# Law of Iterated Logarithms and Fractal Properties of the KPZ equation

Sayan Das  S. Das, Department of Mathematics, Columbia University,
S. Das 2990 Broadway, New York, NY 10027, USA<EMAIL_ADDRESS>and
Promit Ghosal  P. Ghosal, Department of Mathematics, Massachusetts Institute
of Technology,
P. Ghosal 77 Massachusetts Avenue, Cambridge, MA 02139, USA<EMAIL_ADDRESS>
###### Abstract.

We consider the Cole-Hopf solution of the $(1+1)$-dimensional KPZ equation
started from the narrow wedge initial condition. In this article, we ask how
the peaks and valleys of the KPZ height function (centered by time/$24$) at
any spatial point grow as time increases. Our first main result is about the
law of iterated logarithms for the KPZ equation. As time variable $t$ goes to
$\infty$, we show that the limsup of the KPZ height function with the scaling
by $t^{1/3}(\log\log t)^{2/3}$ is almost surely equal to $(3/4\sqrt{2})^{2/3}$
whereas the liminf of the height function with the scaling by
$t^{1/3}(\log\log t)^{1/3}$ is almost surely equal to $-6^{1/3}$. Our second
main result concerns with the _macroscopic_ fractal properties of the KPZ
equation. Under exponential transformation of the time variable, we show that
the peaks of KPZ height function mutate from being monofractal to
multifractal, a property reminiscent of a similar phenomenon in Brownian
motion [KKX17, Theorem 1.4].

The proofs of our main results hinge on the following three key tools: $(1)$ a
_multi-point composition law_ of the KPZ equation which can be regarded as a
generalization of the two point composition law from [CGH19, Proposition 2.9],
$(2)$ the Gibbsian line ensemble techniques from [CH14, CH16, CGH19] and,
$(3)$ the tail probabilities of the KPZ height function in short time and its
spatio-temporal modulus of continuity. We advocate this last tool as one of
our new and important contributions which might garner independent interest.

##  1\. Introduction

We study the Kardar-Parisi-Zhang (KPZ) equation, a stochastic PDE which is
formally written

$\displaystyle\partial_{t}\mathcal{H}=\tfrac{1}{2}\partial_{xx}\mathcal{H}+\tfrac{1}{2}(\partial_{x}\mathcal{H})^{2}+\xi,\qquad\mathcal{H}:=\mathcal{H}(t,x)\qquad(t,x)\in[0,\infty)\times\mathbb{R}.$
(1.1)

Here, $\xi=\xi(t,x)$ is the space time white noise. The KPZ equation was
originally introduced in [KPZ86] for studying the fluctuation of growing
interfaces and since then, it has found links to many systems including
directed polymers, last passage percolation, interacting particle systems, and
random matrices via its connections to the _KPZ universality class_ (see
[FS10, Qua11, Cor12, QS15]).

The KPZ equation, as given in (1.1), is ill-posed as a stochastic PDE due to
the presence of the nonlinear term $(\partial_{x}\mathcal{H})^{2}$. The
physically relevant notion of solution for the KPZ equation is given by the
Cole-Hopf solution which is defined as

$\displaystyle\mathcal{H}(t,x):=\log\mathcal{Z}(t,x)$

where $\mathcal{Z}(t,x)$ is the solution of the stochastic heat equation (SHE)

$\displaystyle\partial_{t}\mathcal{Z}=\tfrac{1}{2}\partial_{xx}\mathcal{Z}+\xi\mathcal{Z},\qquad\mathcal{Z}:=\mathcal{Z}(t,x).$
(1.2)

Throughout this paper, we work with the fundamental solution
$\mathcal{Z}^{\mathbf{nw}}(t,x)$ of (1.2) and the associated Cole-Hopf
solution $\mathcal{H}^{\mathbf{nw}}(t,x):=\log\mathcal{Z}^{\mathbf{nw}}(t,x)$
which corresponds to the SHE being started from the delta initial measure,
i.e., $\mathcal{Z}^{\mathbf{nw}}(0,x)=\delta_{x=0}$. For any positive $t>0$,
$\mathcal{Z}^{\mathbf{nw}}(t,x)$ is strictly positive (see [Flo14]) which
makes the Cole-Hopf solution $\mathcal{H}^{\mathbf{nw}}(t,x)$ well-defined.
The corresponding initial data of the KPZ equation is termed as the _narrow
wedge_ initial data.

The ubiquity of the SHE is discernible in many applications stretching from
modeling the density of the particles diffusing through random environments
[Mol96, Kho14, BC17, CG17] to the partition function of the continuum directed
random polymer model [AKQ14, CDR10, BC14]. The solution theory for the SHE is
standard [Wal86, Qua11, Cor18]; based on Itô integral theory or martingale
problems. The mathematical theory of the KPZ equation however has unleashed
new challenges in recent years. Most notably, the study of the KPZ equation
can now be classified into three broad directions, namely, to understand how
the KPZ equation approximates the interface fluctuation of the random growth
models, to build a robust solution theory of the KPZ equation and to unveil
fine properties and asymptotics of the solution of the KPZ equation. The Cole-
Hopf solution of the KPZ equation coincides with the limits of certain growth
processes [BG97, CT17, CST18, Gho17, CGST20, Lin20]. The KPZ equation being a
testing ground for the nonlinear stochastic PDEs, stirs up intense recent
innovations in the theory of singular PDEs including regularity structures
[Hai13], paracontrolled distributions [GIP15, GP17], energy solution [GJ14]
and renormalisation group [Kup16] methods. In this paper, we seek to pursue
the third direction, i.e., to unravel finer properties of the Cole-Hopf
solution of the KPZ equation.

In this paper, we consider the following $1:2:3$ scaled version of the KPZ
height function:

$\displaystyle\mathfrak{h}_{t}(\alpha,x):=\frac{\mathcal{H}^{\mathbf{nw}}(\alpha
t,t^{2/3}x)+\frac{\alpha t}{24}}{t^{1/3}}.$ (1.3)

where $t$ specifies the time scale and $\alpha$ measures the time judged on
that scale, $x$ measures the space judged on $t^{2/3}$ scale. Although the
presence of $t$ and $\alpha$ bears a stain of redundancy, the notation
introduced in (1.3) will be useful in stating and proving many of our results.
For $\alpha=1$, we will often use the shorthand
$\mathfrak{h}_{t}(x):=\mathfrak{h}_{t}(1,x)$ and
$\mathfrak{h}_{t}:=\mathfrak{h}_{t}(0)$. We will call the stochastic process
$\mathfrak{h}_{t}$ indexed by the time parameter $t$ as the _KPZ temporal
process_. In a seminal work, [ACQ11] showed that

$\displaystyle\mathfrak{h}_{t}\stackrel{{\scriptstyle
d}}{{\to}}2^{-1/3}TW_{\mathrm{GUE}},\quad\text{as }t\to\infty.$

Here, $TW_{\mathrm{GUE}}$ is the Tracy-Widom GUE distribution. The _KPZ
scaling_ of the fluctuation, space and time, i.e., the ratio of the
corresponding scaling exponents being $1:2:3$ and the Tracy-Widom distribution
as the limit of the fluctuations are the characteristics of the models in the
KPZ universality class. Recently, [QS20, Vir20] have announced proofs of the
convergence of the spatial process $\mathfrak{h}_{t}(x)$ (upto a parabola) to
the universal limiting process of the KPZ universality class, namely the _KPZ
fixed point_ as $t$ goes to $\infty$.

Our objects of study are the large peaks and valleys of the KPZ temporal
process as the KPZ equation approaches the KPZ fixed point. Such study for any
generic one-dimensional stochastic process with a macroscopic limiting profile
usually starts up with two questions: _What are the scalings of the large
peaks and valleys_? _Do they converge to any limit under such scaling_? For a
Brownian motion $\mathfrak{B}_{t}$, these questions are answered via the
(Brownian) _law of iterated logarithms_ (LIL). Under the $\sqrt{t}$ scaling,
the fluctuation of the Brownian motion $\mathfrak{B}_{t}$ has the Gaussian
limit. At the onset of this macroscopic Gaussianity, the peaks and valleys of
$\mathfrak{B}_{t}/\sqrt{t}$ under further scaling by $\sqrt{2\log\log t}$
stays in between $-1$ and $1$. The extra scaling by an iterated logarithmic
factor $\sqrt{2\log\log t}$ inflicts the name ‘law of iterated logarithms’.

Our first main result which is stated as follows concerns with the law of
iterated logarithms of the KPZ equation started from the narrow wedge initial
data.

######  Theorem 1.1.

With probability $1$, we have

$\displaystyle\limsup_{t\to\infty}\frac{\mathfrak{h}_{t}}{(\log\log
t)^{2/3}}=\Big{(}\frac{3}{4\sqrt{2}}\Big{)}^{\frac{2}{3}},\quad\mbox{and }\ \
\liminf_{t\to\infty}\frac{\mathfrak{h}_{t}}{(\log\log
t)^{1/3}}=-6^{\frac{1}{3}}.$

The above law of iterated logarithms reveals the scaling of the large peaks
and valleys of $\mathfrak{h}_{t}$. As we may see, the scalings for limsup and
liminf differ from each other. This naturally gives rise to the following two
questions:

$(1)$ _Where are the scaling $(\log\log t)^{2/3}$ and $(\log\log t)^{1/3}$
coming from?_

The scaling of the large peaks and valleys for the KPZ height fluctuation are
in fact orchestrated by the Tracy-Widom GUE distribution. This is in line with
the LIL for the Brownian motion where the exponent $1/2$ of $(\log\log t)$
factor stems from the Gaussian tail decay of the limiting law. For the KPZ
equation, the peaks and valleys have different scaling thanks to the distinct
decay exponents of the upper and lower tail probabilities of the Tracy-Widom
GUE. If $X$ is a Tracy-Widom GUE random variable, then, the probability of $X$
being higher than $s$ (i.e., upper tail probability) decays as
$e^{-4s^{3/2}/3}$ and the probability of lower (i.e., lower tail probability)
than $-s$ decays as $e^{-s^{3}/12}$. So, the upper tail decay exponent is
$3/2$ which induce the scaling $(\log\log t)^{2/3}$ for the peaks of the KPZ
fluctuation whereas the lower tail exponent being $3$ is the source for the
scaling $(\log\log t)^{1/3}$ of the valleys. Interestingly, as one may
observe, the values of the limsup and liminf in Theorem 1.1 are seemingly
connected to the constants $4/3$ and $1/12$ of the respective tail decays of
the Tracy-Widom GUE distribution. This association is commensurate with the
Brownian LIL and predicted in other works (discussed in Section 1.2).

$(2)$ _How the LILs will vary with the initial data?_

Based on the LIL for the narrow wedge solution, one may insinuate that the
scaling of the peaks and valleys of the KPZ solution under other initial
condition will be governed by the tail exponents of the limiting random
variables. It follows from Theorem 1.1 and 1.4 of [CG20a] that for a wide
class of initial data, the upper tail exponents of the limiting r.v. of the
KPZ equation under KPZ scaling is $3/2$ and the lower tail exponent is at
least $3$. By drawing the analogy with the narrow wedge case, we conjecture
that correct scaling of the peaks and valleys of the KPZ height fluctuation
will be $(\log\log t)^{2/3}$ and $(\log\log t)^{1/3}$ respectively. Proving
these claims is beyond the scope of the present paper since some of the major
tools that we use are not available for the KPZ solution under other initial
data. However, we hope to explore this direction in future works.

Our next objective is to quantify how often the peaks and valleys of the KPZ
fluctuation exceed a given level. This entails to studying the _upper level
sets_ $\\{t>t_{0}:\mathfrak{h}_{t}\geq\gamma(\log\log t)^{2/3}\\}$ and _lower
level sets_ $\\{t>t_{0}:\mathfrak{h}_{t}\leq-\gamma(\log\log t)^{1/3}\\}$ for
different values of $\gamma$ where $\gamma>0$ is a tuning parameter and
$t_{0}$ is an arbitrary constant. In particular, we study the _macroscopic_
fractal nature of the level sets. For brevity, we mainly focus on the study of
the upper level sets in this paper.

Fractal nature of the level sets of the KPZ equation is intimately connected
to the moment growth of the SHE which is captured through the _Lyapunov
exponents_ , i.e., the limit of
$t^{-1}\mathbb{E}[(\mathcal{Z}^{\mathbf{nw}}(t,0))^{k}]$ as $t\to\infty$ for
any integer $k$. The nonlinear nature of the Lyapunov exponents of the SHE
(predicted by _Kardar’s formula_ [Kar87]) suggests an abundance of the large
peaks of the SHE. This is manifested through the existence of infinitely many
scales for the peaks, a property often called as _multifractality_. In
contrast, the peaks of a scaled Brownian motion $\mathfrak{B}_{t}/\sqrt{t}$
only show a single scale as time $t$ increases to infinity. This latter
property is named as _monofractality_. In the following, we give a
mathematical definition of these two different natures of the (macroscopic)
fractality.

######  Definition 1.2 (Mono- and Multifractality).

Let $X$ be a stochastic process. Suppose there exists a non-random gauge
function $g$ such that $g(r)$ increases to $\infty$ as $r\to\infty$ and

$\limsup_{r\to\infty}\frac{X(r)}{g(r)}=1\qquad\text{a.s.}$

Fix a scalar $\gamma,t_{0}>0$. Define

$\displaystyle\Xi_{X,g}(\gamma):=\Big{\\{}t>t_{0}:\frac{X(t)}{g(t)}>\gamma\Big{\\}}.$

We denote the (Barlow-Taylor) _macroscopic Hausdorff dimension_ (see
Definition 2.6) of any Borel set $\mathfrak{F}$ by
$\mathrm{Dim}_{\mathbb{H}}(\mathfrak{F})$. The tall peaks of $X$ is
_multifractal_ in gauge $g$ when there exist infinitely many length scales
$\gamma_{1}>\gamma_{2}>\ldots>0$ such that, with probability one,

$\displaystyle\mathrm{Dim}_{\mathbb{H}}\big{(}\Xi_{X,g}(\gamma_{i+1})\big{)}>\mathrm{Dim}_{\mathbb{H}}\big{(}\Xi_{X,g}(\gamma_{i})\big{)},\quad\forall
i\geq 1.$

On the other hand, the peaks of $X$ with gauge function $g$ is _monofractal_
when

$\displaystyle\mathrm{Dim}_{\mathbb{H}}(\Xi_{X,g}(\gamma))=\begin{cases}\mathrm{Constant}&\gamma\leq\gamma_{0}\\\
0&\gamma>\gamma_{0}\end{cases}$

for some $\gamma_{0}>0$.

By the law of iterated logarithms, the gauge function of a scaled Brownian
motion $\mathfrak{B}_{t}/\sqrt{t}$ is dictated as $(2\log\log t)^{1/2}$. It
follows from the works of [KKX17, Str64, Mot58] that the Brownian motion with
such choice of the gauge function is monofractal. However, the macroscopic
nature of the peaks undergoes a transition under the exponential
transformation of the time variable underpinning the Brownian motion. For
instance, the Ornstein-Uhlenbeck process which is defined as
$U(t):=\exp(-t/2)\mathfrak{B}_{e^{t}}$ for $t\in\mathbb{R}$ is multifractal in
the gauge function $(2\log t)^{1/2}$.

Our second main result which is stated below shows that the KPZ tempral
process is monofractal in the gauge function $(\tfrac{3}{4\sqrt{2}}\log\log
t)^{2/3}$. Whereas under the exponential transformation of the time variable,
the peaks of the KPZ temporal process exhibits multifractality.

######  Theorem 1.3.

Consider the rescaled height function $\mathfrak{h}_{t}$ of the KPZ equation
and the exponential time-changed process
$\mathfrak{G}(t):=\mathfrak{h}_{e^{t}}$. Then, we have the following:
$\mathfrak{h}_{t}$ is monofractal with positive probability in gauge function
$(\log\log t)^{2/3}$, i.e., for every $t_{0},\gamma>0$,

$\displaystyle\mathrm{Dim}_{\mathbb{H}}\Big{\\{}t\geq
e^{e}:\frac{\mathfrak{h}_{t}}{(\log\log
t)^{2/3}}\geq\gamma\Big{\\}}\stackrel{{\scriptstyle
a.s.}}{{=}}\begin{cases}1&\text{when
}\gamma\leq\big{(}\frac{3}{4\sqrt{2}}\big{)}^{\frac{2}{3}},\\\ 0&\text{when
}\gamma>\big{(}\frac{3}{4\sqrt{2}}\big{)}^{\frac{2}{3}}.\end{cases}$ (1.4)

In contrast, $\mathfrak{G}(t)$ is multifractal in gauge function
$(3/4\sqrt{2})^{2/3}(\log t)^{2/3}$. In fact,

$\displaystyle\mathrm{Dim}_{\mathbb{H}}\Big{\\{}t\geq
e^{e}:\frac{\mathfrak{G}(t)}{(3/4\sqrt{2})^{2/3}(\log
t)^{2/3}}\geq\gamma\Big{\\}}\stackrel{{\scriptstyle
a.s.}}{{=}}1-\gamma^{3/2},\quad\text{for }\gamma\in[0,1].$ (1.5)

Note that (1.4) show that the peaks of $\mathfrak{h}_{t}$ are monofractal in
the gauge function $(\log\log t)^{2/3}$. On the other hand, the
multifractality of the peaks of $\mathfrak{G}(t)$ is clear from (1.5) since

$\Xi_{\mathfrak{G}(t),(3/4\sqrt{2})^{2/3}(\log
t)^{2/3}}(\gamma_{2})\stackrel{{\scriptstyle
a.s}}{{=}}1-\gamma^{3/2}_{2}<1-\gamma^{3/2}_{1}\stackrel{{\scriptstyle
a.s}}{{=}}\Xi_{\mathfrak{G}(t),(3/4\sqrt{2})^{2/3}(\log t)^{2/3}}(\gamma_{1})$

for $0\leq\gamma_{1}<\gamma_{2}\leq 1$. This raises the following three
interesting questions.

$(1)$ _What is the minimal speed up needed for the time variable to see
transition from monofractality to multifractality of the peaks of the KPZ
equation?_

We are indebted to Davar Khoshnevisan for asking this question. By carefully
studying the outreach of our tools, we expect to see the appearance of
multifractality of the peaks under the transformation
$t\mapsto\mathfrak{h}_{e^{(\log t)^{a}}}$ for any $a>1$. Due to lack of
detailed information on the correlation decay of the KPZ temporal process, we
are unable to make precise prediction of the fractality under the
transformation $t\mapsto\mathfrak{h}_{t^{a}}$ for any $a>1$. Based on our
intuition, we expect that the monofractality will still be survived under such
transformations.

$(2)$ _Is there a similar notion of macroscopic fractality for the valleys?
What are the macroscopic fractal properties of the valleys of the KPZ height
function?_

The fractal properties of the valleys can be studied using the lower level
sets. For instance, if $X$ is a stochastic process such that
$\liminf_{r\to-\infty}X(r)/f(r)=-1$ almost surely for some gauge function $f$,
then, the multifractality and/or monofractality of the valleys of $X$ can be
defined in the same way as in Definition 1.2 using the macroscopic Hausdorff
dimension of the following lower level sets

$\widehat{\Xi}_{X,f}(\gamma):=\Big{\\{}t>t_{0}:\frac{X(t)}{f(t)}<-\gamma\Big{\\}}.$

For studying the valleys of $\mathfrak{h}_{t}$, the natural choice of the
gauge function is $(6\log\log t)^{1/3}$ as shown by Theorem 1.1. Using the
tools of this paper, we expect that one can show monofractality of the valleys
of $\mathfrak{h}_{t}$ in the gauge function $(6\log\log t)^{1/3}$.
Furthermore, drawing the analogy with (1.5), we also expect the following
equality holds

$\mathrm{Dim}_{\mathbb{H}}\Big{(}\widehat{\Xi}_{\mathfrak{G}(t),(6\log
t)^{1/3}}(\gamma)\Big{)}\stackrel{{\scriptstyle a.s}}{{=}}1-\gamma^{3}.$

While the fractal properties of the valleys seem extremely exciting, for
brevity, we restrict ourselves only to exploring the peaks of the KPZ temporal
process in this paper.

$(3)$ _What is expected about the peaks and valleys of the KPZ fixed point in
the temporal direction?_

It is believed that $\mathfrak{h}_{t}(\alpha,x)$ weakly converges as a time-
space process to the KPZ fixed point (started from the narrow wedge data)
which has recently been constructed in [MQR16] via its transition probability
and simultaneously in [DOV18] via the _Airy sheet_. Very recently, [QS20,
Vir20] announced proofs of a special case of this conjecture, namely the weak
convergence of the spatial process
$x\mapsto(2\alpha^{-1})^{1/3}(\mathfrak{h}_{t}(\alpha,x)+\frac{x^{2}}{2})$ to
the Airy2 process (introduced in [PS02]) for any fixed $\alpha>0$. In light of
this conjecture, we expect that the law of iterated logarithms of the KPZ
fixed point in the temporal direction bear the same scaling as in Theorem 1.1.
Moreover, the macroscopic nature of the peaks and valleys of the KPZ equation
as revealed in the above discussion is expected to be reflective of the case
for the KPZ fixed point. Although, our proof techniques which will be touched
on in Section 1.1 are very much likely to be applicable for the KPZ fixed
point, we defer from proving results analogous to Theorem 1.1 and 1.3 for the
KPZ fixed point.

