# Time correlations in KPZ models with diffusive initial conditions

Riddhipratim Basu Xiao Shen International Centre for Theoretical Sciences,
Tata Institute of Fundamental Research, Bengaluru, INDIA.
<EMAIL_ADDRESS>of Mathematics, University of Utah, Utah, USA.
<EMAIL_ADDRESS>
###### Abstract

Temporal correlation for randomly growing interfaces in the KPZ universality
class is a topic of recent interest. Most of the works so far have been
concentrated on the zero temperature model of exponential last passage
percolation, and three special initial conditions, namely droplet, flat and
stationary. We focus on studying the time correlation problem for generic
random initial conditions with diffusive growth. We formulate our results in
terms of the positive temperature exactly solvable model of the inverse-gamma
polymer and obtain up to constant upper and lower bounds for the correlation
between the free energy of two polymers whose endpoints are close together or
far apart. The exponents match the zero temperature results obtained in [18]
in the stationary case following the conjectures of [20]. Our proofs apply
almost verbatim to the zero temperature set-up of exponential LPP and are
valid for a broad class of initial conditions. Our arguments rely on the one
point moderate deviation estimates which have recently been obtained using
stationary polymer techniques and thus do not depend on 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 and main results
    1. 1.1 Polymer models on $\mathbb{Z}^{2}$ with a given initial condition
    2. 1.2 Assumptions on bulk weights and initial condition
    3. 1.3 Main results on time correlation of the free energies
    4. 1.4 Organization of the paper
  2. 2 Preliminaries
    1. 2.1 Notation
    2. 2.2 The shape function for the bulk polymer
    3. 2.3 Estimates for the bulk polymer
    4. 2.4 Estimates for the polymer with boundary
  3. 3 Time correlations for large $r$: Proof of Theorem 1.1
  4. 4 Time correlations for small $r$: Proof of Theorem 1.2
    1. 4.1 Upper bound in Theorem 1.2
    2. 4.2 Lower bound in Theorem 1.2
  5. 5 Stationary inverse-gamma polymer
    1. 5.1 Two special boundaries $\mathcal{S}$ and $\mathcal{S}^{\prime}$
    2. 5.2 Exit time estimate
  6. 6 Temporal correlation for the stationary polymer using duality
  7. 7 Proofs for the estimates in Section 2.3 and Section 2.4
    1. 7.1 Proof of Proposition 2.10
    2. 7.2 Proof of Proposition 2.11
    3. 7.3 Proof of Proposition 2.13
    4. 7.4 Proof of Proposition 2.14
    5. 7.5 Proof of Proposition 2.15
    6. 7.6 Proof of Theorem 2.16
    7. 7.7 Proof of Theorem 2.17
    8. 7.8 Proof of Theorem 2.18
  8. 8 Parallel results in exponential LPP
  9. A Appendix
    1. A.1 Sub-exponential random variables.
    2. A.2 Monotonicity results for the polymer model
    3. A.3 Assumption A implies Assumption B

##  1 Introduction and main results

A large number of planar random growth models are expected to share the
universal features of the so-called Kardar-Parisi-Zhang (KPZ) universality
class; these include planar last passage percolation (LPP) and polymer models
on the plane under mild conditions on the underlying noise. Although the
questions of universality remain out of reach, there are several exactly
solvable lattice models which can be rigorously analyzed, and studying these
models has been an important topic of research over the last twenty-five
years. One of the problems that have attracted a lot of interest over the last
few years is understanding the temporal decay of correlations in such models
with different initial conditions.

We undertake the study of time correlations in (1+1)-dimensional KPZ growth
models started from a class of random initial conditions. For concreteness, we
shall formulate and prove our main results in the framework of the exactly
solvable positive temperature model of inverse-gamma polymer on
$\mathbb{Z}^{2}$. Our arguments, nonetheless, carry over almost verbatim to
the zero temperature model of exponential LPP on $\mathbb{Z}^{2}$ as well; a
short discussion on this appears in Section 8.

Before proceeding further, let us explain the basic formulation of this
problem. For simplicity, we describe this in the setup of the zero temperature
model of exponential LPP. We consider an initial condition $f$, which is a
function defined on the anti-diagonal line $x+y=0$. For
$u,v\in\mathbb{Z}^{2}$, ($u\leq v$ coordinatewise) let $T_{u,v}$ denote the
point-to-point last passage time (i.e., the maximum weight of a directed path
joining $u$ and $v$ in an underlying field of i.i.d. exponential vertex
weights). We then define the last-passage time to $(n,n)$ with the initial
condition $f$, denoted $T_{n}^{f}$ by $\max_{v}T_{v,(n,n)}+f(v)$ where the
maximum is taken over all points $v$ on the line $x+y=0$. Then the two-time
correlation is the correlation between $T^{f}_{r}$ and $T_{n}^{f}$ for
$r,n\in\mathbb{Z}_{>0}$, and one usually studies the asymptotics of this
quantity as $r,n\to\infty$ and $r/n\to 0$ or $1$. The role of the last passage
time is played by the free energy (i.e., the log of the partition function) in
positive temperature models; a precise formulation in the setup of the
inverse-gamma polymer is given later.

Following the experimental and numerical studies by physicists [34, 35, 36],
and precise conjectures formulated by Ferrari and Spohn in [20], a host of
rigorous works have studied this problem in the context of the zero
temperature exactly solvable models (particularly exponential LPP) [7, 8, 18,
19, 20]. We shall not get into a detailed discussion of the existing
literature; rather we refer the interested reader to the introduction of [9]
for a comprehensive discussion of available results and different methods
employed to establish them. Three special initial conditions have been
investigated in particular: (i) droplet initial condition, i.e., point-to-
point passage time, (ii) flat initial condition, i.e., line-to-point passage
time and (iii) stationary initial condition where $f$ is a certain two-sided
random walk which is (increment)-stationary under the growth dynamics (ratio-
stationary in the positive temperature setup; precise definitions later). More
general (deterministic as well as random) initial conditions are also of
interest but those are not very well understood even in the zero temperature
case.

As alluded to above, our focus in this paper is KPZ growth with general random
initial conditions, including the stationary ones. The only rigorous results
for the generic random initial condition so far appear in [18] where non-
negative constant multiples of the stationary initial condition are
considered. The statistic studied in [18] is the limit $\rho(\tau)$ of the
correlation as $r,n\to\infty$ and $r/n\to\tau\in(0,1)$. It was shown that for
these initial conditions $1-\rho(\tau)=\Theta((1-\tau)^{2/3})$ as $\tau\to 1$.
For the special case of stationary initial condition it was also shown that
$\rho(\tau)=\Theta(\tau^{1/3})$ as $\tau\to 0$; in fact and explicit formula
for $\rho(\tau)$ was obtained.

Our work considers a broad class of random initial conditions which satisfy
certain ”diffusive growth” conditions; in particular, these include the
positive temperature analogues of the initial conditions considered in [18].
For this class of initial conditions, and for $r,n$ _large but finite_ , and
for $r/n$ close to $0$ or close to $1$, we establish up to constant upper and
lower bounds for the correlation establishing the same exponents as described
above, see Theorem 1.1 and Theorem 1.2. For the special case of the stationary
initial condition, in the small $r$ regime, we also provide a different
argument for the upper bound in Section 6 using duality which leads to a
quantitatively better estimate.

The reliance of our arguments on the integrable nature of the models is rather
weak, we rely only on the curvature of the limit shape and the one point
moderate deviation estimates for the point-to-point free energy (last passage
time in the zero temperature setting). For the zero temperature set-up, the
one point estimates can be obtained either by analysis of Fredholm determinant
formulae [3] (see also [29, 30, 2] for similar estimates in Poissonian and
Geometric LPP) or using random matrix methods [28]. More recently, these
estimates have been obtained using softer stationary LPP/ coupling techniques
[15, 16]. For various positive temperature models, including the inverse-gamma
polymer, these have recently been obtained in [9, 17, 26, 27] using the
stationary polymer techniques; as far as we know these are the only moderate
deviation results in the positive temperature discrete or semi-discrete models
established so far; similar results are also known for the KPZ equation using
Gibbsian line ensemble techniques (see, e.g. [14]). Using these estimates, the
time correlation in point-to-point inverse-gamma polymer (i.e., droplet
initial condition) has been studied in [9] by the current authors together
with Seppäläinen and the same exponents as described above were obtained.
Although the exponents are the same, and the ingredients are broadly similar,
the arguments in this paper are largely different from [9] for the case
$r/n\ll 1$. The contributing factor to the correlation in both cases are the
same in the $r/n\to 1$ regime, and the arguments are rather similar. The main
technical contribution of this paper is in the estimates in the $r\ll n$
regime. When $r\ll n$, although the correlation in both the droplet and
“diffusive” initial conditions decay like $(r/n)^{1/3}$; the mechanisms are
very different. In the former case, the correlation comes from the fluctuation
in the bulk weights in the region between the lines $x+y=0$ and $x+y=2r$ while
in the latter case, the primary contribution to the correlation is the
fluctuation of the initial condition. Establishing this requires a different
set of geometric arguments which is the heart of the new technical content of
this paper. Notice that we work here in the annealed setting, that is we also
average over the initial condition; one would expect a much smaller
correlation (for $r\ll n$) for a fixed realization of the initial condition;
see [8] for a discussion regarding this.

We wrap up this section with some further comments relating our work to the
relevant results of [18]. The argument in [18] is very different from the
current article; the proof of the stationary correlation therein relies on the
stationarity of the model as well as the convergence of the one point
distribution to the Baik-Rains distribution in that case. The $\tau\to 1$
asymptotics for constant multiples of stationary initial condition considered
there depended on the convergence of the passage time profiles with different
initial conditions to Airy processes. Although one point convergence of the
point-to-point free energy to the GUE Tracy-Widom distribution is known for
the inverse-gamma polymer, finer results such as convergence to Airy processes
appears to not have been established yet, therefore the approach from [18]
does not seem applicable in the inverse-gamma case, for non-stationary initial
conditions. For the stationary model, although we could not find the Baik-
Rains convergence result in the literature in the inverse-gamma case; it is
likely that one could obtain this result by arguing along the same lines as
[22] where the Baik-Rains limit was established for the O’ Connell-Yor
polymer. If so, one could prove an asymptotic version of our results, using
the approach of [18] in the stationary case. Beyond the stationary case, even
for the zero temperature setup, our results apply to a broader class of
initial conditions, give finite size estimates, and also complete the picture
by establishing the correlation for the $r\ll n$ case. For the case $\tau\to
1$ and for the subclass of initial conditions considered in [18], however, our
estimates are quantitatively somewhat weaker than the limiting estimates
there.

We now move towards precise definitions and formulation of the main results in
this article.

###  1.1 Polymer models on $\mathbb{Z}^{2}$ with a given initial condition

First, we define the bulk partition function without an initial condition. Let
$\mathbb{X}_{{\bf a},{\bf b}}$ denote the collection of up-right paths between
${\bf a}$ and ${\bf b}$. Paths ${\bf
x}_{{{\scaleobj{0.6}{\bullet}}}}\in\mathbb{X}_{{\bf a},{\bf b}}$ are of the
form ${\bf x}_{0}={\bf a}$ and ${\bf x}_{|{\bf a}-{\bf b}|_{1}}={\bf b}$. We
attach non-negative weights to the vertices $\\{Y_{\bf z}\\}_{{\bf
z}\in\mathbb{Z}^{2}}$. If $\mathbb{X}_{{\bf a},{\bf b}}$ is non-empty, define

$\mathaccent 869{Z}_{{\bf a},{\bf b}}=\sum_{{\bf
x}_{{{\scaleobj{0.6}{\bullet}}}}\in\mathbb{X}_{{\bf a},{\bf
b}}}\prod_{i=1}^{|{\bf a}-{\bf b}|_{1}}Y_{{\bf x}_{i}}.$

By our convention, we do not pick up any weights at the starting point of the
paths ${\bf x}_{0}={\bf a}$. Thus, we will define $\mathaccent 869{Z}_{{\bf
a},{\bf a}}=1$. And if $\mathbb{X}_{{\bf a},{\bf b}}$ is empty, define
$\mathaccent 869{Z}_{{\bf a},{\bf b}}=0$.

Next, we shall define a polymer model with an initial condition $W$ on the
anti-diagonal line through the origin $\mathcal{L}_{\bf
0}=\\{(j,-j):j\in\mathbb{Z}\\}$. Let $\\{W_{k}\\}_{k\in\mathbb{Z}}$ denote a
sequence of non-negative weights ($W_{k}$ should be interpreted as the
boundary condition at $(k,-k)\in\mathcal{L}_{\bf 0}$) with $W_{0}=1$. We now
define the partition function for the polymer to an endpoint ${\bf v}$ above
$\mathcal{L}_{\bf 0}$, denoted $Z^{W}_{\bf v}$ by

$Z^{W}_{{\bf v}}=\sum_{k\in\mathbb{Z}}W_{k}\cdot\mathaccent
869{Z}_{(k,-k),{\bf v}}.$

The free energy of the polymer between to ${\bf v}$ with initial condition $W$
is then defined as $\log Z^{W}_{{\bf v}}$. Clearly, the above definition can
also obviously be extended to initial conditions defined on $\mathcal{L}_{\bf
a}$ for any ${\bf a}\in\mathbb{Z}^{2}$. 111It is a common practice to define
initial conditions in terms of a class of _boundary weights_ on the edges of
$\mathcal{S}_{0}$, the bi-infinite down-right staircase path $\\{\dots,-{\bf
e}_{1}+\bf e_{2},-\bf e_{1},{\bf 0},\bf e_{1},\bf e_{1}-e_{2},\dots\\}$, but
we shall not make much use of this formulation in the first part of our paper.
This definition will be delayed until Section 5.

Let $\mathbb{X}^{\textup{line}}_{{\bf v}}$ to denote the collection of up-
right paths from $\mathcal{L}_{\bf 0}$ to ${\bf v}$. Fix a directed path ${\bf
x}_{{{\scaleobj{0.6}{\bullet}}}}\in\mathbb{X}^{\textup{line}}_{{\bf v}}$ and
suppose $k\in\mathbb{Z}$ is such that ${\bf
x}_{{{\scaleobj{0.6}{\bullet}}}}\in\mathbb{X}_{(k,-k),{\bf v}}$. The partition
function over this particular path ${\bf x}_{{\scaleobj{0.6}{\bullet}}}$ is
defined by

$Z^{W}_{{\bf v}}({\bf
x}_{{\scaleobj{0.6}{\bullet}}})=W_{k}\cdot\prod_{i=1}^{|{\bf v}|_{1}}Y_{{\bf
x}_{i}}.$

Similarly, for a subset $\mathfrak{A}\subset\mathbb{X}^{\textup{line}}_{{\bf
v}}$, define $Z^{W}_{{\bf v}}(\mathfrak{A})=\sum_{{\bf
x}_{{{\scaleobj{0.6}{\bullet}}}}\in\mathfrak{A}}Z^{W}_{{\bf v}}({\bf
x}_{{\scaleobj{0.6}{\bullet}}}).$ Then, the quenched polymer measure is
defined by $Q^{W}_{{\bf v}}\\{\mathfrak{A}\\}=\frac{Z^{W}_{{\bf
v}}(\mathfrak{A})}{Z^{W}_{{\bf v}}}$, which is a probability measure on
$\mathbb{X}^{\textup{line}}_{{\bf v}}$.

###  1.2 Assumptions on bulk weights and initial condition

Fix an anti-diagonal line $\mathcal{L}_{\bf a}={\bf a}+\mathcal{L}_{\bf 0}$,
and let us set $\mathcal{L}_{\bf a}^{>}=\bigcup_{{\bf b}\in\mathcal{L}_{\bf
a}}({\bf b}+\mathbb{Z}^{2}_{>0})$. As already mentioned we shall work with the
inverse-gamma polymer with an initial condition on $\mathcal{L}_{\bf 0}$,
i.e., our boundary weights $\\{Y_{\bf z}\\}_{{\bf z}\in\mathcal{L}_{\bf
0}^{>}}$ will be distributed as i.i.d. inverse-gamma random variables with
shape parameter $\mu$. Recall the density function of the inverse-gamma
distribution is defined by

$f_{\mu}(x)={\displaystyle{\frac{1}{\Gamma(\mu)}}x^{-\mu-1}e^{-{\frac{1}{x}}}}\quad\text{for
}x>0.$ (1.1)

The shape parameter $\mu\in(0,\infty)$ plays the role of temperature in this
polymer model, which will be fixed in this paper.

Assumptions on the initial condition: We shall assume that the initial
condition $W$ will be such that $\log W$ is a two-sided random walk with
$W_{0}=1$.

Assumption A: The collection of random variables $\\{\log W_{i}-\log
W_{i-1}\\}_{i\in\mathbb{Z}}$ are independent, have mean $0$ with the following
properties:

  1. (i)

$X_{i}:=\log W_{i}-\log W_{i-1}$ are uniformly sub-exponential i.e., there
exist parameters $\lambda_{1},K_{1}>0$ such that
$\log\mathbb{E}(\exp(\lambda|X_{i}-\mathbb{E}X_{i}|))\leq K_{1}\lambda^{2}$
for $\lambda\in[-\lambda_{1},\lambda_{1}]$.

  2. (ii)

$\inf_{i}\textup{$\mathbb{V}$ar}X_{i}>0$.

In fact, we shall use a somewhat weaker assumption in our proofs, but we
recorded Assumption A above since it is more straightforward to verify, and
already contains many interesting cases. Now, let us introduce the specific
assumptions employed in our paper, which shall be referred to as Assumption B.
Assumption B comprises of three distinct components: B1, B2, and B3.

To simplify the notation, we will introduce a collection of random variables
denoted as $\\{X_{i}\\}_{i\in\mathbb{Z}}=\\{\log W_{i}-\log
W_{i-1}\\}_{i\in\mathbb{Z}}$.

  * •

Assumption B1:

Right tail decay: There exist $C_{1},N_{0},t_{0}$ such that for each $N\geq
N_{0}$ and $t\geq t_{0}$

$\mathbb{P}\Big{(}\max_{-N\leq k\leq N}\log W_{k}\geq t\sqrt{N}\Big{)}\leq
e^{-C_{1}t}.$ (1.2)

  * •

Assumption B2:

    1. (i)

_$\rho$ -mixing:_ For each $A,B\subset\mathbb{Z}$ with
$\textup{dist}(A,B)>\ell\geq 1$, we have

$\Big{|}\textup{$\mathbb{C}$orr}(f(X_{i}:i\in A),g(X_{j}:j\in
B))\Big{|}\leq\frac{1}{\ell^{100}};$ (1.3)

for all square-integrable functions $f$ and $g$, i.e., the sequence
$\\{X_{i}\\}_{i\in\mathbb{Z}}$ is fast $\rho$-mixing.222Note that the
$\rho$-mixing is usually defined for stationary sequences, we consider the
obvious extension of this definition in our case. For a precise definition of
$\rho$-mixing coefficient and its relationship with other notions of mixing,
see [11].

    2. (ii)

Left tail decay: There exist $C_{1},N_{0},t_{0}$ such that for each $N\geq
N_{0}$ and $t\geq t_{0}$

$\mathbb{P}\Big{(}\log W_{N}\leq-t\sqrt{N}\Big{)}\leq e^{-C_{1}t}.$ (1.4)

  * •

Assumption B3:

For $N\in\mathbb{Z}_{>0}$, let $\mathcal{F}_{r}=$ denote the $\sigma$-algebra
generated by $\sigma(\\{X_{i}\\}_{i\notin[1,r]})$.

    1. (i)

Conditional right tail decays: There exist $C_{1},r_{0},t_{0}$ such that for
each $r\geq r_{0}$, $N\geq r$ and $t\geq t_{0}$

$\mathbb{P}\Big{(}\mathbb{P}\big{(}\max_{0\leq k\leq N}\log W_{k}\geq
t\sqrt{N}\big{|}\mathcal{F}_{r}\big{)}\leq e^{-C_{1}t}\Big{)}\geq
1-\frac{1}{\log r}.$ (1.5)

    2. (ii)

Conditional left tail decay: There exist $C_{1},r_{0},t_{0}$ such that for
each $r\geq r_{0}$, $N\geq r$ and $t\geq t_{0}$

$\mathbb{P}\Big{(}\mathbb{P}\big{(}\log
W_{N}\leq-t\sqrt{N}\big{|}\mathcal{F}_{r}\big{)}\leq e^{-C_{1}t}\Big{)}\geq
1-\frac{1}{\log r}.$ (1.6)

    3. (iii)

Conditional Variance bound: There exists $C_{1},r_{0}>0$, such that for all
$r\geq r_{0}$,

$\mathbb{P}\Big{(}\textup{$\mathbb{V}$ar}(\log W_{r}|\mathcal{F}_{r})\geq
C_{1}r\Big{)}\geq 1-\frac{1}{\log r}.$ (1.7)

almost surely.

