# Temporal correlation in the inverse-gamma polymer

Riddhipratim Basu Timo Seppäläinen Xiao Shen International Centre for
Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru,
INDIA<EMAIL_ADDRESS>of Mathematics, University of Wisconson-
Madison, Wisconsin, USA<EMAIL_ADDRESS>Department of Mathematics,
University of Utah, Utah, USA<EMAIL_ADDRESS>
###### Abstract

Understanding the decay of correlations in time for (1+1)-dimensional polymer
models in the KPZ universality class has been a challenging topic. Following
numerical studies by physicists, concrete conjectures were formulated by
Ferrari and Spohn [27] in the context of planar exponential last passage
percolation. These have mostly been resolved by various authors. In the
context of positive temperature lattice models, however, these questions have
remained open. We consider the time correlation problem for the exactly
solvable inverse-gamma polymer in $\mathbb{Z}^{2}$. We establish, up to
constant factors, upper and lower bounds on the correlation between free
energy functions for two polymers rooted at the origin (droplet initial
condition) when the endpoints are either close together or far apart. We find
the same exponents as predicted in [27]. Our arguments rely on the
understanding of stationary polymers, coupling, and random walk comparison. We
use recently established moderate deviation estimates for the free energy. In
particular, we do not require asymptotic analysis of complicated exact
formulae.

††footnotetext: Date: ††footnotetext: 2010 Mathematics Subject Classification.
60K35, 60K37††footnotetext: Key words and phrases: correlation, coupling,
directed polymer, Kardar-Parisi-Zhang, random growth model.

###### Contents

  1. 1 Introduction
    1. 1.1 Universality in stochastic growth
    2. 1.2 Methods in the study of the KPZ class
    3. 1.3 Time correlation problem
    4. 1.4 Organization of the paper
  2. 2 Main results
  3. 3 Preliminaries on the polymer model
    1. 3.1 Notation
    2. 3.2 Regularity of the shape function and the characteristic direction
    3. 3.3 Free energy estimates
    4. 3.4 Stationary inverse-gamma polymer
    5. 3.5 Random walk comparison for the free energy profile
  4. 4 Local fluctuations
  5. 5 Proof of Theorem 2.1
  6. 6 Proof of Theorem 2.2
    1. 6.1 Barrier event $\mathcal{B}_{\textup{bar}}$
    2. 6.2 Concentration of the free energy between $(0,0)$ and $\mathcal{L}_{r}$
    3. 6.3 Concentration of the global free energy along $\mathcal{L}_{r}$
    4. 6.4 Expectation bounds
    5. 6.5 Constrained variance lower bound
    6. 6.6 Covariance lower bound
  7. 7 Nonrandom fluctuation lower bound
  8. 8 Moderate deviation bounds for the left tail
    1. 8.1 Upper bound for the left tail
    2. 8.2 Lower bound for the left tail
  9. A Proofs of the free energy estimates of Section 3.3
    1. A.1 Free energy and path fluctuations
    2. A.2 Interval-to-line free energy
    3. A.3 Estimates for the constrained free energy
    4. A.4 Minimum and Maximum of the constrained free energy in a box
  10. B Proof of the random walk comparison in Section 3.5
  11. C Monotonicity for the polymer model
  12. D Sub-exponential random variables.
  13. E Random walk estimate

##  1 Introduction

###  1.1 Universality in stochastic growth

Random growth models have always been at the heart of probability theory. The
simplest example of random growth is a sum of independent and identically
distributed (i.i.d.) random variables. Provided their second moment is finite,
the large-scale behavior of the centered sum is independent of the
distribution of the summands, as described by the central limit theorem. With
the fluctuation exponent $1/2$ and the Gaussian distribution as the central
limit scaling law, this model is a member of the Gaussian universality class.

Following the seminal 1986 physics work of Kardar, Parisi and Zhang [39], one
major goal of recent probability research has been to demonstrate that very
different universal behavior arises in a wide class of stochastic models with
spatial dependence. Extensive computer simulations, non-rigorous physical
arguments, laboratory experiments, and rigorous mathematical results have all
suggested that this Kardar-Parisi-Zhang universality class (KPZ) is rich. It
includes interacting particle systems, percolation models, polymer models,
random tilings, certain stochastic PDEs and more. All known members of the KPZ
class share universal fluctuation exponents and limit distributions from
random matrix theory [18, 48].

In the past twenty-five years, many ground-breaking advances in understanding
KPZ universality have come through the study of exactly solvable or integrable
models. However, for the vast majority of the conjectural members of the KPZ
class, these fine techniques of integrable probability, representation theory,
and algebraic combinatorics do not apply. With the eventual goal of extending
results beyond the integrable cases, a second line of research uses in
principle broadly applicable probabilistic techniques and geometric arguments
to study the integrable models. This paper falls in the latter category. We
study the temporal correlation in the inverse-gamma polymer model, originally
introduced by the second author [50].

In the remainder of this introduction, Section 1.2 gives a brief overview of
the presently used mathematical methods in KPZ study, Section 1.3 discusses
the correlation problem studied in this paper, and Section 1.4 explains the
organization of the rest of the paper.

###  1.2 Methods in the study of the KPZ class

Several different approaches to studying the exactly solvable models of the
KPZ class have emerged over the last 25 years. We describe these methods
briefly on a very general level, mainly in the context of zero-temperature
last-passage percolation (LPP) with exponential or geometric weights, where
mathematical development is farthest along.

####  1.2.1 Integrable probability

For exactly solvable LPP models based on RSK correspondence or similar
remarkable bijections, it is possible to write down explicit formulas for one-
point and multi-point distributions. These formulas typically involve Fredholm
determinants and are rather complicated to work with. Integrable probability
estimates refer to estimates and asymptotics obtained by careful analysis of
these formulas. Beginning with the seminal work of Baik, Deift and Johansson
[1], which established the Tracy-Widom scaling limit for the longest
increasing subsequence problem, this approach has brought much success. This
includes Airy process limits for the last-passage time profile started from
particular initial conditions and more recently the construction of the KPZ
fixed point that admits general initial conditions. Formulas for two-time
distributions have also been obtained but so far these have not yielded useful
asymptotics.

####  1.2.2 Gibbsian line ensembles

A useful approach based on resampling in line ensembles was introduced by
Corwin and Hammond [20]. In the zero temperature model of Brownian last-
passage percolation, where the corresponding line ensemble has the _Brownian
Gibbs property_ , a detailed understanding of the passage time profile was
obtained in a series of works [31, 32, 33, 34]. A similar approach exists for
the positive temperature KPZ equation and has been used recently to great
effect [21]. The Gibbsian line ensemble approach led to the construction of
the _directed landscape_ (DL), the space-time scaling limit of zero
temperature models. Subsequently, DL limits were established for the KPZ
equation [57, 56].

####  1.2.3 Percolation methods with integrable inputs

Yet another suite of techniques uses black-box integrable inputs together with
probabilistic and geometric arguments that can in general be referred to as
_percolation arguments_. The inputs typically used in this line of work
include (i) uniform curvature of the limit shape, (ii) moderate deviation
estimates of the passage time, and (iii) convergence of the one point
distribution to the GUE Tracy-Widom law that has a negative mean [8, 10]. Some
cases require more sophisticated inputs such as the Airy process limit of the
full profile [11]. These inputs are typically obtained from the first approach
above. In cases like exponential and Brownian LPP, one can also exploit random
matrix connections to obtain similar, albeit usually a bit weaker, estimates
[44]. These estimates are then applied to obtain fine information about the
geodesic geometry, which in turn provides further information about the space-
time profile of last-passage times. An axiomatic framework for these types of
arguments has been developed and used in [10].

####  1.2.4 Coupling methods

The most probabilistic approach that minimizes the role of integrability
utilizes couplings with stationary growth processes. In zero temperature the
seminal work was [17], followed by [5], and [50] began this development in
positive temperature polymer models. This effort has been recently
revolutionized by [23] that made possible certain quantitatively optimal
bounds. Presently this approach still relies on a special feature of the
model, namely, that the stationary measure is explicitly known. It has been
most successful in the study of solvable models. Its applicability does extend
to some stochastic processes presently not known to be fully integrable,
namely classes of zero-range processes and interacting diffusion [6, 41].
Through comparisons with the stationary process, many results about the
geodesics, parallel to those developed by the previous approach, have been
proved [3, 51]. Following the optimal bounds of [23], some of the integrable
inputs of the percolation approach can now be supplied by coupling techniques,
thereby reducing dependence on random matrix theory and integrable
probability.

####  1.2.5 The approach of this paper

The current paper uses a combination of the final two approaches discussed
above to study the temporal decay of correlations in the positive-temperature
exactly solvable inverse gamma polymer model. The major barrier to applying
the percolation arguments from [9, 11] has been the lack of one-point moderate
deviation estimates. One advantage of the coupling techniques is that they can
be extended from zero temperature to positive temperature [24, 58]. In the
context of the semi-discrete O’Connell-Yor polymer, one-point estimates have
recently been obtained under stationary initial conditions [43] and more
recently for the point-to-point problem [42]. These techniques carry over to
the inverse gamma polymer model as well. This opens the door for proving
versions of the lattice LPP correlation results obtained through one-point
estimates, now in the context of the positive temperature polymer models. Our
paper provides the first example in this vein, by proving bounds on the time
correlation structure of a lattice polymer model. We expect similar techniques
to be applicable to a number of other related problems. To emphasize, although
our approach here is similar to the one described in § 1.2.3, we only require
the one point moderate deviation inputs, and these are provided by the recent
advances in the coupling/stationary polymer approach of § 1.2.4. Therefore our
work does not rely on the integrable methods described in § 1.2.1-1.2.2.

###  1.3 Time correlation problem

We turn to the precise problem we are studying, its history, and our
contributions.

A central object in KPZ models is the random height function
$h:\mathbb{R}\times\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$. Depending on the
situation studied, $h(x,t)$ can be the height of a randomly moving interface
over spatial location $x$ at time $t$, the passage time on the plane from the
origin to $(x,t)$, or the free energy of point to point polymers between the
origin and location $(x,t)$.

The spatial statistics $x\mapsto h(x,t_{0})$ at a fixed time $t_{0}$ are much
better understood than the temporal process $t\mapsto h(x_{0},t)$. Multi-time
joint distributions for the height function have been obtained in several
exactly solvable models [2, 36, 37, 38, 45, 46]. However, it has remained
difficult to extract useful information from these impressive formulas.

Short of capturing the full distribution of the temporal evolution, a natural
object to study is the two-time correlation function

$\textup{$\mathbb{C}$orr}(h(0,t_{1}),h(0,t_{2})),$ (1.1)

where we have now singled out the origin $x_{0}=0$ as the spatial location.
This correlation was first studied by physicists Takeuchi and Sano [55], who
measured the quantity (1.1) from a turbulent liquid crystal experiment.
Subsequently came numerical simulations [52, 54] that predicted the behavior
of (1.1) by fixing $t_{1}$ and sending $t_{2}$ to infinity.

####  1.3.1 Prior rigorous time correlation results

Ferrari and Spohn [27] studied the large-time behavior of (1.1) from various
initial conditions in the corner growth model (CGM). CGM is the most-studied
zero-temperature KPZ last-passage growth model on the lattice. Taking time to
infinity, they obtained a variational formulation of the (rescaled) height
function in terms of two independent Airy processes. From the variational
problem they derived an explicit formula for the limiting two-time covariance
under the stationary initial distribution, as $t_{1},t_{2}$ both tend to
infinity. For the step and flat initial conditions [27] conjectured
asymptotics in the regimes $t_{1}/t_{2}\rightarrow 0$ and
$t_{1}/t_{2}\rightarrow 1$.

Following the conjectures of [27], several rigorous works studied this problem
under different initial conditions in the zero-temperature setting.

The time correlation problem for the droplet initial condition in exponential
LPP was solved in two parallel works. Both employed a combination of
integrable inputs and a geometric study of geodesics. The results of [25],
which also utilizes comparison with stationary processes, are limiting in
nature and also used the convergence of the passage time profile to the Airy2
process. They also obtained an exact formula for the stationary case and
identified universal behavior with respect to the initial condition when the
two time points are close to one another. In contrast, [9] used one point
estimates, convergence to Tracy-Widom GUE distribution (and the negativity of
its mean), together with geometric arguments, to obtain similar, but
quantitatively weaker, results for the droplet initial condition, but valid
also in the pre-limit setting. When the two time points are far away, the case
of the flat initial condition was dealt with in [11]. This work relied on
strong Brownian comparison results for the Airy2 process, in addition to
convergence to it. The time correlation problem in the half-space exponential
LPP has also been recently studied in [26].

The Gibbsian line ensemble approach has also been useful in this context. In
an unpublished work, Corwin and Hammond solved the time correlation problem in
Brownian LPP with this approach. Subsequently, together with Ghosal, they
extended their work to the positive temperature KPZ equation [19].

####  1.3.2 Our work: temporal correlations in positive temperature on the
lattice

Prior to the present work, there does not appear to be any mathematically
rigorous work on this problem for positive temperature lattice models.
Application of Gibbsian line ensemble techniques appears difficult. Although
the one-point convergence to Tracy-Widom GUE is known for the inverse gamma
polymer [7, 40, 14], the convergence of the free energy profile does not
appear to be known for any positive temperature lattice model (see [56] for
recent results along these lines for the O’ Connell-Yor polymer and the KPZ
equation), and hence the approach of [25] does not seem too feasible in this
case.

Our approach is inspired by [9] and the recent progress in stationary
techniques. One cannot directly apply the techniques of [9] in the positive
temperature set-up, as much of it refers to the fluctuations and coalescence
of _geodesics_ which do not exist in our setting. We modify their definitions
appropriately and construct events in terms of the free energy profile and
restricted free energies, which can serve similar purposes. Certain estimates
are directly proved using stationary techniques. The novel technical
ingredients of our paper are developed in these directions.

For the upper bound, we directly prove the locally diffusive behavior of the
free energy profile (Section 4) instead of utilizing local fluctuations of
geodesics as in [9]. For the lower bound, we use the FKG inequality as in [9],
but in the absence of geodesics, the resampling argument is significantly
different, and, in fact, somewhat simpler. We give a direct proof of a lower
bound of the difference between the expected free energy and its long-term
value, at the standard deviation scale. This way we avoid the need for the
Tracy-Widom limit, and in fact, this provides a new proof of the negativity of
the mean of the Tracy-Widom distribution. Our arguments carry over to the zero
temperature setting as well, thus eliminating the integrable probability
inputs from the LPP results of [9].

To summarize, we establish the exponents that govern the decay of correlations
in the time direction for the inverse gamma polymer model. As expected on
universality grounds, the exponents are the same as in the zero-temperature
case. Ours is the first such result in a lattice polymer model in the KPZ
class. The only special feature of the model we use is the explicit
description of the stationary process. In particular, we do not use any weak
convergence result (to the Tracy-Widom distribution, for example). Our
techniques consist of one-point estimates obtained through stationary
polymers, random walk comparisons, and percolation arguments. Ours is the
first instance where the stationary polymer techniques have been put together
with the percolation arguments in a positive temperature setting. This
combination can be useful for extending many zero-temperature results to the
inverse gamma polymer.

That our approach does not rely on integrable inputs is not only potentially
useful for future extensions, but also necessary in the current state of the
subject. There are fewer integrable tools available for our model in
comparison with exponential LPP, Brownian LPP or the KPZ equation. There is no
determinantal formula for the multi-point joint distribution of the free
energy, and there is no corresponding Brownian Gibbs property in this discrete
setting (unless one takes a certain limit of the model). Lastly, the inverse-
gamma polymer model sits higher up in the hierarchy of the KPZ models. This
means that through appropriate transformations and limits, LPP, BLPP, and the
KPZ equation can be derived from the inverse-gamma polymer. In consequence,
our results should carry over to these other models and thereby remove the
integrable inputs utilized in previous works.

###  1.4 Organization of the paper

The polymer model is defined and our main results on the correlation bounds,
Theorems 2.1 and 2.2, are stated in Section 2. Theorem 2.1 is proved in
Section 5 and Theorem 2.2 in Section 6. Auxiliary results needed for the main
proofs are collected in Sections 3 and 4. We treat the proofs of these
auxiliary results differently depending on their status. Those that require
significant proof are verified in Sections 7 and 8, while those based on
existing work, such as analogous zero-temperature results, are in the
appendices. Next, we explain the organization of the supporting results in
more detail.

Section 3.1 contains additional notation and conventions, in particular, for
various subsets of $\mathbb{Z}^{2}$ and partition functions of restricted
collections of paths. Section 3.2 collects regularity properties of the shape
function. Nothing beyond calculus is used here.

Section 3.3 covers various estimates for the free energy, organized into
several subsections.

  * •

Section 3.3.1 gives moderate deviation estimates for the point-to-point free
energy. Two estimates for the left tail that appear here are used multiple
times in the paper and proved in Section 8.

  * •

Sections 3.3.2–3.3.6 contain a a variety of estimates. These are used only for
the lower bound of Theorem 2.2. Those that have previously appeared in the
zero-temperature setting have their proofs in Appendix A.1.

  * •

Section 3.3.7 gives a lower bound on the discrepancy between the asymptotic
free energy and the finite-volume expected free energy, sometimes called the
non-random fluctuation. It is proved in Section 7 by comparison with the
increment-stationary polymer. This result is used in the proof of the lower
bound of the left tail in Section 3.3.1 and the construction of the Barrier
event $\mathcal{B}_{\textup{bar}}$ in Section 6.1.

Section 3.4 introduces the increment-stationary inverse-gamma polymer and
discusses some of its properties. The proofs for these properties can be found
in Section 7. Among the results here are upper and lower bounds on the free
energy difference between the stationary model and the i.i.d. model.

Section 3.5 presents a random walk comparison of the free energy profile.
Specifically, we establish upper and lower bounds on the free energy along a
down-right path using two random walks. The proof of this comparison, which
relies on the stationary polymer process, can be found in Appendix B.

Section 4 is dedicated to local fluctuations in the free energy profile. The
proofs in this section rely on the moderate deviation estimates and the random
walk comparison. The results obtained here are crucial for the proofs of the
main theorems.

Acknowledgements. RB was partially supported by a MATRICS grant
(MTR/2021/000093) from SERB, Govt. of India, DAE project no. RTI4001 via ICTS,
and the Infosys Foundation via the Infosys-Chandrasekharan Virtual Centre for
Random Geometry of TIFR. TS was partially supported by National Science
Foundation grants DMS-1854619 and DMS-2152362 and by the Wisconsin Alumni
Research Foundation. Part of this work was conducted at International Centre
for Theoretical Sciences (ICTS), Bengaluru, India during the program ”First-
passage percolation and related models” in July 2022 (code: ICTS/fpp-2022/7),
the authors thank ICTS for the hospitality.

##  2 Main results

Let $\\{Y_{\bf z}\\}_{{\bf z}\in\mathbb{Z}^{2}}$ be a collection of positive
weights on the integer lattice $\mathbb{Z}^{2}$. Fix two points ${\bf u},{\bf
v}\in\mathbb{Z}^{2}$, we denote the collection of up-right paths between them
as $\mathbb{X}_{{\bf u},{\bf v}}$. This means if $\gamma\in\mathbb{X}_{{\bf
u},{\bf v}}$, where $\gamma$ is viewed as a sequence of vertices
$\gamma=(\gamma_{0},\gamma_{1},\dots,\gamma_{|{\bf u}-{\bf v}|_{1}})$ such
that $\gamma_{0}={\bf u}$, $\gamma_{|{\bf u}-{\bf v}|_{1}}={\bf v}$ and
$\gamma_{i+1}-\gamma_{i}\in\\{{\bf e}_{1},{\bf e}_{2}\\}$.

The point-to-point polymer partition function between ${\bf u}$ and ${\bf v}$
is defined by

$Z_{{\bf u},{\bf v}}=\sum_{\gamma\in\mathbb{X}_{{\bf u},{\bf
v}}}\prod_{i=1}^{|{\bf u}-{\bf v}|_{1}}Y_{\gamma_{i}},$ (2.1)

provided that ${\bf u}\neq{\bf v}$ and $\mathbb{X}_{{\bf u},{\bf v}}$ is non-
empty. Otherwise, we set $Z_{{\bf u},{\bf v}}=0$. Note the convention here
that the weight $Y_{\bf u}$ at the beginning of the path does not enter into
the definition of the partition function, since the product starts with $i=1$.