Proving the law of iterated logarithms and the fractal properties of the KPZ
equation requires information on the growth of
$\mathfrak{h}_{t_{1}}-\mathfrak{h}_{t_{2}}$ for $t_{1}>t_{2}>0$. When
$t_{1}-t_{2}$ is large, [CGH19, Theorem 1.5] obtained upper and lower bounds
on the tail probabilities of $\mathfrak{h}_{t_{1}}-\mathfrak{h}_{t_{2}}$.
However, controlling the variations of the peaks in a smaller interval
necessitates the study of the tail probabilities of the increments
$\mathfrak{h}_{t_{1}}-\mathfrak{h}_{t_{2}}$ for $t_{1}-t_{2}$ small. One of
the main obstructions for studying the increments of $\mathfrak{h}_{t}$ in a
small interval is the lack of uniform tail bounds of $\mathfrak{h}_{t}$ for
all small $t>0$. In the following two results, we seek to fill this gap. To
state those results, we introduce the following notations:

$\displaystyle\mathfrak{g}_{t}:=\frac{\mathcal{H}^{\mathbf{nw}}(t,0)+\log\sqrt{2\pi
t}}{(\pi t/4)^{1/4}}.$

The first result proves a uniform bound on the upper tail probabilities of
$\mathfrak{h}_{t}$ for all small $t>0$.

######  Theorem 1.4.

Fix $\varepsilon>0$. There exist $t_{0}=t_{0}(\varepsilon)>0$,
$c=c(\varepsilon)>0$, and $s_{0}=s_{0}(\varepsilon)>0$ such that for all
$t\leq t_{0}$ and $s\geq s_{0}$,

$\displaystyle\mathbb{P}(\mathfrak{g}_{t}\geq
s)\leq\exp\Big{(}-\frac{cs^{2}}{1+\sqrt{1+st^{1/4-4\varepsilon}}}\Big{)}.$
(1.6)

######  Remark 1.5.

Note that the right hand side of (1.6) decays like Gaussian tails, i.e.,
$\exp(-cs^{2})$ for some constant $c>0$ as $t\downarrow 0$. This is embraced
by the fact that $\mathfrak{g}_{t}$ weakly converges to a standard Gaussian
distribution as $t$ approaches $0$ (shown in [ACQ11, Proposition 1.8]). On the
other hand, for large $t$, the decay turns to
$\exp(-cs^{3/2}t^{-1/8+2\varepsilon})$. The decay exponent $3/2$ accords with
the finite time upper tail exponent (see [CG20a, Theorem 1.10]) of the KPZ
equation.

For the purpose of latter use, we will only require the following loose bound
which is free of the time variable and follows immediately from Theorem 1.4.

######  Corollary 1.6.

There exists $t_{0}>0$, $c>0$, and $s_{0}>0$ such that for all $t\leq t_{0}$
and $s\geq s_{0}$, we have

$\displaystyle\mathbb{P}(\mathfrak{g}_{t}\geq s)\leq\exp(-cs^{3/2}).$

The second result shows an uniform bound on the lower tail probability of
$\mathfrak{g}_{t}$ for all small $t>0$.

######  Theorem 1.7.

There exist constants $t_{0}\in(0,2]$, $s_{0}>0$ and $c>0$ such that for all
$t\leq t_{0}$, $s\geq s_{0}$,

$\displaystyle\mathbb{P}(\mathfrak{g}_{t}\leq-s)\leq e^{-cs^{2}}.$ (1.7)

######  Remark 1.8.

The decay exponent of the upper bound in (1.7) is consistent with the Gaussian
limit of $\mathfrak{g}_{t}$ as $t$ goes down to zero. It is worthwhile to note
that Theorem 1.7 provides an upper bound to the lower tail probability which
holds uniformly for all small $t>0$. This should be contrasted with the work
of [CG20b, Theorem 1.1] which showed that the lower tail probability at finite
time $t>0$ decays as $\exp(-ct^{1/3}s^{5/2})$ for some constant $c>0$. The
interpolation between the exponents $2$ and $5/2$ as one gradually increases
time $t$ from $0$ to a finite value is not covered in Theorem 1.7.

Short time uniform tail bounds of Theorem 1.4 and 1.7 opens directions to a
plethora of new results. One of such directions is the study of modulus of
continuity of the time-space process $\mathfrak{h}_{t}(\alpha,x)$. Our next
and final main result proves a super-exponential tail bound of the modulus of
continuity of $\mathfrak{h}_{t}(\alpha,x)$.

######  Theorem 1.9.

Fix $t_{0}>0$, $\varepsilon\in(0,\frac{1}{4})$ and any interval
$[a,b]\subset\mathbb{R}$ and $[c,d]\subset\mathbb{R}_{>t_{0}}$. Define
$\mathrm{Norm}:([a,b]\times[c,d])^{2}\to\mathbb{R}_{\geq 0}$

$\displaystyle\mathrm{Norm}(\alpha_{1},x_{1};\alpha_{2},x_{2})=|x_{1}-x_{2}|^{\frac{1}{2}}\Big{(}\log\frac{|b-a|}{|x_{1}-x_{2}|}\Big{)}^{2/3}+|\alpha_{1}-\alpha_{2}|^{\frac{1}{4}-\varepsilon}\Big{(}\log\frac{|d-c|}{|\alpha_{1}-\alpha_{2}|}\Big{)}^{2/3}$
(1.8)

and

$\displaystyle\mathcal{C}:=\sup_{\alpha_{1}\neq\alpha_{2},x_{1}\neq
x_{2}}\frac{1}{\mathrm{Norm}(\alpha_{1},x_{1};\alpha_{2},x_{2})}\big{|}\mathfrak{h}_{t}(\alpha_{1},x_{1})+\frac{x^{2}_{1}}{2\alpha_{1}}-\mathfrak{h}_{t}(\alpha_{2},x_{2})-\frac{x^{2}_{2}}{2\alpha_{2}}\big{|}.$
(1.9)

Then there exist $s_{0}=s_{0}(t_{0},|b-a|,|c-d|,\varepsilon)>0$ and
$c=c(t_{0},|b-a|,|c-d|,\varepsilon)>0$ such that for all $s\geq s_{0}$ and
$t\geq t_{0}$,

$\displaystyle\mathbb{P}(\mathcal{C}>s)\leq e^{-cs^{3/2}}.$ (1.10)

######  Remark 1.10.

It was known (due to [BC95, Theorem 2.2]) that the fundamental solution of the
SHE (i.e., $\mathcal{Z}^{\mathbf{nw}}(t,x)$) as a time-space process is almost
surely Hölder continuous with the spatial and temporal Hölder exponents being
less than $1/2$ and $1/4$ respectively. This indicates Hölder continuity of
$\mathcal{H}^{\mathbf{nw}}(t,x)$ with same spatial and temporal Hölder
exponents as that of $\mathcal{Z}^{\mathbf{nw}}(t,x)$. Theorem 1.9
corroborates to this fact by giving tail bounds to the modulus of continuity.

###  1.1. Proof ideas

We start with discussing what makes our work hard to accomplish using other
approaches. As a testing ground for non-linear SPDE’s, the KPZ equation
embraces a stack of new tools including regularity structures, paracontrolled
distributions, energy solution method. Through its connection with the KPZ
universality class, the KPZ equation is a paramount testament of a playing
field for the techniques from integrable systems and random matrix theory.
While these tools unveiled salient features of the KPZ equation in the past,
many finer properties are still out of reach. One of the basic requirement for
showing the law of iterated logarithms and the fractal nature of the KPZ level
sets is to attain a delicate understanding of the modulus of continuity of the
KPZ temporal process. This entails to knowing multi-point joint distribution
of the KPZ equation. While the seminal paper [ACQ11] derived one point
distribution of the narrow wedge solution of the KPZ equation, the exact
formulas of more than one point does not seem to be on the horizon (see
[Dim20] for some recent progress in other positive temperature models). In
[CGH19, Theorem 1.5], the authors derived near-exponentially decaying bounds
on the tail probabilities of the difference of the KPZ equation at two time
points. Although these tail bounds were useful for setting forth the two time
correlations of the KPZ equation, they fell short of achieving the modulus of
continuity of the KPZ temporal process since those bounds only valid when the
two time points are far apart.

Our approach is mainly probabilistic while some of the key inputs bear an
integrable origin. Two of such examples are the short time (upper) tail bounds
of the KPZ equation (see Theorem 1.4) and the Gibbsian line ensemble. The
short time upper tail will be derived using the integer moments of the SHE
which has the recourse to some amenable contour integral formulas. On the
other hand, while the Gibbsian line ensemble owes it inception to some
integrable system, it has so far been fostered by the probabilistic ideas. One
of the other key tools which we will procure in the due course of this paper
is the short time lower tail bound (see Theorem 1.7) which in contrast to the
upper tail has its chassis made of core probabilistic ideas like Talagrand’s
concentration inequality.

Our first main tool is a multi-point composition law (see Proposition 2.12)
which generalizes the two-point composition law of [CGH19, Proposition 2.9].
In words, for any given set of time points $0<t_{1}<t_{2}<\ldots<t_{k}$, this
law constructs $k$ independent random spatial profiles equivalent in law to
the narrow wedge solution such that the KPZ temporal process at at $t_{i}$ is
obtained by exponential convolution of one of such independent profiles and
$\mathfrak{h}_{t_{i-1}}(\cdot)$ for $i=2,\ldots,k$.

Our second main tool is the Gibbsian line ensemble. More precisely, we use a
special Gibbsian line ensemble called the KPZ line ensemble introduced by
[CH14]. In short, KPZ line ensemble is a set of random curves whose lowest
indexed curve has the same law as the narrow wedge solution of the KPZ
equation. Furthermore, this set of random curves satisfies the _Brownian Gibbs
property_ which ensures that the law of any fixed index curve in an interval
only depends on the boundary value and can be described using the law of a
Brownian bridge conditioned to have same boundary values, a connection
elicited through a very explicit Radon-Nikodym derivative expression. As it
was revealed in [CH14], the Brownian Gibbs property of the KPZ line ensemble
imparts _stochastic monotonicity_ on its lowest indexed curve, a property
amenable to finding exquisite tail bounds of the spatial profile of the KPZ
equation. Furthermore, we also enrich the arsenal of the Gibbsian line
ensemble by introducing and exploring a _short time KPZ line ensemble_ (see
Proposition 2.5) whose lowest indexed curve is the narrow wedge solution with
_short-time KPZ scaling_ , i.e., the scaling exponent of the fluctuation,
space and time follows the ratio $1:2:4$. In order to distinguish, we would
refer the KPZ line ensemble whose lowest indexed curve is narrow wedge
solution with the KPZ scaling as the _long-time KPZ line ensemble_.

Our third main tool is the short time upper and lower tail bounds (Theorem 1.4
and 1.7) and the long time tail bounds of the KPZ equation from [CG20b, CG20a]
(summarized in Proposition 2.15-2.18). The short time upper tail is derived
using the contour integral formulas of the moments of the SHE whereas the
short time lower tail (uniform in time) is obtained via controlling the tail
estimates of the partition function of random polymer model whose continuum
limit solves the SHE. We also improve the bounds available for the long time
upper tail of the KPZ equation (see Proposition A.1), a key input for showing
the fractal nature of the upper level sets in Theorem 1.3.

Now we proceed to discuss how we use those tools to prove our results. The one
point tail estimates of the KPZ equation (from Theorem 1.4, 1.7 and
Proposition 2.15\- 2.18) in conjugation with the tail bounds of the Brownian
bridge fluctuations would allow us to derive delicate tail bounds of the
spatial profile of the narrow wedge solution in finite intervals at the behest
of the Brownian Gibbs property of the long and short time KPZ line ensembles.
All these new tail estimates are detailed in Section 4. For any given
$t_{1}>t_{2}$, the two point composition law relates $\mathfrak{h}_{t_{1}}$
with the narrow wedge profile $\mathfrak{h}_{t_{2}}(1,\cdot)$ via an
exponential convolution with another independent random spatial process which
will be denoted as $\mathfrak{h}_{t_{2}\downarrow t_{1}}(\cdot)$ and has the
same distribution as $\mathfrak{h}_{t_{1}}((t_{2}-t_{1})/t_{1},\cdot)$. Mating
of this convolution principle with the tail bounds of the KPZ spatial process
from Section 4 propagates the one point tail estimates to the tail bounds of
the difference of the KPZ height functions at two time points. These ideas,
inculcated in Proposition 5.1-5.4 of Section 5, will unfold to be a mainstay
on which the proof of Theorem 1.9 rests with.

By the Borel-Cantelli lemmas, the law of iterated logarithm of Theorem 1.1 can
be recasted as showing that the infimum and supremum of the LIL adjusted
temporal processes $\mathfrak{h}_{t}/(\log\log t)^{1/3}$ and
$\mathfrak{h}_{t}/(\log\log t)^{2/3}$ respectively over the intervals
$[\exp(e^{n}),\exp(e^{n+1})]$ cannot stay further away from $-6^{1/3}$ and
$(3/4\sqrt{2})^{2/3}$ infinitely often. For proving these claims, one needs
delicate tail bounds of the supremum and infimum of the KPZ temporal process
which will be obtained in the following two ways. The first way uses the
multi-point composition law of the KPZ equation (from Proposition 2.12) to
find upper bounds to the upper tail probability of the infimum and lower tail
probability of the supremum. For any given set of time points, the multi-point
composition law returns a set of independent random spatial profiles which are
same in law with the narrow wedge solution. By Proposition 5.1-5.4 of Section
5, we give upper bound of the multi-point tail probabilities of the temporal
process by the one-point tail probabilities of those independent spatial
processes upto to some sharply decaying additive terms (Proposition 6.1). The
tail estimates of the multi-point distribution of the KPZ temporal process
which are later turned into the tail probabilities of the infimum and supremum
bring forth a new set of tools, unknown previously and hefty to obtain
otherwise. The second way would find upper bound to the lower tail probability
of the infimum and upper tail probability of the supremum of the KPZ temporal
process using the tail bounds of the modulus of continuity claimed and proved
in Proposition 5.5.

Much akin to the law of iterated logarithms, the proofs of mono- and multi-
fractality of the KPZ equation heavily rely on the tail probabilities of the
supremum and infimum of the KPZ temporal process in compact intervals. In
addition, the mono-fractality result ((1.4) of Theorem 1.3) requires fast
decoupling of the two-point upper tail probabilities of the KPZ equation.
While such decoupling results are obtained for the Brownian motion in [KKX17,
Lemma 3.5-3.6] without much ado, the situation for the KPZ equation is
complicated and hinges on getting fine estimates of the one-point upper tail
probability. Based on similar techniques as in [CG20a, Proposition 4.1],
Proposition A.3 of Section A provides such tail bounds which will be finally
used in Proposition 7.1 for showcasing the decoupling in the KPZ upper tail
probabilities.

Our approach of studying the peaks and valleys of the KPZ equation has the
potential to generalize for other models in the KPZ universality class. As it
was mentioned earlier, our approach stands on the shoulders of three main
components: multi-point composition law, Gibbsian line ensemble and one-point
tail probabilities. For the zero temperature models like the last passage
percolation model, Airy process and many more, the analogues of the multi-
point composition law are easy to obtain and stated in terms of the _maximum
convolution_ instead of the exponential convolution. Gibbsian line ensemble
approach was first introduced by [CH14] for studying the Airy line ensemble
and then, latter been applied in numerous zero temperature models.
Furthermore, precise one-point tail estimates are available for many zero
temperature model including the KPZ fixed point. Some of these technical
appliances are also available for few positive temperature models such as the
asymmetric simple exclusion process (ASEP), stochastic six vertex model,
strictly weak lattice polymer model etc. With the aid of the above three proof
components, the revelation of the landscape of the aforementioned models bears
immense possibilities which we hope to explore in future works.

###  1.2. Previous works

Studying _macroscopic landscapes_ of stochastic processes is one of the most
compelling research directions in probability theory. Starting from the middle
of the previous century to the present time, Brownian motion serves as a
fertile ground for doing alluring predictions on the landscape of the models
in the Gaussian universality class and demonstrating those with lots of
success. One of the main goals of this work is to showcase the KPZ equation as
a representative of the models in the KPZ universality class when it comes to
explaining the macroscopic landscape of its members under the KPZ scaling.
Below, we review some of the previous works on the LIL and fractal properties
of the models in the KPZ universality with the aim of comparing and
contrasting those with our results.

Random matrix theory is intimately connected with the models of the KPZ
universality class. In fact, the Tracy-Widom GUE distribution which became one
of the characteristics of the fixed points of the universality class was born
out [TW94] as a by-product of a random matrix model. To be more precise, the
limiting distribution of the largest eigenvalue $\lambda^{\mathrm{GUE}}_{n}$
of an $n\times n$ Gaussian unitary ensemble under centering by $\sqrt{2n}$ and
scaling by $n^{1/6}$ is essentially known as the Tracy-Widom GUE distribution.
One may also regard $\lambda^{\mathrm{GUE}}_{n}$ as the $n$-th element of the
GUE minor process. From this point of view, it was an interesting open
question to study the law of fractional logarithm of
$\lambda^{\mathrm{GUE}}_{n}$ which was finally solved by [PZ17]. The authors
found the value of the limsup of
$(\lambda^{\mathrm{GUE}}_{n}-\sqrt{2n})/\sqrt{2}n^{1/6}$ under a normalization
by $(\log n)^{2/3}$ when $n$ goes to $\infty$. The authors had shown that the
value of the limsup is almost surely equal to $(1/4)^{2/3}$. On the other
hand, [PZ17] had also shown that the liminf of
$(\lambda^{\mathrm{GUE}}_{n}-\sqrt{2n})/\sqrt{2}n^{1/6}$ under a normalization
by $(\log n)^{1/3}$ is almost surely finite. They had conjectured that the
liminf is almost surely equal to $-4^{1/3}$. To the best of our knowledge, the
macroscopic Hausdorff dimensions of the level sets of
$\lambda^{\mathrm{GUE}}_{n}$ are not known yet. Drawing the analogy with the
KPZ equation, we conjecture that the peaks and valleys of
$\lambda^{\mathrm{GUE}}_{n}$ are multifractal in the gauge functions $(\log
n)^{2/3}$ and $(\log n)^{1/3}$ respectively.

Last passage percolation (LPP) is one of the widely studied models in the KPZ
universality class. Due to the presence of endearing geometric properties, the
study of the LPP model fueled lots of interests in the recent times. [Led18]
had initiated the study on the laws of iterated logarithms in the case of
integrable LPP models. In [Led18], the author had considered the LPP model in
$\mathbb{Z}^{2}_{\geq 0}$ lattice where the weights of the lattice sites are
independent exponential or, geometric random variables. It was shown in
[Led18] that the limsup of point to point last passage percolation time from
$(0,0)$ to $(n,n)$ (centered by $4n$ and scaled by $(2^{4}n)^{1/3}(\log\log
n)^{2/3}$) is almost surely bounded between $\alpha_{\mathrm{sup}}$ and
$(3/4)^{2/3}$ for some $0<\alpha_{\mathrm{sup}}\leq(3/4)^{2/3}$. In fact,
[Led18] had conjectured that $\alpha_{\mathrm{sup}}$ is equal to
$(3/4)^{2/3}$. [Led18] had also investigated the liminf of the LPP model. It
was shown that the LPP time between $(0,0)$ and $(n,n)$ (centered by $4n$ and
scaled by $(2^{4}n)^{1/3}(\log\log n)^{1/3}$) is almost surely lower bounded
by some constant. Recently, [BGHK19] have shown that the value of liminf is
almost surely equal to a constant. However, not much is known about the exact
value.

Fractal properties of the putative distributional limit of the models in the
KPZ universality class, namely the KPZ fixed point has been investigated in
few of the latest works. Recently, [DOV18] gave a probabilistic construction
of the KPZ fixed point as a distributional limit of the point-to-point
Brownian last passage percolation model. The limiting space-time process which
they named as the _directed landscape_ led to a flurry of new discoveries. The
study of the fractal geometry of the directed landscape has lately been
initiated by [BGH+21, BGH19] who considered the problem of fractal dimension
of some exceptional points along the spatial direction. In spite of the recent
developments, the fractal nature of the space-time process of the directed
landscape is still not fully understood. We hope that our results on
fractality of the KPZ equation would shed some light for such study in future.

In the last decade, fractal properties of stochastic partial differential
equations (SPDE) became an active area of research. The main focus of a vast
majority of those works resided on the study of the large peaks of the SPDEs
with multiplicative noise [GM90, CM94, BC95, HHNT15, FK09, CJKS13, CD15, BC16,
Che17, CHN19]. The growth of the large peaks of the SPDEs is attested by the
intermittency property which is the center of attention in the field of the
research of complex multiscale system for last five-six decades. See
introduction of [BC95] and [CM94, Kho14] for a detailed discussion. Recently,
[KKX17] investigated the fractal properties of the stochastic heat equation
started from the constant initial data at the onset of intermittency and
established the multifractal nature of the spatial process. Denote the
solution of the SHE started from the constant initial data (i.e.,
$\mathcal{Z}^{\mathrm{flat}}(0,x)=1$ for all $x\in\mathbb{R}$) by
$\mathcal{Z}^{\mathrm{flat}}(t,x)$. Drawing on an earlier result of [Che15]
which showed a fractional law of logarithm -

$\displaystyle\limsup_{x\to\infty}\frac{\log\mathcal{Z}^{\mathrm{flat}}(t,x)}{t^{1/3}(\frac{3}{4\sqrt{2}}\log_{+}x)^{2/3}}=1\quad\text{a.s.},$
(1.11)

Theorem 1.2 of [KKX17] established the multifractal nature of the spatial
process $\log\mathcal{Z}^{\mathrm{flat}}(t,\cdot)$ for any fixed $t>0$. The
results of [KKX17] is complemented by the study of the spatio-temporal fractal
properties by [KKX18] which showed that there are infinitely many different
stretch scale (in the spatial direction) and time scale such that for any
given stretch and time scale, the peaks of the spatio-temporal process of the
stochastic heat equation attain non-trivial macroscopic Hausdorff dimensions.
The idea of peaks of the stochastic heat equation forming complex multiscale
system were also echoed in [ZTPSM00, GD05b, GD05a]. However, the macroscopic
behavior of the KPZ temporal process as considered in this paper shows a
different nature due to its slow decay of correlations in comparison to the
KPZ equation along the spatial direction. For instance, our first result,
Theorem 1.1 exhibits LIL for the KPZ temporal process as opposed to the
fractional law of logarithm satisfied by the KPZ spatial process demonstrated
in (1.11). In the same spirit, our second result (Theorem 1.3) which is
reminiscent of a similar result in [KKX17, Theorem 1.4] for $1$-dimensional
Brownian motion shows that the peaks of the KPZ temporal process exhibit a
monofractal (see Definition 1.2) nature as time $t$ goes to $\infty$. This is
in contrast to the multifractal nature of the spatial process as shown in
[KKX17]. Nevertheless, Theorem 1.3 shows that the crossover to the
multifractality of the KPZ temporal process happens under exponential
transformation of the time variable. While the complete understanding of the
spatio-temporal landscape of the KPZ equation is far-off to our present reach,
we hope that our results will ignite further interests along this direction.

