ be in the same scale as $L$,
dictated by the appearance of the same factors in the convergence of
Exponential LPP to the directed landscape (to be recorded shortly in Section
2.4). The main result of this paper is the convergence of $L^{(n)}$ to $L$.

######  Theorem 1.2.

As $n\to\infty$, we have $L^{(n)}\to L$ weakly, in the topology of uniform
convergence on compact sets.

We remark that our arguments also go through (essentially verbatim) for other
exact-solvable LPP models, such as Geometric, Poissonian, and Brownian LPP.
For Geometric LPP, one issue that arises is that of multiple geodesics between
vertices owing to the discreteness of vertex weights. In such a situation one
could consider the “rightmost” (or “leftmost”) semi-infinite geodesic from
$(0,0)$. In Poissonian LPP, one considers a Poisson point process on
$\mathbb{R}^{2}$, and the geodesic between two points $p,q\in\mathbb{R}^{2}$
with $p\preceq q$ (i.e., $p\leq q$ in each co-ordinate) is any up/right path
between them, passing through the most number of points. In this setting, the
pre-limiting local time can similarly be defined by counting the number of
intersections of the “rightmost” (or “leftmost”) semi-infinite geodesic from
$(0,0)$ with the diagonal $y=x$. Finally, in Brownian LPP, the random field is
given by a sequence of independent two-sided Brownian motions
$\\{\mathcal{B}_{i}\\}_{i\in\mathbb{Z}}$, and for any two points
$p,q\in\mathbb{R}\times\mathbb{Z}$ with $p\preceq q$ (i.e., $p\leq q$ in each
co-ordinate as before), the geodesic between them is the (almost surely
unique) up/right path in $\mathbb{R}\times\mathbb{Z}$, which accumulates the
largest Brownian motion increments. In this case, the analogous pre-limiting
local time could be taken to be the number of integers $i$ such that $(i,i)$
is on the semi-infinite geodesic from $(0,0)$. For each of these exactly-
solvable LPP models, using minor variations of the arguments in this paper,
one can deduce results analogous to Theorem 1.2.

As another remark on the generality of the methods in this paper, we point out
that (by slightly adapting our arguments) similar convergence of local times
can also be proved for point-to-point geodesics, going beyond semi-infinite
geodesics, with the limit being the local time of the corresponding point-to-
point geodesic in the directed landscape, which can also be constructed using
the same methods employed in [GZ22] for the construction of the semi-infinite
geodesic local time.

We finish this subsection with a brief overview of the more classical
literature on BLT as well as the more recent advances in understanding
geodesics models of random geometry which this paper contributes to as well.
The notion of local time was invented by Paul Lévy, and BLT was constructed as
a continuous process by Trotter [Tro58]. In terms of convergence from discrete
models, it was first shown by Knight in [Kni63] that the simple symmetric
random walk local time converges to BLT. Convergence of the local time of more
general random walks to BLT was later established in [Per82, Bor82]. (See also
[SS65, SS66] whose results plus Tanaka’s formula also imply convergence to BLT
of general random walks, as explained in the introduction of [Bor86]. In the
same paper [Bor86] the difference between the local time of random walks and
BLT was carefully analyzed.)

However, perhaps more relevant to our paper is the recent explosion of
developments around the understanding of planar models of random geometry in
the KPZ universality class (see e.g. [BBS20, BF22, BHS18, BSS14, BSS19, DSV22,
DZ21, JRAS22, SS20, Zha20]). As already mentioned above, a key feature of
these models is coalescence where geodesics between different points tend to
merge with each other. This has naturally led to a recent investigation of
atypical behavior of non-coalescence exhibited by a fractal set of points (see
e.g. [BGH21, BGH19, GH21, Dau21]). In fact, such a recent study in [GZ22]
necessitated the construction of the geodesic local time where it was shown
that the latter is $\frac{1}{3}^{-}$ Hölder regular in contrast to
$\frac{1}{2}^{-}$ Hölder regularity for BLT.

The consequences of coalescence as well as regularity properties of passage
times have found several other applications as well recently. (See [Gan22] for
a short survey).

###  1.4 Idea of proof

Before outlining the key ideas in the proof it might be instructive to discuss
an approach to prove the counterpart statement showing the convergence of the
random walk local time to the Brownian version. For example, in [MP10, Section
6.1]) it is shown that the occupation measure of Brownian motion $\mathcal{B}$
of the cylinder $[0,1]\times[-w,w]$ is approximated well by the number of
times $\mathcal{B}$ crosses the cylinder, and hence one can equivalently take
the limit of the number of such crossings to construct Brownian local time.
The proof of this equivalence relies heavily on the Markovian nature of
$\mathcal{B}$ which provides precise crossing number estimates. The exact
embedding of random walk in $\mathcal{B}$ then allows one to compare the
crossing number of the former to the crossing number of the latter, thereby
implying convergence.

However, for the geodesic, being a rather global object, there is no
Markovianity. Nonetheless, owing to coalescence, it does, albeit in a rather
implicit sense, exhibit some decay of correlation.

Our proof relies on making use of this and the known convergence of
Exponential LPP and the associated geodesics to their counterparts in the
directed landscape (from [DV21]). However, the topology of convergence is in
terms of Hausdorff distance between the geodesics thought of as compact
subsets of $\mathbb{R}^{2}$ which is clearly not refined enough to capture the
local time on the line $y=0.$ Nonetheless, it does help relate the occupation
measures of the geodesics in the limiting and pre-limiting settings. For
instance, the convergence allows us to deduce that the amount of time the
“suitably scaled” pre-limiting geodesic $\Gamma_{0}$ spends in the cylinder
$[-w,w]\times[0,1]$ is close to that spent by $\pi_{(0,0)}$ (relying on an
apriori estimate on the occupation measure developed in [SSZ21] which rules
out $\Gamma_{(0,0)}$ spending a significant amount of time near the boundary
of such a cylinder, i.e. in the narrower cylinders
$[-w-\delta,-w+\delta]\times[0,1]$ or $[w-\delta,w+\delta]\times[0,1]$).

As already stated, the occupation time of the cylinder $[-w,w]\times[0,1]$ for
$\pi_{(0,0)}$ will be denoted by $L_{w}$. The corresponding quantity for
$\Gamma_{(0,0)}$ will be denoted by $L^{(n)}_{w}$ (formal definition appears
in (3.1)).

With the above notation, the preceding discussion implies that $L^{(n)}_{w}\to
L_{w}$ as $n\to\infty$. Further, it was shown in [GZ22, Proposition 2.4] that
$L=\lim_{w\to\infty}\frac{L_{w}}{2w}.$ Thus the proof of Theorem 1.2 will be
complete once we prove the following key estimate (which most of the paper is
devoted to):

$\big{|}L^{(n)}-\frac{L^{(n)}_{w}}{2w}\big{|}=o_{w}(n^{1/3}),$

where $o_{w}(1)$ denotes a quantity that converges to $0$ with $w.$ This will
be shown by establishing that the pre-limiting local time exhibits a degree of
_smoothness_. More precisely, for any $m,$ we will show that the amounts of
time spent by the geodesic $\Gamma_{(0,0)}$ on the lines $\\{x-y=m\\}$ with
$|m|\leq wn^{2/3}$, up to time $n$ (i.e., below the line $\\{x+y=2n\\}$), are
very close to each other. For expository purposes, it would be convenient to
work with unscaled coordinates in the proof in which the unscaled local time
on the line $\\{x-y=m\\}$ will be denoted by $Z^{(n)}[m]$. In this notation it
will suffice to show that
$\frac{\mathbb{E}[(Z^{(n)}[m]-Z^{(n)}[0])^{2}]}{n^{2/3}}=o_{w}(1)$ for all
$|m|\leq wn^{2/3}$ (recall that $Z^{(n)}[0]$ is expected to be of order
$n^{1/3}$ owing to the transversal fluctuations of $\Gamma_{(0,0)}$ being of
order $n^{2/3}$ below the line $\\{x+y=2n\\}$).

Note that it is also natural to speculate that $Z^{(n)}[m]$ varies smoothly as
$m$ varies, while the expectation is maximized at $m=0$. Thus we expect
$\mathbb{E}[(Z^{(n)}[m]-Z^{(n)}[0])^{2}]$ to be of order $w^{4}n^{2/3}$ (due
to a second order effect) when $|m|$ is approximately $wn^{2/3}$. However, we
refrain from pursuing such a refined estimate.

We briefly outline in the remainder of this subsection the key steps involved
in establishing the above-mentioned smoothness across Sections 4 and 5, which
can be considered as the main novelty in the paper. The key input is a
stability estimate which shows geodesics, between arbitrary nice enough
boundary conditions (formal definition appears in Proposition 4.6), are stable
under mild perturbations of the boundary data. This is done in two steps.
First, we show that if the boundary data is perturbed, the original and the
perturbed geodesic must be close to each other during a large fraction of
their journey owing to certain regularity and decay properties of last passage
values as the endpoints are varied. The next step is to show that it is
unlikely that two geodesic paths spending a lot of time next to each other do
not coalesce. This last statement has already been proven and applied in a few
other instances ([GH20, BHS18, MSZ21]).

Given this general stability, the smoothness of the probability of the
geodesic passing through nearby points, and hence the local time, is shown by
shifting the noise environment suitably (causing a perturbation of the
boundary data as depicted in Figure 3). Under the natural coupling of the
corresponding geodesics, the event of one geodesic passing through a given
point and the other geodesic not passing through the corresponding shifted
point would entail a non-coalescence of the two geodesics whose probability is
shown to be low using the arguments in the preceding paragraph.

We end this discussion by remarking on a technical point that in fact to
perform second-moment computations we would need to show the smoothness of
two-point functions, i.e., the probability of the geodesic passing through a
pair of points, as one of the two points is varied in space.

### Organization of the remaining text

The rest of this paper is organized as follows. In Section 2 we set up
notation conventions, and collect useful results about Exponential LPP and the
directed landscape from the literature. The next three sections are devoted to
proving Theorem 1.2. Section 3 gives the general framework by carrying out the
second-moment computations and finishes the proof of Theorem 1.2, modulo the
smoothness of the LPP local time on different lines (Proposition 3.1). In
Section 4 we prove Proposition 3.1, via a key estimate on the two-point
distribution of the geodesic $\Gamma_{(0,0)}$ (Proposition 4.2). In Section 5
we prove the stability of geodesics with respect to boundary conditions, which
is a crucial input in the proof of Proposition 4.2. In Appendix A we prove a
key occupation time estimate (Lemma 2.18), using arguments from [SSZ21].

### Acknowledgement

LZ would like to thank Allan Sly for many discussions on the stability of
geodesics with respect to boundary conditions. SG is partially supported by
NSF grant DMS-1855688, NSF Career grant DMS-1945172, and a Sloan Fellowship.
The research of LZ is supported by the Miller Institute for Basic Research in
Science, at the University of California, Berkeley.

##  2 Preliminaries

###  2.1 Notations

We record some basic notations that will appear throughout the paper (in
addition to those already introduced in the previous section). For any real
numbers $x$ and $y$, let $x\wedge y=\min\\{x,y\\}$ and $x\vee
y=\max\\{x,y\\}$. For $x<y,$ we let $\llbracket
x,y\rrbracket=[x,y]\cap\mathbb{Z}$. For any set $A$ we use $|A|$ to denote its
cardinality while we will use $\mathscr{L}$ to denote the Lebesgue measure on
$\mathbb{R}$. We will also use $\mathbf{0}$ to denote $(0,0)$.

We would consider the space of all non-empty compact subsets of
$\mathbb{R}^{2}$. With the Hausdorff distance, it becomes a metric space, and
the induced topology is referred to as the Hausdorff topology.

For the notation $\operatorname{argmax}$ throughout this paper, when using it
we do not assume that there is a unique maximum (but there must exist at least
one); and when there are multiple ones we take an arbitrary one.

We end by stating a standard practice we will adopt in many of the proofs in
this paper where we will use $c,C>0$ to denote small and large constants.
Their specific values will depend on the context and may change from line to
line. To avoid repetition, we will not be restating this before every proof.

###  2.2 Properties of the directed landscape

In this subsection, we collect some known results about the directed
landscape.

####  2.2.1 Basic symmetries and estimates

We first record some of the various symmetries that the directed landscape
satisfies.

######  Lemma 2.1 ([DOV18, Lemma 10.2]).

As continuous functions on $\mathbb{R}^{4}_{\uparrow}$, $\mathcal{L}$ is equal
in distribution as

  * •

(flip invariance)

$(x,s;y,t)\mapsto\mathcal{L}(-y,-t;-x,-s);$

  * •

(skew-shift invariance)

$(x,s;y,t)\mapsto\mathcal{L}(x+cs+z,s+r;y+ct+z,t+r)+(t-s)^{-1}((x-y-c(t-s))^{2}-(x-y)^{2})$

for any $c,z,r\in\mathbb{R}$ (when $c=0$ this is also referred to as
translation invariance);

  * •

(scaling invariance)

$(x,s;y,t)\mapsto w\mathcal{L}(w^{-2}x,w^{-3}s;w^{-2}y,w^{-3}t);$

for any $w>0$.

Another basic property of the directed landscape (owing to planarity) is the
following quadrangle inequality.

######  Lemma 2.2 ([DOV18, Lemma 9.1][BGH21]).

For any $s<t$, $x_{1}<x_{2}$, $y_{1}<y_{2}$, we have

$\mathcal{L}(x_{1},s;y_{1},t)+\mathcal{L}(x_{2},s;y_{2},t)\geq\mathcal{L}(x_{1},s;y_{2},t)+\mathcal{L}(x_{2},s;y_{1},t).$

We next record some basic estimates on the passage times and its regularities.

######  Lemma 2.3 ([DOV18, Corollary 10.7]).

There is a random number $R$ such that the following is true. First, for any
$M>0$ we have $\mathbb{P}[R>M]<Ce^{-cM^{3/2}}$ for some universal constants
$c,C>0$. Second, for any $u=(x,s;y,t)\in\mathbb{R}_{\uparrow}^{4}$, we have

$\left|\mathcal{L}(x,s;y,t)+\frac{(x-y)^{2}}{t-s}\right|\leq
R(t-s)^{1/3}\log^{4/3}(2(\|u\|+2)^{3/2}/(t-s))\log^{2/3}(\|u\|+2).$

######  Lemma 2.4 ([DOV18, Proposition 1.6]).

Let $K$ be a compact subset of $\mathbb{R}_{\uparrow}^{4}$. There is a random
number $R$ depending only on $K$, such that the following is true. First, for
any $M>0$ we have $\mathbb{P}[R>M]<Ce^{-cM^{3/2}}$ for some universal
constants $c,C>0$. Second, for any
$(x,s;y,t),(x^{\prime},s^{\prime};y^{\prime},t^{\prime})\in K$, we have

$\left|\left(\mathcal{L}(x,s;y,t)+\frac{(x-y)^{2}}{t-s}\right)-\left(\mathcal{L}(x^{\prime},s^{\prime};y^{\prime},t^{\prime})+\frac{(x^{\prime}-y^{\prime})^{2}}{t^{\prime}-s^{\prime}}\right)\right|\\\
\leq
R\left(\tau^{1/3}\log^{2/3}(1+\tau^{-1})+\xi^{1/2}\log^{1/2}(1+\xi^{-1})\right),$

where $\tau=|s-s^{\prime}|\vee|t-t^{\prime}|$ and
$\tau=|x-x^{\prime}|\vee|y-y^{\prime}|$.

####  2.2.2 Airy2 process and local Brownian properties

Another central object in the KPZ universality class is the parabolic Airy2
process on $\mathbb{R}$, denoted as $\mathcal{A}_{2}:\mathbb{R}\to\mathbb{R}$.
It had in fact been rigorously established much before $\mathcal{L}$ [PS02],
and has the same distribution as the function $\mathcal{L}(0,0;\cdot,1)$. On
the other hand, by translation invariance of $\mathcal{L}$ (see Lemma 2.1),
there is $\mathcal{L}(0,0;\cdot,1)\overset{d}{=}\mathcal{L}(y,0;y+\cdot,1)$
for any $y\in\mathbb{R}$. Therefore the landscape $\mathcal{L}$ provides a
natural coupling of infinitely many parabolic Airy2 processes rooted at
different spatial points.

The process $\mathcal{A}_{2}$ also has the following Brownian property. For
$K\in\mathbb{R},d>0$, let $\mathcal{B}^{[K,K+d]}$ denote a Brownian motion on
$[K,K+d]$, with $\mathcal{B}^{[K,K+d]}(K)=0$. Let $\mathcal{A}_{2}^{[K,K+d]}$
denote the random function on $[K,K+d]$ defined by

$\mathcal{A}_{2}^{[K,K+d]}(x):=\mathcal{A}_{2}(x)-\mathcal{A}_{2}(K),\leavevmode\nobreak\
\forall x\in[K,K+d].$

######  Theorem 2.5 ([CHH19, Theorem 1.1]).

There exists a universal constant $G>0$ such that the following holds. For any
fixed $M>0$, there exists $a_{0}=a_{0}(M)$ such that for any interval
$[K,K+d]\subset[-M,M]$ and any measurable set
$A\subset\mathcal{C}\big{(}[K,K+d],\mathbb{R}\big{)}$ (the space of real
continuous functions on $[K,K+d]$) with $0<\mathbb{P}[\mathcal{B}^{[K,K+d]}\in
A]=a\leq a_{0}$, we have

$\mathbb{P}\left(\mathcal{A}_{2}^{[K,K+d]}\in A\right)\leq
a\exp\Big{(}G{M}\big{(}\log a^{-1}\big{)}^{5/6}\Big{)}.$

We will also need the Brownian absolute continuity of the KPZ fixed point with
arbitrary initial conditions.

######  Theorem 2.6 ([SV21, Theorem 1.2]).

Let $t>0$. Take any function $f_{0}:\mathbb{R}\to\mathbb{R}\cup\\{-\infty\\}$,
such that $f(x)\neq-\infty$ for some $x$, and
$\frac{f_{0}(x)-x^{2}/t}{|x|}\to-\infty$ as $|x|\to\infty$. Denote

$f_{t}(y)=\sup_{x\in\mathbb{R}}f_{0}(x)+\mathcal{L}(x,0;y,t),$

then for any $y_{1}<y_{2}$, the law of $y\mapsto f_{t}(y+y_{1})-f(y_{1})$ for
$y\in[0,y_{2}-y_{1}]$ is absolutely continuous with respect to the law of a
Brownian motion on $[0,y_{2}-y_{1}]$.

###  2.3 Properties of Exponential LPP

We next collect some known results on Exponential LPP that will be important
for us.

####  2.3.1 Busemann functions

Busemann functions were first used to study geodesics in Riemannian manifolds.
They were introduced in the context of first passage percolation (FPP) models
by Hoffman [Hof05, Hof08]. We now set up the Busemann function in Exponential
LPP, which has also been intensively studied and widely used (see e.g.
[Sep17]).

By the tree structure of semi-infinite geodesics (already alluded to in the
earlier discussion on Exponential LPP), for each $p\in\mathbb{Z}^{2}$ we let
$G(p)=T_{p,\mathbf{c}}-T_{\mathbf{0},\mathbf{c}}$, where
$\mathbf{c}\in\mathbb{Z}^{2}$ is the coalescing point of $\Gamma_{p}$ and
$\Gamma_{\mathbf{0}}$; i.e., $\mathbf{c}$ is the vertex in
$\Gamma_{p}\cap\Gamma_{\mathbf{0}}$ with the smallest $d(\mathbf{c})$. Such
Busemann function $G$ satisfies the following properties.