The free energy is defined to be $\log Z_{{\bf u},{\bf v}}$ and takes the
value $-\infty$ if $Z_{{\bf u},{\bf v}}=0$. Provided that $Z_{{\bf u},{\bf
v}}>0$, the quenched polymer measure is a probability measure on the set of
paths $\mathbb{X}_{{\bf u},{\bf v}}$ which is defined by

$Q_{{\bf u},{\bf v}}\\{\gamma\\}=\frac{1}{Z_{{\bf u},{\bf
v}}}\prod_{i=1}^{|{\bf u}-{\bf v}|_{1}}Y_{\gamma_{i}}\qquad\textup{ for
$\gamma\in\mathbb{X}_{{\bf u},{\bf v}}$}.$

In general, the positive weights $\\{Y_{\bf z}\\}_{{\bf z}\in\mathbb{Z}^{2}}$
can be chosen as a collection of i.i.d. positive random variables on some
probability space $(\Omega,\mathbb{P})$. Under a mild moment assumption such
as

$\mathbb{E}\big{[}|\log Y_{\bf z}|^{p}\big{]}<\infty\quad\textup{ for some
$p>2$},$

a law of large numbers type result called the shape theorem holds for the free
energy (Section 2.3 of [35]): there exists a concave, positively homogeneous
and deterministic continuous function $\Lambda:\mathbb{R}^{2}_{\geq
0}\rightarrow\mathbb{R}$ that satisfies

$\lim_{n\rightarrow\infty}\sup_{{\bf z}\in\mathbb{Z}^{2}_{\geq 0}:|{\bf
z}|_{1}\geq n}\frac{|\log Z_{(0,0),{\bf z}}-\Lambda({\bf z})|}{|{\bf
z}|_{1}}=0\qquad\textup{$\mathbb{P}$-almost surely}.$

For general i.i.d. weights, regularity properties of $\Lambda$ such as strict
concavity or differentiability, expected to hold at least for continuous
weights, are unknown. There is a special case, first observed in [50], that if
the i.i.d. weights have the inverse-gamma distribution, then $\Lambda$ can be
computed explicitly. The density function of the inverse-gamma distribution is
defined by

$f_{\mu}(x)={\displaystyle{\frac{1}{\Gamma(\mu)}}x^{-\mu-1}e^{-{\frac{1}{x}}}}\quad\text{for
}x>0.$ (2.2)

The shape parameter $\mu\in(0,\infty)$ plays the role of temperature in this
polymer model. We derive several properties for $\Lambda$ in Section 3.2 which
will be used in our proofs later on. In addition, for this inverse-gamma
polymer, many more explicit estimates can be established, hence it is often
referred to as an exactly solvable model.

As is standard, the correlation coefficient of two random variables $\zeta$
and $\eta$ is defined by

$\textup{$\mathbb{C}$orr}(\zeta,\eta)=\frac{\textup{$\mathbb{C}$ov}(\zeta,\eta)}{\textup{$\mathbb{V}$ar}(\zeta)^{1/2}\,\textup{$\mathbb{V}$ar}(\eta)^{1/2}}=\frac{\mathbb{E}[\zeta\eta]-\mathbb{E}\zeta\cdot\mathbb{E}\eta}{\mathbb{E}[\,|\zeta-\mathbb{E}\zeta|^{2}\,]^{1/2}\,\mathbb{E}[\,|\eta-\mathbb{E}\eta|^{2}\,]^{1/2}}.$

Our main result establishes the time correlation exponents $1/3$ and $2/3$ for
two free energies based on the separation of their endpoints.

The bounds in the next two theorems are valid under the assumption that the
weights $\\{Y_{\bf z}\\}$ have the i.i.d. inverse-gamma distribution (2.2) for
some choice of the parameter $\mu\in(0,\infty)$.

######  Theorem 2.1.

There exist positive constants $C_{1},C_{2},c_{0},N_{0}$ such that, whenever
$N\geq N_{0}$ and $N/2\leq r\leq N-c_{0}$, we have

$1-C_{\ref*{thm_r_large_c1}}\Big{(}\frac{N-r}{N}\Big{)}^{2/3}\leq\textup{$\mathbb{C}$orr}\big{(}\log
Z_{(0,0),(r,r)},\log Z_{(0,0),(N,N)}\big{)}\leq
1-C_{\ref*{thm_r_large_c2}}\Big{(}\frac{N-r}{N}\Big{)}^{2/3}.$

######  Theorem 2.2.

There exist positive constants $C_{3},C_{4},c_{0},N_{0}$ such that, whenever
$N\geq N_{0}$ and $c_{0}\leq r\leq N/2$, we have

$C_{\ref*{thm_r_small_c1}}\Big{(}\frac{r}{N}\Big{)}^{1/3}\leq\textup{$\mathbb{C}$orr}\big{(}\log
Z_{(0,0),(r,r)},\log Z_{(0,0),(N,N)}\big{)}\leq
C_{\ref*{thm_r_small_c2}}\Big{(}\frac{r}{N}\Big{)}^{1/3}.$

##  3 Preliminaries on the polymer model

###  3.1 Notation

${\bf a}$${\bf b}$${R}_{{\bf a},{\bf b}}^{k}$${\bf a}$$\mathcal{L}_{\bf
a}^{k}$ Figure 3.1: Illustrations of the segment $\mathcal{L}_{\bf a}^{k}$ and
the parallelogram $R_{{\bf a},{\bf b}}^{k}$. All anti-diagonal segments have
$\ell^{\infty}$-length $2k$.

Generic positive constants will be denoted by $C,C^{\prime}$ in the
calculations of the proofs. They may change from line to line without change
in notation. Other important positive constants appearing in the results will
be numbered in the form $C_{\textup{number}}$.

As shown in Figure 3.1, for any point ${\bf a}\in\mathbb{Z}^{2}$,
$\mathcal{L}_{{\bf a}}=\\{{\bf a}+(j,-j):j\in\mathbb{Z}\\}$ denotes the anti-
diagonal line with slope $-1$ going through the point ${\bf a}$, and for any
positive constant $k$, set

$\mathcal{L}^{k}_{{\bf a}}=\\{{\bf x}\in\mathcal{L}_{{\bf a}}:|{\bf x}-{\bf
a}|_{\infty}\leq k\\}.$

For ${\bf a},{\bf b}\in\mathbb{Z}^{2}$ and $k\in\mathbb{R}_{\geq 0}$, $R_{{\bf
a},{\bf b}}^{k}$ denotes the parallelogram spanned by the four corners ${\bf
a}\pm(-k,k)$ and ${\bf b}\pm(-k,k)$.

For a collection of directed paths $\mathfrak{A}$, let $Z(\mathfrak{A})$ be
the free energy obtained by summing over all the paths in $\mathfrak{A}$. For
$A,B\subset\mathbb{R}^{2}$, let $Z_{A,B}$ denote the partition function
obtained by summing over all directed paths starting from integer points

$A^{\circ}=\\{{\bf a}\in\mathbb{Z}^{2}:{\bf a}+[0,1)^{2}\cap
A\neq\emptyset\\}$

and ending in

$B^{\circ}=\\{{\bf b}\in\mathbb{Z}^{2}:{\bf b}+[0,1)^{2}\cap
B\neq\emptyset\\}.$

Furthermore, set

$Z^{\textup{max}}_{A,B}=\max_{{\bf a}\in A^{\circ},{\bf b}\in
B^{\circ}}Z_{{\bf a},{\bf b}}.$

For $A,B\subset\mathbb{R}^{2}$, ${\bf c},{\bf d}\in\mathbb{Z}^{2}$ and $h>0$
we define two specific partition functions:

$\displaystyle Z_{A,B}^{\textup{in},R^{h}_{{\bf c},{\bf d}}}$
$\displaystyle=\text{sum over directed paths from $A$ to $B$ contained inside
the parallelogram $R^{h}_{{\bf c},{\bf d}}$, }$ $\displaystyle
Z_{A,B}^{\textup{exit},R^{h}_{{\bf c},{\bf d}}}$ $\displaystyle=\text{sum over
directed paths from $A$ to $B$ that exit at least one of }$ the sides of
$R^{h}_{{\bf c},{\bf d}}$ parallel to ${\bf d}-{\bf c}$.

We simplify the notation when the starting and end places of the free energy
match with the parallelogram, for example

$Z_{\mathcal{L}_{{\bf a}}^{s_{1}},\mathcal{L}_{{\bf
b}}^{s_{2}}}^{\textup{in},R^{k}_{{\bf a},{\bf b}}}=Z_{\mathcal{L}_{{\bf
a}}^{s_{1}},\mathcal{L}_{{\bf b}}^{s_{2}}}^{\textup{in},k}\qquad\textup{ and
}\qquad Z_{\mathcal{L}_{{\bf a}}^{s_{1}},\mathcal{L}_{{\bf
b}}^{s_{2}}}^{\textup{exit},R^{k}_{{\bf a},{\bf b}}}=Z_{\mathcal{L}_{{\bf
a}}^{s_{1}},\mathcal{L}_{{\bf b}}^{s_{2}}}^{\textup{exit},k}.$

Integer points on the diagonal are abbreviated as $a=(a,a)\in\mathbb{Z}^{2}$.
Common occurrences of this include $Z_{r,N}=Z_{(r,r),(N,N)}$, $Z_{{\bf
p},N}=Z_{{\bf p},(N,N)}$, $\mathcal{L}_{a}^{k}=\mathcal{L}_{(a,a)}^{k}$ and
$R_{a,b}^{k}=R^{k}(a,b)=R_{(a,a),(b,b)}^{k}$.

The standard gamma function is $\Gamma(s)=\int_{0}^{\infty}x^{s-1}e^{-x}\,dx$
and the polygamma functions are
$\Psi_{k}(s)=\frac{d^{k+1}}{ds^{k+1}}\log\Gamma(s)$ for $k=0,1,2,\dotsc$.

Finally, we point out two conventions. First, we drop the integer floor
function to simplify notation. For example, if we divide the line segment from
$(0,0)$ to $(N,N)$ in $5$ equal pieces, we denote the free energy of the first
segment by $\log Z_{0,N/5}$ even if $N/5$ is not an integer. The second one is
about the dependence of constants on parameters. A statement of the type
“there exists a positive $\theta_{0}$ such that for each
$0<\theta<\theta_{0}$, there exist positive constants $C_{0},N_{0},t_{0}$ such
that …” means that $C_{0},N_{0}$ and $t_{0}$ can (and often necessarily do)
depend on $\theta$.

###  3.2 Regularity of the shape function and the characteristic direction

$-\mu/2$$\mu/2$ Figure 3.2: The graph of $m_{\mu/2}(z)$ for
$z\in(-\mu/2,\mu/2)$. The function $m$ is smooth and has a non-vanishing
derivative on $(-\mu/2,\mu/2)$. The image of $m_{\mu/2}(z)$ is $(0,\infty)$
which corresponds the slopes of the points inside $]{\bf e}_{1},{\bf e}_{2}[$.

Henceforth fix the shape parameter $\mu\in(0,\infty)$ and assume that the
weights $\\{Y_{\bf z}\\}$ have the i.i.d. inverse-gamma distribution (2.2).
Define the characteristic direction as a function of $\rho\in(0,\mu)$

${\boldsymbol{\xi}}[\rho]=\Big{(}\frac{\Psi_{1}(\rho)}{\Psi_{1}(\rho)+\Psi_{1}(\mu-\rho)}\,,\,\frac{\Psi_{1}(\mu-\rho)}{\Psi_{1}(\rho)+\Psi_{1}(\mu-\rho)}\Big{)}.$
(3.1)

The term characteristic direction becomes meaningful when we define the
stationary inverse-gamma polymer in Section 3.4. $\Psi_{1}$ is strictly
decreasing and $C^{\infty}$ on $\mathbb{R}_{>0}$. Thus
${\boldsymbol{\xi}}[\rho]$ is a continuous bijection between $\rho\in(0,\mu)$
and vectors (or directions) on the open line segment between ${\bf e}_{1}$ and
${\bf e}_{2}$. Denote the slope of the vector ${\boldsymbol{\xi}}[\rho+z]$ by

$m_{\rho}(z)=\frac{{\boldsymbol{\xi}}[\rho+z]\cdot{\bf
e}_{2}}{{\boldsymbol{\xi}}[\rho+z]\cdot{\bf
e}_{1}}=\frac{\Psi_{1}(\mu-\rho-z)}{\Psi_{1}(\rho+z)}.$

It is $C^{\infty}$ with non-vanishing derivative on the interval
$z\in(-\rho,\mu-\rho)$. Its inverse function $z_{\rho}(m)$ is $C^{\infty}$ and
has a non-vanishing positive derivative for $m\in(0,\infty)$. The graph of
$m_{\mu/2}(z)$ is illustrated in Figure 3.2. Taylor expansion for
$m_{\rho}(z)$ around $z=0$ gives this estimate:

######  Proposition 3.1 (Lemma 3.1 of [15]).

There exist positive constants $C_{5},C_{6},\epsilon$ such that for each
$z\in[-\epsilon,\epsilon]$ and each $\rho\in[\epsilon,\mu-\epsilon]$, we have

$\big{|}m_{\rho}(z)-(m_{\rho}(0)+C_{\ref*{slope1_c1}}z)\big{|}\leq
C_{\ref*{slope1_c2}}z^{2}.$

The next few results specialize to the diagonal direction $\rho=\mu/2$. We
drop the subscript and write $m=m_{\mu/2}$ and $z=z_{\mu/2}$. Taylor expansion
of $z(m)$ around $m=1$ gives this estimate:

######  Proposition 3.2.

There exist positive constants $C_{7},C_{8},\epsilon$ such that for each
$m\in[1-\epsilon,1+\epsilon]$, we have

$\big{|}z(m)-C_{\ref*{slope_z_c1}}(m-1)\big{|}\leq
C_{\ref*{slope_z_c2}}(m-1)^{2}.$

Next we quantify the dependence of the shape function on $\rho$. Let $f(\rho)$
denote the shape function evaluated at the vector ${\boldsymbol{\xi}}[\rho]$,
and recall from [50] that

$f(\rho)=\Lambda({\boldsymbol{\xi}}[\rho])=-\tfrac{\Psi_{1}({{\rho}})}{\Psi_{1}({{\rho}})+\Psi_{1}({{\mu-\rho}})}\cdot\Psi_{0}(\mu-\rho)-\tfrac{\Psi_{1}({{\mu-\rho}})}{\Psi_{1}({{\rho}})+\Psi_{1}({{\mu-\rho}})}\Psi_{0}(\rho).$
(3.2)

Let $f_{d}=f(\mu/2)$ denote the shape function in the diagonal direction. From
concavity and symmetry, we get this inequality:

######  Proposition 3.3.

For each $\mu>0$ and each $z\in(-\mu/2,\mu/2)$, $f(\mu/2)\geq f(\mu/2+z)$.

The next bound captures the curvature of the shape function.

######  Proposition 3.4.

There exist positive constants $C_{9},C_{10},\epsilon$ such that for each
$z\in[-\epsilon,\epsilon]$, we have

$\Big{|}(f(\mu/2+z)-f(\mu/2))-(-C_{\ref*{time_const_c1}}z^{2})\Big{|}\leq
C_{\ref*{time_const_c2}}z^{4}.$

###### Proof.

From (3.2),

$\displaystyle f(\mu/2+z)-f(\mu/2)$
$\displaystyle=\Big{[}-\Big{(}\tfrac{\Psi_{1}(\mu/2+z)}{\Psi_{1}(\mu/2+z)+\Psi_{1}(\mu/2-z)}\Psi_{0}(\mu/2-z)+\tfrac{\Psi_{1}(\mu/2-z)}{\Psi_{1}(\mu/2+z)+\Psi_{1}(\mu/2-z)}\Psi_{0}(\mu/2+z)\Big{)}\Big{]}$
(3.3)
$\displaystyle\qquad\qquad\qquad\qquad-\Big{[}-\Big{(}\tfrac{\Psi_{1}(\mu/2)}{\Psi_{1}(\mu/2)+\Psi_{1}(\mu/2)}\Psi_{0}(\mu/2)+\tfrac{\Psi_{1}(\mu/2)}{\Psi_{1}(\mu/2)+\Psi_{1}(\mu/2)}\Psi_{0}(\mu/2)\Big{)}\Big{]}.$
(3.4)

Taylor expand (3.3) around $z=0$. The “zeroth” derivative terms and (3.4)
cancel each other. The coefficients of $z$, $z^{3}$, $z^{5}$ are zero. The
coefficient of $z^{2}$ is $\frac{1}{2}\Psi_{2}(\mu/2)<0$. ∎

Our next proposition controls the variation of the shape function on a segment
$\mathcal{L}_{N}^{hN^{2/3}}$.

######  Proposition 3.5.

There exist positive constants $C_{11},N_{0},\epsilon_{0}$ such that for each
$N\geq N_{0}$ , $h\leq\epsilon_{0}N^{1/3}$ and each ${\bf
p}\in\mathcal{L}_{N}^{hN^{2/3}}$, we have

$\big{|}\Lambda({\bf p})-2N{f_{d}}\big{|}\leq
C_{\ref*{reg_shape_c1}}h^{2}N^{1/3}.$

###### Proof.

Since each ${\bf p}\in\mathcal{L}_{N}^{hN^{2/3}}$ has the same $\ell^{1}$-norm
$2N$, let us rewrite ${\bf p}=2N{\boldsymbol{\xi}}[\mu/2+z_{\bf p}]$ for some
real number $z_{\bf p}$. Then, by our definition $\Lambda({\bf
p})=2Nf(\mu/2+z_{\bf p}).$

Since the perpendicular $\ell^{\infty}$-distance from ${\bf p}$ to the
diagonal is at most $hN^{2/3}$, the slope of the characteristic vector
${\boldsymbol{\xi}}[\mu/2+z_{\bf p}]$ satisfies

$|m(z_{\bf p})-1|\leq 2hN^{-1/3}.$

Fix $\epsilon_{0}$ sufficiently small, by Proposition 3.2, we obtain

$|z_{\bf p}|\leq ChN^{-1/3}.$

Finally, applying Proposition 3.4, we obtain that

$|f(\mu/2+z_{\bf p})-{f_{d}}|\leq Ch^{2}N^{-2/3}$

which directly implies the result of our proposition after multiplying by $2N$
on both sides. ∎

###  3.3 Free energy estimates

In this section, we collect a number of estimates used later in the proofs,
organized thematically into subsections. Some results are merely quoted, some
proved later, and in cases where the result has already appeared in the zero
temperature setting the positive temperature proofs are in Appendix A.

####  3.3.1 Moderate deviation estimates for the free energy

There are four moderate deviation estimates: upper and lower bounds for both
left and right tails.

The first theorem gives the upper bound on the right tail of the free energy.
This result for the inverse-gamma polymer was first proved as a combination of
the moderate deviation estimate from [7], which used integrable techniques,
and the large deviation estimate from [30]. The same moderate deviation upper
bound was also recently proven in [42] for the O’Connell-Yor polymer without
input from integrable probability. Also without integrable techniques, the
forthcoming work [24] proves this bound and obtains the sharp leading order
term $\tfrac{4}{3}t^{3/2}$ in the exponent for $t\leq CN^{2/3}$. A version of
this bound can be found in the Ph.D. thesis of one of the authors of [24], as
Theorem 4.3.1 in [58]. The right tail estimate for the KPZ equation with the
sharp leading order term was also recently obtained in [29]. Finally, we note
that both [42] and [24] use the technique of the zero-temperature version of
the estimate that first appeared in [23].

######  Proposition 3.6.

Let $\epsilon\in(0,\mu/2)$. There exist positive constants $C_{12},N_{0}$
depending on $\epsilon$ such that for each $N\geq N_{0}$, $t\geq 1$, and each
$\rho\in[\epsilon,\mu-\epsilon]$, we have

$\mathbb{P}(\log Z_{0,2N{\boldsymbol{\xi}}[\rho]}-2Nf(\rho)\geq tN^{1/3})\leq
e^{-C_{\ref*{up_ub_c1}}\min\\{t^{3/2}\\!,\,tN^{1/3}\\}}.$

The next theorem is the corresponding lower bound for the right tail,
restricted to the diagonal direction. This was recently proved for the
O’Connell-Yor polymer in [42] in the diagonal direction. The proof uses the
subadditivity of the free energy and the Tracy-Widom limit of the free energy.
Since using integrable techniques, the Tracy-Widom limit of the inverse-gamma
polymer is also known [7], the proof for the O’Connell-Yor polymer in Section
9 of [42] can be repeated verbatim for the inverse-gamma polymer. A similar
argument in the zero-temperature setting appeared earlier in [9, 28]. Without
this input from integrable probability, a lower bound with the correct leading
order $\tfrac{4}{3}t^{3/2}$ for $t\leq CN^{2/3}$ over all directions in a
compact interval away from ${\bf e}_{1}$ and ${\bf e}_{2}$ will appear in
[24].

######  Proposition 3.7.

There exist positive constants $C_{13},N_{0},t_{0},\epsilon_{0}$ such that for
each $N\geq N_{0}$, $t_{0}\leq t\leq\epsilon_{0}N^{2/3}$, we have

$\mathbb{P}(\log Z_{0,N}-2N{f_{d}}\geq tN^{1/3})\geq
e^{-C_{\ref*{up_lb_c1}}t^{3/2}}.$

The next theorem is the upper bound for the left tail. A similar result was
stated as Proposition 3.4 in [42] for the O’Connell-Yor polymer. We prove this
estimate for the inverse-gamma polymer in Section 8. Our proof is similar to
[42], based on ideas from the zero-temperature work [22].

######  Proposition 3.8.

Let $\epsilon\in(0,\mu/2)$. There exist positive constants $C_{14},N_{0}$
depending on $\epsilon$ such that for each $N\geq N_{0}$, $t\geq 1$ and each
$\rho\in[\epsilon,\mu-\epsilon]$, we have

$\mathbb{P}(\log Z_{0,2N{\boldsymbol{\xi}}[\rho]}-2Nf(\rho)\leq-tN^{1/3})\leq
e^{-C_{\ref*{up_ub_c1}}\min\\{t^{3/2},tN^{1/3}\\}}.$

######  Remark 3.9.

The correct order of the left tail should be $e^{-C\min\\{t^{3},tN^{4/3}\\}}$
for all $t\geq t_{0}$. This is different from the zero-temperature model where
the left tail behaves as $e^{-Ct^{3}}$. For the O’Connell-Yor polymer, the
authors in [42] also proved an upper bound $e^{-Ct^{3}}$ when $t_{0}\leq t\leq
N^{2/3}(\log N)^{-1}$. This is done by adapting the bootstrapping argument
from the zero-temperature work [28]. We do not pursue this here but expect the
same result.