We end this section with a review on the tail probabilities of the KPZ
equation, one of the key tools of this paper. Study of the KPZ tail
probabilities had been taken up in many works [MN08, CQ13, Flo14] in the past.
One of the recent major advances has been achieved in [CG20b] which proved
tight bounds to the lower tail probability of the KPZ equation started from
the narrow wedge initial data. This sowed the seeds of a series of works
[CGK+18, Tsa18, KL19, CC19, Zho19, CCR20] which studied in details the lower
tail large deviation of the KPZ equation as time goes to $\infty$. The upper
tail probabilities of the KPZ equation has been recently investigated by
[CG20a]. The same paper also initiated the study of the tail probabilities
under general initial data. The upper tail large deviation was later found in
[DT19] for narrow wedge initial data and in [GL20] for general initial data.
In spite of these recent advances, not much were known about the evolution of
the tail probabilities of the KPZ as time $t$ goes to $0$. In a very recent
work, [LT20] showed the large deviation of the KPZ equation as $t$ tends to
$0$. However, this does not shed much light on the uniform tail estimates of
the KPZ height function starting from time equal to $0$ to a finite value.
Such uniform estimates which were reported in Theorem 1.4 and 1.7 will be
instrumental in obtaining our other main results Theorem 1.1, 1.3 and 1.9.

Outline. Section 2 will introduce the basic frameworks of the KPZ line
ensemble and the Barlow-Taylor macroscopic fractal theory. It will also
introduce other useful tools including multipoint composition law, one-point
tail probabilities of the KPZ equation, tail probabilities of the supremum and
infimum of the KPZ spatial process. Section 3 will prove Theorem 1.4 and 1.7.
This will be followed by Section 4 where we derive delicate tail bounds of the
KPZ spatial process for finite and short time. Section 5 will study the
temporal modulus of continuity of the KPZ equation and use it to prove Theorem
1.9. Based on the tools from Section 2-5, the law of iterated logarithms of
Theorem 1.1 will be proved in Section 6. The proof of the mono- and
multifractality results of the KPZ equation from Theorem 1.3 will be given in
Section 7. This last section will use an improved KPZ upper tail probability
estimate which is proved in Proposition A.1 of Appendix A.

Acknowledgements. We are grateful to Ivan Corwin for numerous stimulating
discussions, encouragement, and giving us valuable inputs in an earlier draft
of the paper. We express our gratitude towards Benjamin Landon whose
suggestions helped us to improve Theorem 1.3. We thank Davar Khoshnevisan for
asking interesting questions on our work. We thank Shalin Parekh and Li-Cheng
Tsai for helpful conversations and discussions. We thank Yier Lin for reading
the manuscript carefully and making useful comments which improve the
presentation.

SD’s research was partially supported by Ivan Corwin’s NSF grant DMS-1811143
as well as the Fernholz Foundation’s “Summer Minerva Fellows” program.

##  2\. Basic Framework and Tools

In this section, we will review mainly three topics which are required for our
subsequent analysis. One of the main topics of this section is the KPZ line
ensemble and its Brownian Gibbs property. The KPZ line ensemble is a set of
random curves whose lowest indexed curve is same in distribution with the
narrow wedge solution of the KPZ equation. The Brownian Gibbs property of the
KPZ line ensemble induces stochastic monotonicity of the spatial profile the
KPZ equation, one of the major tools in our analysis. Lemma 2.4 of Section 2.1
will precisely state such monotonicity result. In a similar way as in [CH16],
we will introduce a short-time version of the KPZ line ensemble which would
play a key role in later sections to find the temporal modulus of continuity
of the KPZ equation.

Our second main topic of this section is the Barlow-Taylor theory of
macroscopic fractal properties of a stochastic process. In light of the
expositions in [KKX17, BT89, BT92], the notions of Barlow-Taylor _Hausdorff
content_ and _dimension_ of any Borel set will be recalled. Some of the basic
properties of the Barlow-Taylor Hausdorff dimension are presented in
Proposition 2.7, 2.8 and 2.10 of Section 2.2.

Lastly, we recall some of the known facts about the KPZ equation including its
_multipoint composition law_ and the tail estimates of its one point
distribution in Section 2.3.

###  2.1. KPZ line ensemble

Describing the KPZ line ensemble and its Brownian Gibbs property requires some
notations which are introduced as follows.

######  Definition 2.1 (Brownian Gibbs line ensembles; Definitions 2.1 and 2.2
of [CH16]).

Fix intervals $\Sigma\subset\mathbb{Z}$ and $\Lambda\subset\mathbb{R}$. Let
$X$ be the set of continuous functions $f:\Sigma\times\Lambda\to\mathbb{R}$,
endowed with the topology of uniform convergence on compact subsets;. Denote
the $\sigma$-field generated by $X$ by $\mathcal{C}$.

A $(\Sigma\times\Lambda)$-_indexed line ensemble_ $\mathcal{L}$ is a random
variable $\mathcal{L}$ in a probability space
$(\Omega,\mathcal{B},\mathbb{P})$ taking values in $X$ such that $\mathcal{L}$
is measurable with respect to $(\mathcal{B},\mathcal{C})$. In simple words,
$\mathcal{L}$ is a set of random continuous curves indexed by $\Sigma$ where
each of those curves maps $\Lambda$ to $\mathbb{R}$. An element of $\Sigma$ is
a curve index, and we will write $\mathcal{L}_{k}(x)$ instead of
$\mathcal{L}(k,x)$ for $k\in\Sigma$ and $x\in\Lambda$; we will write
$\mathcal{L}_{k}$ for the entire index $k$ curve.

For any two integers $k_{1}<k_{2}$, two vectors of reals
$\vec{x},\vec{y}\in\mathbb{R}^{k_{1}-k_{2}+1}$, and an interval $(a,b)$, we
say that a $\\{k_{1},\ldots,k_{2}\\}\times(a,b)$-indexed line ensemble is a
_Brownian bridge line ensemble with entrance data $\vec{x}$ and exit data
$\vec{y}$_ if its law, which we denote by
$\mathbb{P}^{k_{1},k_{2},(a,b),\vec{x},\vec{y}}_{\mathrm{free}}$, is equal to
that of $k_{2}-k_{1}+1$ independent Brownian bridges starting at values
$\vec{x}$ at $a$ and ending at values $\vec{y}$ at $b$. We use the notation
$\mathbb{E}^{k_{1},k_{2},(a,b),\vec{x},\vec{y}}_{\mathrm{free}}$ to denote the
expectation with respect to the probability measure
$\mathbb{P}^{k_{1},k_{2},(a,b),\vec{x},\vec{y}}_{\mathrm{free}}$. When
$k_{1}=k_{2}=1$, we write
$\mathbb{P}^{(a,b),\vec{x},\vec{y}}_{\mathrm{free}}$. One may think $a$ and
$b$ as times and $\vec{x}$ and $\vec{y}$ as starting and ending locations for
the Brownian bridges.

Suppose we are given a continuous function
$\mathbf{H}:\mathbb{R}\to[0,\infty)$ which we will call a _Hamiltonian_. We
will consider the following two kinds of Hamiltonians:

$\displaystyle\mathbf{H}^{\mathrm{long}}_{t}(x)=e^{t^{1/3}x}\,\,\,\,\quad\text{and,
}\quad\,\,\,\mathbf{H}^{\mathrm{short}}_{t}(x)=e^{(\pi
t/4)^{1/4}x}\,\,\,\,\textrm{for given $t>0$}\,$ (2.1)

For any given Hamiltonian $\mathbf{H}$ and two measurable functions
$f,g:(a,b)\to\mathbb{R}$, we define a
$\\{k_{1},\ldots,k_{2}\\}\times(a,b)$-indexed $\mathbf{H}$-Brownian bridge
line ensemble with entrance data $\vec{x}$, exit data $\vec{y}$ and boundary
data $(f,g)$ to be a collection of random curves
$\mathcal{L}_{k_{1}},\ldots,\mathcal{L}_{k_{2}}:(a,b)\to\mathbb{R}$ whose law
will be denoted by
$\mathbb{P}^{k_{1},k_{2},(a,b),\vec{x},\vec{y},f,g}_{\mathbf{H}}$ and is
specified by the Radon-Nikodym derivative

$\displaystyle\frac{d\mathbb{P}^{k_{1},k_{2},(a,b),\vec{x},\vec{y},f,g}_{\mathbf{H}}}{d\mathbb{P}^{k_{1},k_{2},(a,b),\vec{x},\vec{y}}_{\mathrm{free}}}(\mathcal{L}_{k_{1}},\ldots,\mathcal{L}_{k_{2}})$
$\displaystyle=\frac{W^{k_{1},k_{2},(a,b),\vec{x},\vec{y},f,g}_{\mathbf{H}}(\mathcal{L}_{k_{1}},\ldots,\mathcal{L}_{k_{2}})}{Z^{k_{1},k_{2},(a,b),\vec{x},\vec{y},f,g}_{\mathbf{H}}}\,,$
$\displaystyle
W^{k_{1},k_{2},(a,b),\vec{x},\vec{y},f,g}_{\mathbf{H}}(\mathcal{L}_{k_{1}},\ldots,\mathcal{L}_{k_{2}})$
$\displaystyle=\exp\bigg{\\{}-\sum_{i=k_{1}}^{k_{2}+1}\int\mathbf{H}\big{(}\mathcal{L}_{i}(x)-\mathcal{L}_{i-1}(x)\big{)}dx\bigg{\\}}\,.$

where in the right-hand side of the preceding display, we use the convention
that $\mathcal{L}_{k_{1}-1}$ is equal to $f$, or to $+\infty$ if
$k_{1}-1\notin\Sigma$; and of $\mathcal{L}_{k_{2}+1}$ is equal to $g$, or to
$-\infty$ if $k_{2}+1\notin\Sigma$. Here,
$Z^{k_{1},k_{2},(a,b),\vec{x},\vec{y},f,g}_{\mathbf{H}}$ is the normalizing
constant which produces a probability measure.

A $(\Sigma\times\Lambda)$-indexed line ensemble $\mathcal{L}$ enjoys the
_$\mathbf{H}$ -Brownian Gibbs property_ if, for all
$K=\\{k_{1},\ldots,k_{2}\\}\subset\Sigma$ and $(a,b)\subset\Lambda$, the
following distributional equality holds:

$\displaystyle\mathrm{Law}\Big{(}\mathcal{L}_{K\times(a,b)}\text{ conditioned
on }\mathcal{L}_{\Sigma\times\Lambda\backslash
K\times(a,b)}\Big{)}=\mathbb{P}^{k_{1},k_{2},\vec{x},\vec{y},f,g}_{\mathbf{H}}\,,$

where $\vec{x}=(\mathcal{L}_{k_{1}}(a),\ldots,\mathcal{L}_{k_{2}}(a))$,
$\vec{y}=(\mathcal{L}_{k_{1}}(b),\ldots,\mathcal{L}_{k_{2}}(b))$, and where
again $f=\mathcal{L}_{k_{1}-1}$ (or $+\infty$ if $k_{1}-1\notin\Sigma$) and
$g=\mathcal{L}_{k_{2}+1}$ (or $-\infty$ if $k_{2}+1\notin\Sigma$). That is,
the influence of the complementary part on the ensemble’s restriction to
$K\times(a,b)$ comes only through the boundary data, i.e., the starting and
ending points and the neighbouring curves.

Like as for Markov processes, there is a _strong_ version of the
$\mathbf{H}$-Brownian Gibbs property which is valid with respect to _stopping
domains_ which we now describe. For a given line ensemble $\mathcal{L}$, let
$\mathfrak{F}_{\textrm{ext}}\big{(}K\times(a,b)\big{)}$ be the $\sigma$-field
generated by the curves $K\times(a,b)$. A pair $(\mathfrak{a},\mathfrak{b})$
of random variables is called a $K$-stopping domain if
$\big{\\{}\mathfrak{a}\leq a,\mathfrak{b}\geq
b\big{\\}}\in\mathfrak{F}_{\textrm{ext}}\big{(}K\times(a,b)\big{)}$. Denote
the set of continuous $K$-indexed functions $(f_{k_{1}},\ldots,f_{k_{2}})$,
each from $(a,b)\to\mathbb{R}$ by $C^{K}(a,b)$ and furthermore, write

$C^{K}:=\Big{\\{}(a,b,f_{k_{1}},\ldots,f_{k_{2}}):a<b\textrm{ and
}(f_{k_{1}},\ldots,f_{k_{2}})\in C^{K}(a,b)\Big{\\}}\,.$

Let $\mathcal{B}(C^{K})$ be the set of all Borel measurable functions from
$C^{K}$ to $\mathbb{R}$. We will say a $K$-stopping domain
$(\mathfrak{a},\mathfrak{b})$ satisfies the _strong $\mathbf{H}$-Brownian
Gibbs property_ if, for all $F\in\mathcal{B}(C^{K})$, $\mathbb{P}$-almost
surely

$\mathbb{E}\Big{[}F\big{(}\mathfrak{a},\mathfrak{b},\mathcal{L}\big{|}_{K\times(\mathfrak{a},\mathfrak{b})}\big{)}\Big{|}\mathfrak{F}_{\textrm{ext}}\big{(}K\times(a,b)\big{)}\Big{]}=\mathbb{E}^{k_{1},k_{2},(\ell,r),\vec{x},\vec{y},f,g}_{\mathbf{H}}\Big{[}F\big{(}\ell,r,\mathcal{L}_{k_{1}},\ldots,\mathcal{L}_{k_{2}}\big{)}\Big{]}\,,$

where, on the right-hand side, $\ell=\mathfrak{a}$, $r=\mathfrak{b}$,
$\vec{x}=(\mathcal{L}_{i}(\mathfrak{a}))_{i\in K}$,
$\vec{y}=(\mathcal{L}_{i}(\mathfrak{b}))_{i\in K}$, $f=\mathcal{L}_{k_{1}-1}$
(or $+\infty$ if $k_{2}+1\notin\Sigma$), $g=\mathcal{L}_{k_{2}+1}$ (or
$-\infty$ if $k_{2}+1\notin\Sigma$), and the curves
$\mathcal{L}_{k_{1}},\ldots,\mathcal{L}_{k_{2}}$ have law
$\mathbb{P}^{k_{1},k_{2},(\ell,r),\vec{x},\vec{y},f,g}_{\mathbf{H}}$.

The following lemma demonstrates a sufficient condition under which the strong
H-Brownian Gibbs property holds.

######  Lemma 2.2 (Lemma 2.5 of [CH16]).

Any line ensemble which enjoys the $\mathbf{H}$-Brownian Gibbs property also
enjoys the strong $\mathbf{H}$-Brownian Gibbs property.

Line ensembles with the $\mathbf{H}$-Brownian Gibbs property benefit from
certain stochastic monotonicities of the underlying measures. The following
definition formally defines such monotonicity of measures. Afterwards, we
state a proposition showing that two line ensembles with the same index set
can be coupled in such a way that if the boundary conditions of one ensemble
dominates the other, then same is true for laws of the restricted curves.

######  Definition 2.3 (Domination of measure).

Let $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ be two
$(\Sigma\times\Lambda)$-indexed line ensembles with respective laws
$\mathbb{P}_{1}$ and $\mathbb{P}_{2}$. We say that $\mathbb{P}_{1}$ dominates
$\mathbb{P}_{2}$ if there exists a coupling of $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$ such that $\mathcal{L}^{j}_{1}(x)\geq\mathcal{L}^{j}_{2}(x)$
for all $j\in\Sigma$ and $x\in\Lambda$.

######  Lemma 2.4 (Stochastic monotonicity: Lemmas $2.6$ and $2.7$ of [CH16]).

Fix finite intervals $K\subset\Sigma$ and $(a,b)\subset\Lambda$; and, for
$i\in\\{1,2\\}$, vectors $\vec{x}_{i}=\big{(}x_{i}^{(k)}:k\in K\big{)}$ and
$\vec{y}_{i}=\big{(}y_{i}^{(k)}:k\in K\big{)}$ in $\mathbb{R}^{K}$ that
satisfy $x^{(k)}_{2}\leq x^{(k)}_{1}$ and $y^{(k)}_{2}\leq y^{(k)}_{1}$ for
$k\in K$; as well as measurable functions
$f_{i}:(a,b)\to\mathbb{R}\cup\\{+\infty\\}$ and
$g_{i}:(a,b)\to\mathbb{R}\cup\\{-\infty\\}$ such that $f_{2}(s)\leq f_{1}(s)$
and $g_{2}(s)\leq g_{1}(s)$ for $s\in(a,b)$. For $i\in\\{1,2\\}$, let
$\mathbb{P}_{i}$ denote the law
$\mathbb{P}^{k_{1},k_{2},(a,b),\vec{x}_{i},\vec{y}_{i},f_{i},g_{i}}_{\mathbf{H}}$,
so that a $\mathbb{P}_{i}$-distributed random variable
$\mathcal{L}_{i}=\\{\mathcal{L}_{i}^{k}(s)\\}_{k\in K,s\in(a,b)}$ is a
$K\times(a,b)$-indexed line ensemble. If $\mathbf{H}:[0,\infty)\to\mathbb{R}$
is convex, then $\mathbb{P}_{1}$ dominates $\mathbb{P}_{2}$ – that is, a
common probability space $(\Omega,\mathcal{B},\mathbb{P})$ may be constructed
on which the two measures are supported such that, almost surely,
$\mathcal{L}_{1}^{k}(s)\geq\mathcal{L}_{2}^{k}(s)$ for $k\in K$ and
$s\in(a,b)$.

Recall that the Hamiltonians $\mathbf{H}^{\mathrm{long}}_{t}(x)$ and
$\mathbf{H}^{\mathrm{short}}_{t}(x)$ in (2.1) are convex. Thus, Lemma 2.4
applies to any $\mathbf{H}^{\mathrm{long}}_{t}$ or,
$\mathbf{H}^{\mathrm{short}}_{t}$-Brownian Gibbs line ensemble.

The following proposition recalls the unscaled and scaled KPZ line ensemble
constructed in [CH16] which satisfies
$\mathbf{H}^{\mathrm{long}}_{t}$-Brownian Gibbs property and introduces the
short time KPZ line ensemble which exhibits
$\mathbf{H}^{\mathrm{short}}_{t}$-Brownian Gibbs property.

######  Lemma 2.5.

Let $t>0$. There exists an $\mathbb{N}\times\mathbb{R}$-indexed line ensemble
$\mathcal{H}_{t}=\\{\mathcal{H}^{(n)}_{t}(x)\\}_{n\in\mathbb{N},x\in\mathbb{R}}$
such that:

  1. (1)

the lowest indexed curve $\mathcal{H}^{(1)}_{t}(x)$ is equal in distribution
(as a process in $x$) to the Cole-Hopf solution
$\mathcal{H}^{\mathbf{nw}}(t,x)$ of KPZ started from the narrow wedge initial
data;

  2. (2)

$\mathcal{H}_{t}$ satisfies the $\mathbf{H}^{\mathrm{long}}_{1}$-Brownian
Gibbs property;

  3. (3)

the scaled KPZ line ensemble
$\\{\mathfrak{h}^{(n)}_{t}(x)\\}_{n\in\mathbb{N},x\in\mathbb{R}}$, defined by

$\displaystyle\mathfrak{h}^{(n)}_{t}(x)\,=\,t^{-1/3}\Big{(}\mathcal{H}^{(n)}_{t}\big{(}t^{2/3}x\big{)}+t/24\Big{)}\,,$

satisfies the $\mathbf{H}^{\mathrm{long}}_{t}$-Brownian Gibbs property.

  4. (4)

and the scaled short time line ensemble
$\\{\mathfrak{g}^{(n)}_{t}(x)\\}_{n\in\mathbb{N},x\in\mathbb{R}}$, defined by

$\displaystyle\mathfrak{g}^{(n)}_{t}(x)\,=\,(\pi
t/4)^{-1/4}\Big{(}\mathcal{H}^{(n)}_{t}\big{(}(\pi
t/4)^{1/2}x\big{)}+\log\sqrt{2\pi t}\Big{)}\,,$ (2.2)

satisfies the $\mathbf{H}^{\mathrm{short}}_{t}$-Brownian Gibbs property.