    4. (iv)

Right tail lower bound: For each fixed $t_{0}$ sufficiently large, there exist
$N_{0},\epsilon_{0}$ (depending on $t_{0}$) such that for each
$i\in\mathbb{Z}$ and for each integer $N$ with $|N|\geq N_{0}$,

$\mathbb{P}\Big{(}\log W_{N+i}-\log W_{i}\geq
t_{0}\sqrt{|N|}\Big{)}\geq\epsilon_{0}>0.$ (1.8)

    5. (v)

FKG inequality: The random sequence
$\mathcal{X}:=\\{\hat{X}_{i}\\}_{i\in\mathbb{Z}}$ defined by
$\hat{X}_{i}=X_{i}$ for $i\geq 1$ and $\hat{X}_{i}=-X_{i}$ otherwise,
satisfies the FKG inequality, i.e., for any two square-integrable coordinate
wise increasing real-valued functions $f$ and $g$,333In our applications, we
only use this hypothesis in the case when the arguments of $f$ is a subset of
the arguments of $g$. Thus, it does not necessarily preclude the case of
Busemann increment initial condition for general KPZ models where it is
predicted that the disjoint Busemann increments along a down-right path are
negatively correlated [1].

$\textup{$\mathbb{C}$orr}(f(\mathcal{X}),g(\mathcal{X}))\geq 0.$ (1.9)

Further, we also require that for all $k\in\mathbb{Z}_{>0}$, the conditional
measure of $\mathcal{X}$, restricted to the coordinates $\\{1,2,\ldots,k\\}$
conditional on the remaining coordinates also satisfies the FKG inequality.
444It might appear that this assumption is rather strong and difficult to
verify, but observe that if $X_{i}$ are independent then this is immediate. We
shall also provide examples beyond the independent case which satisfy this and
the other hypotheses; see Remark 1.6.

Note that the assumptions (1.2), (1.4) and (1.8) give bounds on non-trivial
diffusive growth of $\log W$ (and (1.5), (1.6) and (1.7) gives conditional
versions of the same). Many of the assumptions become trivial if the
independence of $X_{i}$ is assumed and it is not hard to see that Assumption A
implies Assumption B; see Proposition A.7. Note also that, Assumption A
includes that case of stationary inverse-gamma polymer with parameter
$\rho=\mu/2$; see Appendix A.1.1 for a proof of this and Section 5 for more
details on the stationary polymer model. See Remark 1.6 for some examples of
initial conditions that satisfy Assumption B but not Assumption A. We shall
see later that not all parts of the hypotheses are used in all our arguments.
Also, some of the hypotheses are not quantitatively optimal; see Remark 1.7
for a discussion regarding this.

###  1.3 Main results on time correlation of the free energies

We now state our main results for the temporal correlations in the inverse-
gamma polymer with general initial conditions. Recall that 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
a pair of free energies in the inverse-gamma polymer with the above initial
conditions, based on on the separation of their endpoints.

######  Theorem 1.1.

Consider the inverse-gamma polymer with boundary condition $W$ on
$\mathcal{L}_{\bf 0}$ that satisfies Assumption B1. 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

$1-C_{1}\Big{(}\frac{N-r}{N}\Big{)}^{2/3}\leq\textup{$\mathbb{C}$orr}\Big{(}\log
Z^{W}_{(r,r)},\log Z^{W}_{(N,N)}\Big{)}\leq
1-C_{2}\Big{(}\frac{N-r}{N}\Big{)}^{2/3}.$

######  Theorem 1.2.

Consider the inverse-gamma polymer with boundary condition $W$ on
$\mathcal{L}_{\bf 0}$ that satisfies Assumptions B1 and B2. There exist
positive constants $C_{3},c_{0},N_{0}$ such that, whenever $N\geq N_{0}$ and
$c_{0}\leq r\leq N/2$, we have

$\textup{$\mathbb{C}$orr}\Big{(}\log Z^{W}_{(r,r)},\log
Z^{W}_{(N,N)}\Big{)}\leq C_{3}\Big{(}\frac{r}{N}\Big{)}^{1/3}.$

Furthermore, if $W$ satisfies Assumptions B1, B2 and B3, we have a matching
lower bound. There exist positive constants $C_{4},c_{0},N_{0}$ such that,
whenever $N\geq N_{0}$ and $c_{0}\leq r\leq N/2$, we have

$C_{4}\Big{(}\frac{r}{N}\Big{)}^{1/3}\leq\textup{$\mathbb{C}$orr}\Big{(}\log
Z^{W}_{(r,r)},\log Z^{W}_{(N,N)}\Big{)}.$

We shall informally refer to the setups of Theorem 1.1 and Theorem 1.2 by the
large $r$ regime and the small $r$ regime respectively. Notice that since
Assumption A implies Assumption B (Proposition A.7), both Theorem 1.1 and
Theorem 1.2 remain valid for initial conditions satisfying Assumption A as
well. We record the following corollary.

######  Corollary 1.3.

Consider the inverse-gamma polymer with boundary condition $W$ on
$\mathcal{L}_{\bf 0}$ that satisfies Assumption A. There exist positive
constants $C_{1},C_{2},c_{0},N_{0}$ such that,

  * •

for $N\geq N_{0}$ and $N/2\leq r\leq N-c_{0}$,

$1-C_{1}\Big{(}\frac{N-r}{N}\Big{)}^{2/3}\leq\textup{$\mathbb{C}$orr}\Big{(}\log
Z^{W}_{(r,r)},\log Z^{W}_{(N,N)}\Big{)}\leq
1-C_{2}\Big{(}\frac{N-r}{N}\Big{)}^{2/3},$

  * •

for $N\geq N_{0}$ and $c_{0}\leq r\leq N/2$, we have

$C_{2}\Big{(}\frac{r}{N}\Big{)}^{1/3}\leq\textup{$\mathbb{C}$orr}\Big{(}\log
Z^{W}_{(r,r)},\log Z^{W}_{(N,N)}\Big{)}\leq
C_{1}\Big{(}\frac{r}{N}\Big{)}^{1/3}.$

As mentioned before, the initial conditions satisfying Assumption A include
the stationary initial condition with parameter $\rho=\mu/2$, (see Section 5
for precise definitions). A different argument giving the upper bound in the
small $r$ regime for the special case of this stationary initial condition
using duality is given in Theorem 6.1. We finish this subsection with a few
remarks regarding examples and potential extensions of our results.

######  Remark 1.4.

As mentioned already, our arguments carry over verbatim to the case of
exponential last passage percolation on $\mathbb{Z}^{2}$ with a class of
“diffusive” initial conditions upon changing the free energy to the last
passage time and making other appropriate changes to the definitions. In fact,
many of the arguments become simpler in this case, since one can consider
geodesics instead of the quenched polymer measures. To avoid repetition, we
shall not write down the details in this case, but we shall formulate a
precise statement in this setting; see Theorem 8.1 and Theorem 8.2.

######  Remark 1.5.

As mentioned above, Assumption A includes the case of stationary initial
condition with parameter $\mu/2$. This is related to the fact that the
corresponding characteristic direction, in this case, is the diagonal
direction $(1,1)$. One should be able to extend our results to the stationary
initial conditions with parameters $\rho\in(0,\mu),\rho\neq\mu/2$ (see Section
5 for precise definitions) provided the endpoint of the polymers varies along
the corresponding characteristic direction. In this case, our arguments should
apply to initial conditions satisfying Assumption A, but with an appropriate
drift (i.e., $\mathbb{E}X_{i}=-\Psi_{0}(\mu-\rho)+\Psi_{0}(\rho)$). However,
we have not attempted to verify all the details in this case.

######  Remark 1.6.

Assumption B contains many more interesting cases beyond the stationary one;
we discuss a few examples.

  * •

First consider the case when $X_{i}$’s are independent. It is clear that
independent sequences satisfy parts assumptions (1.3), (1.9) from Assumption
B, and the other parts follow under conditions on the tails of
$\\{X_{i}\\}_{i\in\mathbb{Z}}$. In particular, let
$\\{\sigma_{i}\\}_{i\in\mathbb{Z}}$ denote a positive sequence uniformly
bounded away from $0$ and $\infty$. Denoting by $\hat{X}^{\mu/2}_{i}$ the
increment sequence in the stationary case, it is easy to check that
$X_{i}:=\sigma_{i}\hat{X}^{\mu/2}_{i}$ satisfies Assumption B. Compare this to
[18], where the case $\sigma_{i}\equiv\sigma$ was considered in the large $r$
regime.

  * •

Consider next the “mixtures” between deterministic and stationary increments.
We can consider (a) $X_{i}=\hat{X}^{\mu/2}_{i}$ if $i$ is even and 0
otherwise, or (b) $X_{i}=1,-1,\hat{X}^{\mu/2}_{i}$ if $i=0,1,2$ modulo $3$
respectively. It is easy to check that $X_{i}$’s defined as above satisfy all
the hypotheses in Assumption B.

  * •

Finally let us consider $X_{i}$’s which are not independent; the hardest
assumption to verify in this case is (1.9). Although there should be many
examples satisfying these hypotheses, we shall restrict to Gaussian vectors
$X_{i}$ since it is a well-known fact that such a vector satisfies the FKG
inequality if and only if the pairwise correlations are non-negative. For a
sequence $\\{U_{i}\\}_{i\in\mathbb{Z}}$ of i.i.d. standard Gaussians, define
$X_{i}=U_{i}+U_{i+1}$ for all $i\geq 0$, and $X_{i}=U_{i}+U_{i-1}$ for $i<0$.
Since this sequence is $1$-dependent, it satisfies the mixing hypothesis
(1.3). By calculating pairwise correlations (both unconditional and
conditional) using standard formulas for Gaussian vectors, together with the
fact described above, one can easily verify parts (1.7) and (1.9) of the
hypotheses. Parts (1.2), (1.4), (1.5), (1.6), and (1.8) are consequences of
standard estimates and are easy to verify. Therefore, the example described
above (and similar ones) satisfy Assumption B.

######  Remark 1.7.

The quantitative estimates in several parts of Assumption B are not optimal.
The reader can check that the exponent 100 in (1.3) can be replaced by a
smaller number and the exponential bound in (1.2) can be replaced by a
sufficiently high degree polynomial without making any major changes to the
arguments. It is possible that (1.2), (1.8) and (1.7) can be removed by
replacing the mixing condition in (1.3) with a stronger hypothesis, but we
have not managed to locate a result that fits our set-up and hence have not
attempted to find a more concise replacement for these hypotheses.

###  1.4 Organization of the paper

The rest of the paper is organized as follows. In Section 2, we introduce
basic notations and recall certain basic estimates about the point-to-point
polymer from the literature. Section 2.4 contains several important estimates
for polymers with an initial condition whose proofs are provided later.
Section 3 and Sec 4 prove Theorems 1.1 and 1.2 respectively. Section 4 is the
heart of the new technical contributions of this paper; Sections 4.1 and 4.2
provide the proofs of the upper bound and lower bound in Theorem 1.2
respectively. Section 5 introduces the stationary polymer and recalls some of
the known estimates about it. Section 6 provides an alternative proof of the
upper bound in Theorem 1.2 for the stationary polymer using duality; we
believe this is an argument of independent interest. Section 7 provides the
proof of the technical estimates from Section 2.3 and Section 2.4. Finally,
Section 8 precisely formulates time correlation results for exponential LPP
with diffusive initial conditions and describes how the arguments in the
inverse-gamma polymer case carry over to the zero temperature set-up.

### 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. XS
was partially supported by the Wylie research fund from the University of
Utah.

##  2 Preliminaries

###  2.1 Notation

Fix ${\bf a}\in\mathbb{Z}^{2}$, let $\mathcal{L}_{{\bf a}}=\\{{\bf
a}+(j,-j):j\in\mathbb{Z}\\}$. For $k\in\mathbb{R}_{\geq 0}$, define
$\mathcal{L}^{k}_{{\bf a}}=\\{{\bf x}\in\mathcal{L}_{{\bf a}}:|{\bf x}-{\bf
a}|_{\infty}\leq k\\}.$ 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 any
subset $A\subset\mathbb{Z}^{2}$ and any $\triangle\in\\{>,\geq,<,\leq\\}$,
define $A^{\triangle}=\bigcup_{x\in A}(x+\mathbb{Z}^{2}_{\triangle 0})$.

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

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

Let $A,B\subset\mathbb{R}^{2}$, ${\bf c},{\bf d}\in\mathbb{Z}^{2}$ and $h>0$.
We define three 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 ${\bf d}-{\bf c}$. $\displaystyle
Z_{A,B}^{\textup{out},R^{h}_{{\bf c},{\bf d}}}$ $\displaystyle=\text{sum over
directed paths from $A$ to $B$ that avoid $R^{h}_{{\bf c},{\bf d}}$.}$

Integer points on the diagonal or the anti-diagonal are abbreviated as
$a=(a,a)$ and $\underline{{a}}=(a,-a)$. Common occurrences of this include
$Z^{W}_{(r,r)}=Z^{W}_{r}$, $\mathaccent 869{Z}_{\underline{{r}},N}=\mathaccent
869{Z}_{(r,-r),(N,N)}$, $\mathaccent 869{Z}_{{\bf
p},N+\underline{{k}}}=\mathaccent 869{Z}_{{\bf p},(N+k,N-k)}$,
$\mathcal{L}_{a}^{k}=\mathcal{L}_{(a,a)}^{k}$ and
$R_{a,b}^{k}=R^{k}(a,b)=R_{(a,a),(b,b)}^{k}$.

Finally, we point out two conventions about constants and integer rounding.
First, 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. Second, 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.

###  2.2 The shape function for the bulk polymer

When the positive weights $\\{Y_{\bf z}\\}_{{\bf z}\in\mathbb{Z}^{2}}$ are
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 [23]): there exists a concave, positively homogeneous
and deterministic continuous function $\Lambda:\mathbb{R}^{2}_{\geq
0}\rightarrow\mathbb{R}$ that satisfies

$\lim_{n\rightarrow\infty}\sup_{|{\bf z}|_{1}\geq n}\frac{|\log\mathaccent
869{Z}_{0,{\bf z}}-\Lambda({\bf z})|}{|{\bf
z}|_{1}}=0\qquad\textup{$\mathbb{P}$-almost surely}.$ (2.1)

For general i.i.d. weights, regularity properties of $\Lambda$ such as strict
concavity or differentiability, are conjectured to hold at least for all
continuous weights. There is a special case, first observed in [32], that if
the i.i.d. weights have the inverse-gamma distribution (1.1), then $\Lambda$
can be computed explicitly. Using this fact, several estimates were done (only
using calculus) in Section 3.2 of [9], and we recall one of the results from
there.

To simplify the notation, let $\Lambda_{N}=\Lambda((N,N))$.

######  Proposition 2.1 ([9, Proposition 3.5]).

There exist positive constants $C_{1},N_{0},\epsilon_{0}$ 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

$\big{|}\Lambda({\bf p})-\Lambda_{N}\big{|}\leq C_{1}h^{2}N^{1/3}.$

###  2.3 Estimates for the bulk polymer

We recall estimates from Section 3.3 of [9]. Note that these estimates were
originally proved for partition function $Z_{0,{{\scaleobj{0.6}{\bullet}}}}$
which included the weight at the starting point, i.e.

$\log Z_{0,{{\scaleobj{0.6}{\bullet}}}}=\log Y_{{\bf 0}}+\log\mathaccent
869{Z}_{0,{{\scaleobj{0.6}{\bullet}}}}.$

However, because $Y_{{\bf 0}}\sim\textup{Ga}^{-1}(\mu)$, the exact same
estimates also hold for $\mathaccent 869{Z}_{0,{{\scaleobj{0.6}{\bullet}}}}$.

Recall the shape function for the bulk polymer $\Lambda_{N}=\Lambda((N,N))$
from (2.1). Our first two propositions give the upper and lower bounds for the
right tail.

######  Proposition 2.2.

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

$\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{\mathcal{L}_{0}^{hN^{2/3}},\mathcal{L}_{N}}-\Lambda_{N}\geq
tN^{1/3}\Big{)}\leq e^{-C_{2}\min\\{t^{3/2},tN^{1/3}\\}}.$

######  Proposition 2.3.

There exist positive constants $C_{1},N_{0},t_{0},\epsilon_{0}$ such that for
each $N\geq N_{0}$, $t_{0}\leq t\leq\epsilon_{0}N^{2/3}$, we have

$\mathbb{P}(\log\mathaccent 869{Z}_{0,N}-\Lambda_{N}\geq tN^{1/3})\geq
e^{-C_{1}t^{3/2}}.$

The next proposition gives the upper bound for the left tail.

######  Proposition 2.4.

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

$\mathbb{P}(\log\mathaccent 869{Z}_{0,N}-\Lambda_{N}\leq-tN^{1/3})\leq
e^{-C_{1}\min\\{t^{3/2},tN^{1/3}\\}}.$

Then, we have a variance bound for the free energy in the bulk, which follows
from the estimates for the left and right tails above.

######  Proposition 2.5.

There exists positive constants $C_{1},C_{2},N_{0}$ such that for each $N\geq
N_{0}$, we have

$C_{1}N^{2/3}\leq\textup{$\mathbb{V}$ar}(\log\mathaccent 869{Z}_{0,N})\leq
C_{2}N^{2/3}.$

The next two propositions summarize the loss of free energy from paths having
too much transversal fluctuation.

######  Proposition 2.6.

There exist positive constants $C_{1},C_{2},N_{0}$ such that for each $N\geq
N_{0}$, $t\geq 0$ and $s\geq 0$ we have

$\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{\mathcal{L}^{sN^{2/3}}_{0},\mathcal{L}_{N}\setminus\mathcal{L}_{N}^{(s+t)N^{2/3}}}-\Lambda_{N}\geq-
C_{1}t^{2}N^{1/3}\Big{)}\leq e^{-C_{2}t^{3}}.$

######  Proposition 2.7.

There exist positive constants $C_{1},C_{2},N_{0}$ such that for each $N\geq
N_{0}$, $1\leq t\leq N^{1/3}$ and $0<s<e^{t}$, we have

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

The proposition below shows when constrain our paths to a rectangle that obeys
the KPZ scale, the free energy will not be too small.

######  Proposition 2.8.

For each positive $a_{0}$, there exist positive constants $C_{3},t_{0}$ 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

$\mathbb{P}\Big{(}\log\mathaccent 869{Z}^{\textup{in},{\theta
N^{2/3}}}_{0,{\bf p}}-\Lambda_{N}\leq-
tN^{1/3}\Big{)}\leq\tfrac{\sqrt{t}}{\theta}e^{-C_{\ref{wide_similar_c1}}\theta
t}.$

The next two propositions in this section describe the local fluctuations of
the free energy profile along an anti-diagonal line. The left tail estimate in
Proposition 2.10 is new, and its proof will appear in Section 7.1.

######  Proposition 2.9.

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

$\mathbb{P}\Big{(}\log\mathaccent 869{Z}_{0,\mathcal{L}^{a}_{N}}\
-\log\mathaccent 869{Z}_{0,N}\geq t\sqrt{a}\Big{)}\leq e^{-t^{1/10}}.$

######  Proposition 2.10.

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

$\mathbb{P}\Big{(}\min_{|k|\leq t^{1/20}r^{2/3}}\log\mathaccent
869{Z}_{0,{N}+\underline{{k}}}\ -\log\mathaccent 869{Z}_{0,N}\leq-
tr^{1/3}\Big{)}\leq e^{-t^{1/10}}.$

The last two propositions look at the local fluctuation around the starting
point instead of the endpoint. Recall $\mathaccent 869{Z}$ is defined to
include the weight at the end point but not the starting point and we
basically show this does not affect the local fluctuation. The proof of the
following proposition is in Section 7.2.

######  Proposition 2.11.

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

$\mathbb{P}\Big{(}\log\mathaccent 869{Z}_{\mathcal{L}^{a}_{0},N}\
-\log\mathaccent 869{Z}_{0,N}\geq t\sqrt{a}\Big{)}\leq e^{-t^{1/10}}.$

We omit the proof Proposition 2.12 below, as it is similar to the proof of
Proposition 2.11 above.

######  Proposition 2.12.

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

$\mathbb{P}\Big{(}\min_{|k|\leq t^{1/20}r^{2/3}}\log\mathaccent
869{Z}_{\underline{{k}},N}\ -\log\mathaccent 869{Z}_{0,N}\leq-
tr^{1/3}\Big{)}\leq e^{-t^{1/10}}.$

###  2.4 Estimates for the polymer with boundary

This section summarizes the results for the polymer with general initial
conditions. The proofs of these results use the coupling with stationary
polymer, thus they are postponed to Section 7.