Finally, we have the lower bound on the left tail, which we prove in Section
8. The same lower bound was proved in [42] for the O’Connell-Yor polymer. The
idea of the proof follows the zero-temperature work [28].

######  Proposition 3.10.

There exist positive constants $C_{15},N_{0},t_{0},\epsilon_{0}$ such that for
each $N\geq N_{0}$ and each $t_{0}\leq t\leq\epsilon_{0}N^{2/3}/(\log N)^{2}$,
we have

$\mathbb{P}(\log Z_{0,N}-2N{f_{d}}\leq-tN^{1/3})\geq
e^{-C_{\ref*{ptp_low_c1}}t^{3}}.$

####  3.3.2 Free energy and path fluctuations

In this section we state the estimates which capture the loss of free energy
when the paths have high fluctuations.

######  Proposition 3.11.

There exist positive constants $C_{16},C_{17},N_{0}$ such that for each $N\geq
N_{0}$, $h\in\mathbb{Z}$ and $t\geq 0$ we have

$\mathbb{P}\Big{(}\log
Z_{\mathcal{L}_{0}^{N^{2/3}},\mathcal{L}_{(N-2hN^{2/3},N+2hN^{2/3})}^{N^{2/3}}}-2N{f_{d}}\geq(-C_{\ref*{trans_fluc_loss_c1}}h^{2}+t)N^{1/3}\Big{)}\leq
e^{-C_{\ref*{trans_fluc_loss_c2}}(|h|^{3}+\min\\{t^{3/2},tN^{1/3}\\})}.$

Then, by essentially a union bound, we obtain the following proposition.

######  Proposition 3.12.

There exist positive constants $C_{18},C_{19},N_{0}$ such that for each $N\geq
N_{0}$, $t\geq 1$ and $s\geq 0$, we have

$\mathbb{P}\Big{(}\log
Z_{\mathcal{L}^{sN^{2/3}}_{0},\mathcal{L}_{N}\setminus\mathcal{L}_{N}^{(s+t)N^{2/3}}}-2N{f_{d}}\geq-
C_{\ref*{trans_fluc_loss2_c1}}t^{2}N^{1/3}\Big{)}\leq
e^{-C_{\ref*{trans_fluc_loss2_c2}}t^{3}}.$

Following this, we have the next proposition which states that paths with high
fluctuation tend to have much small free energy.

######  Theorem 3.13.

There exist positive constants $C_{20},C_{21},N_{0}$ such that for each $N\geq
N_{0}$, $1\leq t\leq N^{1/3}$ and $0<s<e^{t}$, we have

$\mathbb{P}\Big{(}\log
Z^{\textup{exit},(s+t)N^{2/3}}_{\mathcal{L}_{0}^{sN^{2/3}},\mathcal{L}_{N}^{sN^{2/3}}}-2N{f_{d}}\geq-
C_{\ref*{trans_fluc_loss3_c1}}t^{2}N^{1/3}\Big{)}\leq
e^{-C_{\ref*{trans_fluc_loss3_c2}}t^{3}}.$

From this, we have the following corollary which is a similar bound for point-
to-point free energy that is slightly off the diagonal direction.

######  Corollary 3.14.

There exist positive constants $C_{22},C_{23},N_{0}$ such that for each $N\geq
N_{0}$, $1\leq t\leq N^{1/3}$ and $0<s<t/10$, we have

$\mathbb{P}\Big{(}\log
Z^{\textup{exit},{tN^{2/3}}}_{(-sN^{2/3},sN^{2/3}),N}-2N{f_{d}}\geq-
C_{\ref*{trans_fluc_loss4_c1}}t^{2}N^{1/3}\Big{)}\leq
e^{-C_{\ref*{trans_fluc_loss4_c2}}t^{3}}.$

####  3.3.3 Interval-to-line free energy

In our work, we will also need an upper bound for the right tail of the
interval-to-line free energy.

######  Theorem 3.15.

There exist positive constants $C_{24},C_{25},N_{0}$ such that for each $N\geq
N_{0}$, $t\geq 1$ and $1\leq h\leq
e^{C_{\ref*{ptl_upper_c1}}{\min\\{t^{3/2},tN^{1/3}\\}}}$, we have

$\mathbb{P}\Big{(}\log
Z_{\mathcal{L}_{0}^{hN^{2/3}},\mathcal{L}_{N}}-2N{f_{d}}\geq
tN^{1/3}\Big{)}\leq e^{-C_{\ref*{ptl_upper_c2}}\min\\{t^{3/2},tN^{1/3}\\}}.$

####  3.3.4 Estimates for the constrained free energy

When we constrain the paths, the free energy decreases because we are summing
over a smaller collection of paths in the partition function. The first
theorem captures that the point-to-point free energy can not be too small if
we constrain the paths to a fixed rectangle of size order $N\times N^{2/3}$
which obeys the KPZ transversal fluctuation scale. Our second theorem gives a
lower bound for the probability that a constrained free energy is large.

######  Theorem 3.16.

For each positive $a_{0}$, there exist positive constants $C_{26},t_{0}$ such
that for each $0<\theta\leq 100$, there exists a positive constant $N_{0}$
such that for each $N\geq N_{0}$, $t\geq t_{0}$ and ${\bf
p}\in\mathcal{L}_{N}^{a_{0}\theta N^{2/3}}$, we have

$\mathbb{P}\Big{(}\log Z^{\textup{in},{\theta N^{2/3}}}_{0,{\bf
p}}-2N{f_{d}}\leq-
tN^{1/3}\Big{)}\leq\tfrac{\sqrt{t}}{\theta}e^{-C_{\ref*{wide_similar_c1}}\theta
t}.$

######  Theorem 3.17.

For any positive constant $s$, there exist positive constants
$C_{27},t_{0},N_{0}$ such that for each $N\geq N_{0}$, $t_{0}\leq t\leq
N^{2/3}$,

$\mathbb{P}\Big{(}\log Z^{\textup{in},{sN^{2/3}}}_{0,N}-2N{f_{d}}\geq
tN^{1/3}\Big{)}\geq e^{-C_{\ref*{c_up_lb_c1}}t^{3/2}}.$

####  3.3.5 Minimum and maximum estimate for the free energy

Our first theorem is the box-to-point minimum bound. This was first proved in
the zero-temperature setting which appeared in [13] for the Poissonian LPP
model, then later in [11] for the exponential LPP model. The proof follows the
idea from Section C.4 of [11].

######  Theorem 3.18.

There exist positive constants $C_{28},N_{0},t_{0}$ such that for each $N\geq
N_{0}$ and $t\geq t_{0}$, we have

$\mathbb{P}\Big{(}\min_{{\bf p}\in R^{N^{2/3}}_{0,9N/10}}\log
Z^{\textup{in},R_{0,N}^{N^{2/3}}}_{{\bf p},N}-(2N-|{\bf p}|_{1}){f_{d}}\leq-
tN^{1/3}\Big{)}\leq e^{-C_{\ref*{high_inf_c1}}t}.$

Lastly, we state a box-to-line maximum bound.

######  Theorem 3.19.

There exist positive constants $C_{29},N_{0},t_{0}$ such that for each $N\geq
N_{0}$ and each $t\geq t_{0}$, we have

$\mathbb{P}\Big{(}\max_{{\bf p}\in R^{N^{2/3}}_{0,9N/10}}\log Z_{{\bf
p},\mathcal{L}_{N}}-(2N-|{\bf p}|_{1}){f_{d}}\geq tN^{1/3}\Big{)}\leq
e^{-C_{\ref*{btl_upper_c1}}t}.$

######  Remark 3.20.

In both bounds of Theorem 3.18 and Theorem 3.19, the power $1$ on the exponent
$t^{1}$ is not expected to be optimal.

####  3.3.6 Variance bound for the free energy

We state the variance bound of the free energy which follows directly from the
upper and lower bounds for the left and right tails. We omit its proof. The
upper bound was first shown in [50] where the inverse-gamma polymer was first
introduced.

######  Theorem 3.21.

There exist positive constants $C_{30},C_{31},N_{0}$ such that for each $N\geq
N_{0}$, we have

$C_{\ref*{var_c1}}N^{2/3}\leq\textup{$\mathbb{V}$ar}(\log Z_{0,N})\leq
C_{\ref*{var_c2}}N^{2/3}.$

####  3.3.7 Nonrandom fluctuation

Finally, we record a lower bound for the nonrandom fluctuation of the free
energy in i.i.d. inverse-gamma polymer. This result follows directly from the
Tracy-Widom limit of the inverse-gamma model and the fact that the Tracy-Widom
distribution has a negative mean. Our contribution here is an alternative
proof (in Section 7) without relying on the Tracy-Widom limit.

######  Theorem 3.22.

Let $\epsilon\in(0,\mu/2)$. There exist positive constants $C_{32},N_{0}$ such
that for each $N\geq N_{0}$ and $\rho\in[\epsilon,\mu-\epsilon]$, we have

$2Nf(\rho)-\mathbb{E}[\log Z_{0,2N{\boldsymbol{\xi}}[\rho]}]\geq
C_{\ref*{nr_lb_c1}}N^{1/3}.$

###  3.4 Stationary inverse-gamma polymer

The (increment) stationary inverse-gamma polymer (with southwest boundary) is
defined on a quadrant. To start, we fix a parameter $\rho\in(0,\mu)$ and a
base vertex ${\bf v}\in\mathbb{Z}^{2}$. For each ${\bf z}\in{\bf
v}+\mathbb{Z}_{>0}^{2}$, the (vertex) bulk weights are defined by
$Y_{z}\sim\text{Ga}^{-1}(\mu)$. On the boundary ${\bf v}+k{\bf e}_{1}$, and
${\bf v}+k{\bf e}_{2}$, the (edge) weights have the distributions

$\displaystyle\begin{aligned} I^{{\rho}}_{[\\![v+(k-1)k{\bf e}_{1},v+k{\bf
e}_{1}]\\!]}&\sim\text{Ga}^{-1}(\mu-{{\rho}})\\\ J^{{\rho}}_{[\\![v+(k-1)k{\bf
e}_{2},v+k{\bf e}_{2}]\\!]}&\sim\text{Ga}^{-1}({{\rho}}).\end{aligned}$ (3.5)

And all the weights in the quadrant are independent. We denote the probability
measure for the stationary inverse-gamma polymer by $\mathbb{P}$ and record
the parameter $\rho$ and the base point ${\bf v}$ in the notation of the
partition function. For ${\bf w}\in{\bf v}+\mathbb{Z}_{\geq 0}^{2}$, let us
define

$Z^{\rho}_{{\bf v},{\bf w}}=\sum_{\gamma\in\mathbb{X}_{{\bf v},{\bf
w}}}\prod_{i=0}^{|{\bf v}-{\bf w}|_{1}}\mathaccent
869{Y}_{\gamma_{i}}\qquad\text{ where }\mathaccent
869{Y}_{\gamma_{i}}=\begin{cases}1\quad&\text{if $\gamma_{i}={\bf v}$}\\\
I^{{\rho}}_{[\\![\gamma_{i}-{\bf e}_{1},{\gamma_{i}}]\\!]}\quad&\text{if
$\gamma_{i}\cdot{\bf e}_{2}={\bf v}\cdot{\bf e}_{2}$}\\\
J^{{\rho}}_{[\\![\gamma_{i}-{\bf e}_{2},{\gamma_{i}}]\\!]}\quad&\text{if
$\gamma_{i}\cdot{\bf e}_{1}={\bf v}\cdot{\bf e}_{1}$}\\\
Y_{z}\quad&\text{otherwise.}\\\ \end{cases}$

And for $\gamma\in\mathbb{X}_{{\bf v},{\bf w}}$, the quenched polymer measure
is defined by

$Q^{{{\rho}}}_{{\bf v},{\bf w}}(\gamma)=\frac{1}{Z^{{{\rho}}}_{{\bf v},{\bf
w}}}\prod_{i=0}^{|{\bf v}-{\bf w}|_{1}}\mathaccent 869{Y}_{\gamma_{i}}.$

The name (increment) stationary inverse-gamma polymer is justified by the next
theorem, which first appeared in [50, Theorem 3.3].

######  Theorem 3.23.

For each ${\bf w}\in{\bf v}+\mathbb{Z}^{2}_{>0}$ . We have

$\frac{Z^{\rho}_{{\bf v},{\bf w}}}{Z^{\rho}_{{\bf v},{\bf w}-{\bf
e}_{1}}}\sim\textup{Ga}^{-1}(\mu-\rho)\qquad\text{ and
}\qquad\frac{Z^{\rho}_{{\bf v},{\bf w}}}{Z^{\rho}_{{\bf v},{\bf w}-{\bf
e}_{2}}}\sim\textup{Ga}^{-1}(\rho).$

Furthermore, let $\eta=\\{\eta_{i}\\}$ be any finite or infinite down-right
path in ${\bf v}+\mathbb{Z}^{2}_{\geq 0}$. This means $\eta_{i+1}-\eta_{i}$ is
either ${\bf e}_{1}$ or $-{\bf e}_{2}$. Then, the increments
$\\{Z^{\rho}_{{\bf v},\eta_{i+1}}/Z^{\rho}_{{\bf v},\eta_{i}}\\}$ are
independent.

From Theorem 3.23 above, we have the following identity for the expectation of
the free energy.

$\mathbb{E}\Big{[}\log
Z^{\rho}_{0,(a,b)}\Big{]}=-a\Phi_{0}(\mu-\rho)-b\Phi_{0}(\rho)$ (3.6)

Because the weights appearing on the boundary are stochastically larger than
the bulk weights, the sampled polymer paths tend to stay on the boundary.
However, for each fixed ${{\rho}}\in(0,\mu)$, there is a unique direction for
which this effect between the ${\bf e}_{1}$\- and ${\bf e}_{2}$-boundary is
balanced out, we call this the characteristic direction ${\bf\xi}[\rho]$,
which is defined previously in (3.1).

The first estimate below is the upper bound for the right tail of the free
energy. It first appeared in the Ph.D. Thesis [58]. Then, it was proven again
in [43]. We know that by (3.6), $2Nf(\rho)$ can be thought as the expectation
of $\log Z^{\rho}_{0,2N\bf{\xi}[\rho]}$, if we ignore the error from the
integer rounding.

######  Theorem 3.24.

Let $\epsilon\in(0,\mu/2)$. There exist positive constants $C_{33},N_{0}$ such
that for each $N\geq N_{0}$, $t\geq 1$ and $\rho\in[\epsilon,\mu-\epsilon]$,
we have

$\mathbb{P}\Big{(}\log Z^{\rho}_{0,2N{\boldsymbol{\xi}}[\rho]}-2Nf(\rho)\geq
tN^{1/3}\Big{)}\leq e^{-C_{\ref*{stat_up_ub_c1}}\min\\{t^{3/2},tN^{1/3}\\}}.$

Along the characteristic direction, the sampled paths tend to stay on the
boundary for order $N^{2/3}$ number of steps. Our next result is a corollary
of this fact, which appears as Corollary 4.2 in [49]. Fix ${\bf w}\in{\bf
v}+\mathbb{Z}^{2}_{\geq 0}$ and any $k\in\mathbb{R}_{>0}$. Let $\\{\tau_{{\bf
v},{\bf w}}\geq k\\}$ denote the subset of $\mathbb{X}_{{\bf v},{\bf w}}$ such
that the first $\lfloor k\rfloor$ steps of the path are all ${\bf
e}_{1}$-steps. Similarly, $\\{\tau_{{\bf v},{\bf w}}\leq-k\\}$ is the subset
of $\mathbb{X}_{{\bf v},{\bf w}}$ whose first $\lfloor k\rfloor$ steps are all
${\bf e}_{2}$-steps. When $\tau_{{\bf v},{\bf w}}$ appears inside a quenched
polymer measure as below, we will simplify the notation $\tau_{{\bf v},{\bf
w}}=\tau$ as the starting and end point of the paths are clear.

######  Theorem 3.25.

Let $\epsilon\in(0,\mu/2)$. There exist positive constants
$C_{34},C_{35},N_{0}$ such that for for all
${\rho}\in[\epsilon,\mu-\epsilon]$, $N\geq N_{0}$ and $r\geq 1$, we have

$\mathbb{P}^{{\rho}}(Q_{0,2N{\boldsymbol{\xi}}[\rho]+rN^{2/3}{\bf
e}_{1}}\\{\tau\leq-1\\}\geq e^{-C_{\ref*{exit_time_c1}}r^{3}})\leq
e^{-C_{\ref*{exit_time_c2}}r^{3}}.$

Let $\mathaccent 869{Z}$ denote the version of the free energy that also
include the weight at the beginning of the path. The following is essentially
a lower bound for the difference between the free energies of the stationary
boundary model and i.i.d. bulk polymer. We included an additional boundary
weight with the i.i.d. bulk free energy in the estimate below because this
version will be used to prove Theorem 3.22. Its proof will appear in Section
7.

######  Theorem 3.26.

Let $\epsilon\in(0,\mu/2)$. There exist positive constants $C_{36},N_{0}$ such
that for each $N\geq N_{0}$, $0<\delta\leq 1/2$ and
$\rho\in[\epsilon,\mu-\epsilon]$, we have

$\mathbb{P}\Big{(}\log Z^{\rho}_{-1,2N{\boldsymbol{\xi}}[\rho]}-\Big{(}\log
I^{\rho}_{[\\![(-1,-1),(0,-1)]\\!]}+\log\mathaccent
869{Z}_{0,2N{\boldsymbol{\xi}}[\rho]}\Big{)}\leq\delta N^{1/3}\Big{)}\leq
C_{\ref*{statiid_low_c1}}|\log(\delta\vee N^{-1/3})|\cdot(\delta\vee
N^{-1/3}).$

For completeness, we also record the following upper bound for the difference
between the stationary and i.i.d. free energy. This result follows directly
from Theorem 3.24 and Proposition 3.8 using a union bound, hence we omit its
proof.

######  Theorem 3.27.

Let $\epsilon\in(0,\mu/2)$. There exist positive constants $C_{37},N_{0}$ such
that for each $N\geq N_{0}$, $t\geq 1$ and $\rho\in[\epsilon,\mu-\epsilon]$,
we have

$\mathbb{P}\Big{(}\log Z^{\rho}_{-1,2N{\boldsymbol{\xi}}[\rho]}-\log
Z_{0,{2N\boldsymbol{\xi}}[\rho]}\geq tN^{1/3}\Big{)}\leq
e^{-C_{\ref*{statiid_up_c1}}\min\\{t^{3/2},tN^{1/3}\\}}.$

###  3.5 Random walk comparison for the free energy profile

The stationary polymer allows one to compare the free energy profile along a
segment of a downright path to random walks. This technique has appeared
previously in [3, 5, 15, 49, 50, 51] and many more places.

To start, fix $\rho\in(0,\mu)$ and define

${\bf v}_{N}=2N{\boldsymbol{\xi}}[\rho].$

Let $\Theta_{k}$ denote a down-right path of $k$ (edge) steps that goes
through the vertex ${\bf v}_{N}$. Order the vertices of $\Theta_{k}$ as ${\bf
z}_{0},\dotsc,{\bf z}_{k}$, where ${\bf z}_{0}$ has the largest ${\bf
e}_{2}$-coordinate value. Define the free energy profile to be the following
collection of random variables

$\log Z_{0,{\bf z}_{i}}-\log Z_{0,{\bf z}_{i-1}}\quad\text{ where
$i=1,\dots,k$}.$ (3.7)

The proof of the following theorem appears in Appendix B.

######  Theorem 3.28.

Fix $\epsilon\in(0,\mu/2)$. There exist positive constants
$C_{38},N_{0},s_{0},a_{0},q_{0}$ such that for each
$\rho\in[\epsilon,\mu-\epsilon]$, $N\geq N_{0}$, $s_{0}\leq s\leq
a_{0}N^{1/3}$, $1\leq k\leq sN^{2/3}$ and each down-right path
$\Theta_{k}=\\{{\bf z}_{0},\dotsc,{\bf z}_{k}\\}$, there exist two collections
of random variables $\\{X_{i}\\}$ and $\\{Y_{i}\\}$ such that the following
holds. Set

$\lambda=\rho+q_{0}sN^{-1/3}\qquad\text{ and }\qquad\eta=\rho-q_{0}sN^{-1/3}.$

The random variables $\\{X_{i}\\}$ are mutually independent with marginal
distributions

$\displaystyle X_{i}$ $\displaystyle\sim\log(\textup{Ga}^{-1}(\mu-\lambda))$
if ${\bf z}_{i}-{\bf z}_{i-1}={\bf e}_{1}$ $\displaystyle-X_{i}$
$\displaystyle\sim\log(\textup{Ga}^{-1}(\lambda))$ $\displaystyle\quad\text{
if ${\bf z}_{i}-{\bf z}_{i-1}=-{\bf e}_{2}$ }.$

The random variables $\\{Y_{i}\\}$ are mutually independent with marginal
distributions

$\displaystyle Y_{i}$ $\displaystyle\sim\log(\textup{Ga}^{-1}(\mu-\eta))$ if
${\bf z}_{i}-{\bf z}_{i-1}={\bf e}_{1}$ $\displaystyle-Y_{i}$
$\displaystyle\sim\log(\textup{Ga}^{-1}(\eta))$ $\displaystyle\quad\text{ if
${\bf z}_{i}-{\bf z}_{i-1}=-{\bf e}_{2}$ }.$

Furthermore, $X_{i}$ and $Y_{i}$ bounds the free energy profile with high
probability.