###### Proof.

The part $(1)$, $(2)$ and $(3)$ follow from the part $(1)$, $(2)$ and $(3)$ of
Theorem 2.15 of [CH16] respectively. For the proof of part $(4)$, we rely on
the proof of part $(3)$ in [CH16, Theorem 2.15]. The main ingredients of their
proof were the one point tail probabilities and spatial stationarity of
$\mathfrak{h}^{(1)}_{t}(\cdot)$. These two properties are also present for the
lowest indexed curve for the short time line ensemble
$\\{\mathfrak{g}^{(n)}_{t}(x)\\}_{n\in\mathbb{N},x\in\mathbb{R}}$ (see Theorem
1.4, 1.7 and Lemma 2.11). With these in hand, the part $(4)$ can be proved
exactly in the same way as part $(3)$ of [CH16, Theorem 2.15]. For brevity, we
skip the details. ∎

The above result envisages the Brownian Gibbs property of the ensemble to be
brought to bear as a tool for analysing the spatial profiles
$\mathfrak{h}_{t}(x)$ and $\mathfrak{g}_{t}(x)$ by demonstrating that the
lowest indexed curves $\mathfrak{h}_{t}^{(1)}$ and $\mathfrak{g}_{t}^{(1)}$ in
the scaled long time and short time KPZ line ensemble have the laws of the
centered and scaled narrow wedge solution
$\mathfrak{h}_{t}(x):=\mathfrak{h}_{t}(1,x)$ and
$\mathfrak{g}_{t}(x):=\mathfrak{g}_{t}(1,x)$ of the KPZ equation defined in
(2.4).

###  2.2. Barlow-Taylor’s macroscopic fractal theory

######  Definition 2.6 (Hausdorff content and dimension).

For any Borel set $\mathcal{A}\subset\mathbb{R}$, the $n$_-th shell_ of
$\mathcal{A}$ is defined as
$\mathcal{A}\cap\big{\\{}(-e^{n+1},-e^{n}]\cup[e^{n},e^{n+1})\big{\\}}$. Let
us fix a number $c_{0}>0$, and the set $\mathcal{A}\subset\mathbb{R}$ and
$\rho>0$, define $\rho$-dimensional _Hausdorff content_ of the $n$-th shell of
$\mathcal{A}$ as

$\displaystyle\nu_{n,\rho}(\mathcal{A}):=\inf\sum_{i=1}^{m}\Big{(}\frac{\mathrm{Length}(Q_{i})}{e^{n}}\Big{)}^{\rho}$

where the infimum is taken over all sets of intervals $Q_{1},\ldots,Q_{m}$ of
length greater than $c_{0}$ and covering $n$-th shell of $\mathcal{A}$. Define
the $\rho$-dimensional Hausdorff content of the set $\mathcal{A}$ as a sum
total of $\nu_{n,\rho}(\mathcal{A})$ as $n$ varies over the set of all
positive integers. Then, the Barlow-Taylor _macroscopic Hausdorff dimension_
the set $\mathcal{A}$ is defined as the infimum over all $\rho>0$ such that
the $\rho$-dimensional Hausdorff content of $\mathcal{A}$ is finite, i.e.,

$\displaystyle\mathrm{Dim}_{\mathbb{H}}(\mathcal{A}):=\inf\big{\\{}\rho>0:\sum_{n=0}^{\infty}\nu_{n,\rho}(\mathcal{A})<\infty\big{\\}}.$

From the definition, it follows that the macroscopic Hausdorff dimension of a
bounded set is $0$. Just as in the microscopic case, one has
$\mathrm{Dim}_{\mathbb{H}}(E)\leq\mathrm{Dim}_{\mathbb{H}}(F)$ when $E$ is
contained in $F$. Furthermore, it has been observed in [KKX17, Lemma 2.3] that
the macroscopic Hausdorff dimension does not depend on the value of $c_{0}$.
These observations are summarized in the following proposition.

######  Proposition 2.7 ([BT89, BT92, KKX17]).

Consider $E\subset\mathbb{R}$. Then, $\mathrm{Dim}_{\mathbb{H}}(E)$ does not
depend on the value of $c_{0}$ of Definition 2.6 and
$\mathrm{Dim}_{\mathbb{H}}(E)\leq\mathrm{Dim}_{\mathbb{H}}(F)$ for $F\supset
E$. Moreover, $\mathrm{Dim}_{\mathbb{H}}(E)=0$ if $E$ is bounded.

Since the choice of $c_{0}>0$ does not matter, we will work with the choice of
Barlow and Taylor [BT89, BT92, KKX17] and from now on, we set $c_{0}:=1$.

We next mention a technical estimate on the Hausdorff content of any set. The
following proposition, as stated in [KKX17] is a macroscopic analogue of the
classical Frostman lemma for microscopic Hausdorff dimension.

######  Proposition 2.8 (Lemma 2.5 of [KKX17]).

Fix $n\in\mathbb{R}_{\geq 1}$, and suppose $E$ is a subset of the shell
$[-e^{n+1},-e^{n})\cup(e^{n},e^{n+1}]$. Denote the Lebesgue measure of a Borel
set $B\subset\mathbb{R}$ by $\mathrm{Leb}(B)$. Let $\mu$ be a finite Borel
measure on $\mathbb{R}$ and define for $\rho>0$,

$\displaystyle K_{n,\rho}:=\sup\Big{\\{}\frac{\mu(Q)}{\mathrm{Leb}(Q)}:Q\text{
is a Borel set in
}[-e^{n+1},-e^{n})\cup(e^{n},e^{n+1}],\quad\mathrm{Leb}(Q)\geq 1\Big{\\}}.$
(2.3)

Then, we have $\nu_{n,\rho}(E)\geq K^{-1}_{n,\rho}e^{-n\rho}\mu(E)$.

The above proposition will be used in Section 7 to show lower bound to the
macroscopic Hausdorff dimension of the level sets of the KPZ equation. In the
following, we introduce a notion of _thickness_ of a set, another important
tool to bound the Hausdorff dimension from below.

######  Definition 2.9 ($\theta$-Thickness).

Fix $\theta\in(0,1)$ and define

$\Pi_{n}(\theta):=\bigcup_{\begin{subarray}{c}0\leq j\leq
e^{n(1-\theta)+1}-e^{n(1-\theta)}\\\
j\in\mathbb{Z}\end{subarray}}\\{e^{n}+je^{n\theta}\\}.$

We say $E\subset\mathbb{R}$ is $\theta$-thick if there exist integer
$M=M(\theta)$ such that $E\cap[x,x+e^{\theta n}]\neq\varnothing$ for all
$x\in\Pi_{n}(\theta)$ and for all $n\geq M$.

The following result from [KKX17] provides a lower bound to the Hausdorff
dimension of a given set in terms of its thickness.

######  Proposition 2.10 (Corollary 4.6 in [KKX17]).

If $E\subset\mathbb{R}$ is $\theta$-thick for some $\theta\in(0,1)$, then
$\operatorname{Dim}_{\mathbb{H}}(E)\geq 1-\theta$.

###  2.3. KPZ equation results

We start with introducing the space-time scaling of the KPZ height function
appropriate for the short time regime, i.e., the case when the time variable
goes to $0$.

$\displaystyle\mathfrak{g}_{t}(\alpha,x):=\frac{\mathcal{H}^{\mathbf{nw}}(\alpha
t,(\pi t/4)^{1/2}x)+\log\sqrt{2\pi\alpha t}}{(\pi t/4)^{1/4}}.$ (2.4)

We will often use the shorthand notation
$\mathfrak{g}_{t}(x):=\mathfrak{g}_{t}(1,x)$. In addition, we simply write
$\mathfrak{g}_{t}:=\mathfrak{g}_{t}(1,0)$ when $x=0$. The following lemma
shows the spatial stationarity of the process $\mathfrak{h}_{t}(\cdot)$ and
$\mathfrak{g}_{t}(\cdot)$.

######  Lemma 2.11 (Stationarity).

The one point distribution of $\mathfrak{h}_{t}(x)+\frac{x^{2}}{2}$ is
independent of $x$ and converges weakly to Tracy-Widom GUE distribution as
$t\uparrow\infty$. On the other hand, the one point distribution of
$\mathfrak{g}_{t}(x)+\frac{(\pi t/4)^{3/4}x^{2}}{2t}$ is independent of $x$
and converges weakly to standard Gaussian distribution as $t\downarrow 0$.

###### Proof.

The first part was proved in Proposition 1.7 of [CH16]. By Proposition 1.4 of
[ACQ11], we know $\mathcal{H}^{\mathbf{nw}}(t,z)+\frac{z^{2}}{2t}$ is
stationary in $z$. As a result,

$\displaystyle\mathfrak{g}_{t}(x)+\frac{(\pi t/4)^{3/4}x^{2}}{2t}=(\pi
t/4)^{-1/4}\left[\mathcal{H}^{\mathbf{nw}}(t,(\pi t/4)^{1/2}x)+\frac{(\pi
t/4)x^{2}}{2t}+\log\sqrt{2\pi t}\right]$

is stationary in $x$. From Proposition 1.8 in [ACQ11], it follows that
$\mathfrak{g}_{t}(0)$ converges weakly to standard Gaussian distribution as
$t\downarrow 0$. ∎

Our next result provides a multipoint composition law of the KPZ temporal
process. In latter sections, this will be used to infer properties of
multipoint distributions of $\mathfrak{h}_{t}$. Our proof of the multipoint
composition law resembles the one in [CGH19, Proposition 2.9] which proves the
two point composition law. For stating the law, we introduce the following
notation. For $t>0$, define a $t$-indexed composition map $I_{t}(f,g)$ between
two functions $f(\cdot)$ and $g(\cdot)$ as

$\displaystyle
I_{t}(f,g):=t^{-1/3}\log\int^{\infty}_{-\infty}e^{t^{1/3}\big{(}f(t^{-2/3}y)+g(-t^{-2/3}y)\big{)}}dy\,.$
(2.5)

######  Proposition 2.12.

For any fixed $t>0$, $k\in\mathbb{N}$ and
$1<\alpha_{1}<\alpha_{2}<\ldots<\alpha_{k}$, there exist independent spatial
processes $\mathfrak{h}_{\alpha_{1}t\downarrow
t},\mathfrak{h}_{\alpha_{2}t\downarrow\alpha_{1}t},\cdots,\mathfrak{h}_{\alpha_{k}t\downarrow\alpha_{k-1}t}$
supported on the same probability space as the KPZ equation solution such
that:

  1. (1)

$\mathfrak{h}_{\alpha_{i}t\downarrow\alpha_{i-1}t}(\cdot)$ is distributed
according to the law of the process
$\mathfrak{h}_{\alpha_{i-1}t}((\alpha_{i}-\alpha_{i-1})/\alpha_{i-1},\cdot)$;

  2. (2)

$\mathfrak{h}_{\alpha_{i}t\downarrow\alpha_{i-1}t}(\cdot)$ is independent of
$\mathfrak{h}_{\alpha_{i-1}t}(\cdot)$; and

  3. (3)

$\mathfrak{h}_{\alpha_{i-1}t}(\frac{\alpha_{i}}{\alpha_{i-1}},0)=I_{\alpha_{i-1}t}\big{(}\mathfrak{h}_{\alpha_{i-1}t},\mathfrak{h}_{\alpha_{i}t\downarrow\alpha_{i-1}t}\big{)}$.

###### Proof.

For $s<t$ and $x,y\in\mathbb{R}$, let $\mathcal{Z}^{\mathbf{nw}}_{s,x}(t,y)$
be the solution at time $t$ and position $y$ of the SHE started at time $s$
with Dirac delta initial data at position $x$. We will show that for any
$0<t_{1}<\ldots<t_{k}$ and $y_{1},\ldots,y_{k}\in\mathbb{R}$, there exists
independent spatial processes $\mathcal{Z}_{y_{2}}(t_{2}\downarrow
t_{1},\cdot),\ldots,\mathcal{Z}_{y_{k}}(t_{k}\downarrow t_{k-1},\cdot)$
coupled on a probability space upon which the space-time white noise of the
KPZ equation is defined such that

$\displaystyle\mathcal{Z}^{\mathbf{nw}}(t_{i},y_{i})=\mathcal{Z}^{\mathbf{nw}}_{0,0}(t_{i},y_{i})=\int_{\mathbb{R}}\mathcal{Z}^{\mathbf{nw}}_{0,0}(t_{i-1},x)\mathcal{Z}_{y_{i}}(t_{i}\downarrow
t_{i},x)dx\,,$ (2.6)

and the law of $\mathcal{Z}_{y_{i}}(t_{i}\downarrow t_{i-1},\cdot)$ is same as
that of $\mathcal{Z}^{\mathbf{nw}}(t_{i}-t_{i-1},y_{i}-\cdot)$ for $2\leq
i\leq k$. Expressing the convolution and interchange properties in terms of
$\mathfrak{h}_{t}$ immediately yields the proposition.

We now return to show (2.6). The above convolution formula is known when $k=2$
(see [CGH19]). We extend the proof given in [CGH19] for $k>2$ using the chaos
series for the SHE (see [Cor18, Wal86, AKQ14] for background). Here we have
used the following notations. We write
$\vec{s}=(s_{1},\ldots,s_{\ell})\in\mathbb{R}^{\ell}_{\geq 0}$,
$\vec{x}=(x_{1},\ldots,x_{\ell})\in\mathbb{R}^{\ell}$ and define the set of
ordered times

$\Delta_{\ell}(s,t)=\\{\vec{s}:s\leq s_{1}\leq s_{2}\leq\ldots\leq
s_{\ell}\leq t\\}.$

For any $0\leq s<t$ and $x,y\in\mathbb{R}$,
$\mathcal{Z}^{\mathbf{nw}}_{s,x}(t,y)$ is given as the following chaos series
expansion (see Theorem 2.2 of [Cor18]):

$\displaystyle\mathcal{Z}^{\mathbf{nw}}_{s,x}(t,y)=\sum_{\ell=0}^{\infty}\int_{\Delta_{\ell}(s,t)}\int_{\mathbb{R}^{\ell}}P_{\ell;s,x;t,y}(\vec{s},\vec{x})d\xi^{\otimes_{\ell}}(\vec{s},\vec{x}).$
(2.7)

The integration in (2.7) is a multiple Itô stochastic integral against the
white noise $\xi$ and the term $P_{\ell;s,x;t,y}(\vec{s},\vec{x})$ is the
density function for a one-dimensional Brownian motion starting from $(s,x)$
to go through the time-space points $(s_{1},x_{1}),\ldots,(s_{\ell},x_{\ell})$
and ends up at $(t,y)$. This transition density has the following product
formula using the Gaussian heat kernel $p(s,y):=(2\pi
s)^{-1/2}\exp(-y^{2}/2s)$ and the conventions $s_{0}=s$, $s_{\ell+1}=t$,
$x_{0}=x$ and $x_{\ell+1}=y$:

$\displaystyle
P_{\ell;s,x;t,y}(\vec{s},\vec{x})=\prod_{j=0}^{\ell}p(s_{i+1}-s_{i},x_{i+1}-x_{i}).$

For any $0\leq s<t$, the heat kernel $p(\cdot,\cdot)$ satisfies the simple
convolution identity

$\displaystyle p(t,x)=\int p(s,y)p(t-s,x-y)dy.$ (2.8)

Fix $2\leq i\leq k$. By using the fact that the sum of indicator functions
gives the value one, we may replace $\int_{\Delta_{\ell}(0,t_{i})}$ in (2.7)
by the quantity
$\sum_{j=0}^{\ell}\int_{\Delta_{\ell}(0,t_{i})}\mathbf{1}_{s_{j}\leq
t_{i}<s_{j+1}}$. As a consequence, we get

$\displaystyle\mathcal{Z}^{\mathbf{nw}}_{0,0}(t_{i},y_{i})=\sum_{\ell=0}^{\infty}\sum_{j=0}^{\ell}\int_{\Delta_{\ell}(s_{j},t_{i})}\int_{\mathbb{R}^{k}}\mathbf{1}_{s_{j}\leq
t_{i-1}<s_{j+1}}P_{\ell;0,0;t_{i},y_{i}}(\vec{s},\vec{x})d\xi^{\otimes_{\ell}}(\vec{s},\vec{x}).$
(2.9)

For $1\leq a\leq b\leq k$, $\vec{s}_{[a,b]}$ denotes $(s_{a},\ldots,s_{b})$
and likewise for $\vec{x}$. Using these notations and (2.8), we may write

$\mathbf{1}_{s_{j}\leq
t_{i-1}<s_{j+1}}P_{\ell;0,0;t_{i},y_{i}}(\vec{s},\vec{x})=\mathbf{1}_{s_{j}\leq
t_{i-1}<s_{j+1}}\int_{\mathbb{R}}P_{\ell;0,0;t_{i-1},z}(\vec{s}_{[1,j]},\vec{x}_{[1,j]})P_{\ell-j;t_{i-1},z;t_{i},y}(\vec{s}_{[j+1,\ell]},\vec{x}_{[j+1,\ell]})dz.$

We now insert the above display into (2.9). We also replace
$\int_{\Delta_{\ell}(0,t_{i})}\mathbf{1}_{s_{j}\leq t_{i-1}<s_{j+1}}$ by the
product of the integral
$\int_{\Delta_{j}(0,t_{i-1})}\int_{\Delta_{\ell-j}(t_{i-1},t_{i})}$ and
relabel $\vec{s}_{[1,j]}=\vec{u}$, $\vec{s}_{[j+1,\ell]}=\vec{v}$,
$\vec{x}_{[1,j]}=\vec{a}$, $\vec{x}_{[j+1,\ell]}=\vec{b}$,
$P_{j;0,0;t_{i-1},z}(\vec{u},\vec{a})=P^{0,0}_{j;t_{i-1},z}(\vec{u},\vec{a})=P_{j;t_{i-1},z}(\vec{u},\vec{a})$
and
$P_{\ell-j;t_{i-1},z;t_{i},y}(\vec{v},\vec{b})=P^{t_{i-1},z}_{\ell-j;t_{i},y}(\vec{v},\vec{b})$.
Using of the fact that the white noise integration can be split since the
times range over disjoint intervals, we find

$\mathcal{Z}^{\mathbf{nw}}_{0,0}(t_{i},y_{i})=\sum_{\ell=0}^{\infty}\sum_{j=0}^{\ell}\int_{\Delta_{j}(0,t_{i-1})}\int_{\Delta_{\ell-j}(t_{i-1},t_{i})}\int_{\mathbb{R}^{i}}\int_{\mathbb{R}^{\ell-j}}\int_{\mathbb{R}}P_{j;t_{i-1},z}(\vec{u},\vec{a})P^{t_{i-1},z}_{\ell-j;t_{i},y}(\vec{v},\vec{b})dzd\xi^{\otimes_{j}}(\vec{u},\vec{a})d\xi^{\otimes_{\ell-j}}(\vec{v},\vec{b}).$

By the change of variables $m=\ell-j$, the double sum
$\sum_{\ell=0}^{\infty}\sum_{j=0}^{\ell}$ can be replaced by
$\sum_{j=0}^{\infty}\sum_{m=0}^{\infty}$. We bring the integral in $z$ to the
outside resumming and reordering of integrals is readily justified since all
sums are convergent in $L^{2}$ (with respect to the probability space on which
$\xi$ is defined – see, for example, [Cor18, Theorem 2.2] for details). As a
result, we get

$\displaystyle\mathcal{Z}^{\mathbf{nw}}_{0,0}(t_{i},y_{i})=\int_{\mathbb{R}}dz$
$\displaystyle\Big{(}\sum_{j=0}^{\infty}\int_{\Delta_{j}(0,t_{i-1})}P_{j;t_{i-1},z}(\vec{u},\vec{a})d\xi^{\otimes_{j}}(\vec{u},\vec{a})\Big{)}$
$\displaystyle\times\Big{(}\sum_{m=0}^{\infty}\int_{\Delta_{\ell-j}(t_{i-1},t_{i})}P^{t_{i-1},z}_{\ell-j;t_{i},y}(\vec{v},\vec{b})d\xi^{\otimes_{\ell-j}}(\vec{v},\vec{b})\Big{)}.$

Comparing with (2.7), we may now recognize that

$\displaystyle\mathcal{Z}^{\mathbf{nw}}_{0,0}(t_{i-1},z)=\sum_{j=0}^{\infty}\int_{\Delta_{j}(0,t_{i-1})}P_{j;t_{i-1},z}(\vec{u},\vec{a})d\xi^{\otimes_{j}}(\vec{u},\vec{a}),$

for any $z\in\mathbb{R}$ whereas the stochastic process

$\mathcal{Z}_{y_{i}}(t_{i}\downarrow
t_{i-1},z):=\sum_{m=0}^{\infty}\int_{\Delta_{\ell-j}(t_{i-1},t_{i})}P^{t_{i-1},z}_{\ell-j;t_{i},y_{i}}(\vec{v},\vec{b})d\xi^{\otimes_{\ell-j}}(\vec{v},\vec{b})$

is same in distribution with
$\mathcal{Z}^{\mathbf{nw}}_{t_{i-1},z}(t_{i},y_{i})$. Furthermore,
$\mathcal{Z}^{\mathbf{nw}}_{0,0}(t_{i-1},\cdot)$ and
$\mathcal{Z}_{y_{i}}(t_{i}\downarrow t_{i-1},\cdot)$ are independent since
they are defined with respect to disjoint portions of the space-time white
noise. Due to the same reason, $\mathcal{Z}_{y_{i}}(t_{i}\downarrow
t_{i-1},\cdot)$ and $\mathcal{Z}_{y_{j}}(t_{i}\downarrow t_{j-1},\cdot)$ for
any $1\leq i<j\leq k$. Recall the interchange property of the SHE: namely
that, for $s<t$ and $y\in\mathbb{R}$ fixed,
$\mathcal{Z}^{\mathbf{nw}}_{s,x}(t,y)$ is equal in law as a process in $x$ to
$\mathcal{Z}^{\mathbf{nw}}_{s,y}(t,x)$ – the change between the two
expressions is in the interchange of $x$ and $y$. By the interchange property,
the spatial process $\mathcal{Z}^{\mathbf{nw}}_{t_{i-1},\cdot}(t_{i},y_{i})$
has same law as $\mathcal{Z}^{\mathbf{nw}}_{0,0}(t_{i}-t_{i-1},y_{i}-\cdot)$.
This completes the proof of (2.6). ∎

Our next proposition which is taken from [CGH19, Proposition 2.7] states a FKG
type inequality for the KPZ equation.