Recall the shape function for the bulk polymer $\Lambda((N,N))=\Lambda_{N}$.
For $k\in\mathbb{Z}$, let us define
$\\{\tau_{\mathcal{L}_{N}^{s}}=k\\}=\bigcup_{{\bf
v}\in\mathcal{L}_{N}^{s}}\mathbb{X}_{(k,-k),{\bf v}}$, which will often be
referred to as the collection of paths with exit time $k$. The partition
function over $\\{\tau_{\mathcal{L}_{N}^{s}}=k\\}$ is given by

$Z^{W}_{\mathcal{L}_{N}^{s}}(\tau_{\mathcal{L}_{N}^{s}}=k)=Z^{W}_{\mathcal{L}_{N}^{s}}(\tau=k)=W_{k}\cdot\mathaccent
869{Z}_{(k,-k),{\mathcal{L}_{N}^{s}}},$

where we omit the subscript under $\tau$ when it is clear from the partition
function. The quenched path measure
$Q^{W}_{\mathcal{L}_{N}^{s}}\\{\tau=k\\}=\frac{Z^{W}_{\mathcal{L}_{N}^{s}}(\tau=k)}{Z^{W}_{\mathcal{L}_{N}^{s}}}.$

######  Proposition 2.13.

Fix $\epsilon>0$ and suppose boundary condition $W$ satisfies assumption
(1.2). There exist positive constants $C_{1},C_{2},N_{0},t_{0}$ such that for
each $N\geq N_{0}$ and $t\geq t_{0}$, we have

$\mathbb{P}\Big{(}\log
Z^{W}_{\mathcal{L}_{N}^{tN^{2/3}}}(|\tau|>(1+\epsilon)tN^{2/3})-\Lambda_{N}\geq-
C_{1}t^{2}N^{1/3}\Big{)}\leq e^{-C_{2}t^{3/2}}.$

######  Proposition 2.14.

Fix $\epsilon>0$ and suppose boundary condition $W$ satisfies assumption
(1.2). There exist positive constants $C_{1},N_{0},t_{0}$ such that for each
$N\geq N_{0}$, $t\geq t_{0}$, we have

  1. (i)

$\mathbb{P}\Big{(}\log
Z^{W}_{\mathcal{L}_{N}^{tN^{2/3}}}-\Lambda_{N}\geq\epsilon
t^{2}N^{1/3}\Big{)}\leq e^{-C_{1}t^{3/2}}.$

  2. (ii)

$\mathbb{P}\Big{(}\log
Z^{W}_{\mathcal{L}_{N}^{tN^{2/3}}}-\Lambda_{N}\geq\epsilon tN^{1/3}\Big{)}\leq
e^{-C_{1}t^{1/2}}.$

######  Proposition 2.15.

Fix $\epsilon>0$ and suppose boundary condition $W$ satisfies assumption
(1.2). There exist positive constants $C_{1},N_{0},t_{0}$ such that for each
$N\geq N_{0}$, $t\geq t_{0}$, we have

$\mathbb{P}\Big{(}\log Z^{W}_{N}-\Lambda_{N}\leq-\epsilon tN^{1/3}\Big{)}\leq
e^{-C_{1}\min\\{t^{3/2},tN^{1/3}\\}}.$

######  Theorem 2.16.

Suppose boundary condition $W$ satisfies assumption (1.2). There exists
positive constants $C_{1},C_{2},N_{0}$ such that for each $N\geq N_{0}$, we
have

$C_{1}N^{2/3}\leq\textup{$\mathbb{V}$ar}(\log{Z}^{W}_{N})\leq C_{2}N^{2/3}.$

######  Theorem 2.17.

Fix $\epsilon>0$ and suppose boundary condition $W$ satisfies assumption
(1.2). There exist positive constants $C_{1},C_{2},N_{0},t_{0}$ such that for
each $N\geq N_{0}$, $t\geq t_{0}$ and $0\leq s\leq tN^{2/3}$, we have

$\mathbb{P}\Big{(}Q^{W}_{\mathcal{L}_{N}^{s}}(|\tau|>(1+\epsilon)tN^{2/3})\geq
e^{-C_{1}t^{2}N^{1/3}}\Big{)}\leq e^{-C_{2}t^{3/2}}.$

######  Theorem 2.18.

Suppose boundary condition $W$ satisfies assumption (1.2). 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

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

##  3 Time correlations for large $r$: Proof of Theorem 1.1

The argument proving Theorem 1.1 is similar to the corresponding argument in
[9], which proved an analogous estimate for the point-to-point polymer;
however, some of the estimates for point-to-point polymer needs to be replaced
by corresponding estimates for polymers with an initial condition stated in
Section 2.4.

Let us assume $c_{0}\leq N-r\leq N/2$. Recall the following identity

$\displaystyle\textup{$\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}$ar}(U).$ (3.1)

We will upper and lower bound the quantity
$1-\textup{$\mathbb{C}$orr}^{2}(U,V)$ for $U=\log Z^{W}_{N}$ and $V=\log
Z^{W}_{r}$ using the next two lemmas.

######  Lemma 3.1.

There exists $C>0$ such that in the above setup we have

$\textup{$\mathbb{V}$ar}(\log Z^{W}_{N}-\log Z^{W}_{r})\leq C(N-r)^{2/3}.$
(3.2)

###### Proof.

We apply the inequality $\textup{$\mathbb{V}$ar}(A)\leq
2(\textup{$\mathbb{V}$ar}(B)+\mathbb{E}[(A-B)^{2}])$ to $A=\log Z^{W}_{N}-\log
Z^{W}_{r}$ and $B=\log\mathaccent 869{Z}_{r,N}$. We have
$\textup{$\mathbb{V}$ar}(B)\leq C(N-r)^{2/3}$ by Proposition 2.5, and
$\mathbb{E}[(A-B)^{2}]\leq C(N-r)^{2/3}$ follows from Theorem 2.18. This
completes the proof of the lemma. ∎

######  Lemma 3.2.

There exists $C>0$ such that in the above set-up we have

$\textup{$\mathbb{V}$ar}(\log Z^{W}_{N}-\lambda\log Z^{W}_{r})\geq
C(N-r)^{2/3}$ (3.3)

for all $\lambda\in\mathbb{R}$.

###### Proof.

Let $\mathcal{F}$ be the $\sigma$-algebra of the weights in
$\mathcal{L}_{r}^{\leq}$. Note that $\log Z^{W}_{r}$ is
$\mathcal{F}$-measurable.

Then,

$\displaystyle\textup{$\mathbb{V}$ar}(\log Z^{W}_{N}|\mathcal{F})$
$\displaystyle=\textup{$\mathbb{V}$ar}(\log Z^{W}_{N}-\log
Z^{W}_{r}|\mathcal{F})$ $\displaystyle=\mathbb{E}\Big{[}\Big{(}\log
Z^{W}_{N}-\log Z^{W}_{r}-\mathbb{E}[\log Z^{W}_{N}-\log
Z^{W}_{r}|\mathcal{F}]\Big{)}^{2}\Big{|}\mathcal{F}\Big{]}.$ (3.4)

We develop a lower bound for (3.4) on a positive probability $\mathcal{F}$
measurable set. By Theorem 2.18,

$\Big{|}\mathbb{E}[\log Z^{W}_{N}-\log Z^{W}_{r}]-\mathbb{E}[\log\mathaccent
869{Z}_{r,N}]\Big{|}\leq C(N-r)^{1/3}.$ (3.5)

In Proposition 2.3 the centering $\Lambda_{N}$ can be replaced with
$\mathbb{E}[\log\mathaccent 869{Z}_{0,N}]$ because $\mathbb{E}[\log\mathaccent
869{Z}_{0,N}]\leq\Lambda_{N}$ by superadditivity. Then, Proposition 2.3 and
(3.5) give

$\displaystyle e^{-ct^{3/2}}$
$\displaystyle\leq\mathbb{P}\bigl{(}\log\mathaccent
869{Z}_{r,N}-\mathbb{E}[\log\mathaccent 869{Z}_{r,N}]\geq
t(N-r)^{1/3}\bigr{)}$ $\displaystyle\leq\mathbb{P}\bigl{(}\log\mathaccent
869{Z}_{r,N}-\mathbb{E}[\log Z^{W}_{N}-\log
Z^{W}_{r}]\geq(t-C)(N-r)^{1/3}\bigr{)}.$

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

$A_{r,N}=\bigl{\\{}\log\mathaccent 869{Z}_{r,N}-\mathbb{E}[\log Z^{W}_{N}-\log
Z^{W}_{r}]\geq s_{0}(N-r)^{1/3}\bigr{\\}},$ (3.6)

Then, $A_{r,N}$ is independent of $\mathcal{F}$ and $\mathbb{P}(A_{r,N})$ is
bounded below independently of $r$ and $N$ for $N-r$ sufficiently large. Next,
using Chebyshev’s inequality we get

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

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

$\Big{|}\mathbb{E}[\log Z^{W}_{N}-\log Z^{W}_{r}|\mathcal{F}]-\mathbb{E}[\log
Z^{W}_{N}-\log Z^{W}_{r}]\Big{|}\leq\frac{s_{0}}{10}(N-r)^{1/3}.$ (3.7)

On $A_{r,N}\cap B_{r,N}$ we have the following bound, using first
superadditivity $\log Z^{W}_{N}-\log Z^{W}_{r}\geq\log\mathaccent
869{Z}_{r,N}$, then (3.7) and finally (3.6):

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

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

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

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

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

This completes the proof of the lemma. ∎

We can now complete the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

Using (3.1) together with the bounds on $\textup{$\mathbb{V}$ar}(\log
Z^{W}_{N})$ from Theorem 2.16 it follows from Lemma 3.1 and Lemma 3.2 that
there exists $C^{\prime}_{1},C^{\prime}_{2}>0$ such that in the set-up of
Theorem 1.1

$C^{\prime}_{1}\left(\frac{N-r}{N}\right)^{2/3}\leq
1-\textup{$\mathbb{C}$orr}^{2}(\log Z^{W}_{N},\log Z^{W}_{r})\leq
C_{2}^{\prime}\left(\frac{N-r}{N}\right)^{2/3}.$ (3.8)

Since $\textup{$\mathbb{C}$orr}(\log Z^{W}_{N},\log Z^{W}_{r})\leq 1$, The
lower bound in (3.8) immediately implies the upper bound in Theorem 1.1. To
prove the lower bound in Theorem 1.1, notice that it suffices to prove it only
in the case where $(N-r)/N$ is sufficiently small; the remaining case can be
proved by adjusting the constants as necessary. If $(N-r)/N$ is suffficiently
small, it follows from Lemma 3.1 together with Theorem 2.16 that
$\textup{$\mathbb{C}$orr}(\log Z^{W}_{N},\log Z^{W}_{r})\geq 0$; and the lower
bound in Theorem 1.1 is immediate from the upper bound in (3.8). ∎

######  Remark 3.3.

Note that Theorem 1.1 is valid for any such initial condition satisfying
Assumption B1, random as well as deterministic. In particular, this remains
true for the flat initial condition, i.e., $W\equiv 1$. A similar result, for
a narrower class of initial conditions, was obtained in [18] in the zero
temperature setup; see Section 8 for more details.

##  4 Time correlations for small $r$: Proof of Theorem 1.2

Notice that by Theorem 2.16, Theorem 1.2 is equivalent to showing, for some
$C,C^{\prime}>0$

$C^{\prime}r^{2/3}\leq\textup{$\mathbb{C}$ov}(\log Z^{W}_{r},\log
Z^{W}_{N})\leq Cr^{2/3}.$

Separate proofs of the upper and the lower bound in the display above are
given in Sections 4.1 and 4.2 respectively.

###  4.1 Upper bound in Theorem 1.2

First, note that if $r\geq N/1000$, our estimate follows directly from Cauchy-
Schwarz inequality and Theorem 2.16 as

$\textup{$\mathbb{C}$ov}(\log Z^{W}_{r},\log
Z^{W}_{N})\leq\textup{$\mathbb{V}$ar}(\log
Z^{W}_{r})^{1/2}\textup{$\mathbb{V}$ar}(\log Z^{W}_{N})^{1/2}\leq Cr^{2/3}.$

Thus for the rest of this section, we will always assume that $0\leq r\leq
N/1000$.

We start with defining some notation. Let $Z^{W,\textup{in},jr^{2/3}}_{r}$
denote the partition function where the paths are contained inside
$R_{0,r}^{jr^{2/3}}$. By convention, we set $\log Z^{W,\textup{in},0}_{r}=0$.
Notice also that $Z^{W,\textup{in},r}_{r}=Z^{W}_{r}$ by definition. Let
$W^{\prime}$ be an independent copy of $W$, and define another initial
condition $W^{*,j}$ (which interpolates between $W$ and $W^{\prime}$) such
that

$\displaystyle\log W^{*,j}_{i+1}-\log W^{*,j}_{i}$ $\displaystyle=\log
W^{\prime}_{i+1}-\log W^{\prime}_{i}\text{ for $-jr^{2/3}\leq i\leq
jr^{2/3}-1$}$ $\displaystyle\log W^{*,j}_{i+1}-\log W^{*,j}_{i}$
$\displaystyle=\log W_{i+1}-\log W_{i}\text{ otherwise.}$

Let $Z^{W^{*,j},\textup{out},k}_{N}$ denote the partition function using
initial condition $W^{*,j}$ and the paths inside $\mathbb{Z}^{2}\setminus
R_{0,r}^{k}$. We set $Z^{W^{*,0},\textup{out},0}_{N}=Z^{W}_{N}$ by convention.

Without loss of generality, we shall assume that both $r^{2/3}$ and $r^{1/3}$
are integers. To simplify notation, for $1<j\leq r^{1/3},1<k\leq r^{1/3}$, let

$\displaystyle U_{j}$ $\displaystyle=\log Z^{W,\textup{in},jr^{2/3}}_{r}-\log
Z^{W,\textup{in},(j-1)r^{2/3}}_{r}$ $\displaystyle V_{k}$ $\displaystyle=\log
Z^{W^{*,(k-1)r^{2/3}},\textup{out},(k-1)r^{2/3}}_{N}-\log
Z^{W^{*,kr^{2/3}},\textup{out},kr^{2/3}}_{N}.$

Then we can rewrite the covariance as follows

$\displaystyle\textup{$\mathbb{C}$ov}(\log Z^{W}_{r},\log
Z^{W}_{N})=\Big{(}\sum_{j=1}^{r^{1/3}}\sum_{k=1}^{r^{1/3}}\textup{$\mathbb{C}$ov}(U_{j},V_{k})\Big{)}+\textup{$\mathbb{C}$ov}(Z^{W}_{r},Z^{W^{*,r},\textup{out},r}_{N}).$
(4.1)

A similar but different decomposition of covariance was also used in a related
problem in [6]. The proof shall proceed by bounding the two terms separately.
To bound the first term we shall use bounds on the variances of the random
variables appearing in the above decomposition followed by an application of
the Cauchy-Schwarz inequality and the mixing hypothesis (1.3) in Assumption
B2. The second term shall be bounded by a different decomposition. We shall
need the following propositions which are proved in the subsequent
subsections.

######  Proposition 4.1.

Suppose the boundary condition $W$ satisfies assumption (1.2). There exist
positive constants $C_{1},C_{2},r_{0}$ such that for each $r\geq r_{0}$ and
$0\leq j\leq r^{1/3}-1$

$\textup{$\mathbb{V}$ar}\Big{(}\log Z^{W,\textup{in},(j+1)r^{2/3}}_{r}-\log
Z^{W,\textup{in},jr^{2/3}}_{r}\Big{)}\leq C_{2}e^{-C_{1}j}r^{2/3}.$

######  Proposition 4.2.

Suppose the boundary condition $W$ satisfies assumptions (1.2) and (1.4).
There exist positive constants $C_{1},r_{0},N_{0}$ such that for each $r\geq
r_{0}$, $N\geq N_{0}\vee 1000r$ and $0\leq k\leq r^{1/3}-1$,

$\textup{$\mathbb{V}$ar}\Big{(}\log
Z^{W^{*,kr^{2/3}},\textup{out},kr^{2/3}}_{N}-\log
Z^{W^{*,(k+1)r^{2/3}},\textup{out},(k+1)r^{2/3}}_{N}\Big{)}\leq
C_{1}{(k+1)}^{40}r^{2/3}.$

######  Proposition 4.3.

Suppose the boundary condition $W$ satisfies assumptions (1.2) and (1.4).
There exists $C,r_{0}\geq 0$ such that for all $r>r_{0}$

$\textup{$\mathbb{C}$ov}(Z^{W}_{r},Z^{W^{*,r},\textup{out},r}_{N})\leq
Cr^{2/3}.$

Notice that Proposition 4.3 is trivial in the case where $X_{i}:=\log
W_{i}-\log W_{i-1}$ is an independent sequence; so the proof becomes
substantially simpler. Assuming Propositions 4.1, 4.2 and 4.3 we can now
complete the proof of the upper bound.

###### Proof of Theorem 1.2, upper bound.

Recall the decomposition (4.1). Observe that by Proposition 4.3, it suffices
to upper bound the first sum thereby $Cr^{2/3}$. For this, we further
decompose the first sum in (4.1) as

$\displaystyle\sum_{j=1}^{r^{1/3}}\sum_{k=1}^{r^{1/3}}\textup{$\mathbb{C}$ov}(U_{j},V_{k})=\sum_{j=1}^{r^{1/3}}\sum_{k=1}^{j}\textup{$\mathbb{C}$ov}(U_{j},V_{k})+\sum_{j=1}^{r^{1/3}}\sum_{k=j+1}^{r^{1/3}}\textup{$\mathbb{C}$ov}(U_{j},V_{k}).$
(4.2)

Notice that, by the Cauchy-Schwarz inequality together with Propositions 4.1
and 4.2, the first term on the right of (4.2) is upper bounded by

$\sum_{j=1}^{r^{1/3}}\sum_{k=1}^{j}\sqrt{\textup{$\mathbb{V}$ar}(U_{j})}\sqrt{\textup{$\mathbb{V}$ar}(V_{k})}\leq
Cr^{2/3}\sum_{j=1}^{r^{1/3}}e^{-Cj}\sum_{k=1}^{j}k^{20}\leq Cr^{2/3}.$

Therefore it suffices to prove a similar bound for the second sum. For this,
we shall show that for $r^{1/3}\geq k\geq j+1$,

$\textup{$\mathbb{C}$ov}(U_{j},V_{k})\leq\frac{1}{((k-j-1)r^{2/3}+1)^{100}}\sqrt{\textup{$\mathbb{V}$ar}(U_{j})}\sqrt{\textup{$\mathbb{V}$ar}(V_{k})}.$
(4.3)

Indeed, observe that $(k-j-1)r^{2/3}+1\geq(k-j)$ and also that

$\sum_{k=j+1}^{r^{1/3}}\frac{k^{20}}{(k-j)^{100}}\leq C(j+1)^{21}$

for some $C>0$. Therefore, by (4.3) and Propositions 4.1 and 4.2, it follows
that the second term in (4.2) is upper bounded by $Cr^{2/3}$ for some $C>0$.

It remains to show (4.3). Consider the $\sigma$-algebras,

$\displaystyle\begin{split}\mathcal{F}^{j}_{1}&=\sigma\Big{(}\\{Y_{\bf
z}\\}_{{\bf z}\in R_{0,r}^{jr^{2/3}}}\Big{)},\\\
\mathcal{F}^{k}_{2}&=\sigma\Big{(}\\{Y_{\bf z}\\}_{{\bf
z}\in\mathbb{Z}^{2}\setminus R_{0,r}^{kr^{2/3}}}\bigcup\\{\log
W^{\prime}_{i+1}-\log
W^{\prime}_{i}\\}_{i\in[-kr^{2/3},kr^{2/3}-1]}\Big{)}.\end{split}$ (4.4)

Notice that if $k\geq j+2$, then $\mathcal{F}^{j+1}_{1}$ and
$\mathcal{F}^{k-1}_{2}$ are independent, further $U_{j}$ is
$\mathcal{F}^{j+1}_{1}$ measurable and $V_{k}$ is $\mathcal{F}^{k-1}_{2}$
measurable.

Fix $1\leq j,k\leq r^{1/3}$ with $k\geq j+2$, and set
$\mathcal{F}=\sigma(\mathcal{F}^{j+1}_{1},\mathcal{F}^{k-1}_{2})$. By the
above observation,
$\mathbb{E}[U_{j}|\mathcal{F}]=\mathbb{E}[U_{j}\mid\mathcal{F}^{j+1}_{1}]$ and
$\mathbb{E}[V_{k}|\mathcal{F}]=\mathbb{E}[V_{k}\mid\mathcal{F}^{k-1}_{2}]$.
Hence, these two random variables are independent. Observe also that
conditional on a fixed realisation of $\mathcal{F}$, $U_{j}$ and $V_{k}$ are
functions of $X_{i}=\log W_{i}-\log W_{i-1}$ restricted to disjoint index sets
$[-(j+1)r^{2/3}+1,(j+1)r^{2/3}]$ and
$\mathbb{Z}\setminus[-(k-1)r^{2/3}+1,kr^{2/3}]$ respectively, and these sets
have mutual distance $(k-j-2)r^{2/3}+1$. Therefore, using assumption (1.3)

$\displaystyle\textup{$\mathbb{C}$ov}(U_{j},V_{k})$
$\displaystyle=\mathbb{E}[\textup{$\mathbb{C}$ov}(U_{j},V_{k}|\mathcal{F})]+\textup{$\mathbb{C}$ov}(\mathbb{E}[U_{j}|\mathcal{F}],\mathbb{E}[V_{k}|\mathcal{F}])$
$\displaystyle=\mathbb{E}[\textup{$\mathbb{C}$ov}(U_{j},V_{k}|\mathcal{F})]$
(4.5) by (1.3)
$\displaystyle\leq\mathbb{E}\Big{[}\frac{1}{((k-j-2)r^{2/3}+1)^{100}}\sqrt{\textup{$\mathbb{V}$ar}(U_{j}|\mathcal{F})}\sqrt{\textup{$\mathbb{V}$ar}(V_{k}|\mathcal{F})}\Big{]}$
$\displaystyle\leq\frac{1}{((k-j-2)r^{2/3}+1)^{100}}\sqrt{\textup{$\mathbb{V}$ar}(U_{j})}\sqrt{\textup{$\mathbb{V}$ar}(V_{k})}$
(4.6)

where in the last step we have used the Cauchy-Schwarz inequality together
with the fact the expectation of conditional variance is bounded by the
unconditional variance. This establishes (4.3). With this, we have finished
the proof of the upper bound.