  1. 1.

For any $p=(x,y)\in\mathbb{Z}^{2}$ and $n>y$, if we take
$m_{*}=\operatorname{argmax}_{m\geq x}T_{p,(m,n)}+G(m,n+1)$, then
$\Gamma_{p}=\Gamma_{p,(m_{*},n)}\cup\Gamma_{(m_{*},n+1)}$, and
$G(p)=T_{p,(m_{*},n)}+G((m_{*},n+1))$.

  2. 2.

For any down-right path $\mathcal{U}=\\{u_{k}\\}_{k\in\mathbb{Z}}$, the random
variables $G(u_{k})-G(u_{k-1})$ are independent. The law of
$G(u_{k})-G(u_{k-1})$ is $\operatorname{Exp}(1/2)$ if $u_{k}=u_{k-1}-(0,1)$,
and is $-\operatorname{Exp}(1/2)$ if $u_{k}=u_{k-1}+(1,0)$. They are also
independent of $\omega(p)$ for all $p\in\mathcal{U}_{-}$, where
$\mathcal{U}_{-}$ is the lower part of $\mathbb{Z}^{2}\setminus\mathcal{U}$,

The first property is a consequence of the definition of $G$. For the second
property, its proof can be found in [Sep20].

####  2.3.2 Passage time estimates

The passage time $T_{\mathbf{0},(m,n)}$ has the same law as the largest
eigenvalue of $X^{*}X$, where $X$ is an $(m+1)\times(n+1)$ matrix of i.i.d.
standard complex Gaussian entries (see [Joh00, Proposition 1.4]). Hence we get
the following one-point estimates from [LR10, Theorem 2].

######  Theorem 2.7.

There exist constants $C,c>0$, such that for any $m\geq n\geq 1$ and $x>0$, we
have

$\mathbb{P}[T_{\mathbf{0},(m,n)}-(\sqrt{m}+\sqrt{n})^{2}\geq
xm^{1/2}n^{-1/6}]\leq Ce^{-cx}.$ (2.1)

In addition, for each $\psi>1$, there exist $C^{\prime},c^{\prime}>0$
depending on $\psi$ such that if $\frac{m}{n}<\psi$, we have

$\begin{split}&\mathbb{P}[T_{\mathbf{0},(m,n)}-(\sqrt{m}+\sqrt{n})^{2}\geq
xn^{1/3}]\leq C^{\prime}e^{-c^{\prime}\min\\{x^{3/2},xn^{1/3}\\}},\\\
&\mathbb{P}[T_{\mathbf{0},(m,n)}-(\sqrt{m}+\sqrt{n})^{2}\leq-xn^{1/3}]\leq
C^{\prime}e^{-c^{\prime}x^{3}},\end{split}$ (2.2)

and as a consequence

$|\mathbb{E}T_{\mathbf{0},(m,n)}-(\sqrt{m}+\sqrt{n})^{2}|\leq
C^{\prime}n^{1/3}.$ (2.3)

We also have the following parallelogram estimate.

######  Proposition 2.8 ([BGZ21, Theorem 4.2]).

Let $U$ be the parallelogram whose one pair of sides have length $2n^{2/3}$
and are aligned with $\mathbb{L}_{0}$ and $\mathbb{L}_{n}$ respectively, with
their midpoints being $(m,-m)$ and $\mathbf{n}$. Let $U_{1},U_{2}$ be the
parts of $U$ below $\mathbb{L}_{n/3}$ and above $\mathbb{L}_{2n/3}$
respectively. For each $\psi<1$, there exist constants $C,c>0$ depending only
on $\psi$, such that when $|m|<\psi n$,

$\mathbb{P}\Big{[}\sup_{u\in U_{1},v\in U_{2}}|T_{u,v}-\mathbb{E}T_{u,v}|\geq
xn^{1/3}\Big{]}\leq Ce^{-cx}.$

Such a result was first proved as [BSS14, Proposition 10.1, 10.5], in the
setting of Poissionian LPP. For Exponential LPP, a proof is given in [BGZ21,
Appendix C], following similar ideas.

####  2.3.3 Transversal fluctuation

We will rely on the following transversal fluctuation estimates of geodesics,
which can be proved using Proposition 2.8 and some geometric arguments. Such
estimates have appeared and been used in several works [BGZ21, MSZ21, BSS14,
BSS19, Zha20]. We quote the following versions for point-to-point and semi-
infinite geodesics.

######  Lemma 2.9 ([BGZ21, Proposition C.9]).

For any $0<\psi<1$, there exist constants $C,c>0$ such that the following is
true. Take any $n\in\mathbb{N}$ large enough and any $m\in\mathbb{Z},|m|<\psi
n$, $x>0$. Consider the parallelogram whose four vertices are
$(-xn^{2/3},xn^{2/3})$, $(xn^{2/3},-xn^{2/3})$, and $(n+m-xn^{2/3},n-m-
xn^{2/3})$, $(n+m+xn^{2/3},n-m+xn^{2/3})$. Then with probability
$1-Ce^{-cx^{3}}$, the geodesic $\Gamma_{\mathbf{0},(n+m,n-m)}$ is contained in
that parallelogram.

######  Lemma 2.10 ([MSZ21, Corollary 2.11]).

There exist constants $C,c>0$ such that the following is true. Take any
$n\in\mathbb{N}$ large enough and any $x>0$, and consider the rectangle whose
four vertices are $(-xn^{2/3},xn^{2/3})$, $(xn^{2/3},-xn^{2/3})$, and
$(n-xn^{2/3},n+xn^{2/3})$, $(n+xn^{2/3},n-xn^{2/3})$. Then with probability
$1-Ce^{-cx^{3}}$, the part of the geodesic $\Gamma_{\mathbf{0}}$ below
$\mathbb{L}_{n}$ is contained in that rectangle.

####  2.3.4 Intersections of geodesics

The following estimate from [BHS18] bounds the number of intersections that
geodesics can have with a line.

######  Proposition 2.11 ([BHS18, Proposition 3.10]).

Let $A_{n}$ and $B_{n}$ be line segment along the lines $\mathbb{L}_{0}$ and
$\mathbb{L}_{n}$, with midpoints $(xn^{2/3},-xn^{2/3})$ and $(n,n)$,
respectively, and each has length $2n^{2/3}$. Let $F_{n,m}=\left(\bigcup_{u\in
A_{n},v\in B_{n}}\Gamma_{u,v}\right)\cap\mathbb{L}_{m}$, for $0<m<n$. For any
$\psi\in(0,1)$, $\alpha\in(0,1/2)$, there is a constant $c>0$, such that when
$|x|<\psi n^{1/3}$ and $\alpha n<m<(1-\alpha)n$, we have

$\mathbb{P}[|F_{n,m}|>M]<e^{-cM^{1/128}}$

for any sufficiently large $M<n^{0.01}$ and sufficiently large $n$.

The above statement is a consequence of geodesic coalescence which causes only
a few (order $1$ many) highways passing through the line $\mathbb{L}_{m}$ in
an order $n^{2/3}$ window and all the geodesics ($\Gamma_{u,v}$ for some $u\in
A_{n}$, $v\in B_{n}$) merging with one of them.

We note that Proposition 2.11 is slightly more general than [BHS18,
Proposition 3.10], which is only for $\alpha=1/3$; but the proof in [BHS18]
goes through verbatim for general $\alpha\in(0,1)$.

In Proposition 2.11, we considered geodesics with endpoints varying on some
segments of length in the order of $n^{2/3}$, and their intersections with the
line $\mathbb{L}_{m}$. We next state a different version that allows the upper
end of the geodesics to vary on the whole line $\mathbb{L}_{n}$, but we only
consider their intersections with a segment of the length of the order of
$n^{2/3}$.

######  Lemma 2.12.

Take any $m<n\in\mathbb{N}$ and $\ell>1$, let $F=\left(\bigcup_{|i|<\ell
n^{2/3},j\in\mathbb{Z}}\Gamma_{(i,-i),(n+j,n-j)}\right)\cap\\{(m+i,m-i):|i|<\ell
n^{2/3}\\}|$. Then for each $0<\alpha<1/2$, there exist $c,C>0$, such that
$\mathbb{P}[|F|>M]<Ce^{-cM^{1/12800}}$ when $\ell<cn^{1/3}\wedge M^{1/3}$, and
$\alpha n<m<(1-\alpha)n$.

Transversal fluctuation estimates (from Section 2.3.3) imply that for any
geodesic from some $(i,-i)$ with $|i|<\ell n^{2/3}$ to $\mathbb{L}_{n}$, if it
intersects $\\{(m+i,m-i):|i|<\ell n^{2/3}\\}$, the distance between its upper
end and $(n,n)$ is likely to be of order (at most) $n^{2/3}$. Therefore we can
apply Proposition 2.11.

###### Proof of Lemma 2.12.

We let $c,C>0$ denote small and large constants depending on $\alpha$. We may
also assume that $M<2\ell n^{2/3}+1<n$, since otherwise we would always have
$|F|\leq M$.

Let $\ell^{\prime}=C\ell+M^{1/10}<n^{1/3}/2$. Consider
$F^{\prime}=\left(\bigcup_{|i|<\ell
n^{2/3},|j|<\ell^{\prime}n^{2/3}}\Gamma_{(i,-i),(n+j,n-j)}\right)\cap\\{(m+i,m-i):|i|<\ell
n^{2/3}\\}|$. Then under $|F|>M$, we have either $|F^{\prime}|>M$, or there
are some $|i|<\ell n^{2/3}$ and $|j|\geq\ell^{\prime}n^{2/3}$, such that
$\Gamma_{(i,-i),(n+j,n-j)}$ intersects $\mathbb{L}_{m}$ at some $(m+k,m-k)$,
for $|k|<\ell n^{2/3}$. This would imply that either $\Gamma_{(\lfloor\ell
n^{2/3}\rfloor,-\lfloor\ell
n^{2/3}\rfloor),(n-\lfloor\ell^{\prime}n^{2/3}\rfloor,n+\lfloor\ell^{\prime}n^{2/3}\rfloor)}$
intersects $\mathbb{L}_{m}$ at some $(m+k,m-k)$, for $k>-\ell n^{2/3}$; or
$\Gamma_{(-\lfloor\ell n^{2/3}\rfloor,\lfloor\ell
n^{2/3}\rfloor),(n+\lfloor\ell^{\prime}n^{2/3}\rfloor,n-\lfloor\ell^{\prime}n^{2/3}\rfloor)}$
intersects $\mathbb{L}_{m}$ at some $(m+k,m-k)$, for $k<\ell n^{2/3}$. By
Lemma 2.9, these happen with probability $<Ce^{-cM^{3/10}}$.

Finally, by splitting $\\{(i,-i):|i|<\ell n^{2/3}\\}$ and
$\\{(n+j,n-j):|j|<\ell^{\prime}n^{2/3}\\}$ into segments of length $n^{2/3}$,
we can write $F^{\prime}$ as the union of $\ell\ell^{\prime}$ sets; and for
each set, the probability that its size is $>M^{0.01}$ is at most
$e^{-c(M^{0.01})^{1/128}}$, by Proposition 2.11 (note that $M^{0.01}<n^{0.01}$
by our assumption above). Then since $\ell\ell^{\prime}M^{0.01}<M$, we have

$\mathbb{P}[|F^{\prime}|>M]\leq\mathbb{P}[|F^{\prime}|>\ell\ell^{\prime}M^{0.01}]<\ell\ell^{\prime}e^{-c(M^{0.01})^{1/128}}<Ce^{-cM^{1/12800}}.$

Thus the conclusion follows. ∎

We can get similar results for semi-infinite geodesics (in the
$(1,1)$-direction). In this case, note that the lower endpoint can vary among
all points in $\mathbb{L}_{0}$.

######  Lemma 2.13.

For any $n\in\mathbb{N}$ and $\ell>1$, let
$F=(\bigcup_{i\in\mathbb{Z}}\Gamma_{(i,-i)})\cap\\{(n+i,n-i):|i|<\ell
n^{2/3}\\}$. Then there exist $c,C>0$, such that
$\mathbb{P}[|F|>M]<Ce^{-cM^{1/12800}}$ when $\ell<cn^{1/3}\wedge M^{1/3}$.

###### Proof.

In this proof we let $c,C>0$ denote small and large constants. We can also
assume that $M<2\ell n^{2/3}+1<n$, since otherwise we would always have $F\leq
M$.

Let $\ell^{\prime}=C\ell+M^{1/10}<n^{1/3}/2$. Consider
$F^{\prime}=\left(\bigcup_{|i|,|j|<\ell^{\prime}n^{2/3}}\Gamma_{(i,-i),(2n+j,2n-j)}\right)\cap\\{(n+i,n-i):|i|<\ell
n^{2/3}\\}|$. Then under $|F|>M$, we have either $|F^{\prime}|>M$, or there
are some $|i|\geq\ell^{\prime}n^{2/3}$, such that $\Gamma_{(i,-i)}$ intersects
$\mathbb{L}_{n}$ at some $(n+k,n-k)$, for $|k|<\ell n^{2/3}$. This would imply
that either
$\Gamma_{(\lfloor\ell^{\prime}n^{2/3}\rfloor,-\lfloor\ell^{\prime}n^{2/3}\rfloor)}$
intersects $\mathbb{L}_{n}$ at some $(n+k,n-k)$, for $k>-\ell n^{2/3}$; or
$\Gamma_{(-\lfloor\ell^{\prime}n^{2/3}\rfloor,\lfloor\ell^{\prime}n^{2/3}\rfloor)}$
intersects $\mathbb{L}_{n}$ at some $(n+k,n-k)$, for $k<\ell n^{2/3}$. By
Lemma 2.10, we would have that these happen with probability
$<Ce^{-cM^{3/10}}$.

Finally, as in the proof of Lemma 2.12, we split
$\\{(i,-i):|i|<\ell^{\prime}n^{2/3}\\}$ and
$\\{(2n+j,2n-j):|j|<\ell^{\prime}n^{2/3}\\}$ into segments of length
$n^{2/3}$, and apply Proposition 2.11 to each pair. We then conclude that
$\mathbb{P}[|F^{\prime}|>M]<Ce^{-cM^{1/12800}}$, so the conclusion follows. ∎

###  2.4 Convergence of Exponential LPP to the directed landscape

In [DV21] it is proved that Exponential LPP converges to the directed
landscape after suitable scaling. We record the formal statement in this
section.

To state the convergence, we need a linear transformation of Exponential LPP
to the scaling of the directed landscape. Among other things, the
transformation redirects the orientation from the $(1,1)$-direction to the
vertical direction.

For any $p=(x,y)\in\mathbb{R}^{2}$ we let
$\mathsf{R}_{n}(p)=(2^{-5/3}n^{-2/3}(x-y),n^{-1}y)$. For any set
$A\subset\mathbb{R}^{2}$ and $n\in\mathbb{N}$, we let
$\mathsf{R}_{n}(A)=\\{(\mathsf{R}_{n}(p):p\in A\\}$. We then define a function
$\mathcal{K}_{n}$ on $\mathbb{R}^{4}$, the transformed last-passage times. It
is also a “directed metric” on $\mathbb{R}^{2}$ (as it satisfies a composition
law similar to (1.1)). For any $(x,s;y,t)\in\mathbb{R}^{4}$, we let

$\mathcal{K}_{n}(x,s;y,t)=2^{-4/3}n^{-1/3}F(\mathsf{R}_{n}^{-1}(x,s);\mathsf{R}_{n}^{-1}(y,t)).$

where $F:\mathbb{R}^{4}\to\mathbb{R}$ is the rounded version of last-passage
times, defined as follows. Let $\mathsf{r}:\mathbb{R}^{2}\to\mathbb{Z}^{2}$ be
the “rounding” function where

$\mathsf{r}(x,y)=\begin{cases}(x,y)&x,y\in\mathbb{Z};\\\ (\lfloor
x\rfloor,y)&x\not\in\mathbb{Z},y\in\mathbb{Z};\\\ (x,\lfloor
y\rfloor)&x\in\mathbb{Z},y\not\in\mathbb{Z};\\\ (\lceil x\rceil,\lfloor
y\rfloor)&x\not\in\mathbb{Z},y\not\in\mathbb{Z}.\end{cases}$

We then let $F(x,y;z,w)=T_{(\lceil x\rceil,\lceil
y\rceil),\mathsf{r}(z,w)}-2d(\mathsf{r}(z,w)-\mathsf{r}(x,y))-\omega_{(x,y)}\mathds{1}[x,y\in\mathbb{Z}]$.
Here $2d(\mathsf{r}(z,w)-\mathsf{r}(x,y))$ is the leading linear approximation
of $T_{(\lceil x\rceil,\lceil y\rceil),\mathsf{r}(z,w)}$, and
$\omega_{(x,y)}\mathds{1}[x,y\in\mathbb{Z}]$ is the correction term to make
$F$ satisfy a composition law (similar to (1.1)). We note that
$\mathcal{K}_{n}(x,s;y,t)=-\infty$ if $(\lceil ns+2^{5/3}n^{2/3}x\rceil,\lceil
ns\rceil)\not\preceq\mathsf{r}(nt+2^{5/3}n^{2/3}y,nt)$, as we let
$T_{p,q}=-\infty$ when $p\not\preceq q$.

For any $p,q\in\mathbb{R}^{2}$, a set $A\subset\mathbb{R}^{2}$ is called a
_geodesic set_ in $\mathcal{K}_{n}$ from $p$ to $q$, if there is a total
ordering $\preceq_{A}$ on $A$, such that $p\preceq_{A}p^{\prime}\preceq_{A}q$
for any $p^{\prime}\in A$; and for any
$p_{1}\preceq_{A}p_{2}\preceq_{A}p_{3}\in A$,

$\mathcal{K}_{n}(p_{1};p_{2})+\mathcal{K}_{n}(p_{2};p_{3})=\mathcal{K}_{n}(p_{1};p_{3}).$
(2.4)

Such a set $A$ is called a _maximal geodesic set_ in $\mathcal{K}_{n}$ if it
is not contained in any other geodesic set from $p$ to $q$. From these
definitions, we have that for any $p,q\in\mathbb{Z}^{2}$, the set
$\mathsf{R}_{n}(\Gamma_{p,q})$ is a maximal geodesic set from
$\mathsf{R}_{n}(p)$ to $\mathsf{R}_{n}(q)$. Indeed, obviously
$\mathsf{R}_{n}(\Gamma_{p,q})$ is a geodesic set, and it is maximal because
adding any extra point (with any ordering) would violate (2.4) (while this can
be checked straightforwardly, we omit the somewhat tedious details.)

Finally, for any continuous path $\pi:I\to\mathbb{R}$, where
$I\subset\mathbb{R}$ is any subset, we define its _graph_
$\operatorname{Graph}(\pi)$ as the set $\\{(\pi(t),t):t\in I\\}$. We then have
the following convergence result.

######  Theorem 2.14 ([DV21, Theorem 13.8(2)]).

There is a coupling of $\mathcal{L}$ with $\mathcal{K}_{n}$ for all
$n\in\mathbb{N}$, such that the following holds almost surely. First, for any
compact $K\subset\mathbb{R}^{4}_{\uparrow}$, we have
$\mathcal{K}_{n}\to\mathcal{L}$ uniformly in $K$. Second, for any
$(p_{n};q_{n})\to(p;q)\in\mathbb{R}^{4}_{\uparrow}$, and any $\mathcal{K}_{n}$
maximal geodesic set from $p_{n}$ to $q_{n}$, denoted as $\pi_{n}$, they are
pre-compact in Hausdorff topology, and any subsequential limit is
$\operatorname{Graph}(\pi)$, for some $\mathcal{L}$ geodesic $\pi$ from $p$ to
$q$.