######  Proposition 2.13.

For any $t_{1},t_{2}>\mathbb{R}_{>0}$ and $s_{1},s_{2}\in\mathbb{R}$, we have

$\displaystyle\mathbb{P}(\mathfrak{h}_{t_{1}}\geq
s_{1},\mathfrak{h}_{t_{2}}\geq
s_{2}\big{)}\geq\mathbb{P}(\mathfrak{h}_{t_{1}}\geq
s_{1})\mathbb{P}(\mathfrak{h}_{t_{2}}\geq s_{2}).$

In the following two results, we will see the one point tail probabilities of
the temporal process $\mathfrak{h}_{t}$ which are proved in [CG20b, CG20a]. We
state the results from [CGH19] which has used same notations as ours. These
results hold for any finite time $t>0$. Since the short time scaling of the
KPZ equation has the Gaussian limit, the same tail bounds as in the
forthcoming result does not hold as $t$ goes to $0$. The short time tail
bounds which are tackled in Theorem 1.6 and 1.7 should be contrasted with the
following two propositions.

######  Proposition 2.14 (Proposition 2.12 from [CGH19]).

For any $t_{0}>0$, there exist $s_{0}=s_{0}(t_{0})>0$ and
$c_{1}(t_{0})>c_{2}(t_{0})>0$ such that, for $t\geq t_{0}$, $s>s_{0}$ and
$x\in\mathbb{R}$,

$\displaystyle\exp\big{(}-c_{1}s^{3/2}\big{)}\leq\mathbb{P}\Big{(}\mathfrak{h}_{t}(x)+\frac{x^{2}}{2}\geq
s\Big{)}\leq\exp\big{(}-c_{2}s^{3/2}\big{)}\,.$ (2.10)

######  Proposition 2.15 (Proposition 2.11 from [CGH19]).

For any $t_{0}>0,\varepsilon>0$, there exist $s_{0}=s_{0}(t_{0})>0$ and
$c=c(t_{0})>0$ such that, for $t>t_{0}$, $s>s_{0}$ and $x\in\mathbb{R}$,

$\displaystyle\mathbb{P}\Big{(}\mathfrak{h}_{t}(x)+\frac{x^{2}}{2}\leq-s\Big{)}$
$\displaystyle\leq\exp\big{(}-cs^{5/2}\big{)}\,.$ (2.11)

As one may notice, the constants of the tail bound in the above two
propositions are left imprecise. For deriving tail bounds of Section 4 and 5,
we do not need the exact values of those constants. However, in Section 6 and
7, we require precise description of those constants only in the case when the
time variable $t$ is large. The following proposition quotes relevant tail
bounds from [CG20b, CG20a, CC19, Zho19] for large values of $t$.

######  Proposition 2.16.

Fix $t_{0}>0$ large and $\varepsilon\in(0,1)$. Then, there exist
$s_{0}=s_{0}(t_{0},\varepsilon)>0$ and $c=c(t_{0},\varepsilon)>0$ such that,
for $t>t_{0}$, $c(\log t)^{2/3}>s>s_{0}$ and $x\in\mathbb{R}$,

$\displaystyle\exp\big{(}-\frac{4\sqrt{2}}{3}(1+\varepsilon)s^{3/2}\big{)}\leq\mathbb{P}\Big{(}\mathfrak{h}_{t}(x)+\frac{x^{2}}{2}\geq
s\Big{)}\leq\exp\big{(}-\frac{4\sqrt{2}}{3}(1-\varepsilon)s^{3/2}\big{)}$
(2.12)

and,

$\displaystyle\exp\big{(}-\frac{1}{6}(1+\varepsilon)s^{3}\big{)}\leq\mathbb{P}\Big{(}\mathfrak{h}_{t}(x)+\frac{x^{2}}{2}\leq-s\Big{)}\leq\exp\big{(}-\frac{1}{6}(1-\varepsilon)s^{3}\big{)}.$
(2.13)

###### Proof.

Since $\mathfrak{h}_{t}(x)+\frac{x^{2}}{2}$ is stationary in $x$, it suffices
to prove (2.12) and (2.13) for $x=0$. From the specifications of the upper and
lower bounds of the upper tail probabilities in Theorem 1.10 (part (a)) of
[CG20a], (2.12) follows immediately. It remains to show (2.13). Theorem 1.1 of
[CG20b] which is recently been strengthened in [CC19, Zho19] proves that for
any given $\varepsilon,t_{0}>0$, there exists $s_{0}=(t_{0},\varepsilon)>0$
such that for all $s\geq s_{0}$ and $t\geq t_{0}$,

$\displaystyle\mathbb{P}\big{(}\mathfrak{h}_{t}(0)\leq-s\big{)}\leq\exp\Big{(}-\frac{4\sqrt{2}(1-2\varepsilon)}{15}t^{\frac{1}{3}}s^{\frac{5}{2}}\Big{)}+\exp(-Ks^{3}-\varepsilon
st^{\frac{1}{3}})+\exp\Big{(}-\frac{(1-2\varepsilon)}{6}s^{3}\Big{)}$ (2.14)

and,

$\displaystyle\mathbb{P}\big{(}\mathfrak{h}_{t}(0)\leq-s\big{)}\geq\exp\big{(}-\frac{4\sqrt{2}}{15}(1+\varepsilon)t^{\frac{1}{3}}s^{\frac{5}{2}}\big{)}+\exp\big{(}-\frac{1}{6}(1+\varepsilon)s^{3}\big{)}.$
(2.15)

The first inequality of (2.13) follows from (2.15). Note that
$s^{5/2}t^{1/3}\gg s^{3}$ and $\varepsilon st^{1/3}\gg s^{3}$ when we have
$(\log t)^{2/3}\gg s$. By choosing $s_{0}$ and $c$ large, we may bound the
right hand side of (2.14) by $\exp(-(1-\varepsilon)s^{3}/6)$ for all $s\geq
s_{0}$ satisfying $c(\log t)^{2/3}>s$. This proves (2.13). ∎

The next two results which are proved in [CGH19] provide tail bounds on the
supremum and infimum of the spatial process $\mathfrak{h}_{t}(\cdot)$ for any
fixed time $t>0$.

######  Proposition 2.17 (Proposition 4.1 from [CGH19]).

For any $t_{0}>0$ and $\nu\in(0,1]$, there exist $s_{0}=s_{0}(t_{0},\nu)>0$
and $c=c(t_{0},\nu)>0$ such that, for $t\geq t_{0}$ and $s>s_{0}$,

$\displaystyle\mathbb{P}(\mathsf{B})\leq\exp\big{(}-cs^{5/2}\big{)}\qquad\textrm{where}\quad\mathsf{B}=\Big{\\{}\inf_{x\in\mathbb{R}}\Big{(}\mathfrak{h}_{t}(x)+\frac{(1-\nu)x^{2}}{2}\Big{)}\leq-s\Big{\\}}.$

######  Proposition 2.18 (Proposition 4.2 from [CGH19]).

For any $t_{0}>0$ and $\nu\in(0,1]$, there exist $s_{0}=s_{0}(t_{0},\nu)>0$
and $c_{1}=c_{1}(t_{0},\nu)>c_{2}=c_{2}(t_{0},\nu)>0$ such that, for $t\geq
t_{0}$ and $s>s_{0}$,

$\displaystyle\exp\big{(}-c_{1}s^{3/2}\big{)}\leq\mathbb{P}(\mathsf{A})\leq\exp\big{(}-c_{2}s^{3/2}\big{)}\quad\textrm{where}\quad\mathsf{A}=\Big{\\{}\sup_{x\in\mathbb{R}}\Big{(}\mathfrak{h}_{t}(x)+\frac{(1-\nu)x^{2}}{2}\Big{)}\geq
s\Big{\\}}.$

We end this section with the Paley-Zygmund inequality which is applicable for
any positive random variable.

######  Proposition 2.19 (Paley-Zygmund Inequality).

Fix $\delta\in(0,1)$. For any positive random variable $X$,

$\displaystyle\mathbb{P}\big{(}X\geq\delta\mathbb{E}[X]\big{)}\geq\frac{(1-\delta)^{2}(\mathbb{E}[X])^{2}}{\mathbb{E}[X^{2}]}.$

##  3\. Short Time Tail Bounds

The main goal of this section is to prove Theorem 1.4 and 1.7 which describe
uniform bounds to the one point tail probabilities of the KPZ height function
as time variable $t$ goes to $0$. The proof of Theorem 1.4 which is given in
Section 3.1 will use the exact formulas of the integer moments of the SHE.
These formulas are put forward by Kardar [Kar87] using the techniques of
replica Bethe ansatz. See [Gho18] for a discussion on different approaches to
prove those formulas rigorously. On the other hand, the proof of Theorem 1.7
which is contained in Section 3.2 will be based on core probabilistic aspect
like Gaussian concentration.

###  3.1. Upper Tail

Our starting point which is the content of the following proposition is to
provide upper bounds to the exponential moments of $\mathfrak{g}_{t}$. Using
these moment estimates, the proof of Theorem 1.4 will be completed in the
ensuing subsection.

######  Proposition 3.1.

Fix $\varepsilon>0$. There exist $t_{0}=t_{0}(\varepsilon)>0$,
$C=C(\varepsilon)>0$, and $s_{0}=s_{0}(\varepsilon)>0$, such that for all
$t\leq t_{0}$, $s\geq s_{0}$ and $k:=\lfloor s(\pi t/4)^{-1/4}\rfloor$ we have

$\displaystyle\mathbb{E}\big{[}\exp\big{(}k(\pi
t/4)^{1/4}\mathfrak{g}_{2t}\big{)}\big{]}$
$\displaystyle\leq\exp\big{(}C(s^{3}t^{1/4-4\varepsilon}+s^{2})\big{)}.$ (3.1)

###### Proof.

For any positive integer $k$, we recall the $k$-moment formula for
$\mathcal{Z}^{\mathbf{nw}}(2t,0)$ (see [BC14, Gho18])

$\displaystyle\mathbb{E}\left[\mathcal{Z}^{\mathbf{nw}}(2t,0)^{k}e^{\frac{kt}{12}}\right]$
$\displaystyle=\sum_{\begin{subarray}{c}\lambda\vdash k\\\
\lambda=1^{m_{1}}2^{m_{2}}\ldots\end{subarray}}\frac{k!}{\prod
m_{j}!}\prod_{i=1}^{\ell(\lambda)}\frac{e^{\frac{t\lambda_{i}^{3}}{12}}}{2\pi}\int\limits_{\mathbb{R}}\cdots\int\limits_{\mathbb{R}}\prod_{i=1}^{\ell(\lambda)}\frac{dz_{i}e^{-t^{\frac{1}{3}}\lambda_{i}z_{i}^{2}}}{t^{\frac{1}{3}}\lambda_{i}}\prod_{i<j}^{\ell(\lambda)}\frac{\frac{t^{\frac{2}{3}}(\lambda_{i}-\lambda_{j})^{2}}{4}+(z_{i}-z_{j})^{2}}{\frac{t^{\frac{2}{3}}(\lambda_{i}+\lambda_{j})^{2}}{4}+(z_{i}-z_{j})^{2}}.$

Note that each terms of the product inside the integral is less than $1$.
Bounding those terms by $1$ and evaluating the left over Gaussian integral, we
have

$\displaystyle\mathbb{E}\left[\mathcal{Z}^{\mathbf{nw}}(2t,0)^{k}e^{\frac{kt}{12}}\right]$
$\displaystyle\leq\sum_{\begin{subarray}{c}\lambda\vdash k\\\
\lambda=1^{m_{1}}2^{m_{2}}\ldots\end{subarray}}\frac{k!}{\prod
m_{j}!}\prod_{i=1}^{\ell(\lambda)}\frac{e^{\frac{t\lambda_{i}^{3}}{12}}}{2\pi}\prod_{i=1}^{\ell(\lambda)}\frac{\sqrt{\pi}}{t^{\frac{1}{2}}\lambda_{i}^{3/2}}\leq\sum_{\begin{subarray}{c}\lambda\vdash
k\\\
\lambda=1^{m_{1}}2^{m_{2}}\ldots\end{subarray}}\frac{k!e^{\frac{tk^{3}}{12}}}{(4\pi
t)^{\frac{\ell(\lambda)}{2}}\prod m_{j}!}$

The last inequality in above equation follows by using $\lambda_{i}^{3/2}\geq
1$ and $\sum_{i}\lambda_{i}^{3}\leq k^{3}$. Expressing the left hand side of
the above display in terms of $\mathfrak{g}_{2t}$ we get

$\displaystyle\mathbb{E}e^{k(\pi
t/2)^{1/4}\mathfrak{g}_{2t}}=\mathbb{E}\left[(\mathcal{Z}^{\mathbf{nw}}(2t,0)\sqrt{4\pi
t})^{k}\right]$ $\displaystyle\leq
e^{\frac{tk^{3}-tk}{12}}\sum_{\begin{subarray}{c}\lambda\vdash k\\\
\lambda=1^{m_{1}}2^{m_{2}}\ldots\end{subarray}}(4\pi
t)^{\frac{k-\ell(\lambda)}{2}}\frac{k!}{\prod m_{j}!}$ (3.2)

We choose $t_{0}$ and $s_{0}$ such that
$2^{5/2}t_{0}^{\varepsilon}(\pi/2)^{1/4}\leq\frac{1}{2}$ and $s_{0}\geq 2(\pi
t_{0}/2)^{1/4}$. Then for all $t\leq t_{0}$ and $s\geq s_{0}$, we set
$k=k(t):=\lfloor s(\pi t/2)^{-1/4}\rfloor$. By the condition on $t_{0},s_{0}$
and $k$, we always have $k\geq 2$. We further have $k\leq s(\pi t/2)^{-1/4}$
which implies $t\leq\frac{2s^{4}}{\pi k^{4}}$. Bounding $t$ with this
inequality, combining it with the estimate $k!\leq k^{k-m_{1}}m_{1}!$ and
using those in the right hand side of (3.2) yields

$\displaystyle\mathbb{E}e^{k(\pi t/2)^{1/4}\mathfrak{g}_{2t}}\leq
e^{\frac{s^{4}(k^{-1}-k^{-3})}{6\pi}}\sum_{\begin{subarray}{c}\lambda\vdash
k\\\
\lambda=1^{m_{1}}2^{m_{2}}\ldots\end{subarray}}\left(2^{3/4}s\right)^{2k-2\ell(\lambda)}\frac{k^{k-m_{1}}k^{2\ell(\lambda)-2k}}{\prod\limits_{j\geq
2}m_{j}!}$ (3.3)

Throughout the rest, we provide bound for the right hand side of (3.3). We
separate our analysis into three cases depending on the location of $s$.

Case-1. $s\leq t^{-1/4+\varepsilon}$. Observe that
$k-\ell(\lambda)=\sum_{j\geq 2}(j-1)m_{j}$ and
$2\ell(\lambda)-m_{1}-k=-\sum_{j\geq 3}(j-2)m_{j}$.

We extend the range of $m_{2},m_{3},m_{4},\ldots$ over all non-negative
integers in (3.3). Taking first the sum w.r.t. $m_{2}$ shows

$\displaystyle\mbox{r.h.s.~{}of }\eqref{e:mom-bound1}$ $\displaystyle\leq
e^{\frac{s^{4}(k^{-1}-k^{-3})}{6\pi}}\sum_{j=3}^{\infty}\sum_{m_{j}=0}^{\infty}\left(2^{3/2}s^{2}\right)^{\sum\limits_{j\geq
3}(j-1)m_{j}}\frac{k^{-\sum\limits_{j\geq 3}(j-2)m_{j}}}{\prod\limits_{j\geq
3}m_{j}!}\sum_{m_{2}=0}^{\infty}\frac{(2^{3/2}s^{2})^{m_{2}}}{m_{2}!}$ (3.4)

Note that the inner sum w.r.t. $m_{2}$ is equal to $\exp(2^{3/2}s^{2})$. We
may now write the right hand side of the above display as

$\displaystyle
e^{\frac{s^{4}(k^{-1}-k^{-3})}{6\pi}}e^{2^{3/2}s^{2}}\prod_{j=3}^{\infty}\sum_{m_{j}=0}^{\infty}\left(2^{3/2}s^{2}\right)^{(j-1)m_{j}}\frac{k^{-(j-2)m_{j}}}{m_{j}!}$
$\displaystyle=\exp\left(\frac{s^{4}(k^{-1}-k^{-3})}{6\pi}+2^{3/2}s^{2}+\frac{2^{3}s^{4}k^{-1}}{1-2^{3/2}s^{2}k^{-1}}\right)$

where the equality is obtained by taking sum w.r.t. $m_{3},m_{4},...$
separately and simplifying the product. With this equality, we get

$\displaystyle\mbox{r.h.s.~{}of }\eqref{e:mom-bound3}$
$\displaystyle\leq\exp\left(\frac{2s^{3}(\pi
t/2)^{1/4}}{6\pi}+2^{3/2}s^{2}+\frac{2^{4}s^{3}(\pi t/2)^{1/4}}{1-2^{5/2}s(\pi
t/2)^{1/4}}\right)\leq\exp\big{(}Cs^{3}t^{1/4}+Cs^{2}\big{)}$

where the last inequality is obtained by using the facts $k^{-1}\leq
2s^{-1}(\pi t/2)^{1/4}$, $s\leq t^{-1/4+\varepsilon}$ and $t\leq t_{0}$ with
$2^{5/2}t_{0}^{\varepsilon}(\pi/2)^{1/4}\leq\frac{1}{2}$. This proves (3.1)
for $s\leq t^{-1/4+\varepsilon}$.

Case-2. $s\geq t^{-1/4-\varepsilon}$. We assume $t_{0}\leq\frac{1}{4\pi}$.
Recall the definition of $k$. Since $k+1\geq s/(\pi t/2)^{1/4}$, we may bound
$s^{2k-2\ell(\lambda)}$ by $(\pi
t/2)^{(k-\ell(\lambda))/2}(k+1)^{2k-2\ell(\lambda)}$. Combining this with the
facts $k!\leq k^{k}$ and $\prod m_{j}!\geq 1$, we get

$\displaystyle\mbox{r.h.s.~{}of }\eqref{e:mom-bound4}\leq
e^{\frac{s^{4}(k^{-1}-k^{-3})}{6\pi}}\sum_{\begin{subarray}{c}\lambda\vdash
k\\\ \lambda=1^{m_{1}}2^{m_{2}}\ldots\end{subarray}}\left(4\pi
t\right)^{\frac{k-\ell(\lambda)}{2}}k^{k-m_{1}}\cdot
k!\big{(}1+\frac{1}{k}\big{)}^{2(k-\ell(\lambda))}$ $\displaystyle\leq
e^{\frac{s^{4}(k^{-1}-k^{-3})}{6\pi}}k^{2k}$

where we bound $(1+1/k)^{2(k-\ell(\lambda))}$ by $1$ and the number of
partitions of $k$ by $k^{k}$ to get the last inequality. Since we are in the
case $s\geq t^{-1/4-\varepsilon}$, we have $s^{4}k^{-1}\leq s^{3}(\pi
t/2)^{1/4}$ and $k\ln k\leq cst^{-1/4}\ln(st^{-1/4})\leq cs^{3}t^{1/4}$. Due
to these inequalities, the right hand side of the above display is bounded by
$\exp(cs^{3}t^{1/4})$ for some constant $c>0$. Combining this with (3.3) shows
$\mathbb{E}\exp(k(\pi t/2)^{1/4}\mathfrak{g}_{2t})\leq\exp(cs^{3}t^{1/4}).$

Case-3. $t^{-1/4+\varepsilon}\leq s\leq t^{-1/4-\varepsilon}$. Define
$\tilde{s}=t^{-1/4-\varepsilon}$ and $\tilde{k}:=\lfloor\tilde{s}(\pi
t/2)^{-1/4}\rfloor$. Note that $k\leq\tilde{k}$ since $s\leq
t^{-1/4-\varepsilon}$. Using the Hölder’s inequality, we know
$\mathbb{E}\exp(k(\pi t/2)^{1/4}\mathfrak{g}_{2t})$ is bounded by
$\big{(}\mathbb{E}\exp(\tilde{k}(\pi
t/2)^{1/4}\mathfrak{g}_{2t})\big{)}^{k/\tilde{k}}$. By Case-2, we know
$\mathbb{E}\exp(\tilde{k}(\pi
t/2)^{1/4}\mathfrak{g}_{2t})\leq\exp(c\tilde{s}^{3}t^{1/4})$ for all $t\leq
t_{0}=\frac{1}{4\pi}$. Combining these observations shows

$\displaystyle\mathbb{E}\exp(k(\pi
t/2)^{1/4}\mathfrak{g}_{2t})\leq\exp(ck\tilde{s}^{3}t^{1/4}/\tilde{k})\leq\exp(ct^{-3/4-3\varepsilon}t^{1/4}st^{1/4+\varepsilon})=\exp(cst^{-1/4-2\varepsilon})$

where the second inequality follows from the definition of $\tilde{k}$ and
$\tilde{s}$. Since $s\geq t^{-1/4+\varepsilon}$, the last term of the above
display is bounded by $\exp(cs^{3}t^{1/4-4\varepsilon})$. This completes the
proof for Case-3.

