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#  Brownian bridge limit of path measures
in the upper tail of KPZ models

Shirshendu Ganguly  ,  Milind Hegde  and  Lingfu Zhang

###### Abstract.

For models in the KPZ universality class, such as the zero temperature model
of planar last passage-percolation (LPP) and the positive temperature model of
directed polymers, the upper tail behavior has been a topic of recent
interest, with particular focus on the associated path measures (i.e.,
geodesics or polymers). For Exponential LPP, diffusive fluctuation had been
established in [BG23]. In the directed landscape, the continuum limit of LPP,
the limiting Gaussianity at one point, as well as of related finite-
dimensional distributions of the KPZ fixed point, were established in [Liu22b,
LW22] using exact formulas. It was further conjectured in these works that the
limit of the corresponding geodesic should be a Brownian bridge. We prove this
in both zero as well as positive temperatures; for the latter, neither the
one-point limit nor the scale of fluctuations was previously known. Instead of
relying on formulas (which are still missing in the positive temperature
literature), our arguments are geometric and probabilistic, using results on
the shape of the weight and free energy profiles under the upper tail from
[GH22] as a starting point. Another key ingredient involves novel coalescence
estimates, developed using the recently discovered shift-invariance [BGW22] in
these models. Finally, our proof also yields insight into the structure of the
polymer measure under the upper tail conditioning, establishing a quenched
localization exponent around a random backbone.

Department of Statistics, UC Berkeley, Berkeley, CA, USA. e-mail:
<EMAIL_ADDRESS>
Department of Mathematics, Columbia University, New York, NY, USA. e-mail:
<EMAIL_ADDRESS>
Department of Statistics, UC Berkeley, Berkeley, CA, USA. e-mail:
<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Coalescence and Brownian bridge comparison under upper tail
  4. 4 Tail comparison estimates
  5. 5 Tightness as continuous functions: geodesics and bounds for polymers
  6. 6 Proportionality and estimates on sums
  7. 7 Concentration of polymers
  8. 8 Tightness for polymers
  9. 9 Estimates on free energies under conditionings
  10. 10 Global and segment maximizers
  11. 11 Finite dimensional Brownian bridge limit
  12. 12 Joint comparison of maximizer location and free energy across peaks
  13. A Weak convergence lemma
  14. B Tent Brownian comparison and estimates
  15. C Tail and tent estimates for small time

##  1\. Introduction

Planar last passage percolation (LPP) models are paradigm examples of models
of random planar geometry. In such models, one studies the weight and geometry
of the maximum weight directed path (termed as the geodesic) between two far
away points in a 2D i.i.d. random field, such as $\mathbb{Z}^{2}$ with i.i.d.
random variables at each vertex, or a homogeneous Poisson point process in
$\mathbb{R}^{2}$. The geodesic weight is often also termed as the last passage
time between the corresponding endpoints. A handful of such models, when the
underlying noise is given by i.i.d Exponentials, a Poisson point process
(connected to the problem of the longest increasing subsequence in a random
permutation) or Brownian motions, admit exact-solvability. Pioneered by the
seminal work of Baik, Deift and Johansson [BDK99], for these models exact
formulae stemming from algebraic combinatorics and representation theory have
been employed to rigorously deduce the predicted KPZ behaviors: the weight of
a geodesic between points with Euclidean separation $n$ fluctuates by
$n^{1/3}$ while the geodesic itself deviates from the straight line joining
the endpoints by $n^{2/3}$ (leading to the well-known $1:2:3$ scaling of KPZ).
Moreover, the geodesic weight scaled by $n^{1/3}$ is known to converge to the
GUE Tracy-Widom distribution. More recently, in a breakthrough work [DOV22], a
continuum object known as the directed landscape was constructed as the
scaling limit of Brownian LPP, and further in [DV21b] it was shown that all
the known exactly solvable LPP models indeed converge to it under the KPZ
scaling.

A particularly important topic of study has been tail estimates of the last
passage time. Beyond the above mentioned Tracy-Widom limit, a line of work has
emerged relying on moderate deviations estimates [LR10, LM01, LMR02] that have
proved to be the key in the study of several central problems. This includes,
for instance, the solution of the slow bond problem of TASEP [BSS14, SSZ21]
and the study of the correlation structure in KPZ [BG21, BGZ21, CGH21]. An
important tool featuring in many of these works is the geometry of geodesics.
Some recent work focusing on the study of geodesics include [BHS22, BBS20]
which rules out bi-infinite geodesics in LPP as well as the study of local
statistics around geodesics [MSZ21]. An approach using geometric ideas to
obtain tail estimates was also developed in [GH23].

Going beyond moderate deviation estimates, various attempts have been directed
towards the understanding of the large deviation behavior of passage time. The
study in the setting of exactly solvable LPPs appears across the papers [LS77,
Sep98a, Sep98b, DZ99, Joh00], using a range of methods. More recently there
has been significant progress in studying the large deviations behavior of the
entire passage time profile extending the one point results [OT19, QT21,
GH22].

While the above results primarily considered the last passage time values, in
another line of work the object of focus has been the distribution of the
geodesic under the large deviation conditioning. The behavior of the geodesic
differs starkly across the upper and lower tail regimes. This is already
manifested in the disparity between the upper and lower tail behaviors of the
Tracy-Widom distribution (which has upper and lower tail exponents of
$\frac{3}{2}$ and $3$ respectively). This stems from the basic reasoning that
the upper tail event simply requires an atypically large weighted path while
the lower tail event stipulates that _all_ paths must have low weights,
leading to a more global event and consequently a much lower probability. In
the large deviation scale, for two points with Euclidean distance $n$, the
logarithm of the probability for the geodesic weight to be $\delta n$ larger
(resp. smaller) than its expectation is of order $n$ (resp. $n^{2}$).

Such a discrepancy between the upper and lower tails has been noted in the
rigorous literature even beyond exactly-solvable models going back to Kesten
[Kes86] who studied the problem in the context of first passage percolation
(FPP) (see also [BGS21] for a more recent result about the existence of a rate
function in this context).

Significantly more has been established for exactly solvable models. Under
upper tail large deviation (i.e., the rate $n$ tail), in [DZ99] it was shown
that the transversal fluctuation for the geodesic still stays $o(n)$ for
Poissonian LPP. This was improved significantly in [BG23] where the geodesic
fluctuation exponent was shown to decrease from $\frac{2}{3}$ to
$\frac{1}{2}$. Under the lower tail large deviation of LPP (i.e., the rate
$n^{2}$ tail) for LPP models for a large class of weight distributions,
including and going beyond exactly-solvable models, it was shown in [BGS19]
that the transversal fluctuation of the geodesic becomes order $n$, i.e. the
transversal fluctuation exponent equals $1$ and hence the geodesic becomes
fully delocalized.

The exponents $\frac{1}{2}$ and $1$ could be attributed, at least
heuristically, to the fact that the upper tail event entails local changes
effected by picking a uniformly random directed path connecting the endpoints
and making its weight large, whereas the lower tail event forces every path to
have a small weight; this makes even paths with large transversal fluctuations
competitive.

This reasoning suggests in particular that LPP geodesics in the upper tail
large deviation regime should scale to a Brownian bridge. Evidence in this
direction was provided in [LW22], where it was shown that in the directed
landscape, if the weight between two points is conditioned to be $>L$, for a
process that one expects to be close to the geodesic, as $L\to\infty,$ the
multi-point distribution converge to a joint Gaussian with correlation
structure matching that of a Brownian bridge; we also mention [Liu22b], which
proves the one-point convergence for the geodesic itself. These results led
Liu and Wang to conjecture the Brownian bridge limit for the geodesic. Such a
statement was also previously posed as a question in the prior work [BG23] in
the context of Exponential LPP.

Proving this conjecture is the main focus of this article.

We consider this problem in both the zero and the positive temperature
settings. In the former case, the pre-limiting geodesics in exactly-solvable
LPP were shown in [DOV22] and [DV21b] to converge to their scaling limits,
which may be termed as geodesics in the directed landscape. This is the first
object that we will work with. For these processes various properties have
been established, such as being $\smash{\frac{2}{3}^{-}}$-Hölder regular and
possessing a finite $\smash{\frac{3}{2}}$-variation in [DSV22]. Associated
local time processes were constructed and shown to be
$\smash{\frac{1}{3}^{-}}$-Hölder regular in [GZ22]. Finally, an explicit,
albeit somewhat complicated, expression for the one point distribution of the
geodesic was established in [Liu22a] relying on exact formulae.

The positive temperature case describes the more general directed polymer
models, where one still has a 2D i.i.d. random field, but where one now
considers a Gibbs measure parametrized by temperature on the space of all
directed paths between two points. More precisely, the probability density of
a path $\gamma$ is proportional to $\exp(\beta H(\gamma))$, where $\beta>0$ is
the inverse temperature and $H(\gamma)$ is the weight of the path in the
random field. We henceforth refer to the random path under this measure as the
polymer (between the two points). At least formally, when $\beta\to\infty$,
the polymer degenerates into the corresponding geodesic, which can therefore
be viewed as the zero-temperature polymer.

Certain integrable features persist even for positive temperature models, such
as the log-gamma polymer and the O’Connell-Yor polymer [OY01, Sep12].
Exploiting such special properties, KPZ behavior has also been established, at
least to some degree, for these examples as well. In particular, for the KPZ
equation from the original paper of Kardar, Parisi, and Zhang [KPZ86], the
Cole-Hopf solution turns out to be the free energy of the continuum directed
random polymer (CDRP) model [AKQ14], and the KPZ scaling convergence to the
directed landscape has been established in a series of recent works [QS23,
DM21, Wu21, Wu23]. The polymer measure in the CDRP is the second object we
work with.

The tail probabilities of these positive temperature models have also been
studied. In particular, there have been extensive works on tails of the KPZ
equation: its one-point upper tail probability is similar to that of the
Tracy-Widom distribution, as established in [KKX17, CG20a, DT21, GH22]; the
lower tail probability is much more involved with a cross-over behavior, as
has been shown in [CG20b, CG20a, CC22, Tsa22]. The behavior of the profile
under large deviations has also been studied [LT23, LT22, GH22]. Besides the
KPZ equation, estimates on both tails have also been obtained for the
O’Connell-Yor polymer in [LS22], adapting zero temperature techniques from
[GH23] to the polymer setting, as well as the log gamma model [BCD21, RASS23].

The decision behind working with the directed landscape and the CDRP is guided
by the various symmetries and scaling invariance properties that are absent in
the pre-limit, which help make our arguments more transparent. However, we
emphasize that we rely on probabilistic and geometric arguments rather than
exact formulae, in contrast to [LW22] and [Liu22b]. In particular, our proofs
primarily rely on the line ensemble representation through the RSK and
geometric RSK correspondences, and hence should be adaptable to any pre-
limiting exact-solvable LPP or polymer models, where similar line ensembles
given by different Brownian motions or random walks are available [DNV23].

We now move on to the main results of this paper which prove the full Brownian
bridge conjecture for both the directed landscape (zero temperature) and the
continuum directed random polymer (positive temperature). Further, in the
latter case, our results also establish a quenched localization phenomenon,
where the polymer localizes around a random backbone, the law of the latter
being a Brownian bridge.

###  1.1. Main results

We start by defining some of the key objects to help set up the framework to
state our main results. The directed landscape $\mathcal{L}$, constructed in
[DOV22], is a continuous random function from the parameter space

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

to $\mathbb{R}$; $(x,s)$ and $(y,t)$ should be thought of as specifying a pair
of space-time coordinates. It satisfies the composition law

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

for any $r<s<t$, yielding the ‘reverse triangle inequality’
$\mathcal{L}(x,r;z,t)\geq\mathcal{L}(x,r;y,s)+\mathcal{L}(y,s;z,t)$ and thus
making it a ‘directed metric’.

We next describe the ‘directed geometry’ induced by $\mathcal{L}$ and record
some facts about it, following [DOV22, Section 12]. For any $s<t$, and $x,y$,
a path from $(x,s)$ to $(y,t)$ is a continuous function
$\pi:[s,t]\to\mathbb{R}$ with $\pi(s)=x$ and $\pi(t)=y$; and its length is
given by

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

A path $\pi$ is a geodesic if $\|\pi\|_{\mathcal{L}}$ is maximal among all
paths with the same start and endpoints. Equivalently, a geodesic between
$(x,s)$ and $(y,t)$ is any path $\pi$ with
$\|\pi\|_{\mathcal{L}}=\mathcal{L}(x,s;y,t)$. Almost surely, geodesics exist
between every pair $(x,s),(y,t)$ with $s<t$, and we use $\pi_{(x,s;y,t)}$ to
denote any such a geodesic. Moreover, there is almost surely a unique geodesic
between any fixed pair $(x,s),(y,t)$. Let $\pi_{0}:[0,1]\to\mathbb{R}$ denote
the geodesic from $(0,0)$ to $(0,1)$.

We now arrive at our first main result concerning its limit, under the upper
tail event.

######  Theorem 1.1.

As $L\to\infty$, $2L^{1/4}\pi_{0}$ conditioned on $\mathcal{L}(0,0;0,1)>L$
converges to a standard Brownian bridge, weakly in the topology of uniform
convergence.

While the source of the exponent $\frac{1}{4}$ will be reviewed shortly in the
upcoming idea of proofs section, momentarily we switch to our positive
temperature ($\beta=1$) model of the continuum directed random polymer (CDRP).

For $(x,s;y,t)\in\mathbb{R}_{4}^{\uparrow}$, let
$(x,s;y,t)\mapsto\mathcal{Z}(x,s;y,t)$ be the (mild) solution to the
multiplicative stochastic heat equation (SHE) defined by requiring, for all
$x,s\in\mathbb{R}$,

$\begin{cases}\partial_{t}\mathcal{Z}(x,s;y,t)=\frac{1}{4}\partial_{yy}\mathcal{Z}(x,s;y,t)+\mathcal{Z}(x,s;y,t)\xi(x,s;y,t)&s<t\\\
\mathcal{Z}(x,s;\cdot,s)=\delta_{x},\end{cases}$ (1.2)

where $\xi$ is a space-time white noise (which is the same for all choices of
initial coordinates $(x,s)$) and $\delta_{x}$ is the delta mass at $x$. The
initial condition is understood in the weak sense, i.e., with probability one
$\lim_{t\to s}\int f(y)\mathcal{Z}(x,s;y,t)\,\,\mathrm{d}y=f(x)$ for all
smooth functions $f$ of compact support.

This random field was constructed in [AKQ14, Theorem 3.1] and is a continuous
process. (More precisely, the field constructed in [AKQ14] satisfies (1.2)
with coefficient $\frac{1}{2}$ in place of $\frac{1}{4}$, but this is related
to our solution by a simple scaling by constants; we will discuss this a bit
more in Section 2. We adopt this variant of the SHE in order to match more
closely with the directed landscape.)

The KPZ equation is the stochastic PDE

$\displaystyle\partial_{t}\mathcal{H}(x,t)=\tfrac{1}{4}(\partial_{x}\mathcal{H}(x,t))^{2}+\tfrac{1}{4}\partial_{xx}\mathcal{H}(x,t)+\xi(x,t)$

(again we have adopted coefficients of $\frac{1}{4}$ in place of the more
standard $\frac{1}{2}$). It is related to the SHE by the _Cole-Hopf_
transform: the solution $\mathcal{H}$ to the KPZ equation is defined to be
$\log\mathcal{Z}$; this solution, which corresponds to delta initial
conditions for the SHE, is called the _narrow-wedge_ solution.

A key property of $\mathcal{Z}$, allowing one to define the CDRP, is the
following semi-group property, which can be understood as a positive
temperature analogue of the composition law (1.1): almost surely, for any
$x,y\in\mathbb{R}$ and $s<r<t$,

$\mathcal{Z}(x,s;y,t)=\int\mathcal{Z}(x,s;z,r)\mathcal{Z}(z,r;y,t)\,\mathrm{d}z$
(1.3)

(see e.g. [AJRAS22, Theorem 2.6(v)]). As in [AKQ14], conditional on
$\mathcal{Z}$ we can define the random polymer from $(x,s)$ to $(y,t)$ (for
any $s<t$), as the continuous random function $\Gamma:[s,t]\to\mathbb{R}$,
such that for any $k\in\mathbb{N}$ and $s=s_{0}<s_{1}<\cdots<s_{k}<s_{k+1}=t$,
and $x=x_{0},x_{1},\ldots,x_{k},x_{k+1}=y$, the probability density for
$\Gamma(s_{1})=x_{1},\ldots,\Gamma(s_{k})=x_{k}$ is proportional to

$\prod_{i=0}^{k}\mathcal{Z}(x_{i},s_{i};x_{i+1},s_{i+1}).$ (1.4)

We will use $\boldsymbol{\mathscr{P}}$ to denote the (quenched) measure of the
polymer. Thus $\mathcal{Z}$ is the partition function of this polymer model
and $\mathcal{H}$ is the free energy.

Letting $\Gamma_{0}$ denote the (annealed) random polymer from $(0,0)$ to
$(0,1)$, the counterpart of Theorem 1.1 in the setting of the CDRP is the
following.

######  Theorem 1.2.

As $L\to\infty$, $2L^{1/4}\Gamma_{0}$ conditioned on
$\log\mathcal{Z}(0,0;0,1)>L$ converges to a standard Brownian bridge, weakly
in the topology of uniform convergence.

Further, our proof will show that in this case a further exponent arises in
the form of a quenched structural result. More precisely, under the upper tail
event and conditional on the environment, the polymer fluctuates on a scale
$L^{-1/2}$ (up to logarithmic factors), which is much smaller than $L^{-1/4}$,
around a random path which we call the random backbone. The latter when scaled
by $L^{-1/4},$ converges to a Brownian bridge. We postpone a further
discussion of this to later in the article, once the relevant arguments have
already been presented (see 8.3).

Our arguments involve several ingredients which we now provide an overview of.
Due to the length of the paper and the technical nature of some of the
arguments, we have chosen to give a reasonably detailed discussion.

###  1.2. Idea of proofs: zero temperature

We start by outlining our proof for Theorem 1.1, and then move on in Section
1.3 to the extra ingredients involved in proving the positive temperature
case, Theorem 1.2.

The arguments consist of two components: (i) tightness of the paths and (ii)
the convergence of the finite dimensional distributions to that of the
Brownian bridge. But we start off by introducing a phenomenon that is the
driving force of many of the arguments.

####  1.2.1. The tent picture

The basic perspective of our proof relies on a result recently proved by the
first two authors in [GH22] on the shape of the profile
$\mathcal{L}(0,0;\cdot,1)$ conditional on $\mathcal{L}(0,0;0,1)>L$. First we
recall that $x\mapsto\mathcal{L}(0,0;x,1)+x^{2}$ is stationary, so when there
is no conditioning $\mathcal{L}(0,0;x,1)$ fluctuates around the parabola
$-x^{2}$. (This is a consequence of the mentioned shear invariance of
$\mathcal{L}$, and a fuller form of this invariance will play a crucial role
in our arguments; we expand on it in the next subsection.)

In short, the result in [GH22] says that the conditioned profile adopts a
_tent_ -like shape (see Figure 1) on the interval $[-L^{1/2},L^{1/2}]$ (which
is determined by locating the points of tangency of the tangent lines to
$-x^{2}$ which pass through $(0,L)$). Further, the fluctuations of the profile
around the tent are essentially Brownian, in particular with a fluctuation
scale of $L^{1/4}$.

This description then allows to produce sharp upper tail asymptotics; while
the one-point estimate in the zero temperature case that we will need was
already known, [GH22] also provides these estimates for the KPZ equation.

Figure 1. An illustration of the profile $\mathcal{L}(0,0;\cdot,1)$
conditional on it equaling $L$ at $0$, and the parabola $-x^{2}$ that it
fluctuates around when there is no conditioning.