The convergence of the Exponential LPP semi-infinite geodesic
$\Gamma_{\mathbf{0}}$ to $\pi_{\mathbf{0}}$ is also established, in the
Hausdorff topology.

######  Proposition 2.15 ([GZ22, Proposition 5.1]).

For any $h>0$,
$\mathsf{R}_{n}(\Gamma_{\mathbf{0}})\cap\mathbb{R}\times[0,h]\to\operatorname{Graph}(\pi_{\mathbf{0}})\cap\mathbb{R}\times[0,h]$
in distribution in the Hausdorff topology, as $n\to\infty$.

This is also obtained in [RV21, Lemma 4.2].

###  2.5 Occupation estimates

We will rely on the following crucial tail estimate established in [GZ22] on
the occupation time of a semi-infinite geodesic in the directed landscape
based on an earlier similar estimate for Exponential LPP obtained in [SSZ21].

######  Lemma 2.16 ([GZ22, Lemma 2.7]).

There are universal constants $C,c>0$ such that the following is true. For any
closed interval $I\subset\mathbb{R}$ and $J\subset\mathbb{R}_{\geq 0}$, we
have

$\mathbb{P}\left[\int_{J}\mathds{1}[\pi_{\mathbf{0}}(t)\in
I]dt>M\mathscr{L}(I)\mathscr{L}(J)^{1/3}\right]<Ce^{-cM}$

for any $M>0$.

One obtains the following continuity result as a straightforward consequence
of the above.

######  Lemma 2.17.

Almost surely the following holds. For any $h\geq 0$, $w\mapsto L_{w}(h)$ is
continuous for $w\geq 0$.

###### Proof.

By Lemma 2.16, we have that there exists a random $R>0$, such that for any
$i,a,b\in\mathbb{Z}$, we have

$\int_{0}^{2^{a}}\mathds{1}[\pi_{\mathbf{0}}(t)\in[i2^{b},(i+2)2^{b}]]dt<R(|a|+|b|+|i|^{1/2}+1)2^{b+a/3}.$

Indeed, for any fixed $i,a,b\in\mathbb{Z}$ and $R>0$, the probability for this
inequality to fail is exponentially small in $R(|a|+|b|+|i|^{1/2}+1)$. Thus it
is summable when taking a union bound over $i,a,b$.

For any $h>0$, $w>0$, and $|\delta|<w$, by taking $a=\lceil\log_{2}(h)\rceil$,
$b=\lceil\log_{2}(\delta)\rceil$, and making appropriate choices of $i$, we
have that

$|L_{w}(h)-L_{w+\delta}(h)|<16R\delta
h^{1/3}(|\log_{2}(h)|+|\log_{2}(\delta)|+(|w|/|\delta|)^{1/2}+4).$

So the conclusion follows. ∎

We will also use a version of the occupation time bound in the Exponential LPP
setting. For any $a<b\in\mathbb{Z}$, we let $W[a,b]=|\\{a\leq k\leq
b:(k,k)\in\Gamma_{(a,a),(b,b)}\\}|$, and let $W_{a,b}=\max_{a\leq
a^{\prime}\leq b^{\prime}\leq b}W[a^{\prime},b^{\prime}]$.

######  Lemma 2.18.

There are universal constants $C,c>0$ such that the following is true. Take
any large enough $A$. For any $a<b\in\mathbb{Z}$, there is

$\mathbb{E}[W_{a,b}]<A^{2/3+0.01}(b-a)^{1/3},$ (2.5)

and for any $M>0$,

$\mathbb{P}[W_{a,b}>AM(b-a)^{1/3}]<Ce^{-cM}.$ (2.6)

This result in a slightly different setting has appeared in [SSZ21]. For
completeness, we give a proof of Lemma 2.18 in the appendix, following a
similar strategy.

##  3 Convergence to the local time

We start by recalling briefly our strategy to prove Theorem 1.2. The key input
is Proposition 2.15, which is the convergence of the semi-infinite geodesic in
Exponential LPP to the one in the directed landscape, in the Hausdorff
topology. However, the scaling of Proposition 2.15 is too coarse to get
information on a line. Thus, to use the convergence result, we need to relate
the time spent in a strip with the local time on a line. This will be shown by
establishing certain smoothness properties of the local time spent on a line
as the latter moves in the spatial direction. Towards this, we consider the
time spent in a strip, defined as follows: for any $w>0$ and $n\in\mathbb{N}$,
we define

$L^{(n)}_{w}(h)=(2n)^{-1}|\\{(x,y):0\leq x+y\leq 2hn,|x-y|\leq
2^{5/3}n^{2/3}w,(x,y)\in\Gamma_{\mathbf{0}}\\}|.$ (3.1)

This is a discrete analog of the function $L_{w}$, and the scalings match
those in Section 2.4 to make it converge to $L_{w}$ as $n\to\infty$.

The proof of Theorem 1.2 consists of the following components.

  1. 1.

$(2w)^{-1}L_{w}(h)\to L(h)$ as $w\to 0$ almost surely.

  2. 2.

For any $\delta>0$, and any $w>0$ is small enough (depending on $\delta$), we
have $|(2w)^{-1}L^{(n)}_{w}(h)-L^{(n)}(h)|<\delta$ for any $n$ large enough
(depending on $\delta,w$).

  3. 3.

For any $w>0$, $L^{(n)}_{w}(h)\to L_{w}(h)$ as $n\to\infty$ (in a sense that
will be clear later).

Given the above statements, by first taking $w$ small enough, then $n$ large
enough, it follows that $L^{(n)}\to L$ in probability.

The first step has been accomplished in [GZ22] and is recorded as Proposition
1.1. The third step will involve applying Proposition 2.15, and is stated as
Lemma 3.2 below. The main task that remains is to carry out the second step,
which is formulated as the following result.

######  Proposition 3.1.

For each $h\geq 0$, we have $\lim_{w\to
0}\limsup_{n\to\infty}\mathbb{E}[((2w)^{-1}L^{(n)}_{w}(h)-L^{(n)}(h))^{2}]=0$.

We leave the proof of this proposition for later and perform the third step
now. The argument will not depend on the exact value of $h$ and hence for
concreteness, we shall assume $h=1$.

By Proposition 2.15 and using Skorokhod’s representation theorem (see e.g.
[Bil99, Chapter 1, Section 6], for the space of all compact subsets of
$\mathbb{R}^{2}$ with the Hausdorff distance metric, which is separable), we
couple a sequence of $\mathbb{Z}^{2}$ Exponential LPP models and the directed
landscape, such that if we denote by $\Gamma_{\mathbf{0}}^{(n)}$ the
$(1,1)$-direction semi-infinite geodesic from $\mathbf{0}$ in the $n$-th
Exponential LPP, we have
$\mathsf{R}_{n}(\Gamma_{\mathbf{0}}^{(n)})\cap\mathbb{R}\times[0,2]\to\operatorname{Graph}(\pi_{\mathbf{0}})\cap\mathbb{R}\times[0,2]$
in the Hausdorff metric. We then let $L^{(n)}_{w}$ be defined using
$\Gamma_{\mathbf{0}}^{(n)}$. Then the following convergence holds.

######  Lemma 3.2.

For any $w>0$, almost surely we have $L^{(n)}_{w}\to L_{w}$ uniformly in any
compact set, as $n\to\infty$.

To prove this lemma, it suffices to use the convergence given by Proposition
2.15, and bound the time that the geodesic $\pi_{\mathbf{0}}$ spends in
$[-w-\delta,-w+\delta]\cup[w-\delta,w+\delta]$, as illustrated in Figure 1.
For the latter task, we use Lemma 2.17.

$\mathbf{0}$$t=1$$\pi_{\mathbf{0}}$$J_{\delta}^{in}$$J_{\delta}^{in}$$J_{\delta}$$J_{\delta}$$J_{\delta}$$J_{\delta}^{out}$$J_{\delta}^{out}$
Figure 1: An illustration of the proof of Lemma 3.2: to bound the time
$\pi_{\mathbf{0}}$ spends near the left and right boundaries of
$[-w,w]\times[0,1]$. The green, blue, and red segments illustrate the time
intervals of $J_{\delta}^{in}$, $J_{\delta}$, $J_{\delta}^{out}$,
respectively.

###### Proof of Lemma 3.2.

Without loss of generality we show that $L^{(n)}_{w}\to L_{w}$ uniformly in
$[0,1]$. Given the coupling and the convergence in the Hausdorff topology, the
main thing to consider is the set $\operatorname{Graph}(\pi_{\mathbf{0}})$
near the boundary of $[-w,w]\times[0,1]$. More precisely, for the limit of
$L^{(n)}_{w}$ as $n\to\infty$, we will upper bound it by $L_{w+\delta}$ and
lower bound it by $L_{w-\delta}$, for any small $\delta>0$. Then the
conclusion follows by the continuity of $L_{w}$ in $w$.

To connect $L^{(n)}_{w}$ to $L_{w}$, we start by re-writing

$L^{(n)}_{w}(h)=(2n)^{-1}|\\{(x,s)\in\mathsf{R}_{n}(\Gamma_{\mathbf{0}}^{(n)}):0\leq
2^{5/3}n^{2/3}x+2ns\leq 2hn,|x|\leq w\\}|,$

for any $h\geq 0$. This directly follows from (3.1) and the fact that
$\mathsf{R}_{n}^{-1}(x,s)=(2^{5/3}n^{2/3}x+ns,ns)$. We note that for any two
different $(x,s),(y,r)\in\mathsf{R}_{n}(\Gamma_{\mathbf{0}}^{(n)})$, the
numbers $2^{5/3}n^{2/3}x+2ns=d(\mathsf{R}_{n}^{-1}(x,s))$ and
$2^{5/3}n^{2/3}y+2nr=d(\mathsf{R}_{n}^{-1}(y,r))$ are different non-negative
integers.

We now define

$J^{(n)}=\\{0\leq t\leq
1:\exists(x,s)\in\mathsf{R}_{n}(\Gamma_{\mathbf{0}}^{(n)}),|2nt-(2^{5/3}n^{2/3}x+2ns)|\leq
1/2,|x|\leq w\\}.$

Then it equals

$[0,1]\cap\bigcup_{(x,s)\in\mathsf{R}_{n}(\Gamma_{\mathbf{0}}^{(n)}),|x|\leq
w}[2^{2/3}n^{-1/3}x+s-(4n)^{-1},2^{2/3}n^{-1/3}x+s+(4n)^{-1}].$

Recall that $\mathscr{L}$ denotes the Lebesgue measure on $\mathbb{R}$. Also
note that for all $(x,s)\in\mathsf{R}_{n}(\Gamma_{\mathbf{0}}^{(n)})$,
$2^{5/3}n^{2/3}x+2ns$ are mutually different integers. Then

$|\\{(x,s)\in\mathsf{R}_{n}(\Gamma_{\mathbf{0}}^{(n)}):0\leq
2^{2/3}n^{-1/3}x+s\leq h,|x|\leq w\\}|\in\\{\lceil
2n\mathscr{L}(J^{(n)}\cap[0,h])\rceil,\lfloor
2n\mathscr{L}(J^{(n)}\cap[0,h])\rfloor\\}.$

Therefore $|L^{(n)}_{w}(h)-\mathscr{L}(J^{(n)}\cap[0,h])|\leq(2n)^{-1}$ for
$h\in[0,1]$. It then suffices to prove the convergence of
$\mathscr{L}(J^{(n)}\cap[0,\cdot])$ to $L_{w}$ (uniformly in $[0,1]$). For
this, we consider the following quantities. Take any $0<\delta<w$. We denote

$\displaystyle J_{\delta}$ $\displaystyle=\\{t\geq 0:\exists
t^{\prime},|t^{\prime}-t|\leq\delta,w-\delta\leq|\pi_{\mathbf{0}}(t^{\prime})|\leq
w+\delta\\}$ $\displaystyle J_{\delta}^{out}$ $\displaystyle=\\{t\geq
0:\exists
t^{\prime},|t^{\prime}-t|\leq\delta,|\pi_{\mathbf{0}}(t^{\prime})|\geq
w+\delta\\}$ $\displaystyle J_{\delta}^{in}$ $\displaystyle=\\{t\geq 0:\exists
t^{\prime},|t^{\prime}-t|\leq\delta,|\pi_{\mathbf{0}}(t^{\prime})|\leq
w-\delta\\}.$

See Figure 1 for an illustration of these sets. We next show that for any
$\delta>0$,

$J^{(n)}\subset J_{\delta}\cup J_{\delta}^{in},\quad[0,1]\setminus
J^{(n)}\subset J_{\delta}\cup J_{\delta}^{out},$ (3.2)

for $n$ large enough. For any $t\in J^{(n)}$, we take
$(x,s)\in\mathsf{R}_{n}(\Gamma_{\mathbf{0}}^{(n)})$ such that $|x|\leq w$ and
$|2nt-(2^{5/3}n^{2/3}x+2ns)|\leq 1/2$. When $n$ is large enough, we have
$|2^{2/3}n^{-1/3}x|<\delta/3$, therefore $|s-t|<\delta/3+(4n)^{-1}$; and we
can find $(x^{\prime},s^{\prime})\in\operatorname{Graph}(\pi_{\mathbf{0}})$
such that $|x-x^{\prime}|,|s-s^{\prime}|<\delta/3$ (by almost sure convergence
under the coupling). Thus

$|s^{\prime}-t|\leq|s-s^{\prime}|+|s-t|<2\delta/3+(4n)^{-1}$

and

$|\pi_{\mathbf{0}}(s^{\prime})|=|x^{\prime}|\leq|x|+|x-x^{\prime}|<w+\delta/3.$

So we have $t\in J_{\delta}\cup J_{\delta}^{in}$, and we conclude that
$J^{(n)}\subset J_{\delta}\cup J_{\delta}^{in}$. By a similar argument we have
that $[0,1]\setminus J^{(n)}\subset J_{\delta}\cup J_{\delta}^{out}$.

By (3.2), when $n$ is large enough, we have that

$\mathscr{L}(J^{(n)}\cap[0,h])\leq\mathscr{L}((J_{\delta}\cup
J_{\delta}^{in})\cap[0,h])=L_{w+\delta}(h),$

and

$\mathscr{L}(J^{(n)}\cap[0,h])\geq h-\mathscr{L}((J_{\delta}\cup
J_{\delta}^{out})\cap[0,h])=\int_{0}^{h}\mathds{1}[\pi_{\mathbf{0}}(t)\in(-w+\delta,w-\delta)]dt{\geq}L_{w-2\delta}(h),$

for any $h\in[0,1]$. Note that $0\leq L_{w+\delta}(h)-L_{w}(h)\leq
L_{w+\delta}(1)-L_{w}(1)$ and $0\geq L_{w-2\delta}(h)-L_{w}(h)\geq
L_{w-2\delta}(1)-L_{w}(1)$ for any $h\in[0,1]$. Thus by sending $\delta\to 0$,
and using Lemma 2.17, we conclude that almost surely,
$\lim_{n\to\infty}\mathscr{L}(J^{(n)}\cap[0,h])=L_{w}(h)$ uniformly for
$h\in[0,1]$, and the conclusion follows. ∎

We now assemble the three steps to prove Theorem 1.2.

###### Proof of Theorem 1.2.

Without loss of generality, we prove that $L^{(n)}\to L$ as random functions
on $[0,1]$, weakly in the topology of uniform convergence.

Take the above coupling between a sequence of Exponential LPP models and the
directed landscape, and consider the induced coupling between
$\Gamma_{\mathbf{0}}^{(n)}$, $L^{(n)}$, $L^{(n)}_{w}$ and $\pi_{\mathbf{0}}$,
$L$, $L_{w}$, for all $n\in\mathbb{N}$ and $w>0$. It is worth remarking that
under this coupling, the convergence $L^{(n)}\to L$ can be proved in even
stronger senses than the theorem states.

By Lemma 3.2, we assume the almost sure event that for each $m\in\mathbb{N}$,
$\lim_{n\to\infty}L^{(n)}_{1/m}=L_{1/m}$ uniformly as functions on $[0,1]$.

For any $m\in\mathbb{N}$ and $0\leq h\leq 1$ we have

$\displaystyle\mathbb{E}[(L^{(n)}(h)-L(h))^{2}]\leq$ $\displaystyle
3\mathbb{E}[(L^{(n)}(h)-(m/2)L^{(n)}_{1/m}(h))^{2}]+3(m/2)^{2}\mathbb{E}[(L^{(n)}_{1/m}(h)-L_{1/m}(h))^{2}]$
$\displaystyle+3\mathbb{E}[(L(h)-(m/2)L_{1/m}(h))^{2}].$

Let
$c_{m}=\limsup_{n\to\infty}\mathbb{E}[(L^{(n)}(h)-(m/2)L^{(n)}_{1/m}(h))^{2}]$.
By Lemma 3.2, $\mathbb{E}[(L^{(n)}_{1/m}(h)-L_{1/m}(h))^{2}]\to 0$ as
$n\to\infty$, since almost surely $L^{(n)}_{1/m}(h),L_{1/m}(h)\leq 1$. Thus we
have

$\limsup_{n\to\infty}\mathbb{E}[(L^{(n)}(h)-L(h))^{2}]\leq
3c_{m}+3\mathbb{E}[(L(h)-(m/2)L_{1/m}(h))^{2}].$

Now we send $m\to\infty$ on the right-hand side. We have
$\lim_{m\to\infty}c_{m}=0$ by Proposition 3.1. We also have that
$\lim_{m\to\infty}\mathbb{E}[(L(h)-(m/2)L_{1/m}(h))^{2}]=0$, since
$(L(h)-(m/2)L_{1/m}(h))^{2}\to 0$ almost surely by Proposition 1.1, and these
random variables are uniformly integrable in $m$, by Lemma 2.16. Thus we
conclude that $\lim_{n\to\infty}\mathbb{E}[(L^{(n)}(h)-L(h))^{2}]=0$. This
implies that $L^{(n)}\to L$ in $[0,1]$, in finite-dimensional distributions.

We next upgrade the result to weak convergence in the topology of uniform
convergence in $[0,1]$, using that $L^{(n)}$ and $L$ are non-decreasing.

For any $f:[0,1]\to\mathbb{R}$ and $k\in\mathbb{N}$, we denote by $M_{k}f$ the
function where $(M_{k}f)(i/k)=f(i/k)$ for any $i\in\llbracket 0,k\rrbracket$,
and linear between these points. Then $M_{k}L^{(n)}$ converges to $M_{k}L$ as
$n\to\infty$ weakly in the topology of uniform convergence on compact sets, by
the convergence of finite-dimensional distributions. For any $k\in\mathbb{N}$,
we let $D_{k}=\max_{i\in\llbracket 1,k\rrbracket}L(i/k)-L((i-1)/k)$, and
$D_{n,k}=\max_{i\in\llbracket 1,k\rrbracket}L^{(n)}(i/k)-L^{(n)}((i-1)/k)$.
Then for any $h\in[0,1]$ we have $|M_{k}L(h)-L(h)|\leq D_{k}$ and
$|M_{k}L^{(n)}(h)-L^{(n)}(h)|\leq D_{n,k}$, since $L^{(n)}$ and $L$ are non-
decreasing. Also by the convergence of finite-dimensional distributions, we
have $D_{n,k}\to D_{k}$ in distribution as $n\to\infty$, for any
$k\in\mathbb{N}$; and by Proposition 1.1 we have $D_{k}\to 0$ in distribution
as $k\to\infty$. Thus the conclusion follows. ∎

The remainder of this paper is devoted to proving Proposition 3.1.