$\mathbb{P}\Big{(}\log\tfrac{9}{10}+Y_{i}\leq\log Z_{0,{\bf z}_{i}}-\log
Z_{0,{\bf z}_{i-1}}\leq\log\tfrac{10}{9}+X_{i}\text{ for each $i=1,2,\dots
k$}\Big{)}\geq 1-e^{-C_{\ref*{rwrw_c1}}s^{3}}.$

We also note that when $\Theta_{k}$ is vertical or horizontal, then $X_{i}$
and $Y_{i}$ can be coupled together with an explicit joint distribution that
allows calculations, see [3, 4, 15, 49, 51]. However, we will not use this
fact in this paper.

##  4 Local fluctuations

In this section, we look at fluctuations for the polymer near $0$ or $(N,N)$
where the time scale can be much smaller than the full scale $N$. We start
with an estimate for the fluctuation of the free energy profile along the
anti-diagonal line. This result was first proved for a zero-temperature model
(Brownian last-passage percolation) in [34] using the Brownian Gibbs property.
Other related results and extensions for the various zero-temperature models
have appeared in [9, 11, 16]. Compared to these, our proof does not rely on
integrable probability which was used in [16, 34], and we improve the tail
estimate from [9, 11] to optimal order.

######  Proposition 4.1.

There exist positive constants $C_{39},C_{40},c_{0},N_{0}$ such that for each
$N\geq N_{0}$, $1\leq t\leq c_{0}N^{1/2}$, and each $a\in\mathbb{Z}_{\geq 0}$,
we have

$\mathbb{P}\Big{(}\log Z_{0,\mathcal{L}^{a}_{N}}-\log Z_{0,N}\geq
C_{\ref*{compare_max_c1}}t\sqrt{a}\Big{)}\leq
e^{-C_{\ref*{compare_max_c2}}\min\\{t^{2}\\!,\,t\sqrt{a}\\}}.$

######  Remark 4.2.

Since the free energy profile is expected to be locally Brownian after
dividing by $N^{1/3}$, the difference of the free energies in the probability
above should approximate the running maximum of a two-sided random walk. Thus
the tail bound above is optimal.

###### Proof.

The case $a=0$ is trivial, so we will always assume $a\in\mathbb{Z}_{>0}$. We
may prove the proposition with the maximum version of the free energy since

$\log Z_{0,N}\leq\log Z_{0,\mathcal{L}^{a}_{N}}\leq\log
Z^{\textup{max}}_{0,\mathcal{L}^{a}_{N}}+10\log(a+1).$

Let us also note that when $a\geq t^{2/3}N^{2/3}$, the estimate is
straightforward. It holds that

$\displaystyle\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,\mathcal{L}^{a}_{N}}-\log Z_{0,N}\geq Ct\sqrt{a}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,\mathcal{L}_{N}}-\log Z_{0,N}\geq
C\tfrac{\sqrt{a}}{t^{1/3}N^{1/3}}t^{4/3}N^{1/3}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,\mathcal{L}_{N}}-\log Z_{0,N}\geq Ct^{4/3}N^{1/3}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,\mathcal{L}_{N}}-2N{f_{d}}\geq\tfrac{C}{2}t^{4/3}N^{1/3}\Big{)}$
$\displaystyle\qquad\qquad\qquad+\mathbb{P}\Big{(}\log
Z_{0,N}-2N{f_{d}}\leq-\tfrac{C}{2}t^{4/3}N^{1/3}\Big{)}\leq e^{-Ct^{2}},$

where the last inequality comes from Proposition A.2 and Proposition 3.8.

From now on, we will assume that the integer $a$ satisfies $1\leq a\leq
t^{2/3}N^{2/3}$. In addition, note that our estimate for the difference of two
free energies does not change if we included the weight $Y_{(0,0)}$ in both
partition functions. For the remaining part of the proof, we will also assume
this without introducing a new notation for this version of the partition
function.

By a union bound, it suffices to prove our estimate for

$\mathbb{P}\Big{(}\log Z^{\textup{max}}_{0,\mathcal{L}^{a,+}_{N}}-\log
Z_{0,N}\geq C^{\prime}t\sqrt{a}\Big{)}$ (4.1)

where $\mathcal{L}^{a,+}_{N}$ is part of $\mathcal{L}^{a}_{N}$ above $(N,N)$.
For any fixed $k=0,\dots,a$, let us rewrite

$\log Z_{0,(N-k,N+k)}-\log Z_{0,N}=\sum_{i=1}^{k}\log Z_{0,(N-i,N+i)}-\log
Z_{0,(N-(i-1),N+(i-1))}=S_{k}.$

This allows us to work with a running maximum of the walk $S_{k}$ since

$\eqref{goal1}=\mathbb{P}\Big{(}\max_{0\leq k\leq a}S_{k}\geq
C^{\prime}t\sqrt{a}\Big{)}.$ (4.2)

The steps of $S_{k}$ are not i.i.d., however, Theorem 3.28 allows us to work
with an i.i.d. random walk $\mathaccent 869{S}_{k}$ which upper bounds $S_{k}$
with high probability. More precisely, the down-right path $\Theta_{2a}$ will
be the staircase from $(N-a,N+a)$ to $(N,N)$. Because the steps of $S_{k}$ and
the free energy profile defined in (3.7) differ by a negative sign, the
perturbed parameter will be $\eta=\mu/2-q_{0}t^{2/3}N^{-1/3}$, and the
distribution of the steps of $\mathaccent 869{S}_{k}$ is given by
$\log(\textup{Ga}^{-1}(\eta))-\log(\textup{Ga}^{-1}(\mu-\eta))$.

Let $A$ denote the event that $\log\tfrac{10}{9}+\mathaccent 869{S}_{k}\geq
S_{k}$ for each $k=0,1\dots,a$. Then, we have

$\displaystyle\eqref{max_walk}$
$\displaystyle\leq\mathbb{P}\Big{(}\Big{\\{}\max_{0\leq k\leq a}S_{k}\geq
C^{\prime}\sqrt{a}t^{3/4}\Big{\\}}\cap A\Big{)}+\mathbb{P}(A^{c})$
$\displaystyle\leq\mathbb{P}\Big{(}\Big{\\{}\log\tfrac{10}{9}+\max_{0\leq
k\leq a}\mathaccent 869{S}_{k}\geq
C^{\prime}\sqrt{a}t^{3/4}\Big{\\}}\Big{)}+\mathbb{P}(A^{c}).$

From Theorem 3.28, we know $\mathbb{P}(A^{c})\leq e^{-Ct^{2}}$. Absorb the
constant $\log({10}/{9})$ into the constant $C$, and it suffices to obtain the
upper bound

$\mathbb{P}\Big{(}\max_{0\leq k\leq a}\mathaccent 869{S}_{k}\geq
C^{\prime}t\sqrt{a}\Big{)}\leq e^{-C\min\\{t^{2},t\sqrt{a}\\}}.$ (4.3)

This is a standard running maximum estimate for an i.i.d. random walk whose
steps are sub-exponential. We omit the details here and postpone the proof of
(4.3) to the end of Appendix D.

∎

Next, we extend the value of $t$ in the previous proposition from $t_{0}\leq
t\leq c_{0}N^{1/2}$ to all $t\geq t_{0}$. The cost of this is a non-optimal
exponent appearing in the exponential bound.

######  Proposition 4.3.

There exist positive constants $t_{0},N_{0}$ such that for each $N\geq N_{0}$,
$t\geq t_{0}$, and each $a\in\mathbb{Z}_{\geq 0}$, we have

$\mathbb{P}\Big{(}\log Z_{0,\mathcal{L}^{a}_{N}}\ -\log Z_{0,N}\geq
t\sqrt{a}\Big{)}\leq e^{-t^{1/10}}.$

###### Proof.

The case $a=0$ is trivial, so we will always assume $a\geq 1$. Due to
Proposition 4.1, we only have to show the estimate when
$t\geq{C_{\ref*{compare_max_c1}}}c_{0}N^{1/2}$ where both constants
${C_{\ref*{compare_max_c1}}}$ and $c_{0}$ are from Proposition 4.1. Suppose
$t=zN^{1/2}$ where $z\geq{C_{\ref*{compare_max_c1}}}c_{0}$. Then,

$\displaystyle\mathbb{P}\Big{(}\log Z_{0,\mathcal{L}^{a}_{N}}\ -\log
Z_{0,N}\geq t\sqrt{a}\Big{)}\leq\mathbb{P}\Big{(}\log
Z_{0,\mathcal{L}^{a}_{N}}\ -\log Z_{0,N}\geq t\Big{)}$
$\displaystyle\quad=\mathbb{P}\Big{(}\log Z_{0,\mathcal{L}^{a}_{N}}\ -\log
Z_{0,N}\geq(zN^{1/6})N^{1/3}\Big{)}$
$\displaystyle\quad\leq\mathbb{P}\Big{(}\log\mathbb{Z}_{0,\mathcal{L}^{a}_{N}}\
-2Nf_{d}\geq(\tfrac{1}{2}zN^{1/6})N^{1/3}\Big{)}+\mathbb{P}\Big{(}\log
Z_{0,N}-2Nf_{d}\leq-(\tfrac{1}{2}zN^{1/6})N^{1/3}\Big{)}$
$\displaystyle\quad\leq e^{-t^{1/10}}.$

The last inequality comes from Proposition A.2 and Proposition 3.8. ∎

Fix $0\leq r\leq N/2$. Recall that $\mathcal{L}_{r}$ is the anti-diagonal
through the point $(r,r)$. Let ${\bf p}_{*}$ denote the random maximizer in

$\max_{{\bf p}\in\mathcal{L}_{r}}\big{\\{}\log Z_{0,{\bf p}}+\log Z_{{\bf
p},N}\big{\\}}=\log Z_{0,{\bf p}_{*}}+\log Z_{{\bf p}_{*},N}.$

The proposition below captures the KPZ transversal fluctuation which says that
the maximizer ${\bf p}_{*}$ cannot be too far from the diagonal on the local
scale $r^{2/3}$. This can of course be much smaller than the global
fluctuation scale $N^{2/3}$. This result was first proved in the zero-
temperature model in [12].

######  Proposition 4.4.

There exist positive constants $C_{41},c_{0},t_{0},N_{0}$ such that for each
$N\geq N_{0}$, $c_{0}\leq r\leq N/2$ and $t\geq t_{0}$, we have

$\mathbb{P}(|{\bf p}_{*}-(r,r)|_{\infty}>tr^{2/3})\leq
e^{-C_{\ref*{fluc_bound_l_c1}}{t}^{3}}.$

###### Proof.

Abbreviate $J^{h}=\mathcal{L}^{r^{2/3}}_{(r-2hr^{2/3},r+2hr^{2/3})}$. We bound
the probability as follows.

$\displaystyle\mathbb{P}(|{\bf p}_{*}-(r,r)|_{\infty}>tr^{2/3})$
$\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}\setminus\mathcal{L}_{r}^{tr^{2/3}}}\Big{\\{}\log
Z_{0,{\bf p}}+\log Z_{{\bf p},N}\Big{\\}}>\log Z_{0,r}+\log Z_{r,N}\Big{)}$
$\displaystyle\leq\sum_{|h|=\lfloor t/2\rfloor}^{r^{1/3}}\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,J^{h}}+\log Z^{\textup{max}}_{J^{h},N}>\log Z_{0,r}+\log
Z_{r,N}\Big{)}$ $\displaystyle=\sum_{|h|=\lfloor
t/2\rfloor}^{r^{1/3}}\mathbb{P}\Big{(}\Big{[}\log
Z^{\textup{max}}_{0,J^{h}}-\log Z_{0,r}\Big{]}+\Big{[}\log
Z^{\textup{max}}_{J^{h},N}-\log Z_{r,N}\Big{]}>0\Big{)}$
$\displaystyle=\sum_{|h|=\lfloor t/2\rfloor}^{r^{1/3}}\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,J^{h}}-\log Z_{0,r}\geq-Dh^{2}r^{1/3}\Big{)}$ (4.4)
$\displaystyle\qquad\qquad\qquad\qquad+\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{J^{h},N}-\log Z_{r,N}\geq Dh^{2}r^{1/3}\Big{)}.$ (4.5)

where $D$ is a small positive constant that we will fix.

For (4.4), provided $t_{0}$ is fixed sufficiently large, we may upper bound
(4.4) using Proposition 3.11 and Proposition 3.8 as the following

$\displaystyle\eqref{close_max}$ $\displaystyle=\mathbb{P}\Big{(}[\log
Z^{\textup{max}}_{0,J^{h}}-2r{f_{d}}]-[\log Z_{0,r}-2r{f_{d}}]\geq-
Dh^{2}r^{1/3}\Big{)}$ $\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,J^{h}}-2r{f_{d}}\geq-2Dh^{2}r^{1/3}\Big{)}+\mathbb{P}\Big{(}\log
Z_{0,r}-2r{f_{d}}\leq-Dh^{2}r^{1/3}\Big{)}\leq e^{-C|h|^{3}}$

provided $D\leq\tfrac{1}{10}C_{\ref*{trans_fluc_loss_c1}}$ from Proposition
3.11.

For (4.5), we will split the value of $r$ into two cases, whether
$r\leq\epsilon_{0}(N-r)$ or $r\geq\epsilon_{0}(N-r)$, for $\epsilon_{0}$ which
we will fix below (between (4.7) and (4.8)). When $r\leq\epsilon_{0}(N-r)$, we
upper bound (4.5) by

$\eqref{far_max}\leq\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{\mathcal{L}_{r}^{4|h|r^{2/3}},N}-\log Z_{r,N}\geq
Dh^{2}r^{1/3}\Big{)},$ (4.6)

and we would like to apply Proposition 4.1. From there, We let $a=4|h|r^{2/3}$
and $t=8|h|^{3/2}\epsilon_{0}^{3/2}$. Then, continue from (4.6),

$\displaystyle\eqref{bound_small_r}$ $\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{\mathcal{L}_{r}^{4|h|r^{2/3}},N}-\log
Z_{r,N}\geq\tfrac{1}{10}D|h|^{3/2}\sqrt{4|h|r^{2/3}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{\mathcal{L}_{r}^{a},N}-\log
Z_{r,N}\geq\tfrac{\frac{1}{10}D}{8\epsilon_{0}^{3/2}}t\sqrt{a}\Big{)}.$ (4.7)

Next, we fix $\epsilon_{0}$ sufficiently small so that $t\leq\mathaccent
869{c}_{0}r^{1/2}$, where $\mathaccent 869{c}_{0}$ is the constant $c_{0}$
from Proposition 4.1. Then, we lower the value of $D$ to get

$\tfrac{\frac{1}{10}D}{8\epsilon_{0}^{3/2}}\leq C_{\ref*{compare_max_c1}}$
(4.8)

where $C_{\ref*{compare_max_c1}}$ is the constant appearing in Proposition
4.1. Finally, by Proposition 4.1, the above probability in (4.7) will always
be bounded by $e^{-Ct^{2}}=e^{-C|h|^{3}}$.

On the other hand, when $r\geq\epsilon_{0}(N-r)$, if the maximizer ${\bf
p}_{*}$ is located more than $|h|r^{2/3}$ away from the diagonal, it means it
is more than $(\epsilon_{0}/2)^{2/3}|h|(N-r)^{2/3}$ away from the diagonal,
which is on the full distance scale $N-r$. Provided that $t_{0}$ is fixed
sufficiently large depending on $\epsilon_{0}$, (4.5) can be upper bounded
with a similar argument as in (4.4).

To summarize, the arguments above show that

$\sum_{|h|=\lfloor
t/2\rfloor}^{r^{1/3}}\eqref{close_max}+\eqref{far_max}\leq\sum_{|h|=\lfloor
t/2\rfloor}^{\infty}e^{-C|h|^{3}}\leq e^{-Ct^{3}},$

with this, we have finished the proof of this proposition. ∎

By symmetry, similar results also hold for the case when $N/2\leq r\leq N$. We
record this in the following proposition. Let ${\bf p}^{*}$ denote the random
maximizer of

$\max_{{\bf p}\in\mathcal{L}_{r}}\\{\log Z_{0,{\bf p}}+\log Z_{{\bf p},N}\\}.$

######  Proposition 4.5.

There exist positive constants $C_{42},c_{0},N_{0},t_{0}$ such that for each
$N\geq N_{0}$, $N/2\leq r\leq N-c_{0}$ and $t\geq t_{0}$, we have

$\mathbb{P}(|{\bf p}^{*}-(r,r)|_{\infty}>t(N-r)^{2/3})\leq
e^{-C_{\ref*{fluc_bound_u_c1}}{t}^{3}}.$

The next estimate quantifies the effect of letting the crossing point of the
path on an antidiagonal fluctuate in the KPZ scale versus forcing the path to
go through a fixed point.

######  Proposition 4.6.

There exist positive constants $c_{0},t_{0},N_{0},\epsilon_{0}$ such that for
each $N\geq N_{0}$, $N/2\leq r\leq N-c_{0}$, $t\geq t_{0}$, we have

$\displaystyle\mathbb{P}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}^{t(N-r)^{2/3}}}\Big{\\{}\log Z_{0,{\bf p}}+\log Z_{{\bf
p},N}\Big{\\}}-\Big{[}\log Z_{0,r}+\log Z_{r,N}\Big{]}\geq
t(N-r)^{1/3}\Big{)}\leq e^{-t^{1/10}}.$

###### Proof.

Let us start by rewriting

$\displaystyle\mathbb{P}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}^{t(N-r)^{2/3}}}\Big{\\{}\log Z_{0,{\bf p}}+\log Z_{{\bf
p},N}\Big{\\}}-\Big{[}\log Z_{0,r}+\log Z_{r,N}\Big{]}\geq
t(N-r)^{1/3}\Big{)}$ $\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,\mathcal{L}_{r}^{t(N-r)^{2/3}}}-\log
Z_{0,r}\geq\tfrac{1}{2}t(N-r)^{1/3}\Big{)}$ (4.9)
$\displaystyle\qquad\qquad+\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{\mathcal{L}_{r}^{t(N-r)^{2/3}},N}-\log
Z_{r,N}\geq\tfrac{1}{2}t(N-r)^{1/3}\Big{)}.$ (4.10)

Both probabilities (4.9) and (4.10) can be upper bounded by $e^{-Ct^{1/10}}$
using Proposition 4.3. ∎

We combine the previous propositions into the following statement.

######  Proposition 4.7.

There exist positive constants $c_{0},t_{0},N_{0},\epsilon_{0}$ such that for
each $N\geq N_{0}$, $N/2\leq r\leq N-c_{0}$, $t\geq t_{0}$, we have

$\displaystyle\mathbb{P}\Big{(}\max_{{\bf p}\in\mathcal{L}_{r}}\Big{\\{}\log
Z_{0,{\bf p}}+\log Z_{{\bf p},N}\Big{\\}}-\Big{[}\log Z_{0,r}+\log
Z_{r,N}\Big{]}\geq t(N-r)^{1/3}\Big{)}\leq e^{-t^{1/10}}.$

###### Proof.

By a union bound, we split the above maximum over ${\bf p}\in\mathcal{L}_{r}$
to ${\bf p}\not\in\mathcal{L}_{r}^{t(N-r)^{2/3}}$ and ${\bf
p}\in\mathcal{L}_{r}^{t(N-r)^{2/3}}$. Propositions 4.5 and 4.6 show that in
both cases the probability can be upper bounded by $e^{-Ct^{1/10}}$. ∎

The development culminates in the following theorem.