A guiding principle in many of our arguments will be that the conditioning
$\mathcal{L}(0,0;0,1)>L$ corresponds to the existence (at some location at
height $s$ for any fixed $s$) of such a peak of height approximately $sL$.

####  1.2.2. Tightness

We prove tightness of $\pi_{0}$, conditioned on the upper tail at depth $L$ of
the path weight/free energy and as a family of continuous random functions on
$[0,1]$ indexed by $L$, using the Kolmogorov-Chentsov criterion for tightness
(e.g. [Kal22, Theorem 23.7]). This reduces the task to estimating the
transversal fluctuation (on the $\smash{L^{-1/4}}$ scale) at two points. More
precisely, we will prove an upper bound on the probability that
$|\pi_{0}(s)-\pi_{0}(t)|\smash{(t-s)^{-1/2}L^{1/4}}>M$, conditional on
$\mathcal{L}(0,0;0,1)>L$, that is uniform in $L$ and $0<s<t<1$.

The strategy uses a fuller form of the earlier mentioned shear invariance
property of the $\mathcal{L}$ field in a crucial way. The event under
consideration implies that there exist some $x$, $y$ such that

$|x-y|>M(t-s)^{1/2}L^{-1/4}\quad\text{and}\quad\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;y,t)+\mathcal{L}(y,t;0,1)>L.$

By doing a trivial bound on the conditional probability, we must show that the
unconditional probability of this event is much smaller than the probability
of $\mathcal{L}(0,0;0,1)>L$. Shear invariance of $\mathcal{L}$ says that, for
any fixed $z\in\mathbb{R}$

$\mathcal{L}(x,s;y,t)\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}(x,s;y+z,t)+\frac{(y+z-x)^{2}}{t-s}-\frac{(y-x)^{2}}{t-s}$

as processes in $(x,s;y,t)$ (see Section 2.1 below). Using this, the
probability of the displayed event can be reduced to that of
$\mathcal{L}(0,0;0,1)>L+\frac{(M(t-s)^{1/2}L^{-1/4})^{2}}{t-s}=L+M^{2}L^{-1/2}$.
So, essentially, shear invariance yields that the geodesic moving out by an
amount $\varepsilon$ between times $s$ and $t$ corresponds to a loss in weight
of $\varepsilon^{2}/(t-s)$ for any $\varepsilon>0$ and $s<t$, which must be
made up in order to achieve the upper tail conditioning.

To implement this strategy, we must obtain a relatively sharp comparison of
the probabilities of $\mathcal{L}(0,0;0,1)>L+\delta$ and that of
$\mathcal{L}(0,0;0,1)>L$ for $\delta>0$. The distribution of
$\mathcal{L}(0,0;0,1)$ is the same as that of the GUE Tracy-Widom
distribution, for which extremely precise tail asymptotics are available using
the exact formulas it is described by: indeed, for instance, it is known that
$\mathbb{P}(\mathcal{L}(0,0;0,1)>L)=(32\pi)^{-1}L^{-3/2}\exp(-\frac{4}{3}L^{3/2}+O(L^{-3/2}))$
([BBD08, eq. (25)]) which immediately implies that
$\mathbb{P}(\mathcal{L}(0,0;0,1)>L+\delta)=\exp(-2\delta\smash{L^{1/2}}+O(L^{-1/2}))\mathbb{P}(\mathcal{L}(0,0;0,1)>L)$.

However, our proof must also work for the positive temperature case, and there
such precise asymptotics are not available (the discussion above using shear
invariance will not apply verbatim to the KPZ equation as we will explain in
Section 1.3, but we will still need a tail comparison statement). Indeed, the
sharpest upper tail asymptotics currently available for $\mathcal{H}(0,1)$ are
from [GH22] and state that
$\mathbb{P}(\mathcal{H}(0,1)>L)=\exp(-\frac{4}{3}L^{3/2}+O(L^{3/4}))$; the
error term is too large to see that the ratio is $\exp(-2\delta
L^{1/2}+o(L^{1/2}))$. However, the methods from [GH22] can be used to couple
together the tent pictures coming from the events
$\mathcal{L}(0,0;0,1)>L+\delta$ and the same with $\delta=0$ to directly
obtain a bound on the ratio which also holds for the positive temperature
case:

######  Theorem 1.3.

Let $L\geq 2$ and $0<\delta<L^{1/4}$. Then

$\displaystyle\frac{\mathbb{P}\left(\mathcal{L}(0,0;0,1)>L+\delta\right)}{\mathbb{P}\left(\mathcal{L}(0,0;0,1)>L\right)}=\exp\left(-2\delta
L^{1/2}+O(\delta L^{-1/4}\log L)\right),$

and the same bound also holds in the positive temperature case of the KPZ
equation.

This theorem is restated and proven in a more precise and quantified form as
Theorem 4.1 in Section 4.

Applying this ratio estimate with $\delta=M^{2}L^{-1/2}$ thus yields the
desired transversal fluctuation bound with a tail of $\exp(-2M^{2})$ (which we
note matches the upper tail asymptotic of $\frac{1}{2}(s(1-s))^{-1/2}B(s)$),
which more than suffices to invoke the Kolmogorov-Chentsov criterion.

Thus in our argument, the fluctuation scale of $L^{-1/4}$ of the geodesic is
obtained as a consequence of the shear invariance of $\mathcal{L}$ and the
precise one-point upper tail asymptotics. However, the implicit source of the
exponent $\smash{\frac{1}{4}}$ is the underlying Brownian behavior of the
passage time profiles. Ignoring $s$-dependent constants, from the tent
picture, $\mathcal{L}(0,0;\cdot,s)+\mathcal{L}(\cdot,s;0,1)$ is essentially a
sum of independent rate two Brownian bridges on $[-L^{1/2},L^{1/2}]$ which
reach order $L$ above their starting points, and we are interested in the
maximizer of the sum. The maximizer density at $x$ is essentially the density
that $L$ will be reached at $x$ (ignoring fluctuations above $L$). Doing a
Taylor expansion, we see that at $x$ (where the variance is
$(L^{1/2}+x)(L^{1/2}-x)/L^{1/2}=L^{1/2}-x^{2}L^{-1/2}$), the one-point density
is proportional to

$\displaystyle\exp\left(-c\frac{L^{2}}{L^{1/2}-x^{2}L^{-1/2}}\right)=\exp\left(-cL^{3/2}-cx^{2}L^{1/2}+O(L^{-1/2})\right).$

The first term is the same for all $x$ and gets removed in the normalization,
and thus we see that the distribution of the maximizer is on scale $L^{-1/4}$.

However, we do not use this intuition explicitly in our arguments. The details
of our argument will be given in Section 5. The argument in the positive
temperature case is substantially more complicated, and we will discuss it in
Section 1.3 below.

For the moment we turn to proving the Gaussianity of the finite dimensional
distributions. We proceed by setting up the primary tool we rely on in this
endeavor.

Figure 2. A depiction of the parabolic Airy line ensemble.

####  1.2.3. Line ensembles

First, consider the one point distribution, say $\pi_{0}(\frac{1}{2})$.
Observe that it is precisely the argmax of
$x\mapsto\mathcal{L}(0,0;x,\frac{1}{2})+\mathcal{L}(x,\frac{1}{2};0,1)$,
whereas the conditioning of $\mathcal{L}(0,0;0,1)>L$ is precisely
$\max_{x}\mathcal{L}(0,0;x,\frac{1}{2})+\mathcal{L}(x,\frac{1}{2};0,1)>L$. The
two processes $\mathcal{L}(0,0;\cdot,\frac{1}{2})$ and
$\mathcal{L}(\cdot,\frac{1}{2};0,1)$ (without conditioning on the upper tail)
are independent, and up to a rescaling each has the distribution of the
(parabolic) Airy2 process (see [QR14] for a survey about it). For us an
important property of the parabolic Airy2 process is the earlier mentioned
fact that it is equal to a stationary process minus a parabola $x^{2}$.

This process is also the top (or lowest indexed) line of the Airy line
ensemble, a $\mathbb{N}$-indexed family of continuous non-intersecting
processes on $\mathbb{R}$ constructed by Corwin and Hammond [CH14] who also
showed that it admits a crucial resampling invariance property which they
termed as the Brownian Gibbs property. Essentially, this states that, for any
fixed interval $[a,b]$, conditional on the second line of the Airy line
ensemble and the values of the Airy2 process at the boundary points of this
interval, inside $[a,b]$ the distribution of the process is that of a (rate
two) Brownian bridge conditioned to stay above the second line (which can be
thought of as the negative parabola $-x^{2}$ for the purposes of this
discussion, ignoring the stationary fluctuations of it around the parabola).

It is using this tool that [GH22] deduces the tent description of the shape of
the top curve conditioned on the upper tail mentioned in Section 1.2.1, which
we recall briefly. Conditional on the Airy2 process at $0$ equaling $h$ for
some large value $h$, the process on either side of 0 behaves like two
independent Brownian bridges, one each on $[-h^{1/2},0]$ and $[0,h^{1/2}]$,
with values $-h$ (plus a random fluctuation term of lower order) at $\pm
h^{1/2}$ (see Figure 1). In particular, the slope of the lines around which
the Brownian bridges fluctuate are respectively $\pm h^{1/2}$ to first order.
By stationarity, there is a similar picture when the Airy2 process is
conditioned to be large at any other point.

In the positive temperature case, the narrow wedge solution to the KPZ
equation (the logarithm of (1.2)) can also be embedded as the lowest indexed
curve in an $\mathbb{N}$-indexed family of continuous curves (though they are
no longer non-intersecting) known as the KPZ line ensemble. This ensemble too
has a resampling property in terms of a rate two Brownian bridge, though
instead of being conditioned to not intersect the next curve, it is reweighted
by an energetic penalization for intersection with the lower curve (see
(2.6)). The arguments in [GH22] also established such a tent picture in the
setting of the KPZ line ensemble, which was then further used as a key input
to prove the mentioned sharp upper tail estimates for the KPZ equation.

####  1.2.4. One-point Gaussianity

Discretizing space using a fine mesh, and considering the closest mesh point
to the geodesic, leads to the following simplified form of the basic idea
which drives this argument. For any $s>0$, the conditioning that
$\mathcal{L}(0,0;0,1)>L$ can be realized as a union of the events that
$\mathcal{L}(0,0;x,s)=h_{1},\mathcal{L}(x,s;0,1)=h_{2}$ over a fine mesh of
points $x$ and a collection of heights $h_{1}$ and $h_{2}$ (which we will
often refer to as _peak heights_ in the discussion) such that their sum is
close to $L$ and they are approximately proportionate, i.e., $h_{1}\approx sL$
and $h_{2}\approx(1-s)L$. Conditioning on one of the latter events where $x$,
$h_{1}$, $h_{2}$ are fixed then allows us to make use of the just described
tent picture. We also note that the tent picture for $\mathcal{L}(0,0;\cdot
s)$ corresponds to a rescaled parabolic Airy2 process such that the parabola
is $-x^{2}/s$, and thus the tangent lines under
$\mathcal{L}(0,0;\cdot,s)=h_{1}$ will have slope $\smash{\pm
s^{-1/2}h_{1}^{1/2}(1+o(1))}$; so if $h_{1}\approx sL$, the slope is to first
order $\pm L^{1/2}$ independent of $s$, and similarly for the time interval
$[s,1]$.

With this idea in mind, we want to use the tent picture to understand the
distribution of
$\operatorname*{argmax}_{x}\mathcal{L}(0,0;x,\frac{1}{2})+\mathcal{L}(x,\frac{1}{2};0,1)$,
conditional on $\mathcal{L}(0,0;x_{*},1/2)=h_{1}$ and
$\mathcal{L}(x_{*},1/2;0,1)=h_{2}$, for given $h_{1}$, $h_{2}$ and $x_{*}$.
Then we would like to average over $h_{1}$ and $h_{2}$ such that
$h_{1}+h_{2}>L$ to obtain the distribution of $\pi_{0}(\frac{1}{2})$. However,
in the above strategy, the tent picture will only be useful if $h_{1}$ and
$h_{2}$ are such that $\pi_{0}(\frac{1}{2}),$ with high probability, lands in
the mesh interval adjacent to $x_{*}$. Even then, this as just described is
technically very delicate; indeed, apart from obtaining the correct expression
in the exponent for the density of $L^{1/4}\pi_{0}(s)$ as $L\to\infty$, one
also needs to obtain the correct constant prefactor of
$(2\pi_{0}\cdot\frac{1}{4}s(1-s))^{-1/2}$ (as we are trying to prove
$L^{1/4}\pi_{0}(s)\stackrel{{\scriptstyle\smash{d}}}{{\to}}\frac{1}{2}B(s)=N(0,\frac{1}{4}s(1-s))$).

To obtain such sharp estimates, instead of directly obtaining probability
bounds, we instead study the ratio of the probability that $L^{1/4}\pi_{0}(s)$
is close to $x_{*}$ vs that it is close to $y_{*}$. In this approach, we
simply need to obtain the correct expression in the exponent, since the
constant pre-factor then gets determined by normalization considerations. By a
decomposition based on the location and height of the tent, we essentially
must compare the two probabilities

$\displaystyle\mathbb{P}\Bigl{(}\pi_{0}(s)\approx
z,\mathcal{L}(0,0;0,1)>L,\mathcal{L}(0,0;z,s)\approx
h_{1},\mathcal{L}(z,s;0,1)\approx h_{2}\Bigr{)}$ (1.5)

with $z=x_{*}L^{-1/4}$ or $y_{*}L^{-1/4}$ and $h_{i}$ in a nice set. By shear
invariance of $\mathcal{L}$, it holds that

$\displaystyle\mathcal{L}(0,0;x_{*}L^{-1/4},s)$
$\displaystyle\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}(0,0;y_{*}L^{-1/4},s)+\left(\frac{y_{*}^{2}}{s}-\frac{x_{*}^{2}}{s}\right)L^{-1/2}$
$\displaystyle\mathcal{L}(x_{*}L^{-1/4},s;0,1)$
$\displaystyle\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}(y_{*}L^{-1/4},s;0,1)+\left(\frac{y_{*}^{2}}{1-s}-\frac{x_{*}^{2}}{1-s}\right)L^{-1/2}.$

Thus by Theorem 1.3 and the independence of $\mathcal{L}$ across disjoint
temporal strips, it holds that

$\displaystyle\mathbb{P}\left(\mathcal{L}(0,0;x_{*}L^{-1/4},s)\approx
h_{1},\mathcal{L}(x_{*}L^{-1/4},s;0,1)\approx h_{2}\right)$
$\displaystyle=\exp\left(-2\frac{y_{*}^{2}-x_{*}^{2}}{s(1-s)}+o(1)\right)\cdot\mathbb{P}\left(\mathcal{L}(0,0;y_{*}L^{-1/4},s)\approx
h_{1},\mathcal{L}(y_{*}L^{-1/4},s;0,1)\approx h_{2}\right).$

So to complete the comparison of the probabilities in (1.5), by a Bayes’
argument, it remains to show that

$\displaystyle\mathbb{P}\Bigl{(}\pi_{0}(s)\approx z,\mathcal{L}(0,0;0,1)>L\
\Big{|}\ \mathcal{L}(0,0;z,s)\approx h_{1},\mathcal{L}(z,s;0,1)\approx
h_{2}\Bigr{)}$

is the same up to a $1+o(1)$ factor for $z=x_{*}L^{-1/4}$ and $y_{*}L^{-1/4}$
for $h_{i}$ in a nice set. Towards this, the tent description tells us that
conditional on $\mathcal{L}(0,0;z,s)\approx h_{1}$ and if $h_{1}\approx sL$,
$\mathcal{L}(0,0;\cdot,s)$ is essentially a pair of independent Brownian
bridges with slope $\pm 2L^{1/2}$ on either side of $0$ (shifting coordinates
so that the new origin corresponds to $z$). The event $\pi_{0}(s)\approx
z,\mathcal{L}(0,0;0,1)>L$ is that the sum of these Brownian bridges has
maximizer near $0$ and maximum at least $L$. Now, a Brownian bridge with slope
$-L^{1/2}$ drops by order $L^{-1/2}$ in a neighborhood of order $L^{-1}$,
which is the same order as its fluctuations on the interval; this means that
such a Brownian bridge typically achieves its maximum in an order $L^{-1}$
neighbourhood of zero, and its maximum is greater than the value at zero by
$O(L^{-1/2})$.

The effect of $z=x_{*}L^{-1/4}$ vs $y_{*}L^{-1/4}$ is manifest by an
$O(L^{1/4})$ perturbation of the endpoint value of the Brownian bridge at the
tangency locations at location $\pm\Theta(L^{1/2})$ which does not
significantly affect the distribution of the bridges on the size $L^{-1}$ size
interval. This essentially establishes that the comparison in (1.5) is exactly
the ratio of Gaussian densities.

To make these arguments precise is actually a somewhat delicate task,
particularly due to complexities introduced in the positive temperature case
(that we discuss shortly). It is done in Sections 11 and 12.

####  1.2.5. Multi-point: coalescence of geodesics

We next discuss the two-point joint distribution of $\pi_{0}$, namely that of
$\pi_{0}(s)$ and $\pi_{0}(t)$, which are the argmax of
$(x,y)\mapsto\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;y,t)+\mathcal{L}(y,t;0,1)$.
Even without conditioning on the upper tail, in contrast to the one-point
distribution of $\pi_{0}$ which can be described as the argmax of the sum of
two independent scaled Airy2 processes, the two-point joint distribution does
not have such a simple or accessible description. The reason is that there is
no exact formula for the two-variable process $\mathcal{L}(\cdot,s;\cdot,t)$.
However, in the upper tail regime, we manage to decouple the argmax problem,
i.e., we are able to write $\mathcal{L}(\cdot,s;\cdot,t)$ as a sum of two one-
argument processes, using a strong coalescence phenomenon which is brought
about by the upper tail conditioning that we explain next

A key observation we make is that, assuming $\mathcal{L}(0,0;0,1)>L$ for a
large $L$, all the geodesics from $(x,s)$ to $(y,t)$ for any $|x|,|y|$ of
order smaller than $L^{1/2}$ would tend to coalesce (see the left panel of
Figure 3); that the window of coalescence is an interval of size of order
$L^{1/2}$ is closely related to the fact that the tent profile under the upper
tail holds on an interval with size of the same order, as can be seen in the
proof in Section 3. The coalescence phenomenon can be described via the
following quadrangle equality (see also the right panel of Figure 3 as well as
Lemma 2.4 below): with high probability it holds that

$\mathcal{L}(x,s;y,t)=\mathcal{L}(x^{\prime},s;y,t)+\mathcal{L}(x,s;y^{\prime},t)-\mathcal{L}(x^{\prime},s;y^{\prime},t),$
(1.6)

for all $|x|,|y|$ and $|x^{\prime}|,|y^{\prime}|$ that are of order smaller
than $L^{1/2}$. Such a quantitative description of coalescence can also be
generalized to the positive temperature setting of CDRP, to be discussed
shortly. Therefore, to have $\pi_{0}(s)=x_{*}$ and $\pi_{0}(t)=y_{*}$, roughly
one just needs to ensure that

$x_{*}=\operatorname*{argmax}_{x}\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;0,t),\quad
y_{*}=\operatorname*{argmax}_{y}\mathcal{L}(0,s;y,t)+\mathcal{L}(y,t;0,1).$

We further have that these two events are nearly independent. This is again
owing to the above mentioned coalescence which ensures that when $x$ and $y$
are varied, the changes in $\mathcal{L}(x,s;0,t)$ and $\mathcal{L}(0,s;y,t)$
respectively are local and hence are determined by disjoint parts of the
environment and the argmax only depends on the changes in the processes. We
can then estimate the ratio of the probabilities of these events using the
same arguments as the one-point case above.

The proof of coalescence and the induced independence is proven using tools
such as the multi-point passage times (which in the setting of the directed
landscape is studied in [DZ21]) and a shift-invariance symmetry of the
directed landscape from [BGW22] (see also [Gal21, Dau22, Zha23]). The details
are presented in Section 3.