Combining all cases we get (3.1). This completes the proof. ∎

####  3.1.1. Proof of Theorem 1.4

We introduce the notations
$f_{t,s}:=\frac{1}{C+\sqrt{C^{2}+3Cst^{1/4-4\varepsilon}}}$,
$\tilde{s}:=sf_{t,s}$ and $\tilde{k}:=\lfloor\tilde{s}(\pi t/2)^{-1/4}\rfloor$
where the constant $C$ is same as in (3.1). By Markov’s inequality,

$\displaystyle\mathbb{P}(\mathfrak{g}_{2t}\geq s)=\mathbb{P}(e^{\tilde{k}(\pi
t/2)^{1/4}\mathfrak{g}_{2t}}\geq e^{\tilde{k}{s}(\pi t/2)^{1/4}})$
$\displaystyle\leq\exp(-\tilde{k}{s}(\pi
t/2)^{1/4})\mathbb{E}\big{[}\exp(\tilde{k}(\pi
t/2)^{1/4}\mathfrak{g}_{2t})\big{]}$ $\displaystyle\leq$
$\displaystyle\exp\big{(}Cs^{3}f_{t,s}^{3}t^{1/4-4\varepsilon}+Cs^{2}f_{t,s}^{2}-\tilde{k}{s}(\pi
t/2)^{1/4}\big{)}$ (3.5)

where the last inequality follows from Proposition 3.1. We choose $s_{0}$
large enough such that for all $s\geq s_{0}$ and $t\leq t_{0}$ we have
$\tilde{k}{s}(\pi t/2)^{1/4}\geq\frac{11}{12}s^{2}f_{t,s}$. From the
definition of $f_{t,s}$, it follows

$Cf_{t,s}\leq C\frac{1}{2C}=\frac{1}{2},\qquad
Cst^{1/4-4\varepsilon}f_{t,s}^{2}\leq
Cst^{1/4-4\varepsilon}\frac{1}{3Cst^{1/4-4\varepsilon}}=\frac{1}{3}.$

Plugging all these inequalities in the right side of (3.5) yields

$\displaystyle\mathbb{P}(\mathfrak{g}_{2t}\geq
s)\leq\exp\big{(}-\frac{s^{2}f_{t,s}}{12}\big{)}\leq\exp\Big{(}-\frac{C^{\prime}s^{2}}{1+\sqrt{1+st^{1/4-4\varepsilon}}}\Big{)}.$

for all $t\leq t_{0}$, $s\geq s_{0}$ and some constant $C^{\prime}>0$. This
completes the proof.

###  3.2. Lower Tail

Our proof of Theorem 1.7 will utilize ideas from [Flo14]. In [Flo14], the
author provided an upper bound to the lower tail probability of
$\mathcal{H}^{\mathbf{nw}}$. However, it was not clear whether the same bound
holds for $\mathfrak{g}_{t}$, i.e., centering $\mathcal{H}^{\mathbf{nw}}$ with
$\log\sqrt{2\pi t}$ and scaling by $(\pi t/4)^{1/4}$. Our analysis will
demonstrate that it is indeed possible to derive similar tail bound for
$\mathfrak{g}_{t}$.

The main tool of our proof of Theorem 1.7 are some properties of the directed
random polymer partition functions and its convergence to the solution of the
SHE. Below, we introduce relevant notations.

Let $\Xi:=\\{\mathcal{E}(i,x):i\in\mathbb{N},x\in\mathbb{Z}\\}$ be a
collection of independent standard normal random variables. We call such
collections as _lattice environment_. Let $\\{S_{i}\\}_{i\geq 0}$ be a simple
symmetric random walk on $\mathbb{Z}$ starting at $S_{0}=0$ independent of
$\Xi$. Denote the law of $\\{S_{i}\\}_{i\geq 0}$ by $\mathbb{P}_{S}$. At
inverse temperature $\beta>0$, the directed polymer partition function
$Z_{n}^{(\Xi)}(\beta)$ is defined as

$\displaystyle
Z^{(\Xi)}_{n}(\beta)=\mathbb{E}_{S}\left[\exp\left\\{{\beta\sum_{i=1}^{n}\mathcal{E}(i,S_{i})}\right\\}\mathbf{1}_{S_{n}=0}\right].$

where the expectation $\mathbb{E}_{S}$ is taken w.r.t. $\mathbb{P}_{S}$. From
[AKQ14], we know as $n\to\infty$

$\displaystyle\frac{1}{(\pi
t/2)^{1/4}}\left[\log\sqrt{\frac{n\pi}{2}}+\log\frac{Z_{n}^{(\Xi)}((t/2n)^{1/4})}{\mathbb{E}Z_{n}^{(\Xi)}((t/2n)^{1/4})}\right]\Rightarrow\mathfrak{g}_{t},\quad\text{for
each }t>0$ (3.6)

Here, ‘$\Rightarrow$’ denotes the weak convergence. To complete the proof of
Theorem 1.7, we need the following two lemmas. Lemma 3.2 is originally from a
part of the proof of Theorem 1.5 in [CH02].

######  Lemma 3.2 (Lemma 1 of [Flo14]).

Let $\Xi$ and $\Xi^{\prime}$ be two independent lattice environments. Let
$S^{(1)}$ and $S^{(2)}$ be two independent simple symmetric random walks
starting at origin. Denote the expectation w.r.t. the joint law of $S^{(1)}$
and $S^{(2)}$ by $\mathbb{E}_{S^{(1)},S^{(2)}}$.Then, we have

$\displaystyle\log Z_{n}^{(\Xi)}(\beta)\geq\log
Z_{n}^{(\Xi^{\prime})}(\beta)-\beta
d_{n}(\Xi,\Xi^{\prime})\sqrt{\mathfrak{Overlap}_{\Xi^{\prime}}(S^{(1)},S^{(2)})}$

where $d_{n}(\Xi,\Xi^{\prime})^{2}:=\sum_{i=1}^{n}\sum_{|x|\leq
i}(\mathcal{E}(i,x)-\mathcal{E}^{\prime}(i,x))^{2}$ and,

$\displaystyle\mathfrak{Overlap}_{\Xi^{\prime}}(S^{(1)},S^{(2)}):=\frac{1}{Z_{n}^{(\Xi^{\prime})}(\beta)^{2}}\mathbb{E}_{S^{(1)},S^{(2)}}\left[\sum_{i=1}^{n}\mathbf{1}_{S_{i}^{(1)}=S_{i}^{(2)}}e^{\beta\sum_{i=1}^{n}(\mathcal{E}^{\prime}(i,S_{i}^{(1)})+\mathcal{E}^{\prime}(i,S_{i}^{(2)}))}\mathbf{1}_{S_{n}^{(1)}=S_{n}^{(2)}=0}\right].$

The next lemma is similar to Lemma 2 of [Flo14]. To state the lemma, we
introduce for any $n\in\mathbb{N}$, $t>0$ and $C>0$

$\displaystyle
A_{n,t,C}:=\left\\{\Xi^{\prime}:Z_{n}^{(\Xi^{\prime})}((t/2n)^{1/4})\geq\sqrt{\frac{2}{n\pi}}\mathbb{E}Z_{n}^{(\Xi)}((t/2n)^{1/4}),\mathfrak{Overlap}_{\Xi^{\prime}}(S^{(1)},S^{(2)})\leq
C\sqrt{n}\right\\}.$

######  Lemma 3.3.

For any given $\varepsilon>0$, there exist constants
$t_{0}=t_{0}(\varepsilon)\in(0,2]$ and $C=C(\varepsilon)>0$ satisfying the
following: for any $t\leq t_{0}$, there exists $n_{t}\in\mathbb{N}$ such that
for all $n\geq n_{t}$, we have
$\mathbb{P}(A_{n,t,C})\geq\frac{1}{2}-\varepsilon$.

Our proof of the above lemma uses some of the ideas from the proof of Lemma 2
of [Flo14]. However, there is a major difference between these two results.
Unlike Lemma 2 of [Flo14], Lemma 3.3 provides a lower bound to
$\mathbb{P}(A_{n,t,C})$ which does not depend on $t$. On the other hand, the
lower bound of Lemma 2 of [Flo14] is valid for all $n\geq 1$ which is not the
case in Lemma 3.3. Since we are interested in the evolution of tail
probabilities of $Z_{n}^{(\Xi)}((t/2n)^{1/4})$ as $n$ grows large, the
probability bound of $A_{n,t,C}$ for large $n$ is more relevant to our
analysis than a uniform bound for all $n\geq 1$. Furthermore, the independence
of the lower bound of $\mathbb{P}(A_{n,t,C})$ from $t$ enables us in Theorem
1.7 to derive bounds on the lower tail probability of $\mathfrak{g}_{t}$
uniform in $t$. Before proceeding to the proof of Lemma 3.3, we will show
Theorem 1.7 by assuming Lemma 3.3.

####  3.2.1. Proof of Theorem 1.7

Fix $\epsilon\in(0,\frac{1}{2})$. We choose $t_{0}=t_{0}(\varepsilon)\in(0,2]$
as defined in Lemma 3.3. Fix $t\leq t_{0}$. From Lemma 3.3 we pick $C>0$ and
$n_{t}\in\mathbb{N}$ such that for all $n\geq n_{t}$,
$\mathbb{P}(A_{n,t,C})\geq\frac{1}{4}$. Fix $n\geq n_{t}$. Consider any
$\Xi^{\prime}\in A_{n,t,C}$. By Lemma 3.2, we have

$\displaystyle\log Z_{n}^{(\Xi)}((t/2n)^{1/4})$ $\displaystyle\geq\log
Z_{n}^{(\Xi^{\prime})}((t/2n)^{1/4})-(t/2n)^{1/4}d_{n}(\Xi,\Xi^{\prime})\sqrt{\mathfrak{Overlap}_{\Xi^{\prime}}(S^{(1)},S^{(2)})}$
$\displaystyle\geq\log\sqrt{\frac{2}{n\pi}}+\log\mathbb{E}Z_{n}^{(\Xi)}((t/2n)^{1/4})-(t/2n)^{1/4}d_{n}(\Xi,\Xi^{\prime})\sqrt{C\sqrt{n}}.$

where the second inequality follows since $\Xi^{\prime}\in A_{n,t,C}$.
Rearranging the above inequality and using the fact that it holds for any
$\Xi^{\prime}\in A_{n,t,C}$ shows

$\displaystyle\frac{1}{(\pi
t/2)^{1/4}}\left[\log\sqrt{\frac{n\pi}{2}}+\log\frac{Z_{n}^{(\Xi)}((t/2n)^{1/4})}{\mathbb{E}Z_{n}^{(\Xi)}((t/2n)^{1/4})}\right]$
$\displaystyle\geq-C^{1/2}\pi^{-1/4}\inf_{\Xi^{\prime}\in
A_{n,t,C}}d_{n}(\Xi,\Xi^{\prime}).$

Thus, for all $s>0$,

$\displaystyle\mathbb{P}\left(\frac{1}{(\pi
t/2)^{1/4}}\left[\log\sqrt{\frac{n\pi}{2}}+\log\frac{Z_{n}^{(\Xi)}((t/2n)^{1/4})}{\mathbb{E}Z_{n}^{(\Xi)}((t/2n)^{1/4})}\right]\leq-s\right)\leq\mathbb{P}(d_{n}(\Xi,A_{n,t,C})\geq
s\pi^{1/4}C^{-1/2})$ (3.7)

where $d_{n}(\Xi,A_{n,t,C}):=\inf_{\Xi^{\prime}\in
A_{n,t,C}}d_{n}(\Xi,\Xi^{\prime})$. Since
$\mathbb{P}(A_{n,t,C})\geq\frac{1}{2}-\varepsilon$, applying Theorem 3 of
[Flo14] ( Talagrand’s inequality) shows $\mathbb{P}(d_{n}(\Xi,A_{n,t,C})\geq
u+\sqrt{4\log 2})\leq e^{-u^{2}/2}$. Applying this probability bound into the
right hand side of the above display yields

$\displaystyle\mbox{r.h.s.~{}of }\eqref{eq:dist}$
$\displaystyle\leq\exp\left({-\frac{1}{2}\left\\{s\pi^{-1/4}C^{-1/2}-\sqrt{4\log
2}\right\\}^{2}}\right)\leq e^{-cs^{2}}$ (3.8)

for some positive constant $c>0$ and for all $s\geq s_{0}$ where neither
$s_{0}$ nor $c$ does depend on $n$ or $t$. Due to the weak convergence of
(3.6), we have

$\displaystyle\mathbb{P}\left(\frac{1}{(\pi
t/2)^{1/4}}\left[\log\sqrt{\frac{n\pi}{2}}+\log\frac{Z_{n}^{(\Xi)}((t/2n)^{1/4})}{\mathbb{E}Z_{n}^{(\Xi)}((t/2n)^{1/4})}\right]\leq-s\right)\stackrel{{\scriptstyle
n\to\infty}}{{\to}}\mathbb{P}(\mathfrak{g}_{t}\leq-s)$

Combining this convergence with (3.7) and (3.8) shows the desired conclusion.

####  3.2.2. Proof of Lemma 3.3

Define $\beta_{n}=(t/2n)^{1/4}$. By Proposition of [ACQ11], $\mathfrak{g}_{t}$
converges weakly to the standard Gaussian distribution implying
$\lim\limits_{t\to 0}\mathbb{P}(\mathfrak{g}_{t}\geq 0)=\frac{1}{2}.$ We
choose the largest $t_{0}=t_{0}(\varepsilon)\in(0,2]$ such that
$\mathbb{P}(\mathfrak{g}_{t}\geq 0)\geq\frac{1}{2}-\frac{\varepsilon}{2}$ for
all $t\leq t_{0}$. For simplicity in notations, we set

$\displaystyle\mathfrak{L}_{n}:=\sum_{i=1}^{n}\mathbf{1}_{S_{i}^{(1)}=S_{i}^{(2)}}\cdot\mathbf{1}_{S_{n}^{(1)}=S_{n}^{(2)}=0}\cdot
e^{\beta_{n}\sum_{i=1}^{n}(\mathcal{E}^{\prime}(i,S_{i}^{(1)})+\mathcal{E}^{\prime}(i,S_{i}^{(2)}))},\quad
L_{n}=\sum_{i=1}^{n}\mathbf{1}_{S_{i}^{(1)}=S_{i}^{(2)}}.$

Recall that $\mathfrak{Overlap}_{\Xi^{\prime}}(S^{(1)},S^{(2)})$ is equal to
$\mathbb{E}_{S^{(1)},S^{(2)}}[\mathfrak{L}_{n}]/(Z_{n}^{(\Xi^{\prime})}(\beta))^{2}$.
By simple probability bounds, we get

$\displaystyle\mathbb{P}(A_{n,t,C})$
$\displaystyle\geq\mathbb{P}\left(Z_{n}^{(\Xi^{\prime})}(\beta_{n})\geq\sqrt{\frac{2}{n\pi}}\mathbb{E}Z_{n}^{(\Xi)}(\beta_{n}),\mathbb{E}_{S^{(1)}S^{(2)}}(\mathfrak{L}_{n})\leq\frac{2C}{\sqrt{n\pi^{2}}}(\mathbb{E}Z_{n}^{(\Xi)}(\beta_{n}))^{2}\right)$
$\displaystyle\geq\mathbb{P}\left(Z_{n}^{(\Xi^{\prime})}(\beta_{n})\geq\sqrt{\frac{2}{n\pi}}\mathbb{E}Z_{n}^{(\Xi)}(\beta_{n})\right)-\mathbb{P}\left(\frac{\mathbb{E}_{S^{(1)}S^{(2)}}(\mathfrak{L}_{n})}{(\mathbb{E}Z_{n}^{(\Xi)}(\beta_{n}))^{2}}>\frac{2C}{\sqrt{n\pi^{2}}}\right)$
(3.9)

We claim that for any $t\leq t_{0}$, there exists $n_{t}\in\mathbb{N}$ such
that for all $n\geq n_{t}$,

$\displaystyle\mathbb{P}\left(Z_{n}^{(\Xi^{\prime})}(\beta_{n})\geq\sqrt{\frac{2}{n\pi}}\mathbb{E}Z_{n}^{(\Xi)}(\beta_{n})\right)\geq\frac{1}{2}-\frac{3\varepsilon}{4},\quad\mathbb{P}\left(\frac{\mathbb{E}_{S^{(1)}S^{(2)}}(\mathfrak{L}_{n})}{(\mathbb{E}Z_{n}^{(\Xi)}(\beta_{n}))^{2}}>\frac{2C}{\sqrt{n\pi^{2}}}\right)\leq\frac{\varepsilon}{4}.$
(3.10)

Substituting the above inequalities into the right hand side of (3.9)
completes the proof of Lemma 3.3. Thus, it suffices to show that the above
inequalities hold for all large $n$. To see the first inequality of (3.10), we
first note that
$\mathbb{E}Z_{n}^{(\Xi)}(\beta_{n})=\mathbb{E}Z_{n}^{(\Xi^{\prime})}(\beta_{n})$
and write

$\displaystyle\mathbb{P}\left(Z_{n}^{(\Xi^{\prime})}(\beta_{n})\geq\sqrt{\frac{2}{n\pi}}\mathbb{E}Z_{n}^{(\Xi^{\prime})}(\beta_{n})\right)=\mathbb{P}\left(\frac{1}{(\pi
t/2)^{\frac{1}{4}}}\left[\log\sqrt{\frac{n\pi}{2}}+\log\frac{Z_{n}^{(\Xi^{\prime})}(\beta_{n})}{\mathbb{E}Z_{n}^{(\Xi^{\prime})}(\beta_{n})}\right]\geq
0\right).$

By the weak convergence in (3.6) and $\mathbb{P}(\mathfrak{g}_{t}\geq
0)\geq\frac{1}{2}-\frac{\varepsilon}{2}$, it follows that the right side of
the above display is greater than $\frac{1}{2}-\frac{3\epsilon}{4}$ for all
large $n$. This proves the first inequality of (3.10).

Now, we show the second inequality of (3.10). Note that
$\mathbb{E}Z_{n}^{(\Xi)}(\beta_{n})=e^{n\beta^{2}_{n}/2}$. By Fubini, we have

$\displaystyle\mathbb{E}_{\Xi}\mathbb{E}_{S^{(1)}S^{(2)}}[\mathfrak{L}_{n}]$
$\displaystyle=\mathbb{E}_{S^{(1)}S^{(2)}}\Big{[}\sum_{i=1}^{n}\mathbf{1}_{S_{i}^{(1)}=S_{i}^{(2)}}\cdot\mathbf{1}_{S_{n}^{(1)}=S_{n}^{(2)}=0}\cdot\prod_{j=1}^{n}\mathbb{E}_{\Xi}\Big{(}e^{\beta_{n}(\mathcal{E}(j,S_{j}^{(1)})+\mathcal{E}(j,S_{j}^{(2)}))}\Big{)}\Big{]}$
$\displaystyle=e^{n\beta^{2}_{n}}\mathbb{E}_{S^{(1)}S^{(2)}}\Big{[}\sum_{i=1}^{n}\mathbf{1}_{S_{i}^{(1)}=S_{i}^{(2)}}\cdot\mathbf{1}_{S_{n}^{(1)}=S_{n}^{(2)}=0}\cdot\exp\big{(}\beta^{2}_{n}\sum\limits_{i=1}^{n}\mathbf{1}_{S_{i}^{(1)}=S_{i}^{(2)}}\big{)}\Big{]}$
$\displaystyle=(\mathbb{E}Z_{n}^{(\Xi)}(\beta)_{n})^{2}\mathbb{E}_{S^{(1)}S^{(2)}}\big{[}L_{n}e^{\beta^{2}_{n}L_{n}}\mathbf{1}_{S_{n}^{(1)}=S_{n}^{(2)}=0}\big{]}$

Applying Markov’s inequality and using the above expression of
$\mathbb{E}_{\Xi}\mathbb{E}_{S^{(1)}S^{(2)}}[\mathfrak{L}_{n}]$ shows

$\displaystyle\mathbb{P}\left(\frac{\mathbb{E}_{S^{(1)}S^{(2)}}(\mathfrak{L}_{n})}{(\mathbb{E}Z_{n}^{(\Xi)}(\beta_{n}))^{2}}>\frac{2C}{\sqrt{n\pi^{2}}}\right)$
$\displaystyle\leq\frac{\sqrt{n\pi^{2}}}{2C}\mathbb{E}_{S^{(1)}S^{(2)}}\left[L_{n}e^{\beta^{2}_{n}L_{n}}\mathbf{1}_{S_{n}^{(1)}=S_{n}^{(2)}=0}\right]$
$\displaystyle\leq\frac{\sqrt{n\pi^{2}}}{2C}\mathbb{P}(S_{n}^{(1)}=S_{n}^{(2)}=0)\mathbb{E}_{S^{(1)}S^{(2)}}\left[L_{n}e^{\beta^{2}_{n}L_{n}}\mid
S_{n}^{(1)}=S_{n}^{(2)}=0\right]$

By Stirling’s approximation, there exists constant $a>0$ such that
$\mathbb{P}(S_{n}^{(1)}=S_{n}^{(2)}=0)=\frac{1}{2^{2n}}\binom{n}{n/2}^{2}\leq\frac{a}{n}$
for all $n$. Since $\beta_{n}=(t/2n)^{1/4}$, we have
$L_{n}e^{\beta^{2}_{n}L_{n}}=L_{n}e^{(t/2n)^{1/2}L_{n}}\leq
L_{n}e^{n^{-1/2}L_{n}}$ for all $t\leq t_{0}\leq 2$. Furthermore, Lemma 3 in
[Flo14] proves

$\displaystyle\sup_{N\geq
1}\frac{1}{\sqrt{n}}\mathbb{E}_{S^{(1)}S^{(2)}}\left[L_{n}e^{n^{-1/2}L_{n}}\mid
S_{n}^{(1)}=S_{n}^{(2)}=0\right]=K<\infty.$

Thus for all $t\leq t_{0}$ we have a constant $K^{\prime}>0$ (free of $t$) so
that

$\displaystyle\mathbb{P}\left(\frac{\mathbb{E}_{S^{(1)}S^{(2)}}(\mathfrak{L}_{n})}{(\mathbb{E}Z_{n}^{(\Xi)}(\beta_{n}))^{2}}>\frac{2C}{\sqrt{n\pi^{2}}}\right)\leq\frac{K^{\prime}}{C}$

Letting $C$ large shows the second inequality of (3.10) for all large $n$.
This completes the proof.