######  Theorem 4.8.

There exist positive constants $c_{0},t_{0},N_{0},$ such that for each $N\geq
N_{0}$, $N/2\leq r\leq N-c_{0}$, $t\geq t_{0}$, we have

$\displaystyle\mathbb{P}\Big{(}\log Z_{0,N}-[\log Z_{0,r}+\log Z_{r,N}]\geq
t(N-r)^{1/3}\Big{)}\leq e^{-t^{1/10}}.$

###### Proof.

This follows directly from the fact that

$\log Z_{0,N}\leq\max_{{\bf p}\in\mathcal{L}_{r}}\\{\log Z_{0,{\bf p}}+\log
Z_{{\bf p},N}\\}+2\log(N-r)$

and Proposition 4.7. ∎

##  5 Proof of Theorem 2.1

This section proves Theorem 2.1. Throughout, $c_{0}\leq N-r\leq N/2$ is
assumed. First observe,

$\displaystyle\textup{$\mathbb{V}$ar}(U-V)$
$\displaystyle\geq\inf_{\lambda\in\mathbb{R}}\textup{$\mathbb{V}$ar}(U-\lambda
V)=(1-\textup{$\mathbb{C}$orr}^{2}(U,V))\textup{$\mathbb{V}$ar}(U)$ (5.1)
$\displaystyle=(1-\textup{$\mathbb{C}$orr}(U,V))(1+\textup{$\mathbb{C}$orr}(U,V))\textup{$\mathbb{V}$ar}(U).$

Apply this to bound $1-\textup{$\mathbb{C}$orr}(U,V)$ for $U=\log Z_{0,N}$ and
$V=\log Z_{0,r}$. By the FKG inequality, $\textup{$\mathbb{C}$orr}(\log
Z_{0,N},\log Z_{0,r})\in[0,1]$. (5.1) gives

$\frac{\inf_{\lambda\in\mathbb{R}}\textup{$\mathbb{V}$ar}(\log
Z_{0,N}-\lambda\log Z_{0,r})}{2\textup{$\mathbb{V}$ar}(\log Z_{0,N})}\leq
1-\textup{$\mathbb{C}$orr}(\log Z_{0,N},\log
Z_{0,r})\leq\frac{\textup{$\mathbb{V}$ar}(\log Z_{0,N}-\log
Z_{0,r})}{\textup{$\mathbb{V}$ar}(\log Z_{0,N})}.$ (5.2)

Since Theorem 3.21 gives $\textup{$\mathbb{V}$ar}(\log Z_{0,N})\geq CN^{2/3}$,
the lower bound of Theorem 2.1 follows from the second inequality of (5.2) and

$\textup{$\mathbb{V}$ar}(\log Z_{0,N}-\log Z_{0,r})\leq C(N-r)^{2/3}.$ (5.3)

To show (5.3), apply the inequality $\textup{$\mathbb{V}$ar}(A)\leq
2(\textup{$\mathbb{V}$ar}(B)+\mathbb{E}[(A-B)^{2}])$ to $A=\log Z_{0,N}-\log
Z_{0,r}$ and $B=\log Z_{r,N}$. $\textup{$\mathbb{V}$ar}(B)\leq C(N-r)^{2/3}$
follows from Theorem 3.21, and $\mathbb{E}[(A-B)^{2}]\leq C(N-r)^{2/3}$
follows from Theorem 4.8. Proof of the lower bound of Theorem 2.1 is complete.

We turn to prove the upper bound of Theorem 2.1, by bounding a conditional
variance. Recall that $[\\![0,(N,N)]\\!]$ is the square with lower left corner
at $(0,0)$ and upper right corner at $(N,N)$. Let $\mathcal{F}$ be the
$\sigma$-algebra of the weights in $[\\![0,(N,N)]\\!]$ that lie on or below
the anti-diagonal line $\mathcal{L}_{r}$. Note that $\log Z_{0,r}$ is
$\mathcal{F}$-measurable.

$\displaystyle\textup{$\mathbb{V}$ar}(\log Z_{0,N}|\mathcal{F})$
$\displaystyle=\textup{$\mathbb{V}$ar}(\log Z_{0,N}-\log Z_{0,r}|\mathcal{F})$
$\displaystyle=\mathbb{E}\Big{[}\Big{(}\log Z_{0,N}-\log
Z_{0,r}-\mathbb{E}[\log Z_{0,N}-\log
Z_{0,r}|\mathcal{F}]\Big{)}^{2}\Big{|}\mathcal{F}\Big{]}.$ (5.4)

We develop a lower bound for the last conditional expectation above.

By Theorem 4.8,

$\Big{|}\mathbb{E}[\log Z_{0,N}-\log Z_{0,r}]-\mathbb{E}[\log
Z_{r,N}]\Big{|}\leq C(N-r)^{1/3}.$ (5.5)

In Proposition 3.7 the centering $2N{f_{d}}$ can be replaced with
$\mathbb{E}[\log Z_{0,N}]$ because $\mathbb{E}[\log Z_{0,N}]\leq 2N{f_{d}}$ by
superadditivity. Thus altered, Proposition 3.7 and (5.5) give

$\displaystyle e^{-C_{\ref*{up_lb_c1}}t^{3/2}}$
$\displaystyle\leq\mathbb{P}\bigl{(}\log Z_{r,N}-\mathbb{E}[\log Z_{r,N}]\geq
t(N-r)^{1/3}\bigr{)}$ $\displaystyle\leq\mathbb{P}\bigl{(}\log
Z_{r,N}-\mathbb{E}[\log Z_{0,N}-\log Z_{0,r}]\geq(t-C)(N-r)^{1/3}\bigr{)}.$

Let $s_{0}$ be a large constant and define the event

$A_{r,N}=\bigl{\\{}\log Z_{r,N}-\mathbb{E}[\log Z_{0,N}-\log Z_{0,r}]\geq
s_{0}(N-r)^{1/3}\bigr{\\}}.$ (5.6)

$A_{r,N}$ is independent of $\mathcal{F}$ and $\mathbb{P}(A_{r,N})$ is bounded
below independently of $r$ and $N$.

Next, using Chebyshev’s inequality we get

$\displaystyle\mathbb{P}\Big{(}\Big{|}\mathbb{E}[\log Z_{0,N}-\log
Z_{0,r}|\mathcal{F}]-\mathbb{E}[\log Z_{0,N}-\log
Z_{0,r}]\Big{|}>t(N-r)^{1/3}\Big{)}$
$\displaystyle\leq\frac{\textup{$\mathbb{V}$ar}(\mathbb{E}[\log Z_{0,N}-\log
Z_{0,r}|\mathcal{F}])}{t^{2}(N-r)^{2/3}}$
$\displaystyle\leq\frac{\textup{$\mathbb{V}$ar}(\log Z_{0,N}-\log
Z_{0,r})}{t^{2}(N-r)^{2/3}}\leq C/t^{2}\qquad\textup{by \eqref{nest_var}}.$

By choosing $t$ and $s_{0}$ large enough, there is an event
$B_{r,N}\in\mathcal{F}$, with positive probability bounded below independently
of $N$ and $r$, on which

$\Big{|}\mathbb{E}[\log Z_{0,N}-\log Z_{0,r}|\mathcal{F}]-\mathbb{E}[\log
Z_{0,N}-\log Z_{0,r}]\Big{|}\leq\frac{s_{0}}{10}(N-r)^{1/3}.$ (5.7)

On $A_{r,N}\cap B_{r,N}$ we have the following bound, using first
superadditivity $\log Z_{0,N}-\log Z_{0,r}\geq\log Z_{r,N}$, then (5.7) and
last (5.6):

$\displaystyle\log Z_{0,N}-\log Z_{0,r}-\mathbb{E}[\log Z_{0,N}-\log
Z_{0,r}|\mathcal{F}]$ $\displaystyle\geq\log Z_{r,N}-\mathbb{E}[\log
Z_{0,N}-\log
Z_{0,r}]-\frac{s_{0}}{10}(N-r)^{1/3}\geq\frac{9s_{0}}{10}(N-r)^{1/3}.$

Square this bound and insert it inside the conditional expectation on line
(5.4). Continuing from that line, we then have

$\displaystyle\textup{$\mathbb{V}$ar}(\log Z_{0,N}|\mathcal{F})\geq
C(N-r)^{2/3}\,\mathbb{E}[\mathbf{1}_{A_{r,N}}\mathbf{1}_{B_{r,N}}|\mathcal{F}]\geq
C(N-r)^{2/3}\,\mathbf{1}_{B_{r,N}}.$

By the law of total variance, for all $\lambda\in\mathbb{R}$,

$\displaystyle\textup{$\mathbb{V}$ar}(\log Z_{0,N}-\lambda\log Z_{0,r})$
$\displaystyle=\mathbb{E}\big{[}\textup{$\mathbb{V}$ar}(\log
Z_{0,N}-\lambda\log
Z_{0,r}|\mathcal{F})\big{]}+\textup{$\mathbb{V}$ar}\big{[}\mathbb{E}(\log
Z_{0,N}-\lambda\log Z_{0,r}|\mathcal{F})\big{]}$
$\displaystyle\geq\mathbb{E}\big{[}\textup{$\mathbb{V}$ar}(\log
Z_{0,N}-\lambda\log Z_{0,r}|\mathcal{F})\big{]}$
$\displaystyle=\mathbb{E}\big{[}\textup{$\mathbb{V}$ar}(\log
Z_{0,N}|\mathcal{F})\big{]}\geq C(N-r)^{2/3}\,\mathbb{P}(B_{r,N})\geq
C(N-r)^{2/3}.$

Apply this lower bound to the numerator of the first member of (5.2) and apply
Theorem 3.21 to the denominator. The upper bound of Theorem 2.1 has been
established.

##  6 Proof of Theorem 2.2

We assume throughout that $c_{0}\leq r\leq N/2$. First, we prove the upper
bound. By the Cauchy-Schwarz inequality and the independence of $Z_{0,r}$ and
$Z_{r,N}$,

$\displaystyle\textup{$\mathbb{C}$ov}(\log Z_{0,r},\log Z_{0,N})$
$\displaystyle=\textup{$\mathbb{C}$ov}(\log Z_{0,r},\log Z_{0,N}-\log
Z_{r,N})$ $\displaystyle\leq\textup{$\mathbb{V}$ar}(\log
Z_{0,r})^{1/2}\cdot\textup{$\mathbb{V}$ar}(\log Z_{0,N}-\log Z_{r,N})^{1/2}.$

It therefore suffices to show that both variances above have upper bounds of
the order $r^{2/3}$. The first variance satisfies
$\textup{$\mathbb{V}$ar}(\log Z_{0,r})\leq Cr^{2/3}$ by Theorem 3.21. The
second variance can be bounded again using the inequality
$\textup{$\mathbb{V}$ar}(A)\leq
2(\textup{$\mathbb{V}$ar}(B)+\mathbb{E}[(A-B)^{2}])$ with $A=\log Z_{0,N}-\log
Z_{r,N}$ and $B=\log Z_{0,r}$. $\textup{$\mathbb{V}$ar}(B)\leq Cr^{2/3}$
follows from Theorem 3.21, and $\mathbb{E}[(A-B)^{2}]\leq Cr^{2/3}$ from
Proposition 4.8 with the parameters $r$ and $N-r$ swapped. This finishes the
proof of the upper bound. The remainder of this section is dedicated to the
lower bound of Theorem 2.2.

Our approach follows ideas from [9, 11] that we now describe. For $\theta>0$,
let $\mathcal{F}_{\theta}$ denote the $\sigma$-algebra generated by the
weights in the set $[\\![(0,0),(N,N)]\\!]\setminus R^{\theta r^{2/3}}_{0,r}$.
In Section 6.6, we will show that there exists an event
$\mathcal{E}_{\theta}\in\mathcal{F}_{\theta}$ with
$\mathbb{P}(\mathcal{E}_{\theta})\geq\epsilon_{0}>0$ ($\epsilon_{0}$
independent of $r$ and $N$) such that

$\textup{$\mathbb{C}$ov}(\log Z_{0,N},\log
Z_{0,r}|\mathcal{F}_{\theta})(\omega)\geq Cr^{2/3}\quad\textup{ for
}\omega\in\mathcal{E}_{\theta}.$ (6.1)

Then, by (6.1) and applying the FKG inequality twice, we have

$\displaystyle\mathbb{E}[\log Z_{0,N}\log Z_{0,r}]$
$\displaystyle=\mathbb{E}\Big{[}\mathbb{E}[\log Z_{0,N}\log
Z_{0,r}|\mathcal{F}_{\theta}]\Big{]}$
$\displaystyle=\int_{\mathcal{E}_{\theta}}\mathbb{E}[\log Z_{0,N}\log
Z_{0,r}|\mathcal{F}_{\theta}]\,d\mathbb{P}+\int_{\mathcal{E}_{\theta}^{c}}\mathbb{E}[\log
Z_{0,N}\log Z_{0,r}|\mathcal{F}_{\theta}]\,d\mathbb{P}$
$\displaystyle\geq\int_{\mathcal{E}_{\theta}}\mathbb{E}[\log
Z_{0,N}|\mathcal{F}_{\theta}]\mathbb{E}[\log
Z_{0,r}|\mathcal{F}_{\theta}]\,d\mathbb{P}+C\epsilon_{0}r^{2/3}+\int_{\mathcal{E}_{\theta}^{c}}\mathbb{E}[\log
Z_{0,N}|\mathcal{F}_{\theta}]\mathbb{E}[\log
Z_{0,r}|\mathcal{F}_{\theta}]\,d\mathbb{P}$
$\displaystyle=\mathbb{E}\Big{[}\mathbb{E}[\log
Z_{0,N}|\mathcal{F}_{\theta}]\mathbb{E}[\log
Z_{0,r}|\mathcal{F}_{\theta}]\Big{]}+C\epsilon_{0}r^{2/3}$
$\displaystyle\geq\mathbb{E}[\log Z_{0,N}]\mathbb{E}[\log
Z_{0,r}]+C\epsilon_{0}r^{2/3}.$

This shows $\textup{$\mathbb{C}$ov}(\log Z_{0,N},\log Z_{0,r})\geq Cr^{2/3}$,
hence the lower bound in our theorem. In the next few sections, we prove
(6.1).

###  6.1 Barrier event $\mathcal{B}_{\textup{bar}}$

$\phi_{2}r^{2/3}$$U_{1}$$U_{2}$$\theta
r^{2/3}$${U_{2}^{k}}$$(0,0)$$(r,r)$$\overline{U_{2}^{k}}$$\underline{U_{2}^{k}}$$r/L^{30}$$r^{2/3}/L^{20}$$v$$u$$R(u,v)$$\mathcal{L}(u)$
Figure 6.1: Left: A demonstration of the region used to define the barrier
event $\mathcal{B}_{\textup{bar}}$ in (6.2). Right: construction used in the
event $A$ in (6.4).

This section defines a barrier event
$\mathcal{B}_{\textup{bar}}\in\mathcal{F}_{\theta}$ and investigates
consequences of conditioning on it. Fix the parameters
$0<\theta<1/2,\phi_{1}=\theta^{-10},\phi_{2}=\theta^{-100}$, and
$L=\theta^{-1000}.$ We have the freedom to decrease $\theta$ if necessary, so
that $0<\theta\leq\theta_{0}$.

Next, we define a barrier event around the rectangle $R^{\theta
r^{2/3}}_{0,r}$. This part of the construction is illustrated on the left of
Figure 6.1. The region $R^{\phi_{2}r^{2/3}}_{0,r}\setminus R^{\theta
r^{2/3}}_{0,r}$ is formed by two disjoint rectangles, $U_{1}$ above the
diagonal and $U_{2}$ below the diagonal.

The anti-diagonal lines $\\{\mathcal{L}_{kr/L}\\}_{k=1}^{L-1}$ cut each of
$U_{1}$ and $U_{2}$ into $L$ small rectangles. If we fix $r$ sufficiently
large depending on $L$, these rectangles are not degenerate. Denote these
small rectangles by $U^{k}_{i}$ for $i=1,2$ and $k=1,\dots,L$. Let
$\underline{U}_{i}^{k}$ and $\overline{U}_{i}^{k}$ denote the top and bottom
sides (with slope $-1$) of the rectangle $U_{i}^{k}$. We define the event

$\mathcal{B}_{\textup{bar}}=\bigcap_{i=1}^{2}\bigcap_{k=1}^{L}\Big{\\{}\log
Z^{\textup{in},U_{i}}_{\underline{U_{i}^{k}},\overline{U_{i}^{k}}}-2(r/L){f_{d}}\leq-
Lr^{1/3}\Big{\\}}.$ (6.2)

######  Lemma 6.1.

There exists a positive constant $\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exists a positive constants $c_{0}$ depending
on $\theta$ such that for each $r\geq c_{0}$, we have

$\mathbb{P}(\mathcal{B}_{\textup{bar}})\geq e^{-e^{L^{100}}}.$

###### Proof.

Note $\mathcal{B}_{\textup{bar}}$ is the intersection of $2L$ events, of equal
probability by translation invariance. It therefore suffices to lower bound
the probability

$\mathbb{P}\Big{(}\log
Z^{\textup{in},U_{1}}_{\underline{U_{1}^{1}},\overline{U_{1}^{1}}}-2(r/L){f_{d}}\leq-
Lr^{1/3}\Big{)}.$ (6.3)

The following construction is illustrated on the right of Figure 6.1. Using
diagonal and anti-diagonal lines, we cut the rectangle $U^{1}_{1}$ into
smaller rectangles whose diagonal $\ell^{\infty}$-length is
$\tfrac{r}{L^{30}}$ and anti-diagonal $\ell^{\infty}$-length
$\tfrac{r^{2/3}}{L^{20}}$. Then, the number of rectangles in this grid (see
the right of Figure 6.1) along the diagonal will be $L^{29}$. And the number
of rectangles in the anti-diagonal direction is no more than $L^{21}$. Let us
use $R(u,v)$ to enumerate these small rectangles, where the index $u=1,2,\dots
L^{29}$ records the position along the diagonal direction, and
$v=1,2,\dots,v_{L}\leq L^{21}$ enumerates the small rectangles along the each
anti-diagonal line. Let us also use $\mathcal{L}(u)$ to denote the anti-
diagonal line which contains the upper anti-diagonal side of ${R(u,v)}$, and
let $\underline{R(u,v)}$ to denote the lower anti-diagonal side of ${R(u,v)}$.

Let $D$ be the small constant $C_{\ref*{itl_bound_c1}}$ from Proposition 8.3,
and we define the event

$A=\bigcap_{u,v}\Big{\\{}\log
Z_{\underline{R(u,v)},\mathcal{L}(u)}-2(r/L^{30})f_{d}\leq-D(r/L^{30})^{1/3}\Big{\\}}.$
(6.4)

Using the FKG inequality and Proposition 8.3, we have
$\mathbb{P}(A)\geq{e}^{-e^{L^{99}}}$.

Next, the constrained free energy can be upper bounded by

$\displaystyle\log
Z^{\textup{in},U_{1}}_{\underline{U_{1}^{1}},\overline{U_{1}^{1}}}$
$\displaystyle\leq\sum_{u=1}^{L^{29}}\Big{(}100\log L+\max_{v}\log
Z_{\underline{R(u,v)},\mathcal{L}(u)}\Big{)}.$ $\displaystyle\leq
L^{29}\Big{(}100\log L+\max_{u,v}\log
Z_{\underline{R(u,v)},\mathcal{L}(u)}\Big{)}$ restrict to the event $A$
$\displaystyle\leq L^{29}\Big{(}100\log
L+2(r/L^{30})f_{d}-Dr^{1/3}/L^{10}\Big{)}$ $\displaystyle\leq
2(r/L)f_{d}-DL^{19}r^{1/3}+L^{30}$ $\displaystyle\leq 2(r/L)f_{d}-Lr^{1/3}$

provided that $\theta_{0}$ is sufficiently small (which makes $L$ large) and
then increase $c_{0}$. With this, we have shown

$\eqref{one_term_lb}\geq e^{-e^{L^{100}}}.$

and finished the proof of the lemma.

∎

###  6.2 Concentration of the free energy between $(0,0)$ and
$\mathcal{L}_{r}$

Our goal in this section is to show that when conditioned on
$\mathcal{B}_{\textup{bar}}$ the free energy $\log Z_{0,\mathcal{L}_{r}}$ is
concentrated on paths that go from $(0,0)$ to $\mathcal{L}_{r}^{r^{2/3}}$ and
are contained between the diagonal sides of the rectangle
$\mathcal{R}_{0,r-r/L}^{3\theta r^{2/3}}$. This is stated in Proposition 6.9
at the end of this subsection.