An analogous analysis can also be done for the multi-point joint
distributions. This accounts for a significant part of the technical effort,
particularly in the positive temperature case. The last two sections of this
paper are devoted to them.

Figure 3. Left: An illustration of the coalescence phenomenon in time
$[\frac{1}{3},\frac{2}{3}]$, under upper tail. The geodesic from $(0,0)$ to
$(0,1)$ is shown in brown except for the portion common to all the paths,
which is in dark green. Right: a depiction of how coalescence implies equality
in the quadrangle inequality. It is clear that under coalescence the points
covered by the geodesics from $x$ to $y$ and from $x^{\prime}$ to $y^{\prime}$
equals (with multiplicity) those covered by the geodesics from $x$ to
$y^{\prime}$ and $x^{\prime}$ to $y$.

###  1.3. Complications in positive temperature

As has been mentioned several times, the positive temperature case introduces
several significant complications. We explain the key differences in the proof
of Theorem 1.2 (versus Theorem 1.1) next.

We will work with the free energy landscape defined by
$\mathcal{L}^{\beta}(x,s;y,t)=\log\mathcal{Z}(x,s;y,t)+(t-s)/12$.

While it should be thought of as largely the same as the directed landscape
$\mathcal{L}$ in terms of properties such as shear invariance and independence
in disjoint temporal strips, a significant difficulty already appears in the
first step regarding tightness, which we discuss next.

####  1.3.1. Tightness and polymer concentration

The first difficulty can be seen by the fact that the zero temperature path
measure, conditional on the environment and at any given height, is a delta
mass. In contrast, a priori we have no non-trivial concentration of the
quenched polymer measure. For this reason, though the shear invariance
argument described above also applies in the positive temperature case, it
only yields a bound on the transversal fluctuations of the polymer
$\Gamma_{0}$ of order $1$, not order $L^{-1/4}$ as in the zero temperature
case.

The key new idea to obtain the correct scale $L^{-1/4}$ of fluctuations is to
show that the quenched polymer measure in fact concentrates on an interval of
size of lower order than $L^{-1/4}$, which is the quenched localization result
mentioned after Theorem 1.2.

More specifically, the polymer measure concentrates in an order $L^{-1/2}$ (up
to a $\log L$ factor) window around a random backbone, which is
$\pi(s):=\operatorname*{argmax}_{x}\mathcal{L}^{\beta}(0,0;x,s)+\mathcal{L}^{\beta}(x,s;0,1)$
at level $s$:

######  Proposition 1.4.

There exists $M_{0}>0$ such that for any $L\geq 2$, $M>M_{0}$ and $s\in(0,1)$,

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\pi(s)|>ML^{-1/2}\log
L\right)>L^{-2M}\ \Big{|}\
\mathcal{L}^{\beta}(0,0;0,1)>L\right)<C\exp(-c(\log(L))^{2}).$

We have not been careful with the quantifiers on $s$ in this statement as we
are stating it for illustration; the technically precise statement that we
will actually prove and use is given as Proposition 7.1.

The exponent of $-\frac{1}{2}$ is due to the fact that, away from $\pi(s)$ and
by the tent picture, the free energy profile
$\smash{\mathcal{L}^{\beta}}(0,0;\cdot,s)+\smash{\mathcal{L}^{\beta}}(\cdot,s;0,1)$
decays with a slope of order $\smash{L^{1/2}}$. Then the window of order
$\smash{L^{-1/2}}$ around $\pi(s)$ is precisely where the free energy profile
is within order one of its maximum, and so the polymer measure density
(defined via the partition function, i.e., the exponential of the free energy)
is uniformly positive there.

The complete proof of this concentration result will be given in Section 7,
with preparations in the two sections before that. This concentration of the
polymer measure will allow us to upgrade the crude bounds coming from shear
invariance in Section 5 to tightness in Section 8; for technical reasons, we
only bound $|\Gamma_{0}(s)-\Gamma_{0}(t)|$ by order $(t-s)^{1/11}L^{-1/4}$
(rather than $(t-s)^{1/2}L^{-1/4}$ in the zero temperature setting).

####  1.3.2. Quantitative coalescence

A second difficulty working with polymers is establishing the coalescence
phenomenon, which we need to obtain the multi-point distributions. Indeed,
unlike geodesics with different yet close by end points, which actually
coalesce with high probability, the notion of coalescence for the CDRP
certainly has to be qualified with an associated coupling of the corresponding
Gibbs measures. For instance, independent samples of polymers with different
endpoints will in fact stay disjoint for almost all of their journey.

And indeed, while there have been a number of works studying coalescence in
zero temperature (e.g. [Pim16, Zha20, BSS19, SS20, BF22]), there has been only
one previous studies on polymer coalescence in the positive temperature
setting, namely [RASS23].

However, ultimately the zero temperature arguments outlined above only use
coalescence by way of its effect on the weight profile, i.e., in the sense of
(1.6) holding with high probability. Fortunately, this relation has the
potential to be generalized more directly to positive temperature. In fact, we
show that it still holds up to a error term which is exponentially small in
$L$, as captured in the following statement.

######  Proposition 1.5.

There exists $C,c$ such that for any $L>0$ and with $\delta=10^{-6}$,
conditionally on $\mathcal{L}^{\beta}(0,0;0,1)>L$, it holds with probability
at least $1-C\exp(-cL^{3/2})$ that

$\sup_{|x|,|y|\leq\delta
L^{1/2}}\left|\mathcal{L}^{\beta}(x,0;y,1)-\bigl{(}\mathcal{L}^{\beta}(x,0;0,1)+\mathcal{L}^{\beta}(0,0;y,1)-\mathcal{L}^{\beta}(0,0;0,1)\bigr{)}\right|<C\exp(-cL).$

This is proven in a more technically flexible form as Proposition 3.4, and
suffices for the remaining arguments go through. (Note however that the
conditioning and the coalescence is for free energies between heights $0$ and
$1$, while we will need it for intermediate temporal intervals; Proposition
3.4 allows this flexibility and we obtain the conditioning on the intermediate
free energies using the decomposition into tents with proportional peak
heights mentioned above.)

In proving this coalescence, we make connections to multi-point partition
functions of CDRP (analogs of the mentioned above multi-point passage times,
studied in e.g. [OW16, Nic21]), and use geometric arguments of performing
surgeries on the polymers. We note that such surgery techniques have appeared
several times in zero temperature settings (in e.g. [Ham16]), but do not seem
to have been introduced in positive temperature settings.

####  1.3.3. Multi-point Gaussianity

The CDRP setting also faces extra complexities in computing the joint multi-
point Gaussian limit. From the concentration result, it suffices to deduce the
joint multi-point Gaussian limit of the backbone $\pi$.

As in the zero temperature setting, the upper tail conditioning on
$\mathcal{L}^{\beta}(0,0;0,1)$ shall be realized as conditioning on tents of
certain height and certain locations, and we then show that these peaks are
close to the random backbone $\pi$ (and this is implemented in Section 10).

Both tasks of changing the conditioning and deducing the closeness would
require more careful treatment in the CDRP setting. There are two basic
sources of complication. The first is that entropy effects must be taken into
account, since in positive temperature we must take integrals over partition
function profiles instead of maximums as in the zero temperature setting. The
second is that in positive temperature the tent picture obtained by
decomposing into peaks at different locations is not fully accurate on certain
small scales.

The complication from entropy is simple to see: in zero temperature, the peak
heights of the tents were essentially such that their sum was at least $L$,
but in positive temperature there is an extra correction term of order $\log
L$ which must be added. This is because, by the tent picture, the slopes
around the peak locations are $\pm 2L^{1/2}$, which means almost all the
contribution to the integral in the composition formula (1.3) comes from an
interval of order $L^{-1/2}$ around the peak. On taking logarithms, the small
size of the interval leads to a loss of order $\log L$ which must be made up
by the peak heights.

The inaccuracy of the tent picture on small scales (in fact, on scale
$L^{-1/2}$) is slightly more subtle. Recall that the integral of the free
energy profile
$\smash{\mathcal{L}^{\beta}}(0,0;\cdot,s)+\smash{\mathcal{L}^{\beta}}(\cdot,s;0,1)$
has dominant contributions on scale $\smash{L^{-1/2}}$ around the peak
location (due to the fact that that the slopes in the tent picture are of
order $\smash{L^{1/2}}$). Further, as in zero temperature and from Theorem
1.3, the integral (i.e., the total free energy) has fluctuations of order
$\smash{L^{-1/2}}$. However, though the latter fluctuation scale matches with
zero temperature, the fact that the integral is on a $O(\smash{L^{-1/2}})$
window and a Brownian computation indicate that a significant contribution to
the partition function instead comes from the event of the Brownian bridges
atypically rising up by $O(1)$ on a $\smash{L^{-1/2}}$ window, with the peak
heights being slightly lower by $O(1)$. Therefore, conditional on the total
free energy upper tail, the Brownian bridges should exhibit behaviors of
probability $\exp(-cL^{1/2})$; thus the tents are flat in an order $L^{-1/2}$
interval around the peak location. This is in contrast to zero temperature
where under the analogous conditioning the free energy profiles to the left
and right of the peak location are with high probability independent Brownian
bridges, and thus are flat only on order $L^{-1}$ intervals (where Brownian
fluctuations match with slope loss).

######  Remark 1.6.

To make our proofs as short as possible, we will use notation which allows us
to unify the positive temperature and zero temperature arguments as much as
possible.

### Organization of the remaining text

In Section 2 we give the formal setup and quote existing tools and estimates
that we will use. Section 3 establishes coalescence under the upper tail, and
Section 4 proves the estimate on the ratio of upper tail probabilities. The
next four sections prove the tightness; in particular, the zero temperature
tightness is done in Section 5, while the positive temperature tightness is
much more involved and is proved in Section 8, along with the localization
around a random backbone. The last four sections are devoted to proving the
finite-point Gaussian limit.

### Acknowledgement

SG was partially supported by NSF grant DMS-1855688, NSF CAREER grant
DMS-1945172, and a Sloan Research Fellowship. MH is partially supported by NSF
grant DMS-1937254. LZ is supported by the Miller Institute for Basic Research
in Science, at the University of California, Berkeley, and NSF grant
DMS-2246664. The authors are grateful to Zhipeng Liu for many useful
discussions and explaining the ideas in [Liu22b, LW22].

##  2\. Preliminaries

Notations. Throughout this paper, we will use $C,c>0$ to denote large and
small constants, whose values may and often change from line to line. We will
also use the Bachmann-Landau notations: for $A>0$, $O(A)$ denotes a number $B$
such that $|B|<CA$, and $\Omega(A)$ denotes a number $B$ such that $B>cA$. For
any $x,y\in\mathbb{R}\cup\\{-\infty,\infty\\}$, $x\leq y$, we denote
$[\\![{x,y}]\\!]=[x,y]\cap\mathbb{Z}$.

For any $m\in\mathbb{N}$, we introduce the following simplexes,

$\Lambda_{m}=\\{(x_{1},\cdots,x_{m}):x_{1}\leq\cdots\leq x_{m}\\},$

and for any $a<b$,

$\Lambda_{m}([a,b])=\\{(x_{1},\cdots,x_{m}):a\leq x_{1}\leq\cdots\leq
x_{m}\leq b\\}.$

We use $\mathring{\Lambda}_{m}$ and $\mathring{\Lambda}_{m}([a,b])$ to denote
the interiors of $\Lambda_{m}$ and $\Lambda_{m}([a,b])$, respectively. For any
$\mathbf{x}=(x_{1},\cdots,x_{m})\in\Lambda_{m}$, let
$\Delta(\mathbf{x})=\prod_{i<j}(x_{i}-x_{j})$.

For any $\mu$ and $\sigma$, we let $\mathcal{N}(\mu,\sigma^{2})$ denote the
normal distribution with mean $\mu$ and variance $\sigma^{2}$.

For any topological space $X$, we use $\mathcal{C}(X,\mathbb{R})$ to denote
the space of real continuous functions on $X$ with the uniform topology.

We will occasionally use notation such as $\mathbb{P}(\cdot\mid
X\in(K,K+\mathrm{d}K))$ for a random variable $X$ and real number $K$ to
denote the conditional probability distribution given that $X=K$. The precise
meaning of this is to consider the regular conditional distribution
$\mathbb{P}(\cdot\mid X)$ (which exists when conditioning on real random
variables by well-known abstract results such as [Kal22, Theorem 8.5]) and
evaluate the associated probability kernel at $K$.

Similarly the notation $\mathbb{P}(X\in(K,K+\mathrm{d}K))/\mathrm{d}K$ is
simply the density of the random variable $X$ with respect to Lebesgue
measure, evaluated at $K$. It is well-known that the one-point distributions
of objects such as $\mathcal{L}$ and $\mathcal{L}^{\beta}$ have densities,
e.g. from the Brownian Gibbs properties [CH14, CH16].

We next move on to some properties of the directed landscapes and the
associated geodesics therein which will appear in our arguments repeatedly.

###  2.1. The directed landscape and geodesics

The directed landscape $\mathcal{L}$, which is a random continuous function on
$\mathbb{R}^{4}_{\uparrow}$, is shift, shear, reflection, and $1:2:3$ scaling
invariant. More precisely, we have the following:

######  Lemma 2.1 (Lemma 10.2, [DOV22]).

$\mathcal{L}$ has the same distribution as

  * $\bullet$

(Shift and shear) $(x,s;y,t)\mapsto\mathcal{L}(x+\nu s+\alpha,s+\eta;y+\nu
t+\alpha,t+\eta)+2\nu(y-x)+\nu^{2}(t-s)$, for any
$\nu,\alpha,\eta\in\mathbb{R}$;

  * $\bullet$

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

  * $\bullet$

(Scaling) $(x,s;y,t)\mapsto w\mathcal{L}(w^{-2}x,w^{-3}s;w^{-2}y,w^{-3}t)$,
for any $w>0$.

Further, for any disjoint time intervals $\\{(s_{i},t_{i})\\}_{i=1}^{k}$, the
functions $\mathcal{L}(\cdot,s_{i};\cdot,t_{i})$ are independent.

We next describe the multi-point passage times, studied in [DZ21]. For any
$k\in\mathbb{N}$, $\mathbf{x}=(x_{1},\cdots,x_{k})\in\Lambda_{k}$,
$\mathbf{y}=(y_{1},\cdots,y_{k})\in\Lambda_{k}$, and $s<t$, we define

$\mathcal{L}(\mathbf{x},s;\mathbf{y},t)=\sup_{\pi_{1},\dots,\pi_{k}}\sum_{i=1}^{k}\|\pi_{i}\|_{\mathcal{L}},$

where the supremum is over all $k$-tuples of paths
$\pi=(\pi_{1},\dots,\pi_{k})$ where each $\pi_{i}$ is a path from $(x_{i},s)$
to $(y_{i},t)$, satisfying the disjointness condition $\pi_{i}(r)<\pi_{j}(r)$
for all $i<j$ and $r\in(s,t)$. It is shown (in [DZ21, Theorem 1.7]) that,
almost surely, for every set of endpoints the supremum is achieved by some
paths satisfying the disjointness condition. Therefore the following statement
holds.

######  Lemma 2.2 ([DZ21, Corollary 1.11]).

Almost surely the following holds. For any $k\in\mathbb{N}$,
$\mathbf{x}=(x_{1},\cdots,x_{k}),\mathbf{y}=(y_{1},\cdots,y_{k})\in\Lambda_{k}$,
and $s<t$, we have

$\mathcal{L}(\mathbf{x},s;\mathbf{y},t)=\sum_{i=1}^{k}\mathcal{L}(x_{i},s;y_{i},t),$

if and only if there exist geodesics $\pi_{1},\dots,\pi_{k}$, where $\pi_{i}$
is from $(x_{i},s)$ to $(y_{i},t)$, satisfying $\pi_{i}(r)<\pi_{i+1}(r)$ for
each $1\leq i<k$ and $r\in(s,t)$.

For any $s<t$ and $x,y$, we also denote

$\mathcal{L}_{k}(x,s;y,t)=\mathcal{L}(x\mathbf{1}_{k},s;y\mathbf{1}_{k},t),$

where $\mathbf{1}_{k}\in\mathbb{R}^{k}$ is the vector with each coordinate
equal $1$.

A key property of the directed landscape is the following inequality due to
planarity.

######  Lemma 2.3 ([DZ21, Lemma 5.7]).

The following holds almost surely. Take any $k\in\mathbb{N}$ and
$\mathbf{x},\mathbf{y},\mathbf{x}^{\prime},\mathbf{y}^{\prime}\in\Lambda^{k}$.
Define
$\mathbf{x}^{\ell},\mathbf{y}^{\ell},\mathbf{x}^{r},\mathbf{y}^{r}\in\Lambda^{k}$
by setting $x^{\ell}_{i}=x_{i}\wedge x_{i}^{\prime}$,
$y^{\ell}_{i}=y_{i}\wedge y_{i}^{\prime}$, and $x^{r}_{i}=x_{i}\vee
x_{i}^{\prime}$, $y^{r}_{i}=y_{i}\vee y_{i}^{\prime}$, for each $1\leq i\leq
k$. Take any $s<t$. Then we have

$\mathcal{L}(\mathbf{x}^{\ell},s;\mathbf{y}^{\ell},t)+\mathcal{L}(\mathbf{x}^{r},s;\mathbf{y}^{r},t)\geq\mathcal{L}(\mathbf{x},s;\mathbf{y},t)+\mathcal{L}(\mathbf{x}^{\prime},s;\mathbf{y}^{\prime},t).$

In the case of $k=1$, this is the quadrangle inequality: for any $s<t$,
$x_{1}<x_{2}$, $y_{1}<y_{2}$, we have

$\mathcal{L}(x_{1},s;y_{1},t)+\mathcal{L}(x_{2},s;y_{2},t)\geq\mathcal{L}(x_{1},s;y_{2},t)+\mathcal{L}(x_{2},s;y_{1},t).$
(2.1)

See e.g. [DOV22, Lemma 9.1]. Besides, the strict inequality is known to be
equivalent to the disjointness of geodesics.

######  Lemma 2.4 ([GZ22, Lemma 3.15]).

For any fixed $s<t$, $x_{1}<x_{2}$, $y_{1}<y_{2}$, almost surely the
inequality

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

is equivalent to that the geodesics $\pi_{(x_{1},s;y_{1},t)}$ and
$\pi_{(x_{2},s;y_{2},t)}$ are disjoint; i.e.,
$\pi_{(x_{1},s;y_{1},t)}(r)<\pi_{(x_{2},s;y_{2},t)}(r)$, $\forall r\in[s,t]$.

Another degeneration of Lemma 2.3 is the following inequality which will be
used later. For any $x$ and $y_{1}\leq y_{2}\leq y_{3}$, and $s<t$, we have
almost surely,

$\mathcal{L}((x,x),s;(y_{1},y_{3}),t)+\mathcal{L}(x,s;y_{2},t)\geq\mathcal{L}((x,x),s;(y_{1},y_{2}),t)+\mathcal{L}(x,s;y_{3},t).$
(2.2)

We note that by the reflection symmetry of $\mathcal{L}$, this also holds for
$y_{1}\geq y_{2}\geq y_{3}$.