##  4 Continuity of occupation measure

In this section, we prove Proposition 3.1. Without loss of generality, we
assume $h=1$.

As already indicated, our general strategy is to write $L^{(n)}_{w}(1)$ as an
average of local times of $\Gamma_{\mathbf{0}}$, on different lines parallel
to $\\{(i,i):i\in\mathbb{Z}\\}$; then we bound the differences of these local
times (on different lines), in terms of a second moment bound. More precisely,
for each $n\in\mathbb{N}$ and $m\in\mathbb{Z}$, we define

$Z^{(n)}[m]=|\\{0\leq i\leq n:(i+\lceil m/2\rceil,i-\lfloor
m/2\rfloor)\in\Gamma_{\mathbf{0}}\\}|.$

Then we have

$L^{(n)}(1)=2^{2/3}n^{-2/3}Z^{(n)}[0].$ (4.1)

For the function $L^{(n)}_{w}$, we have

$2nL^{(n)}_{w}(1)=\sum_{|m|\leq 2^{5/3}n^{2/3}w}Z^{(n)}[m]-\sum_{|2k+1|\leq
2^{5/3}n^{2/3}w}\mathds{1}[(n+k+1,n-k)\in\Gamma_{\mathbf{0}}].$ (4.2)

The second term occurs due to a rounding issue from (3.1) and is at most $1$
and hence negligible. We will prove the following result.

######  Proposition 4.1.

We have $\lim_{w\to 0}\limsup_{n\to\infty}\max_{|m|\leq
wn^{2/3}}n^{-2/3}\mathbb{E}[(Z^{(n)}[m]-Z^{(n)}[0])^{2}]=0$.

As indicated in Section 1.4, we expect
$\mathbb{E}[(Z^{(n)}[m]-Z^{(n)}[0])^{2}]$ to be of order $w^{4}n^{2/3}$ when
$|m|$ is approximately $wn^{2/3}$. However, Proposition 4.1 will already
suffice for our application, i.e., proving Proposition 3.1.

###### Proof of Proposition 3.1.

By (4.1) and (4.2), we have

$\left|2nL^{(n)}_{w}(1)-\sum_{|m|\leq 2^{5/3}n^{2/3}w}Z^{(n)}[m]\right|\leq
1.$

Thus we have

$\displaystyle\left|(2w)^{-1}L^{(n)}_{w}(1)-L^{(n)}(1)\right|\leq$
$\displaystyle\left|(4nw)^{-1}\sum_{|m|\leq
2^{5/3}n^{2/3}w}Z^{(n)}[m]-2^{2/3}n^{-1/3}Z^{(n)}[0]\right|+(4nw)^{-1}$
$\displaystyle\leq$
$\displaystyle\left|\left((4nw)^{-1}-\frac{2^{2/3}n^{-1/3}}{2\lfloor
2^{5/3}n^{2/3}w\rfloor+1}\right)\sum_{|m|\leq
2^{5/3}n^{2/3}w}Z^{(n)}[m]\right|$ $\displaystyle+\left|\sum_{|m|\leq
2^{5/3}n^{2/3}w}\frac{2^{2/3}n^{-1/3}}{2\lfloor
2^{5/3}n^{2/3}w\rfloor+1}(Z^{(n)}[m]-Z^{(n)}[0])\right|+(4nw)^{-1}.$

For the term in the second line, since $\sum_{m\in\mathbb{Z}}Z^{(n)}[m]\leq
2n+2$, we have

$\left|\left((4nw)^{-1}-\frac{2^{2/3}n^{-1/3}}{2\lfloor
2^{5/3}n^{2/3}w\rfloor+1}\right)\sum_{|m|\leq
2^{5/3}n^{2/3}w}Z^{(n)}[m]\right|\leq n^{-2/3}w^{-2}.$

For the term in the final line, by Cauchy-Schwarz inequality, we have

$\displaystyle\mathbb{E}\left[\left(\sum_{|m|\leq
2^{5/3}n^{2/3}w}\frac{2^{2/3}n^{-1/3}}{2\lfloor
2^{5/3}n^{2/3}w\rfloor+1}(Z^{(n)}[m]-Z^{(n)}[0])\right)^{2}\right]$
$\displaystyle\leq$ $\displaystyle\sum_{|m|\leq
2^{5/3}n^{2/3}w}\frac{2^{4/3}n^{-2/3}}{2\lfloor
2^{5/3}n^{2/3}w\rfloor+1}\mathbb{E}\left[(Z^{(n)}[m]-Z^{(n)}[0])^{2}\right],$

which decays to $0$ as $n\to\infty$ then $w\to 0$, by Proposition 4.1. Thus
the conclusion follows by combining the above two estimates. ∎

It remains to prove Proposition 4.1. The idea is to decompose
$\mathbb{E}[(Z^{(n)}[m]-Z^{(n)}[0])^{2}]$ into sums of two-point functions of
$\Gamma_{\mathbf{0}}$. The next step then involves establishing a certain
smoothness property of the two-point function as the points move about in the
spatial direction. It will suffice to simply consider the case when one point
remains the same, and another point moves by $m$ in the spatial direction. In
this direction, the following is the key estimate. Recall that $d(x,y)=x+y$
and $ad(x,y)=x-y$ for any $(x,y)\in\mathbb{Z}^{2}$.

######  Proposition 4.2.

For any $0<\theta,w<1$, take any $p_{1},p_{2},p_{3}\in\mathbb{Z}^{2}$ such
that

  1. 1.

$\theta n\leq d(p_{1})<d(p_{2})\leq d(p_{3})\leq 2n+1$,

  2. 2.

$d(p_{2})-d(p_{1})>\theta n$, $d(p_{3})-d(p_{2})\leq 1$,

  3. 3.

$|ad(p_{1})|,|ad(p_{2})|,|ad(p_{3})|\leq wn^{2/3}$.

Then, when $n$ is large enough (depending on $w,\theta$) we have
$|\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}}]-\mathbb{P}[p_{1},p_{3}\in\Gamma_{\mathbf{0}}]|<c_{w,\theta}n^{-4/3}$,
where $c_{w,\theta}$ depends only on $w,\theta$, and $\lim_{w\to
0}c_{w,\theta}=0$ for any $\theta$.

The heuristic behind the upper bound $c_{w,\theta}n^{-4/3}$ is as follows.
Since the spatial fluctuation of $\Gamma_{\mathbf{0}}$ is in the order of
$n^{2/3}$, we expect that each of $\mathbb{P}[p_{1}\in\Gamma_{\mathbf{0}}]$,
$\mathbb{P}[p_{2}\in\Gamma_{\mathbf{0}}]$, and
$\mathbb{P}[p_{3}\in\Gamma_{\mathbf{0}}]$ is in the order of $n^{-2/3}$.
Further, the separation lower bound on $d(p_{2})-d(p_{1})>\theta n$ implies a
degree of independence leading to both
$\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}}]$ and
$\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}}]$ being in the order of
$n^{-4/3}$. The fact that the distance between $p_{2}$ and $p_{3}$ is at most
$2wn^{2/3}$ produces the factor of $c_{w,\theta}$.

Postponing the proof of Proposition 4.2 till later, momentarily we prove
Proposition 4.1 assuming it.

$\mathbf{0}$$(i,i)$$(j,j)$$(i+\lceil m/2\rceil,i-\lfloor
m/2\rfloor)$$(j+\lceil m/2\rceil,j-\lfloor m/2\rfloor)$$\Gamma_{\mathbf{0}}$

$\mathbf{0}$$(i,i)$$(j,j)$$(i+\lceil m/2\rceil,i-\lfloor
m/2\rfloor)$$(j+\lceil m/2\rceil,j-\lfloor m/2\rfloor)$$\Gamma_{\mathbf{0}}$

$\mathbf{0}$$(i,i)$$(j,j)$$(i+\lceil m/2\rceil,i-\lfloor
m/2\rfloor)$$(j+\lceil m/2\rceil,j-\lfloor m/2\rfloor)$$\Gamma_{\mathbf{0}}$

Figure 2: An illustration of the proof of Proposition 4.1: we decompose the
second moment of $Z^{(n)}[m]-Z^{(n)}[0]$ as the sum of two-point probabilities
of $\Gamma_{\mathbf{0}}$. To apply Proposition 4.2, we further divide the sum
over the two points into three parts, which are respectively illustrated in
this figure.

###### Proof of Proposition 4.1.

In this proof we use $C,c>0$ to denote large and small constants.

We can write

$\displaystyle\mathbb{E}[(Z^{(n)}[0]-Z^{(n)}[m])^{2}]=$
$\displaystyle\mathbb{E}[(Z^{(n)}[0])^{2}]-2\mathbb{E}[Z^{(n)}[0]Z^{(n)}[m]]+\mathbb{E}[(Z^{(n)}[m])^{2}]$
$\displaystyle=$
$\displaystyle\sum_{i,j=0}^{n}\Big{(}\mathbb{P}[(i,i),(j,j)\in\Gamma_{\mathbf{0}}]-2\mathbb{P}[(i,i),(j+\lceil
m/2\rceil,j-\lfloor m/2\rfloor)\in\Gamma_{\mathbf{0}}]$
$\displaystyle+\mathbb{P}[(i+\lceil m/2\rceil,i-\lfloor m/2\rfloor),(j+\lceil
m/2\rceil,j-\lfloor m/2\rfloor)\in\Gamma_{\mathbf{0}}]\Big{)}.$

For the sum on $i,j$ from $0$ to $n$, Proposition 4.2 only applies to the case
where both $i,j$ are far from $0$, and $|i-j|$ is not small. There are two
other parts in the sum: where $i$ and $j$ are close to each other, and where
$i$ or $j$ is near $0$. For these two cases, we just bound the sum of each
probability (instead of the differences), using Lemma 2.18 and translation
invariant arguments.

More precisely, we consider the following three parts (see Figure 2):

  1. 1.

$\theta n/2\leq i,j\leq n,|i-j|>\theta n/2$,

  2. 2.

$0\leq i,j\leq n,|i-j|\leq\theta n/2$,

  3. 3.

$0\leq i\leq\theta n/2,0\leq j\leq n$ or $0\leq i\leq n,0\leq j\leq\theta
n/2$.

For Part 1, we use Proposition 4.2. Indeed, take any $0<\theta,w<1$. Since
$|m|\leq wn^{2/3}$, we have

$\displaystyle\sum_{\theta n/2\leq i,j\leq n,|i-j|>\theta
n/2}\Big{(}\mathbb{P}[(i,i),(j,j)\in\Gamma_{\mathbf{0}}]-2\mathbb{P}[(i,i),(j+\lceil
m/2\rceil,j-\lfloor m/2\rfloor)\in\Gamma_{\mathbf{0}}]$
$\displaystyle+\mathbb{P}[(i+\lceil m/2\rceil,i-\lfloor m/2\rfloor),(j+\lceil
m/2\rceil,j-\lfloor m/2\rfloor)\in\Gamma_{\mathbf{0}}]\Big{)}$
$\displaystyle<$ $\displaystyle 2c_{w,\theta}n^{2/3},$

since by Proposition 4.2, each summand is bounded by $2c_{w,\theta}n^{-4/3}$.

We next bound the sums for $i,j$ in Part 2 and Part 3. We write the sums as
the number of points in the intersection of $\Gamma_{\mathbf{0}}$ with some
short segments of length $\theta n$, and use the bound of Lemma 2.18. For Part
3, the only such short segment we would consider is the one near the origin.
For Part 2, we would consider short segments far from the origin as well, and
we would need to also use the fact that such segments would be disjoint from
$\Gamma_{\mathbf{0}}$ with high probability. Eventually, we will bound the
sums over $i,j$ in Part 2 and Part 3 by
$C\log(\theta^{-1})^{4}\theta^{1/3}n^{2/3}$ and $C\theta^{1/3}n^{2/3}$,
respectively. Combining this with the above bound for Part 1, we conclude that

$n^{-2/3}\mathbb{E}[(Z^{(n)}[m]-Z^{(n)}[0])^{2}]\leq
2c_{w,\theta}+C\log(\theta^{-1})^{4}\theta^{1/3}.$

By sending $w\to 0$, and then noting that $\theta>0$ is arbitrary, the
conclusion follows.

We now finish the bounds for Part 2 and Part 3. For this we let
$w<\theta^{2/3}$. For each $k\in\llbracket 1,\lceil 2/\theta\rceil\rrbracket$,
and $m\in\mathbb{Z}$, $|m|\leq wn^{2/3}$, we define

$Z^{(n,k)}[m]=|\\{(k-1)\theta n/2\leq i\leq(k+1)\theta n/2:(i+\lceil
m/2\rceil,i-\lfloor m/2\rfloor)\in\Gamma_{\mathbf{0}}\\}|.$

Part 2: We have

$\begin{split}&\sum_{0\leq i,j\leq n,|i-j|\leq\theta
n/2}\Big{(}\mathbb{P}[(i,i),(j,j)\in\Gamma_{\mathbf{0}}]\\\
&+\mathbb{P}[(i+\lceil m/2\rceil,i-\lfloor m/2\rfloor),(j+\lceil
m/2\rceil,j-\lfloor m/2\rfloor)\in\Gamma_{\mathbf{0}}]\Big{)}\\\
\leq&\sum_{k=1}^{\lceil
2/\theta\rceil}\mathbb{E}[(Z^{(n,k)}[0])^{2}]+\mathbb{E}[(Z^{(n,k)}[m])^{2}].\end{split}$
(4.3)

We next bound each $\mathbb{E}[(Z^{(n,k)}[m])^{2}]$. The general idea is to
use Lemma 2.18; but to get the desired bound, we also need to bound
$\mathbb{P}[Z^{(n,k)}[m]>0]$, since $Z^{(n,k)}[m]=0$ with high probability,
when $k$ is large. In other words, we need to bound the probability that
$\Gamma_{\mathbf{0}}$ intersects a line segment, which is in the
$(1,1)$-direction and has length $\theta n$. Our strategy is to reduce this to
the probability that $\Gamma_{\mathbf{0}}$ intersects a line segment in the
$(1,-1)$-direction with length of order $(\theta n)^{2/3}$, using transversal
fluctuation estimates of geodesics.

Take $j_{*}\in\mathbb{Z}$ such that $(\lceil(k-1)\theta
n/2\rceil+j_{*},\lceil(k-1)\theta n/2\rceil-j_{*})\in\Gamma_{\mathbf{0}}$.
Denote

$p_{*,1}=(\lceil(k-1)\theta n/2\rceil+\lfloor m/2+\log(k)^{2}(\theta
n)^{2/3}\rfloor,\lceil(k-1)\theta n/2\rceil-\lfloor m/2+\log(k)^{2}(\theta
n)^{2/3}\rfloor),$ $p_{*,2}=(\lceil(k-1)\theta n/2\rceil+\lfloor
m/2-\log(k)^{2}(\theta n)^{2/3}\rfloor,\lceil(k-1)\theta n/2\rceil-\lfloor
m/2-\log(k)^{2}(\theta n)^{2/3}\rfloor).$

The line segment connecting $p_{*,1}$ and $p_{*,2}$ is the $(1,-1)$-direction
line segment that we will use. If $Z^{(n,k)}[m]>0$, and $(\lceil(k-1)\theta
n/2\rceil+j_{*},\lceil(k-1)\theta n/2\rceil-j_{*})$ is not contained in the
line segment connecting $p_{*,1}$ and $p_{*,2}$, then (by the tree structure
of semi-infinite geodesics) one of $\Gamma_{p_{*,1}}$ and $\Gamma_{p_{*,2}}$
must intersect the set $\\{(i+\lceil m/2\rceil,i-\lfloor
m/2\rfloor):(k-1)\theta n/2\leq i\leq(k+1)\theta n/2\\}$. We thus have

$\mathbb{P}[Z^{(n,k)}[m]>0]\leq\mathbb{P}[|j_{*}-m/2|<2\log(k)^{2}(\theta
n)^{2/3}]\\\
+\mathbb{P}[(\Gamma_{p_{*,1}}\cup\Gamma_{p_{*,2}})\cap\\{(i+\lceil
m/2\rceil,i-\lfloor m/2\rfloor):(k-1)\theta n/2\leq i\leq(k+1)\theta
n/2\\}\neq\emptyset].$

By Lemma 2.10, the last line is bounded by $Ce^{-c\log(k)^{6}}$. By
translation invariance we have

$\mathbb{P}[|j_{*}-m/2|<2\log(k)^{2}(\theta
n)^{2/3}]<\frac{2\log(k)^{2}(\theta n)^{2/3}}{(k\theta
n)^{2/3}}\mathbb{E}[X],$

where

$X=|\\{p\in\mathbb{L}_{2\lceil(k-1)\theta n/2\rceil}:|ad(p)|\leq 2(k\theta
n)^{2/3}+|m|+4\log(k)^{2}(\theta n)^{2/3},p\in\cup_{|i|<(k\theta
n)^{2/3}}\Gamma_{(i,-i)}\\}|.$

Note that $|m|\leq wn^{2/3}<(\theta n)^{2/3}$. By Lemma 2.13 we have
$\mathbb{E}[X]<C$. Thus we conclude that

$\mathbb{P}[Z^{(n,k)}[m]>0]<C\log(k)^{2}k^{-2/3}.$

On the other hand, by Lemma 2.18 we also have that
$\mathbb{P}[Z^{(n,k)}[m]>M(\theta n)^{1/3}]<Ce^{-cM}$, for any $M>0$.
Therefore we have

$\begin{split}\mathbb{E}[(Z^{(n,k)}[m])^{2}]&=(\theta
n)^{2/3}\int_{0}^{\infty}2M\mathbb{P}[Z^{(n,k)}[m]>Mk^{-1/3}(\theta
n)^{1/3}]dM\\\ &\leq C(\theta
n)^{2/3}\int_{0}^{\infty}M(\log(k)^{2}k^{-2/3}\wedge e^{-cM})dM\\\ &\leq
C\log(k)^{4}k^{-2/3}(\theta n)^{2/3}.\end{split}$

For $k=1$ we have that $\mathbb{E}[(Z^{(n,1)}[m])^{2}]\leq C(\theta n)^{2/3}$
by Lemma 2.18. Thus we can bound the last line of (4.3) by
$C\log(\theta^{-1})^{4}\theta^{1/3}n^{2/3}$, as desired.