##  4\. Tail Bounds of the KPZ Spatial Process

In this section, we prove delicate tail bounds on several functionals of the
long and short time spatial processes $\mathfrak{h}_{t}(\cdot)$ and
$\mathfrak{g}_{t}(\cdot)$ respectively. Four propositions will be proved in
this section; two of them are about the supremum and the infimum of the
spatial process $\mathfrak{h}_{t}$ and other two are devoted on similar
results about $\mathfrak{g}_{t}$. One may notice similarities between
Proposition 4.1, 4.2 and Theorem 1.3 of [CGH19] since both bound the tail
probabilities of the supremum and/or infimum of the KPZ height differences
between spatial points. However, in comparison to [CGH19, Theorem 1.3], the
bounds on the tail probabilities in Proposition 4.1 and 4.2 improve on
multiple aspects (e.g., decay exponents) which turn out to be extremely useful
for proving the results of Section 5. The main ingredients of the proofs of
this sections are: $(1)$ tail bounds from Section 3 and $(2)$ Brownian Gibbs
property of the line ensemble discussed in Section 2. From this time forth, we
will denote complement of any set $\mathsf{B}$ by $\neg\mathsf{B}$.

######  Proposition 4.1.

Fix $\kappa>0$ and $\alpha\in[\frac{3}{2},2]$. There exist constant $c>0$,
$t_{0}>0$ such that for all $t\geq t_{0}$ and $\beta\in(0,1]$ and $s\geq
s_{0}(t_{0})$ we have

$\displaystyle\mathbb{P}\Big{(}\inf_{|y|\leq\beta^{2\kappa}s^{2-\alpha}}(\mathfrak{h}_{t}(y)-\mathfrak{h}_{t}(0))\leq-\frac{7}{8}\beta^{\kappa}s\Big{)}$
$\displaystyle\leq e^{-cs^{\alpha}}.$ (4.1)

###### Proof.

Let us define

$\displaystyle\mathsf{A}:=\left\\{\inf_{y\in[0,\beta^{2\kappa}s^{2-\alpha}]}(\mathfrak{h}_{t}(y)-\mathfrak{h}_{t}(0))\leq-\frac{7}{8}\beta^{\kappa}s\right\\},\
\
\mathsf{B}:=\left\\{\mathfrak{h}_{t}(\beta^{\kappa}s^{1-\frac{\alpha}{3}})-\mathfrak{h}_{t}(0)\leq-\frac{3s^{2\alpha/3}}{4}\right\\}.$

We seek to show that $\mathbb{P}(\mathsf{A})$ is bounded above by
$\exp(-cs^{\alpha})$ for all large $s$ with some constant $c>0$. Observe that
$\mathbb{P}(\mathsf{A})\leq\mathbb{P}(\mathsf{A}\cap\neg\mathsf{B})+\mathbb{P}(\mathsf{B})$.
In what follows, we show that there exists $s_{0}=s_{0}(t_{0})$, $c>0$ such
that for all $s\geq s_{0}$ and $t>t_{0}$,

$\displaystyle\underbrace{\mathbb{P}(\mathsf{B})\leq\exp(-cs^{\alpha})}_{(\mathbf{I})},\qquad\underbrace{\mathbb{P}(\mathsf{A}\cap\neg\mathsf{B})\leq\exp(-cs^{\alpha})}_{(\mathbf{II})}.$

Combining $(\mathbf{I})$ and $(\mathbf{II})$ will bound
$\mathbb{P}(\mathsf{A})$. By repeating the same argument for the interval
$[-\beta^{2\kappa}s^{2-\alpha},0]$, one can show

$\mathbb{P}\Big{(}\inf_{y\in[-\beta^{2\kappa}s^{2-\alpha},0]}(\mathfrak{h}_{t}(y)-\mathfrak{h}_{t}(0))\leq-\frac{7}{8}\beta^{\kappa}s\Big{)}\leq
e^{-cs^{\alpha}}.$

Combining this inequality with the upper bound on $\mathbb{P}(\mathsf{A})$
will complete the proof of this proposition. Throughout the rest, we prove the
inequalities $(\mathbf{I})$ and $(\mathbf{II})$.

We first show the inequality $(\mathbf{I})$. Note that $\mathsf{B}$ is
contained in the union of
$\\{\mathfrak{h}_{t}(\beta^{\kappa}s^{1-\alpha/3})\leq-5s^{2\alpha/3}/8\\}$
and $\\{\mathfrak{h}_{t}(0)\geq s^{2\alpha/3}/8\\}$. By the union bound,

$\displaystyle\mathbb{P}(\mathsf{B})$
$\displaystyle\leq\mathbb{P}\Big{(}\mathfrak{h}_{t}(\beta^{\kappa}s^{1-\frac{\alpha}{3}})+\frac{\beta^{2\kappa}s^{2-\frac{2\alpha}{3}}}{2}\leq-\frac{5s^{2\alpha/3}}{8}+\frac{\beta^{2\kappa}s^{2-\frac{2\alpha}{3}}}{2}\Big{)}+\mathbb{P}\Big{(}\mathfrak{h}_{t}(0)\geq\frac{s^{2\alpha/3}}{8}\Big{)}$
(4.2)

Due to the stationarity,
$\mathfrak{h}_{t}(\beta^{\kappa}s^{1-\frac{\alpha}{3}})+\beta^{2\kappa}s^{2-\frac{2\alpha}{3}}/2$
is same in distribution with $\mathfrak{h}_{t}(0)$. Furthermore we have the
inequality
$-5s^{2\alpha/3}8+\beta^{2\kappa}s^{2-\frac{2\alpha}{3}}/2\leq-s^{2\alpha/3}/8$
because $\alpha\geq 3/2$ and $\beta\leq 1$. Combining we get

$\displaystyle\text{r.h.s. of
\eqref{eq:BBound}}\leq\mathbb{P}\Big{(}\mathfrak{h}_{t}(0)\leq-\frac{s^{2\alpha/3}}{8}\Big{)}+\mathbb{P}\Big{(}\mathfrak{h}_{t}(0)\geq\frac{s^{2\alpha/3}}{8}\Big{)}.$

Using (2.11) of Proposition 2.15 and (2.10) of Proposition 2.14, we bound
$\mathbb{P}(\mathfrak{h}_{t}(0)\leq-s^{2\alpha/3}/8)$ and
$\mathbb{P}(\mathfrak{h}_{t}(0)\geq s^{2\alpha/3}/8)$ by $\exp(-cs^{\alpha})$
for some constant $c>0$. Substituting these bound into the right side of the
above display yields $\mathbb{P}(\mathsf{B})\leq 2\exp(-cs^{\alpha})$.

Next we show $(\mathbf{II})$. For this, we use the Brownian Gibbs Property of
the KPZ line ensemble. Let us denote
$\mathcal{I}_{s,\beta}:=(0,\beta^{2\kappa}s^{1-\alpha/3})$. Recall that
$\mathfrak{h}_{t}$ is the lowest indexed curve $\mathfrak{h}^{(1)}_{t}$ of the
KPZ line ensemble $\\{\mathfrak{h}^{(n)}_{t}\\}_{n\in\mathbb{N}}$. Let
$\mathcal{F}_{s}:=\mathcal{F}_{\mathbf{ext}}(\\{1\\},\mathcal{I}_{s,\beta})$
be the $\sigma$-algebra generated by
$\\{\mathfrak{h}^{(1)}_{t}(x):x\in\mathbb{R}\backslash\mathcal{I}_{s,\beta}\\}$
and $\\{\mathfrak{h}_{t}^{(n)}(x):x\in\mathbb{R}\\}_{n\in\mathbb{N}_{\geq
2}}$. Note that $\neg\mathsf{B}$ is measurable w.r.t. $\mathcal{F}_{s}$. Thus,
we may write

$\displaystyle\mathbb{P}\big{(}\mathsf{A}\cap\neg\mathsf{B}\big{)}=\mathbb{E}\left[\mathbf{1}_{\neg\mathsf{B}}\mathbb{E}[\mathbf{1}_{\mathsf{A}}|\mathcal{F}_{s}]\right]=\mathbb{E}\left[\mathbf{1}_{\neg\mathsf{B}}\mathbb{P}_{s}(\mathsf{A})\right].$
(4.3)

where
$\mathbb{P}_{s}:=\mathbb{P}_{\mathbf{H}^{\mathrm{long}}_{t}}^{1,1,(0,\beta^{\kappa}s^{1-\frac{\alpha}{3}}),\mathfrak{h}_{t}^{(1)}(0),\mathfrak{h}_{t}^{(1)}(\beta^{\kappa}s^{1-\frac{\alpha}{3}}),+\infty,\mathfrak{h}_{t}^{(2)}}$.
By the monotone coupling (Lemma 2.4)
$\mathbb{P}_{s}(\mathsf{A})\leq\mathbb{P}_{\mathbf{free}}(\mathsf{A})$, where
$\mathbb{P}_{\mathbf{free}}:=\mathbb{P}_{\mathbf{H}^{\mathrm{long}}_{t}}^{1,1,(0,\beta^{\kappa}s^{1-\frac{\alpha}{3}}),\mathfrak{h}_{t}^{(1)}(0),\mathfrak{h}_{t}^{(1)}(\beta^{\kappa}s^{1-\frac{\alpha}{3}}),+\infty,-\infty}$
is the law of a Brownian Bridge $\mathfrak{B}$ on
$[0,\beta^{\kappa}s^{1-\frac{\alpha}{3}}]$ with
$\mathfrak{B}(0):=\mathfrak{h}_{t}(0)$ and
$\mathfrak{B}(\beta^{\kappa}s^{1-\frac{\alpha}{3}}):=\mathfrak{h}_{t}(\beta^{\kappa}s^{1-\frac{\alpha}{3}})$.
Since $\beta\in(0,1]$ and $\alpha\geq 3/2$, we have
$\beta^{\kappa}s^{1-\alpha/3}\geq\beta^{2\kappa}s^{2-\alpha}$. By the affine
equivariance of the law of Brownian bridges,

$\big{\\{}\mathfrak{B}(x):x\in\mathcal{I}_{s,\beta}\big{\\}}\stackrel{{\scriptstyle
d}}{{=}}\big{\\{}\widetilde{\mathfrak{B}}(x)+\frac{\mathfrak{h}_{t}(\beta^{\kappa}s^{1-\frac{\alpha}{3}})-\mathfrak{h}_{t}(0)}{\beta^{\kappa}s^{1-\frac{\alpha}{3}}}y:x\in\mathcal{I}_{s,\beta}\big{\\}}$

where $\widetilde{\mathfrak{B}}$ is a Brownian Bridge on
$[0,\beta^{\kappa}s^{1-\frac{\alpha}{3}}]$ starting and ending at $0$.
Combining these observations with (4.3) shows

$\displaystyle\mathbb{P}(\mathsf{A}\cap\neg\mathsf{B})$
$\displaystyle\leq\mathbb{E}[\mathbf{1}_{\neg\mathsf{B}}\mathbb{P}_{s}(\mathsf{A})]$
$\displaystyle=\mathbb{E}\Big{[}\mathbf{1}_{\neg\mathsf{B}}\mathbb{P}\big{(}\inf_{y\in[0,\beta^{2\kappa}s^{2-\alpha}]}\big{[}\widetilde{\mathfrak{B}}(y)+\frac{\mathfrak{h}_{t}(\beta^{\kappa}s^{1-\frac{\alpha}{3}})-\mathfrak{h}_{t}(0)}{\beta^{\kappa}s^{1-\frac{\alpha}{3}}}y\big{]}\leq-\frac{7}{8}\beta^{\kappa}s\big{|}\mathcal{F}_{s}\big{)}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\mathbf{1}_{\neg\mathsf{B}}\mathbb{P}\big{(}\inf_{y\in[0,\beta^{2\kappa}s^{2-\alpha}]}\widetilde{\mathfrak{B}}(y)-\frac{3s^{2\alpha/3}\beta^{2\kappa}s^{2-\alpha}}{4\beta^{\kappa}s^{1-\frac{\alpha}{3}}}\leq-\frac{7}{8}\beta^{\kappa}s\big{|}\mathcal{F}_{s}\Big{)}\Big{]}$
$\displaystyle\leq\mathbb{P}\Big{(}\inf_{y\in[0,\beta^{2\kappa}s^{2-\alpha}]}\widetilde{\mathfrak{B}}(y)\leq-\frac{\beta^{\kappa}s}{8}\Big{)}=\mathbb{P}\Big{(}\frac{1}{\beta^{\kappa}s^{1-\frac{\alpha}{2}}}\inf_{y\in[0,\beta^{2\kappa}s^{2-\alpha}]}\widetilde{\mathfrak{B}}(y)\leq-\frac{s^{\alpha/2}}{8}\Big{)}$

The inequality in the third line follows since

$\inf_{y\in[0,\beta^{2\kappa}s^{2-\alpha}]}\Big{\\{}\widetilde{\mathfrak{B}}(y)+\frac{\mathfrak{h}_{t}(\beta^{\kappa}s^{1-\frac{\alpha}{3}})-\mathfrak{h}_{t}(0)}{\beta^{\kappa}s^{1-\frac{\alpha}{3}}}y\Big{\\}}\geq\inf_{y\in[0,\beta^{2\kappa}s^{2-\alpha}]}\widetilde{\mathfrak{B}}(y)-\frac{3s^{2\alpha/3}\beta^{2\kappa}s^{2-\alpha}}{4\beta^{\kappa}s^{1-\frac{\alpha}{3}}}$

on the event $\neg\mathsf{B}$. The next inequality follows by neglecting the
indicator. The last probability is clearly bounded by $\exp(-cs^{\alpha})$ by
tail estimates of Brownian motion. This proves $(\mathbf{II})$ and hence,
completes the proof of this proposition. ∎

######  Proposition 4.2.

Fix $\kappa>0$. There exist constant $c>0$, $t_{0}>0$ such that for all $t\geq
t_{0}$ and $\beta\in(0,1]$ and $s\geq s_{0}(t_{0})$ we have

$\displaystyle\mathbb{P}\Big{(}\sup_{|y|\leq\frac{1}{16}\beta^{2\kappa}\sqrt{s}}(\mathfrak{h}_{t}(y)-\mathfrak{h}_{t}(0))\geq\beta^{\kappa}s\Big{)}$
$\displaystyle\leq e^{-cs^{3/2}}.$ (4.4)

###### Proof.

Let $\mathsf{Sup}_{1}$ and $\mathsf{Sup}_{2}$ be the supremum value of
$\mathfrak{h}_{t}(y)-\mathfrak{h}_{t}(0)$ for
$y\in[0,\frac{1}{16}\beta^{2\kappa}\sqrt{s}]$ and
$y\in[-\frac{1}{16}\beta^{2\kappa}\sqrt{s},0]$ respectively. In what follows,
we only bound $\mathbb{P}(\mathsf{Sup}_{1}\geq\beta^{\kappa}s)$. One can bound
$\mathbb{P}(\mathsf{Sup}_{2}\geq\beta^{\kappa}s)$ analogously. Let $\chi$ be
the infimum of $y$ in $[0,\frac{1}{16}\beta^{2\kappa}\sqrt{s}]$ such that
$\mathfrak{h}_{t}(y)-\mathfrak{h}_{t}(0)\geq\beta^{\kappa}s$. If there is no
such $y$, define $\chi$ to be $+\infty$. Note that
$\mathbb{P}(\mathsf{Sup}_{1}\geq\beta^{\kappa}s)=\mathbb{P}(\chi\leq\frac{1}{16}\beta^{2\kappa}\sqrt{s})$.
We can write the event $\\{\chi\leq\frac{1}{16}\beta^{2\kappa}\sqrt{s}\\}$ as
a disjoint union of $\mathsf{A}_{1}$ and $\mathsf{A}_{2}$ which are defined as

$\mathsf{A}_{1}:=\Big{\\{}\chi\leq\frac{\beta^{2\kappa}\sqrt{s}}{16},\Big{(}\mathfrak{h}_{t}(\chi)-\mathfrak{h}_{t}\big{(}\frac{\beta^{2\kappa}\sqrt{s}}{16}\big{)}\Big{)}<\frac{\beta^{\kappa}s}{8}\Big{\\}},\quad\mathsf{A}_{2}:=\Big{\\{}\chi\leq\frac{\beta^{2\kappa}\sqrt{s}}{16},\Big{(}\mathfrak{h}_{t}(\chi)-\mathfrak{h}_{t}\big{(}\frac{\beta^{2\kappa}\sqrt{s}}{16}\big{)}\Big{)}\geq\frac{\beta^{\kappa}s}{8}\Big{\\}}.$

In what follows, we show there exist $s_{0}=s_{0}(t_{0})>0$ and constant $c>0$
such that for all $s\geq s_{0}$ and $t\geq t_{0}$, we have

$\displaystyle\mathbb{P}(A_{1})\leq\exp(-cs^{3/2}),\qquad\qquad\mathbb{P}(A_{2})\leq\frac{1}{2}\mathbb{P}(\chi\leq\frac{1}{16}\beta^{2\kappa}\sqrt{s})+\exp(-cs^{3/2}).$
(4.5)

Since
$\mathbb{P}(\chi\leq\frac{1}{16}\beta^{2\kappa}\sqrt{s})=\mathbb{P}(\mathsf{A}_{1})+\mathbb{P}(\mathsf{A}_{2})$,
combining the above two inequalities shows
$2^{-1}\mathbb{P}(\chi\leq\frac{1}{16}\beta^{2\kappa}\sqrt{s})\leq
2\exp(-cs^{\alpha})$. Thus, proving (4.4) boils down to showing (4.5).

We first prove $\mathbb{P}(\mathsf{A}_{1})\leq\exp(-cs^{3/2})$. By the
continuity of the spatial process $\mathfrak{h}_{t}(\cdot)$, we have
$\mathfrak{h}_{t}(\chi)=\mathfrak{h}_{t}(0)+\beta^{\kappa}s$ on the event
$\\{\chi\leq\frac{1}{16}\beta^{2\kappa}\sqrt{s}\\}$. Thus

$\displaystyle\mathbb{P}(A_{1})$
$\displaystyle\leq\mathbb{P}\Big{(}\mathfrak{h}_{t}(0)-\mathfrak{h}_{t}(\tfrac{\beta^{2\kappa}\sqrt{s}}{16})\leq-\frac{7\beta^{\kappa}s}{8}\Big{)}\leq\mathbb{P}\left(\inf_{y\in[0,\beta^{2\kappa}\sqrt{s}/16]}\left(\mathfrak{h}_{t}(y)-\mathfrak{h}_{t}(\tfrac{\beta^{2\kappa}\sqrt{s}}{16})\right)\leq-\frac{7\beta^{\kappa}s}{8}\right)$

The right hand side of the above inequality is bounded by $\exp(-cs^{3/2})$
due to Proposition 4.1 and the stationarity of spatial process
$\mathfrak{h}_{t}(x)+\frac{x^{2}}{2}$. This proves the first inequality of
(4.5).

Now we turn to show the second inequality of (4.5). Consider the following
event

$\mathsf{B}:=\Big{\\{}\mathfrak{h}_{t}(0)\in[-s/4,s/4],\mathfrak{h}_{t}(\beta^{\kappa}\sqrt{s})\in[-3s/4,s/4]\Big{\\}}.$

Observe that
$\mathbb{P}(\mathsf{A}_{2})\leq\mathbb{P}(\mathsf{A}_{2}\cap\mathsf{B})+\mathbb{P}(\neg\mathsf{B})$.
By Proposition 2.15 and 2.14, we get
$\mathbb{P}(\neg\mathsf{B})\leq\exp(-cs^{3/2})$ for some constant $c>0$ and
all large $s$ and $t$. It suffices to show

$\displaystyle\mathbb{P}(\mathsf{A}_{2}\cap\mathsf{B})\leq
2^{-1}\mathbb{P}(\chi\leq\beta^{2\kappa}\sqrt{s}/16)$ (4.6)

which is proven below.