###### Proof of (2.2).

We take $z<x\wedge y_{1}$ and apply Lemma 2.3 with $k=2$, $\mathbf{x}=(x,x)$,
$\mathbf{x}^{\prime}=(z,x)$, $\mathbf{y}=(y_{1},y_{2})$,
$\mathbf{y}^{\prime}=(z,y_{3})$, to get

$\mathcal{L}((x,x),s;(y_{1},y_{3}),t)+\mathcal{L}((z,x),s;(z,y_{2}),t)\geq\mathcal{L}((x,x),s;(y_{1},y_{2}),t)+\mathcal{L}((z,x),s;(z,y_{3}),t).$
(2.3)

We note that by Lemma 2.2, and the fact that any geodesic is almost surely
continuous (therefore bounded), as well as from the stationarity of
$\mathcal{L}$ (so that the geodesic from $(z,s)$ to $(z,t)$ has the same
distribution as that from $(0,s)$ to $(0,t)$, shifted by $z$), we have

$\lim_{z\to-\infty}\mathbb{P}\bigl{(}\mathcal{L}(x,s;y_{2},t)=\mathcal{L}((z,x),s;(z,y_{2}),t)-\mathcal{L}(z,s;z,t)\bigr{)}=1,$
$\lim_{z\to-\infty}\mathbb{P}\bigl{(}\mathcal{L}(x,s;y_{3},t)=\mathcal{L}((z,x),s;(z,y_{3}),t)-\mathcal{L}(z,s;z,t)\bigr{)}=1.$

Plugging these into (2.3) we get (2.2). ∎

###  2.2. Continuum directed random polymer

We define the multi-line continuum partition functions through the chaos
expansion, as done in [OW16]. For CDRP, we take the inverse temperature
$\beta=1$ throughout this paper for simplicity of notations, while our
arguments go through verbatim for any fixed $\beta$.

####  2.2.1. Partition function

Let $W$ be a cylindrical Brownian motion on $L^{2}(\mathbb{R})$, and $\dot{W}$
be the space-time white noise associated with $W$. Denote

$p_{t}(x)=\frac{1}{\sqrt{2\pi t}}\exp(-x^{2}/2t).$

For any $s<t$, $x,y\in\mathbb{R}$, and $k\in\mathbb{Z}_{+}$, we let
$\tilde{\mathcal{Z}}_{k}(x,s;y,t)$ be defined by

$p_{t-s}(y-x)^{k}\left(1+\sum_{m=1}^{\infty}\int_{\Lambda_{m}([s,t])}\int_{\mathbb{R}^{m}}R((x_{1},t_{1}),\ldots,(x_{m},t_{m}))W(dt_{1},dx_{1})\cdots
W(dt_{m},dx_{m})\right),$

where $R$ denotes the $m$ point correlation function for a collection of $k$
non-intersecting Brownian bridges which all start at $x$ at time $s$ and end
at $y$ at time $t$. We write $\tilde{\mathcal{Z}}=\tilde{\mathcal{Z}}_{1}$,
which, historically, was introduced as the solution to the multiplicative
stochastic heat equation (SHE) with Dirac delta initial condition, via the
Feynman-Kac representation. Specifically,
$\tilde{u}(x,t)=\tilde{\mathcal{Z}}(0,0;x,t)$ satisfies

$\partial_{t}\tilde{u}=\frac{1}{2}\partial_{x}^{2}\tilde{u}+\tilde{u}\dot{W},$

with $\tilde{u}(\cdot,0)$ being the delta mass at $0$.

In what sense is $\tilde{\mathcal{Z}}_{k}(x,s;y,t)$ defined? In [OW16] this is
defined for any fixed $k$ and $x,s,y,t$, by proving the $L^{2}(W)$ convergence
of the chaos expansion. In [Nic21], it is shown that

$(y,k)\mapsto\log(\tilde{\mathcal{Z}}_{k}(0,0;y,t)/\tilde{\mathcal{Z}}_{k-1}(0,0;y,t))$

is a (scaled) KPZt line ensemble, as defined in [CH16, Theorem 2.15];
therefore $\tilde{\mathcal{Z}}_{k}(x,s;y,t)$ can be thought of as a continuous
function of $y$, for any fixed $k$ and $x,s,t$ (see [Nic21, Corollary 1.9,
1.11]). In [LW20], it is further shown that
$(y,t)\mapsto\tilde{\mathcal{Z}}_{k}(x,s;y,t)$ can be defined as a continuous
function, for any fixed $k,x,s$.

Moreover, $\tilde{\mathcal{Z}}$ can be defined as a four-parameter random
continuous function. It is also shift, shear, and reflection invariant (in
distribution), as recorded ahead in Lemma 2.5.

Scaling. Under certain limiting transitions (either $t\to\infty$ or
$\beta\to\infty$) and appropriate scaling, the logarithm of
$\tilde{\mathcal{Z}}$, which can be understood as a solution to the KPZ
equation, converges to the directed landscape [QS23, Wu23]. While we do not
actually use this convergence in this paper, in light of the scaling involved
in this limit transition, and for the purposes of being consistent with the
directed landscape setting and reducing notations (which will be clear
shortly), we denote
$\mathcal{Z}_{k}(x,s;y,t)=2^{k}\tilde{\mathcal{Z}}_{k}(2x,2s;2y,2t)$ and
$\mathcal{Z}=\mathcal{Z}_{1}$.111The reason for the outside factor of 2 in
$\mathcal{Z}(x,s;y,t)=2\tilde{\mathcal{Z}}(2x,2s;2y,2t)$ is to ensure that
indeed $\mathcal{Z}(x,s;\cdot,t)\to\delta_{x}$ as $t\to s$ in the weak sense.
Now $\mathcal{Z}$ satisfies the SHE of (1.2). For the shift, shear, and
reflection invariances of $\mathcal{Z}$, we have the following.

######  Lemma 2.5 (Theorem 3.1, [AKQ14] or Proposition 2.3, [AJRAS22]).

$\mathcal{Z}$ has the same distribution as

  * $\bullet$

(Shift and shear) $(x,s;y,t)\mapsto\mathcal{Z}(x+\nu s+\alpha,s+\eta;y+\nu
t+\alpha,t+\eta)\exp(\nu^{2}(t-s)+2\nu(y-x))$, for any
$\nu,\alpha,\eta\in\mathbb{R}$;

  * $\bullet$

(Reflection) $(x,s;y,t)\mapsto\mathcal{Z}(-x,s;-y,t)$, and
$(x,s;y,t)\mapsto\mathcal{Z}(y,-t;x,-s)$.

Further, for any disjoint time intervals $\\{(s_{i},t_{i})\\}_{i=1}^{k}$, the
functions $\mathcal{Z}(\cdot,s_{i};\cdot,t_{i})$ are independent. Also, with
probability one $\mathcal{Z}(x,s;y,t)>0$ for all $x,y\in\mathbb{R}$ and
$0<s<t$

As already mentioned in the introduction, for any $s<t$ and $x,y$, in [AKQ14]
a measure (denoted by $\boldsymbol{\mathscr{P}}$) on
$\mathcal{C}([s,t],\mathbb{R})$ is defined and gives the random polymer from
$(x,s)$ to $(y,t)$, with finite-dimensional distribution given by (1.4).

####  2.2.2. Multi-point partition function with distinct endpoints

For any
$\mathbf{x}=(x_{1},\cdots,x_{k}),\mathbf{y}=(y_{1},\cdots,y_{k})\in\mathring{\Lambda}_{k}$,
and $s<t$, we define

$\mathcal{M}(\mathbf{x},s;\mathbf{y},t)=\det[\mathcal{Z}(x_{i},s;y_{j},t)]_{i,j=1}^{k}\Delta(\mathbf{x})^{-1}\Delta(\mathbf{y})^{-1}$

where recall $\Delta(\mathbf{x})=\prod_{i<j}(x_{i}-x_{j})$. Then from the
continuity of $\mathcal{Z}=\mathcal{Z}_{1}$, we have that
$\mathcal{M}(\mathbf{x},s;\mathbf{y},t)$ is almost surely continuous in all
the variables.

Positivity and implications. Using the Karlin-McGregor theorem, it is
straightforward to deduce that $\mathcal{M}$ is non-negative, as shown in
[OW16, Proposition 5.5]. The simultaneous strict inequality is proved in
[LW20, Theorem 1.4], and also in [AJRAS22, Theorem 2.17] with a different
method.

######  Lemma 2.6.

Almost surely, for any $s<t$, $k\in\mathbb{N}$, and
$\mathbf{x},\mathbf{y}\in\mathring{\Lambda}_{k}$, there is
$\mathcal{M}(\mathbf{x},s;\mathbf{y},t)>0$.

The case of Lemma 2.6 where $k=2$ can be viewed as an analog to Equation 2.1;
namely, for any $s<t$, $x_{1}<x_{2}$, $y_{1}<y_{2}$, we have

$\mathcal{Z}(x_{1},s;y_{1},t)\mathcal{Z}(x_{2},s;y_{2},t)>\mathcal{Z}(x_{1},s;y_{2},t)\mathcal{Z}(x_{2},s;y_{1},t).$
(2.4)

Another useful statement that can be deduced from Lemma 2.6 is the following
monotonicity.

######  Lemma 2.7.

Almost surely the following is true. For any $s<t$, $x_{1}<x_{2}<x_{3}$ and
$y_{1}<y_{2}<y_{3}$, we have

$\displaystyle(y_{3}-y_{1})\mathcal{M}((x_{1},x_{3}),s;(y_{1},y_{3}),t)\mathcal{Z}(x_{2},s;y_{2},t)$
$\displaystyle>(y_{2}-y_{1})\mathcal{M}((x_{1},x_{3}),s;(y_{1},y_{2}),t)\mathcal{Z}(x_{2},s;y_{3},t)$
$\displaystyle\qquad+(y_{3}-y_{2})\mathcal{M}((x_{1},x_{3}),s;(y_{2},y_{3}),t)\mathcal{Z}(x_{2},s;y_{1},t).$

###### Proof.

This is equivalent to
$\mathcal{M}((x_{1},x_{2},x_{3}),s;(y_{1},y_{2},y_{3}),t)>0$, which holds by
Lemma 2.6. ∎

Composition. There is also a composition law of $\mathcal{M}$, which can be
obtained from (1.3) and the Cauchy-Binet formula: almost surely, for any
$k\in\mathbb{N}$, $\mathbf{x},\mathbf{y}\in\mathring{\Lambda}_{k}$ and
$s<r<t$, we have

$\mathcal{M}(\mathbf{x},s;\mathbf{y},t)=\int_{\mathring{\Lambda}_{k}}\mathcal{M}(\mathbf{x},s;\mathbf{z},r)\mathcal{M}(\mathbf{z},r;\mathbf{y},t)\Delta(\mathbf{z})^{2}d\mathbf{z}.$
(2.5)

Continuous extension. The function $\mathcal{M}$ is connected to the multi-
layer partition function, through the following extension of $\mathcal{M}$ to
the boundary of $\Lambda_{k}\times\Lambda_{k}$.

######  Lemma 2.8 ([OW16, Lemma 6.1]).

For any $s<t$ and $k\in\mathbb{N}$, the function
$\mathbf{x},\mathbf{y}\mapsto\mathcal{M}(\mathbf{x},s;\mathbf{y},t)$ extends
continuously in $L^{2}(W)$ to $\Lambda_{k}\times\Lambda_{k}$, and the
extension satisfies

$2^{-k(k-1)/2}(t-s)^{k(k-1)/2}\prod_{i=1}^{k-1}i!\mathcal{M}(x\mathbf{1},s;y\mathbf{1},t)=\mathcal{Z}_{k}(x,s;y,t),$

where $\mathbf{1}$ is the vector in $\mathbb{R}^{k}$ where each entry equals
$1$.

###  2.3. Line ensembles and Gibbs properties

As already indicated, a tool widely used in the study of the directed
landscape is the Airy line ensemble constructed in [CH14]. Analogously, a tool
widely used in the study of the KPZ equation and the free energy of the CDRP
is the related KPZt line ensemble from [CH16], and its Gibbs property. To be
concise, instead of giving the complete line ensemble setups, we quote some
useful results from these connections.

For any $t>0$, $x\in\mathbb{R}$, we denote

$\mathfrak{h}^{\beta=1}_{t,1}(x)=\log\mathcal{Z}(0,0;x,t)+t/12,\quad\mathfrak{h}^{\beta=1}_{t,2}(x)=\log(\mathcal{Z}_{2}(0,0;x,t)/\mathcal{Z}(0,0;x,t))+t/12,$

and

$\mathfrak{h}^{\beta=\infty}_{t,1}(x)=\mathcal{L}(0,0;x,t),\quad\mathfrak{h}^{\beta=\infty}_{t,2}(x)=\mathcal{L}_{2}(0,0;x,t)-\mathcal{L}(0,0;x,t).$

We note that $\mathcal{H}(x,t)=\mathfrak{h}^{\beta=1}_{t,1}(x)$ solves the KPZ
equation

$\displaystyle\partial_{t}\mathcal{H}=\tfrac{1}{4}\partial_{x}^{2}\mathcal{H}+\tfrac{1}{4}(\partial_{x}\mathcal{H})^{2}+\dot{W}.$

(Recall that $\dot{W}$ is the space-time white noise.)

Thanks to the scaling in defining $\mathcal{Z}$,
$\smash{\mathfrak{h}^{\beta=1}_{t,1}}$ has the same parabolic decay of
$-x^{2}/t$ as $\smash{\mathfrak{h}^{\beta=\infty}_{t,1}}$. More precisely,
$\smash{\mathfrak{h}^{\beta}_{t,1}(x)}+x^{2}/t$ for both $\beta=1$ and
$\infty$ are stationary (which can be deduced from the shear invariance of
$\mathcal{Z}$ and $\mathcal{L}$). We also denote

$\hat{\mathfrak{h}}^{\beta}_{t,1}(x):=t^{-1/3}\mathfrak{h}^{\beta}_{t,1}(t^{2/3}x),\quad\hat{\mathfrak{h}}^{\beta}_{t,2}(x):=t^{-1/3}\mathfrak{h}^{\beta}_{t,2}(t^{2/3}x).$

As will be clear later on, using $\smash{\hat{\mathfrak{h}}^{\beta}_{t,1}}$
and $\smash{\hat{\mathfrak{h}}^{\beta}_{t,2}}$ instead of
$\smash{{\mathfrak{h}}^{\beta}_{t,1}}$ and
$\smash{{\mathfrak{h}}^{\beta}_{t,2}}$ will reduce some notation, since
$\smash{\hat{\mathfrak{h}}^{\beta}_{t,1}}$ and
$\smash{\hat{\mathfrak{h}}^{\beta}_{t,2}}$ have the parabolic decay of
$-x^{2}$ independent of $t$.

Another consequence of the scaling in defining $\mathcal{Z}$ is that
$\smash{\mathfrak{h}^{\beta=1}_{t,1}}$ is locally absolutely continuous with
respect to a rate $2$ Brownian bridge, which is also true for
$\smash{\mathfrak{h}^{\beta=\infty}_{t,1}}$. To be more precise, we quote the
following Gibbs properties of $\mathfrak{h}^{\beta}_{t,1}$ given
$\mathfrak{h}^{\beta}_{t,2}$. For any $a<b$, denote by
$\mathcal{F}_{\mathrm{ext}}([a,b])$ the $\sigma$-algebra generated by
$\smash{\mathfrak{h}^{\beta}_{t,1}}$ in $\mathbb{R}\setminus(a,b)$, and
$\smash{\mathfrak{h}^{\beta}_{t,2}}$.

######  Lemma 2.9.

Take any $a<b$ and $t>0$. Conditional on $\mathcal{F}_{\mathrm{ext}}([a,b])$,
for (1) law of $\mathfrak{h}^{\beta}_{t,1}$ in $[a,b]$, (2) the rate $2$
Brownian bridge connecting $\smash{\mathfrak{h}^{\beta}_{t,1}(a)}$ and
$\smash{\mathfrak{h}^{\beta}_{t,1}(b)}$, the former is absolutely continuous
with respect to the latter, with Radon-Nikodym derivative (for a path $B$)
proportional to $W(B,\mathfrak{h}^{\beta}_{t,2})$, where

$W(f,g)=\begin{cases}\exp\Big{(}-2\int_{a}^{b}\exp(f(x)-g(x))dx\Big{)}&\text{
when }\beta=1,\\\ \mathds{1}[g(x)\leq f(x),\forall x\in[a,b]]&\text{ when
}\beta=\infty.\end{cases}$ (2.6)

For the case where $\beta=\infty$ (the Airy line ensemble setting), this was
established in [CH14]. For $\beta=1$ (the KPZt line ensemble setting), such a
Gibbs property was first introduced in [CH16]; and the connection between the
KPZt line ensemble and CDRP was formally established in [Nic21]. The form of
the Gibbs property presented here is from [GH22, Proposition 2.6, Theorem
2.8].

A useful consequence of the Gibbs property is the monotonicity recorded below.

######  Lemma 2.10 (Monotonicity in boundary data).

Fix $a<b$, real numbers $\smash{w^{*},z^{*}}\in\mathbb{R}$ and measurable
functions $\smash{g^{*}}:[a,b]\to\mathbb{R}\cup\\{-\infty\\}$ for
$*\in\\{\uparrow,\downarrow\\}$ such that $\smash{w^{\downarrow}\leq
w^{\uparrow}}$, $\smash{z^{\downarrow}\leq z^{\uparrow}}$, and, for all
$s\in(a,b)$, $\smash{g^{\downarrow}(s)\leq g^{\uparrow}(s)}$.

For $*\in\\{\uparrow,\downarrow\\}$, let $\smash{\mathcal{Q}^{*}}$ be a
process on $[a,b]$ such that $\smash{\mathcal{Q}^{*}}(a)=\smash{w^{*}}$ and
$\smash{\mathcal{Q}^{*}}(b)=\smash{z^{*}}$, with Radon-Nikodym derivative with
respect to Brownian bridge given by $W(\mathcal{Q}^{*},g)$ for $\beta=1$. Then
there exists a coupling of the laws of $\smash{\mathcal{Q}^{\uparrow}}$ and
$\smash{\mathcal{Q}^{\downarrow}}$ such that almost surely
$\smash{\mathcal{Q}^{\downarrow}}(s)\leq\smash{\mathcal{Q}^{\uparrow}_{j}(s)}$
for all $s\in(a,b)$.

The same is true in the $\beta=\infty$ (zero-temperature) case if additionally
$w^{*}>\smash{g^{*}(a)}$ and $z^{*}>\smash{g^{*}(b)}$, for
$*\in\\{\uparrow\,\downarrow\\}$.

The positive temperature ($\beta=1$) statements are Lemmas 2.6 and 2.7 of
[CH16]. The zero temperature ($\beta=\infty$) statements are Lemmas 2.6 and
2.7 of [CH14]. See also [DM21] and [Dim22] for more detailed proofs of the
respective cases.

The following two correlation inequalities can be deduced using the
monotonicity property. They are contained in [GH22, Theorem 2.8], and their
proofs can be found in [GH22, Appendix A].

######  Lemma 2.11.

For any $t>0$, $a<b$, $\beta=1$ or $\infty$, and any pair of increasing events
$A$ and $B$ in the space of all real continuous functions on $[a,b]$,

$\mathbb{P}(\mathfrak{h}^{\beta}_{t,1}|_{[a,b]}\in
A,\,\mathfrak{h}^{\beta}_{t,1}|_{[a,b]}\in
B)\geq\mathbb{P}(\mathfrak{h}^{\beta}_{t,1}|_{[a,b]}\in
A)\cdot\mathbb{P}(\mathfrak{h}^{\beta}_{t,1}|_{[a,b]}\in B).$

This is the FKG inequality for $\mathfrak{h}^{\beta}_{t,1}|_{[a,b]}$; and for
discrete models such as the exponential LPP, such a result follows from the
classical FKG inequality.

For any $a<b$, an event $A$ in the space of all real continuous functions on
$[a,b]$ is called ‘increasing’, if for any $f\in A$ and $f\leq g$ point-
wisely, there is also $g\in A$.

######  Lemma 2.12.

For any $t>0$, $a<b$, $\beta=1$ or $\infty$, and any increasing event $A$ in
the space of all real continuous functions on $[a,b]$,

$\mathbb{P}\left(\mathfrak{h}^{\beta}_{t,2}|_{[a,b]}\in
A\mid\mathfrak{h}^{\beta}_{t,1}|_{[a,b]}\right)\leq\mathbb{P}\left(\mathfrak{h}^{\beta}_{t,1}|_{[a,b]}\in
A\right).$

This is also called the BK inequality for (reweighted) Brownian bridge
ensemble, due to the connection between line ensembles and disjoint paths
through the RSK correspondence (see also the footnote in [GH22, Page 14]).