∎

####  4.1.1 Proof of Proposition 4.1

Since $\log Z^{W}_{r}\geq\log Z^{W,\textup{in},(j+1)r^{2/3}}_{r}\geq\log
Z^{W,\textup{in},jr^{2/3}}_{r}$, it suffices for us to show for $0\leq j\leq
r^{1/3}$,

$\mathbb{E}\Big{[}\Big{(}\log Z^{W}_{r}-\log
Z^{W,\textup{in},jr^{2/3}}_{r}\Big{)}^{2}\Big{]}\leq e^{-Cj}r^{2/3}.$

To show this, we will first prove show that

$\mathbb{P}\Big{(}\log Z^{W}_{r}-\log Z^{W,\textup{in},jr^{2/3}}_{r}\geq
e^{-C^{\prime}j^{2}r^{1/3}}\Big{)}\leq e^{-Cj^{3}}.$ (4.7)

Let us rewrite the event inside (4.7). Let $Z^{W,\textup{exit},jr^{2/3}}_{r}$
denote the partition function over paths from $\mathcal{L}_{0}$ to $(r,r)$
that exit the two diagonal sides of $R_{0,r}^{jr^{2/3}}$. In our calculation
below, the third equality holds because
$Z^{W}_{r}=Z^{W,\textup{in},jr^{2/3}}_{r}+Z^{W,\textup{exit},jr^{2/3}}_{r}$.
Also note Taylor’s theorem states that $1-e^{z}\approx-z$ when $|z|$ is small,
so the last set inclusion below holds provided that that $r\geq r_{0}$ is
sufficiently large. We have

$\displaystyle\\{\text{the event in \eqref{event_A}}\\}$
$\displaystyle=\Big{\\{}\frac{Z^{W}_{r}}{Z^{W,\textup{in},jr^{2/3}}_{r}}\geq\exp(e^{-C^{\prime}j^{2}r^{1/3})}\Big{\\}}$
$\displaystyle=\Big{\\{}\frac{Z^{W,\textup{in},jr^{2/3}}_{r}}{Z^{W}_{r}}\leq\exp(-e^{-C^{\prime}j^{2}r^{1/3})}\Big{\\}}$
$\displaystyle=\Big{\\{}\frac{Z^{W,\textup{exit},jr^{2/3}}_{r}}{Z^{W}_{r}}\geq
1-\exp(-e^{-C^{\prime}j^{2}r^{1/3})}\Big{\\}}\subset\Big{\\{}\frac{Z^{W,\textup{exit},jr^{2/3}}_{r}}{Z^{W}_{r}}\geq\tfrac{1}{2}e^{-C^{\prime}j^{2}r^{1/3}}\Big{\\}}.$

Thus, to show (4.7), it suffices to show

$\mathbb{P}\Big{(}\frac{Z^{W,\textup{exit},jr^{2/3}}_{r}}{Z^{W}_{r}}\geq
e^{-C^{\prime}j^{2}r^{1/3}}\Big{)}\leq e^{-Cj^{3}}.$ (4.8)

Note here we drop the constant $\tfrac{1}{2}$ by modifying $C^{\prime}$.

To get (4.8), let $\mathfrak{A}$ be the collection of polymer paths from
$\mathcal{L}_{0}$ to $(r,r)$ which exit $R_{0,r}^{jr^{2/3}}$. We break
$\mathfrak{A}$ into two groups:

$\mathfrak{A}_{1}=\bigcup_{|k|\geq\tfrac{j}{2}r^{2/3}}\mathfrak{A}\cap\mathbb{X}_{\underline{{k}},N}\qquad\text{and}\qquad\mathfrak{A}_{2}=\bigcup_{|k|<\tfrac{j}{2}r^{2/3}}\mathfrak{A}\cap\mathbb{X}_{\underline{{k}},N}.$

We will show that for $i=1,2$ and $C^{\prime}>0$ sufficiently small,

$\mathbb{P}\big{(}\log Z^{W}_{r}(\mathfrak{A}_{i})-\log
Z^{W}_{r}\geq-C^{\prime}j^{2}r^{1/3}\big{)}\leq e^{-Cj^{3}}.$ (4.9)

Note for $i=1$, this estimate follows directly from Theorem 2.17. For $i=2$,
by a union bound,

$\displaystyle\eqref{est_i12}$ $\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{W}_{r}(\mathfrak{A}_{2})-\Lambda_{N}\geq-2C^{\prime}j^{2}r^{1/3}\Big{)}+\mathbb{P}\Big{(}\log
Z^{W}_{r}-\Lambda_{N}\leq-C^{\prime}j^{2}r^{1/3}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\log\mathaccent
869{Z}^{\textup{exit},jr^{2/3}}_{\mathcal{L}_{0}^{\frac{j}{2}r^{2/3}},r}-\Lambda_{N}\geq-3C^{\prime}j^{2}r^{1/3}\Big{)}+\mathbb{P}\Big{(}\max_{|k|\leq\frac{j}{2}r^{2/3}}\log
W_{k}\geq\tfrac{1}{3}C^{\prime}j^{2}r^{1/3}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad+\mathbb{P}\Big{(}\log
Z^{W}_{r}-\Lambda_{N}\leq-C^{\prime}j^{2}r^{1/3}\Big{)},$

and all three probabilities are bounded by $e^{-Cj^{3}}$ using Proposition
2.7, assumption (1.2) and Proposition 2.15. Then, because
$Z^{W,\textup{exit},jr^{2/3}}_{r}=Z^{W}_{r}(\mathfrak{A}_{1})+Z^{W}_{r}(\mathfrak{A}_{2})$,
(4.9) implies (4.8). And we have finished the proof for (4.7).

Next, let us denote the event inside the probability of (4.7) as $A$. Then,

$\displaystyle\mathbb{E}\Big{[}\big{(}\log Z^{W}_{r}-\log
Z^{W,\textup{in},jr^{2/3}}_{r}\big{)}^{2}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\big{(}\log Z^{W}_{r}-\log
Z^{W,\textup{in},jr^{2/3}}_{r}\big{)}^{2}\mathbbm{1}_{A^{c}}\Big{]}+\mathbb{E}\Big{[}\big{(}\log
Z^{W}_{r}-\log
Z^{W,\textup{in},jr^{2/3}}_{r}\big{)}^{2}\mathbbm{1}_{A}\Big{]}$
$\displaystyle\leq e^{-Cj}+\mathbb{E}\Big{[}\big{(}\log Z^{W}_{r}-\log
Z^{W,\textup{in},jr^{2/3}}_{r}\big{)}^{4}\Big{]}^{1/2}\mathbb{P}(A)^{1/2}.$

To finish the proof, it remains to show that

$\mathbb{E}\Big{[}\big{(}\log Z^{W}_{r}-\log
Z^{W,\textup{in},jr^{2/3}}_{r}\big{)}^{4}\Big{]}\leq Cr^{4/3}.$

And this follows directly from Proposition 2.8 and Proposition 2.14 which
essentially provide an exponential upper bound for the right tails of $\log
Z^{W}_{r}-\log Z^{W,\textup{in},jr^{2/3}}_{r}$.

####  4.1.2 Proof of Proposition 4.2

We shall first prove the following proposition.

######  Proposition 4.4.

Suppose the boundary condition $W$ satisfies assumption (1.2) and (1.4). Then
for $0\leq k\leq r^{1/3}$ we have

$\mathbb{E}[(\log Z^{W}_{N}-\log Z^{W,\textup{out},kr^{2/3}}_{N})^{2}]\leq
C(k+1)^{40}r^{2/3}.$ (4.10)

###### Proof.

First, note that when $k=0$, the expectation is zero by our convention. We may
now assume that $k\geq 1$.

Let $Z^{W,\textup{touch},kr^{2/3}}_{N}$ denote the partition function that
sums over all paths from $\mathcal{L}_{0}$ to $(N,N)$ which intersect the
rectangle $R^{kr^{2/3}}_{0,r}$, then

$\max\Big{\\{}\log Z^{W,\text{out},kr^{2/3}}_{N},\log
Z^{W,\textup{touch},kr^{2/3}}_{N}\Big{\\}}\leq\log
Z^{W}_{N}\leq\max\Big{\\{}\log Z^{W,\text{out},kr^{2/3}}_{N},\log
Z^{W,\textup{touch},kr^{2/3}}_{N}\Big{\\}}+2.$ (4.11)

Let us define the event

$B=\Big{\\{}\log Z^{W,\text{out},kr^{2/3}}_{N}<\log
Z^{W,\textup{touch},kr^{2/3}}_{N}\Big{\\}}.$

And we split the expectation in (4.10) according to $B$,

$\eqref{hard_est}=\mathbb{E}[(\log Z^{W}_{N}-\log
Z^{W,\textup{out},kr^{2/3}}_{N})^{2}\mathbbm{1}_{B}]+\mathbb{E}[(\log
Z^{W}_{N}-\log Z^{W,\textup{out},kr^{2/3}}_{N})^{2}\mathbbm{1}_{B^{c}}].$

Now, using (4.11), the expectation term above with $B^{c}$ is bounded by
$2^{2}$. Since $r$ is sufficiently large, to get (4.10), it remains to prove
that

$\mathbb{E}[(\log Z^{W,\textup{touch},kr^{2/3}}_{N}-\log
Z^{W,\textup{out},kr^{2/3}}_{N})^{2}\mathbbm{1}_{B}]\leq C(k+1)^{40}r^{2/3}.$
(4.12)

Because we are on the event $B$, it suffices for us to show that for each
$t\geq t_{0}$ sufficiently large and $1\leq k\leq r^{1/3}$

$\mathbb{P}\Big{(}\log Z^{W,\textup{touch},kr^{2/3}}_{N}-\log
Z^{W,\textup{out},kr^{2/3}}_{N}\geq{t}k^{20}r^{1/3}\Big{)}\leq
e^{-Ct^{1/100}}.$ (4.13)

First, let us focus on the term $\log Z^{W,\textup{touch},kr^{2/3}}_{N}$
inside the probability above. Because of the paths from $\log
Z^{W,\textup{touch},kr^{2/3}}_{N}$ must touch the rectangle
$R_{0,r}^{kr^{2/3}}$ by definition, we may replace $\log
Z^{W,\textup{touch},kr^{2/3}}_{N}$ by

$\max_{{\bf p}_{1}\in\mathcal{L}_{0}^{100r}}\max_{{\bf
p}_{2}\in\mathcal{L}_{8r}^{100r}}\log Z^{W}_{{\bf p}_{1}}+\log\mathaccent
869{Z}^{\textup{touch},kr^{2/3}}_{{\bf p}_{1},{\bf p}_{2}}+\log\mathaccent
869{Z}_{{\bf p}_{2},N},$ (4.14)

because their absolute difference is at most $100\log r$. Note here that the
last free energy $\log\mathaccent 869{Z}_{{\bf p}_{2},N}$ is well defined
since $8r\leq N/10$, and this inequality holds because we are considering the
case $r\leq N/1000$.

Next, let us denote the two maximizers in (4.14) as ${\bf p}_{1}^{*}$ and
${\bf p}_{2}^{*}$ and define

$H=\Big{\\{}\max_{i\in\\{1,2\\}}|{\bf p}_{i}^{*}\cdot({\bf e}_{1}-{\bf
e}_{2})|\geq tkr^{2/3}\Big{\\}}.$

Lemma 4.5 at the end of the subsection states that $\mathbb{P}(H)\leq
e^{-Ct^{1/100}}.$ With this, let us start the estimate for (4.13), we have

right side of (4.13) $\displaystyle\leq\mathbb{P}\Big{(}\Big{\\{}\log
Z^{W,\textup{touch},kr^{2/3}}_{N}-\log
Z^{W,\textup{out},kr^{2/3}}_{N}\geq{t}kr^{1/3}\Big{\\}}\cap
H^{c}\Big{)}+e^{-Ct^{1/100}}.$

Looking at the probability term above,

$\displaystyle\mathbb{P}\Big{(}\Big{\\{}\log
Z^{W,\textup{touch},kr^{2/3}}_{N}-\log
Z^{W,\textup{out},kr^{2/3}}_{N}\geq{t}kr^{1/3}\Big{\\}}\cap H^{c}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\Big{[}\max_{{\bf
p_{1}}\in\mathcal{L}_{0}^{tkr^{2/3}}}\max_{{\bf
p_{2}}\in\mathcal{L}_{8r}^{tkr^{2/3}}}\log Z^{W}_{{\bf p}_{1}}+\log\mathaccent
869{Z}_{{\bf p}_{1},{\bf p}_{2}}+\log\mathaccent 869{Z}_{{\bf
p}_{2},N}\Big{]}$ $\displaystyle-\Big{[}\log
Z^{W}_{\underline{{2kr^{2/3}}}}+\log\mathaccent
869{Z}^{\textup{in},R_{\underline{{2kr^{2/3}}},r+\underline{{2kr^{2/3}}}}^{kr^{2/3}}}_{\underline{{2kr^{2/3}}},r+\underline{{2kr^{2/3}}}}+\log\mathaccent
869{Z}_{r+\underline{{2kr^{2/3}}},8r}+\log\mathaccent
869{Z}_{8r,N}\Big{]}\geq\tfrac{1}{2}{t}k^{20}r^{1/3}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf
p_{1}}\in\mathcal{L}_{0}^{tkr^{2/3}}}\log Z^{W}_{{\bf p}_{1}}-\log
Z^{W}_{\underline{{2kr^{2/3}}}}\geq\tfrac{1}{6}tk^{20}r^{1/3}\Big{)}$ (4.15)
$\displaystyle+\mathbb{P}\Big{(}\max_{{\bf
p_{1}}\in\mathcal{L}_{0}^{tkr^{2/3}}}\max_{{\bf
p_{2}}\in\mathcal{L}_{{t}}^{tkr^{2/3}}}\log\mathaccent 869{Z}_{{\bf
p}_{1},{\bf p}_{2}}-\log\mathaccent
869{Z}^{\textup{in},R_{\underline{{2kr^{2/3}}},r+\underline{{2kr^{2/3}}}}^{kr^{2/3}}}_{\underline{{2kr^{2/3}}},r+\underline{{2kr^{2/3}}}}-\log\mathaccent
869{Z}_{r+\underline{{2kr^{2/3}}},8r}\geq\tfrac{1}{6}tk^{20}r^{1/3}\Big{)}$
(4.16) $\displaystyle\qquad+\mathbb{P}\Big{(}\max_{{\bf
p_{2}}\in\mathcal{L}_{{t}}^{tkr^{2/3}}}\log\mathaccent 869{Z}_{{\bf
p}_{2},N}-\log\mathaccent
869{Z}_{8r,N}\geq\tfrac{1}{6}k^{20}{t}r^{1/3}\Big{)}$ (4.17)

Note that by assumptions (1.2) and (1.4),

$\eqref{item111}\leq\mathbb{P}\Big{(}\max_{{\bf
p_{1}}\in\mathcal{L}_{0}^{tkr^{2/3}}}\log Z^{W}_{{\bf
p}_{1}}\geq\tfrac{1}{12}tk^{20}r^{1/3}\Big{)}+\mathbb{P}\Big{(}\log
Z^{W}_{\underline{{2kr^{2/3}}}}\leq-\tfrac{1}{12}tk^{20}r^{1/3}\Big{)}\leq
e^{-Ct^{1/10}}.$

And (4.17) $\leq e^{-Ct^{1/10}}$ following from Proposition 2.11. Finally, we
estimate (4.16),

$\displaystyle\eqref{item222}$
$\displaystyle\leq\mathbb{P}\Big{(}\Big{[}\max_{{\bf
p_{1}}\in\mathcal{L}_{0}^{tkr^{2/3}}}\max_{{\bf
p_{2}}\in\mathcal{L}_{8r}^{tkr^{2/3}}}\log\mathaccent 869{Z}_{{\bf p}_{1},{\bf
p}_{2}}-\Lambda_{8r}\geq\tfrac{1}{12}tk^{20}r^{1/3}\Big{)}$ (4.18)
$\displaystyle\qquad\qquad+\mathbb{P}\Big{(}\log\mathaccent
869{Z}^{\textup{in},R_{\underline{{2kr^{2/3}}},r+\underline{{2rk^{2/3}}}}^{kr^{2/3}}}_{\underline{{2kr^{2/3}}},r+\underline{{2kr^{2/3}}}}-\Lambda_{r}\leq-\tfrac{1}{24}tk^{20}r^{1/3}\Big{)}$
(4.19) $\displaystyle\qquad\qquad+\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{r+\underline{{2kr^{2/3}}},8r}-\Lambda_{7r}\leq-\tfrac{1}{24}tk^{20}r^{1/3}\Big{)}$
(4.20)

The terms (4.18) and (4.19) are upper bounded by $e^{-t^{1/10}}$ following
from Proposition 2.2 and Proposition 2.8. For (4.20), we split the estimate
into two cases, depending on the value of $k$.

Note that if $k\geq r^{1/4}$, then

$\eqref{it3}\leq\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{r+\underline{{2kr^{2/3}}},r}\leq-{t}k^{4}r^{4}\Big{)}.$

We can chose an up-right path $\gamma$ between $(r+{2kr^{2/3}},r-{2kr^{2/3}})$
and $(8r,8r)$ in some deterministic ordering, and then

$\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{r+\underline{{2kr^{2/3}}},8r}\leq-{t}k^{4}r^{4}\Big{)}\leq\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{r+\underline{{2kr^{2/3}}},8r}(\\{\gamma\\})\leq-{t}k^{4}r^{4}\Big{)}\leq
e^{-t^{1/10}},$

where the last inequality holds by Theorem A.1 since $\log\mathaccent
869{Z}_{r+\underline{{2kr^{2/3}}},8r}(\\{\gamma\\})$ is just a sum of i.i.d.
inverse-gamma random variables.

On the other hand if $k\leq r^{1/4}$, then for $r\geq r_{0}$ sufficiently
large, $kr^{2/3}\leq\epsilon_{0}r$ where $\epsilon_{0}$ is the constant from
Proposition 2.1. And from the same proposition, we obtain

$\Big{|}\Lambda((7r-{2kr^{2/3}},7r+{2kr^{2/3}}))-\Lambda_{7r}\Big{|}\leq
Ck^{2}r^{1/3}.$

Then, we obtain

$\eqref{it3}\leq\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{r+\underline{{2kr^{2/3}}},8r}-\Lambda((7r-{2kr^{2/3}},7r+{2kr^{2/3}}))\leq-\tfrac{1}{48}tk^{20}r^{1/3}\Big{)}\leq
e^{-t^{1/10}}$

where the last inequality comes from Proposition 2.4. With this, we have
finished the proof of Proposition 4.4.

∎

We can now give a proof of Proposition 4.2.

###### Proof of Proposition 4.2.

First, we note that by a union bound $W^{*,j}$ also satisfies assumptions
(1.2) and (1.4). Let $K\in\mathbb{Z}_{>0}$ be given, if $jr^{2/3}\geq K$, then
$W^{*,j}_{K}=W^{\prime}_{K}$ and the claim is clear. Suppose $jr^{2/3}<K$,
then

$\displaystyle\log W^{*,j}_{K}$ $\displaystyle=\log W_{K}-\log
W_{jr^{2/3}}+\log W^{\prime}_{jr^{2/3}}$ $\displaystyle\max_{0\leq k\leq
K}\log W^{*,j}_{k}$ $\displaystyle\leq\max_{0\leq k\leq jr^{2/3}}\log
W^{\prime}_{k}+\max_{0\leq k\leq K}\log W_{k}-\log W_{jr^{2/3}},$

and the claim follows from a union bound.

Then, using Proposition 4.4 and a triangle inequality we get

$\displaystyle\mathbb{E}\Big{[}\Big{(}\log$ $\displaystyle
Z^{W^{*,kr^{2/3}},\textup{out},kr^{2/3}}_{N}-\log
Z^{W,\textup{out},kr^{2/3}}_{N}\Big{)}^{2}\Big{]}^{1/2}$
$\displaystyle\leq\mathbb{E}\Big{[}\Big{(}\log
Z^{W^{*,kr^{2/3}},\textup{out},kr^{2/3}}_{N}-\log
Z^{W^{*,kr^{2/3}}}_{N}\Big{)}^{2}\Big{]}^{1/2}+\mathbb{E}\Big{[}\Big{(}\log
Z^{W}_{N}-\log Z^{W,\textup{out},kr^{2/3}}_{N}\Big{)}^{2}\Big{]}^{1/2}$
$\displaystyle\qquad\qquad+\mathbb{E}\Big{[}\Big{(}\log
Z^{W^{*,kr^{2/3}}}_{N}-\Lambda_{N}\Big{)}^{2}\Big{]}^{1/2}+\mathbb{E}\Big{[}\Big{(}\log
Z^{W}_{N}-\Lambda_{N}\Big{)}^{2}\Big{]}^{1/2}.$ (4.21)

Since $W^{*,j}$ and $W$ both satisfy the hypothesis of Proposition 4.4, it
follows that the first two terms above are bounded by $C(k+1)^{20}r^{1/3}$
while the last two terms are bounded by $Cr^{1/3}$ using Proposition 2.14 and
Proposition 2.15. It therefore follows that

$\mathbb{E}[(\log Z^{W^{*,kr^{2/3}},\textup{out},kr^{2/3}}_{N}-\log
Z^{W,\textup{out},kr^{2/3}}_{N})^{2}]\leq C(k+1)^{40}r^{2/3}$ (4.22)

for some $C>0$. Combining (4.22) with Proposition 4.4 gives

$\textup{$\mathbb{V}$ar}\Big{(}\log
Z^{W^{*,kr^{2/3}},\textup{out},kr^{2/3}}_{N}-\log Z^{W}_{N}\Big{)}\leq
C_{1}(k+1)^{40}r^{2/3}$

for each $k\leq r^{1/3}$. Using the fact

$\textup{$\mathbb{V}$ar}(A+B)\leq\textup{$\mathbb{V}$ar}(A)+\textup{$\mathbb{V}$ar}(B)+2\sqrt{\textup{$\mathbb{V}$ar}(A)}\sqrt{\textup{$\mathbb{V}$ar}(B)}\leq
4\max\\{\textup{$\mathbb{V}$ar}(A),\textup{$\mathbb{V}$ar}(B)\\}$

where $A=\log Z^{W^{*,kr^{2/3}},\textup{out},kr^{2/3}}_{N}-\log Z^{W}_{N}$ and
$B=\log Z^{W}_{N}-\log Z^{W^{*,(k+1)r^{2/3}},\textup{out},(k+1)r^{2/3}}_{N}$,
this immediately completes the proof of Proposition 4.2. ∎

Recall ${\bf p}_{1}^{*}$ and ${\bf p}_{2}^{*}$ are the two maximizers in
(4.14). The following lemma was used in the proof of Proposition 4.4.