Before stating Proposition 6.9, we define our high probability events. To
start, split the collection of paths from $(0,0)$ to $\mathcal{L}_{r}$ as
follows. First, let

$\displaystyle\mathfrak{A}$ $\displaystyle=\textup{all paths from $(0,0)$ to
$\mathcal{L}_{r}^{\phi_{1}r^{2/3}}$ that stay inside
$R_{0,r}^{\phi_{2}r^{2/3}}$}.$ $\displaystyle\mathfrak{B}$
$\displaystyle=\text{all other paths from $(0,0)$ to $\mathcal{L}_{r}$}.$

Then among $\mathfrak{A}$, let us further split
$\mathfrak{A}=\mathfrak{A}_{1}\cup\mathfrak{A}_{2}\cup\mathfrak{A}_{3}\cup\mathfrak{A}_{4}$
where

$\displaystyle\begin{split}\mathfrak{A}_{1}&=\textup{paths from $(0,0)$ to
$\mathcal{L}_{r}^{r^{2/3}}$ that stay between the diagonal sides of
$R_{0,r-r/L}^{3\theta r^{2/3}}$ and }\\\ &\qquad\qquad\qquad\textup{touch each
$R_{ir/L,(i+1)r/L}^{\theta r^{2/3}}$ for $i=0,1,\dots,L-2$},\\\
\mathfrak{A}_{2}&=\textup{paths that avoid at least one of
$R_{ir/L,(i+1)r/L}^{\theta r^{2/3}}$ completely for $i=0,1,\dots,L-2$}.\\\
\mathfrak{A}_{3}&=\textup{paths that exit from the diagonal sides of
$R_{0,r}^{3\theta r^{2/3}}$ and }\\\ &\qquad\qquad\qquad\textup{intersect
$R_{ir/L,(i+1)r/L}^{\theta r^{2/3}}$ for all $i=0,1,\dots,L-2$}.\\\
\mathfrak{A}_{4}&=\textup{paths from $(0,0)$ to
$\mathcal{L}_{r}^{\phi_{1}r^{2/3}}\setminus\mathcal{L}_{r}^{r^{2/3}}$ that
stay between the diagonal sides of $R_{0,r-r/L}^{3\theta r^{2/3}}$ and }\\\
&\qquad\qquad\qquad\textup{touch each $R_{ir/L,(i+1)r/L}^{\theta r^{2/3}}$ for
$i=0,1,\dots,L-2$},\\\ \end{split}$ (6.5)

And among $\mathfrak{B}$, we write
$\mathfrak{B}=\mathfrak{A}_{5}\cup\mathfrak{A}_{6}$ where

$\displaystyle\mathfrak{A}_{5}$ $\displaystyle=\textup{all paths from $(0,0)$
to $\mathcal{L}_{r}\setminus\mathcal{L}_{r}^{\phi_{1}r^{2/3}}$}$
$\displaystyle\mathfrak{A}_{6}$ $\displaystyle=\text{all paths from $(0,0)$ to
$\mathcal{L}_{r}^{\phi_{1}r^{2/3}}$ that exit $R_{0,r}^{\phi_{2}r^{2/3}}$}$

Let $\log Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{i})$ denote the free energy when
we sum over only the paths inside $\mathfrak{A}_{i}$, and define the following
events:

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=\big{\\{}\log
Z^{\text{in},\theta r^{2/3}}_{0,r}-2r{f_{d}}\geq-\theta^{-5}r^{1/3}\big{\\}},$
$\displaystyle\mathcal{A}_{2}$ $\displaystyle=\big{\\{}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{2})-2r{f_{d}}\leq-\theta^{-900}r^{1/3}\big{\\}}$
$\displaystyle\mathcal{A}_{3}$ $\displaystyle=\big{\\{}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{3})-2r{f_{d}}\leq-\theta^{-900}r^{1/3}\big{\\}}.$
$\displaystyle\mathcal{A}_{4}$ $\displaystyle=\big{\\{}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{4})-2r{f_{d}}\leq-\theta^{-900}r^{1/3}\big{\\}}.$
$\displaystyle\mathcal{A}_{5}$ $\displaystyle=\big{\\{}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{5})-2rf_{d}\leq-\theta^{-10}r^{1/3}\big{\\}}$
$\displaystyle\mathcal{A}_{6}$ $\displaystyle=\big{\\{}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{6})-2rf_{d}\leq-\theta^{-100}r^{1/3}\big{\\}}$

Next, we show that all six events are likely events.

######  Proposition 6.2.

For $i=1,\dots,6$. There exists a positive constant $\theta_{0}$ such that for
each $0<\theta\leq\theta_{0}$, there exists a positive constant $c_{0}$ such
that for each $c_{0}\leq r$,

$\mathbb{P}(\mathcal{A}_{i}|\mathcal{B}_{\textup{bar}})\geq
1-e^{-\theta^{-2}}.$

To prove Proposition 6.2, we will split it into six separated lemmas,
according to $i=1,\dots,6.$

######  Lemma 6.3.

There exists a positive constant $\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exists a positive constant $c_{0}$ such that
for each $c_{0}\leq r$,

$\mathbb{P}(\mathcal{A}_{1}|\mathcal{B}_{\textup{bar}})\geq
1-e^{-\theta^{-2}}.$

###### Proof.

We upper bound $\mathbb{P}(\mathcal{A}_{1}^{c}|\mathcal{B}_{\textup{bar}})$.
By independence and Theorem 3.16,

$\mathbb{P}(\mathcal{A}_{1}^{c}|\mathcal{B}_{\textup{bar}})=\mathbb{P}\Big{(}\log
Z^{\textup{in},{\theta
r^{2/3}}}_{0,r}-2r{f_{d}}\leq-\theta^{-5}r^{1/3}\Big{)}\leq e^{-\theta^{-2}}.$

∎

######  Lemma 6.4.

There exists a positive constant $\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exists a positive constant $c_{0}$ such that
for each $c_{0}\leq r$,

$\mathbb{P}(\mathcal{A}_{2}|\mathcal{B}_{\textup{bar}})\geq
1-e^{-\theta^{-10}}.$

###### Proof.

Let us further rewrite $\mathfrak{A}_{2}$ as a non-disjoint union of paths
$\bigcup_{k=1}^{L-2}\mathfrak{A}_{2}^{k,+}\cup\mathfrak{A}_{2}^{k,-}$ where
$\mathfrak{A}_{2}^{k,+}$ and $\mathfrak{A}_{2}^{k,-}$ are the collections of
paths which avoid $R_{kr/L,(k+1)r/L}^{\theta r^{2/3}}$ by going above or
below. For simplicity of the notation, let

$U_{k}^{+}=\mathcal{L}_{(\frac{kr}{L}-\frac{\phi_{2}+\theta}{2}r^{2/3},\frac{kr}{L}+\frac{\phi_{2}+\theta}{2}r^{2/3})}^{\frac{\phi_{2}-\theta}{2}r^{2/3}}\quad\textup{
and }\quad
U_{k}^{-}=\mathcal{L}_{(\frac{kr}{L}+\frac{\phi_{2}+\theta}{2}r^{2/3},\frac{kr}{L}-\frac{\phi_{2}+\theta}{2}r^{2/3})}^{\frac{\phi_{2}-\theta}{2}r^{2/3}}.$

For each fixed $k\in[\\![1,L-2]\\!]$ and $\Box\in\\{+,-\\}$, we have the upper
bound

$\log Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{2}^{k,\Box})\leq\log
Z_{0,\mathcal{L}_{kr/L}}+\log Z_{U^{\Box}_{k},U^{\Box}_{k+1}}+\log
Z_{\mathcal{L}_{(k+1)r/L},\mathcal{L}_{r}}.$ (6.6)

Since we are conditioned on the event $\mathcal{B}_{\textup{bar}}$, then

$\log Z_{U^{\Box}_{k},U^{\Box}_{k+1}}-2(r/L){f_{d}}\leq-Lr^{1/3}.$ (6.7)

Using the FKG inequality and the interval to line estimate from Theorem 3.15,
we have

$\displaystyle\begin{split}&\mathbb{P}\Big{(}\log
Z_{0,\mathcal{L}_{kr/L}}-2(kr/L){f_{d}}\geq\sqrt{L}r^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}\\\
&\qquad\qquad\qquad\leq\mathbb{P}\Big{(}\log
Z_{0,\mathcal{L}_{kr/L}}-2(kr/L){f_{d}}\geq\sqrt{L}r^{1/3}\Big{)}\leq
e^{-\theta^{-100}}\\\
&\mathbb{P}\Big{(}\log(Z_{\mathcal{L}_{(k+1)r/L},\mathcal{L}_{r}})-2((L-k-1)r/L){f_{d}}\geq\sqrt{L}r^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}\\\
&\qquad\qquad\qquad\leq\mathbb{P}\Big{(}\log
Z_{\mathcal{L}_{(k+1)r/L},\mathcal{L}_{r}}-2((L-k-1)r/L){f_{d}}\geq\sqrt{L}r^{1/3}\Big{)}\leq
e^{-\theta^{-100}}.\end{split}$ (6.8)

From (6.6), (6.7), (LABEL:3_ptl_est), We obtain the following estimate for
each fixed $k=1,2,\dots,L-2$

$\mathbb{P}\Big{(}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{2}^{k,\Box})-2r{f_{d}}\geq-\tfrac{L}{2}r^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}\leq
e^{-\theta^{-90}}.$ (6.9)

Using (6.9), a union bound will give us the desired result from our lemma,

$\displaystyle\mathbb{P}\Big{(}\mathcal{A}_{2}^{c}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\Big{\\{}\log\Big{(}\sum_{k=1}^{L-2}Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{2}^{k,+})+Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{2}^{k,-})\Big{)}-2r{f_{d}}\geq-\tfrac{L}{10}r^{1/3}\Big{\\}}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\Big{\\{}\max_{\begin{subarray}{c}k\in[\\![1,L-2]\\!]\\\
\Box\in\\{+,-\\}\end{subarray}}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{2}^{k,\Box})+2\log
L-2r{f_{d}}\geq-\tfrac{L}{10}r^{1/3}\Big{\\}}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq\sum_{k=1}^{L-2}\sum_{\Box\in\\{+,-\\}}\mathbb{P}\Big{(}\Big{\\{}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{2}^{k,\Box})-2r{f_{d}}\geq-\tfrac{L}{9}r^{1/3}\Big{\\}}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq e^{-\theta^{-10}}$

where the last inequality comes from (6.9) and provided $\theta_{0}$ is
sufficiently small. ∎

$\mathcal{L}_{kr/L}$$\mathcal{L}_{(k+1)r/L}$$(0,0)$$(r,r)$$3\theta
r^{2/3}$$\mathcal{L}_{kr/L}$$\mathcal{L}_{(k+1)r/L}$$(0,0)$$(r,r)$$3\theta
r^{2/3}$$\mathcal{L}_{kr/L}$$\mathcal{L}_{(k+1)r/L}$$(0,0)$$(r,r)$$3\theta
r^{2/3}$ Figure 6.2: An illustration of the paths from the collection
$\mathfrak{A}_{3}^{k,+}$. The path in this collection must intersect the two
gray rectangle shown in the picture. The top two paths crosses the lines
$\mathcal{L}_{kr/L}$ and $\mathcal{L}_{(k+1)r/L}$ at neighboring segments (the
case $|i-j|\leq 1$ from the proof of Lemma 6.5). They must have a high
transversal fluctuation between $\mathcal{L}_{kr/L}$ and
$\mathcal{L}_{(k+1)r/L}$ because they have to reach the gray rectangles. The
bottom picture is a path that crosses the lines $\mathcal{L}_{kr/L}$ and
$\mathcal{L}_{(k+1)r/L}$ at non-neighboring positions (the case $|i-j|\geq
2$). This path has a high transversal fluctuation between $\mathcal{L}_{kr/L}$
and $\mathcal{L}_{(k+1)r/L}$ because of these non-neighboring crossing
positions.

######  Lemma 6.5.

There exists a positive constant $\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exists a positive constant $c_{0}$ such that
for each $c_{0}\leq r$,

$\mathbb{P}\Big{(}\mathcal{A}_{3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}\geq
1-e^{-\theta^{-10}}.$

###### Proof.

As before, let us rewrite $\mathcal{A}_{3}$ as a non-disjoint union of paths
$\bigcup_{k=0}^{L-2}\mathfrak{A}_{3}^{k,+}\cup\mathfrak{A}_{3}^{k,-}$ where
$\mathfrak{A}_{3}^{k,+}$ and $\mathfrak{A}_{3}^{k,-}$ are the collections of
paths which exit from the upper and lower diagonal sides of the rectangle
$R_{kr/L,(k+1)r/L}^{3\theta r^{2/3}}$.

Let us fix $k$ and look at $\mathfrak{A}_{3}^{k,+}$. We will show that all
paths in this collection must have high transversal fluctuations. This fact is
illustrated in Figure 6.2. First, we further break up this collection of paths
by where they cross the lines $\mathcal{L}_{kr/L}$ and
$\mathcal{L}_{(k+1)r/L}$. For $i,j\in\mathbb{Z}_{\geq 0}$, let

$\displaystyle I^{i}_{kr/L}$
$\displaystyle=\mathcal{L}_{(kr/L-(i+\frac{1}{2})\theta
r^{2/3},kr/L+(i+\frac{1}{2})\theta r^{2/3})}^{\frac{1}{2}\theta r^{2/3}}$
$\displaystyle J^{j}_{(k+1)r/L}$
$\displaystyle=\mathcal{L}_{((k+1)r/L-(j+\frac{1}{2})\theta
r^{2/3},(k+1)r/L+(j+\frac{1}{2})\theta r^{2/3})}^{\frac{1}{2}\theta r^{2/3}}.$

Then any path in $\mathfrak{A}_{3}^{k,+}$ must cross $I^{i}_{kr/L}$ and
$J^{j}_{(k+1)r/L}$ for some $i,j\in[\\![0,\phi_{2}\theta^{-1}]\\!]$. Thus we
may rewrite $\mathfrak{A}_{3}^{k,+}$ as a non-disjoint union
$\bigcup_{i,j=0}^{\phi_{2}\theta^{-1}}\mathfrak{A}_{3}^{k,+}(i,j)$ where
$\mathfrak{A}_{3}^{k,+}(i,j)$ is the collection of paths inside
$\mathfrak{A}_{3}^{k,+}$ that goes through $I^{i}_{kr/L}$ and
$J^{j}_{(k+1)r/L}$. Next, we will split the case of $i$ and $j$ into two
cases, when $|i-j|\leq 1$ or otherwise.

By our assumption, the paths inside $\mathfrak{A}_{3}^{k,+}$ must intersect
$R^{\theta r^{2/3}}_{kr/L,(k+1)r/L}$ while also exit the upper diagonal side
of $R^{3\theta r^{2/3}}_{kr/L,(k+1)r/L}$. If $|i-j|\leq 1$, there must be an
unusually large transversal of size at least $\theta
r^{2/3}=({\theta}{L^{2/3}})(r/L)^{2/3}$ for the segment of the path
$\mathfrak{A}_{3}^{k,+}(i,j)$ between $I^{i}_{kr/L}$ and $J^{j}_{(k+1)r/L}$.
We may invoke Theorem 3.13 and obtain that for $|i-j|\leq 1$,

$\mathbb{P}\Big{(}\log
Z_{I^{i}_{kr/L},J^{j}_{(k+1)r/L}}(\mathfrak{A}_{3}^{k,+}(i,j))-2(r/L){f_{d}}\geq-C(\theta
L^{2/3})^{2}(r/L)^{1/3}\Big{)}\leq e^{-\theta^{-100}}$ (6.10)

for some small constant $D$.

Next, when $|i-j|\geq 2$, then there is already a large transversal
fluctuation of size at least $\theta r^{2/3}=({\theta}{L^{2/3}})(r/L)^{2/3}$
between for the segment of the path $\mathfrak{A}_{3}^{k,+}(i,j)$ between
$I^{i}_{kr/L}$ and $J^{j}_{(k+1)r/L}$. By Proposition 3.12, we obtain that for
$|i-j|\geq 2$,

$\mathbb{P}\Big{(}\log
Z_{I^{i}_{kr/L},J^{j}_{(k+1)r/L}}(\mathfrak{A}_{3}^{k,+}(i,j))-2(r/L){f_{d}}\geq-D(\theta
L^{2/3})^{2}(r/L)^{1/3}\Big{)}\leq e^{-\theta^{-100}}$ (6.11)

for some small constant $D$.

By the FKG inequality, (6.10) and (6.11) still hold if we replace the
probability measure $\mathbb{P}$ with
$\mathbb{P}(\boldsymbol{\cdot}|\mathcal{B}_{\textup{bar}})$. Now as before, we
upper bound the value of the free energy of the paths outside the region
between $\mathcal{L}_{kr/L}$ and $\mathcal{L}_{(k+1)r/L}$ by
(LABEL:3_ptl_est). We obtain

$\mathbb{P}\Big{(}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{3}^{k,\Box}(i,j))-2r{f_{d}}\geq-L^{0.99}r^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}\leq
e^{-\theta^{-90}}.$ (6.12)

We have

$\displaystyle\mathbb{P}\Big{(}\mathcal{A}_{3}^{c}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\Big{\\{}\log\Big{(}\sum_{k=0}^{L-2}\sum_{\Box\in\\{+,-\\}}\sum_{i,j=0}^{\phi_{2}\theta^{-1}}Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{3}^{k,\Box}(i,j))\Big{)}-2r{f_{d}}\geq-{L^{0.9}}r^{1/3}\Big{\\}}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\Big{\\{}\max_{\begin{subarray}{c}k\in[\\![0,L-2]\\!]\\\
\Box\in\\{+,-\\}\\\ i,j\in[\\![0,\phi_{2}\theta^{-1}]\\!]\end{subarray}}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{3}^{k,\Box}(i,j))+100\log
L-2r{f_{d}}\geq-L^{0.9}r^{1/3}\Big{\\}}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq\sum_{k=1}^{L-2}\sum_{\Box\in\\{+,-\\}}\sum_{i,j=0}^{\phi_{2}\theta^{-1}}\mathbb{P}\Big{(}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{2}^{k,\Box})-2r{f_{d}}\geq-L^{0.95}r^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq e^{-\theta^{-80}}$

where the last inequality uses (6.12) and provided $\theta_{0}$ is
sufficiently small. ∎

######  Lemma 6.6.

There exists a positive constant $\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exists positive constant $c_{0}$ such that for
each $c_{0}\leq r$,

$\mathbb{P}\Big{(}\mathcal{A}_{4}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}\geq
1-e^{-\theta^{-2}}.$

###### Proof.

This simply because the last part of the paths have a very large transversal
fluctuation,

$\displaystyle\mathbb{P}\Big{(}\log
Z_{0,\mathcal{L}_{r}^{\phi_{1}r^{2/3}}\setminus\mathcal{L}_{r}^{r^{2/3}}}(\mathfrak{A}_{4})-2r{f_{d}}\geq-L^{0.9}r^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\log Z_{0,\mathcal{L}_{r-r/L}}+\log
Z_{\mathcal{L}_{r-r/L}^{3\theta
r^{2/3}},\mathcal{L}_{r}^{\phi_{1}r^{2/3}}\setminus\mathcal{L}_{r}^{r^{2/3}}}-2r{f_{d}}\geq-L^{0.9}r^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\log
Z_{0,\mathcal{L}_{r-r/L}}-2(r-r/L){f_{d}}\geq(L-L^{0.9})r^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
(6.13) $\displaystyle\qquad\qquad\qquad+\mathbb{P}\Big{(}\log
Z_{\mathcal{L}_{r-r/L}^{3\theta
r^{2/3}},\mathcal{L}_{r}^{\phi_{1}r^{2/3}}\setminus\mathcal{L}_{r}^{r^{2/3}}}-2(r/L){f_{d}}\geq-
Lr^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}.$ (6.14)

By the FKG inequality and Theorem 3.15,

$\eqref{longterm}\leq\mathbb{P}\Big{(}\log
Z_{0,\mathcal{L}_{r-r/L}}-2(r-r/L){f_{d}}\geq(L-L^{0.9})r^{1/3}\Big{)}\leq
e^{-L^{0.1}}.$

By the FKG inequality and Proposition 3.12,

$\eqref{shortterm}\leq\mathbb{P}\Big{(}\log Z_{\mathcal{L}_{r-r/L}^{3\theta
r^{2/3}},\mathcal{L}_{r}^{\phi_{1}r^{2/3}}\setminus\mathcal{L}_{r}^{r^{2/3}}}-2(r/L){f_{d}}\geq-
Lr^{1/3}\Big{)}\leq e^{-L^{0.1}}.$

With these, we have finished the proof of this theorem.

∎

######  Lemma 6.7.

There exists a positive constant $\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exists positive constant $c_{0}$ such that for
each $c_{0}\leq r$,

$\mathbb{P}\Big{(}\mathcal{A}_{5}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}\geq
1-e^{-\theta^{-2}}.$

###### Proof.

By the FKG inequality, it suffices to show
$\mathbb{P}(\mathcal{A}_{5}^{c})\leq e^{-\theta^{-2}}.$ Then, this estimate
follows directly from Proposition 3.12.

∎

######  Lemma 6.8.

There exists a positive constant $\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exists positive constant $c_{0}$ such that for
each $c_{0}\leq r$,

$\mathbb{P}\Big{(}\mathcal{A}_{6}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}\geq
1-e^{-\theta^{-2}}.$

###### Proof.

By the FKG inequality, it suffices to show $\mathbb{P}(\mathcal{A}_{6})\leq
e^{-\theta^{-2}}.$ Then, this estimate follows directly from Theorem 3.13.

∎

With these lemmas, we have shown Proposition 6.2. Finally, we have the
following proposition.

######  Proposition 6.9.

On the event $\cap_{i=1}^{6}\mathcal{A}_{i}$, we have

$\log Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{1})\leq\log
Z_{0,\mathcal{L}_{r}}\leq\log Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{1})+\log 6.$

###### Proof.

This follows directly from the definition of our events $\mathcal{A}_{i}$ and
the fact

$\max_{j}\\{\log Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{j})\\}\leq\log
Z_{0,\mathcal{L}_{r}}\leq\max_{j}\\{\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{j})\\}+\log 6$

and $\log Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{1})\geq\log
Z^{\textup{in},{\theta r^{2/3}}}_{0,r}$. ∎

###  6.3 Concentration of the global free energy along $\mathcal{L}_{r}$

Define ${\bf p}^{*}$ to be the maximizer in

$\max_{{\bf p}\in\mathcal{L}_{r}}\bigl{\\{}\log Z_{0,{\bf p}}+\log Z_{{\bf
p},N}\bigr{\\}}.$

Our goal in this section is to show that when conditioned on
$\mathcal{B}_{\textup{bar}}$, with high probability, ${\bf
p}^{*}\in\mathcal{L}_{r}^{r^{2/3}}$. This is stated as Proposition 6.12 at the
end of this subsection.