###  2.4. Existing estimates

In this subsection, we list some existing estimates of the line ensembles,
which hold in both zero and positive temperature settings. For the convenience
of notations, we state them in terms of the scaled version
$\hat{\mathfrak{h}}^{\beta}_{t,1}$ and $\hat{\mathfrak{h}}^{\beta}_{t,2}$.

We next state the one-point upper- and lower-tails for
$\hat{\mathfrak{h}}^{\beta}_{t,1}$. We fix arbitrary $\varepsilon>0$ in this
subsection.

######  Theorem 2.13 ([GH22, Theorem 1 and Proposition 9.5]).

There exists $L_{0}>0$ such that, for any $t>0$ and
$L>(t^{-1/3-\varepsilon}\vee 1)L_{0}$,

$\exp\Big{(}-\tfrac{4}{3}L^{3/2}-CL^{3/4}\Big{)}<\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L,L+\,\mathrm{d}L)\right)/\,\mathrm{d}L<\exp\Big{(}-\tfrac{4}{3}L^{3/2}+CL^{3/4}\Big{)}.$

In [GH22] this estimate is actually proved for $t$ bounded away from zero.
This is because the arguments in [GH22] take as input a priori tail estimates
from [CG20a], which hold for $t>t_{0}$ and $L>L_{0}$ with $L_{0}$ possibly
depending on $t_{0}$. However, analogous estimates are also available for
arbitrary $t>0$ from [DG23], namely Theorems 1.4 (upper bound on upper tail)
and 1.7 (upper bound on lower tail) there. Using these inputs the tail
estimate in [GH22] can be upgraded to cover small $t>0$. More details will be
given in Appendix C.

Using Theorem 2.13 and stochastic domination, one can deduce an upper-tail
bound for one-point distribution of $\hat{\mathfrak{h}}^{\beta}_{t,2}$, which
will be useful later.

######  Lemma 2.14.

There exists $L_{0}>0$ such that, for any $t>0$ and
$L>(t^{-1/3-\varepsilon}\vee 1)L_{0}$,

$\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L,\hat{\mathfrak{h}}^{\beta}_{t,1}(0)+\hat{\mathfrak{h}}^{\beta}_{t,2}(0)>2L\right)<\exp\Big{(}-\tfrac{8}{3}L^{3/2}+CL^{3/4}\Big{)}.$

###### Proof.

We can write the left-hand side as

$\int_{L}^{\infty}\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,2}(0)>2L-\vartheta\mid\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(\vartheta,\vartheta+\,\mathrm{d}\vartheta)\right)\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(\vartheta,\vartheta+\,\mathrm{d}\vartheta)\right).$

By Lemma 2.12, this is upper bounded by

$\int_{L}^{\infty}\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>2L-\vartheta\right)\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(\vartheta,\vartheta+\,\mathrm{d}\vartheta)\right).$

Then the conclusion follows by using Theorem 2.13. ∎

We have the following estimate on the one-point lower tail of
$\hat{\mathfrak{h}}^{\beta}_{t,1}$.

######  Theorem 2.15.

There exists $L_{0}>0$ such that, for any $0<t\leq 1$ and $L>(t^{-1/6}\vee
1)L_{0}$,

$\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)<-L\right)\leq\exp(-cL^{2}t^{1/6}).$

If we instead assume $t>t_{0}$ for some $t_{0}>0$, and $L>L_{0}$, then

$\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)<-L\right)<\exp(-cL^{5/2}),$

with the constant $c$ depending on $t_{0}$.

For $\beta=1$ these two estimates can be deduced from [DG23, Theorem 1.7] and
[CG20b, Theorem 1] respectively. For $\beta=\infty$,
$\hat{\mathfrak{h}}^{\beta=\infty}_{t,1}(0)$ has GUE Tracy-Widom distribution,
whose lower tail is more classical (see e.g., [TW94] or [CG20b, Proposition
5.1])

The process $\hat{\mathfrak{h}}^{\beta}_{t,1}$ is also $1/2$-Hölder.

######  Proposition 2.16.

There exists $L_{0}>0$ such that, for any $t>0$ and $M^{2}>(t^{-1/6}\vee
1)L_{0}$, and $0<d\leq 1$, we have

$\mathbb{P}\Bigl{(}\sup_{x\in[0,d]}|\hat{\mathfrak{h}}^{\beta}_{t,1}(x)-\hat{\mathfrak{h}}^{\beta}_{t,1}(0)|>Md^{1/2}\Bigr{)}<C\exp(-cM^{2}).$

For $\beta=\infty$ this is proved in [DV21a, Lemma 6.1] and [Dau23, Lemma
3.4], and their arguments carry over to the $\beta=1$ setting. We omit the
details here.

We next quote the following tent behavior of
$\hat{\mathfrak{h}}^{\beta}_{t,1}$, under the one-point upper-tail event.

######  Theorem 2.17 ([GH22, Theorem 9]).

There exists $L_{0}>0$, such that for any $t>0$ and
$L>(t^{-1/3-\varepsilon}\vee 1)L_{0}$, we have

$\mathbb{P}\left(\sup_{x\in[-L^{1/2},L^{1/2}]}\left|\hat{\mathfrak{h}}^{\beta}_{t,1}(x)-L+2L^{1/2}|x|\right|>ML^{1/4}\
\Big{|}\
\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L,L+\,\mathrm{d}L)\right)<\exp(-cM^{2}),$

for any $0<M<cL^{3/4}$. The same is true under the conditioning
$\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L$.

As Theorem 2.13, in [GH22] this is proved for $t$ bounded away from zero, and
we will give explanations on how to upgrade it in Appendix C.

A version of this tent behavior with the conditioning replaced by
$\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L$ also holds, and we will only need that
for $t$ bounded away from zero.

######  Lemma 2.18.

For any $t_{0}>0$, there exists $L_{0}>0$, such that for any $t>t_{0}$ and
$L>L_{0}$, we have

$\mathbb{P}\left(\sup_{x\in[-L^{1/2},L^{1/2}]}\left|\hat{\mathfrak{h}}^{\beta}_{t,1}(x)-L+2L^{1/2}|x|\right|>ML^{1/4}\
\Big{|}\ \hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)<\exp(-cM^{2}),$

for any $0<M<cL^{3/4}$. Here the constant $c$ can depend on $t_{0}$.

We will give a proof of this result in Appendix B.

We next give a more precise comparison of the tent with Brownian bridges.

######  Lemma 2.19.

For $t$ and $L$ as in Theorem 2.17, the following is true. Consider the
following processes, each defined on $[0,L^{1/2}/2]$,

$x\mapsto\hat{\mathfrak{h}}^{\beta}_{t,1}(x)-L,\quad
x\mapsto\hat{\mathfrak{h}}^{\beta}_{t,1}(-x)-L,$

conditional on $\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L,L+\,\mathrm{d}L)$.
Also, consider the processes (each defined on $[0,L^{1/2}/2]$ as well)

$x\mapsto
B_{1}(x)+2xL^{-1/2}(\hat{\mathfrak{h}}^{\beta}_{t,1}(L^{1/2}/2)-L),\quad
x\mapsto B_{2}(x)+2xL^{-1/2}(\hat{\mathfrak{h}}^{\beta}_{t,1}(-L^{1/2}/2)-L,$

also conditional on
$\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L,L+\,\mathrm{d}L)$, where
$B_{1},B_{2}$ are rate $2$ Brownian bridges in $[0,L^{1/2}/2]$, independent of
each other and independent of $\hat{\mathfrak{h}}^{\beta}_{t,1}$. They can be
coupled so that under an event with probability $>1-C\exp(-cL^{3/2})$ for
both, their Radon-Nikodym derivative is $1+O(\exp(-ct^{1/3}L))$.

The proof of this as well as the following lemma will be given in Appendix B.

######  Lemma 2.20.

For $t$ and $L$ as in Theorem 2.17, any
$I\subseteq[-\frac{1}{2}L^{1/2},\frac{1}{2}L^{1/2}]$ and
$\sigma_{I}=\sup_{x\in I}|x|^{1/2}$, and $0<M<L^{3/4}$,

$\mathbb{P}\left(\sup_{x\in
I}|\hat{\mathfrak{h}}^{\beta}_{t,1}(x)-(L-2L^{1/2}|x|)|>M\sigma_{I}\ \Big{|}\
\hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L,L+\,\mathrm{d}L)\right)<C\exp(-cM^{2}).$

If we in addition assume $t>t_{0}$ for some $t_{0}>0$, then we have

$\mathbb{P}\left(\sup_{x\in
I}|\hat{\mathfrak{h}}^{\beta}_{t,1}(x)-(L-2L^{1/2}|x|)|>M\sigma_{I}\ \Big{|}\
\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)<C\exp\left(-c(M^{2}\wedge
M\sigma_{I}L^{1/2})\right),$

with the constant $c$ depending on $t_{0}$.

The estimates above quickly lead to the following.

######  Corollary 2.21.

For $t$ and $L$ as in Theorem 2.17, for any $0<M<L^{3/4}$, and any $a>0$, we
have

$\mathbb{P}\left(\sup_{|x|\leq
L^{1/2}}\left|\hat{\mathfrak{h}}^{\beta}_{t,1}(x)-(L-2L^{1/2}|x|)\right|(|\log(|x|/a)|+1)^{-1}|x|^{-1/2}>M\
\Big{|}\ \hat{\mathfrak{h}}^{\beta}_{t,1}(0)\in(L,L+\,\mathrm{d}L)\right)\\\
<C\exp(-cM^{2}).$

This estimate is obtained via a union bound over dyadic scales, using Theorem
2.17 or Lemma 2.20 at each scale. The parameter $a$ represents the scale of
$x$ at which the logarithm becomes of constant order, which will provide some
convenient flexibility in applications.

###  2.5. Gaussian estimate

Here we recall a standard bound on the tail of centered Gaussian random
variables.

######  Lemma 2.22.

For $\sigma>0$ and $x>0$,

$\frac{1}{\sqrt{2\pi}}\cdot\frac{\sigma}{x}\left(1-\frac{\sigma^{2}}{x^{2}}\right)\exp\left(-\frac{x^{2}}{2\sigma^{2}}\right)\leq\mathbb{P}\left(\mathcal{N}(0,\sigma^{2})\geq
x\right)\leq\frac{1}{\sqrt{2\pi}}\cdot\frac{\sigma}{x}\exp\left(-\frac{x^{2}}{2\sigma^{2}}\right).$

###### Proof.

We set $\sigma=1$ without loss of generality. Now we write
$\mathbb{P}\left(\mathcal{N}(0,\sigma^{2})\geq x\right)$ as an integral of the
normal density and obtain the claimed bounds by doing integration by parts one
for the upper bound and again for the lower bound. ∎

##  3\. Coalescence and Brownian bridge comparison under upper tail

This section develops the key coalescence estimates that our analysis relies
on. It will be convenient to denote (within this section)

$\mathcal{S}^{\beta=1}_{t}(x,y)=t^{-1/3}(\log\mathcal{Z}(t^{2/3}x,0;t^{2/3}y,t)+t/12),$

for any $t>0$. One may interpret this as a ‘rescaled KPZ sheet’. The zero
temperature analog, that is the Airy sheet, will be denoted by

$\mathcal{S}^{\beta=\infty}_{t}(x,y)=t^{-1/3}\mathcal{L}(t^{2/3}x,0;t^{2/3}y,1).$

Note that actually the law of $\mathcal{S}^{\beta=\infty}_{t}$ is the same for
any $t>0$.

From these definitions, we see that for any $t>0$ and $\beta=1,\infty$, we
have
$\mathcal{S}^{\beta}_{t}(0,x)=\hat{\mathfrak{h}}^{\beta}_{t,1}(x)=t^{-1/3}\mathfrak{h}^{\beta}_{t,1}(t^{2/3}x)$.
By the shear, shift, and reflection invariance properties of $\mathcal{L}$ and
$\mathcal{Z}$ (introduced in Section 2.1 and Section 2.2.1), we have that
$\mathcal{S}^{\beta}_{t}$ has the same law as

$(x,y)\mapsto\mathcal{S}^{\beta}_{t}(x+a,y+b)+(x+a-y-b)^{2}-(x-y)^{2},\quad(x,y)\mapsto\mathcal{S}^{\beta}_{t}(-x,-y),\quad(x,y)\mapsto\mathcal{S}^{\beta}_{t}(y,x).$

We next deduce two uniform bounds of $\mathcal{S}^{\beta}_{t}$, which are for
both $\beta=1,\infty$ and any $t>0$. The first is an Hölder estimate.

######  Lemma 3.1.

There exists $L_{0}>0$ such that, for any $t>0$ and $M^{2}>(t^{-1/6}\vee
1)L_{0}$, and $0<d\leq 1$, we have

$\mathbb{P}\left(\sup_{x,y\in[0,d]}|\mathcal{S}^{\beta}_{t}(x,y)-\mathcal{S}^{\beta}_{t}(0,0)|>Md^{1/2}\right)<C\exp(-cM^{2}).$

###### Proof.

Using (2.1) or (2.4), for any $x,y\in[0,d]$, we have

$-\mathcal{S}^{\beta}_{t}(0,0)+\mathcal{S}^{\beta}_{t}(0,y)+\mathcal{S}^{\beta}_{t}(x,0)\leq\mathcal{S}^{\beta}_{t}(x,y)\leq-\mathcal{S}^{\beta}_{t}(d,0)+\mathcal{S}^{\beta}_{t}(d,y)+\mathcal{S}^{\beta}_{t}(x,0).$

By Proposition 2.16, and symmetries of $\mathcal{S}^{\beta}_{t}$, we have

$\mathbb{P}\left(\sup_{x\in[0,d]}|\mathcal{S}^{\beta}_{t}(0,x)-\mathcal{S}^{\beta}_{t}(0,0)|>Md^{1/2}\right),\mathbb{P}\left(\sup_{x\in[0,d]}|\mathcal{S}^{\beta}_{t}(x,0)-\mathcal{S}^{\beta}_{t}(0,0)|>Md^{1/2}\right)<C\exp(-cM^{2}),$

and

$\mathbb{P}\left(\sup_{x\in[0,d]}|\mathcal{S}^{\beta}_{t}(d,x)-\mathcal{S}^{\beta}_{t}(d,0)|>Md^{1/2}\right)<C\exp(-cM^{2});$

therefore the conclusion holds. ∎

######  Lemma 3.2.

Fix $\varepsilon>0$. There exist $M_{0}>0$ and a random variable $H>0$, such
that $\mathbb{P}(H>M)<C\exp(-cM^{3/2})$ for any $M>(t^{-1/3+\varepsilon}\vee
1)M_{0}$, and

$|\mathcal{S}^{\beta}_{t}(x,y)+(x-y)^{2}|<H+\log(|x|+|y|+2),\quad\forall
x,y\in\mathbb{R}.$

###### Proof.

It suffices to show that, for any $M>0$,

$\mathbb{P}\left(\sup_{x,y\in[0,1]}|\mathcal{S}^{\beta}_{t}(x,y)+(x-y)^{2}|>M\right)<C\exp(-cM^{3/2}).$
(3.1)

Then the conclusion follows by splitting $\mathbb{R}$ into intervals of length
$1$, using the shear and shift invariance properties of
$\mathcal{S}^{\beta}_{t}$, and taking a union bound.

As for (3.1), we just apply Lemma 3.1 for $d=1$, and use that
$\mathbb{P}(|\mathcal{S}^{\beta}_{t}(0,0)|>M)<C\exp(-cM^{3/2})$, which can be
obtained from Theorems 2.13 and 2.15. ∎

Figure 4. An illustration of the shift-invariance: the joint distributions of
the three passage times/partition functions in the left and right panels are
the same, under the condition that both endpoints of each path are shifted by
the same amount, and the endpoints and shifts are such that the paths are all
forced by planarity to intersect both before and after the shift.

The following remarkable shift-invariance property (illustrated in Figure 4)
proved in [BGW22] will also be a key input. It follows from [BGW22, Theorems
7.8, 7.10] immediately.

######  Lemma 3.3.

Take any $m\in\mathbb{N}$, and $x_{1}\leq\cdots\leq x_{m}$,
$y_{1}\geq\cdots\geq y_{m}$, and $x_{1}^{\prime}\leq\cdots\leq
x_{m}^{\prime}$, $y_{1}^{\prime}\geq\cdots\geq y_{m}^{\prime}$, such that
$x_{i}-y_{i}=x_{i}^{\prime}-y_{i}^{\prime}$ for any $i\in[\\![{1,m}]\\!]$.
Then $\\{\mathcal{S}^{\beta}_{t}(x_{i},y_{i})\\}_{i=1}^{m}$ and
$\\{\mathcal{S}^{\beta}_{t}(x_{i}^{\prime},y_{i}^{\prime})\\}_{i=1}^{m}$ are
equal in distribution.

We next give the behavior of $\mathcal{S}^{\beta}_{t}$ under upper tail
events. To be concise, for the rest of this section we fix $t>0$. All the
constants (including all $C,c>0$) are allowed to depend on $t$. And $\beta$ is
taken to equal either $1$ or $\infty$.

###  3.1. Coalescence and independent tents with Brownian bridges

We now give our main coalescence estimate, in the form of stating that the
quadrangle inequalities from (2.1) and (2.4) are sharp under the upper tail.
Its proof will be given in Section 3.2.

######  Proposition 3.4.

Take any $L>0$ and $L^{+}>L+\exp(-0.001L^{3/2})$ (including $L^{+}=\infty$),
and denote $H=10^{-6}L^{1/2}$. When $\beta=\infty$,

$\mathbb{P}\left(\mathcal{S}^{\beta}_{t}(x,y)=\mathcal{S}^{\beta}_{t}(x,0)+\mathcal{S}^{\beta}_{t}(0,y)-\mathcal{S}^{\beta}_{t}(0,0),\;\forall|x|,|y|\leq
H\ \Big{|}\ L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}\right)>1-C\exp(-cL^{3/2}).$

When $\beta=1$, the same bound holds when the event is replaced by

$|\mathcal{S}^{\beta}_{t}(x,y)-(\mathcal{S}^{\beta}_{t}(x,0)+\mathcal{S}^{\beta}_{t}(0,y)-\mathcal{S}^{\beta}_{t}(0,0))|<C\exp(-cL),\quad|x|,|y|\leq
H.$

The lower bound on $L^{+}$ above is to ensure a lower bound on
$\mathbb{P}(L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+})$ which will be needed in the
proof.

The connection between the above estimate and coalescence is that, at zero
temperature ($\beta=\infty$), the equality is equivalent to the coalescence of
a family of geodesics, according to Lemma 2.4. At positive temperature, almost
surely the quadrangle inequality is strict (see (2.4)), so the equality is
replaced by an upper bound of $C\exp(-cL)$, which is roughly the probability
for the corresponding polymers to be disjoint given the field $\mathcal{Z}$.

As we have seen from e.g. Theorem 2.17, there are tent behaviors under the
upper tail event. The following proposition states that conditional on the
upper tail event, the two tents seen from both positive and negative
directions are roughly independent, and are close to Brownian bridges. It is a
two-sided version of Lemma 2.19.