######  Lemma 4.5.

Let $k\in\mathbb{Z}_{>0}$ and $H=\\{\max_{i\in\\{1,2\\}}|{\bf
p}_{i}^{*}\cdot({\bf e}_{1}-{\bf e}_{2})|\geq tkr^{2/3}\\}.$ There exist
positive constants $C_{1},t_{0}$ such that for each $t\geq t_{0}$, it holds
that

$\mathbb{P}(H)\leq e^{-C_{1}t^{1/100}}.$ (4.23)

###### Proof.

Recall $\underline{{a}}=(a,-a)$, and for integers $|u|,|v|\geq t$, let us
define

$\displaystyle
I_{u}=\mathcal{L}_{\underline{{ukr^{2/3}}}}^{kr^{2/3}}\quad\text{ and }\quad
J_{v}=\mathcal{L}_{8r+\underline{{vkr^{2/3}}}}^{kr^{2/3}}.$

And we will show that

$\mathbb{P}({\bf p}_{1}^{*}\in I_{u},{\bf p}_{2}^{*}\in J_{v})\leq
e^{-C(k(|u|+|v|))^{1/10}}.$ (4.24)

Once we have (4.24), by a union bound, we obtain (4.23) since

$\mathbb{P}(H)\leq\sum_{|u|,|v|\geq t}\mathbb{P}({\bf p}_{1}^{*}\in I_{u},{\bf
p}_{2}^{*}\in J_{v})\leq\sum_{|u|,|v|\geq t}e^{-C(k(|u|+|v|))^{1/10}}\leq
e^{-Ct^{1/100}}.$

To show (4.24), let $c_{0}$ be some positive constant which we fix later when
estimating (4.26) , and it holds that

$\displaystyle\mathbb{P}({\bf p}_{1}\in I_{u},{\bf p}_{2}\in J_{v})$
$\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf p}_{1}\in I_{u}}\\{\log
Z^{W}_{{\bf p}_{1}}\\}+\max_{{\bf p}_{1}\in I_{u}}\max_{{\bf p}_{2}\in
J_{v}}\\{\log\mathaccent 869{Z}^{\textup{touch},kr^{2/3}}_{{\bf p}_{1},{\bf
p}_{2}}\\}+\max_{{\bf p}_{2}\in J_{v}}\\{\log\mathaccent 869{Z}_{{\bf
p}_{2},N}\\}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\geq\log\mathaccent
869{Z}_{0,8r}+\log\mathaccent 869{Z}_{8r,N}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf p}_{1}\in I_{u}}\log Z^{W}_{{\bf
p}_{1}}\geq c_{0}(|u|+|v|)kr^{1/3}\Big{)}$ (4.25)
$\displaystyle\qquad+\mathbb{P}\Big{(}\max_{{\bf p}_{1}\in I_{u}}\max_{{\bf
p}_{2}\in J_{v}}\mathaccent 869{Z}^{\textup{touch},kr^{2/3}}_{{\bf p}_{1},{\bf
p}_{2}}-\log\mathaccent 869{Z}_{0,8r}\geq-2c_{0}(|u|+|v|)kr^{1/3}\Big{)}$
(4.26) $\displaystyle\qquad+\mathbb{P}\Big{(}\max_{{\bf p}_{2}\in
J_{v}}\mathaccent 869{Z}_{{\bf p}_{2},N}-\log\mathaccent 869{Z}_{8r,N}\geq
c_{0}(|u|+|v|)kr^{1/3}\Big{)}.$ (4.27)

First, we see that (4.25) $\leq e^{-C(k(|u|+|v|))^{1/10}}$ from assumption
(1.2). And (4.27) $\leq e^{-C(k(|u|+|v|))^{1/10}}$ following from Proposition
2.11. It remains to show (4.26). First, using Proposition 2.4 and a union
bound, it suffices for us to upper bound

$\mathbb{P}\Big{(}\max_{{\bf p}_{1}\in I_{u}}\max_{{\bf p}_{2}\in
J_{v}}\mathaccent 869{Z}^{\textup{touch},kr^{2/3}}_{{\bf p}_{1},{\bf
p}_{2}}-\Lambda_{8r}\geq-3c_{0}k(|u|+|v|)r^{1/3}\Big{)}.$ (4.28)

If $u$ and $v$ from (4.28) have opposite signs, then the paths between $I_{u}$
and $J_{v}$ must have high transversal fluctuation of order at least
$\tfrac{1}{10}k(|u|+|v|)r^{2/3}$. Thus provided $c_{0}$ is fixed sufficiently
small, by Proposition 2.6, we have (4.28) $\leq e^{-C(k(|u|+|v|))^{1/10}}$.
Without the loss of generality, we will assume that $u$ and $v$ are both
positive. We will split the remaining estimate into two cases, whether
$|u-v|\geq\frac{u+v}{10}$ or $|u-v|\leq\frac{u+v}{10}$. In the first case, if
$|u-v|\geq\frac{u+v}{10}$, again the paths between $I_{u}$ and $J_{v}$ must
have a high transversal fluctuation of order at least
$\tfrac{1}{10}k(|u|+|v|)r^{2/3}$. And by Proposition 2.6, (4.28) $\leq
e^{-C(k(|u|+|v|))^{1/10}}$. Lastly, if $|u-v|\leq\frac{u+v}{10}$, because the
paths between $I_{u}$ and $J_{v}$ must touch $R^{kr^{2/3}}_{0,r}$, they must
also have a high transversal fluctuation of order at least
$\tfrac{1}{10}k(|u|+|v|)r^{2/3}$. In this case, Proposition 2.7 gives (4.28)
$\leq e^{-C(k(|u|+|v|))^{1/10}}$. With this, we have finished the proof of
(4.24). ∎

####  4.1.3 Proof of Proposition 4.3

Recall the boundary condition $W^{\prime}$ which was an independent copy of
$W$. Notice that $Z_{N}^{W^{\prime},\textup{out},r}$ is independent of $U_{j}$
for all $j\leq r^{1/3}$,

$\textup{$\mathbb{C}$ov}(Z^{W}_{r},Z^{W^{*,r},\textup{out},r}_{N})=\sum_{j=1}^{r^{1/3}}\textup{$\mathbb{C}$ov}(U_{j},Z^{W^{*,r},\textup{out},r}_{N}-Z^{W^{\prime},\textup{out},r}_{N}).$
(4.29)

We shall show that for $j\leq r^{1/3}$

$\textup{$\mathbb{C}$ov}(U_{j},Z^{W^{*,r},\textup{out},r}_{N}-Z^{W^{\prime},\textup{out},r}_{N})\leq\frac{1}{(r-jr^{2/3}+1)^{100}}\sqrt{\textup{$\mathbb{V}$ar}(U_{j})}\sqrt{\textup{$\mathbb{V}$ar}(Z^{W^{*,r},\textup{out},r}_{N}-Z^{W^{\prime},\textup{out},r}_{N})}.$
(4.30)

This indeed suffices. Note that by an argument identical to the one used to
obtain (4.21), we get
$\textup{$\mathbb{V}$ar}(Z^{W^{*,r},\textup{out},r}_{N}-Z^{W^{\prime},\textup{out},r}_{N})\leq
C(r^{1/3}+1)^{20}r^{2/3}$, and we get that (4.29) is upper bounded by
$Cr^{2/3}$ for some $C>0$ upon using Proposition 4.1 and (4.30).

It remains to show (4.30), and the argument for this is similar to (4.3).
Recall the $\sigma$-algebras $\mathcal{F}_{1}^{j}$ and $\mathcal{F}_{2}^{k}$
defined in (4.4). To establish (4.30), we repeat the argument for (4.3) but
with $\mathcal{F}_{2}^{k-1}$ replaced by $\mathcal{F}_{2}$ which is generated
by $\mathcal{F}^{r^{1/3}}_{2}$ together with $\log W^{\prime}_{i+1}-\log
W^{\prime}_{i}$ for all $i\in\mathbb{Z}$. Observe that conditional on
$\mathcal{F}^{\prime}:=\sigma(\mathcal{F}^{j+1}_{1},\mathcal{F}_{2})$, now,
$Z^{W^{*,r},\textup{out},r}_{N}-Z^{W^{\prime},\textup{out},r}_{N}$ is a
function of $X_{i}$ restricted to the index set $\mathbb{Z}\setminus[-r+1,r]$
which has distance $r-jr^{2/3}+1$ from the set of indices that $U_{j}$ is a
function of. By employing an argument identical to the one presented for
(4.3), we establish (4.30), albeit without providing specific details. This
completes the proof of the upper bound of Theorem 1.2.

###  4.2 Lower bound in Theorem 1.2

Given a polymer with boundary $\mathcal{L}_{0}$, let us define a
$\sigma$-algebra $\mathcal{F}$, generated by the weights $\\{Y_{\bf
z}\\}_{{\bf z}\in\mathcal{L}_{0}^{>}}$ and $\\{X_{k}=\log W_{k}-\log
W_{k-1}\\}_{k\not\in[1,r^{2/3}]}$. Next, we will define the following events
which are $\mathcal{F}$-measurable.

In the definition below, $M$ is a large positive constant, and the parameters
$r\geq r_{0}$ and $N\geq r$. The events $\mathcal{B}_{1}$ and
$\mathcal{B}_{2}$ say that the bulk free energy from $(0,0)$ to $(r,r)$ and
$(N,N)$ can not be too large or too small

$\displaystyle\mathcal{B}_{1}$ $\displaystyle=\\{|\log\mathaccent
869{Z}_{0,r}-\Lambda_{r}|\leq M^{20}r^{1/3}\\};$
$\displaystyle\mathcal{B}_{2}$ $\displaystyle=\\{|\log\mathaccent
869{Z}_{0,N}-\Lambda_{N}|\leq M^{20}N^{1/3}\\}.$

The events $\mathcal{B}_{3}$ and $\mathcal{B}_{4}$ say that the bulk free
energy starting anywhere in a segment around the origin can not be too
different from starting from the origin,

$\displaystyle\mathcal{B}_{3}$ $\displaystyle=\Big{\\{}\max_{0\leq k\leq
Mr^{2/3}}\log\mathaccent 869{Z}_{\underline{{k}},r}-\log\mathaccent
869{Z}_{0,r}\leq M^{20}r^{1/3}\Big{\\}};$ $\displaystyle\mathcal{B}_{4}$
$\displaystyle=\Big{\\{}\max_{0\leq k\leq MN^{2/3}}\log\mathaccent
869{Z}_{\underline{{k}},N}-\log\mathaccent 869{Z}_{0,N}\leq
M^{20}N^{1/3}\Big{\\}}.$

We also have control on the local fluctuation of $\log\mathaccent
869{Z}_{{{\scaleobj{0.6}{\bullet}}},N}$ on the scale $r$,

$\displaystyle\mathcal{B}_{5}$ $\displaystyle=\Big{\\{}\max_{0\leq
k\leq{r^{2/3}}}\log\mathaccent 869{Z}_{\underline{{k}}.N}-\min_{0\leq j\leq
Mr^{2/3}}\log\mathaccent 869{Z}_{\underline{{j}},N}\leq
M^{20}r^{1/3}\Big{\\}}.$

Recall the definition of $\tau$ at the beginning of Section 2.4. The next two
events $\mathcal{B}_{6}$ and $\mathcal{B}_{7}$ say that the restricted free
energy for the paths with non-positive exit times (with boundary) can not be
too large.

$\displaystyle\mathcal{B}_{6}$ $\displaystyle=\\{\log Z^{W}_{r}(\tau\leq
0)-\Lambda_{r}\leq M^{20}r^{1/3}\\};$ $\displaystyle\mathcal{B}_{7}$
$\displaystyle=\\{\log Z^{W}_{N}(\tau\leq 0)-\Lambda_{N}\leq
M^{20}N^{1/3}\\}.$

Finally, we define two rare events $\mathcal{B}_{8}$ and $\mathcal{B}_{9}$
which say the boundary weights along parts of the boundary are large:

$\displaystyle\mathcal{B}_{8}$ $\displaystyle=\Big{\\{}\log W_{Mr^{2/3}}-\log
W_{r^{2/3}}\geq M^{20.1}r^{1/3}\Big{\\}};$ $\displaystyle\mathcal{B}_{9}$
$\displaystyle=\Big{\\{}\log W_{MN^{2/3}}-\log W_{r^{2/3}}\geq
M^{20.1}N^{1/3}\Big{\\}}.$

Let us denote the intersection of the nine events defined above as

$\mathcal{B}=\bigcap_{i=1}^{9}\mathcal{B}_{i},$ (4.31)

and the following lemma lower bounds the probability of the intersection.

######  Lemma 4.6.

There exists a positive constant $M_{0}$ such that for each $M\geq M_{0}$,
there exist positive constants $r_{0}$ and $\epsilon$ such that for each
$r\geq r_{0}$ and $N\geq r$, we have

$\mathbb{P}(\mathcal{B})\geq\epsilon.$

###### Proof.

Note that $\mathcal{B}_{i}$, for $i=1,\dots,8$ are all high probability events
if $M$ is large. To see this fact,
$\mathbb{P}(\mathcal{B}_{1}^{c}\cup\mathcal{B}_{2}^{c})$ can be upper bounded
using Proposition 2.2 and Proposition 2.4. The probability
$\mathbb{P}(\mathcal{B}_{3}^{c}\cup\mathcal{B}_{4}^{c})$ can be upper bounded
using Proposition 2.11. The probability $\mathbb{P}(\mathcal{B}_{5}^{c})$ can
be upper bounded using the triangle inequality, Proposition 2.11 and
Proposition 2.12. The probability
$\mathbb{P}(\mathcal{B}_{6}^{c}\cup\mathcal{B}_{7}^{c})$ can be upper bounded
using Proposition 2.14. Thus,

$\mathbb{P}\Big{(}\bigcap_{i=1}^{7}\mathcal{B}_{i}\Big{)}\geq 0.9$ (4.32)

provided that $M$ is fixed large enough.

Next, we are going to use the assumptions (1.3), (1.9) and (1.8) to finish the
proof of the lower bound. Let $\mathcal{F}^{\prime}$ denote the
$\sigma$-algebra generated by all bulk weights $\\{Y_{\bf
z}\\}_{z\in\mathcal{L}_{0}^{>}}$. For simplicity of the notation, let us
denote by $\mathcal{B}_{i,j}=\mathcal{B}_{i}\cap\mathcal{B}_{j}$,
$\mathcal{B}_{i,j,k}=\mathcal{B}_{i}\cap\mathcal{B}_{j}\cap\mathcal{B}_{k}$
and so on. Then we have

$\displaystyle\mathbb{P}(\mathcal{B}_{1,\dots
5}\cap\mathcal{B}_{6,7}\cap\mathcal{B}_{8,9})$
$\displaystyle=\mathbb{E}[\mathbb{P}(\mathcal{B}_{1,\dots
5}\cap\mathcal{B}_{6,7}\cap\mathcal{B}_{8,9}|\mathcal{F}^{\prime})]$
$\displaystyle=\mathbb{E}[\mathbbm{1}_{\mathcal{B}_{1,\dots
5}}\mathbb{P}(\mathcal{B}_{6,7}\cap\mathcal{B}_{8,9}|\mathcal{F}^{\prime})].$
(4.33)

Now, notice that when conditioned on $\mathcal{F}^{\prime}$,
$\mathbbm{1}_{\mathcal{B}_{6,7}}$ and $\mathbbm{1}_{\mathcal{B}_{8,9}}$ are
functions of the increments $\\{X_{i}\\}_{i\leq 0}$ and $\\{X_{i}\\}_{i\geq
r+1}$, respectively. Furthermore, the conditional variances
$\textup{$\mathbb{V}$ar}(\mathbbm{1}_{\mathcal{B}_{6,7}}|\mathcal{F}^{\prime})$
and
$\textup{$\mathbb{V}$ar}(\mathbbm{1}_{\mathcal{B}_{8,9}}|\mathcal{F}^{\prime})$
are bounded above by $1$. Then, by assumption (1.3), we have

$\displaystyle\eqref{cond_F'}$
$\displaystyle\leq\mathbb{E}[\mathbbm{1}_{\mathcal{B}_{1,\dots
5}}(\mathbb{P}(\mathcal{B}_{6,7}|\mathcal{F}^{\prime})\mathbb{P}(\mathcal{B}_{8,9}|\mathcal{F}^{\prime})-1/r^{50})]\qquad\text{
}$ $\displaystyle\leq\mathbb{E}[\mathbbm{1}_{\mathcal{B}_{1,\dots
5}}(\mathbb{P}(\mathcal{B}_{6,7}|\mathcal{F}^{\prime})\mathbb{P}(\mathcal{B}_{8,9})]-1/r^{50}$
$\displaystyle\leq\mathbb{P}(\mathcal{B}_{8,9})\mathbb{P}(\mathcal{B}_{1,\dots,7})-1/r^{50}$
$\displaystyle\leq\epsilon_{0}^{2}\cdot 0.9-1/r^{50}$

where the the last inequality follows from (4.32) and the fact that
$\mathbb{P}(\mathcal{B}_{8,9})\geq\epsilon_{0}^{2}$ by assumptions (1.8) and
(1.9). Finally, by fixing $r_{0}$ sufficiently large, we have completed the
proof. ∎

Next, we have the following variance bound for the difference of two free
energies when we condition on $\mathcal{F}$.

######  Lemma 4.7.

There exist positive constants $C_{1},C_{2},M_{0}$ such that for each $M\geq
M_{0}$, there exists a positive constant $r_{0}$ and
$\mathcal{B}^{\prime}\subset\mathcal{B}$ with
$\mathbb{P}(\mathcal{B}^{\prime})\geq\tfrac{1}{2}\mathbb{P}(\mathcal{B})$ such
that for each $r\geq r_{0}$, $N\geq r$ and each
$\omega\in\mathcal{B}^{\prime}$, we have

$\displaystyle\textup{$\mathbb{V}$ar}\big{(}\log Z^{W}_{N}-\log
Z^{W}_{N}(\tau\geq r^{2/3})\big{|}\mathcal{F}\big{)}(\omega)$
$\displaystyle\leq e^{-C_{1}M}r^{2/3};$
$\displaystyle\textup{$\mathbb{V}$ar}\big{(}\log
W_{r^{2/3}}\big{|}\mathcal{F}\big{)}(\omega)$ $\displaystyle\geq
C_{2}r^{2/3}.$

###### Proof.

First, let us define $\mathcal{B}^{\prime}$. By assumptions (1.5), (1.6) and
(1.7), there exists a subset $\mathcal{B}^{\prime}\subset\mathcal{B}$ with
probability
$\mathbb{P}(\mathcal{B}^{\prime})\geq\mathbb{P}(\mathcal{B})-100/\log r$ such
that for each $\omega\in\mathcal{B}^{\prime}$, we have

$\displaystyle\mathbb{P}\big{(}\max_{0\leq k\leq N^{2/3}}\log W_{k}\geq
tN^{1/3}\big{|}\mathcal{F}\big{)}(\omega)$ $\displaystyle\leq
e^{-C^{\prime}t}$ $\displaystyle\mathbb{P}\big{(}\log W_{N^{2/3}}\leq-
tN^{1/3}\big{|}\mathcal{F}\big{)}(\omega)$ $\displaystyle\leq
e^{-C^{\prime}t}$ $\displaystyle\textup{$\mathbb{V}$ar}\big{(}\log
W_{r^{2/3}}|\mathcal{F}\big{)}(\omega)$ $\displaystyle\geq Cr^{2/3}.$

By fixing $r_{0}$ sufficiently large, we have that
$\mathbb{P}(\mathcal{B}^{\prime})\geq\tfrac{1}{2}\mathbb{P}(\mathcal{B})$, and
the second conditional variance lower bound from the lemma follows directly.