Again, we start by defining our high probability events,

$\displaystyle\mathcal{E}_{1}$
$\displaystyle=\bigcap_{j=1}^{\phi_{2}}\Big{\\{}\log
Z^{\textup{max}}_{\mathcal{L}_{r}^{jr^{2/3}},N}-\log
Z_{r,N}\leq\theta^{-1}\sqrt{jr^{2/3}}\Big{\\}}$ $\displaystyle\mathcal{E}_{2}$
$\displaystyle=\Big{\\{}\max_{{\bf
p}\in\mathcal{L}_{r}\setminus\mathcal{L}_{r}^{\phi_{1}r^{2/3}}}\log Z_{0,{\bf
p}}+\log Z_{{\bf p},N}\leq\log Z^{\textup{in},{\theta r^{2/3}}}_{0,r}+\log
Z_{r,N}-\theta^{-1}r^{1/3}\Big{\\}}.$

The next two lemmas show that $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are high
probability events.

######  Lemma 6.10.

There exists a positive constant $\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exist positive constants $c_{0},N_{0}$ such
that for each $N\geq N_{0}$, $c_{0}\leq r\leq N/2$, we have

$\mathbb{P}(\mathcal{E}_{1}|\mathcal{B}_{\textup{bar}})\geq
1-e^{-\theta^{-1/100}}.$

###### Proof.

By independence, it suffices to prove this estimate for
$\mathbb{P}(\mathcal{E}_{1}).$ We fix $\theta>0$ small. We will upper bound
$\mathbb{P}(\mathcal{E}_{1}^{c})$ using Proposition 4.3. In our application of
this proposition, the variables $a=\sqrt{jr^{2/3}}$ and $t=\theta^{-1}$. By
Proposition 4.3,

$\mathbb{P}(\mathcal{E}_{1}^{c})\leq\sum_{j=1}^{\phi_{2}}\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{\mathcal{L}_{r}^{jr^{2/3}},N}-\log
Z_{r,N}\geq\theta^{-1}\sqrt{jr^{2/3}}\Big{)}\leq\sum_{j=1}^{\phi_{2}}e^{-\theta^{-1/50}}\leq
e^{-\theta^{-1/100}}.$

∎

######  Lemma 6.11.

There exists a positive constant $\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exist positive constants $c_{0},N_{0}$ such
that for each $N\geq N_{0}$, $c_{0}\leq r\leq N/2$, we have

$\mathbb{P}(\mathcal{E}_{2}|\mathcal{B}_{\textup{bar}})\geq
1-e^{-\theta^{-2}}.$

###### Proof.

By the FKG inequality, it suffices to show $\mathbb{P}(\mathcal{E}_{2})\geq
1-e^{-\theta^{-2}}.$ To do this, we upper bound
$\mathbb{P}(\mathcal{E}_{2}^{c})$. For simplicity, let us denote
$\mathcal{L}^{r^{2/3}}_{(r-2hr^{2/3},r+2hr^{2/3})}$ simply as $J^{h}$.

$\displaystyle\mathbb{P}(\mathcal{E}_{2}^{c})$
$\displaystyle\leq\sum_{|h|=\phi_{1}/2}^{r^{1/3}}\mathbb{P}\Big{(}\max_{{\bf
p}\in J^{h}}\log Z_{0,{\bf p}}+\log Z_{{\bf p},N}\geq\log
Z^{\textup{in},{\theta r^{2/3}}}_{0,r}+\log Z_{r,N}-\theta^{-1}r^{1/3}\Big{)}$
$\displaystyle\leq\sum_{|h|=\phi_{1}/2}^{r^{1/3}}\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,J^{h}}+\log Z^{\textup{max}}_{J^{h},N}-\log
Z^{\textup{in},{\theta r^{2/3}}}_{0,r}-\log
Z_{r,N}\geq-\theta^{-1}r^{1/3}\Big{)}$
$\displaystyle\leq\sum_{|h|=\phi_{1}/2}^{r^{1/3}}\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,J^{h}}-\log Z^{\textup{in},{\theta
r^{2/3}}}_{0,r}\geq-C^{\prime}h^{2}r^{1/3}\Big{)}$ (6.15)
$\displaystyle\qquad\qquad\qquad\qquad+\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{J^{h},N}-\log
Z_{r,N}\geq(C^{\prime}h^{2}-\theta^{-1})r^{1/3}\Big{)}$ (6.16)

where $C^{\prime}$ is a positive constant which we will fix (independent of
$\theta$).

Next, since $h^{2}\geq\theta^{-5}$, we see that (6.16) is bounded by
$e^{-C|h|^{3}}$ as it is exactly the same as (4.5) appearing in the proof of
Proposition 4.4.

The probability in (6.15) can be bounded as

$\displaystyle\eqref{E4_1}$ $\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{\textup{max}}_{0,J^{h}}-2r{f_{d}}\geq-2C^{\prime}h^{2}r^{1/3}\Big{)}$
$\displaystyle\qquad\qquad\qquad+\mathbb{P}\Big{(}\log Z^{\textup{in},{\theta
r^{2/3}}}_{0,r}-2r{f_{d}}\leq-C^{\prime}h^{2}r^{1/3}\Big{)}.$

Provided that $C^{\prime}$ is fixed sufficiently small, the two probabilities
above are upper bounded by $e^{-Ch^{2}}$ using Proposition 3.11 and Theorem
3.16. To summarize, we have shown that

$\mathbb{P}(\mathcal{E}_{2}^{c})\leq\sum_{|h|=\phi_{1}/2}^{\infty}e^{-Ch^{2}}\leq
e^{-\theta^{-2}},$

and this finishes the proof of the lemma. ∎

######  Proposition 6.12.

On the event
$(\cap_{i=1}^{6}\mathcal{A}_{i})\bigcap(\cap_{j=1}^{2}\mathcal{E}_{j})$, we
have ${\bf p}^{*}\in\mathcal{L}_{r}^{r^{2/3}}.$

###### Proof.

The main idea of the proof is the following. First, we know that the
inequality below must hold

$\max_{{\bf p}\in\mathcal{L}_{r}}\\{\log Z_{0,{\bf p}}+\log Z_{{\bf
p},N}\\}-\log Z_{r,N}-\log Z^{\textup{in},{\theta r^{2/3}}}_{0,r}\geq 0.$

And we will show that if ${\bf p}\not\in\mathcal{L}_{r}^{r^{2/3}}$, then on
the event
$(\cap_{i=1}^{6}\mathcal{A}_{i})\bigcap(\cap_{j=1}^{2}\mathcal{E}_{j})$ ,

$\log Z_{0,{\bf p}}+\log Z_{{\bf p},N}-\log Z_{r,N}-\log
Z^{\textup{in},{\theta r^{2/3}}}_{0,r}<-r^{2/3}<0,$

hence the it must be true that the maximizer ${\bf
p}^{*}\in\mathcal{L}_{r}^{r^{2/3}}$.

First, we note that because we are on the event $\mathcal{E}_{2}$, then ${\bf
p}^{*}$ must be in $\mathcal{L}_{r}^{\phi_{1}r^{2/3}}$. Then, within
$\mathcal{L}_{r}^{\phi_{1}r^{2/3}}$, the event $\cap_{i=1}^{6}\mathcal{A}_{i}$
says we would lose more than $\theta^{-50}r^{1/3}$ amount of free energy
comparing with going from $(0,0)$ to ${\bf
p}\in\mathcal{L}_{r}^{\phi_{1}r^{2/3}}\setminus\mathcal{L}_{r}^{\phi_{1}r^{2/3}}$
instead of going to $(r,r)$. And above $\mathcal{L}_{r}$, $\mathcal{E}_{1}$
says for any ${\bf p}$ inside $\mathcal{L}_{r}^{\phi_{1}r^{2/3}}$, we gain at
most $\theta^{-1}\sqrt{\phi_{1}}r^{1/3}$ amount of free energy comparing with
going from $(r,r)$ to $(N,N)$. Thus, the loss $\theta^{-50}r^{1/3}$ is greater
than the gain $\theta^{-1}\sqrt{\phi_{1}}r^{1/3}$, hence, ${\bf
p}^{*}\in\mathcal{L}_{r}^{r^{2/3}}.$

∎

###  6.4 Expectation bounds

In this subsection, we prove two propositions about the expected difference of
free energies when conditioning on $\mathcal{B}_{\textup{bar}}$.

######  Proposition 6.13.

There exist positive constants $C_{43},\theta_{0}$ such that for
$0<\theta<\theta_{0}$, there exists a positive constant $c_{0}$ such that for
each $r\geq c_{0}$, we have

$\mathbb{E}\Big{[}\Big{(}\log Z_{0,\mathcal{L}_{r}}-\log
Z^{\textup{in},{3\theta
r^{2/3}}}_{0,r}\Big{)}^{2}\Big{|}\mathcal{B}_{\textup{bar}}\Big{]}\leq
C_{\ref*{2term_bound_c1}}r^{2/3}.$

###### Proof.

Let us denote the high probability event

$\mathcal{D}=\cap_{i=1}^{6}\mathcal{A}_{i},$

and we have
$\mathbb{P}(\mathcal{D}^{c}|\mathcal{B}_{\textup{bar}})\leq\mathbb{P}(\mathcal{D}^{c})\leq
e^{-\theta^{-2}}$ which is the statement of Proposition 6.2.

Now let us look at the expectation on the event $\mathcal{D}$. Using
Proposition 6.9, we obtain the first inequality below. The second inequality
follows from that $Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{1})+\log 6\geq\log
Z^{\textup{in},{3\theta r^{2/3}}}_{0,r}$ on the event $\mathcal{D}$, so
reducing the value of $Z^{\textup{in},{3\theta r^{2/3}}}_{0,r}$ makes the
expectation bigger. We have

$\displaystyle\mathbb{E}\Big{[}\Big{(}\log Z_{0,\mathcal{L}_{r}}-\log
Z^{\textup{in},{3\theta
r^{2/3}}}_{0,r}\Big{)}^{2}\mathbbm{1}_{\mathcal{D}}\Big{|}\mathcal{B}_{\textup{bar}}\Big{]}\leq\mathbb{E}\Big{[}\Big{(}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{1})-\log Z^{\textup{in},{3\theta
r^{2/3}}}_{0,r}+\log
6\Big{)}^{2}\mathbbm{1}_{\mathcal{D}}\Big{|}\mathcal{B}_{\textup{bar}}\Big{]}$
$\displaystyle\qquad\leq\mathbb{E}\Big{[}\Big{(}\log
Z_{0,\mathcal{L}_{r}}(\mathfrak{A}_{1})-\max_{{\bf p}\in
R_{r-\theta^{3/2}r,r-\theta^{3/2}r+r/L}^{\theta r^{2/3}}}\Big{\\{}\log
Z^{\textup{in},R_{0,r}^{3\theta r^{2/3}}}_{0,{\bf p}}+\log
Z^{\textup{in},R_{0,r}^{\theta r^{2/3}}}_{{\bf p},r}\Big{\\}}+\log
6\Big{)}^{2}\Big{|}\mathcal{B}_{\textup{bar}}\Big{]}$ (6.17)

To simplify the notation, let us denote the rectangle under the maximum above
as

$R^{*}=R_{r-\theta^{3/2}r,r-\theta^{3/2}r+r/L}^{\theta r^{2/3}}.$

Now, recall the definition of $\mathfrak{A}_{1}$ in (6.5), every paths must
touch $R^{*}.$ If we let ${\bf p}^{*}$ to be the maximizer for $\max_{{\bf
p}\in R^{*}}\log Z^{\text{in},R_{0,r}^{3\theta r^{2/3}}}_{0,{\bf p}}+\log
Z^{\text{in},R_{0,r}^{\theta r^{2/3}}}_{{\bf p},r},$ then

$\displaystyle\log Z_{0,\mathcal{L}}(\mathfrak{A}_{1})$ $\displaystyle\leq\log
Z^{\textup{in},{3\theta r^{2/3}}}_{0,{\bf p}^{*}}+\log Z_{{\bf
p}^{*},\mathcal{L}_{r}}.$

Therefore, the expectation (6.17) can be upper bounded by

$\displaystyle\eqref{exp_max}$ $\displaystyle\leq
10\cdot\mathbb{E}\Big{[}\max_{{\bf p}\in R^{*}}\Big{(}\log Z_{{\bf
p},\mathcal{L}_{r}}-\log Z^{\textup{in},R_{0,r}^{\theta r^{2/3}}}_{{\bf
p},r}\big{|}\mathcal{B}_{\textup{bar}}\Big{)}^{2}\Big{]}$ $\displaystyle\leq
10\cdot\mathbb{E}\Big{[}\max_{{\bf p}\in R^{*}}\Big{(}\log Z_{{\bf
p},\mathcal{L}_{r}}-\log Z^{\textup{in},R_{0,r}^{\theta r^{2/3}}}_{{\bf
p},r}\Big{)}^{2}\Big{]}.$ (6.18)

Finally, to bound (6.18), from monotonicity

$\log Z_{{\bf p},\mathcal{L}_{r}^{r^{2/3}}}\geq\log
Z^{\textup{in},R_{0,r}^{\theta r^{2/3}}}_{{\bf p},r},$

and Theorem 3.18 and Theorem 3.19 (note $h=1$ in these theorems), we have

$\displaystyle\begin{split}\mathbb{E}\Big{[}\max_{{\bf p}\in R^{*}}\Big{(}\log
Z_{{\bf p},\mathcal{L}_{r}^{r^{2/3}}}-(2r-|{\bf
p}|_{1})f_{d}\Big{)}^{2}\Big{]}&\leq C\theta r^{2/3}\\\
\mathbb{E}\Big{[}\max_{{\bf p}\in R^{*}}\Big{(}\log
Z^{\textup{in},R_{0,r}^{\theta r^{2/3}}}_{{\bf p},r}-(2r-|{\bf
p}|_{1})f_{d}\Big{)}^{2}\Big{]}&\leq C\theta r^{2/3}.\end{split}$ (6.19)

The estimate (6.19) above implies $\eqref{lastineq}\leq C\theta r^{2/3}$.

Next, for $\mathcal{D}^{c}$, we use the FKG inequality and Cauchy-Schwartz
inequality,

$\displaystyle\mathbb{E}\Big{[}\Big{(}\log Z_{0,\mathcal{L}_{r}}-\log
Z^{\textup{in},{3\theta
r^{2/3}}}_{0,r}\Big{)}^{2}\mathbbm{1}_{\mathcal{D}^{c}}\Big{|}\mathcal{B}_{\textup{bar}}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\Big{(}\log Z_{0,\mathcal{L}_{r}}-\log
Z^{\textup{in},{\theta
r^{2/3}}}_{0,r}\Big{)}^{2}\mathbbm{1}_{\mathcal{D}^{c}}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\Big{(}\log Z_{0,\mathcal{L}_{r}}-\log
Z^{\textup{in},{\theta
r^{2/3}}}_{0,r}\Big{)}^{4}\Big{]}^{1/2}\mathbb{P}(\mathcal{D}^{c})^{1/2}.$

Again, because

$\log Z_{0,\mathcal{L}_{r}}-\log Z^{\textup{in},{\theta r^{2/3}}}_{0,r}\geq
0,$

the fourth moment term can be bounded by $C\theta^{4}r^{4/3}$ using Theorem
3.15 and Theorem 3.16, which provide both right and left tail upper bound for
both. Combined with the fact that $\mathbb{P}(\mathcal{D}^{c})^{1/2}\leq
e^{-\theta^{-1}}$, we have finished the proof. ∎

######  Proposition 6.14.

There exist positive constants $C_{44},\theta_{0}$ such that for
$0<\theta<\theta_{0}$, there exists a positive constant $c_{0}$ such that for
each $r\geq c_{0}$, we have

$\mathbb{E}\Big{[}\Big{(}\log Z_{0,N}-\log Z_{r,N}-\log
Z^{\textup{in},{3\theta
r^{2/3}}}_{0,r}\Big{)}^{2}\Big{|}\mathcal{B}_{\textup{bar}}\Big{]}\leq
C_{\ref*{4term_bound_c1}}r^{2/3}.$

###### Proof.

First, note we may replace $\log Z_{0,N}$ by $\max_{{\bf
p}\in\mathcal{L}_{r}}\\{\log Z_{0,{\bf p}}+\log Z_{{\bf p},N})\\}$ because

$\log Z_{0,N}\geq\log Z_{r,N}+\log Z^{\textup{in},R_{0,r}^{3\theta
r^{2/3}}}_{0,r}$

and

$\log Z_{0,N}\leq\max_{{\bf p}\in\mathcal{L}_{r}}\\{\log Z_{0,{\bf p}}+\log
Z_{{\bf p},N}\\}+100\log r.$

Next, let us define the event

$\mathcal{D}=(\cap_{i=1}^{6}\mathcal{A}_{i})\bigcap(\cap_{j=1}^{2}\mathcal{E}_{j}).$

We have
$\mathbb{P}(\mathcal{D}^{c}|\mathcal{B}_{\textup{bar}})\leq\mathbb{P}(\mathcal{D}^{c})\leq
e^{-\theta^{-1/100}}$ from Lemma 6.10 and Lemma 6.11. We again will split the
expectation into two parts according to $\mathcal{D}$.

On the event $\mathcal{D}$, by Proposition 6.12, the maximizer ${\bf p}^{*}$
is contained inside $\mathcal{L}_{r}^{r^{2/3}}$,

$\displaystyle\mathbb{E}\Big{[}\Big{(}\max_{{\bf p}\in\mathcal{L}_{r}}\\{\log
Z_{0,{\bf p}}+\log Z_{{\bf p},N}\\}-\log Z_{r,N}-\log Z^{\textup{in},{3\theta
r^{2/3}}}_{0,r}\Big{)}^{2}\mathbbm{1}_{\mathcal{D}}\Big{|}\mathcal{B}_{\textup{bar}}\Big{]}$
$\displaystyle\leq 2\mathbb{E}\Big{[}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}}\log Z_{0,{\bf p}}-\log Z^{\textup{in},{3\theta
r^{2/3}}}_{0,r}\Big{)}^{2}\Big{|}\mathcal{B}_{\textup{bar}}\Big{]}+2\mathbb{E}\Big{[}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}^{r^{2/3}}}\log Z_{{\bf p},N}-\log
Z_{r,N}\Big{)}^{2}\Big{]}$ $\displaystyle\leq Cr^{2/3}.$

where the last bound follows from Proposition 6.13 and Proposition 4.3.

On the event $\mathcal{D}^{c}$, we again just bound with the FKG and the
Cauchy-Schwartz inequality,

$\displaystyle\mathbb{E}\Big{[}\Big{(}\max_{{\bf p}\in\mathcal{L}_{r}}\\{\log
Z_{0,{\bf p}}+\log Z_{{\bf p},N}\\}-\log Z_{r,N}-\log Z^{\textup{in},{3\theta
r^{2/3}}}_{0,r}\Big{)}^{2}\mathbbm{1}_{\mathcal{D}^{c}}\Big{|}\mathcal{B}_{\textup{bar}}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}}\\{\log Z_{0,{\bf p}}+\log Z_{{\bf p},N}\\}-\log
Z_{r,N}-\log Z^{\textup{in},{\theta
r^{2/3}}}_{0,r}\Big{)}^{2}\mathbbm{1}_{\mathcal{D}^{c}}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}}\\{\log Z_{0,{\bf p}}+\log Z_{{\bf p},N}\\}-\log
Z_{r,N}-\log Z^{\textup{in},{\theta
r^{2/3}}}_{0,r}\Big{)}^{4}\Big{]}^{1/2}\mathbb{P}(\mathcal{D}^{c})^{1/2}.$
(6.20)

The fourth moment term above can be bounded as

$\displaystyle\mathbb{E}\Big{[}\Big{(}\max_{{\bf p}\in\mathcal{L}_{r}}\\{\log
Z_{0,{\bf p}}+\log Z_{{\bf p},N}\\}-\log Z_{r,N}-\log Z^{\textup{in},{\theta
r^{2/3}}}_{0,r}\Big{)}^{4}\Big{]}$ $\displaystyle\leq
C\mathbb{E}\Big{[}\Big{(}\max_{{\bf p}\in\mathcal{L}_{r}}\\{\log Z_{0,{\bf
p}}+\log Z_{{\bf p},N}\\}-\log Z_{r,N}-\log
Z_{0,r}\Big{)}^{4}\Big{]}+C\mathbb{E}\Big{[}\Big{(}\log Z_{0,r}-\log
Z^{\textup{in},{\theta r^{2/3}}}_{0,r}\Big{)}^{4}\Big{]}$ $\displaystyle\leq
C\mathbb{E}\Big{[}\Big{(}\max_{{\bf p}\in\mathcal{L}_{r}}\\{\log Z_{0,{\bf
p}}+\log Z_{{\bf p},N}\\}-\log Z_{r,N}-\log
Z_{0,r}\Big{)}^{4}\Big{]}+C\mathbb{E}\Big{[}\Big{(}\log
Z_{0,r}-2rf_{d}\Big{)}^{4}\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+C\mathbb{E}\Big{[}\Big{(}2rf_{d}-\log
Z^{\textup{in},{\theta r^{2/3}}}_{0,r}\Big{)}^{4}\Big{]}$ $\displaystyle\leq
Cr^{4/3}+Cr^{4/3}+C\theta^{-4}r^{4/3}$

using Proposition 4.7, Proposition 3.6, and Theorem 3.16. Since
$\mathbb{P}(\mathcal{D}^{c})^{1/2}\leq e^{-\theta^{-1/200}}$, the expectation
on $\mathcal{D}^{c}$ is also upper bounded by $Cr^{2/3}$. With this, we have
finished the proof.