######  Proposition 3.5.

Take any $L>0$ and $L^{+}>L+\exp(-0.001L^{3/2})$ (including $L^{+}=\infty$).
Let $H=10^{-6}L^{1/2}$. Consider the following processes, each defined on
$[0,H]$,

$x\mapsto\mathcal{S}^{\beta}_{t}(0,x)-\mathcal{S}^{\beta}_{t}(0,0),\quad
x\mapsto\mathcal{S}^{\beta}_{t}(0,-x)-\mathcal{S}^{\beta}_{t}(0,0),$
$x\mapsto\mathcal{S}^{\beta}_{t}(x,0)-\mathcal{S}^{\beta}_{t}(0,0),\quad
x\mapsto\mathcal{S}^{\beta}_{t}(-x,0)-\mathcal{S}^{\beta}_{t}(0,0),$

conditional on $L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}$. Further, also consider
the processes (each defined on $[0,H]$ as well)

$x\mapsto
B_{1}(x)+x(\mathcal{S}^{\beta}_{t}(0,H)-\mathcal{S}^{\beta}_{t}(0,0))/H,\quad
x\mapsto
B_{2}(x)+x(\mathcal{S}^{\beta}_{t}(0,-H)-\mathcal{S}^{\beta}_{t}(0,0))/H,$
$x\mapsto
B_{3}(x)+x(\mathcal{S}^{\beta}_{t}(H,0)-\mathcal{S}^{\beta}_{t}(0,0))/H,\quad
x\mapsto
B_{4}(x)+x(\mathcal{S}^{\beta}_{t}(-H,0)-\mathcal{S}^{\beta}_{t}(0,0))/H,$

also conditional on $L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}$, where
$B_{1},B_{2},B_{3},B_{4}$ are four rate $2$ Brownian bridges in $[0,H]$,
independent of each other and independent of $\mathcal{S}^{\beta}_{t}$.

There is another measure on $\mathcal{C}([0,H],\mathbb{R})^{4}$, such that,
with probability $>1-C\exp(-cL^{3/2})$, (1) its Radon-Nikodym derivative over
the first set of processes is $1+O(\exp(-cL))$ and (2) it can be coupled with
the second set of processes such that, the $L^{\infty}$ distance between them
is $<C\exp(-cL)$.

The proof of Proposition 3.5 involves reducing it to the following statement,
using the shift-invariance of Lemma 3.3 and Proposition 3.4.

######  Lemma 3.6.

Take any $L>0$ and denote $H=10^{-6}L^{1/2}$. Consider the following
processes, each defined on $[0,H]$,

$x\mapsto\mathcal{S}^{\beta}_{t}(0,x)-\mathcal{S}^{\beta}_{t}(0,0),\quad
x\mapsto\mathcal{S}^{\beta}_{t}(0,-x)-\mathcal{S}^{\beta}_{t}(0,0),$
$x\mapsto\mathcal{S}^{\beta}_{t}(0,-x-H)-\mathcal{S}^{\beta}_{t}(0,-H),\quad
x\mapsto\mathcal{S}^{\beta}_{t}(0,x+H)-\mathcal{S}^{\beta}_{t}(0,H),$

conditional on $\mathcal{S}^{\beta}_{t}(0,0)\in(L,L+\,\mathrm{d}L)$. Also,
consider the processes (each defined on $[0,H]$ as well)

$x\mapsto
B_{1}(x)+x(\mathcal{S}^{\beta}_{t}(0,H)-\mathcal{S}^{\beta}_{t}(0,0))/H,\quad
x\mapsto
B_{2}(x)+x(\mathcal{S}^{\beta}_{t}(0,-H)-\mathcal{S}^{\beta}_{t}(0,0))/H,$
$x\mapsto
B_{3}(x)+x(\mathcal{S}^{\beta}_{t}(0,-2H)-\mathcal{S}^{\beta}_{t}(0,-H))/H,\quad
x\mapsto
B_{4}(x)+x(\mathcal{S}^{\beta}_{t}(0,2H)-\mathcal{S}^{\beta}_{t}(0,H))/H,$

also conditional on $\mathcal{S}^{\beta}_{t}(0,0)\in(L,L+\,\mathrm{d}L)$,
where $B_{1},B_{2},B_{3},B_{4}$ are four rate $2$ Brownian bridges in $[0,H]$,
independent of each other and independent of $\mathcal{S}^{\beta}_{t}$. They
can be coupled so that under an event with probability $>1-C\exp(-cL^{3/2})$
for both, their Radon-Nikodym derivative is $1+O(\exp(-cL))$.

This lemma directly follows from Lemma 2.19, and we omit the details. We now
prove Proposition 3.5 assuming Proposition 3.4 and Lemma 3.6.

Figure 5. An illustration of transforming the weights of varying one side
(left panel) into varying both sides (middle panel) using shift-invariance,
then to two tents (right panel) by coalescence.

###### Proof of Proposition 3.5.

By shift-invariance (Lemma 3.3), in Lemma 3.6 we can replace
$\mathcal{S}^{\beta}_{t}(0,x+H)$ by $\mathcal{S}^{\beta}_{t}(-x,H)$ and
$\mathcal{S}^{\beta}_{t}(0,-x-H)$ by $\mathcal{S}^{\beta}_{t}(x,-H)$ (see
Figure 5). Note that Lemma 3.3 is stated in terms of finitely many points, but
we can do the replacement for each $x\in[0,H]$ simultaneously since
$\mathcal{S}^{\beta}_{t}$ is continuous. Therefore we get the following
statement. Consider the processes

$x\mapsto\mathcal{S}^{\beta}_{t}(0,x)-\mathcal{S}^{\beta}_{t}(0,0),\quad
x\mapsto\mathcal{S}^{\beta}_{t}(0,-x)-\mathcal{S}^{\beta}_{t}(0,0),$
$x\mapsto\mathcal{S}^{\beta}_{t}(x,-H)-\mathcal{S}^{\beta}_{t}(0,-H),\quad
x\mapsto\mathcal{S}^{\beta}_{t}(-x,H)-\mathcal{S}^{\beta}_{t}(0,H),$ (3.2)

on $[0,H]$, conditional on $L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}$; and the
processes (also on $[0,H]$)

$x\mapsto
B_{1}(x)+x(\mathcal{S}^{\beta}_{t}(0,H)-\mathcal{S}^{\beta}_{t}(0,0))/H,\quad
x\mapsto
B_{2}(x)+x(\mathcal{S}^{\beta}_{t}(0,-H)-\mathcal{S}^{\beta}_{t}(0,0))/H,$
$x\mapsto
B_{3}(x)+x(\mathcal{S}^{\beta}_{t}(H,-H)-\mathcal{S}^{\beta}_{t}(0,-H))/H,\quad
x\mapsto
B_{4}(x)+x(\mathcal{S}^{\beta}_{t}(-H,H)-\mathcal{S}^{\beta}_{t}(0,H))/H,$
(3.3)

conditional on $L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}$, where
$B_{1},B_{2},B_{3},B_{4}$ are four rate $2$ Brownian bridges in $[0,H]$,
independent of each other and independent of $\mathcal{S}^{\beta}_{t}$. They
can be coupled so that under an event with probability $>1-C\exp(-cL^{3/2})$
for both, their Radon-Nikodym derivative is $1+O(\exp(-cL))$.

By Proposition 3.4, with probability $>1-C\exp(-cL^{3/2})$, we have

$|(\mathcal{S}^{\beta}_{t}(x,-H)-\mathcal{S}^{\beta}_{t}(0,-H))-(\mathcal{S}^{\beta}_{t}(x,0)-\mathcal{S}^{\beta}_{t}(0,0))|<C\exp(-cL),$
$|(\mathcal{S}^{\beta}_{t}(-x,H)-\mathcal{S}^{\beta}_{t}(0,H))-(\mathcal{S}^{\beta}_{t}(-x,0)-\mathcal{S}^{\beta}_{t}(0,0))|<C\exp(-cL),$

for any $x\in[0,H]$; in particular,

$|(\mathcal{S}^{\beta}_{t}(H,-H)-\mathcal{S}^{\beta}_{t}(0,-H))-(\mathcal{S}^{\beta}_{t}(H,0)-\mathcal{S}^{\beta}_{t}(0,0))|<C\exp(-cL),$
$|(\mathcal{S}^{\beta}_{t}(-H,H)-\mathcal{S}^{\beta}_{t}(0,H))-(\mathcal{S}^{\beta}_{t}(-H,0)-\mathcal{S}^{\beta}_{t}(0,0))|<C\exp(-cL).$

By plugging these estimates into (3.2) and (3.3) respectively, we get the
conclusion. ∎

It remains to prove Proposition 3.4, which we accomplish in the next
subsection.

###  3.2. Coalescence of polymers

Using the quadrangle inequalities (2.1) and (2.4), Proposition 3.4 can be
reduced to the following lemma.

######  Lemma 3.7.

Take any $L>0$ and $L^{+}>L+\exp(-0.001L^{3/2})$, and denote
$H=10^{-6}L^{1/2}$. Then when $\beta=\infty$, we have

$\mathbb{P}\left(\mathcal{S}^{\beta}_{t}(-H,-H)+\mathcal{S}^{\beta}_{t}(H,H)>\mathcal{S}^{\beta}_{t}(-H,H)+\mathcal{S}^{\beta}_{t}(H,-H)\mid
L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}\right)<C\exp(-cL^{3/2}).$

And when $\beta=1$, the same estimate holds with the event replaced by

$\mathcal{S}^{\beta}_{t}(-H,-H)+\mathcal{S}^{\beta}_{t}(H,H)-\mathcal{S}^{\beta}_{t}(-H,H)-\mathcal{S}^{\beta}_{t}(H,-H)>C\exp(-cL).$

In the $\beta=\infty$ setting, in light of Lemma 2.4, the event whose
probability we wish to bound is equivalent to that (in the directed landscape)
the geodesic from $(-t^{2/3}H,0)$ to $(-t^{2/3}H,t)$ and the geodesic from
$(t^{2/3}H,0)$ to $(t^{2/3}H,t)$ are disjoint. Such disjointness is unlikely
to happen under the upper large since then geodesics tend to merge into the
geodesic from $(0,0)$ to $(0,t)$, shortly away from the endpoints. When
$\beta=1$, the event is instead interpreted as that the multi-point partition
function from the spatial coordinates $-t^{2/3}H$ and $t^{2/3}H$ at time $0$
to $-t^{2/3}H$ and $t^{2/3}H$ at time $t$ is comparable to the product of the
two individual ones. This can be then understood as a positive temperature
form of disjointness.

For multi-point passage times and multi-point partition functions of size $2$,
we can only estimate them when each side of the endpoints is at a single
point, using Lemma 2.14. In the proof of Lemma 3.7 we will need to upper bound
those from $-t^{2/3}H,t^{2/3}H$ at time $0$ to $-t^{2/3}H,t^{2/3}H$ at time
$t$. Our strategy is to do surgeries around time $0$ and time $1$ (see Figure
6): we instead upper bound those from $0,0$ at time $-\varepsilon t$ to $0,0$
at time $(1+\varepsilon)t$. Here $\varepsilon>0$ is a small constant. Then we
use the composition laws and lower bound various passage times and partition
functions between time $-\varepsilon t$ and $0$, and time $t$ and
$(1+\varepsilon)t$. We note that such surgery arguments have appeared in the
directed landscape and LPP models, see e.g. [Ham20].

Figure 6. An illustration of the surgery, bringing the endpoints of multi-
point passage times and multi-point partition functions together.

We now give the details. For concreteness, we take $\varepsilon=10^{-6}$ in
the rest of this section. First, we note that by taking $C$ large and $c$
small, it suffices to prove Lemma 3.7 for large enough $L$. By Lemma 3.3,
$\mathcal{S}^{\beta}_{t}(-H,H)+\mathcal{S}^{\beta}_{t}(H,-H)$ and
$\mathcal{S}^{\beta}_{t}(0,0)$ have the same joint distribution as
$\mathcal{S}^{\beta}_{t}(0,2H)+\mathcal{S}^{\beta}_{t}(0,-2H)$ and
$\mathcal{S}^{\beta}_{t}(0,0)$. Then by Theorem 2.17 we have

$\mathbb{P}\left(\mathcal{S}^{\beta}_{t}(-H,H)+\mathcal{S}^{\beta}_{t}(H,-H)<(2-0.001)L-8L^{1/2}H\mid
L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}\right)<\exp(-cL^{3/2}).$ (3.4)

Also by Theorem 2.13, for large $L$ we have

$\mathbb{P}\left(L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}\right)>c\exp(-(4/3+0.005)L^{3/2}).$
(3.5)

####  3.2.1. Directed landscape ($\beta=\infty$) setting

In this case, we can assume $t=1$ since the law of
$\mathcal{S}^{\beta=\infty}_{t}$ is independent of $t$. Recall that
$\mathcal{S}^{\beta=\infty}_{1}=\mathcal{L}(\cdot,0;\cdot,1)$. On the event in
Lemma 3.7 whose probability we wish to bound, Lemma 2.2 and Lemma 2.4 imply
that

$\mathcal{L}(-H,0;H,1)+\mathcal{L}(H,0;-H,1)<\mathcal{L}(-H,0;-H,1)+\mathcal{L}(H,0;H,1)=\mathcal{L}((-H,H),0;(-H,H),1).$

Therefore, since we have lower bounded the LHS in (3.4), to prove Lemma 3.7 it
suffices to prove the following estimate.

######  Lemma 3.8.

For $L>0$ large enough and $H=10^{-6}L^{1/2}$, we have

$\mathbb{P}\big{[}\mathcal{L}((-H,H),0;(-H,H),1)>1.99L\big{]}\\\
<C\exp\big{(}-(4/3+0.01)L^{3/2}\big{)}.$

Using this lemma and (3.5), we can bound the probability of the same event
conditional on $L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}$, by
$C\exp(-0.005L^{3/2})$. Then by (3.4), and noting that
$(2-0.001)L-8L^{1/2}H>1.99L$, we get Lemma 3.7.

###### Proof of Lemma 3.8.

By using (2.2) twice (first with $y_{1}=-H$, $y_{2}=0$, $y_{3}=H$ and then
with $y_{1}=y_{2}=0$ and $y_{3}=-H$), we have

$\mathcal{L}((0,0),-\varepsilon;(-H,H),0)\geq\mathcal{L}((0,0),-\varepsilon;(0,0),0)+\mathcal{L}(0,-\varepsilon;-H,0)+\mathcal{L}(0,-\varepsilon;H,0)-2\mathcal{L}(0,-\varepsilon;0,0).$

Denote $w:=\mathbb{P}(\mathcal{L}((0,0),-\varepsilon;(0,0),0)>0)$. There
exists $c_{*}>0$, such that

$\mathbb{P}\Bigl{(}\mathcal{L}(0,-\varepsilon;-H,0)+H^{2}/\varepsilon<-c_{*}\Bigr{)},\;\mathbb{P}\Bigl{(}\mathcal{L}(0,-\varepsilon;H,0)+H^{2}/\varepsilon<-c_{*}\Bigr{)},\;\mathbb{P}\Bigl{(}\mathcal{L}(0,-\varepsilon;0,0)>c_{*}\Bigr{)}<w/4.$

We note that $w,c_{*}$ are independent of $L$. Therefore

$\mathbb{P}\Bigl{(}\mathcal{L}((0,0),-\varepsilon;(-H,H),0)>-2H^{2}/\varepsilon-4c_{*}\Bigr{)}>w/4.$

Similarly,

$\mathbb{P}\Bigl{(}\mathcal{L}((-H,H),1;(0,0),1+\varepsilon)>-2H^{2}/\varepsilon-4c_{*}\Bigr{)}>w/4.$

Note that

$\mathcal{L}_{2}(0,-\varepsilon;0,1+\varepsilon)\\\
\geq\mathcal{L}((0,0),-\varepsilon;(-H,H),0)+\mathcal{L}((-H,H),0;(-H,H),1)+\mathcal{L}((-H,H),1;(0,0),1+\varepsilon).$

So we have

$\mathbb{P}\left(\mathcal{L}_{2}(0,-\varepsilon;0,1+\varepsilon)>1.99L-4H^{2}/\varepsilon-8c_{*}\right)\geq\mathbb{P}\Bigl{(}\mathcal{L}((-H,H),0;(-H,H),1)>1.99L\Bigr{)}w^{2}/16.$

Using Lemma 2.14 and the fact that
$\mathcal{L}_{2}(0,-\varepsilon;0,1+\varepsilon)\leq
2\mathcal{L}(0,-\varepsilon;0,1+\varepsilon)$ almost surely, we can bound the
left-hand side by $\exp\big{(}-(4/3+0.01)L^{3/2}\big{)}$. Thus the conclusion
follows. ∎

####  3.2.2. Positive temperature ($\beta=1$) setting

Recall from Section 2.2.2 the notation $\mathcal{M}$ for multi-point partition
functions. We can then write

$\exp\big{(}t^{1/3}(\mathcal{S}^{\beta}_{t}(-H,-H)+\mathcal{S}^{\beta}_{t}(H,H)-\mathcal{S}^{\beta}_{t}(-H,H)-\mathcal{S}^{\beta}_{t}(H,-H))\big{)}-1\\\
=\frac{(2t^{2/3}H)^{2}\mathcal{M}((-t^{2/3}H,t^{2/3}H),0;(-t^{2/3}H,t^{2/3}H),t)}{\exp\big{(}t^{1/3}(\mathcal{S}^{\beta}_{t}(-H,H)+\mathcal{S}^{\beta}_{t}(H,-H))-t/6\big{)}}.$
(3.6)

Therefore, to prove Lemma 3.7, the main task is to prove the following
estimate.

######  Lemma 3.9.

For $L>0$ large enough and $H=10^{-6}L^{1/2}$, we have

$\mathbb{P}\Bigl{(}\mathcal{M}\bigl{(}(-t^{2/3}H,t^{2/3}H),0;(-t^{2/3}H,t^{2/3}H),t)\bigr{)}>\exp\bigl{(}t^{1/3}\cdot
1.99L\bigr{)}\Bigr{)}<C\exp\big{(}-(4/3+0.01)L^{3/2}\big{)}.$

Using this lemma and (3.5), we can bound the probability of the same event
conditional on $L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}$, by
$C\exp(-0.005L^{3/2})$. Then by (3.4), conditional on
$L<\mathcal{S}^{\beta}_{t}(0,0)<L^{+}$, with probability $>1-C\exp(-cL^{3/2})$
the expression (3.6) is upper bounded by

$\frac{(2t^{2/3}H)^{2}\exp\big{(}t^{1/3}\cdot
1.99L\big{)}}{\exp\big{(}t^{1/3}\cdot(2L-8L^{1/2}H-0.001L)-t/6\big{)}}<C\exp(-cL).$

Thus we get Lemma 3.7.

The rest of this section is devoted to proving Lemma 3.9. We let $\mathcal{E}$
denote the event whose probability we wish to bound, in the statement of Lemma
3.9. By Lemma 2.14, we have

$\mathbb{P}\Big{(}\mathcal{Z}(0,-\varepsilon
t;0,(1+\varepsilon)t)>\exp\big{(}t^{1/3}\cdot
0.98L\big{)},\;\mathcal{Z}_{2}(0,-\varepsilon
t;0,(1+\varepsilon)t)>\exp\big{(}t^{1/3}\cdot 1.96L\big{)}\Big{)}\\\
<C\exp\big{(}-(8/3-0.1)L^{3/2}\big{)}.$

By the continuity of $\mathcal{M}$ (Lemma 2.8), we have almost surely

$2^{-1}(1+2\varepsilon)t\mathcal{M}((-\delta,\delta),-\varepsilon
t;(-\delta,\delta),(1+\varepsilon)t)\to\mathcal{Z}_{2}(0,-\varepsilon
t;0,(1+\varepsilon)t),$

as $\delta\to 0$ from the right. Then there exists small enough
$\delta_{L}>0$, such that
$\mathbb{P}(\mathcal{E}^{+})<C\exp\big{(}-(8/3-0.1)L^{3/2}\big{)}$, with
$\mathcal{E}^{+}$ being the event where

$\displaystyle\mathcal{Z}(0,-\varepsilon t;0,(1+\varepsilon)t)$
$\displaystyle>\exp\big{(}t^{1/3}\cdot 0.98L\big{)},$
$\displaystyle\mathcal{M}\bigl{(}(-\delta_{L},\delta_{L}),-\varepsilon
t;(-\delta_{L},\delta_{L}),(1+\varepsilon)t\bigr{)}$
$\displaystyle>\exp\big{(}t^{1/3}\cdot 1.98L\big{)}.$

We will show that conditional on $\mathcal{E}$, with positive probability
(independent of $L$), $\mathcal{E}^{+}$ holds. For this, we define the
following events.