Part 3: We have that

$\begin{split}&\sum_{0\leq i\leq\theta n/2,0\leq j\leq
n}\Big{(}\mathbb{P}[(i,i),(j,j)\in\Gamma_{\mathbf{0}}]\\\
&+\mathbb{P}[(i+\lceil m/2\rceil,i-\lfloor m/2\rfloor),(j+\lceil
m/2\rceil,j-\lfloor m/2\rfloor)\in\Gamma_{\mathbf{0}}]\Big{)}\\\
\leq&\mathbb{E}[Z^{(n,1)}[0]Z^{(n)}[0]]+\mathbb{E}[Z^{(n,1)}[m]Z^{(n)}[m]]\\\
\leq&\sqrt{\mathbb{E}[(Z^{(n,1)}[0])^{2}]\mathbb{E}[(Z^{(n)}[0])^{2}]}+\sqrt{\mathbb{E}[(Z^{(n,1)}[m])^{2}]\mathbb{E}[(Z^{(n)}[m])^{2}]}.\end{split}$
(4.4)

By Lemma 2.18, $\mathbb{E}[(Z^{(n)}[m])^{2}]\leq Cn^{2/3}$ and
$\mathbb{E}[(Z^{(n,1)}[m])^{2}]\leq C(\theta n)^{2/3}$. Thus we can bound the
last line of (4.4) by $C\theta^{1/3}n^{2/3}$, as desired. ∎

###  4.1 Smoothness of the two-point distribution

The rest of this section is devoted to proving Proposition 4.2, the key
estimate on the two-point distribution of the geodesic $\Gamma_{\mathbf{0}}$.

We now explain the general strategy. We cut $\mathbb{Z}^{2}$ into two parts by
a line $\mathbb{L}_{m}$, which separates $p_{1}$ and $p_{2},p_{3}$. For the
random field $\\{\omega_{p}:p\in\mathbb{Z}^{2}\\}$, we translate the part
above $\mathbb{L}_{m}$, sending $p_{2}$ to $p_{3}$. It then remains to show
that, given $p_{1},p_{2}\in\Gamma_{\mathbf{0}}$, with a high probability this
translation would not affect the local structures of $\Gamma_{\mathbf{0}}$,
and hence after the translation, we have $p_{1},p_{3}\in\Gamma_{\mathbf{0}}$
(see Figure 3, left panel).

$p_{1}$$p_{2}$$p_{3}$$\Gamma_{\mathbf{0}}$$\Gamma_{\mathbf{0}}^{\prime}$$\mathbf{0}$$\mathbb{L}_{m}$$\mathbb{L}_{m_{-}}$

$\mathbb{L}_{m}$$\mathbb{L}_{m}$$d(p_{3})=d(p_{2})+1$$d(p_{3})=d(p_{2})$

Figure 3: An illustration of the proof of Proposition 4.2: by shifting the
random field above $\mathbb{L}_{m}$, it is likely that the geodesics
$\Gamma_{\mathbf{0}}$ and $\Gamma_{\mathbf{0}}^{\prime}$ are the same below
$\mathbb{L}_{m_{-}}$ (left panel). The right panel is on the details of
shifting the random field in two cases. The random field at the blue vertices
is shifted to the random field at the brown vertices. In the case of
$d(p_{3})=d(p_{2})+1$, the random variables
$\\{\omega_{p}^{\prime}:d(p)=2m\\}$ are independent of
$\\{\omega_{p}:p\in\mathbb{Z}^{2}\\}$.

To get the desired stability of the local structures, we will use properties
of the directed landscape, via the limiting transition of Theorem 2.14. This
way we would obtain results for $m/n$ converging to a constant. However, our
ultimate goal is to get estimates that are uniform over $p_{1},p_{2},p_{3}$
satisfying the three conditions in the statement of this proposition. To
address this issue, we will take $m$ among a mesh of numbers, whose
cardinality does not grow as $n\to\infty$, while covering all possible choices
of $p_{1},p_{2},p_{3}$.

###### Proof of Proposition 4.2.

Take $m=\lfloor\kappa\theta n/100\rfloor$ for some $\kappa\in\llbracket
1,100\theta^{-1}\rrbracket$, such that $d(p_{1})+\theta n/3<2m<d(p_{2})-\theta
n/3$. Note that the number of possible values of $m$ is $\lfloor
100\theta^{-1}\rfloor$. Such $m$ always exists due to the first two conditions
on $p_{1},p_{2},p_{3}$ in the statement of the proposition.

We consider another random field
$\\{\omega_{p}^{\prime}:p\in\mathbb{Z}^{2}\\}$ of i.i.d.
$\operatorname{Exp}(1)$ random variables, coupled with the original field
$\\{\omega_{p}:p\in\mathbb{Z}^{2}\\}$, by letting
$\omega_{p}=\omega_{p}^{\prime}$ for each $p\in\mathbb{Z}^{2},d(p)<2m$, and
$\omega_{p}=\omega_{p+p_{3}-p_{2}}^{\prime}$ for each
$p\in\mathbb{Z}^{2},d(p)\geq 2m$. In the case where $d(p_{3})=d(p_{2})+1$, the
weights $\\{\omega_{p}^{\prime}:d(p)=2m\\}$ remain undetermined, and we let
them be i.i.d. $\operatorname{Exp}(1)$ and independent of
$\\{\omega_{p}:p\in\mathbb{Z}^{2}\\}$ (see Figure 3, right panel). For each
$q\in\mathbb{Z}^{2}$ we let $\Gamma_{q}^{\prime}$ be the $(1,1)$-direction
semi-infinite geodesic in the field
$\\{\omega_{p}^{\prime}:p\in\mathbb{Z}^{2}\\}$.

We then have that

$\displaystyle\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}}]-\mathbb{P}[p_{1},p_{3}\in\Gamma_{\mathbf{0}}]$
$\displaystyle=\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}}]-\mathbb{P}[p_{1},p_{3}\in\Gamma^{\prime}_{\mathbf{0}}]$
$\displaystyle\leq\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}},\\{p_{1},p_{3}\\}\not\subset\Gamma_{\mathbf{0}}^{\prime}]$
$\displaystyle\leq\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}},p_{1}\not\in\Gamma_{\mathbf{0}}^{\prime}]+\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}},p_{3}\not\in\Gamma_{\mathbf{0}}^{\prime}].$

We next bound
$\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}},p_{1}\not\in\Gamma_{\mathbf{0}}^{\prime}]$.
The bound for the other term follows the same arguments. The general idea is
to decompose the event into independent events whose probabilities are easier
to control.

Take $m_{-}=\lfloor\kappa_{-}\theta n/100\rfloor$ for some
$\kappa_{-}\in\llbracket 1,100\theta^{-1}\rrbracket$, such that
$d(p_{1})+\theta n/7<2m_{-}<2m-\theta n/7$. In words, $2m_{-}$ is order
$\theta n$ separated from $d(p_{1})$ and $2m$, as illustrated in Figure 3
(left panel). Like $m$, this $m_{-}$ is also chosen among a finite mesh of
numbers.

Note that on and below $\mathbb{L}_{m_{-}+1}$, both $\Gamma_{\mathbf{0}}$ and
$\Gamma_{\mathbf{0}}^{\prime}$ are geodesics in the random field
$\\{\omega_{p}:p\in\mathbb{Z}^{2}\\}$. Thus
$p_{1}\in\Gamma_{\mathbf{0}},p_{1}\not\in\Gamma_{\mathbf{0}}^{\prime}$ would
imply that
$\Gamma_{\mathbf{0}}\cap\mathbb{L}_{m_{-}+1}\neq\Gamma_{\mathbf{0}}^{\prime}\cap\mathbb{L}_{m_{-}+1}$
(see Figure 3, left panel). It now suffices to bound

$\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}},\;\Gamma_{\mathbf{0}}\cap\mathbb{L}_{m_{-}+1}\neq\Gamma_{\mathbf{0}}^{\prime}\cap\mathbb{L}_{m_{-}+1}].$
(4.5)

We would like to exploit the fact that owing to their separation, the three
events $p_{1}\in\Gamma_{\mathbf{0}}$, $p_{2}\in\Gamma_{\mathbf{0}}$, and
$\Gamma_{\mathbf{0}}\cap\mathbb{L}_{m_{-}+1}\neq\Gamma_{\mathbf{0}}^{\prime}\cap\mathbb{L}_{m_{-}+1}$
mostly rely on some local randomness: below $\mathbb{L}_{m_{-}}$, above
$\mathbb{L}_{m}$, or between these two lines which can be used to construct
independent proxy events. We design these next.

Take $\beta>0$ to be a small constant, which will be used to quantify the
regularities of (1) the passage times from $\mathbf{0}$ to
$\mathbb{L}_{m_{-}}$, and (2) the Busemann function along $\mathbb{L}_{m}$.

  1. 1.

Define $\overline{\mathcal{E}}_{1}$ to be the event that
$p_{1}\in\cup_{|i|\leq m_{-}}\Gamma_{\mathbf{0},(m_{-}-i,m_{-}+i)}$. A reason
behind this definition is that this event is measurable with respect to
$\\{\omega_{p}:d(p)\leq 2m_{-}\\}$, the random field on and below
$\mathbb{L}_{m_{-}}$. Let $\mathcal{E}_{1}$ be the event that
$\overline{\mathcal{E}}_{1}$ holds along with the following regularity
conditions:

$|T_{\mathbf{0},(m_{-}-i,m_{-}+i)}-T_{\mathbf{0},(m_{-}-j,m_{-}+j)}|<\beta^{-2}|i-j|^{1/2}(n^{2/3}/|i-j|)^{0.01},\\\
\forall|i|,|j|<\beta^{-1}n^{2/3},wn^{2/3}<|i-j|<n^{2/3},$ (4.6)
$|T_{\mathbf{0},(m_{-}-i,m_{-}+i)}-\mathbb{E}[T_{\mathbf{0},(m_{-}-i,m_{-}+i)}]|<\beta^{-1}n^{1/3}+\beta
i^{2}/n,\quad\forall i\in\mathbb{Z}.$ (4.7)

Then $\mathcal{E}_{1}$ is also measurable with respect to
$\\{\omega_{p}:d(p)\leq 2m_{-}\\}$. For the passage times from $\mathbf{0}$ to
$\mathbb{L}_{m_{-}}$, the local fluctuation should actually exhibit random
walk behavior (similar to how the Airy2 process is locally Brownian like as
stated in Section 2.2.2; this is also indicated by the random walk behavior of
the Busemann function stated in Section 2.3.1), so the left-hand side of (4.6)
should be of order $|i-j|^{1/2}$. For the bound to hold uniformly in $i$ and
$j$, an extra factor of $(n^{2/3}/|i-j|)^{0.01}$ is added to the right-hand
side. For (4.7), the term $\beta i^{2}/n$ is added to make this bound hold for
all $i\in\mathbb{Z}$. As $\beta$ is small, (4.7) ensures that
$T_{\mathbf{0},(m_{-}-i,m_{-}+i)}$ decays as $|i|$ increases, since the
expectation $\mathbb{E}[T_{\mathbf{0},(m_{-}-i,m_{-}+i)}]$ already has a
parabolic decay in $i$ (see (2.3)).

  2. 2.

Similarly, let $\overline{\mathcal{E}}_{2}$ be the event where
$p_{2}\in\cup_{|i|\leq m}\Gamma_{(m-i,m+i)}$. Let $\mathcal{E}_{2}$ be the
event that $\overline{\mathcal{E}}_{2}$ as well as the following regularity
conditions hold:

$|G((m-i,m+i))-G((m-j,m+j))|<\beta^{-2}|i-j|^{1/2}(n^{2/3}/|i-j|)^{0.01},\\\
\forall|i|,|j|<\beta^{-1}n^{2/3},wn^{2/3}<|i-j|<n^{2/3},$
$|G((m-i,m+i))-G((m,m))|<\beta^{-1}n^{1/3}+\beta i^{2}/n,\quad\forall
i\in\mathbb{Z}.$ (4.8)

Here $G$ is the Busemann function from Section 2.3.1. The reasoning behind the
design of these bounds is similar to those for (4.6) and (4.7). Then
$\overline{\mathcal{E}}_{2}$ and $\mathcal{E}_{2}$ are measurable with respect
to $\\{\omega_{p}:d(p)\geq 2m\\}$, the random field on and above
$\mathbb{L}_{m}$.

  3. 3.

Let $\mathcal{E}_{3}$ be the event where
$\Gamma_{\mathbf{0}}\cap\mathbb{L}_{m_{-}+1}=\Gamma_{\mathbf{0}}^{\prime}\cap\mathbb{L}_{m_{-}+1}$.

Note that the definition of $\overline{\mathcal{E}}_{1}$ involves not the
geodesic $\Gamma_{\mathbf{0}}$ but the geodesics $\cup_{|i|\leq
m_{-}}\Gamma_{\mathbf{0},(m_{-}-i,m_{-}+i)}$ and similarly for
$\overline{\mathcal{E}}_{2}$. This helps make
$\overline{\mathcal{E}}_{1},\mathcal{E}_{1}$ independent of
$\mathcal{E}_{2},\overline{\mathcal{E}}_{2}$. The probability (4.5) can now be
written as
$\mathbb{P}[\\{p_{1},p_{2}\in\Gamma_{\mathbf{0}}\\}\cap\mathcal{E}_{3}^{c}]$,
which is bounded by
$\mathbb{P}[\overline{\mathcal{E}}_{1}\cap\overline{\mathcal{E}}_{2}\cap\mathcal{E}_{3}^{c}]$.
The additional regularity conditions are added to allow us to invoke a crucial
geodesic stability result that we will prove shortly (Proposition 4.6 below)
to bound the probability of $\mathcal{E}_{3}^{c}$ (see Claim 4.5 below).

We assume the following claims when $n$ is large enough (depending on
$w,\beta,\theta$).

######  Claim 4.3.

We have
$\mathbb{P}[\overline{\mathcal{E}}_{1}],\mathbb{P}[\overline{\mathcal{E}}_{2}]<C_{\theta}n^{-2/3}$,
where $C_{\theta}>0$ is a constant depending only on $\theta$.

######  Claim 4.4.

For each $0<\beta,\theta<1$, there exists a constant $c_{\beta,\theta}^{bad}$,
such that
$\mathbb{P}[\overline{\mathcal{E}}_{1}\setminus\mathcal{E}_{1}],\mathbb{P}[\overline{\mathcal{E}}_{2}\setminus\mathcal{E}_{2}]<c_{\beta,\theta}^{bad}n^{-2/3}$
for $n$ large enough (depending on $\beta,\theta$), and $\lim_{\beta\to
0}c_{\beta,\theta}^{bad}=0$ for fixed $\theta>0$.

######  Claim 4.5.

For each $0<w,\beta,\theta<1$, there exists a constant
$c_{w,\beta,\theta}^{disj}$, such that
$\mathbb{P}[\mathcal{E}_{3}^{c}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}]<c_{w,\beta,\theta}^{disj}$,
and $\lim_{\beta\to 0}\limsup_{w\to 0}c_{w,\beta,\theta}^{disj}=0$ for fixed
$\theta>0$.

Thus we must have that

$\displaystyle\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}},p_{1}\not\in\Gamma_{\mathbf{0}}^{\prime}]\leq$
$\displaystyle\mathbb{P}[\\{p_{1},p_{2}\in\Gamma_{\mathbf{0}}\\}\cap\mathcal{E}_{3}^{c}]$
$\displaystyle\leq$
$\displaystyle\mathbb{P}[\overline{\mathcal{E}}_{1}\cap\overline{\mathcal{E}}_{2}\cap\mathcal{E}_{3}^{c}]$
$\displaystyle\leq$
$\displaystyle\mathbb{P}[\overline{\mathcal{E}}_{1}\setminus\mathcal{E}_{1}]\mathbb{P}[\overline{\mathcal{E}}_{2}]+\mathbb{P}[\overline{\mathcal{E}}_{1}]\mathbb{P}[\overline{\mathcal{E}}_{2}\setminus\mathcal{E}_{2}]+\mathbb{P}[\mathcal{E}_{1}]\mathbb{P}[\mathcal{E}_{2}]\mathbb{P}[\mathcal{E}_{3}^{c}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}]$
$\displaystyle<$
$\displaystyle(2C_{\theta}c_{\beta,\theta}^{bad}+C_{\theta}^{2}c_{w,\beta,\theta}^{disj})n^{-4/3}.$

Let $c_{w,\theta}$ (in the statement of Proposition 4.2) be
$\inf_{\beta>0}2C_{\theta}c_{\beta,\theta}^{bad}+C_{\theta}^{2}c_{w,\beta,\theta}^{disj}$.
For fixed $\theta>0$, we have

$\limsup_{w\to 0}c_{w,\theta}\leq
2C_{\theta}c_{\beta,\theta}^{bad}+\limsup_{w\to
0}C_{\theta}^{2}c_{w,\beta,\theta}^{disj}$

for any $\beta>0$ in the right-hand side. Then by sending $\beta\to 0$ in the
right-hand side, by Claims 4.4 and 4.5 we have $c_{w,\theta}\to 0$ as $w\to
0$. Thus we get the desired upper bound for
$\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}},p_{1}\not\in\Gamma_{\mathbf{0}}^{\prime}]$.
By a similar argument we can bound
$\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}},p_{3}\not\in\Gamma_{\mathbf{0}}^{\prime}]$,
thus we can upper bound
$\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}}]-\mathbb{P}[p_{1},p_{3}\in\Gamma_{\mathbf{0}}]$.
By symmetry we can also upper bound
$\mathbb{P}[p_{1},p_{3}\in\Gamma_{\mathbf{0}}]-\mathbb{P}[p_{1},p_{2}\in\Gamma_{\mathbf{0}}]$,
thereby finish the proof. ∎

We now prove the claims. For Claim 4.3 which bounds
$\mathbb{P}[\overline{\mathcal{E}}_{1}]$ and
$\mathbb{P}[\overline{\mathcal{E}}_{2}]$, we use translation invariance of
Exponential LPP, and the bound on the number of disjoint geodesics (Lemma
2.12).

###### Proof of Claim 4.3.

For each $j\in\mathbb{Z}$, we let $\overline{\mathcal{E}}_{1,j}$ be the event
$\overline{\mathcal{E}}_{1}$ translated by $(-j,j)$. Then we have

$\mathbb{P}[\overline{\mathcal{E}}_{1}]=(2\lfloor
n^{2/3}\rfloor+1)^{-1}\sum_{|j|\leq
n^{2/3}}\mathbb{P}[\overline{\mathcal{E}}_{1,j}]\leq(2\lfloor
n^{2/3}\rfloor+1)^{-1}\mathbb{E}[X],$

where

$X=|\\{p\in\mathbb{Z}^{2}:d(p)=d(p_{1}),|ad(p)|\leq 3n^{2/3},p\in\cup_{|i|\leq
n^{2/3},j\in\mathbb{Z}}\Gamma_{(-i,i),(m_{-}-j,m_{-}+j)}\\}|.$

By Lemma 2.12, $\mathbb{E}[X]$ is bounded by a constant depending only on
$\theta$. Thus we get the desired bound for
$\mathbb{P}[\overline{\mathcal{E}}_{1}]$.

Similarly, we have

$\mathbb{P}[\overline{\mathcal{E}}_{2}]\leq(2\lfloor
n^{2/3}\rfloor+1)^{-1}\mathbb{E}[Y],$

where

$Y=|\\{p\in\mathbb{Z}^{2}:d(p)=d(p_{2}),|ad(p)|\leq
3n^{2/3},p\in\cup_{j\in\mathbb{Z}}\Gamma_{(m-j,m+j)}\\}|.$

By Lemma 2.13, $\mathbb{E}[Y]$ is bounded by a constant depending only on
$\theta$ which then implies the sought bound for
$\mathbb{P}[\overline{\mathcal{E}}_{2}]$. ∎

We next prove Claim 4.4. In addition to the translation invariance arguments,
to prove regularity and bound
$\mathbb{P}[\overline{\mathcal{E}}_{2}\setminus\mathcal{E}_{2}]$, we use the
random walk description of the Busemann function on $\mathbb{L}_{m}$. For
$\mathbb{P}[\overline{\mathcal{E}}_{1}\setminus\mathcal{E}_{1}]$, we instead
send $n\to\infty$ and use regularities of the directed landscape, because of
the absence of exact random walk descriptions of the joint distribution of
passage times for point-to-point geodesics in the pre-limit.