Next, we show the first conditional variance upper bound in the lemma. Note
that because of the event $\mathcal{B}^{\prime}$, it holds that

$\displaystyle\log Z^{W}_{N}(\tau\leq 0)$
$\displaystyle\leq\Lambda_{N}+M^{20}N^{1/3}$ $\displaystyle\log
Z^{W}_{N}(\tau\geq r^{2/3})$ $\displaystyle\geq\log
W_{MN^{2/3}}+\log\mathaccent
869{Z}_{\underline{{MN^{2/3}}},N}\geq\Lambda_{N}-2M^{20}N^{1/3}+M^{30}N^{1/3}.$

Thus, we have

$\mathbb{P}\Big{(}\log Z^{W}_{N}(\tau\leq 0)<\log Z^{W}_{N}(\tau\geq
r^{2/3})\Big{|}\mathcal{F}\Big{)}(\omega)=1\qquad\text{ for
}\omega\in\mathcal{B}^{\prime}.$ (4.34)

Let us define

$\mathcal{L}_{0}^{r^{2/3},+}=\Big{\\{}{\bf a}\in\mathcal{L}_{0}^{r^{2/3}}:{\bf
a}\cdot{\bf e}_{1}\geq 0\Big{\\}}.$

We have the following bound on the conditional variance. For the calculation
below, the fact

$\log Z^{W}_{N}\leq\max\Big{\\{}\log Z^{W}_{N}(\tau\geq 1),\log
Z^{W}_{N}(\tau\leq 0)\Big{\\}}+\log 2$

and (4.34) gives us the second inequality. For each $\omega\in\mathcal{B}$, it
holds that

$\displaystyle\textup{$\mathbb{V}$ar}\Big{(}\log Z^{W}_{N}-\log
Z^{W}_{N}(\tau\geq r^{2/3})\Big{|}\mathcal{F}\Big{)}(\omega)$
$\displaystyle\leq\mathbb{E}\Big{(}\Big{(}\log Z^{W}_{N}-\log
Z^{W}_{N}(\tau\geq r^{2/3})\Big{)}^{2}\Big{|}\mathcal{F}\Big{)}(\omega)$
$\displaystyle\leq\mathbb{E}\Big{(}\Big{(}\log Z^{W}_{N}(\tau\geq 1)-\log
Z^{W}_{N}(\tau\geq r^{2/3})+\log
2\Big{)}^{2}\Big{|}\mathcal{F}\Big{)}(\omega)$
$\displaystyle\leq\mathbb{E}\Big{(}\Big{(}\max\Big{\\{}\log
Z^{W}_{N}(1\leq\tau<r^{2/3}),\log Z^{W}_{N}(\tau\geq r^{2/3})\Big{\\}}-\log
Z^{W}_{N}(\tau\geq r^{2/3})+2\log
2\Big{)}^{2}\Big{|}\mathcal{F}\Big{)}(\omega)$
$\displaystyle=\mathbb{E}\Big{(}\Big{(}\max\Big{\\{}\log
Z^{W}_{N}(1\leq\tau<r^{2/3})-\log Z^{W}_{N}(\tau\geq
r^{2/3}),0\Big{\\}}+4\Big{)}^{2}\Big{|}\mathcal{F}\Big{)}(\omega)$ (4.35)

Let us define the event

$\mathcal{A}=\Big{\\{}\log Z^{W}_{N}(1\leq\tau<r^{2/3})>\log
Z^{W}_{N}(\tau\geq r^{2/3})\Big{\\}}.$

then (4.35) can be upper bounded as as

$\displaystyle\eqref{max_zero}$ $\displaystyle\leq\mathbb{E}\Big{(}\Big{(}\log
Z^{W}_{N}(1\leq\tau<r^{2/3})-\log Z^{W}_{N}(\tau\geq
r^{2/3})+4\Big{)}^{2}\mathbbm{1}_{\mathcal{A}}\Big{|}\mathcal{F}\Big{)}(\omega)+16.$
(4.36)

Next, using the general fact

$\mathbb{E}[(X+4)^{2}]\leq\mathbb{E}[X^{2}]+8\mathbb{E}[X^{2}]^{1/2}+16,$

it suffices for us to ignore the constants $4$ and $16$ in (4.36) and directly
bound the expression (4.37) below. In the calculation for (4.37), the second
inequality follows because on our event $\mathcal{B}$,

$\max_{1\leq k\leq r^{2/3}}\log\mathaccent
869{Z}_{\underline{{k}},N}-\log\mathaccent
869{Z}_{\underline{{r^{2/3}}},N}\leq M^{20}r^{1/3},$

and for each $\omega\in\mathcal{B}^{\prime}$,

$\displaystyle\mathbb{E}\Big{(}\Big{(}\log Z^{W}_{N}(1\leq\tau<r^{2/3})-\log
Z^{W}_{N}(\tau\geq
r^{2/3})\Big{)}^{2}\mathbbm{1}_{\mathcal{A}}\Big{|}\mathcal{F}\Big{)}(\omega)$
(4.37) $\displaystyle\leq\mathbb{E}\Big{(}\Big{(}\max_{1\leq k\leq
r^{2/3}}\log W_{k}+\max_{1\leq k\leq r^{2/3}}\log\mathaccent
869{Z}_{\underline{{k}},N}-\log
Z^{W}_{N}(\tau=r^{2/3})\Big{)}^{2}\mathbbm{1}_{\mathcal{A}}\Big{|}\mathcal{F}\Big{)}(\omega)$
$\displaystyle\leq\mathbb{E}\Big{(}\max_{1\leq k\leq r^{2/3}}\log W_{k}-\log
W_{r^{2/3}}+2M^{2}r^{1/3}\Big{)}^{2}\mathbbm{1}_{\mathcal{A}}\Big{|}\mathcal{F}\Big{)}(\omega)$
$\displaystyle\leq\sqrt{\mathbb{E}\Big{(}\Big{(}\max_{1\leq k\leq r^{2/3}}\log
W_{k}-\log
W_{r^{2/3}}+2M^{2}r^{1/3}\Big{)}^{4}\Big{|}\mathcal{F}\Big{)}(\omega)}\sqrt{\mathbb{P}(\mathcal{A}|\mathcal{F})(\omega)}.$
(4.38)

Next, we will bound the conditional expectation and the conditional
probability in (4.38). For the conditional expectation, by the definition of
$\mathcal{B}^{\prime}$ (defined at the beginning of the proof) and the fact
that the term inside the fourth power is non-negative, the square root of the
conditional expectation above is bounded by $CM^{4}r^{2/3}$. For the
conditional probability, let us define another set $\mathcal{A}^{\prime}$
which contains $\mathcal{A}$,

$\mathcal{A}^{\prime}=\Big{\\{}\max_{1\leq k\leq r^{2/3}}\log
W_{k}+\max_{1\leq k\leq r^{2/3}}\log\mathaccent
869{Z}_{\underline{{k}},N}+\log r>\log W_{Mr^{2/3}}+\log\mathaccent
869{Z}_{\underline{{Mr^{2/3}}},N}\Big{\\}}.$

On the event $\mathcal{B}^{\prime}$, it holds that

$\displaystyle\log W_{Mr^{2/3}}-\log W_{r^{2/3}}$ $\displaystyle\geq
M^{30}r^{1/3},$ $\displaystyle\max_{1\leq k\leq r^{2/3}}\log\mathaccent
869{Z}_{\underline{{k}},N}-\log\mathaccent 869{Z}_{\underline{{Mr^{2/3}}},N}$
$\displaystyle\leq M^{20}r^{1/3}.$

Again by the definition of $\mathcal{B}^{\prime}$, for
$\omega\in\mathcal{B}^{\prime}$, we have

$\displaystyle\mathbb{P}(\mathcal{A}|\mathcal{F})(\omega)$
$\displaystyle\leq\mathbb{P}(\mathcal{A}^{\prime}|\mathcal{F})(\omega)\leq\mathbb{P}\Big{(}\max_{1\leq
k\leq r^{2/3}}\log W_{k}-\log
W_{r^{2/3}}\geq\tfrac{1}{2}M^{30}r^{1/3}|\mathcal{F}\Big{)}(\omega)\leq
e^{-CM^{30}}.$

With this, we have finished the proof of our lemma. ∎

Recall the $\sigma$-algebra $\mathcal{F}$ which was defined at the beginning
of this section and the $\mathcal{F}$-measurable event $\mathcal{B}^{\prime}$
defined in Lemma 4.7. We shall show the following proposition.

######  Proposition 4.8.

There exists positive constants $C,M_{0}$ such that for each $M\geq M_{0}$,
there exists a $r_{0}$ such that for each $r_{0}\leq r\leq N$,

$\textup{$\mathbb{C}$ov}(\log Z^{W}_{r},\log
Z^{W}_{N}|\mathcal{F})(\omega)\geq Cr^{2/3}\qquad\text{ for each
$\omega\in\mathcal{B}^{\prime}$}.$

###### Proof.

We bound the conditional covariance using the Cauchy-Schwarz inequality as
follows.

$\displaystyle\textup{$\mathbb{C}$ov}(\log Z^{W}_{r},\log
Z^{W}_{N}|\mathcal{F})$ $\displaystyle=\textup{$\mathbb{C}$ov}\Big{(}\log
Z^{W}_{r}(\tau\geq r^{2/3}),\log Z^{W}_{N}(\tau\geq
r^{2/3})\Big{|}\mathcal{F}\Big{)}$
$\displaystyle\qquad\qquad+\textup{$\mathbb{C}$ov}\Big{(}\log Z^{W}_{r}-\log
Z^{W}_{r}(\tau\geq r^{2/3}),\log Z^{W}_{N}-\log Z^{W}_{N}(\tau\geq
r^{2/3})\Big{|}\mathcal{F}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad+\textup{$\mathbb{C}$ov}\Big{(}\log
Z^{W}_{r}-\log Z^{W}_{r}(\tau\geq r^{2/3}),\log Z^{W}_{N}(\tau\geq
r^{2/3})\Big{|}\mathcal{F}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad+\textup{$\mathbb{C}$ov}\Big{(}\log
Z^{W}_{r}(\tau\geq r^{2/3}),\log Z^{W}_{N}-\log Z^{W}_{N}(\tau\geq
r^{2/3})\Big{|}\mathcal{F}\Big{)}$
$\displaystyle\geq\textup{$\mathbb{C}$ov}\Big{(}\log Z^{W}_{r}(\tau\geq
r^{2/3}),\log Z^{W}_{N}(\tau\geq r^{2/3})\Big{|}\mathcal{F}\Big{)}$ (4.39)
$\displaystyle\qquad-\sqrt{\textup{$\mathbb{V}$ar}\Big{(}\log Z^{W}_{r}-\log
Z^{W}_{r}(\tau\geq
r^{2/3})\Big{|}\mathcal{F}\Big{)}}\sqrt{\textup{$\mathbb{V}$ar}\Big{(}\log
Z^{W}_{N}-\log Z^{W}_{N}(\tau\geq r^{2/3})\Big{|}\mathcal{F}\Big{)}}$ (4.40)
$\displaystyle\qquad\qquad-\sqrt{\textup{$\mathbb{V}$ar}\Big{(}\log
Z^{W}_{r}-\log Z^{W}_{r}(\tau\geq
r^{2/3})\Big{|}\mathcal{F}\Big{)}}\sqrt{\textup{$\mathbb{V}$ar}\Big{(}\log
Z^{W}_{N}(\tau\geq r^{2/3})\Big{|}\mathcal{F}\Big{)}}$ (4.41)
$\displaystyle\qquad\qquad-\sqrt{\textup{$\mathbb{V}$ar}\Big{(}\log
Z^{W}_{r}(\tau\geq
r^{2/3})\Big{|}\mathcal{F}\Big{)}}\sqrt{\textup{$\mathbb{V}$ar}\Big{(}\log
Z^{W}_{N}-\log Z^{W}_{N}(\tau\geq r^{2/3})\Big{|}\mathcal{F}\Big{)}}$ (4.42)

We note that (4.39) is lower bounded by $Cr^{2/3}$, since both free energies
can be rewritten as

$\log W_{r^{2/3}}+[\text{some $\mathcal{F}$-measurable r.v.}],$

and hence the covariance is equal to $\textup{$\mathbb{V}$ar}(\log
W_{r^{2/3}}|\mathcal{F})$ which is lower bounded by $Cr^{2/3}$ on
$\mathcal{B}^{\prime}$ by Lemma 4.7.

Again by Lemma 4.7, on the event $\mathcal{B}^{\prime}$, (4.40), (4.41) and
(4.42) (without the negative signs in the front) are all upper bounded by
$Ce^{-C^{\prime}M}r^{2/3}$. Hence,

$\textup{$\mathbb{C}$ov}(\log Z^{W}_{r},\log
Z^{W}_{N}|\mathcal{F})(\omega)\geq
Cr^{2/3}-C^{\prime}e^{-C^{\prime\prime}M}r^{2/3}\qquad\text{ for each
$\omega\in\mathcal{B}^{\prime}$},$

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

We can finally complete the proof of the lower bound in Theorem 1.2.

###### Proof of Theorem 1.2, lower bound.

Using Lemma 4.6, Proposition 4.8 and applying the FKG inequality assumption in
Assumption B3 twice, we have

$\displaystyle\mathbb{E}[\log Z^{W}_{N}\log Z^{W}_{r}]$
$\displaystyle=\mathbb{E}\Big{[}\mathbb{E}[\log Z^{W}_{N}\log
Z^{W}_{r}|\mathcal{F}]\Big{]}$
$\displaystyle=\int_{\mathcal{B}^{\prime}}\mathbb{E}[\log Z^{W}_{N}\log
Z^{W}_{r}|\mathcal{F}]\,d\mathbb{P}+\int_{\mathcal{B}^{\prime
c}}\mathbb{E}[\log Z^{W}_{N}\log Z^{W}_{r}|\mathcal{F}]\,d\mathbb{P}$
$\displaystyle\geq\int_{\mathcal{B}^{\prime}}\mathbb{E}[\log
Z^{W}_{N}|\mathcal{F}]\mathbb{E}[\log
Z^{W}_{r}|\mathcal{F}]\,d\mathbb{P}+C\epsilon
r^{2/3}+\int_{\mathcal{B}^{\prime c}}\mathbb{E}[\log
Z^{W}_{N}|\mathcal{F}]\mathbb{E}[\log Z^{W}_{r}|\mathcal{F}]\,d\mathbb{P}$
$\displaystyle=\mathbb{E}\Big{[}\mathbb{E}[\log
Z^{W}_{N}|\mathcal{F}]\mathbb{E}[\log Z^{W}_{r}|\mathcal{F}]\Big{]}+C\epsilon
r^{2/3}$ $\displaystyle\geq\mathbb{E}[\log Z^{W}_{N}]\mathbb{E}[\log
Z^{W}_{r}]+C\epsilon r^{2/3}.$

Note that in the first inequality above we use the fact that conditional on
$\mathcal{F}$, $\log Z_{N}^{W}$ and $\log Z^{W}_{r}$ are both increasing in
$\\{X_{i}=\log W_{i}-\log W_{i-1}\\}_{i=1}^{r^{2/3}}$. By assumption (1.9),
the conditional law of $\\{X_{i}\\}_{i=1}^{r^{2/3}}$ satisfied the FKG
inequality and therefore $\mathbb{E}[\log Z^{W}_{N}\log
Z^{W}_{r}|\mathcal{F}]\geq\mathbb{E}[\log
Z^{W}_{N}|\mathcal{F}]\mathbb{E}[\log Z^{W}_{r}|\mathcal{F}]$. For the second
inequality above, recall the sequence $\mathcal{X}$ as defined in (1.9). Note
that $\mathbb{E}[\log Z_{N}^{W}|\mathcal{F}]$ and $\mathbb{E}[\log
Z_{r}^{W}|\mathcal{F}]$ are both increasing functions of
$\\{Y_{\mathbf{v}}\\}_{{\bf v}\in\mathcal{L}_{0}^{>}}$ as well as
$\mathcal{X}$. Note $\\{Y_{\mathbf{v}}\\}_{{\bf v}\in\mathcal{L}_{0}^{>}}$ are
independent, so they satisfy the FKG inequality. By assumption (1.9), the
marginal on $\mathcal{X}$ also satisfies the FKG inequality. Because
$\\{Y_{\mathbf{v}}\\}_{{\bf v}\in\mathcal{L}_{0}^{>}}$ and $\mathcal{X}$ are
independent, using the well-known fact that product of two measures each of
which satisfy the FKG inequality also satisfies the FKG inequality (see e.g.
[25]), it follows that

$\mathbb{E}\big{[}\mathbb{E}[\log Z^{W}_{N}|\mathcal{F}]\mathbb{E}[\log
Z^{W}_{r}|\mathcal{F}]\big{]}\geq\mathbb{E}[\mathbb{E}[\log
Z^{W}_{N}|\mathcal{F}]]\cdot\mathbb{E}[\mathbb{E}[\log
Z^{W}_{r}|\mathcal{F}]]=\mathbb{E}[\log Z^{W}_{N}]\mathbb{E}[\log Z^{W}_{r}].$

This shows $\textup{$\mathbb{C}$ov}(\log Z^{W}_{N},\log Z^{W}_{r})\geq
Cr^{2/3}$, hence the lower bound in our theorem holds. ∎

##  5 Stationary inverse-gamma polymer

We start by defining the (ratio) stationary inverse-gamma polymer with a
general down-right boundary. Let $\mathcal{Y}_{\bf 0}=\\{{\bf
y}_{i}\\}_{i\in\mathbb{Z}}$ be a bi-infinite down-right path going through
$(0,0)$, which means the increment ${\bf y}_{i}-{\bf y}_{i-1}\in\\{{\bf
e}_{1},-{\bf e}_{2}\\}$. And without the loss of generality, let ${\bf
y}_{0}=(0,0)$.

Next, we define the weights for the stationary polymer. Let $\\{Y_{\bf
z}\\}_{{\bf z}\in\mathcal{Y}_{\bf 0}^{>}}$ be a collection of i.i.d. random
variables with distribution $\textup{Ga}^{-1}(\mu)$, and they will be the bulk
weights of the polymer. Independent of the bulk weights, the boundary edge
weights $G=\\{G_{{\bf y}_{i},{\bf y}_{i-1}}\\}_{i\in\mathbb{Z}}$ are attached
to the collection of unit edges $\\{[\\![{\bf y}_{i},{\bf
y}_{i-1}]\\!]\\}_{i\in\mathbb{Z}}$. The distribution of $G$ is given by

$\displaystyle G_{{\bf y}_{i},{\bf y}_{i-1}}$
$\displaystyle\sim\textup{Ga}^{-1}(\mu-\rho)$ $\displaystyle\qquad\text{if
}{\bf y}_{i}-{\bf y}_{i-1}={\bf e}_{1}$ $\displaystyle G_{{\bf y}_{i},{\bf
y}_{i-1}}$ $\displaystyle\sim\textup{Ga}^{-1}(\rho)$
$\displaystyle\qquad\text{if }{\bf y}_{i}-{\bf y}_{i-1}=-{\bf e}_{2},$

and all $G_{{\bf y}_{i},{\bf y}_{i-1}}$ are independent. This special choice
of weights is referred to as $\rho$-boundary weights.

Next, we will define the partition function. Set $H^{\mathcal{Y}_{\bf
0},\rho}_{\bf 0}=1$,. For each ${\bf y}_{m}$ with $m>0$, define

$H^{\mathcal{Y}_{\bf 0},\rho}_{0,{\bf y}_{m}}=\prod_{n=1}^{m}\mathaccent
869{G}_{{\bf y}_{n-1},{\bf y}_{n}}\qquad\text{where }\mathaccent 869{G}_{{\bf
y}_{n-1},{\bf y}_{n}}=\begin{cases}G_{{\bf y}_{n-1},{\bf y}_{n}}\quad&\text{if
${\bf y}_{n}-{\bf y}_{n-1}={\bf e}_{1}$,}\\\ 1/G_{{\bf y}_{n-1},{\bf
y}_{n}}\quad&\text{if ${\bf y}_{n}-{\bf y}_{n-1}=-\bf e_{2}$.}\end{cases}$

For each ${\bf y}_{m}$ with $m<0$, define

$H^{\mathcal{Y}_{\bf 0},\rho}_{0,{\bf y}_{m}}=\prod_{n=0}^{m+1}\mathaccent
869{G}_{{\bf y}_{n},{\bf y}_{n-1}}\qquad\text{where }\mathaccent 869{G}_{{\bf
y}_{n},{\bf y}_{n-1}}=\begin{cases}1/G_{{\bf y}_{n},{\bf
y}_{n-1}}\quad&\text{if ${\bf y}_{n}-{\bf y}_{n-1}=\bf e_{1}$,}\\\ G_{{\bf
y}_{n},{\bf y}_{n-1}}\quad&\text{if ${\bf y}_{n}-{\bf y}_{n-1}=-\bf
e_{2}$.}\end{cases}$

Recall the bulk partition function $\mathaccent 869{Z}$ defined in Section
1.1, the partition function with the $\rho$-boundary weights is then defined
by

$Z^{\mathcal{Y}_{\bf 0},\rho}_{0,{\bf
v}}=\sum_{k\in\mathbb{Z}}H^{\mathcal{Y}_{\bf 0},\rho}_{{\bf
y}_{k}}\cdot\mathaccent 869{Z}_{{\bf y}_{k},{\bf v}}\qquad\text{ for ${\bf
v}\in\mathcal{Y}_{\bf 0}^{\geq}$}.$

The above definition can be generalized to arbitrary starting point ${\bf a}$
and used to define the partition function $Z^{\mathcal{Y}_{\bf a},\rho}_{{\bf
a},{\bf v}}$.