By Lemma 2.8, there exists a small number $w>0$, such that

$\mathbb{P}\left(\mathcal{M}((-x,x),-\varepsilon t;(-w,w),0)>w\right)>w$

for any $|x|\leq w$. We let $\mathcal{E}_{1}$ be the event where

$\mathcal{M}\bigl{(}(-\delta_{L},\delta_{L}),-\varepsilon
t;(-w,w),0\bigr{)}>w,\quad\mathcal{M}\bigl{(}(-w,w),t;(-\delta_{L},\delta_{L},(1+\varepsilon)t\bigr{)}>w.$

Then $\mathbb{P}(\mathcal{E}_{1})>w^{2}$.

We let $\mathcal{E}_{2}$ be the event where for any $x,y\in\mathbb{R}$,

$\left|(\varepsilon t)^{-1/3}(\log\mathcal{Z}((\varepsilon
t)^{2/3}x,-\varepsilon t;(\varepsilon t)^{2/3}y,0)+\varepsilon
t/12)+(x-y)^{2}\right|<10^{-3}L+\log(|x-y|+2),$

and

$\left|(\varepsilon t)^{-1/3}(\log\mathcal{Z}((\varepsilon
t)^{2/3}x,t;(\varepsilon t)^{2/3}y,(1+\varepsilon)t)+\varepsilon
t/12)+(x-y)^{2}\right|<10^{-3}L+\log(|x-y|+2).$

Then by Lemma 3.2, we have $\mathbb{P}(\mathcal{E}_{2})>1-C\exp(-cL^{3/2})$.

By Lemma 3.1 and the shift invariance of $\mathcal{Z}$, we have
$\mathbb{P}(\mathcal{E}^{*})>1-C\exp(-cL^{2})$, for $\mathcal{E}^{*}$ denoting
the event where

$\sup_{x,y\in[-H-1,-H]}\bigl{|}\mathcal{S}^{\beta}_{t}(x,y)-\mathcal{S}^{\beta}_{t}(-H,-H)\bigr{|},\;\sup_{x,y\in[H,H+1]}\bigl{|}\mathcal{S}^{\beta}_{t}(x,y)-\mathcal{S}^{\beta}_{t}(H,H)\bigr{|}<10^{-10}L.$

######  Lemma 3.10.

We have
$\mathcal{E}_{1}\cap\mathcal{E}_{2}\cap\mathcal{E}^{*}\cap\mathcal{E}\subset\mathcal{E}^{+}$.

###### Proof.

In this proof we always take

$x,x^{\prime}\in[-t^{2/3}(H+1),-t^{2/3}H],\quad
y,y^{\prime}\in[t^{2/3}H,t^{2/3}(H+1)].$

We will use $\mathcal{E}_{1}\cap\mathcal{E}_{2}$ to lower bound
$\mathcal{M}((-\delta_{L},\delta_{L}),-\varepsilon t;(x,y),0)$ and
$\mathcal{M}((x^{\prime},y^{\prime}),t;(-\delta_{L},\delta_{L}),(1+\varepsilon)t)$,
and $\mathcal{Z}(0,-\varepsilon t;x,0)$,
$\mathcal{Z}(x^{\prime},t;0,(1+\varepsilon)t)$; then we will lower bound
$\mathcal{M}((x,y),0;(x^{\prime},y^{\prime}),t)$ and
$\mathcal{Z}(x,0;x^{\prime},t)$ for these $x,x^{\prime},y,y^{\prime}$, using
$\mathcal{E}\cap\mathcal{E}^{*}$. Then using the composition laws (1.3) and
(2.5), we obtain $\mathcal{E}^{+}$.

Step 1. Using Lemma 2.7, we have that

$(y-x)\mathcal{M}((-\delta_{L},\delta_{L}),-\varepsilon
t;(x,y),0)>2w\mathcal{M}((-\delta_{L},\delta_{L}),-\varepsilon t;(-w,w),0)\\\
\times\frac{\mathcal{Z}(-\delta_{L},-\varepsilon
t;x,0)\mathcal{Z}(\delta_{L},-\varepsilon
t;y,0)}{\mathcal{Z}(-\delta_{L},-\varepsilon
t;-w,0)\mathcal{Z}(\delta_{L},-\varepsilon t;w,0)}.$

Under $\mathcal{E}_{1}$, the first factor in the right-hand side is $>2w^{2}$;
and under $\mathcal{E}_{2}$, the second factor in the right hand side is
$>\exp(-t^{1/3}10^{-3}L)$, when $L$ is large enough. Therefore, we have

$\mathcal{M}((-\delta_{L},\delta_{L}),-\varepsilon
t;(x,y),0)>c\exp(-t^{1/3}10^{-3}L).$ (3.7)

By symmetry, the same lower bound holds for
$\mathcal{M}((x^{\prime},y^{\prime}),t;(-\delta_{L},\delta_{L}),(1+\varepsilon)t)$.

The event $\mathcal{E}_{2}$ also implies that

$\mathcal{Z}(0,-\varepsilon
t;x,0),\mathcal{Z}(x^{\prime},2t;0,(2+\varepsilon)t)>\exp(-t^{1/3}10^{-3}L).$
(3.8)

Step 2. Using Lemma 2.7, we have that

$2\mathcal{M}((x,y),0;(x^{\prime},y^{\prime}),t)>\mathcal{M}((-t^{2/3}H,t^{2/3}H),0;(-t^{2/3}H,t^{2/3}H),t)\\\
\times\frac{\mathcal{Z}(x^{\prime},0;x,t)\mathcal{Z}(y^{\prime},-0;y,t)}{\mathcal{Z}(-t^{2/3}H,0;-t^{2/3}H,t)\mathcal{Z}(t^{2/3}H,-0;t^{2/3}H,t)}.$

By $\mathcal{E}$, the first factor in the right-hand side is
$>\exp(t^{1/3}\cdot 1.99L)$; and by $\mathcal{E}^{*}$, the second factor in
the right-hand side is $>\exp(-t^{1/3}10^{-9}L)$. Therefore we have

$\mathcal{M}((x,y),0;(x^{\prime},y^{\prime}),t)>c\exp(t^{1/3}\cdot(1.99-10^{-9})L).$
(3.9)

From $\mathcal{E}$, we also have that

$\mathcal{Z}(-t^{2/3}H,0;-t^{2/3}H,t)\vee\mathcal{Z}(t^{2/3}H,0;t^{2/3}H,t)>\exp(t^{1/3}\cdot
0.99L).$

Without loss of generality, we assume that

$\mathcal{Z}(-t^{2/3}H,0;-t^{2/3}H,t)>\exp(t^{1/3}\cdot 0.99L).$

Then by $\mathcal{E}^{*}$, we have

$\mathcal{Z}(x,0;x^{\prime},t)>\exp(t^{1/3}\cdot(0.99-10^{-9})L).$ (3.10)

Step 3. By the composition law (2.5), and (3.7), (3.9), we have

$\mathcal{M}((-\delta_{L},\delta_{L}),-\varepsilon
t;(-\delta_{L},\delta_{L}),(1+\varepsilon)t)>c\exp(t^{1/3}\cdot(1.99-2\cdot
10^{-3}-10^{-9})L).$

By the composition law (1.3), and (3.8), (3.10), we have

$\mathcal{Z}(0,-\varepsilon
t;0,(1+\varepsilon)t)>c\exp(t^{1/3}\cdot(0.99-2\cdot 10^{-3}-10^{-9})L).$

These imply the event $\mathcal{E}^{+}$. ∎

###### Proof of Lemma 3.9.

By Lemma 3.10, and the fact that $\mathcal{E}_{1},\mathcal{E}_{2}$ are
independent of $\mathcal{E}^{*},\mathcal{E}$, we have
$\mathbb{P}(\mathcal{E}_{1}\cap\mathcal{E}_{2})\mathbb{P}(\mathcal{E}^{*}\cap\mathcal{E})\leq\mathbb{P}(\mathcal{E}^{+})$.
Then by the bounds on $\mathbb{P}(\mathcal{E}_{1})$,
$\mathbb{P}(\mathcal{E}_{2})$, $\mathbb{P}(\mathcal{E}^{*})$,
$\mathbb{P}(\mathcal{E})$, and $\mathbb{P}(\mathcal{E}^{+})$, we have that

$\mathbb{P}(\mathcal{E})\leq
C\exp\big{(}-(8/3-0.1)L^{3/2}\big{)}(w^{2}-C\exp(-cL^{3/2}))^{-1}-C\exp(-cL^{2}),$

and this is bounded by $C\exp\big{(}-(4/3+0.01)L^{3/2}\big{)}$ for $L$ large
enough. ∎

##  4\. Tail comparison estimates

To reduce notations, in the rest of this paper we denote

$\mathcal{L}^{\beta=1}(x,s;y,t)=\log\mathcal{Z}(x,s;y,t)+(t-s)/12,\quad\mathcal{L}^{\beta=\infty}(x,s;y,t)=\mathcal{L}(x,s;y,t).$

We take $\beta=1$ or $\infty$ systematically unless otherwise noted. From the
above definition we have
$\mathfrak{h}^{\beta}_{t,1}(x)=\mathcal{L}^{\beta}(0,0;x,t)$.

In this section, $t$ is taken to be any $t>t_{0}$, where $t_{0}$ is an
arbitrary positive number. All the constants may depend on $t_{0}$, but are
uniform in $t$.

As indicated in Section 1.2, the following tail ratio estimate would be
central in much of our analysis.

######  Theorem 4.1.

For any $L\geq 2$, and $0<\delta<L^{1/4}$,

$\frac{\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)}{\mathbb{P}(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L)}=\exp(-2\delta
L^{1/2}+O(\delta L^{-1/4}\log(L)+L^{-3/2})).$ (4.1)

For $\delta\geq L^{1/4}$, the same ratio equals $\exp(-\Omega(\delta
L^{1/2}))$.

The main term $\exp(-2\delta L^{1/2})$ comes from the following: from the tent
behavior, one considers a Brownian bridge in $[-L^{-1/2},L^{1/2}]$ that equals
$-L$ at the two ends; then the ratio is roughly the probability that it is
$>L+\delta$ at $0$ versus the probability that it is $>L$ at $0$. As mentioned
in the idea of proofs Section 1.2, it can also be understood by Taylor
expanding $\frac{4}{3}x^{3/2}$ around $x=L$.

The general strategy will be to use the Gibbs property (Lemma 2.9) to resample
$\smash{\hat{\mathfrak{h}}^{\beta}_{t,1}}$ in an interval
$[-L^{1/2}+M,L^{1/2}-M]$, with some $M$ large but much smaller than $L^{1/2}$.
The choice of $M$ is such that, conditioned on the upper tail large deviation
event, $\smash{\hat{\mathfrak{h}}^{\beta}_{t,1}}$ in the interval
$[-L^{1/2}+M,L^{1/2}-M]$ is not much affected by the second line
$\smash{\hat{\mathfrak{h}}^{\beta}_{t,2}}$ since, by the tent behavior, it
will have obtained some separation from the second line by that time. Thus
$\smash{\hat{\mathfrak{h}}^{\beta}_{t,1}}$ can be analyzed as a Brownian
bridge on that interval. For this, we need the following estimate of
$\smash{\hat{\mathfrak{h}}^{\beta}_{t,1}}$ at the end point $-L^{1/2}+M$ (and
also for $L^{1/2}-M$ by symmetry).

######  Lemma 4.2.

For any $0<M<L^{1/2}$ and $L\geq 2$,

$\displaystyle\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(-L^{1/2}+M)\leq-(L^{1/2}-M)^{2}+\tfrac{1}{2}M^{2}\
\Big{|}\ \hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)<C\exp(-cM^{5/2}).$

By the tent behavior, we expect the expectation of
$\smash{\hat{\mathfrak{h}}^{\beta}_{t,1}(\pm(L^{1/2}-M))}$ to be around
$-L+2ML^{1/2}$, with a Gaussian fluctuation of order $M^{1/2}$. This estimate
bounds the probability that it is order $M^{2}$ smaller than its mean. The
reason the upper bound is $\exp(-cM^{5/2})$ rather than
$\exp(-c(M^{2})^{2}/M)=\exp(-cM^{3})$ is due to interactions with
$\smash{\hat{\mathfrak{h}}^{\beta}_{t,2}}$.

In the following proof we will use the notation
$\mathbb{E}_{\mathcal{F}}[\cdot]:=\mathbb{E}[\,\cdot\mid\mathcal{F}]$ and
$\mathbb{P}_{\mathcal{F}}(\cdot):=\mathbb{P}(\cdot\mid\mathcal{F})$ for a
$\sigma$-algebra $\mathcal{F}$. The existence of the regular conditional
distribution is justified by the fact that the $\sigma$-algebras we consider
will be generated by random variables taking values in Borel spaces and then
invoking abstract existence results such as [Kal22, Theorem 8.5].

###### Proof of Lemma 4.2.

Let $\mathcal{F}=\mathcal{F}_{\mathrm{ext}}([-t^{2/3}L^{1/2},0])$ (recall the
definition of $\mathcal{F}_{\mathrm{ext}}$ from Section 2.3) and
$L_{M}=(L^{1/2}-M)^{2}$. Then we can write the probability of the complement
of the event in the LHS in the statement of the lemma as

$\displaystyle\frac{\mathbb{E}\left[\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(-L_{M}^{1/2})>-L_{M}+\tfrac{1}{2}M^{2}\right)\mathbbm{1}_{\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L}\right]}{\mathbb{P}(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L)}.$
(4.2)

Let us focus on the conditional probability in the numerator. Let $\tilde{B}$
be a Brownian bridge from
$(-L^{1/2},\hat{\mathfrak{h}}^{\beta}_{t,1}(-L^{1/2}))$ to
$(0,\hat{\mathfrak{h}}^{\beta}_{t,1}(0))$, such that $x\mapsto
t^{1/3}\tilde{B}(t^{-2/3}x)$ in $[-t^{2/3}L^{1/2},0]$ interacts with
$\mathfrak{h}^{\beta}_{t,2}$ by the Radon Nikodym derivative reweighting
(2.6). Then the Brownian Gibbs property says that the conditional probability
in the previous display equals

$\displaystyle\mathbb{P}_{\mathcal{F}}\left(\tilde{B}(-L_{M}^{1/2})>-L_{M}+\tfrac{1}{2}M^{2}\right).$
(4.3)

Next let $B$ be a Brownian bridge from $(-L^{1/2},-L-M)$ to $(0,L)$ (with no
lower boundary conditioning). Then, on the $\mathcal{F}$-measurable event

$A(L,M)=\left\\{\hat{\mathfrak{h}}^{\beta}_{t,1}(-L^{1/2})>-L-M,\;\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right\\},$
(4.4)

it holds that $\tilde{B}$ dominates $B$ by monotonicity (Lemma 2.10), so (4.3)
is lower bounded by

$\displaystyle\mathbb{P}_{\mathcal{F}}\left(B(-L_{M}^{1/2})>-L_{M}+\tfrac{1}{2}M^{2}\right)\mathbbm{1}_{A(L,M)}.$
(4.5)

Now $B(-L_{M}^{1/2})$ is distributed as a normal random variable with mean

$-L-M+M\cdot\frac{L-(-L-M)}{L^{1/2}}=-L-M+2L^{1/2}M+M^{2}L^{-1/2}$

and variance $\sigma^{2}=\frac{2M(L^{1/2}-M)}{L^{1/2}}\leq 2M$. Since
$-L_{M}=-(L^{1/2}-M)^{2}=-L+2L^{1/2}M-M^{2}$, we see that (4.5) equals, on
$A(L,M)$,

$\displaystyle\mathbb{P}\left(\mathcal{N}(0,\sigma^{2})>-M^{2}(\tfrac{1}{2}+L^{-1/2})+M\right)\geq
1-\exp\left(-cM^{3}\right)$

using standard tail bounds for the normal distribution (Lemma 2.22). Putting
this back in (4.2) and recalling the definition (4.4) of $A(L,M)$ , we see
that the LHS in the lemma is upper bounded by

$\displaystyle
1-(1-\exp(-cM^{3}))\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(-L^{1/2})>-L-M\
\Big{|}\ \hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right).$

By the FKG inequality (Lemma 2.11) and Theorem 2.15, we have

$\displaystyle\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(-L^{1/2})>-L-M\
\Big{|}\ \hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)>1-\exp(-cM^{5/2})).$

This completes the proof. ∎

###### Proof of Theorem 4.1.

We can assume that $L$ is large since otherwise the conclusion follows from
Theorem 2.13. For some constant $C_{0}$ sufficiently large, the case of
$\delta>C_{0}L^{1/4}$ also follows from Theorem 2.13. In the remainder of the
proof, we prove (4.1) for $\delta<C_{0}L^{1/4}$.

Denote $M=\log(L)$. It suffices to show that

$\displaystyle\frac{\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)}{\mathbb{P}(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L)}\cdot\exp(2\delta
L^{1/2})$ $\displaystyle<(1+CL^{-3/2})\exp(C\delta ML^{-1/4})\qquad\text{and}$
$\displaystyle\frac{\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)}{\mathbb{P}(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L)}\cdot\exp(2\delta
L^{1/2})$ $\displaystyle>(1-CL^{-3/2})\exp(-C\delta ML^{-1/4}).$

We let $L_{M}=(L^{1/2}-M)^{2}$, and
$\mathcal{F}=\mathcal{F}_{\mathrm{ext}}([-t^{2/3}L_{M}^{1/2},t^{2/3}L_{M}^{1/2}])$.
We start by considering the ratio of conditional probabilities

$\displaystyle\frac{\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)}{\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)}.$

We adopt the notation $\mathcal{B}$ for the law of a Brownian bridge $B$ from
$(-t^{2/3}L_{M}^{1/2},\mathfrak{h}^{\beta}_{t,1}(-t^{2/3}L_{M}^{1/2}))$ to
$(t^{2/3}L_{M}^{1/2},\mathfrak{h}^{\beta}_{t,1}(t^{2/3}L_{M}^{1/2}))$, as well
as the associated expectation. With this notation, by the Brownian Gibbs
property, the previous display equals

$\frac{\mathcal{B}\bigl{(}\mathbbm{1}_{B(0)>t^{1/3}(L+\delta)}W(B,\mathfrak{h}^{\beta}_{t,2})\bigr{)}}{\mathcal{B}\bigl{(}\mathbbm{1}_{B(0)>t^{1/3}L}W(B,\mathfrak{h}^{\beta}_{t,2})\bigr{)}}=\frac{\mathcal{B}\bigl{(}B(0)>t^{1/3}(L+\delta)\bigr{)}}{\mathcal{B}\bigl{(}B(0)>t^{1/3}L\bigr{)}}\cdot\frac{\mathcal{B}\bigl{(}W(B,\mathfrak{h}^{\beta}_{t,2})\mid
B(0)>t^{1/3}(L+\delta)\bigr{)}}{\mathcal{B}\bigl{(}W(B,\mathfrak{h}^{\beta}_{t,2})\mid
B(0)>t^{1/3}L\bigr{)}},$

where $W(B,\mathfrak{h}^{\beta}_{t,2})$ is the weight factor from (2.6). Now
the second ratio of terms in the previous display is lower bounded by $1$
using stochastic monotonicity properties of Brownian bridges and that $W$ is
increasing in $B$. To upper bound the second ratio, we note that, since
$W(B,\mathfrak{h}^{\beta}_{t,2})\leq 1$, it suffices to lower bound the
denominator $\mathcal{B}\bigl{(}W(B,\mathfrak{h}^{\beta}_{t,2})\mid
B(0)>t^{1/3}L\bigr{)}$. To do this we consider the $\mathcal{F}$-measurable
event $\mathrm{BdyCtrl}=\mathrm{BdyCtrl}(L,M)$ defined by

$\displaystyle\left\\{\hat{\mathfrak{h}}^{\beta}_{t,1}(\pm L_{M}^{1/2})\geq-
L_{M}+\tfrac{1}{2}M^{2}\right\\}\cap\bigcap_{i=0}^{M^{-1}L_{M}^{1/2}}\left\\{\sup_{|x|\in[L_{M}^{1/2}-iM,L_{M}^{1/2}-(i+1)M]}\hat{\mathfrak{h}}^{\beta}_{t,2}(x)+x^{2}\leq(i+1)M\right\\}.$

By Lemma 4.2, the BK inequality (Lemma 2.12), and the upper bound on
$\sup\hat{\mathfrak{h}}^{\beta}_{t,1}$ (following from Theorem 2.13 and
Proposition 2.16), we have

$\displaystyle\mathbb{P}\left(\mathrm{BdyCtrl}^{c}\mid\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)\leq
C\exp(-cM^{5/2})+\sum_{i=0}^{M^{-1}L_{M}^{1/2}}C\exp\left(-ci^{3/2}M^{3/2}\right)<C\exp(-cM^{3/2}).$
(4.6)

Let $\ell^{-}$ be the line joining $(-L_{M}^{1/2},-L_{M}+\frac{1}{4}M^{2})$
and $(0,L-\frac{1}{4}M^{2})$, $\ell^{+}$ be the line joining
$(0,L-\frac{1}{4}M^{2})$ and $(L_{M}^{1/2},-L_{M}+\frac{1}{4}M^{2})$, and
$\ell$ be their concatenation. Consider the high corridor event

$\mathrm{HighCorr}=\left\\{t^{-1/3}B(t^{2/3}x)\geq\ell(x)-L_{M}^{1/2}+|x|\text{
for all }x\in[-L_{M}^{1/2},L_{M}^{1/2}]\right\\};$

it is easy to obtain by standard Brownian bridge estimates that, on
$\mathrm{BdyCtrl}$, $\mathcal{B}[\mathrm{HighCorr}\mid
B(0)>t^{1/3}L]>1-C\exp(-cM^{2})$.