###### Proof of Claim 4.4.

For each $j\in\mathbb{Z}$, recall the events $\overline{\mathcal{E}}_{1,j}$,
$\overline{\mathcal{E}}_{2,j}$ from the proof of Claim 4.3. We also let
$\mathcal{E}_{1,j}$ and $\mathcal{E}_{2,j}$ be the event $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ translated by $(-j,j)$, respectively. Also recall the random
variables $X$ and $Y$ from the proof of Claim 4.3.

By translation invariance we have

$\mathbb{P}[\overline{\mathcal{E}}_{1}\setminus\mathcal{E}_{1}]=(2\lfloor
n^{2/3}\rfloor+1)^{-1}\sum_{|j|\leq
n^{2/3}}\mathbb{P}[\overline{\mathcal{E}}_{1,j}\setminus\mathcal{E}_{1,j}]\leq(2\lfloor
n^{2/3}\rfloor+1)^{-1}\mathbb{E}[X\mathds{1}[\mathcal{E}_{1}^{\prime}\cup\mathcal{E}_{1}^{\prime\prime}]],$
(4.9)

where $\mathcal{E}_{1}^{\prime}$ is the event where there exist
$i,j,k\in\mathbb{Z}$, such that $|i|,|j|<2\beta^{-1}n^{2/3}$, $|k|\leq
n^{2/3}$, $wn^{2/3}<|i-j|<n^{2/3}$, and
$|T_{(-k,k),(m_{-}-i,m_{-}+i)}-T_{(-k,k),(m_{-}-j,m_{-}+j)}|\geq\beta^{-2}|i-j|^{1/2}(n^{2/3}/|i-j|)^{0.01}$;
and $\mathcal{E}_{1}^{\prime\prime}$ is the event where there exist
$i,k\in\mathbb{Z}$, such that $|k|\leq n^{2/3}$, and
$|T_{(-k,k),(m_{-}-i,m_{-}+i)}-\mathbb{E}[T_{(-k,k),(m_{-}-i,m_{-}+i)}]|\geq\beta^{-1}n^{1/3}+\beta(i-k)^{2}/n$.
Namely, $\mathcal{E}_{1}^{\prime}$ (resp. $\mathcal{E}_{1}^{\prime\prime}$) is
the “translation union version” (in an order $n^{2/3}$ window) of the event
that the regularity condition (4.6) (resp. (4.7)) does not hold.

By Theorem 2.14, for fixed $\beta,\theta,\kappa_{-}$ (recall that
$m_{-}=\lfloor\kappa_{-}\theta n/100\rfloor$), we have that
$\mathbb{P}[\mathcal{E}_{1}^{\prime}]$ converges as $n\to\infty$. The limit is
the probability of the following event: there exist
$x,y\in[-2^{1/3}\beta^{-1},2^{1/3}\beta^{-1}]$ with
$w2^{-2/3}<|x-y|<2^{-2/3}$, and $z\in[-2^{-2/3},2^{-2/3}]$, such that

$|\mathcal{L}(z,0;x,\kappa_{-}\theta/100)-\mathcal{L}(z,0;y,\kappa_{-}\theta/100)|\geq
2^{-2}\beta^{-2}|x-y|^{1/2-0.01}.$

Note that by the quadrangle inequality (Lemma 2.2), if the above inequality
holds, it must also hold when $z$ is replaced by either $-2^{-2/3}$ or
$2^{-2/3}$. Then by covering $[-2^{1/3}\beta^{-1},2^{1/3}\beta^{-1}]$ with
overlapping intervals, we have that the above probability is bounded by the
probability of the following event: there exist $i\in\mathbb{Z}$,
$|i|<2\beta^{-1}$, and $x,y\in[(i-1)2^{-2/3},(i+1)2^{-2/3}]$ such that

$|\mathcal{L}(2^{-2/3},0;x,\kappa_{-}\theta/100)-\mathcal{L}(2^{-2/3},0;y,\kappa_{-}\theta/100)|\geq
2^{-2}\beta^{-2}|x-y|^{1/2-0.01},$

or

$|\mathcal{L}(-2^{-2/3},0;x,\kappa_{-}\theta/100)-\mathcal{L}(-2^{-2/3},0;y,\kappa_{-}\theta/100)|\geq
2^{-2}\beta^{-2}|x-y|^{1/2-0.01}.$

Since
$x\mapsto(\kappa_{-}\theta/100)^{-1/3}\mathcal{L}(2^{-2/3},0;x+2^{-2/3},\kappa_{-}\theta/100)$
is the Airy2 process, we can bound this probability for each $i$, using that
the Airy2 process plus a parabola is stationary and Theorem 2.5, and take a
union bound for all $i$. We conclude that (for fixed $\beta,\theta$)
$\lim_{n\to\infty}\mathbb{P}[\mathcal{E}_{1}^{\prime}]<C\beta^{-1}e^{-c\beta^{-2}}$,
where $C,c>0$ are constants depending on $\theta,\kappa_{-}$.

For $\overline{\mathcal{E}}_{1}^{\prime\prime}$, we use estimates in Section
2.3.2 to bound its probability. A key thing to notice is the slope conditions
in these estimates. When $|i|\leq m_{-}/2$, the slope of
$(m_{-}-i+k,m_{-}+i-k)$ is always bounded away from $0$ and $\infty$. Thus we
split $\\{(m_{-}-i,m_{-}+i):|i|\leq m_{-}/2\\}$ and $\\{(-k,k):|k|\leq
n^{2/3}\\}$ into segments of length $m_{-}^{2/3}$, and apply Proposition 2.8
to each one of them and take a union bound. When $|i|>m_{-}/2$, the slope can
be close to $0$ or very large, in which cases Proposition 2.8 cannot be
applied, so we just apply (2.1) in Theorem 2.7 to each
$T_{(-k,k),(m_{-}-i,m_{-}+i)}$ and take a union bound. In the end, we conclude
that (for fixed $\beta,\theta$)
$\limsup_{n\to\infty}\mathbb{P}[\mathcal{E}_{1}^{\prime\prime}]<C\beta^{-1/2}e^{-c\beta^{-1}}$,
where $C,c>0$ are constants depending on $\theta,\kappa_{-}$.

By considering all $\kappa_{-}\in\llbracket 1,100\theta^{-1}\rrbracket$, we
have
$\mathbb{P}[\mathcal{E}_{1}^{\prime}\cup\mathcal{E}_{1}^{\prime\prime}]<C\beta^{-1/2}e^{-c\beta^{-1}}$
for constants $C,c>0$ depending on $\theta$. Besides, by Lemma 2.12 we also
have that $\mathbb{P}[X>M]<Ce^{-cM^{1/12800}}$ for any $M>0$, with $C,c>0$
being constants that depend on $\theta$. Thus by (4.9), we get the desired
bound for $\mathbb{P}[\overline{\mathcal{E}}_{1}\setminus\mathcal{E}_{1}]$.

For $\mathbb{P}[\overline{\mathcal{E}}_{2}\setminus\mathcal{E}_{2}]$, we
similarly have

$\mathbb{P}[\overline{\mathcal{E}}_{2}\setminus\mathcal{E}_{2}]\leq(2\lfloor
n^{2/3}\rfloor+1)^{-1}\mathbb{E}[Y\mathds{1}[\mathcal{E}_{2}^{\prime}\cap\mathcal{E}_{2}^{\prime\prime}]],$

where $\mathcal{E}_{2}^{\prime}$ is the event where there exist
$i,j\in\mathbb{Z}$, such that $|i|,|j|<2\beta^{-1}n^{2/3}$,
$wn^{2/3}<|i-j|<n^{2/3}$, and
$|G((m-i,m+i))-G((m-j,m+j))|\geq\beta^{-2}|i-j|^{1/2}(n^{2/3}/|i-j|)^{0.01}$;
and $\mathcal{E}_{2}^{\prime\prime}$ is the event where there exists
$i\in\mathbb{Z}$, such that
$|G((m-i,m+i))-G((m,m))|\geq\beta^{-1}n^{1/3}+\beta i^{2}/n$. We note that
$j\mapsto G((m-j,m+j))-G((m,m))$ is a random walk where each step has
Exponential tails (see the second property in Section 2.3.1). Then we have
that for fixed $\beta,\theta,\kappa$, the probabilities
$\mathbb{P}[\mathcal{E}_{2}^{\prime}]$ and
$\mathbb{P}[\mathcal{E}_{2}^{\prime\prime}]$ converge as $n\to\infty$; and for
the limits, they decay to zero as $\beta\to 0$, for any fixed $\theta,\kappa$.
Thus with the Exponential tail of $Y$ obtained from Lemma 2.13, we get the
desired bound for
$\mathbb{P}[\overline{\mathcal{E}}_{2}\setminus\mathcal{E}_{2}]$. ∎

Finally, for Claim 4.5, we use the following stability result on geodesics.

Let $H_{1},H_{2}>0$. For each $w>0$ and $n\in\mathbb{N}$, we let
$\mathscr{F}_{w,n}$ be the collection of all functions $f:\llbracket-
H_{1}n^{2/3},H_{1}n^{2/3}\rrbracket\to\mathbb{R}$, such that $|f|\leq
H_{2}n^{1/3}$, and $|f(i)-f(j)|<H_{2}|i-j|^{1/2}(n^{2/3}/|i-j|)^{0.01}$ for
any $|i|,|j|\leq H_{1}n^{2/3}$, $wn^{2/3}<|i-j|<n^{2/3}$. For each
$f_{1},f_{2}\in\mathscr{F}_{w,n}$, suppose

$i_{*},j_{*}=\operatorname{argmax}_{i,j\in\llbracket-
H_{1}n^{2/3},H_{1}n^{2/3}\rrbracket}f_{1}(i)+f_{2}(j)+T_{(-i,i),(n-j,n+j)},$

then we let
$\Gamma_{f_{1},f_{2}}^{n}=\Gamma_{(-i_{*},i_{*}),(n-j_{*},n+j_{*})}$. Note
that almost surely, such $i_{*}$ and $j_{*}$ are unique, because the passage
times $T_{(-i,i),(n-j,n+j)}$ have continuous distributions. The next result
states that $\Gamma_{f_{1},f_{2}}^{n}$ overlaps significantly with
$\Gamma_{f_{1},f_{3}}^{n}$

######  Proposition 4.6.

For each $\epsilon>0$, the following is true for any small enough $w,g>0$
(depending on $H_{1},H_{2}$) and any $n\in\mathbb{N}$ large enough (depending
on $w,g$ and $H_{1},H_{2}$). Take any $f_{1},f_{2},f_{3}\in\mathscr{F}_{w,n}$
with $|f_{2}-f_{3}|\leq gn^{1/3}$, we have

$\mathbb{P}[\Gamma_{f_{1},f_{2}}^{n}\cap\mathbb{L}_{\lfloor
n/2\rfloor}=\Gamma_{f_{1},f_{3}}^{n}\cap\mathbb{L}_{\lfloor
n/2\rfloor}]>1-\epsilon.$

We leave its proof to the next section and prove Claim 4.5 now.

###### Proof of Claim 4.5.

Denote $l=m-m_{-}-1$. Let $H_{1}>0$ be a constant depending only on
$\beta,\theta$, such that we always have
$\beta^{-0.9}n^{2/3}<H_{1}l^{2/3}<\beta^{-1}n^{2/3}$. Such an $H_{1}$ exists
whenever $\beta$ is small enough (depending on $\theta$). Let $H_{2}>0$ be
large enough depending on $\beta,\theta$.

We want to apply Proposition 4.6 to the strip between $\mathbb{L}_{m_{-}}$ and
$\mathbb{L}_{m}$. The regularity conditions (of Proposition 4.6) are given by
$\mathcal{E}_{1}\cap\mathcal{E}_{2}$. Also, Proposition 4.6 considers boundary
functions in $\mathscr{F}_{w,n}$, which are defined on the finite interval
$\llbracket-H_{1}n^{2/3},H_{1}n^{2/3}\rrbracket$. To apply it we need to bound
transversal fluctuations of $\Gamma_{\mathbf{0}}$ and
$\Gamma_{\mathbf{0}}^{\prime}$.

Towards this, we first express parts of $\Gamma_{\mathbf{0}}$ and
$\Gamma_{\mathbf{0}}^{\prime}$ in the strip between $\mathbb{L}_{m_{-}}$ and
$\mathbb{L}_{m}$ in terms of boundary conditions. We let $G^{\prime}$ be the
Busemann function for the random field
$\\{\omega_{p}^{\prime}:p\in\mathbb{Z}^{2}\\}$. Because of the shift by
$p_{3}-p_{2},$ we then have that
$G(p)-G(q)=G^{\prime}(p+p_{3}-p_{2})-G^{\prime}(q+p_{3}-p_{2})$, for any
$p,q\in\mathbb{Z}^{2}$ on or above $\mathbb{L}_{m}$. Next, define
$f_{1},f_{2},f_{3}:\mathbb{Z}\to\mathbb{R}$ as

$f_{1}(i)=T_{\mathbf{0},(m_{-}+i,m_{-}-i)}\vee
T_{\mathbf{0},(m_{-}+i+1,m_{-}-i-1)}-4m_{-},$ $f_{2}(i)=G((m+i,m-i))\vee
G((m+i+1,m-i-1))-G((m,m)),$ $f_{3}(i)=G^{\prime}((m+i,m-i))\vee
G^{\prime}((m+i+1,m-i-1))-G^{\prime}((m,m)).$

In these definitions, we take the maximum over two nearest points, because for
$\Gamma_{\mathbf{0}}$ or $\Gamma_{\mathbf{0}}^{\prime}$, the immediate steps
beyond the strip (between $\mathbb{L}_{m_{-}}$ and $\mathbb{L}_{m}$) can take
one of two possible vertices. We let
$\Gamma_{*}=\Gamma_{(m_{-}+i_{*}+1,m_{-}-i_{*}),(m+j_{*},m-j_{*}-1)}$, where

$i_{*},j_{*}=\operatorname{argmax}_{i,j\in\mathbb{Z}}f_{1}(i)+f_{2}(j)+T_{(m_{-}+i+1,m_{-}-i),(m+j,m-j-1)};$

and define
$\Gamma_{*}^{\prime}=\Gamma_{(m_{-}+i_{*}^{\prime}+1,m_{-}-i_{*}^{\prime}),(m+j_{*}^{\prime},m-j_{*}^{\prime}-1)}$
the same way as $\Gamma_{*}$, but using $f_{3}$ in place of $f_{2}$. We then
have that $\Gamma_{*}$ is the part of $\Gamma_{\mathbf{0}}$ between
$\mathbb{L}_{m_{-}}$ and $\mathbb{L}_{m}$, and similarly $\Gamma_{*}^{\prime}$
is the part of $\Gamma_{\mathbf{0}}^{\prime}$ between $\mathbb{L}_{m_{-}}$ and
$\mathbb{L}_{m}$.

For the regularity conditions, we need (1) the functions $f_{1},f_{2},f_{3}$
(restricted to finite intervals) to be in $\mathscr{F}_{w,n}$, and (2) $f_{2}$
to be close to $f_{3}$. Specifically, for (1), we let $\mathcal{E}_{reg}$ be
the event where for each $\iota=1,2,3$, $f_{\iota}$ on $\llbracket-
H_{1}l^{2/3},H_{1}l^{2/3}\rrbracket$ is in $\mathscr{F}_{w,n}$ (i.e.,
$|f_{\iota}|<H_{2}l^{1/3}$ on $\llbracket-
H_{1}l^{2/3},H_{1}l^{2/3}\rrbracket$; and
$|f_{\iota}(i)-f_{\iota}(j)|<H_{2}|i-j|^{1/2}(l^{2/3}/|i-j|)^{0.01}$ for any
$|i|,|j|\leq H_{1}l^{2/3}$, $wl^{2/3}<|i-j|<l^{2/3}$). For (2), we let
$\mathcal{E}_{diff}$ be the event where $|f_{2}-f_{3}|\leq
H_{2}w^{1/2-0.01}l^{1/3}$ on $\llbracket-H_{1}l^{2/3},H_{1}l^{2/3}\rrbracket$.
We have that these events are essentially implied by
$\mathcal{E}_{1}\cap\mathcal{E}_{2}$. Indeed, when $d(p_{2})=d(p_{3})$, one
precisely has
$\mathcal{E}_{1}\cap\mathcal{E}_{2}\subset\mathcal{E}_{reg}\cap\mathcal{E}_{diff}$
(since $H_{1}l^{2/3}<\beta^{-1}n^{2/3}$ by the choice of $H_{1}$). When
$d(p_{3})=d(p_{2})+1$, $\mathcal{E}_{1}\cap\mathcal{E}_{2}$ does not imply
$\mathcal{E}_{reg}\cap\mathcal{E}_{diff}$, because of the extra random weights
$\\{\omega_{(m+i,m-i)}^{\prime}:|i|\leq H_{1}l^{2/3}\\}$ (which are
independent of $\mathcal{E}_{1}\cap\mathcal{E}_{2}$). However, as long as each
$\omega_{(m+i,m-i)}^{\prime}$ is not too large (i.e., $\leq
w^{1/2-0.01}l^{1/3}$), the event $\mathcal{E}_{reg}\cap\mathcal{E}_{diff}$
still holds. Therefore we have

$\mathbb{P}[\mathcal{E}_{reg}^{c}\cup\mathcal{E}_{diff}^{c}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}]<\mathbb{P}[\max_{|i|\leq
H_{1}l^{2/3}}\omega_{(m+i,m-i)}^{\prime}>w^{1/2-0.01}l^{1/3}],$

and the right hand side $\to 0$ as $n\to\infty$, for fixed $w,\beta,\theta$.
Then we have

$\lim_{n\to\infty}\mathbb{P}[\mathcal{E}_{reg}^{c}\cup\mathcal{E}_{diff}^{c}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}]=0$
(4.10)

for any $w,\beta,\theta$.