The name (ratio) stationary inverse-gamma polymer is justified by the next
theorem, which was stated originally for s southwest boundary (where
$\mathcal{Y}_{\bf 0}$ is formed by the boundary of the first quadrant). The
proof uses a now well-known “corner flipping” induction, and the exact same
proof also applies to the general down-right boundary.

######  Theorem 5.1 ([32, Thm. 3.3] and [21, Eqn. (3.6)]).

Fix $\rho\in(0,\mu)$. For each ${\bf v}\in\mathcal{\mathcal{Y}_{\bf
a}}^{\geq}$, we have

$\frac{Z^{\mathcal{Y}_{\bf a},\rho}_{{\bf a},{\bf v}+{\bf
e}_{1}}}{Z^{\mathcal{Y}_{\bf a},\rho}_{{\bf a},{\bf
v}}}\sim\textup{Ga}^{-1}(\mu-\rho),\quad\frac{Z^{\mathcal{Y}_{\bf
a},\rho}_{{\bf a},{\bf v}+{\bf e}_{2}}}{Z^{\mathcal{Y}_{\bf a},\rho}_{{\bf
a},{\bf v}}}\sim\textup{Ga}^{-1}(\rho),$

and the dual weights $\\{\widecheck{Y}^{\rho}_{\bf z}\\}_{{\bf
z}\in\mathcal{Y}_{\bf a}^{\geq}}$ is defined as

$\widecheck{Y}^{\rho}_{\bf v}=\frac{1}{Z^{\mathcal{Y}_{\bf a},\rho}_{{\bf
a},{\bf v}+{\bf e}_{1}}/Z^{\mathcal{Y}_{\bf a},\rho}_{{\bf a},{\bf
v}}+Z^{\mathcal{Y}_{\bf a},\rho}_{{\bf a},{\bf v}+{\bf
e}_{2}}/Z^{\mathcal{Y}_{\bf a},\rho}_{{\bf a},{\bf
v}}}\sim\textup{Ga}^{-1}(\mu).$

Let $\mathcal{W}=\\{{\bf w}_{i}\\}_{i\in\mathcal{I}}$ be any finite or
infinite down-right path in $\mathcal{Y}_{\bf a}^{>}$, indexed by an interval
$\mathcal{I}\subset\mathbb{Z}$. Then, the nearest-neighbor ratios
$\\{Z^{\mathcal{Y}_{\bf a},\rho}_{{\bf a},{\bf w}_{i+1}}/Z^{\mathcal{Y}_{\bf
a},\rho}_{{\bf a},{\bf w}_{i}}\\}$ along the path and the dual weights below
$\mathcal{W}$ $\bigl{\\{}\widecheck{Y}^{\rho}_{\bf v}:{\bf
v}\in\mathcal{Y}_{\bf a}^{\geq}\cap\mathcal{W}^{<}\bigr{\\}}$ are mutually
independent.

###  5.1 Two special boundaries $\mathcal{S}$ and $\mathcal{S}^{\prime}$

$\bf a$$\mathcal{S}_{\bf a}$$\mathcal{L}_{\bf a}$$\mathcal{S}^{\prime}_{\bf
a}$ Figure 5.1: An illustration of $\mathcal{L}_{\bf a}$, $\mathcal{S}_{\bf
a}$ and $\mathcal{S}^{\prime}_{\bf a}$.

Let $\mathcal{S}_{\bf a}$ (and $\mathcal{S}^{\prime}_{\bf a}$) denote the bi-
infinite down-right staircase path that has ${\bf a}$ as one of its upper
right (or lower left) corners (illustrated in Figure 5.1),

$\displaystyle\mathcal{S}_{\bf a}$ $\displaystyle=\\{\dots,{\bf a}-{\bf
e}_{1}+\bf e_{2},{\bf a}-\bf e_{1},{\bf a},{\bf a}+\bf e_{1},{\bf a}+\bf
e_{1}-e_{2},\dots\\},$ $\displaystyle\mathcal{S}_{\bf a}^{\prime}$
$\displaystyle=\\{\dots,{\bf a}-{\bf e}_{1}+\bf e_{2},{\bf a}+\bf e_{2},{\bf
a}+\bf e_{1},{\bf a}+\bf e_{1}-{\bf e}_{2},\dots\\}.$

For the rest of the paper, the general down-right boundary $\mathcal{Y}_{\bf
a}$ will be chosen to be $\mathcal{S}_{\bf a}$. The other boundary
$\mathcal{S}_{\bf a}^{\prime}$ will only appear in the proof of Theorem 6.1.
With this in mind, let us simplify our notation by omitting the boundary
$\mathcal{S}_{\bf a}$ in the superscript of the partition function,

$Z^{\rho}_{\bf a,{{\scaleobj{0.6}{\bullet}}}}=Z^{\mathcal{S}_{\bf
a},\rho}_{\bf a,{{\scaleobj{0.6}{\bullet}}}}.$

And the corresponding quenched measure will be denoted as $Q^{\rho}_{\bf
a,{{\scaleobj{0.6}{\bullet}}}}.$

Lastly, we note that $Z^{\rho}_{\bf a,{{\scaleobj{0.6}{\bullet}}}}$ matches
the definition of a polymer with a given initial condition defined in Section
1.1. Let us define $\mathcal{S}_{\bf 0}=\\{{\bf s_{i}}\\}_{i\in\mathbb{Z}}$
with $\bf s_{0}=0$, then the initial condition $\\{W_{k}\\}_{k\in\mathbb{Z}}$
define in Section 1.1 is given by

$W_{k}=H^{\mathcal{S}_{\bf 0},\rho}_{{\bf s}_{2k}}\qquad\text{ for
}k\in\mathbb{Z}.$

Because of this, the stationary boundary is also called the stationary initial
condition. And we will also often refer to the staircase boundary
$\mathcal{S}_{\bf 0}$ as the anti-diagonal boundary $\mathcal{L}_{0}$, which
is different from the $\mathcal{S}_{\bf 0}^{\prime}$ staircase boundary.

###  5.2 Exit time estimate

Define the characteristic direction as a function of $\rho\in(0,\mu)$

${\boldsymbol{\xi}}[\rho]=\Big{(}\frac{\Psi_{1}(\rho)}{\Psi_{1}(\rho)+\Psi_{1}(\mu-\rho)}\,,\,\frac{\Psi_{1}(\mu-\rho)}{\Psi_{1}(\rho)+\Psi_{1}(\mu-\rho)}\Big{)}$
(5.1)

where $\Psi_{1}$ is the trigamma function defined by
$\Psi_{1}(z)=\frac{d^{2}}{dz^{2}}\Gamma(z)$. The sampled paths between $(0,0)$
and $N\boldsymbol{\xi}[\rho]$ tend to stay on the anti-diagonal boundary for
at most order $N^{2/3}$ number of steps. This is the statement of the next
proposition obtained from [31]. Note the stationary polymer defined in [31]
has a southwest boundary, but the following result is actually proved for the
anti-diagonal boundary and then showed that it implies the same result for the
southwest boundary.

######  Proposition 5.2 ([31, Lemma 4.6]).

Fix $\varepsilon\in(0,\mu/2)$. There exist positive constants $C_{1},C_{2}$,
$r_{0}$, $N_{0}$ depending only on $\varepsilon$ such that for all
${{\rho}}\in[\varepsilon,\mu-\varepsilon]$, $N\geq N_{0}$ and $r\geq r_{0}$,
we have

$\displaystyle\mathbb{P}\Big{(}Q^{\rho}_{0,N\boldsymbol{\xi}[\rho]}\\{|\tau|>rN^{2/3}\\}\geq
e^{-C_{1}r^{2}N^{1/3}}\Big{)}\leq e^{-C_{2}r^{3}}.$

##  6 Temporal correlation for the stationary polymer using duality

In this section, we give a stronger upper bound for the time correlation for
the stationary polymer when $r$ is smaller than $N/2$. The availability of
this result (with simpler proof) can be attributed to a special duality
arising from the stationary boundary. Essentially, the dual weights
$\widecheck{Y}^{\mu/2}$ defined in Theorem 5.1 follow the i.i.d. inverse-gamma
distribution.

######  Theorem 6.1.

For any $r\leq N/2$, we have

$\textup{$\mathbb{C}$ov}\Big{(}\log Z^{\mu/2}_{0,r},\log
Z^{\mu/2}_{0,N}\Big{)}\leq\textup{$\mathbb{V}$ar}\Big{(}\log
Z^{\mu/2}_{0,r}\Big{)}.$

This is a stronger version of the upper bound in Theorem 1.2 since by Theorem
2.16, it holds that $\textup{$\mathbb{V}$ar}\big{(}\log
Z^{\mu/2}_{0,r}\big{)}\leq Cr^{2/3}$.

$(r,r)$$(N,N)$$\log\mathaccent
869{Z}_{r,N}^{\mathcal{S}_{r}^{\prime},\mu/2,(0)}$$\log\widecheck{Z}_{0,r}^{\mathcal{S}_{r}^{\prime},\mu/2}$$\widecheck{Y}^{\mu/2}_{\bf
z}$$Y_{\bf z}$$\mathcal{S}_{r}^{\prime}$ Figure 6.1: An illustration of

###### Proof.

Fix $0\leq r\leq N/2$. First, we will define a nested free energy. Fix the
staircase $\mathcal{S}_{r}^{\prime}=\\{\bf s^{\prime}_{i}\\}_{i\in\mathbb{Z}}$
with ${\bf s}_{0}=(r,r)$. Let us define the boundary weight $G$ on
$\mathcal{S}_{r}^{\prime}$ to be

$\displaystyle G_{{\bf s}^{\prime}_{i},{\bf s}^{\prime}_{i-1}}$
$\displaystyle={Z^{\mu/2}_{0,{\bf s}^{\prime}_{i}}}/{Z^{\mu/2}_{0,{\bf
s}^{\prime}_{i-1}}}$ $\displaystyle\qquad\text{if }{\bf s}^{\prime}_{i}-{\bf
s}^{\prime}_{i-1}={\bf e}_{1}$ $\displaystyle G_{{\bf s}^{\prime}_{i},{\bf
s}^{\prime}_{i-1}}$ $\displaystyle={Z^{\mu/2}_{0,{\bf
s}^{\prime}_{i-1}}}/{Z^{\mu/2}_{0,{\bf s}^{\prime}_{i}}}$
$\displaystyle\qquad\text{if }{\bf s}^{\prime}_{i}-{\bf
s}^{\prime}_{i-1}=-{\bf e}_{2},$

Theorem 5.1 states that the weights $G$ are independent
$\text{Ga}^{-1}(\mu/2)$ random variables, attached to the $e_{1}$ and $e_{2}$
edges in $\mathcal{S}_{r}^{\prime}$. In addition, $G$, $\\{Y_{\bf
z}\\}_{z\in{\mathcal{S}_{r}^{\prime}}^{>}}$ and $\\{\widecheck{Y}^{\mu/2}_{\bf
z}\\}_{z\in{\mathcal{S}_{r}^{\prime}}^{<}}$ are all independent.

With $G$ and bulk weights $\\{Y_{\bf
z}\\}_{z\in{\mathcal{S}_{r}^{\prime}}^{>}}$, we can define a stationary
polymer between $(r,r)$ and $(N,N)$, whose partition function is denoted by
$Z^{\mathcal{S}^{\prime}_{r},\mu/2,(0)}_{r,N}$. Then, Lemma A.6 from [31]
precisely states that

$\log Z^{\mu/2}_{0,N}=\log Z^{\mu/2}_{0,r}+\log
Z^{\mathcal{S}^{\prime}_{r},\mu/2,(0)}_{r,N}.$ (6.1)

With the same boundary $G$ and the dual weights $\\{\widecheck{Y}^{\mu/2}_{\bf
z}\\}_{z\in{\mathcal{S}_{r}^{\prime}}^{<}}$, we can define another stationary
polymer between $(0,0)$ and $(r,r)$, but the boundary and the bulk are rotated
by $180^{\circ}$. We denote its partition function as
$\widecheck{Z}^{\mathcal{S}^{\prime}_{r},\mu/2}_{0,r}$. Lemma 4.3, originally
presented in [21] for a stationary polymer with a southwest boundary, can be
directly applied in this context with an identical proof, yielding the
following result:

$\log Z^{\mu/2}_{0,r}=\log
Z^{\mathcal{S}_{0},\mu/2}_{0,r}=\log\widecheck{Z}^{\mathcal{S}^{\prime}_{r},\mu/2}_{0,r}.$
(6.2)

With (6.1) and (6.2), we see that

$\displaystyle\textup{$\mathbb{C}$ov}(\log Z^{\mu/2}_{0,r},\log
Z^{\mu/2}_{0,N})$ $\displaystyle=\textup{$\mathbb{C}$ov}(\log
Z^{\mu/2}_{0,r},\log Z^{\mu/2}_{0,r}+\log
Z^{\mathcal{S}_{r}^{\prime},\mu/2,(0)}_{r,N})$ (6.3)
$\displaystyle=\textup{$\mathbb{V}$ar}(\log
Z^{\mu/2}_{0,r})+\textup{$\mathbb{C}$ov}({\log{Z}}^{\mu/2}_{0,r},\log
Z^{\mu/2,(0)}_{r,N})$ $\displaystyle=\textup{$\mathbb{V}$ar}(\log
Z^{\mu/2}_{0,r})+\textup{$\mathbb{C}$ov}(\log\widecheck{Z}^{\mathcal{S}^{\prime}_{r},\mu/2}_{0,r},\log
Z^{\mathcal{S}^{\prime}_{r},\mu/2,(0)}_{r,N}).$

The two partition functions from the last covariance term are illustrated in
Figure 6.1. To finish the proof, we will show that

$\textup{$\mathbb{C}$ov}(\log\widecheck{Z}^{\mathcal{S}_{r}^{\prime},{\mu/2}}_{0,r},\log
Z^{\mathcal{S}_{r}^{\prime},{\mu/2},(0)}_{r,N})\leq 0.$ (6.4)

Note once we have this, we obtain our desired result from (6.3),

$\textup{$\mathbb{C}$ov}(\log Z^{\mu/2}_{0,r},\log
Z^{\mu/2}_{0,N})\leq\textup{$\mathbb{V}$ar}(\log Z^{\mu/2}_{0,r}).$

So now it remains to show (6.4), let $\mathcal{F}$ and
$\widecheck{\mathcal{F}}$ denote the $\sigma$-algebras generated by $\\{Y_{\bf
z}\\}_{{\bf z}\in\mathcal{S}_{r}^{>}}$ and $\\{\widecheck{Y}^{\mu/2}_{\bf
z}\\}_{{\bf z}\in\mathcal{S}_{r}^{<}}$. By the law of total covariance,

$\displaystyle\textup{$\mathbb{C}$ov}(\log\widecheck{Z}^{\mathcal{S}_{r}^{\prime},,{\mu/2}}_{0,r},\log
Z^{\mathcal{S}_{r}^{\prime},{\mu/2},(0)}_{r,N})$
$\displaystyle=\mathbb{E}[\textup{$\mathbb{C}$ov}(\log\widecheck{Z}^{\mathcal{S}_{r}^{\prime},{\mu/2}}_{0,r},\log
Z^{\mathcal{S}_{r}^{\prime},{\mu/2},(0)}_{r,N}|\mathcal{F},\widecheck{\mathcal{F}})]+\textup{$\mathbb{C}$ov}(\mathbb{E}[\log\widecheck{Z}^{\mathcal{S}_{r}^{\prime},{\mu/2}}_{0,r}|\mathcal{F},\widecheck{\mathcal{F}}],\mathbb{E}[\log
Z^{\mathcal{S}_{r}^{\prime},{\mu/2},(0)}_{r,N}|\mathcal{F},\widecheck{\mathcal{F}}]).$
(6.5)

Since, $\log\widecheck{Z}^{\mathcal{S}_{r}^{\prime},{\mu/2}}_{0,r}$ is
independent of $\mathcal{F}$,

$\mathbb{E}[\log\widecheck{Z}^{\mathcal{S}_{r}^{\prime},{\mu/2}}_{0,r}|\mathcal{F},\widecheck{\mathcal{F}}]=\mathbb{E}[\log\widecheck{Z}^{\mathcal{S}_{r}^{\prime},{\mu/2}}_{0,r}|\widecheck{\mathcal{F}}],$

and similarly $\log Z^{\mathcal{S}_{r}^{\prime},{\mu/2},(0)}_{r,N}$ is
independent of $\widecheck{\mathcal{F}}$, so

$\mathbb{E}[\log
Z^{\mathcal{S}_{r}^{\prime},{\mu/2},(0)}_{r,N}|\mathcal{F},\widecheck{\mathcal{F}}]=\mathbb{E}[\log
Z^{\mathcal{S}_{r}^{\prime},{\mu/2},(0)}_{r,N}|\mathcal{F}].$

Then, by independence, the second covariance term from (6.5) is zero. And it
remains to show that

$\mathbb{E}[\textup{$\mathbb{C}$ov}(\log\widecheck{Z}^{\mathcal{S}_{r}^{\prime},{\mu/2}}_{0,r},\log
Z^{\mathcal{S}_{r}^{\prime},{\mu/2},(0)}_{r,N}|\mathcal{F},\widecheck{\mathcal{F}})]\leq
0.$

This holds because by definition,
$\log\widecheck{Z}^{\mathcal{S}_{r}^{\prime},{\mu/2}}_{0,r}$ and $\log
Z^{\mathcal{S}_{r}^{\prime},{\mu/2},(0)}_{r,N}$ are negatively correlated in
their boundary weights. This means if we increase the value of one of the edge
weights along the boundary $S_{r}^{\prime}$, one of the two free energies
increases while the other one decreases. Thus, by the FKG inequality,

$\textup{$\mathbb{C}$ov}(\log\widecheck{Z}^{\mathcal{S}_{r}^{\prime},{\mu/2}}_{0,r},\log
Z^{\mathcal{S}_{r}^{\prime},{\mu/2},(0)}_{r,N}|\mathcal{F},\widecheck{\mathcal{F}})\leq
0.$

We have finished showing (6.4), and this finishes the proof.

∎

##  7 Proofs for the estimates in Section 2.3 and Section 2.4

In this section, we prove the various estimates from Section 2.3 and Section
2.4. The proofs use techniques from the stationary polymer.

###  7.1 Proof of Proposition 2.10

###### Proof.

By symmetry and a union bound, it suffices to consider the case when $k$ is
non-negative and upper bound

$\mathbb{P}\Big{(}\min_{0\leq k\leq t^{1/20}r^{2/3}}\log\mathaccent
869{Z}_{0,{N}+\underline{{k}}}-\log\mathaccent 869{Z}_{0,N}\leq-
tr^{1/3}\Big{)}.$ (7.1)

First, let us define $s=t^{1/20}$ and look at the case that
$s\leq\tfrac{\mu}{100}N^{1/3}$. We will use the random walk comparison
technique and turn this into an estimate for random walks. Recall Theorem 3.28
from [9] which states that there exists an event $A$ with $\mathbb{P}(A)\geq
1-e^{-Cs^{3}}$, such that for each $0\leq k\leq sr^{2/3}$,

$\log\mathaccent 869{Z}_{0,{N}+\underline{{k}}}\ -\log\mathaccent
869{Z}_{0,N}\geq\log\tfrac{9}{10}+\sum_{i=1}^{k}H_{i}\qquad\text{ on the event
}A,$

where $\\{H_{i}\\}$ are i.i.d. random variables whose distribution is given by
the difference of two independent log-gamma random variables. More precisely,
set $\eta=\mu/2-sN^{-1/3}$, then

$H_{i}\stackrel{{\scriptstyle d}}{{=}}U-V$

where $U\sim\log\textup{Ga}^{-1}(\mu-\eta),V\sim\log\textup{Ga}^{-1}(\eta)$
and $U,V$ are independent.

By an application of Taylor’s theorem to the digamma functions, we obtain that
$\mathbb{E}[H_{i}]=-\Psi_{0}(\mu-\rho)+\Psi_{0}(\eta)\geq-C^{\prime}sN^{-1/3}$
for some positive constant $C^{\prime}$. Then, it holds that

$\displaystyle\eqref{k_est}\leq e^{-t^{3/20}}+\mathbb{P}\Big{(}\min_{0\leq
k\leq
t^{1/20}r^{2/3}}\sum_{i=1}^{k}(H_{i}-\mathbb{E}[H_{i}])\leq-\tfrac{1}{2}tr^{1/3}\Big{)}\leq
e^{-t^{1/10}}$

where bounding the probability uses Theorem A.1 since $\pm H_{i}$’s are sub-
exponential random variables (this is verified in Appendix A.1.1).