∎

###  6.5 Constrained variance lower bound

The main purpose of this section is to prove the following theorem on the
lower bound of the constrained variance. Recall $\mathcal{F}_{\theta}$ as the
$\sigma$-algebra generated by the weights in the set
$[\\![(0,0),(N,N)]\\!]\setminus R^{\theta r^{2/3}}_{0,r}$.

######  Theorem 6.15.

There exist positive constants $C_{45},\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exists a positive constant $c_{0}$ and an
event $\mathcal{B}^{\prime}\subset\mathcal{B}_{\textup{bar}}$ with
$\mathbb{P}(\mathcal{B}^{\prime}|\mathcal{B}_{\textup{bar}})\geq 1/2$ such
that for each $r\geq c_{0}$, we have

$\textup{$\mathbb{V}$ar}\Big{(}\log Z^{\textup{in},{3\theta
r^{2/3}}}_{0,r}\Big{|}\mathcal{F}_{\theta}\Big{)}(\omega)\geq
C_{\ref*{con_var_c1}}\theta^{-1/2}r^{2/3}\qquad\textup{ for each
$\omega\in\mathcal{B}^{\prime}$.}$

First, let us define the following sequence of events. For
$i\in(\tfrac{1}{3}{\theta^{3/2}},\tfrac{2}{3}{\theta^{3/2}})$ and a positive
constant $q^{*}$ which we fix in Proposition 6.16 below,

$\displaystyle\mathcal{U}_{i}$ $\displaystyle=\Big{\\{}\log
Z_{0,\mathcal{L}_{i\theta^{3/2}r}}-\log Z^{\textup{in},{3\theta
r^{2/3}}}_{0,i\theta^{3/2}r}\leq q^{*}\sqrt{\theta}r^{1/3}\Big{\\}}$
$\displaystyle\mathcal{V}_{i}$ $\displaystyle=\Big{\\{}\log
Z_{\mathcal{L}_{(i+1)\theta^{3/2}r},r}-\log Z^{\textup{in},{3\theta
r^{2/3}}}_{(i+1)\theta^{3/2}r,r}\leq q^{*}\sqrt{\theta}r^{1/3}\\}$
$\displaystyle\mathcal{W}_{i}$ $\displaystyle=\Big{\\{}\log
Z_{\mathcal{L}_{i\theta^{3/2}r}^{3\theta
r^{2/3}},\mathcal{L}_{(i+1)\theta^{3/2}r}^{3\theta
r^{2/3}}}-2\theta^{3/2}r{f_{d}}\leq q^{*}\sqrt{\theta}r^{1/3}\Big{\\}}$

On the event $\mathcal{B}_{\textup{bar}}$, the events defined above happen
with high probability.

######  Proposition 6.16.

There exist positive constants $q^{*},\theta_{0}$ such that for each
$0<\theta\leq\theta_{0}$, there exists a positive constant $c_{0}$ such that
for each $r\geq c_{0}$,

$\mathbb{P}(\mathcal{U}_{i}\cap\mathcal{V}_{i}\cap\mathcal{W}_{i}|\mathcal{B}_{\textup{bar}})\geq
0.9999.$

###### Proof.

We upper bound $\mathcal{U}_{i}^{c}$ and $\mathcal{W}_{i}^{c}$. And by
symmetry, the estimate for $\mathcal{V}_{i}^{c}$ is the same as
$\mathcal{U}_{i}^{c}$.

First, by the FKG inequality and Theorem 3.15,

$\mathbb{P}(\mathcal{W}_{i}^{c}|\mathcal{B}_{\textup{bar}})\leq\mathbb{P}\Big{(}\log
Z^{\textup{in},{3\theta r^{2/3}}}_{\mathcal{L}_{i\theta^{3/2}r}^{3\theta
r^{2/3}},\mathcal{L}_{(i+1)\theta^{3/2}r}^{3\theta
r^{2/3}}}-2\theta^{3/2}r{f_{d}}\leq q^{*}\sqrt{\theta}r^{1/3}\Big{)}\leq
e^{-Cq^{*}}.$

Next, we upper bound
$\mathbb{P}(\mathcal{U}_{i}^{c}|\mathcal{B}_{\textup{bar}})$. Let
$\mathfrak{A}$ denote the collection of paths going from $(0,0)$ to
$\mathcal{L}_{i\theta^{3/2}r}^{3\theta r^{2/3}}$ such that they stay within
the diagonal sides of $R_{0,(i-1)\theta^{3/2}r}^{3\theta r^{2/3}}$ and they
touch the box $R_{(i-2)\theta^{3/2}r,(i-1)\theta^{3/2}r}^{\theta r^{2/3}}$.
(Note the $\mathfrak{A}$ here plays the role of $\mathfrak{A}_{1}$ from
(6.5).) Applying Proposition 6.9 (for the free energy from $0$ to
$i\theta^{3/2}r$ instead of from $0$ to $r$), we know that for $\theta$
sufficiently small.

$\mathbb{P}\Big{(}\log Z_{0,\mathcal{L}_{i\theta^{3/2}r}}\leq\log
Z_{0,\mathcal{L}_{i\theta^{3/2}r}}(\mathfrak{A})+\log
6\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}\geq 0.99999999.$

Then, it suffices for us to upper bound the event

$\Big{\\{}\log Z_{0,\mathcal{L}_{i\theta^{3/2}r}}(\mathfrak{A})-\log
Z^{\textup{in},{3\theta
r^{2/3}}}_{0,i\theta^{3/2}r}>q^{*}\sqrt{\theta}r^{1/3}-\log 6\Big{\\}}.$
(6.21)

Now, since all the paths in $\mathfrak{A}$ enter the box

$R^{*}=R^{\theta r^{2/3}}_{(i-2)\theta^{3/2}r,(i-1)\theta^{3/2}r},$

let $\mathfrak{A}_{\bf p}$ denote the collection of paths in $\mathfrak{A}$
that go through ${\bf p}\in R^{*}$. Let ${\bf p}^{*}$ be the maximizer of

$\max_{{\bf p}\in R^{*}}\Big{\\{}\log
Z_{0,\mathcal{L}_{i\theta^{3/2}r}}(\mathfrak{A}_{{\bf p}})\Big{\\}}.$

And it holds that

$\log Z_{0,\mathcal{L}_{i\theta^{3/2}r}}(\mathfrak{A})\leq\log
Z_{0,\mathcal{L}_{i\theta^{3/2}r}}(\mathfrak{A}_{{\bf p}^{*}})+100\log r.$

Now, to bound (6.21), it suffices for us to upper bound

$\mathbb{P}\Big{(}\log Z_{0,\mathcal{L}_{i\theta^{3/2}r}}(\mathfrak{A}_{\bf
p^{*}})-\log Z^{\textup{in},{3\theta
r^{2/3}}}_{0,i\theta^{3/2}r}>\tfrac{1}{2}q^{*}\sqrt{\theta}r^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}.$

We will replace these two free energies appearing above. Since

$\displaystyle\log Z_{0,\mathcal{L}_{i\theta^{3/2}r}}(\mathfrak{A}_{\bf
p^{*}})=\log Z_{0,{\bf p}^{*}}+\log Z_{{\bf
p}^{*},\mathcal{L}_{i\theta^{3/2}r}}$ $\displaystyle\log
Z^{\textup{in},{3\theta r^{2/3}}}_{0,i\theta^{3/2}r}\geq\log
Z^{\textup{in},R_{0,r}^{3\theta r^{2/3}}}_{0,{\bf p}^{*}}+\log
Z^{\textup{in},R_{0,r}^{\theta r^{2/3}}}_{{\bf p}^{*},i\theta^{3/2}r},$

and it suffices for us to upper bound

$\displaystyle\mathbb{P}\Big{(}\log Z_{{\bf
p}^{*},\mathcal{L}_{i\theta^{3/2}r}}-\log Z^{\textup{in},R_{0,r}^{\theta
r^{2/3}}}_{{\bf
p}^{*},i\theta^{3/2}r}>\tfrac{1}{2}q^{*}\sqrt{\theta}r^{1/3}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\log Z_{{\bf
p}^{*},\mathcal{L}_{i\theta^{3/2}r}}-\log Z^{\textup{in},R_{0,r}^{\theta
r^{2/3}}}_{{\bf
p}^{*},i\theta^{3/2}r}>\tfrac{1}{2}q^{*}\sqrt{\theta}r^{1/3}\Big{)}.$ (6.22)

And we may bound (6.22) as

$\displaystyle\eqref{finalbd}$ $\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf
p}\in R^{*}}\log Z_{{\bf
p},\mathcal{L}_{i\theta^{3/2}r}}-(2i\theta^{3/2}r-|{\bf
p}|_{1}){f_{d}}>\tfrac{1}{4}q^{*}\sqrt{\theta}r^{1/3}\Big{)}$
$\displaystyle\qquad\qquad+\mathbb{P}\Big{(}\min_{{\bf p}\in R^{*}}\log
Z^{\textup{in},R_{0,r}^{\theta r^{2/3}}}_{{\bf
p},i\theta^{3/2}r}-(2i\theta^{3/2}r-|{\bf
p}|_{1}){f_{d}}<-\tfrac{1}{4}q^{*}\sqrt{\theta}r^{1/3}\Big{)}.$

Both of these probabilities are bounded by $e^{-{q^{*}}^{1/10}}$ by Theorem
3.18 and Theorem 3.19 . Finally, by fixing $q^{*}$ sufficiently large, this
completes the proof of this proposition. ∎

We say an index $i\in(\tfrac{1}{3}{\theta^{3/2}},\tfrac{2}{3}{\theta^{3/2}})$
is good if

$\mathbb{P}(\mathcal{U}_{i}\cap\mathcal{V}_{i}\cap\mathcal{W}_{i}|\mathcal{F}_{\theta})(\omega)\geq
0.99\qquad\textup{ where $\omega\in\mathcal{B}_{\textup{bar}}$}.$

Note for a given $\omega\in\mathcal{B}_{\textup{bar}}$, the set of good
indices is deterministic.

######  Lemma 6.17.

Let $\mathcal{B}^{\prime}\subset\mathcal{B}_{\textup{bar}}$ denote the event
that the number of good indices is at least $\tfrac{1}{6}\theta^{-3/2}$. Then
$\mathbb{P}(\mathcal{B}^{\prime}|\mathcal{B}_{\textup{bar}})\geq 1/2.$

###### Proof.

Since $\mathcal{B}_{\textup{bar}}$ is $\mathcal{F}_{\theta}$-measurable, by
Markov’s inequality,

$\mathbb{P}(\textup{$i$ is
bad}|\mathcal{B}_{\textup{bar}})\leq\mathbb{P}\Big{(}\\{\omega:\mathbb{P}(\mathcal{U}_{i}^{c}\cup\mathcal{V}_{i}^{c}\cup\mathcal{W}_{i}^{c}|\mathcal{F}_{\theta})(\omega)>0.01\\}\Big{|}\mathcal{B}_{\textup{bar}}\Big{)}\leq
100\mathbb{P}(\mathcal{U}_{i}^{c}\cup\mathcal{V}_{i}^{c}\cup\mathcal{W}_{i}^{c}|\mathcal{B}_{\textup{bar}})\leq
1/10.$

Then, the expected number of bad indices (conditional on
$\mathcal{B}_{\textup{bar}}$) is upper bounded by
$\tfrac{1}{10}\cdot\tfrac{1}{3}\theta^{-3/2}$, and a further application of
Markov’s inequality completes the proof. ∎

Next we prove Theorem 6.15 using the $\mathcal{B}^{\prime}$ obtained in Lemma
6.17. Let us fix an $\omega^{\prime}\in\mathcal{B}^{\prime}$ for the remainder
of this proof; projection of every configuration we shall define on to
vertices outside $R_{0,r}^{\theta r^{2/3}}$ will agree with $\omega^{\prime}$.
Given this $\omega^{\prime}$, the collection of good vertices is known. Let us
fix an enumeration of a portion of the good vertices

$J=\\{i_{1},i_{2},\dots,i_{K}\\}$

where $K\geq\tfrac{1}{6}\theta^{-3/2}$. Now, define a sequence of
$\sigma$-algebras
$\mathcal{S}_{0}\subset\mathcal{S}_{1}\subset\mathcal{S}_{2}\subset\dots\subset\mathcal{S}_{K}$
where $\mathcal{S}_{0}$ is generated by the configuration
$\omega^{\prime}\in\mathcal{B}^{\prime}$ together with the configuration on
$R^{\theta r^{2/3}}_{i\theta^{3/2}r,(i+1)\theta^{3/2}r}$ for all $i\not\in J$,
and for $j\geq 1$, $\mathcal{S}_{j}$ is the $\sigma$-algebra generated by
$\mathcal{S}_{j-1}$ and the configuration inside $R^{\theta
r^{2/3}}_{i_{j}\theta^{3/2}r,(i_{j}+1)\theta^{3/2}r}$. Note that $S_{K}$ is
the $\sigma$-algebra of the entire weight configuration.

Consider the Doob martingale

$M_{j}=\mathbb{E}\big{[}\log Z_{0,r}^{\text{in},3\theta
r^{2/3}}\big{|}\mathcal{S}_{j}\big{]}.$

By the variance decomposition of a Doob martingale, it follows that

$\textup{$\mathbb{V}$ar}\Big{(}\log Z^{\textup{in},{3\theta
r^{2/3}}}_{0,r}\Big{|}\mathcal{F}_{\theta}\Big{)}(\omega^{\prime})\geq\sum_{j=1}^{K}\mathbb{E}[(M_{j}-M_{j-1})^{2}|\mathcal{F}_{\theta}](\omega^{\prime}).$

Theorem 6.15 follows directly from the lemma below. The proof of the lemma
goes by a re-sampling argument. This idea first appeared in the zero-
temperature set-up in [9, 11], although the setup there is different from
ours.

######  Lemma 6.18.

There exist positive constants $\theta_{0},C_{46}$ such that for each
$0<\theta<\theta_{0}$, there exists a positive constant $c_{0}$ such that for
each $r\geq c_{0}$, the following holds: for each $i_{j}\in J$, there is an
$\mathcal{S}_{j-1}$ measurable event $\mathcal{G}_{j}$ with
$\mathbb{P}(\mathcal{G}_{j}|\mathcal{F}_{\theta})(\omega^{\prime})\geq 1/2$
for each $\omega^{\prime}\in\mathcal{B}^{\prime}$, and for each
$\omega\in\mathcal{G}_{j}$ we have

$\mathbb{E}[(M_{j}-M_{j-1})^{2}|\mathcal{S}_{j-1}](\omega)\geq
C_{\ref*{c_var_lem_c1}}\theta r^{2/3}.$

###### Proof.

Define the event

$F=\Big{\\{}\log Z^{\textup{in},{\theta
r^{2/3}}}_{i_{j}\theta^{3/2}r,(i_{j}+1)\theta^{3/2}r}-2(\theta^{3/2}r){f_{d}}\geq
100q^{*}\sqrt{\theta}r^{1/3}\Big{\\}}$

where the fixed constant $q^{*}$ is from Proposition 6.16. By Theorem 3.17,
$\mathbb{P}(F)\geq c>0$. Let $\omega_{1}$ denote the configuration on
$R_{i_{j}\theta^{3/2}r,(i_{j}+1)\theta^{3/2}r}^{\theta r^{2/3}}$ drawn from
i.i.d. inverse-gamma distribution. And let $\mathaccent 869{\omega}_{1}$
denote the configuration on
$R_{i_{j}\theta^{2/3}r,(i_{j}+1)\theta^{3/2}r}^{\theta r^{2/3}}$ which is
drawn from i.i.d. inverse-gamma distribution but conditioned on $F$. By the
FKG inequality and Strassen’s Theorem [53], there exists a coupling measure of
the joint distribution $(\omega_{1},\mathaccent 869{\omega}_{1})$ such that
$\omega_{1}\leq\mathaccent 869{\omega}_{1}$ coordinatewise. Let $\beta$ denote
such a coupling measure.

Let $\omega_{0}$ denote the configuration on all vertices that are revealed in
$\mathcal{S}_{j-1}$, and recall the projection of $\omega_{0}$ outside of
$R^{\theta r^{2/3}}_{0,r}$ is the same as
$\omega^{\prime}\in\mathcal{B}^{\prime}$, the environment which we have fixed
previously. And $\omega_{2}$ is the remaining weight configurations besides
$\omega_{0},\omega_{1},\mathaccent 869{\omega}_{1}$. Let
$\underline{\omega}=(\omega_{0},\omega_{1},\omega_{2})$ and $\mathaccent
869{\underline{\omega}}=(\omega_{0},\mathaccent 869{\omega}_{1},\omega_{2})$
to denote the two coupled environments under $\beta$, which was defined in the
previous paragraph. We have

$\frac{1}{\mathbb{P}(F)}\mathbb{E}[M_{j}\mathbbm{1}_{F}|\mathcal{S}_{j-1}]-M_{j-1}=\int\log(Z_{0,r}^{\text{in},3\theta
r^{2/3}})(\mathaccent 869{\underline{\omega}})-\log Z_{0,r}^{\text{in},3\theta
r^{2/3}}({\underline{\omega}})\beta(d\omega_{1},d\mathaccent
869{\omega}_{1})\mathbb{P}(d\omega_{2}).$ (6.23)

Since $F$ is independent of $\mathcal{S}_{j-1}$ and $M_{j-1}$, we also have

$\begin{split}\mathbb{E}[(M_{j}-M_{j-1})^{2}|\mathcal{S}_{j-1}]&\geq\mathbb{P}(F)\frac{1}{\mathbb{P}(F)}\mathbb{E}[(M_{j}-M_{j-1})^{2}\mathbbm{1}_{F}|\mathcal{S}_{j-1}]\\\
&\geq\mathbb{P}(F)\Big{(}\frac{1}{\mathbb{P}(F)}\mathbb{E}[M_{j}\mathbbm{1}_{F}|\mathcal{S}_{j-1}]-M_{j-1}\Big{)}^{2}.\end{split}$
(6.24)

Next, we construct the event $\mathcal{G}_{j}$. Note that
$\mathcal{U}_{i_{j}}$ is $\mathcal{S}_{j-1}$ measurable, but
$\mathcal{V}_{i_{j}}$ is not. Define another $\mathcal{S}_{j-1}$ measurable
event

$\mathaccent
869{\mathcal{V}}_{i_{j}}=\\{\omega_{0}:\mathbb{P}(\mathcal{V}_{i_{j}}|\mathcal{S}_{j-1})(\omega_{0})\geq
0.9\\}.$

We set $\mathcal{G}_{j}=\mathaccent 869{\mathcal{U}}_{i_{j}}\cap\mathaccent
869{\mathcal{V}}_{i_{j}}$ where

$\mathaccent
869{\mathcal{U}}_{i_{j}}=\\{\omega_{0}:\mathbbm{1}_{\mathcal{U}_{i_{j}}}=1\\}.$

By the definition that $i_{j}$ is good,
$\mathbb{P}(\mathcal{U}_{i_{j}}|\mathcal{F}_{\theta})(\omega^{\prime})\geq
0.99$. By Markov’s inequality,

$\mathbb{P}(\mathaccent
869{\mathcal{V}}_{i_{j}}^{c}|\mathcal{F}_{\theta})(\omega^{\prime})=\mathbb{P}(\\{\omega_{0}:\mathbb{P}(\mathcal{V}_{i_{j}}^{c}|S_{j-1})(\omega_{0})\geq
0.1\\}|\mathcal{F}_{\theta})(\omega^{\prime})\leq
10\mathbb{P}(\mathcal{V}_{i_{j}}^{c}|\mathcal{F}_{\theta})(\omega^{\prime})\leq
0.1,$

which implies $\mathbb{P}(\mathaccent
869{\mathcal{V}}_{i_{j}}^{c}|\mathcal{F}_{\theta})(\omega^{\prime})\geq 0.9$.
Hence, $\mathcal{G}_{j}$ satisfies the requirement

$\mathbb{P}(\mathcal{G}_{j}|\mathcal{F}_{\theta})(\omega^{\prime})\geq 1/2.$

For $\omega_{0}\in\mathcal{G}_{j}$, combining (6.23) and (6.24), we obtain
that

$\mathbb{E}[(M_{j}-M_{j-1})^{2}|\mathcal{S}_{j-1}](\omega_{0})\geq\mathbb{P}(F)\Big{(}\int\log
Z_{0,r}^{\text{in},3\theta r^{2/3}}(\mathaccent 869{\underline{\omega}})-\log
Z_{0,r}^{\text{in},3\theta
r^{2/3}}({\underline{\omega}})\beta(d\omega_{1},d\mathaccent
869{\omega}_{1})\mathbb{P}(d\omega_{2})\Big{)}^{2}.$ (6.25)

Since the integrand above is non-negative, we further lower bound the integral
in the right-hand side of (6.25) by

$\int_{\omega_{1}\in\mathaccent 869{\mathcal{W}}_{i_{j}},\omega_{2}\in
D}\left(\log Z_{0,r}^{\text{in},3\theta r^{2/3}}(\mathaccent