Now on $\mathrm{HighCorr}$ and $\mathrm{BdyCtrl}$, we have
$t^{-1/3}B(t^{2/3}x)\geq\hat{\mathfrak{h}}^{\beta}_{t,2}(x)+\frac{1}{4}M^{2}-M$;
and it is easy to check with the formula (2.6) that

$W(B,\mathfrak{h}^{\beta}_{t,2})>1-Ct^{2/3}L_{M}^{1/2}\exp(-ct^{1/3}M^{2})>1-C\exp(-cM^{2}).$

Thus, these bounds on $W(B,\mathfrak{h}^{\beta}_{t,2})$ and
$\mathcal{B}(\mathrm{HighCorr}\mid B(0)>t^{1/3}L)$ yield that, on
$\mathrm{BdyCtrl}$,

$\displaystyle\mathcal{B}\bigl{(}W(B,\mathfrak{h}^{\beta}_{t,2})\mid
B(0)>t^{1/3}L\bigr{)}>1-C\exp(-cM^{2}).$

So we see that, on $\mathrm{BdyCtrl}(L,M)$,

$\displaystyle\frac{\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)}{\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)}=(1+O(e^{-cM^{2}}))\cdot\frac{\mathcal{B}\bigl{(}B(0)>t^{1/3}(L+\delta)\bigr{)}}{\mathcal{B}\bigl{(}B(0)>t^{1/3}L\bigr{)}}.$
(4.7)

Let
$\mu=\frac{1}{2}(\hat{\mathfrak{h}}^{\beta}_{t,1}(-L_{M}^{1/2})+\hat{\mathfrak{h}}^{\beta}_{t,1}(L_{M}^{1/2}))$,
and observe that $B(0)$ under $\mathcal{B}$ is distributed as a normal random
variable with mean $t^{1/3}\mu$ and variance $t^{2/3}L_{M}^{1/2}$. Consider
the $\mathcal{F}$-measurable event

$\mathrm{MeanCtrl}(L)=\Bigl{\\{}\mu\in[-L+2L^{1/2}M-ML^{1/4},-L+2L^{1/2}M+ML^{1/4}]\Bigr{\\}}.$

We know from Theorem 2.17 that

$\mathbb{P}(\mathrm{MeanCtrl}(L)^{c}\mid\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta)<\exp(-cM^{2}).$
(4.8)

Using standard bounds on the tail of the normal distribution (Lemma 2.22), on
$\mathrm{MeanCtrl}(L)$ (and on $\mathrm{BdyCtrl}(L,M)$ as we are assuming
throughout), the RHS of (4.7) equals

$\displaystyle(1+O(e^{-cM^{2}}))(1+O(L^{-3/2}))\cdot\frac{L-\mu}{L+\delta-\mu}\cdot\exp\left(-\frac{1}{2L_{M}^{1/2}}\Bigl{[}(L+\delta-\mu)^{2}-(L-\mu)^{2}\Bigr{]}\right)$

and the last factor can be further written as

$\displaystyle\exp\left(-\frac{2\delta(L-\mu)+\delta^{2}}{2L_{M}^{1/2}}\right)$
$\displaystyle=\exp\left(-\frac{2\delta(L+L-2L^{1/2}M+O(ML^{1/4}))+\delta^{2}}{2L_{M}^{1/2}}\right)$
$\displaystyle=\exp\left(-2\delta L^{1/2}+O(\delta ML^{-1/4})\right).$

Overall, we have at this point established that, on
$\mathrm{BdyCtrl}(L,M)\cap\mathrm{MeanCtrl}(L)$,

$\displaystyle\frac{\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)}{\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)}=(1+O(e^{-cM^{2}}))(1+O(L^{-3/2}))\cdot\frac{L-\mu}{L+\delta-\mu}\exp\left(-2\delta
L^{1/2}+O(\delta ML^{-1/4})\right).$ (4.9)

We next convert this into upper and lower bounds on
$\mathbb{P}(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta)$, respectively.

Upper bound. We see from (4.9) that

$\displaystyle\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)$
$\displaystyle=\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)\left(\mathbbm{1}_{\mathrm{BdyCtrl}(L,M)\cap\mathrm{MeanCtrl}(L)}+\mathbbm{1}_{(\mathrm{BdyCtrl}(L,M)\cap\mathrm{MeanCtrl}(L))^{c}}\right)$
$\displaystyle\leq(1+Ce^{-cM^{2}})(1+CL^{-3/2})\exp\left(-2\delta
L^{1/2}+C\delta
ML^{-1/4}\right)\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)$
$\displaystyle\qquad+\mathbb{P}_{\mathcal{F}}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)\mathbbm{1}_{(\mathrm{BdyCtrl}(L,M)\cap\mathrm{MeanCtrl}(L))^{c}},$

so that, by taking expectations,

$\displaystyle\mathbb{P}\bigl{(}\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\bigr{)}$
$\displaystyle\leq(1+Ce^{-cM^{2}})(1+CL^{-3/2})\exp\left(-2\delta
L^{1/2}+C\delta
ML^{-1/4}\right)\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right)$
$\displaystyle\qquad+\mathbb{P}\Bigl{(}\left\\{\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right\\}\cap\bigl{(}\mathrm{BdyCtrl}(L,M)\cap\mathrm{MeanCtrl}(L)\bigr{)}^{c}\Bigr{)}.$

We focus on the last term. It equals

$\displaystyle\mathbb{P}\Bigl{(}\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\Bigr{)}\cdot\mathbb{P}\Bigl{(}\bigl{(}\mathrm{BdyCtrl}(L,M)\cap\mathrm{MeanCtrl}(L)\bigr{)}^{c}\
\Big{|}\ \hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\Bigr{)}.$

If we can show that the second factor is small, we will be done. By a union
bound, it is at most

$\displaystyle\mathbb{P}(\mathrm{BdyCtrl}(L,M)^{c}\mid\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta)+\mathbb{P}(\mathrm{MeanCtrl}(L)^{c}\mid\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta),$

which, from (4.6) and (4.8), is upper bounded by
$C\exp(-cM^{3/2})+\exp(-cM^{2})$. This completes the proof of the upper bound
on $\mathbb{P}(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta)$.

Lower bound. We observe that, on $\mathrm{MeanCtrl}$,
$\frac{L-\mu}{L+\delta-\mu}=\frac{2L+O(ML^{1/2})}{2L+O(ML^{1/2})+\delta}>1-C\delta
L^{-1}$. Then we have, using (4.9),

$\displaystyle\mathbb{P}_{\mathcal{F}}(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta)$
$\displaystyle\geq\mathbb{P}_{\mathcal{F}}(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta)\cdot\mathbbm{1}_{\mathrm{BdyCtrl}(L,M)\cap\mathrm{MeanCtrl}(L)}$
$\displaystyle\geq(1-Ce^{-cM^{2}})(1-CL^{-3/2})\exp\left(-2\delta
L^{1/2}-C\delta
ML^{-1/4}\right)\mathbb{P}_{\mathcal{F}}(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L)\cdot\mathbbm{1}_{\mathrm{BdyCtrl}(L,M)\cap\mathrm{MeanCtrl}(L)},$

absorbing the factor of $1-C\delta L^{-1}$ into $\exp(-C\delta ML^{-1/4})$.
Taking expectations yields

$\displaystyle\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L+\delta\right)$
$\displaystyle\geq(1-Ce^{-cM^{2}})(1-CL^{-3/2})\exp\left(-2\delta
L^{1/2}-C\delta ML^{-1/4}\right)$
$\displaystyle\qquad\times\mathbb{P}\bigl{(}\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L,\mathrm{BdyCtrl}(L,M),\mathrm{MeanCtrl}(L)\bigr{)}$
$\displaystyle\geq(1-Ce^{-cM^{2}})(1-CL^{-3/2})\exp\left(-2\delta
L^{1/2}-C\delta ML^{-1/4}\right)$
$\displaystyle\qquad\times\mathbb{P}\bigl{(}\mathrm{BdyCtrl}(L,M),\mathrm{MeanCtrl}(L)\mid\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\bigr{)}\cdot\mathbb{P}\left(\hat{\mathfrak{h}}^{\beta}_{t,1}(0)>L\right).$

As we saw above, the latter conditional probability is lower bounded by
$1-C\exp(-cM^{3/2})$, thus completing the proof of the lower bound. ∎

##  5\. Tightness as continuous functions: geodesics and bounds for polymers

We next establish the following tightness of the relevant path measures.

As in Theorems 1.1 and 1.2, let $\pi_{0}$ be the geodesic from $(0,0)$ to
$(0,1)$, in the directed landscape $\mathcal{L}^{\beta=\infty}$; and
$\Gamma_{0}$ be sampled from the annealed polymer measure from $(0,0)$ to
$(0,1)$, under $\mathcal{L}^{\beta=1}$.

######  Proposition 5.1 (Tightness).

As random elements in $\mathcal{C}([0,1],\mathbb{R})$, $L^{1/4}\pi_{0}$ or
$L^{1/4}\Gamma_{0}$ conditional on $\mathcal{L}^{\beta}(0,0;0,1)>L$ for all
$L\geq 2$ are tight.

As mentioned in Section 1.2, to prove this requires tail bounds on two-point
deviations, which rely on shear invariance of the directed landscape and the
CDRP free energy field. The proof in zero temperature is much simpler than in
positive temperature as in the former shear invariance alone suffices to give
tightness on the $L^{-1/4}$ scale. In positive temperature, the analogous
argument only yields tightness on the $O(1)$ scale, and additional arguments
are needed to obtain the correct scale. The reason is that in zero
temperature, given the environment, the path location at a given height is
determined; while in positive temperature there is no a priori concentration
of the polymer location.

In this section, we give the zero temperature proof and some rough bounds for
the transversal fluctuation of polymers. The positive temperature part of
Proposition 5.1 will be proved in Section 8.

###  5.1. Geodesic tightness

We start with a one-point estimate and later give the two-point estimate.

######  Lemma 5.2.

For all $K>0$, $L\geq 2$, and $s\in(0,1)$,

$\displaystyle\mathbb{P}\left(|\pi_{0}(s)|>K(s(1-s))^{1/2}L^{-1/4}\ \Big{|}\
\mathcal{L}(0,0;0,1)>L\right)<C\exp\left(-cK^{2}\right).$

In fact, for $K$ up to $L^{1/2}$ we will obtain a tail of $\exp(-2K^{2})$ (via
the first case of the comparison statement Theorem 4.1). Note that it exactly
corresponds to our ultimate goal, namely that $L^{-1/4}\pi_{0}(s)$ converges
to $\frac{1}{2}N(0,s(1-s))$, which on scale $(s(1-s))^{1/2}$ would have a tail
at depth $x$ that satisfies the approximate asymptotics of $\exp(-2x^{2})$.

###### Proof of Lemma 5.2.

Observe that, for any $K$,
$\mathbb{P}(|\pi_{0}(s)|>K\mid\mathcal{L}(0,0;0,1)>L)$ is upper bounded by

$\displaystyle\mathbb{P}\left(\sup_{|x|>K}\Bigl{(}\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;0,1)\Bigr{)}>L\
\Big{|}\ \mathcal{L}(0,0;0,1)>L\right)$ $\displaystyle\leq
2\cdot\mathbb{P}\left(\sup_{x>K}\Bigl{(}\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;0,1)\Bigr{)}>L\
\Big{|}\ \mathcal{L}(0,0;0,1)>L\right)$ $\displaystyle\leq
2\cdot\frac{\mathbb{P}\left(\sup_{x>K}\Bigl{(}\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;0,1)\Bigr{)}>L\right)}{\mathbb{P}(\mathcal{L}(0,0;0,1)>L)},$
(5.1)

the factor of 2 coming from a union bound and using the distributional
symmetry of $\mathcal{L}(0,0;x,s)$ and $\mathcal{L}(x,s;0,1)$ under
$x\mapsto-x$ from Lemma 2.1 (as well as the independence of the two processes)
to remove the absolute value in the condition under the supremum. Then using
the shear invariance and independence properties of $\mathcal{L}$ and that
$x>K$ for the inequality in the second line,

$\displaystyle\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;0,1)$
$\displaystyle\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}(0,0;x-K,s)+\mathcal{L}(x-K,s;0,1)+(s(1-s))^{-1}\left[(x-K)^{2}-x^{2}\right]$
$\displaystyle\leq\mathcal{L}(0,0;x-K,s)+\mathcal{L}(x-K,s;0,1)-(s(1-s))^{-1}K^{2}$

as a process in $x$. Thus we see that the RHS of (5.1) is upper bounded by

$\displaystyle\frac{\mathbb{P}\left(\sup_{x>0}\Bigl{(}\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;0,1)\Bigr{)}>L+(s(1-s))^{-1}K^{2}\right)}{\mathbb{P}(\mathcal{L}(0,0;0,1)>L)}.$

Now using that
$\mathcal{L}(0,0;0,1)=\sup_{x\in\mathbb{R}}\Bigl{(}\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;0,1)\Bigr{)}$
from (1.1), it follows that the previous display is upper bounded by

$\displaystyle\frac{\mathbb{P}\left(\mathcal{L}(0,0;0,1)>L+(s(1-s))^{-1}K^{2}\right)}{\mathbb{P}(\mathcal{L}(0,0;0,1)>L)}.$

Applying Theorem 4.1 now bounds the previous display by

$C\exp\left(-cK^{2}(s(1-s))^{-1}L^{1/2}\right).$

Replacing $K$ by $K(s(1-s))^{1/2}L^{-1/4}$ completes the proof. ∎

We next give a two-point estimate. Combining it with the Kolmogorov-Chentsov
criterion for tightness (see e.g. [Kal22, Theorem 23.7]), the $\beta=\infty$
case of Proposition 5.1 follows.

######  Proposition 5.3.

For all $K>0$, $L\geq 2$, and $0<s<t<1$,

$\displaystyle\mathbb{P}\left(|\pi_{0}(s)-\pi_{0}(t)|>K(t-s)^{1/2}L^{-1/4}\mid\mathcal{L}(0,0;0,1)>L\right)\leq
C\exp(-cK^{2}).$

###### Proof.

We give the proof under the assumption that $t-s\in(0,\frac{1}{2})$. The case
where $t-s\in[\frac{1}{2},1]$ follows from Lemma 5.2 easily.

Similar to the previous proof, the LHS of the display in the lemma is upper
bounded by

$\displaystyle\mathbb{P}\left(\sup_{|x-y|>K}\Bigl{(}\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;y,t)+\mathcal{L}(y,t;0,1)\Bigr{)}>L\
\Big{|}\ \mathcal{L}(0,0;0,1)>L\right)$ $\displaystyle\leq
2\cdot\mathbb{P}\left(\sup_{x-y>K}\Bigl{(}\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;y,t)+\mathcal{L}(y,t;0,1)\Bigr{)}>L\
\Big{|}\ \mathcal{L}(0,0;0,1)>L\right)$ $\displaystyle\leq
2\cdot\frac{\mathbb{P}\left(\sup_{x-y>K}\Bigl{(}\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;y,t)+\mathcal{L}(y,t;0,1)\Bigr{)}>L\right)}{\mathbb{P}\left(\mathcal{L}(0,0;0,1)>L\right)}.$
(5.2)

Now, using the stationarity (and independence) properties of $\mathcal{L}$,

$\displaystyle\mathcal{L}(0,0;x,s)+\mathcal{L}(x,s;y,t)+\mathcal{L}(y,t;0,1)$
$\displaystyle\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}(-K,0;x-K,s)+\mathcal{L}(x-K,s;y,t)+\mathcal{L}(y,t;0,1)+(t-s)^{-1}\left[(x-y-K)^{2}-(x-y)^{2}\right]$

as a process in $(x,y)$. Since $x-y>K$, we have that

$(t-s)^{-1}\left[(x-y-K)^{2}-(x-y)^{2}\right]<-(t-s)^{-1}K^{2}.$

Thus we see that the RHS of (5.2) is upper bounded by

$\displaystyle
2\cdot\frac{\mathbb{P}\left(\sup_{x-y>K}\Bigl{(}\mathcal{L}(-K,0;x-K,s)+\mathcal{L}(x-K,s;y,t)+\mathcal{L}(y,t;0,1)\Bigr{)}>L+(t-s)^{-1}K^{2}\right)}{\mathbb{P}\left(\mathcal{L}(0,0;0,1)>L\right)}$

Now using that
$\mathcal{L}(-K,0;0,1)=\sup_{x,y\in\mathbb{R}}\Bigl{(}\mathcal{L}(-K,0;x,s)+\mathcal{L}(x,s;y,t)+\mathcal{L}(y,t;0,1)\Bigr{)}$,
and that $\mathcal{L}(-K,0;0,1)\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}(0,0;0,1)-K^{2}$, it follows that the previous display is
upper bounded by

$\displaystyle\frac{\mathbb{P}\left(\mathcal{L}(-K,0;0,1)>L+(t-s)^{-1}K^{2}\right)}{\mathbb{P}(\mathcal{L}(0,0;0,1)>L)}$
$\displaystyle=\frac{\mathbb{P}\left(\mathcal{L}(0,0;0,1)>L+((t-s)^{-1}+1)K^{2}\right)}{\mathbb{P}(\mathcal{L}(0,0;0,1)>L)}.$

By Theorem 4.1, this is upper bounded by

$\displaystyle C\exp\left(-c(t-s)^{-1}K^{2}L^{1/2}\right).$

Replacing $K$ by $K(t-s)^{1/2}L^{-1/4}$ completes the proof. ∎

###  5.2. Polymer transversal estimates

We now adapt the zero temperature arguments above to the positive temperature
setting. Although the bounds in this subsection are not sufficient to derive
the $\beta=1$ case of Proposition 5.1, they will be used in the proof to be
given in Section 8.