On the other hand, if $t^{1/20}\geq\tfrac{\mu}{100}N^{1/3}$, then

$\displaystyle\eqref{k_est}$ $\displaystyle\leq\mathbb{P}\Big{(}\min_{0\leq
k\leq t^{1/20}r^{2/3}}\log\mathaccent
869{Z}_{0,{N}+\underline{{k}}}-\log\mathaccent
869{Z}_{0,N}\leq-\sqrt{t}N^{3}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\min_{0\leq k\leq
t^{1/20}r^{2/3}}\log\mathaccent
869{Z}_{0,{N}+\underline{{k}}}\leq-\tfrac{1}{2}\sqrt{t}N^{3}\Big{)}+\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{0,N}\geq\tfrac{1}{2}\sqrt{t}N^{3}\Big{)}$
$\displaystyle\leq\sum_{k=0}^{t^{1/20}r^{2/3}}\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{0,{N}+\underline{{k}}}\leq-\tfrac{1}{2}\sqrt{t}N^{3}\Big{)}+\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{0,N}\geq\tfrac{1}{2}\sqrt{t}N^{10}\Big{)}$ $\displaystyle\leq
t^{1/20}r^{2/3}e^{-t^{1/10}N}+e^{-t^{1/10}},$

where the last inequality comes from [9, Proposition 3.8] (a general version
of Proposition 2.4 which holds in all directions bounded away from the axes),
and Proposition 2.3. With this, we have finished the proof of this
proposition. ∎

###  7.2 Proof of Proposition 2.11

###### Proof.

It suffices for us to get the upper bound

$\mathbb{P}\Big{(}\max_{{\bf x}\in\mathcal{L}^{a}_{0}}\log\mathaccent
869{Z}_{{\bf x},N}-\log\mathaccent 869{Z}_{0,N}\geq t\sqrt{a}\Big{)}\leq
e^{-t^{1/10}}.$

Let $\log Z_{{\bf x},N}=\log Y_{\bf x}+\log\mathaccent 869{Z}_{{\bf x},N}$,
and applying a union bound, we have

$\displaystyle\mathbb{P}\Big{(}\max_{{\bf
x}\in\mathcal{L}^{a}_{0}}\log\mathaccent 869{Z}_{{\bf x},N}-\log\mathaccent
869{Z}_{0,N}\geq t\sqrt{a}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf
x}\in\mathcal{L}^{a}_{0}}\log{Z}_{{\bf x},N}-\min_{{\bf
x}\in\mathcal{L}^{a}_{0}}\log Y_{\bf x}-\log{Z}_{0,N}+\log Y_{(0,0)}\geq
t\sqrt{a}\Big{)}$ $\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf
x}\in\mathcal{L}^{a}_{0}}\log{Z}_{{\bf
x},N}-\log{Z}_{0,N}\geq\tfrac{1}{2}t\sqrt{a}\Big{)}+\mathbb{P}\Big{(}-\min_{{\bf
x}\in\mathcal{L}^{a}_{0}}\log Y_{\bf x}+\log
Y_{(0,0)}\geq\tfrac{1}{2}t\sqrt{a}\Big{)}.$

The first probability is bounded by $e^{-t^{1/10}}$ using Proposition 2.9
after a $180^{\circ}$ rotation of the picture. The second probability is
bounded by $e^{-t^{1/10}}$ as well, because as shown in Appendix A.1.1,
$\pm\log Y_{\bf x}$ are sub-exponential random variables. ∎

###  7.3 Proof of Proposition 2.13

###### Proof.

We may assume that $t\leq N^{1/3}/\epsilon$, since otherwise the collection of
paths
$\big{\\{}|\tau_{\mathcal{L}_{N}^{tN^{2/3}}}|>(1+\epsilon)tN^{2/3}\big{\\}}$
is empty. Also, since

$\log
Z^{W}_{\mathcal{L}_{N}^{tN^{2/3}}}(|\tau|>(1+\epsilon)tN^{2/3})\leq\max_{{\bf
x}\in\mathcal{L}_{N}^{tN^{2/3}}}\log Z^{W}_{{\bf
x}}(|\tau|>(1+\epsilon)tN^{2/3})+10\log N,$

we may prove the estimate for $\max_{{\bf x}\in\mathcal{L}_{N}^{tN^{2/3}}}\log
Z^{W}_{{\bf x}}(|\tau|>2tN^{2/3})$. Finally, by a union bound, we may replace
$|\tau|$ with just $\tau$.

The rest of the proof follows from a union bound. First, we will show that
there exist two positive constants $C,C^{\prime}$ such that for each
$(1+\epsilon)t+1\leq s\leq(t+N^{1/3}/\epsilon)-1$,

$\mathbb{P}\Big{(}\max_{{\bf x}\in\mathcal{L}_{N}^{tN^{2/3}}}\log Z^{W}_{{\bf
x}}((s-1)N^{2/3}\leq\tau\leq(s+1)N^{2/3})-\Lambda_{N}\geq-C^{\prime}s^{2}N^{1/3}\Big{)}\leq
e^{-Cs^{3/2}}.$ (7.2)

This holds because we have

$\displaystyle\mathbb{P}\Big{(}\max_{{\bf x}\in\mathcal{L}_{N}^{tN^{2/3}}}\log
Z^{W}_{{\bf
x}}((s-1)N^{2/3}\leq\tau\leq(s+1)N^{2/3})-\Lambda_{N}\geq-C^{\prime}s^{2}N^{1/3}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf
x}\in\mathcal{L}_{N}^{tN^{2/3}}}\log\mathaccent
869{Z}_{\mathcal{L}_{\underline{{sN^{2/3}}}}^{N^{2/3}},{\bf
x}}-\Lambda_{N}\geq-2C^{\prime}s^{2}N^{1/3}\Big{)}$ (7.3)
$\displaystyle\qquad\qquad+\mathbb{P}\Big{(}\max_{1\leq k\leq(s+1)N^{2/3}}\log
Z^{W}_{\underline{{k}}}\geq\tfrac{1}{2}C^{\prime}s^{2}N^{1/3}\Big{)}.$ (7.4)

Since $s\geq(1+\epsilon)t+1$, (7.3) is bounded by $e^{-Cs^{3}}$ using
Proposition 2.6, provided $C^{\prime}$ in (7.3) is fixed sufficiently small.
And (7.4) is bounded by $e^{-Cs^{3/2}}$ using assumption (1.2). Thus, have
shown that $\eqref{goal}.$

Finally, by a union bound we complete the proof

$\displaystyle\mathbb{P}\Big{(}\log
Z^{W}_{\mathcal{L}_{N}^{tN^{2/3}}}(\tau>(1+\epsilon)tN^{2/3})-\Lambda_{N}\geq-C^{\prime}t^{2}N^{1/3}\Big{)}$
$\displaystyle\leq\sum_{s=(1+\epsilon)t+1}^{(t+N^{1/3})-1}\mathbb{P}\Big{(}\log
Z^{W}_{\mathcal{L}_{N}^{tN^{2/3}}}((s-1)N^{2/3}\leq\tau\leq(s+1)N^{2/3})-\Lambda_{N}\geq-C^{\prime}s^{2}N^{1/3}\Big{)}$
$\displaystyle\leq\sum_{s=(1+\epsilon)t+1}^{(t+N^{1/3})-1}e^{-Cs^{3/2}}\leq
e^{-Ct^{3/2}}.$

∎

###  7.4 Proof of Proposition 2.14

###### Proof.

First, we prove $(i)$. By Proposition 2.13, we have

$\mathbb{P}\Big{(}\log
Z^{W}_{\mathcal{L}_{N}^{tN^{2/3}}}(|\tau|>2tN^{2/3})-\Lambda_{N}\geq-C^{\prime}t^{2}N^{1/3}\Big{)}\leq
e^{-Ct^{3/2}}.$

Then, it suffices for us to bound

$\mathbb{P}\Big{(}\log Z^{W}_{\mathcal{L}_{N}^{tN^{2/3}}}(|\tau|\leq
2tN^{2/3})-\Lambda_{N}\geq\epsilon t^{2}N^{1/3}\Big{)}.$ (7.5)

By a union bound,

$\displaystyle\eqref{in10tN}$
$\displaystyle\leq\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{\mathcal{L}_{0},\mathcal{L}_{N}^{tN^{2/3}}}-\Lambda_{N}\geq\tfrac{1}{3}\epsilon
t^{2}N^{1/3}\Big{)}$ (7.6)
$\displaystyle\qquad\qquad+\mathbb{P}\Big{(}\max_{1\leq|k|\leq 2tN^{2/3}}\log
Z^{W}_{\underline{{k}}}\geq\tfrac{1}{3}\epsilon
t^{3/2}(t^{1/2}N^{1/3})\Big{)}.$ (7.7)

Then, (7.6) is bounded by $e^{-C\min\\{t^{3},t^{2}N^{1/3}\\}}$ from
Proposition 2.2, and (7.7) is bounded by $e^{-Ct^{3/2}}$ by assumption (1.2).
With this, we have shown (7.5) and finished proving the first estimate $(i)$
in our proposition.

The second estimate follows from the exact same argument, the only changes are
that the “$t^{2}$” in (7.6) becomes “$t$”, and “$t^{3/2}$” from (7.7) becomes
“$t^{1/2}$”. So we omit the details. ∎

###  7.5 Proof of Proposition 2.15

###### Proof.

This follows directly from the monotonicity that $\log
Z^{W}_{N}\geq\log\mathaccent 869{Z}_{0,N}$ and the upper bound of the left
tail for $\log\mathaccent 869{Z}_{0,N}$ from Proposition 1.4. ∎

###  7.6 Proof of Theorem 2.16

By Proposition 2.14 and Proposition 2.15,

$\big{|}\mathbb{E}[\log Z^{W}_{N}]-\Lambda_{N}\big{|}\leq CN^{1/3}.$

Then, these propositions give the upper bound on the variance.

The lower bound on the variance follows from the fact $\log
Z^{W}_{N}\geq\log\mathaccent 869{Z}_{0,N}$ and Proposition 2.3.

###  7.7 Proof of Theorem 2.17

By a union bound,

$\displaystyle\mathbb{P}\Big{(}\log
Z^{W}_{\mathcal{L}_{N}^{s}}(|\tau|>(1+\epsilon)tN^{2/3})-\log
Z^{W}_{\mathcal{L}_{N}^{s}}\geq-C^{\prime}t^{2}N^{1/3}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{W}_{\mathcal{L}_{N}^{tN^{2/3}}}(|\tau|>(1+\epsilon)tN^{2/3})-\Lambda_{N}\geq-{2C^{\prime}}t^{2}N^{1/3}\Big{)}$
$\displaystyle\qquad\qquad+\mathbb{P}\Big{(}\log
Z^{W}_{N}-\Lambda_{N}\leq-{C^{\prime}}t^{2}N^{1/3}\Big{)}.$

By Proposition 2.13 and Proposition 2.15, the two probabilities above are
bounded by $e^{-Ct^{3/2}}$ provided $C^{\prime}$ is sufficiently small. We
have finished the proof of the theorem.

###  7.8 Proof of Theorem 2.18

The proof of this Theorem is similar to the proof of Theorem 4.8 from [9],
except here we have to handle the addition of the boundary weights. We start
with a proposition that captures the local fluctuation of the free energy
profile.

######  Proposition 7.1.

Suppose the boundary condition $W$ satisfies assumption (1.2). There exist
positive constants $C_{1},C_{2},c_{0},N_{0},t_{0}$ such that for each $N\geq
N_{0}$, $t_{0}\leq t\leq c_{0}N^{1/2}$, and each integer $0\leq
a\leq\tfrac{1}{2}t^{2/3}N^{2/3}$, we have

$\mathbb{P}\Big{(}\log Z^{W}_{\mathcal{L}^{a}_{N}}-\log Z^{W}_{N}\geq
C_{1}t\sqrt{a}\Big{)}\leq e^{-C_{2}t}.$

###### Proof.

First, by Theorem 2.17, on an event with probability at least $1-e^{-Ct}$, we
have

$Q^{W}_{\mathcal{L}^{a}_{N}}(|\tau|>t^{2/3}N^{2/3})<1/10,$

which implies

$\log Z^{W}_{\mathcal{L}^{a}_{N}}(|\tau|\leq t^{2/3}N^{2/3})\geq\log
Z^{W}_{\mathcal{L}^{a}_{N}}(|\tau|>t^{2/3}N^{2/3}).$

Hence, on the same event, it holds that

$\displaystyle\log Z^{W}_{\mathcal{L}^{a}_{N}}\leq\log
Z^{W}_{\mathcal{L}^{a}_{N}}(|\tau|\leq t^{2/3}N^{2/3})+\log 2.$

Then, to prove the proposition, it suffices for us to bound the following
probability

$\mathbb{P}\Big{(}\log Z^{W}_{\mathcal{L}^{a}_{N}}(|\tau|\leq
t^{2/3}N^{2/3})-\log Z^{W}_{N}(|\tau|\leq t^{2/3}N^{2/3})\geq
C^{\prime}t\sqrt{a}\Big{)}.$ (7.8)

Next, let

$\lambda=\frac{\mu}{2}+q_{0}t^{2/3}N^{-1/3}$ (7.9)

where the constant $q_{0}$ will be chosen later below (7.15). Once $q_{0}$ is
fixed, we may lower the value of $c_{0}$ from the assumption of our
proposition so that $\lambda\in[\mu/2,2\mu/3]$. We enrich our probability
space by introducing another set of independent $\lambda$-boundary weights
along $\mathcal{L}_{0}$, which defines a (ratio) stationary polymer whose free
energy is $\log Z^{\lambda}_{0,{{\scaleobj{0.6}{\bullet}}}}$.

Let us introduce another free energy $\log Z^{*}$ which uses these new
$\lambda$-boundary weights on
$\mathcal{L}_{0}\setminus\mathcal{L}_{0}^{t^{2/3}N^{2/3}}$. Note that by
definition,

$\displaystyle\log Z^{W}_{\mathcal{L}^{a}_{N}}(|\tau|\leq t^{2/3}N^{2/3})-\log
Z^{W}_{N}(|\tau|\leq t^{2/3}N^{2/3})$ $\displaystyle\qquad\qquad\qquad=\log
Z^{*}_{\mathcal{L}^{a}_{N}}(|\tau|\leq t^{2/3}N^{2/3})-\log
Z^{*}_{N}(|\tau|\leq t^{2/3}N^{2/3}).$

Using a union bound and the fact that

$\log Z^{*}_{\mathcal{L}^{a}_{N}}(|\tau|\leq t^{2/3}N^{2/3})\leq\max_{-a\leq
k\leq a}\log Z^{*}_{N+\underline{{k}}}(|\tau|\leq t^{2/3}N^{2/3})+10\log a,$

in order to show (7.8), it suffices for us to bound

$\mathbb{P}\Big{(}\max_{0\leq k\leq a}\log
Z^{*}_{N+\underline{{k}}}(|\tau|\leq t^{2/3}N^{2/3})-\log Z^{*}_{N}(|\tau|\leq
t^{2/3}N^{2/3})\geq C^{\prime}t\sqrt{a}\Big{)}.$ (7.10)

To bound this, we will compare the probability above with the right tail of
the running maximum of a random walk with i.i.d. steps.

Recall $\log Z^{\lambda}$ is the free energy with the $\lambda$-boundary
weights. In the calculation below, the inequality (7.11) follows from
Proposition A.4 and (7.12) follows from Proposition A.5. For each integer $k$
with $0\leq k\leq a$, it holds that

$\displaystyle\log Z^{*}_{N+\underline{{k}}}(|\tau|\leq t^{2/3}N^{2/3})-\log
Z^{*}_{0,N}(|\tau|\leq t^{2/3}N^{2/3})$ $\displaystyle\leq\log
Z^{*}_{N+\underline{{k}}}(\tau\geq 0)-\log Z^{*}_{N}(\tau\geq 0)$ (7.11)
$\displaystyle\leq\log Z^{*}_{N+\underline{{k}}}(\tau\geq t^{2/3}N^{2/3})-\log
Z^{*}_{N}(\tau\geq t^{2/3}N^{2/3})$ (7.12) $\displaystyle=\log
Z^{\lambda}_{0,N+\underline{{k}}}(\tau\geq t^{2/3}N^{2/3})-\log
Z^{\lambda}_{0,N}(\tau\geq t^{2/3}N^{2/3}).$ (7.13)

Then, we shall show that there exists an event $A$ with probability at least
$1-e^{-Ct}$ such that on the event $A$, for each $0\leq k\leq a$ it holds that

$\eqref{lam_eq1}\leq\log Z^{\lambda}_{0,N+\underline{{k}}}-\log
Z^{\lambda}_{0,N}+\log 2.$ (7.14)

Before proving (7.14), we see that (7.10) can then be bounded as

$\eqref{star_est}\leq\mathbb{P}\Big{(}\max_{0\leq k\leq a}\log
Z^{\lambda}_{0,N+\underline{{k}}}-\log Z^{\lambda}_{0,N}\geq
C^{\prime}t\sqrt{a}\Big{)}+\mathbb{P}(A^{c}).$

By Theorem 5.1, $\\{\log Z^{\lambda}_{0,N+\underline{{k}}}-\log
Z^{\lambda}_{0,N}\\}_{0\leq k\leq a}$ is a random walk with i.i.d. steps,
whose step distribution is
$\log(\text{Ga}^{-1}(\mu-\lambda))-\log(\text{Ga}^{-1}(\lambda))$. By Taylor’s
theorem, the expectation of the steps of this random walk is bounded inside a
closed interval
$[-c^{\prime}q_{0}t^{2/3}N^{-1/3},c^{\prime}q_{0}t^{2/3}N^{-1/3}]$, where
$c^{\prime}$ is a constant depending only on $\mu$. We fix the constant
$C^{\prime}$ in (7.10) sufficiently large (depending on $q_{0}$ and $\mu$) so
that

$ac^{\prime}q_{0}t^{2/3}N^{2/3}\leq\tfrac{1}{2}C^{\prime}t\sqrt{a}.$

Then, in the calculation below, we normalize the random walk $\\{\log
Z^{\lambda}_{0,N+\underline{{k}}}-\log Z^{\lambda}_{0,N}\\}_{0\leq k\leq a}$
to be mean zero and apply Theorem A.1,

$\displaystyle\mathbb{P}\Big{(}\max_{0\leq k\leq a}\log
Z^{\lambda}_{0,N+\underline{{k}}}-\log Z^{\lambda}_{0,N}\geq
C^{\prime}t\sqrt{a}\Big{)}$ $\displaystyle\leq\mathbb{P}\Big{(}\max_{0\leq
k\leq a}\Big{\\{}\log Z^{\lambda}_{0,N+\underline{{k}}}-\log
Z^{\lambda}_{0,N}-\mathbb{E}[\log Z^{\lambda}_{0,N+\underline{{k}}}-\log
Z^{\lambda}_{0,N}]\Big{\\}}\geq\tfrac{1}{2}C^{\prime}t\sqrt{a}\Big{)}$
$\displaystyle\leq e^{-C\min\\{t^{2}\\!,\,t\sqrt{a}\\}}.$

This finishes the proof if we assume (7.14) holds on a large probability
event.

$(0,0)$$(t^{2/3}N^{2/3},-t^{2/3}N^{2/3})$$(2t^{2/3}N^{2/3},-2t^{2/3}N^{2/3})$$\boldsymbol{\xi}[\lambda]$$(N,\
N)$${\bf x}$ Figure 7.1: The distance between $(0,0)$ and ${\bf x}$ is lower
bounded by $c^{\prime\prime}q_{0}t^{2/3}N^{2/3}$. By choosing $q_{0}$
sufficiently large, the ${\bf x}$ will be to the right and below the point
$(2t^{2/3}N^{2/3},-2t^{2/3}N^{2/3})$.

Finally, to complete the proof, we show that there is a large probability
event $A$ with $\mathbb{P}(A)\geq 1-e^{-Ct}$ such that (7.14) holds. Note that

$\log Z^{\lambda}_{0,N+\underline{{k}}}(\tau\geq t^{2/3}N^{2/3})-\log
Z^{\lambda}_{0,N}(\tau\geq t^{2/3}N^{2/3})\leq\log
Z^{\lambda}_{0,N+\underline{{k}}}-\log Z^{\lambda}_{0,N}(\tau\geq
t^{2/3}N^{2/3}),$

then it suffices to show that on a high probability event $A$,

$\log Z^{\lambda}_{0,N}\leq\log Z^{\lambda}_{0,N}(\tau\geq
t^{2/3}N^{2/3})+\log 2.$

To obtain the inequality above, it suffices to show that

$\log Z^{\lambda}_{0,N}(\tau\geq t^{2/3}N^{2/3})>\log
Z^{\lambda}_{0,N}(\tau<t^{2/3}N^{2/3})$ (7.15)

with high probability. This follows from the standard exit estimate provided
that the $q_{0}$ in (7.9) is fixed sufficiently large. So first, we will fix
$q_{0}$.

Let ${\bf x}$ denote the point where the $-\boldsymbol{\xi}[\lambda]$-directed
ray from $(N,N)$ crosses $\mathcal{L}_{0}$, as shown in Figure 7.1. By
applying Taylor’s theorem to the slope of the vector
$\boldsymbol{\xi}[\lambda]$ (defined through (5.1)), the distance between
$(0,0)$ and the point ${\bf x}$ (shown in Figure 7.1) is lower bounded by
$c^{\prime\prime}q_{0}t^{2/3}N^{2/3}$ for some constant $c^{\prime\prime}$
depending on $\mu$. Then, we may fix $q_{0}$ such that ${\bf x}$ is to the
right and below the point $(2t^{2/3}N^{2/3},-2t^{2/3}N^{2/3})$.

Next, we will define the event $A$. In the calculation below, Proposition A.6
gives us the first equality. The next inequality is obtained by translating
${\bf x}$ and $(N,N)$ to ${\bf 0}$ and $N\boldsymbol{\xi}[\lambda]$, then
observing that the shifted paths $-{\bf x}+\\{\tau<t^{2/3}N^{2/3}\\}$ are
contained inside $\\{\tau<-t^{2/3}N^{2/3}\\}$. And the third inequality
follows from Theorem 5.2. With these, we have