To apply Proposition 4.6, we need that the optimal paths with endpoints in
$\\{(m_{-}+i+1,m_{-}-i):i\in\llbracket-H_{1}l^{2/3},H_{1}l^{2/3}\rrbracket\\}$
and $\\{(m+j+1,m-j):j\in\llbracket-H_{1}l^{2/3},H_{1}l^{2/3}\rrbracket\\}$ are
the same as the overall optimal paths $\Gamma_{*}$ and $\Gamma_{*}^{\prime}$.
For this, we need to bound
$\mathbb{P}[\mathcal{E}_{fluc}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}]$, for
$\mathcal{E}_{fluc}$ being the event that
$|i_{*}|\vee|i_{*}^{\prime}|\vee|j_{*}|\vee|j_{*}^{\prime}|>H_{1}l^{2/3}$. For
this, we show that when $n$ is large enough, given any $\\{\omega_{p}:d(p)\leq
2m_{-}\\}$ and $\\{\omega_{p}:d(p)\leq 2m_{-}\\}$ such that
$\mathcal{E}_{1}\cap\mathcal{E}_{2}$ holds, the probability of the event where

$\max_{i,j\in\mathbb{Z},|i|\vee|j|>H_{1}l^{2/3}}f_{1}(i)+f_{2}(j)+T_{(m_{-}+i+1,m_{-}-i),(m+j,m-j-1)}\\\
>f_{1}(0)+f_{2}(0)+T_{(m_{-}+1,m_{-}),(m,m-1)}$ (4.11)

or

$\max_{i,j\in\mathbb{Z},|i|\vee|j|>H_{1}l^{2/3}}f_{1}(i)+f_{3}(j)+T_{(m_{-}+i+1,m_{-}-i),(m+j,m-j-1)}\\\
>f_{1}(0)+f_{3}(0)+T_{(m_{-}+1,m_{-}),(m,m-1)}$ (4.12)

is bounded by $c^{div}_{\beta,\theta}$, which depends only on $\beta,\theta$
and satisfies that $\lim_{\beta\to 0}c^{div}_{\beta,\theta}=0$. By (4.7) and
(4.8) from the event $\mathcal{E}_{1}\cap\mathcal{E}_{2}$, and the expectation
estimate (2.3), we have that for any $i,j\in\mathbb{Z}$,

$(f_{1}(i)+i^{2}/m_{-}-\beta^{-1}n^{1/3}-\beta
i^{2}/n)+f_{2}(j)+(\mathbb{E}[T_{(m_{-}+i+1,m_{-}-i),(m+j,m-j-1)}]+(i-j)^{2}/l)\\\
<(f_{1}(0)+\beta^{-1}n^{1/3}+\beta i^{2}/n)+(f_{2}(0)+\beta^{-1}n^{1/3}+\beta
j^{2}/n)+\mathbb{E}[T_{(m_{-}+1,m_{-}),(m,m-1)}]+Cn^{1/3},$

where $C>0$ is a universal constant. Recall from the beginning of this proof
that $H_{1}$ is chosen such that $H_{1}l^{2/3}>\beta^{-0.9}n^{2/3}$. Thus when
$|i|\vee|j|>H_{1}l^{2/3}>\beta^{-0.9}n^{2/3}$, we must have

$f_{1}(i)+f_{2}(j)+\mathbb{E}[T_{(m_{-}+i+1,m_{-}-i),(m+j,m-j-1)}]\\\
<f_{1}(0)+f_{2}(0)+\mathbb{E}[T_{(m_{-}+1,m_{-}),(m,m-1)}]-c((i-j)^{2}+\beta^{-1.8})/n,$

where $c>0$ is another universal constant. Then we can bound the probability
of (4.11) conditioned on $\mathcal{E}_{1}\cap\mathcal{E}_{2}$, by using
Theorem 2.7 and Proposition 2.8 and spitting $\mathbb{Z}$ into intervals of
length $l^{2/3}$. The probability of (4.12) conditioned on
$\mathcal{E}_{1}\cap\mathcal{E}_{2}$ can be bounded similarly. We then
conclude that

$\mathbb{P}[\mathcal{E}_{fluc}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}]<c^{div}_{\beta,\theta}$
(4.13)

when $n$ is large enough.

Unless $\mathcal{E}_{fluc}$ happens, the optimal paths $\Gamma_{*}$ and
$\Gamma_{*}^{\prime}$ are the same as the optimal paths with endpoints
restricted to two intervals of length $2H_{1}l^{2/3}$. Therefore, given any
$f_{1},f_{2},f_{3}$ such that $\mathcal{E}_{reg}\cap\mathcal{E}_{diff}$ holds,
Proposition 4.6 applies. It implies that with high probability, either
$\Gamma_{*}\cap\mathbb{L}_{m_{-}+1}=\Gamma_{*}^{\prime}\cap\mathbb{L}_{m_{-}+1}$
(which is equivalent to $\mathcal{E}_{3}$) or $\mathcal{E}_{fluc}$ happens.
More precisely, we have

$\lim_{w\to
0}\liminf_{n\to\infty}\mathbb{P}[\mathcal{E}_{3}\cup\mathcal{E}_{fluc}\mid\mathcal{E}_{reg}\cap\mathcal{E}_{diff}\cap\mathcal{E}_{1}\cap\mathcal{E}_{2}]>c^{sta}_{w,\beta,\theta},$
(4.14)

when $n$ is large enough; here $c^{sta}_{w,\beta,\theta}$ depends on
$w,\beta,\theta$ and satisfies that $\lim_{w\to 0}c^{sta}_{w,\beta,\theta}=1$.

By the elementary inequality of $x+(1-y)\geq(x\wedge y)/y$ for any
$x,y\in(0,1)$, we have

$\mathbb{P}[\mathcal{E}_{3}\cup\mathcal{E}_{fluc}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}]+\mathbb{P}[\mathcal{E}_{reg}^{c}\cup\mathcal{E}_{diff}^{c}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}]\geq\mathbb{P}[\mathcal{E}_{3}\cup\mathcal{E}_{fluc}\mid\mathcal{E}_{reg}\cap\mathcal{E}_{diff}\cap\mathcal{E}_{1}\cap\mathcal{E}_{2}].$

Thus we have

$\mathbb{P}[\mathcal{E}_{3}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}]\geq\mathbb{P}[\mathcal{E}_{3}\cup\mathcal{E}_{fluc}\mid\mathcal{E}_{reg}\cap\mathcal{E}_{diff}\cap\mathcal{E}_{1}\cap\mathcal{E}_{2}]\\\
-\mathbb{P}[\mathcal{E}_{reg}^{c}\cup\mathcal{E}_{diff}^{c}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}]-\mathbb{P}[\mathcal{E}_{fluc}\mid\mathcal{E}_{1}\cap\mathcal{E}_{2}].$

Then by plugging (4.10), (4.13), (4.14) to the right-hand side, we have that
the conclusion holds. ∎

##  5 Stability with respect to boundary conditions

In this section, we prove Proposition 4.6. Throughout this section, we take
$H_{1},H_{2}>0$ as two fixed constants, and all other constants are allowed to
depend on them.

As already indicated in Section 1.4, the general idea to prove Proposition 4.6
involves two steps. In the first, we show that with high probability, the
geodesics $\Gamma_{f_{1},f_{2}}^{n}$ and $\Gamma_{f_{1},f_{3}}^{n}$ share a
significant part of their journey being quite close to each other; in the
second step we show that the conclusion of the first step implies that the
geodesics coalesce with high probability. The two steps are carried out in
Sections 5.1 and 5.2 respectively.

###  5.1 Closeness of geodesics at one time

######  Proposition 5.1.

For each $\epsilon,\lambda>0$, $0.1<\alpha<0.9$, the following is true for any
small enough $w,g>0$ and any $n\in\mathbb{N}$ large enough (depending on
$w,g$). Take any $f_{1},f_{2},f_{3}\in\mathscr{F}_{w,n}$ with
$|f_{2}-f_{3}|\leq gn^{1/3}$, we have

$\mathbb{P}[|\Gamma_{f_{1},f_{2}}^{n}\cap\mathbb{L}_{\lfloor\alpha
n\rfloor}-\Gamma_{f_{1},f_{3}}^{n}\cap\mathbb{L}_{\lfloor\alpha
n\rfloor}|<\lambda n^{2/3}]>1-\epsilon.$

To prove this, we first prove a limiting version in the directed landscape.
While this is in principle not necessary, taking advantage of the local
Brownian property (Theorem 2.6) in the limit is convenient for the exposition
and helps make the proof cleaner.

Overall the underlying observation is that the non-closeness of the geodesic
implies a “twin-peaks” event, i.e., where certain passage time profiles on
$\mathbb{L}_{\lfloor\alpha n\rfloor}$ are nearly maximized at two points with
distance $\geq\lambda n^{2/3}$. In the directed landscape, this is essentially
the same as a Brownian motion having a “near-maxima” significantly away from
the global maximum, an event whose probability is small. The limiting twin-
peaks event estimate is stated as follows. We let $\mathscr{F}$ be the
collection of all upper semi-continuous functions
$f:[-2^{-2/3}H_{1},2^{-2/3}H_{1}]\to\mathbb{R}$, such that $|f|\leq
2^{-4/3}H_{2}$. For each $f\in\mathscr{F}$ we denote its hypograph
$\operatorname{hypo}(f)$ as

$\operatorname{hypo}(f)=\\{(x,y):x\in[-2^{-2/3}H_{1},2^{-2/3}H_{1}],-2^{-4/3}H_{2}\leq
y\leq f(x)\\},$

which is a closed subset of
$[-2^{-2/3}H_{1},2^{-2/3}H_{1}]\times[-2^{-4/3}H_{2},2^{-4/3}H_{2}]$. Since
the space of all compact subsets of
$[-2^{-2/3}H_{1},2^{-2/3}H_{1}]\times[-2^{-4/3}H_{2},2^{-4/3}H_{2}]$ is
complete (with the Hausdorff distance metric), for any sequence of functions
in $\mathscr{F}$, we can find a subsequence $\\{f_{k}\\}_{k\in\mathbb{N}}$,
such that $\operatorname{hypo}(f_{k})$ converges in the Hausdorff topology as
$k\to\infty$. It is straightforward to check that the limit must be
$\operatorname{hypo}(f)$ for some function $f\in\mathscr{F}$.

Below we let $\mathcal{H}(\cdot,\cdot)$ denote the Hausdorff distance between
compact sets in $\mathbb{R}^{2}$.

######  Lemma 5.2.

For each $\epsilon,\lambda>0$, $0.1<\alpha<0.9$, and $H^{\prime}>0$, there is
$\delta>0$, such that the following is true.

Take any $f_{1},f_{2}\in\mathscr{F}$, define $f_{*}:\mathbb{R}\to\mathbb{R}$
as

$f_{*}(x)=\sup_{|y|,|z|\leq
2^{-2/3}H_{1}}\mathcal{L}(y,0;x,\alpha)+\mathcal{L}(x,\alpha;z,1)+f_{1}(y)+f_{2}(z),$

and we let $\mathsf{NoTP}_{\delta}(f_{1},f_{2})$ denote the following event:
for any $x_{1},x_{2}\in[-2^{-2/3}H^{\prime},2^{-2/3}H^{\prime}]$ with
$x_{2}-x_{1}\geq 2^{-2/3}\lambda$ we have $f_{*}(x_{1})\wedge
f_{*}(x_{2})\leq\sup_{x\in[-2^{-2/3}H^{\prime},2^{-2/3}H^{\prime}]}f_{*}(x)-2^{-4/3}\delta$
(i.e., the event of no-twin-peaks). Then we have
$\mathbb{P}[\mathsf{NoTP}_{\delta}(f_{1},f_{2})]>1-\epsilon$.

###### Proof.

It suffices to show that $\lim_{\delta\to
0}\inf_{f_{1},f_{2}\in\mathscr{F}}\mathbb{P}[\mathsf{NoTP}_{\delta}(f_{1},f_{2})]=1$.
We argue by contradiction, and suppose that for each $k\in\mathbb{N}$ we can
find some $\delta_{k}>0$, and $f_{1,k},f_{2,k}\in\mathscr{F}$, such that
$\lim_{k\to\infty}\delta_{k}=0$, and
$\lim_{k\to\infty}\mathbb{P}[\mathsf{NoTP}_{\delta_{k}}(f_{1,k},f_{2,k})]=\gamma<1$.
By taking a subsequence we can assume that
$\operatorname{hypo}(f_{1,k})\to\operatorname{hypo}(f_{1,\infty})$ and
$\operatorname{hypo}(f_{2,k})\to\operatorname{hypo}(f_{2,\infty})$ in the
Hausdorff topology, for some $f_{1,\infty},f_{2,\infty}\in\mathscr{F}$. In
other words, if we let
$\tau_{k}=\mathcal{H}(\operatorname{hypo}(f_{1,k}),\operatorname{hypo}(f_{1,\infty}))\vee\mathcal{H}(\operatorname{hypo}(f_{2,k}),\operatorname{hypo}(f_{2,\infty}))$,
then $\tau_{k}\to 0$ as $k\to\infty$.

We now explain how we will reach a contradiction. Define
$f_{*,k}:\mathbb{R}\to\mathbb{R}$ by

$f_{*,k}(x)=\sup_{|y|,|z|\leq
2^{-2/3}H_{1}}\mathcal{L}(y,0;x,\alpha)+\mathcal{L}(x,\alpha;z,1)+f_{1,k}(y)+f_{2,k}(z),$

for $k\in\mathbb{N}\cup\\{\infty\\}$. On one hand, $f_{*,\infty}$ is local
Brownian (by Theorem 2.6), so almost surely it has a unique maximum in any
compact interval. However, we will use
$\lim_{k\to\infty}\mathbb{P}[\mathsf{NoTP}_{\delta_{k}}(f_{1,k},f_{2,k})]=\gamma<1$
to deduce that with probability $\geq 1-\gamma$, the maximum of $f_{*,\infty}$
in $[-2^{-2/3}H^{\prime},2^{-2/3}H^{\prime}]$ is not unique. This can be
achieved by bounding the difference between $f_{*,k}$ and $f_{*,\infty}$,
using Lemma 2.4.

We next fill in the details of this outlined strategy. Take $R$ as given by
Lemma 2.4, where the compact set $K$ is taken as the union of

$([-2^{-2/3}H_{1},2^{-2/3}H_{1}]\times[0,\alpha/3])\times([-2^{-2/3}H^{\prime},2^{-2/3}H^{\prime}]\times[2\alpha/3,2\alpha/3+1/3])$

and

$([-2^{-2/3}H^{\prime},2^{-2/3}H^{\prime}]\times[2\alpha/3,2\alpha/3+1/3])\times([-2^{-2/3}H_{1},2^{-2/3}H_{1}]\times[\alpha/3+2/3,1]).$

We take $K$ this way so that we can now bound
$|\mathcal{L}(y,0;x,\alpha)-\mathcal{L}(y^{\prime},0;x,\alpha)|$ and
$|\mathcal{L}(x,\alpha;z,1)-\mathcal{L}(x,\alpha;z^{\prime},1)|$, for any
$|x|\leq 2^{-2/3}H^{\prime}$ and $|y|,|z|,|y^{\prime}|,|z^{\prime}|\leq
2^{-2/3}H_{1}$.

Using $\tau_{k}$ defined above, for any $|x|\leq 2^{-2/3}H^{\prime}$ and
$|y|,|z|\leq 2^{-2/3}H_{1}$, we can find $y^{\prime},z^{\prime}$ such that
$|y^{\prime}|,|z^{\prime}|\leq 2^{-2/3}H_{1}$,
$|y-y^{\prime}|,|z-z^{\prime}|\leq\tau_{k}$, and $f_{1,k}(y^{\prime})\geq
f_{1,\infty}(y)-\tau_{k}$, $f_{2,k}(z^{\prime})\geq f_{2,\infty}(z)-\tau_{k}$.
Thus

$f_{*,k}(x)-f_{*,\infty}(x)\geq-(C\tau_{k}+2R\tau_{k}^{1/2}\log^{1/2}(1+\tau_{k}^{-1/2})),$

where $C>0$ is a constant depending on $H^{\prime}$ and $H_{1}$. Similarly,
the same lower bound holds for $f_{*,\infty}(x)-f_{*,k}(x)$. So we have

$|f_{*,k}(x)-f_{*,\infty}(x)|\leq
C\tau_{k}+2R\tau_{k}^{1/2}\log^{1/2}(1+\tau_{k}^{-1/2}).$

Given that $R\leq\tau_{k}^{-1/4}\log^{-1/2}(1+\tau_{k}^{-1/2})$, the right-
hand side is $\leq 2\tau_{k}^{1/4}+C\tau_{k}$, and we have that
$\mathsf{NoTP}_{\delta_{k}+2\tau_{k}^{1/4}+C\tau_{k}}(f_{1,\infty},f_{2,\infty})$
implies $\mathsf{NoTP}_{\delta_{k}}(f_{1,k},f_{2,k})$. Therefore we have

$\mathbb{P}[\mathsf{NoTP}_{\delta_{k}+2\tau_{k}^{1/4}+C\tau_{k}}(f_{1,\infty},f_{2,\infty})]\leq\mathbb{P}[\mathsf{NoTP}_{\delta_{k}}(f_{1,k},f_{2,k})]+\mathbb{P}[R>\tau_{k}^{-1/4}\log^{-1/2}(1+\tau_{k}^{-1/2})].$

Thus
$\limsup_{k\to\infty}\mathbb{P}[\mathsf{NoTP}_{\delta_{k}+2\tau_{k}^{1/4}+C\tau_{k}}(f_{1,\infty},f_{2,\infty})]\leq\gamma$.

If $f_{*,\infty}$ has a unique maximum in
$[-2^{-2/3}H^{\prime},2^{-2/3}H^{\prime}]$, for any sufficiently large $k$ the
event
$\mathsf{NoTP}_{\delta_{k}+2\tau_{k}^{1/4}+C\tau_{k}}(f_{1,\infty},f_{2,\infty})$
holds. So the probability of $f_{*,\infty}$ having a unique maximum in
$[-2^{-2/3}H^{\prime},2^{-2/3}H^{\prime}]$ is at most $\gamma$.

By Theorem 2.6 the functions

$x\mapsto\sup_{|y|\leq
2^{-2/3}H_{1}}\mathcal{L}(y,0;x,\alpha)+f_{1,k}(y),\quad x\mapsto\sup_{|z|\leq
2^{-2/3}H_{1}}\mathcal{L}(x,\alpha;z,1)+f_{2,k}(z)$

are absolutely continuous with respect to Brownian motions in any compact
interval. (Note that the suprema are taken over compact intervals.) Then we
have that their sum $f_{*,\infty}$ in
$[-2^{-2/3}H^{\prime},2^{-2/3}H^{\prime}]$ is absolutely continuous with
respect to a Brownian motion, thus has a unique maximum almost surely. Thus we
get a contradiction. ∎

We next use this estimate on the twin-peaks event in the limiting setting to
prove Proposition 5.1. Note that the estimate in Proposition 5.1 is required
to hold uniformly for any $f_{1},f_{2},f_{3}\in\mathscr{F}_{w,n}$. To achieve
this we will take a finite family of functions so that any function in
$\mathscr{F}_{w,n}$ is close to one in this family.

From now on, let $T_{p,q}^{*}=T_{p,q}-\omega_{p}$ for any $\mathbb{Z}^{2}$
vertices $p\preceq q$. Removing the weight of the first vertex as such helps
to get useful independence between passage times, as we will soon see.

###### Proof of Proposition 5.1.

We first define the no-twin-peaks event in Exponential LPP. Let $H^{\prime}>0$
be a large number depending on $\epsilon,\alpha$, and the requirement for its
value will be evident from the arguments below. For each
$f_{1},f_{2}:\llbracket-H_{1}n^{2/3},H_{1}n^{2/3}\rrbracket\to\mathbb{R}$, we
let $T_{f_{1},f_{2}}:\mathbb{Z}\to\mathbb{R}$ be the function where

$T_{f_{1},f_{2}}(i)=\max_{|i_{1}|,|i_{2}|\leq
H_{1}n^{2/3}}T_{(i_{1},-i_{1}),(\lfloor\alpha n\rfloor+i,\lfloor\alpha
n\rfloor-i)}+T^{*}_{(\lfloor\alpha n\rfloor+i,\lfloor\alpha
n\rfloor-i),(n+i_{2},n-i_{2})}+f_{1}(i_{1})+f_{2}(i_{2}).$

Let $\mathsf{NoTP}_{\delta,n}(f_{1},f_{2})$ be the event where for any
$i_{1},i_{2}\in\llbracket-H^{\prime}n^{2/3},H^{\prime}n^{2/3}\rrbracket$ with
$i_{2}-i_{1}\geq\lambda n^{2/3}$, one has $T_{f_{1},f_{2}}(i_{1})\wedge
T_{f_{1},f_{2}}(i_{2})\leq\max_{|i|\leq
H^{\prime}n^{2/3}}T_{f_{1},f_{2}}(i)-\delta n^{1/3}$.

As the no-twin-peaks event only involves the function $T_{f_{1},f_{2}}$ on the
interval $\llbracket-H^{\prime}n^{2/3},H^{\prime}n^{2/3}\rrbracket$, we need
to bound the transversal fluctuation of geodesics. By Lemma 2.9, by taking
$H^{\prime}>0$ large, for any $n$ large enough we have
$\mathbb{P}[\mathcal{E}_{trans}]>1-\epsilon/2$, where $\mathcal{E}_{trans}$ is
the event

$\Gamma_{(-\lfloor H_{1}n^{2/3}\rfloor,\lfloor H_{1}n^{2/3}\rfloor),(n-\lfloor
H_{1}n^{2/3}\rfloor,n+\lfloor
H_{1}n^{2/3}\rfloor)}\cap\mathbb{L}_{\lfloor\alpha
n\rfloor}\subset\\{(\lfloor\alpha n\rfloor-i,\lfloor\alpha n\rfloor+i):|i|\leq
H^{\prime}n^{2/3}\\},$

and

$\Gamma_{(\lfloor H_{1}n^{2/3}\rfloor,-\lfloor H_{1}n^{2/3}\rfloor),(n+\lfloor
H_{1}n^{2/3}\rfloor,n-\lfloor
H_{1}n^{2/3}\rfloor)}\cap\mathbb{L}_{\lfloor\alpha
n\rfloor}\subset\\{(\lfloor\alpha n\rfloor-i,\lfloor\alpha n\rfloor+i):|i|\leq
H^{\prime}n^{2/3}\\}.$

Then under $\mathcal{E}_{trans}$, for any $f_{1},f_{2}\in\mathscr{F}_{w,n}$ we
must have that

$\Gamma_{f_{1},f_{2}}^{n}\cap\mathbb{L}_{\lfloor\alpha
n\rfloor}\subset\\{(\lfloor\alpha n\rfloor-i,\lfloor\alpha n\rfloor+i):|i|\leq
H^{\prime}n^{2/3}\\},$

since any two geodesics cannot cross each other twice.

Now by taking $w,g$ small and then $n$ large, we have that for any
$f_{1},f_{2},f_{3}\in\mathscr{F}_{w,n}$ with $|f_{2}-f_{3}|\leq gn^{1/3}$,

$\begin{split}&\mathbb{P}[|\Gamma_{f_{1},f_{2}}^{n}\cap\mathbb{L}_{\lfloor\alpha
n\rfloor}-\Gamma_{f_{1},f_{3}}^{n}\cap\mathbb{L}_{\lfloor\alpha
n\rfloor}|<\lambda n^{2/3}]\\\
\geq&\mathbb{P}[\mathsf{NoTP}_{g,n}(f_{1},f_{2})]-(1-\mathbb{P}[\mathcal{E}_{trans}])\\\
>&\mathbb{P}[\mathsf{NoTP}_{g,n}(f_{1},f_{2})]-\epsilon/2.\end{split}$ (5.1)

Here the first inequality is due to that, if the intersections
$\Gamma_{f_{1},f_{2}}^{n}\cap\mathbb{L}_{\lfloor\alpha n\rfloor}$ and
$\Gamma_{f_{1},f_{3}}^{n}\cap\mathbb{L}_{\lfloor\alpha n\rfloor}$ are
$\geq\lambda n^{2/3}$ from each other, and the transversal fluctuation of
geodesics is bounded (i.e., $\mathcal{E}_{trans}$ happens), then the twin-
peaks event (i.e., the complement of $\mathsf{NoTP}_{g,n}(f_{1},f_{2})$) must
occur. Then it remains to bound the probability of the latter, for which we
use the limiting estimate given by Lemma 5.2.

For any $f:\llbracket-H_{1}n^{2/3},H_{1}n^{2/3}\rrbracket\to\mathbb{R}$, let
$\mathsf{S}_{n}f$ be the function on $[-2^{-2/3}H_{1},2^{-2/3}H_{1}]$, such
that $\mathsf{S}_{n}f(x)=2^{-4/3}n^{-1/3}f(\lfloor(2n)^{2/3}x\rfloor)$. Note
that this is well defined only when $\lfloor(2n)^{2/3}x\rfloor\geq-
H_{1}n^{2/3}$. For $x$ near $-2^{-2/3}H_{1}$ (i.e., when
$\lfloor(2n)^{2/3}x\rfloor<-H_{1}n^{2/3}$), we instead let
$\mathsf{S}_{n}f(x)=2^{-4/3}n^{-1/3}f(\lceil(2n)^{2/3}x\rceil)$.

To bound the twin-peaks probability uniformly in the boundary functions, we
take a finite family of functions in $\mathscr{F}_{w,n}$ to approximate every
function in $\mathscr{F}_{w,n}$, and do limit transition for each function in
this finite family to use Lemma 5.2.

Towards this, for any $w,g>0$ and $n\in\mathbb{N}$, we can find
$\\{f_{w,g,n;k}\\}_{k=1}^{m_{w,g}}\subset\mathscr{F}_{w,n}$, and
$\\{f_{w,g;k}\\}_{k=1}^{m_{w,g}}\subset\mathscr{F}$, where
$m_{w,g}\in\mathbb{N}$ depends only on $w,g$, such that

  1. 1.

$\mathsf{S}_{n}f_{w,g,n;k}\to f_{w,g;k}$ uniformly as $n\to\infty$;

  2. 2.

for each $f\in\mathscr{F}_{w,n}$, there is $f_{w,g,n;k}$, such that
$|f-f_{w,g,n;k}|\leq gn^{1/3}$.

One way to choose such functions is as follows. First, take some $\iota>0$
small enough (depending on $w,g$). Let $\\{f_{w,g;k}\\}_{k=1}^{m_{w,g}}$
contain all functions on $[-2^{-2/3}H_{1},2^{-2/3}H_{1}]$, which (1) take
values in $\iota\mathbb{Z}\cap[-2^{-4/3}H_{2},2^{-4/3}H_{2}]$ at
$(\iota\mathbb{Z}\cap[-2^{-2/3}H_{1},2^{-2/3}H_{1}])\cup\\{-2^{-2/3}H_{1},2^{-2/3}H_{1}\\}$,
(2) are linear on the rest of the domain, and (3) are in the closure (in the
uniform convergence topology) of
$\cup_{n\in\mathbb{N}}\\{\mathsf{S}_{n}f:f\in\mathscr{F}_{w,n}\\}$. Then we
let $f_{w,g,n;k}$ be the function in $\mathscr{F}_{w,n}$ with the smallest
$\|\mathsf{S}_{n}f_{w,g,n;k}-f_{w,g;k}\|_{\infty}$.

By Lemma 5.2, we can take $\delta>0$ small enough, such that
$\mathbb{P}[\mathsf{NoTP}_{\delta}(f_{1},f_{2})]>1-\epsilon/4$ for any
$f_{1},f_{2}\in\mathscr{F}$. By Theorem 2.14, we have

$\lim_{n\to\infty}\mathbb{P}[\mathsf{NoTP}_{\delta,n}(f_{w,g,n;k_{1}},f_{w,g,n;k_{2}})]=\mathbb{P}[\mathsf{NoTP}_{\delta}(f_{w,g;k_{1}},f_{w,g,k_{2}})]>1-\epsilon/4,$
(5.2)

for any $w,g>0$, $1\leq k_{1},k_{2}\leq m_{w,g}$. We next extend this bound to
all boundary functions in $\mathscr{F}_{w,n}$.

We note that for any
$f_{1},f_{2},f_{1}^{\prime},f_{2}^{\prime}\in\mathscr{F}_{w,n}$,

$\max_{i\in\mathbb{Z}}|T_{f_{1},f_{2}}(i)-T_{f_{1}^{\prime},f_{2}^{\prime}}(i)|\leq\max_{|i|\leq
H_{1}n^{2/3}}|f_{1}(i)-f_{1}^{\prime}(i)|+\max_{|i|\leq
H_{1}n^{2/3}}|f_{2}(i)-f_{2}^{\prime}(i)|.$

Then by the second property of $\\{f_{w,g,n;k}\\}_{k=1}^{m_{w,g}}$ from above,
for any $f_{1},f_{2}\in\mathscr{F}_{w,n}$ we can find $f_{w,g,n;k_{1}}$ and
$f_{w,g,n;k_{2}}$ such that

$\max_{i\in\mathbb{Z}}|T_{f_{1},f_{2}}(i)-T_{f_{w,g,n;k_{1}},f_{w,g,n;k_{2}}}(i)|\leq
2gn^{1/3}.$

Then $\mathsf{NoTP}_{3g,n}(f_{w,g,n;k_{1}},f_{w,g,n;k_{2}})$ implies
$\mathsf{NoTP}_{g,n}(f_{1},f_{2})$. Therefore, by taking $g$ small enough
(such that $3g<\delta$), we have

$\displaystyle\lim_{n\to\infty}\inf_{f_{1},f_{2}\in\mathscr{F}_{w,n}}\mathbb{P}[\mathsf{NoTP}_{g,n}(f_{1},f_{2})]$
$\displaystyle\geq$ $\displaystyle\lim_{n\to\infty}\inf_{1\leq k_{1},k_{2}\leq
m_{w}}\mathbb{P}[\mathsf{NoTP}_{\delta,n}(f_{w,g,n;k_{1}},f_{w,g,n;k_{2}})]$
$\displaystyle\geq$ $\displaystyle 1-\epsilon/4.$

Here the second inequality is due to (5.2). By plugging this into (5.1) the
conclusion follows. ∎

###  5.2 From closeness to coalescence

We now deduce Proposition 4.6 from Proposition 5.1. The approach involves
taking a set of different $\alpha$ in Proposition 5.1, and showing that it is
unlikely for $\Gamma_{f_{1},f_{2}}^{n}$ and $\Gamma_{f_{1},f_{3}}^{n}$ to be
close at their intersections with several $\mathbb{L}_{\lfloor\alpha
n\rfloor}$, while remaining disjoint (see Figure 4).

We will need the following lemma, which says that any path passing through a
given sequence of short segments is likely to have a small weight and thereby
unlikely to be the geodesic. Similar estimates have appeared in the literature
in studying Exponential or other integrable LPP models. See e.g. [BHS18, Lemma
3.5], [GH20, Theorem 1.9], [MSZ21, Lemma 7.4].

######  Lemma 5.3.

Given any $\epsilon>0$, any $M\in\mathbb{N}$ large enough (depending on
$\epsilon$), and any $n\in\mathbb{N}$ large enough (depending on $\epsilon$
and $M$), the following is true.

Take any $\alpha_{0}\in\mathbb{R}$, and let
$\alpha_{i}=\alpha_{0}+\frac{i}{10M}$ for each $i=1,\ldots,M$. For any
$r_{0},\ldots,r_{M}\in\mathbb{Z}$ with each $|r_{i}|<M^{0.01}n^{2/3}$, we have

$\mathbb{P}\left[\max_{k_{0},\ldots,k_{M}\in\llbracket
0,n^{2/3}/M^{4}\rrbracket}\sum_{i=0}^{M-1}T^{*}_{w_{i}[r_{i}+k_{i}],w_{i+1}[r_{i+1}+k_{i+1}]}>\frac{4n}{10}-M^{0.05}n^{1/3}\right]<\epsilon,$

where $w_{i}[k]=(\lfloor\alpha_{i}n\rfloor+k,\lfloor\alpha_{i}n\rfloor-k)$ for
each $i\in\llbracket 1,M\rrbracket$ and $k\in\mathbb{Z}$. (Here recall that
$T^{*}_{p,q}=T_{p,q}-\omega_{p}$ for any $p,q\in\mathbb{Z}^{2}$, $p\preceq
q$.)

###### Proof.

As before, in this proof we let $C,c>0$ denote large and small enough
constants.

For each $i$, one has (for $n$ large enough)

$\mathbb{P}\left[\max_{k_{i},k_{i+1}\in\llbracket
0,n^{2/3}/M^{4}\rrbracket}T^{*}_{w_{i}[r_{i}+k_{i}],w_{i+1}[r_{i+1}+k_{i+1}]}-T^{*}_{w_{i}[r_{i}],w_{i+1}[r_{i+1}]}>M^{-1}n^{1/3}\right]<Ce^{-cM}.$

This is by Theorem 2.14, which says that this probability converges to the
corresponding probability in the directed landscape (uniformly in
$r_{i},r_{i+1}$), which is bounded using scaling and skew-shift invariance of
the directed landscape (Lemma 2.1), and the modulus of continuity of the
directed landscape (Lemma 2.4). Next we take a union bound over all $0\leq
i\leq M-1$, and take $M$ large, to conclude that

$\mathbb{P}\left[\sum_{i=0}^{M-1}\max_{k_{i},k_{i+1}\in\llbracket
0,n^{2/3}/M^{4}\rrbracket}T^{*}_{w_{i}[r_{i}+k_{i}],w_{i+1}[r_{i+1}+k_{i+1}]}-T^{*}_{w_{i}[r_{i}],w_{i+1}[r_{i+1}]}>n^{1/3}\right]<\epsilon/2.$

We next estimate $T^{*}_{w_{i}[r_{i}],w_{i+1}[r_{i+1}]}$ for each $i$. By the
convergence to the directed landscape (Theorem 2.14), the GUE Tracy-Widom
marginal of the directed landscape, and that the GUE Tracy-Widom distribution
has negative expectation 222see [BGHH22, Lemma A.4] for a proof by Ivan Corwin
or [GH20, Proposition 4.1] for another proof., we have that for each $0\leq
i\leq M-1$,

$\mathbb{E}[T^{*}_{w_{i}[r_{i}],w_{i+1}[r_{i+1}]}]<\frac{4n}{10M}-c\left(\frac{n}{M}\right)^{1/3},$

when $n$ is large enough. Then since $T^{*}_{w_{i}[r_{i}],w_{i+1}[r_{i+1}]}$
are independent for all $i$ (due to that the weight of the first vertex
$w_{i}[r_{i}]$ is removed from the passage time
$T_{w_{i}[r_{i}],w_{i+1}[r_{i+1}]}$), using Theorem 2.7, we have that for
large enough $n$,

$\operatorname{Var}\left(\sum_{i=0}^{M-1}T^{*}_{w_{i}[r_{i}],w_{i+1}[r_{i+1}]}\right)=\sum_{i=0}^{M-1}\operatorname{Var}(T^{*}_{w_{i}[r_{i}],w_{i+1}[r_{i+1}]})<CM\left(\frac{n}{M}\right)^{2/3}.$

Thus by taking $M$ large, we have

$\mathbb{P}\left[\sum_{i=0}^{M-1}T^{*}_{w_{i}[r_{i}],w_{i+1}[r_{i+1}]}>\frac{4n}{10}-M^{0.1}n^{1/3}\right]<\epsilon/2.$

and hence since for $M$ large enough $-M^{0.1}+1<-M^{0.05}$, our conclusion
follows. ∎

$f_{1}$$f_{2}$, $f_{3}$$\mathbb{L}_{0}$$\mathbb{L}_{\lfloor
n/2\rfloor}$$\mathbb{L}_{n}$$\Gamma^{n}_{f_{1},f_{2}}$$\Gamma^{n}_{f_{1},f_{3}}$$p_{0}$$q_{0}$$p_{M}$$q_{M}$
Figure 4: An illustration of the proof of Proposition 4.6: for the geodesics
$\Gamma_{f_{1},f_{2}}^{n}$ and $\Gamma_{f_{1},f_{3}}^{n}$, if their
intersections with each $\mathbb{L}_{\lfloor\alpha_{i}n\rfloor}$ are close, it
is unlikely for them to remain disjoint between
$\mathbb{L}_{\lfloor\alpha_{0}n\rfloor}$ and
$\mathbb{L}_{\lfloor\alpha_{M}n\rfloor}$.

We now finish proving the stability result, by upgrading Proposition 5.1 using
Lemma 5.3.

###### Proof of Proposition 4.6.

Let $C,c>0$ denote large and small enough constants in this proof.

We now explain the strategy. From Proposition 5.1 we know that
$\Gamma_{f_{1},f_{2}}^{n}$ and $\Gamma_{f_{1},f_{3}}^{n}$ are likely to stay
close to each other at several given times. To deduce coalescence from this,
we will show that $\Gamma_{f_{1},f_{3}}^{n}$ is unlikely to be competitive if
it were disjoint from $\Gamma_{f_{1},f_{2}}^{n}.$ To this end we will employ
Lemma 5.3. However a key difference is that in the lemma the path is steered
through fixed intervals whereas in the current situation one can view
$\Gamma_{f_{1},f_{3}}^{n}$ as passing through intervals adjacent to
$\Gamma_{f_{1},f_{2}}^{n}$ which are of course random. To address this we fix
any upper-right path $\Gamma$ with two ending points on $\mathbb{L}_{0}$ and
$\mathbb{L}_{n}$, and condition on the event
$\Gamma_{f_{1},f_{2}}^{n}=\Gamma$. This “fixes” the intervals on one hand, but
on the other hand, forces us to work with the conditioned environment.
Fortunately, however, the FKG inequality comes to our aid and asserts that the
above conditioning (being decreasing in nature) only makes the remainder of
the environment lower than what it typically is and hence the estimate from
Lemma 5.3 continues to hold true.

Therefore we conclude that it is unlikely for $\Gamma_{f_{1},f_{3}}^{n}$ to
stay close to while being disjoint from $\Gamma$, given that
$\Gamma_{f_{1},f_{2}}^{n}=\Gamma$. Averaging over $\Gamma$ finishes the proof.

To implement the above plan, we first obtain closeness from Proposition 5.1.
Let $M\in\mathbb{N}$ be a large constant to be determined. Take
$\alpha_{0}=0.6$, and for each $i=0,\cdots,M$, let
$\alpha_{i}=\alpha_{0}+\frac{i}{10M}$. Then each $\alpha_{i}$ is contained in
$[0.6,0.7]$, thus away from $0$ and $1$. We let
$p_{i}=(\lfloor\alpha_{i}n\rfloor+x_{i},\lfloor\alpha_{i}n\rfloor-x_{i})$,
$q_{i}=(\lfloor\alpha_{i}n\rfloor+y_{i},\lfloor\alpha_{i}n\rfloor-y_{i})$ be
the intersections of $\Gamma_{f_{1},f_{2}}^{n}$ and $\Gamma_{f_{1},f_{3}}^{n}$
with $\mathbb{L}_{\lfloor\alpha_{i}n\rfloor}$, where
$x_{i},y_{i}\in\mathbb{Z}$. By Proposition 5.1, and taking a union bound over
$i\in\llbracket 0,M\rrbracket$, we have

$\lim_{w,g\to 0}\limsup_{n\to\infty}\mathbb{P}\Big{[}\max_{i\in\llbracket
0,M\rrbracket}|x_{i}-y_{i}|\geq n^{2/3}/M^{4}\Big{]}=0,$ (5.3)

for any fixed $M\in\mathbb{N}$.

Next, we denote $\mathcal{E}_{M}$ as the event where $|x_{i}-y_{i}|\leq