To bound the $\mathbb{P}(\mathsf{A}_{2}\cap\mathsf{B})$ we use the strong
Brownian Gibbs property of the KPZ line ensemble. Let
$\mathcal{F}_{s}=\mathcal{F}_{\mathbf{ext}}(\\{1\\},(\chi,\beta^{\kappa}\sqrt{s}))$
be the $\sigma$-algebra generated by
$\\{\mathfrak{h}_{t}^{(n)}(x)\\}_{n\in\mathbb{N},x\in\mathbb{R}}$ outside
$\\{\mathfrak{h}_{t}^{(1)}(x)\\}_{x\in(\chi,\beta^{\kappa}\sqrt{s})}$. By the
tower property of the conditional expectation, we have

$\displaystyle\mathbb{P}(\mathsf{A}_{2}\cap\mathsf{B})=\mathbb{E}\left[\mathbf{1}_{\left\\{\chi\leq\frac{1}{16}\beta^{2\kappa}\sqrt{s}\right\\}\cap\mathsf{B}}\mathbb{E}(\mathbf{1}_{\mathsf{D}}|\mathcal{F}_{s})\right]=\mathbb{E}\left[\mathbf{1}_{\left\\{\chi\leq\frac{1}{16}\beta^{2\kappa}\sqrt{s}\right\\}\cap\mathsf{B}}\mathbb{P}_{s}(\mathsf{D})\right].$
(4.7)

where
$\mathsf{D}:=\\{\mathfrak{h}_{t}(\chi)-\mathfrak{h}_{t}(\frac{1}{16}\beta^{2\kappa}\sqrt{s})\geq\frac{1}{8}\beta^{\kappa}s\\}$
and
$\mathbb{P}_{s}:=\mathbb{P}_{\mathbf{H}^{\mathrm{long}}_{t}}^{1,1,(\chi,\beta^{\kappa}\sqrt{s}),\mathfrak{h}_{t}^{(1)}(\chi),\mathfrak{h}_{t}^{(1)}(\beta^{\kappa}\sqrt{s}),+\infty,\mathfrak{h}_{t}^{(2)}}$.
We show that $\mathbb{P}_{s}(\mathsf{D})\leq\frac{1}{2}$ on the event
$\\{\chi\leq\frac{1}{16}\beta^{2\kappa}\sqrt{s}\\}\cap\mathsf{B}$. By Lemma
2.4, $\mathbb{P}_{s}(\mathsf{D})\leq\mathbb{P}_{\mathbf{free}}(\mathsf{D})$,
where
$\mathbb{P}_{\mathbf{free}}:=\mathbb{P}_{\mathbf{H}^{\mathrm{long}}_{t}}^{1,1,(\chi,\beta^{\kappa}\sqrt{s}),\mathfrak{h}_{t}^{(1)}(\chi),\mathfrak{h}_{t}^{(1)}(\beta^{\kappa}\sqrt{s}),+\infty,-\infty}$
is the law of a Brownian Bridge $\mathfrak{B}(\cdot)$ on
$[\chi,\beta^{\kappa}\sqrt{s}]$ with
$\mathfrak{B}(\chi):=\mathfrak{h}_{t}(\chi)$ and
$\mathfrak{B}(\beta^{\kappa}\sqrt{s}):=\mathfrak{h}_{t}(\beta^{\kappa}\sqrt{s})$.
Let us define

$\mathfrak{B}_{\mathrm{Interpole}}(y)=\frac{\beta^{\kappa}\sqrt{s}-y}{\beta^{\kappa}\sqrt{s}-\chi}\mathfrak{h}_{t}(\chi)+\frac{y-\chi}{\beta^{\kappa}\sqrt{s}-\chi}\mathfrak{h}_{t}(\beta^{\kappa}\sqrt{s})\quad\text{for
}y\in[\chi,\beta^{\kappa}\sqrt{s}].$

Note that $\mathfrak{B}_{\mathrm{Interpole}}(y)-\mathfrak{h}_{t}(\chi)$ is
equal to
$(y-\chi)(\mathfrak{h}_{t}(\beta^{\kappa}\sqrt{s})-\mathfrak{h}_{t}(\chi))/(\beta^{\kappa}\sqrt{s}-\chi)$.
On the event $\\{\chi\leq\beta^{\kappa}s/16\\}\cap\mathsf{B}$, we have
$\mathfrak{h}_{t}(\chi)=\mathfrak{h}_{t}(0)+\beta^{\kappa}s$ by the continuity
of the spatial process $\mathfrak{h}_{t}$ and hence,
$\mathfrak{h}_{t}(\beta^{\kappa}\sqrt{s})-\mathfrak{h}_{t}(\chi)$ is bounded
below by $\frac{-3s}{4}-\mathfrak{h}_{t}(0)-\beta^{\kappa}s$ which is further
lower bounded by $-2s$. This shows
$\mathfrak{B}_{\mathrm{Interpole}}(\beta^{2\kappa}s/16)\geq\mathfrak{h}_{t}(\chi)-\beta^{\kappa}s/8$.
However, we know that
$\mathfrak{B}(\beta^{2\kappa}s/16)\leq\mathfrak{h}_{t}(\chi)-\beta^{\kappa}s/8$
on the event $\mathsf{D}$. This shows
$\mathfrak{B}(\beta^{2\kappa}s/16)\leq\mathfrak{B}_{\mathrm{Interpole}}(\beta^{2\kappa}s/16)$
on the event $\\{\chi\leq\beta^{\kappa}s/16\\}\cap\mathsf{B}$. However, since
$\mathfrak{B}$ is a Brownian bridge and $\mathfrak{B}_{\mathrm{Interpole}}$ is
the linear interpolation of the end points of $\mathfrak{B}$, the probability
of the event
$\\{\mathfrak{B}(\beta^{2\kappa}s/16)\leq\mathfrak{B}_{\mathrm{Interpole}}(\beta^{2\kappa}s/16)\\}$
is equal to $1/2$. This implies $\mathbb{P}_{s}(\mathsf{D})\leq\frac{1}{2}$ on
the event $\\{\chi\leq\beta^{2\kappa}\sqrt{s}/16\\}\cap\mathsf{B}$.
Substituting this bound into the right hand side of (4.7) shows
$\mathbb{P}(\mathsf{A}_{2}\cap\mathsf{B})\leq\mathbb{P}(\chi\leq\beta^{2\kappa}\sqrt{s}/16)/2$.
This completes the proof.

∎

######  Proposition 4.3.

Fix $a\in\mathbb{R}$ and $\delta>0$. There exist $t_{0}\in(0,1)$ and an
absolute constant $c>0$ such that for all $t\leq t_{0}$, $s\geq s_{0}(t_{0})$
satisfying $(|a|+|\delta|)^{2}-a^{2}\leq\frac{s}{2^{8}}$, we have

$\displaystyle\mathbb{P}\Big{(}\sup_{x\in[a,a+\delta]}\Big{(}\mathfrak{g}_{t}((4^{3}t/\pi^{3})^{1/8}x)+\frac{x^{2}}{2}\Big{)}\geq
s\Big{)}\leq e^{-cs^{3/2}}.$ (4.8)
$\displaystyle\mathbb{P}\Big{(}\inf_{x\in[a,a+\delta]}\Big{(}\mathfrak{g}_{t}((4^{3}t/\pi^{3})^{1/8}x)+\frac{x^{2}}{2}\Big{)}\leq-s\Big{)}\leq
e^{-cs^{2}}+e^{-cs^{2}t^{-1/8}\delta^{-1}}.$ (4.9)

###### Proof.

We introduce the shorthand
$\tilde{\mathfrak{g}}_{t}(x):=\mathfrak{g}_{t}((4^{3}t/\pi^{3})^{1/8}x)$ which
will be used throughout this proof. We divide the proof of this proposition in
two stages. We prove (4.8) and (4.9) in _Stage-1_ and _Stage-2_ respectively.

_Stage-1: Proof of ( 4.8)._ Assume $[a,a+\delta]\subset\mathbb{R}_{\geq 0}$.
Define

$\displaystyle{\mathsf{C}}_{[a,a+\delta]}:=\Big{\\{}\sup_{x\in[a,a+\delta]}(\tilde{\mathfrak{g}}_{t}(x)+\frac{x^{2}}{2})\geq
s\Big{\\}},$
$\displaystyle\quad\mathsf{C}^{\prime}_{[a,a+\delta]}:=\Big{\\{}\sup_{x\in[a,a+\delta]}(\tilde{\mathfrak{g}}_{t}(x)-\tilde{\mathfrak{g}}_{t}(a))\geq\frac{s}{4}\Big{\\}},$
$\displaystyle{\mathsf{D}}_{w}:=\Big{\\{}-\frac{s}{32}\leq$
$\displaystyle\tilde{\mathfrak{g}}_{t}(w)+\frac{w^{2}}{2}\leq\frac{s}{32}\Big{\\}}$

where $w\in\mathbb{R}$. We seek to show that
$\mathbb{P}({\mathsf{C}}_{[a,a+\delta]})\leq\exp(-cs^{3/2})$ for all large $s$
and small $t$. Combining the stationarity in $x$ of the process
$\tilde{\mathfrak{g}}_{t}(x)+\frac{x^{2}}{2}$ (follows from Lemma 2.11) with
Corollary 1.6 and Theorem 1.7 yields
$\mathbb{P}(\neg\mathsf{D}_{w})\leq\exp(-cs^{3/2})$ for all $w\in\mathbb{R}$.
This will be used throughout the proof. On the event
$\mathsf{C}_{[a,a+\delta]}\cap\mathsf{D}_{a}$, there exists $x\in[a,a+\delta]$
such that

$\tilde{\mathfrak{g}}_{t}(x)\geq s-\frac{x^{2}}{2}\geq
s-\frac{(a+\delta)^{2}}{2}\geq\frac{31s}{32}+\tilde{\mathfrak{g}}_{t}(a)+\frac{a^{2}}{2}-\frac{(a+\delta)^{2}}{2}\geq\frac{s}{4}+\tilde{\mathfrak{g}}_{t}(a)$

where the second inequality follows since $x\leq a+\delta$, the third
inequality follows since $\tilde{\mathfrak{g}}_{t}(a)+\frac{a^{2}}{2}\leq
s/32$ on $\mathsf{D}_{a}$ and the last inequality holds since
$(a+\delta)^{2}-a^{2}\leq s/2^{8}$. The above inequalities shows
$\mathsf{C}_{[a,a+\delta]}\cap\mathsf{D}_{a}\subset\mathsf{C}^{\prime}_{[a,a+\delta]}$
which implies

$\displaystyle\mathbb{P}({\mathsf{C}}_{[a,a+\delta]})\leq\mathbb{P}(\neg\mathsf{D}_{a})+\mathbb{P}(\mathsf{C}^{\prime}_{[a,a+\delta]}).$

Recall that $\mathbb{P}(\neg\mathsf{D}_{a})\leq\exp(-cs^{3/2})$. To complete
the proof, it suffices to show that
$\mathbb{P}(\mathsf{C}^{\prime}_{[a,a+\delta]})\leq\exp(-cs^{3/2})$ for large
$s$ and small $t$. This we do as follows.

Let $\sigma$ be the infimum of $y\in[a,a+\delta]$ such that
$\tilde{\mathfrak{g}}_{t}(y)-\tilde{\mathfrak{g}}_{t}(a)\geq\frac{s}{4}$, with
the understanding that $\sigma$ is equal to $+\infty$ if no such point exists.
Let us define
$\mathsf{B}:=\\{\tilde{\mathfrak{g}}_{t}(a+\delta)-\tilde{\mathfrak{g}}_{t}(\sigma)\leq-\frac{s}{8}\\}$
and write

$\displaystyle\mathbb{P}(\mathsf{C}^{\prime}_{[a,a+\delta]})=\mathbb{P}(\sigma\leq
a+\delta)=\mathbb{P}(\\{\sigma\leq
a+\delta\\}\cap\mathsf{B})+\mathbb{P}(\\{\sigma\leq
a+\delta\\}\cap\neg\mathsf{B})$

On the event $\\{\sigma\leq a+\delta\\}$, we have
$\tilde{\mathfrak{g}}_{t}(\sigma)=\tilde{\mathfrak{g}}_{t}(a)+\frac{s}{4}$.
This implies
$\tilde{\mathfrak{g}}_{t}(a+\delta)-\tilde{\mathfrak{g}}_{t}(a)=\tilde{\mathfrak{g}}_{t}(a+\delta)-\tilde{\mathfrak{g}}_{t}(\sigma)+s/4\geq-s/8$
on $\\{\sigma\leq a+\delta\\}\cap\neg\mathsf{B}$ and hence,

$\displaystyle\mathbb{P}(\\{\sigma\leq a+\delta\\}\cap\neg\mathsf{B})$
$\displaystyle\leq\mathbb{P}\big{(}\tilde{\mathfrak{g}}_{t}(a+\delta)+\frac{(a+\delta)^{2}}{2}-\tilde{\mathfrak{g}}_{t}(a)-\frac{a^{2}}{2}\geq-\frac{s}{8}\big{)}$
$\displaystyle\leq\mathbb{P}\big{(}\tilde{\mathfrak{g}}_{t}(a+\delta)+\frac{(a+\delta)^{2}}{2}>\frac{s}{16}\big{)}+P\big{(}\tilde{\mathfrak{g}}_{t}(a)+\frac{a^{2}}{2}\leq-\frac{s}{16}\big{)}\leq\exp(-cs^{3/2}).$
(4.10)

where the second inequality follows from the union bound and the last
inequality follows by combining the stationarity of
$\tilde{\mathfrak{g}}_{t}(x)+\frac{x^{2}}{2}$ with Corollary 1.6 and Theorem
1.7.

Now we proceed to bound $\mathbb{P}(\\{\sigma\leq a+\delta\\}\cap\mathsf{B})$.
By the union bound, we have

$\mathbb{P}(\\{\sigma\leq
a+\delta\\}\cap\mathsf{B})\leq\mathbb{P}(\\{\sigma\leq
a+\delta\\}\cap\mathsf{B}\cap\mathsf{D}_{a}\cap\mathsf{D}_{a+4\delta})+\mathbb{P}(\neg\mathsf{D}_{a})+\mathbb{P}(\neg\mathsf{D}_{a+4\delta}).$
(4.11)

We know
$\mathbb{P}(\neg\mathsf{D}_{a})+\mathbb{P}(\neg\mathsf{D}_{a+4\delta})$ is
bounded above by $\exp(-cs^{3/2})$ for some constant $c>0$. In what follows,
we show that

$\displaystyle\mathbb{P}(\\{\sigma\leq
a+\delta\\}\cap\mathsf{B}\cap\mathsf{D}_{a}\cap\mathsf{D}_{a+4\delta})\leq\frac{1}{2}\mathbb{P}(\sigma\leq
a+\delta).$ (4.12)

Combing this inequality with (4.11) and (4.10) show that
$\mathbb{P}(\mathsf{C}^{\prime}_{[a,a+\delta]})\leq
2^{-1}\mathbb{P}(\mathsf{C}^{\prime}_{[a,a+\delta]})+\exp(-cs^{3/2})$ for all
large $s$ and small $t$. By simplifying this aforementioned inequality, we get
the desired result.

It remains to show (4.12) whose proof is similar to that of (4.6). To avoid
the repetition, we sketch the underlying idea without details. The main tool
that we use is the Brownian Gibbs property of the short time KPZ line ensemble
$\\{\mathfrak{g}^{(n)}_{t}\\}_{n\in\mathbb{N}}$ (Recall its definition from
(2.2)). By the tower property, we write the left hand side of (4.12) as
$\mathbb{E}[\mathbf{1}_{\left\\{\sigma\leq
a+\delta\right\\}\cap\mathcal{D}_{a}\cap\mathsf{D}_{a+4\delta}}\mathbb{P}_{s}(\mathsf{B})]$
where

$\mathbb{P}_{s}:=\mathbb{P}_{\mathbf{H}^{\mathrm{short}}_{t}}^{1,1,(4^{3}t/\pi^{3})^{1/8}(\sigma,a+4\delta),\tilde{\mathfrak{g}}_{t}^{(1)}((4^{3}t/\pi^{3})^{1/8}\sigma),\tilde{\mathfrak{g}}_{t}^{(1)}((4^{3}t/\pi^{3})^{1/8}(a+4\delta)),+\infty,\tilde{\mathfrak{g}}_{t}^{(2)}}.$

By monotone coupling,
$\mathbb{P}_{s}(\mathsf{B})\leq\mathbb{P}_{\mathbf{free}}(\mathsf{B})$ where
$\mathbb{P}_{\mathbf{free}}$ is the law of a free Brownian bridge between
$(4^{3}t/\pi^{3})^{1/8}\sigma$ and $(4^{3}t/\pi^{3})^{1/8}(a+4\delta)$ with
the value of the end points being $\tilde{\mathfrak{g}}_{t}(\sigma)$ and
$\tilde{\mathfrak{g}}_{t}(a+4\delta)$. On the event $\\{\sigma\leq
a+\delta\\}\cap\mathsf{D}_{a}\cap\mathsf{D}_{a+4\delta}\cap\mathsf{B}$, the
value of the Brownian bridge at $(4^{3}t/\pi^{3})^{1/8}(a+\delta)$ has to be
lower than the value of the line joining two end points of the Brownian
bridge. The probability of this is bounded by $1/2$ which shows
$\mathbb{P}_{{\mathbf{free}}}(\mathsf{B})\leq 1/2$ on $\left\\{\sigma\leq
a+\delta\right\\}\cap\mathsf{D}_{a}\cap\mathsf{D}_{a+4\delta}$. Hence, we get
$\mathbb{E}[\mathbf{1}_{\left\\{\sigma\leq
a+\delta\right\\}\cap\mathsf{D}_{a}\cap\mathsf{D}_{a+4\delta}}\mathbb{P}_{s}(\mathsf{B})]$
is less than $\mathbb{P}(\sigma\leq a+\delta)/2$. This shows (4.12) and hence,
completes the proof of (4.8).

_Stage-2: Proof of ( 4.9)._ Let us define the following two events:

$\displaystyle\mathsf{B}_{[a,a+\delta]}=\Big{\\{}\frac{a^{2}}{2}+\inf_{x\in[a,a+\delta]}\tilde{\mathfrak{g}}_{t}(x)\leq-s\Big{\\}},\qquad\mathsf{E}_{w}:=\Big{\\{}\tilde{\mathfrak{g}}_{t}(w)+\frac{w^{2}}{2}\geq-\frac{s}{4}\Big{\\}}$

for $w\in\mathbb{R}$. Note that

$\displaystyle\mathbb{P}\big{(}\inf_{x\in[a,a+\delta]}\big{(}\tilde{\mathfrak{g}}_{t}(x)+\frac{x^{2}}{2}\big{)}\leq-s\big{)}\leq\mathbb{P}(\mathsf{B}_{[a,a+\delta]})\leq\mathbb{P}(\neg\mathsf{E}_{a})+\mathbb{P}(\neg\mathsf{E}_{a+\delta})+\mathbb{P}(\mathsf{B}_{[a,a+\delta]}\cap\mathsf{E}_{a}\cap\mathsf{E}_{a+\delta}).$

Due to the spatial stationarity of the process
$\tilde{\mathfrak{g}}_{t}(x)+x^{2}/2$ (see Lemma 2.11) and Theorem 1.7, we
have
$\mathbb{P}(\neg\mathsf{E}_{a+\delta})=\mathbb{P}(\neg\mathsf{E}_{0})\leq\exp(-cs^{2})$
for all large $s$ and small $t$. To complete the proof of (4.9), it suffices
to show

$\displaystyle\mathbb{P}(\mathsf{B}_{[a,a+\delta]}\cap\mathsf{E}_{a}\cap\mathsf{E}_{a+\delta})\leq\exp(-cs^{2}t^{-1/8}\delta^{-1}).$
(4.13)

To show the above inequality, we use the Brownian-Gibbs property of the short
time KPZ line ensemble. Recall from (2.4) and (2.2) that
$\left\\{\tilde{\mathfrak{g}}_{t}((4^{3}t/\pi^{3})^{-1/8}w)\right\\}_{w\in\mathbb{R}}$
is same in distribution with $\mathfrak{g}_{t}^{(1)}(\cdot)$ where
$\mathfrak{g}_{t}^{(1)}$ is the lowest indexed curve of the short-time KPZ
line ensemble defined in (4) of Lemma 2.5. Let us set
$a^{\prime}:=(4^{3}t/\pi^{3})^{1/8}a$ and
$\delta^{\prime}:=(4^{3}t/\pi^{3})^{1/8}\delta$ for convenience. Let
$\mathcal{F}_{s}:=\mathcal{F}_{\mathbf{ext}}(\\{1\\},(a^{\prime},a^{\prime}+\delta^{\prime}))$
be the $\sigma$-algebra generated by
$\\{\tilde{\mathfrak{g}}_{t}^{(n)}(x)\\}_{n\in\mathbb{N}_{\geq
2},x\in\mathbb{R}}$ outside
$\\{\tilde{\mathfrak{g}}_{t}^{(1)}(x)\\}_{x\in\mathbb{R}\backslash(a^{\prime},a^{\prime}+\delta^{\prime})}$.
Consider the following two measures

$\mathbb{P}_{s}:=\mathbb{P}_{\mathbf{H}^{\mathrm{short}}_{t}}^{1,1,(a^{\prime},a^{\prime}+\delta^{\prime}),\tilde{\mathfrak{g}}_{t}(a),\tilde{\mathfrak{g}}_{t}(a+\delta),\infty,\tilde{\mathfrak{g}}_{t}^{(2)}},\quad\mathbb{P}_{\mathbf{free}}:=\mathbb{P}_{\mathbf{H}^{\mathrm{short}}_{t}}^{1,1,(a^{\prime},a^{\prime}+\delta^{\prime}),\tilde{\mathfrak{g}}_{t}(a),\tilde{\mathfrak{g}}_{t}(a+\delta),\infty,-\infty}$

where $\mathbb{P}_{\mathbf{free}}$ denotes the law of a Brownian bridge on
$[a^{\prime},a^{\prime}+\delta^{\prime}]$ with the boundary values
$\tilde{\mathfrak{g}}_{t}(a)$ and $\tilde{\mathfrak{g}}_{t}(a+\delta)$
respectively. By the strong Brownian Gibbs property for the short-time KPZ
line ensemble,

$\displaystyle\mathbb{P}(\mathsf{B}_{[a,a+\delta]}\cap\mathsf{E}_{a}\cap\mathsf{E}_{a+\delta})=\mathbb{E}\left[\mathbf{1}_{\mathsf{E}_{a}}\mathbf{1}_{\mathsf{E}_{a+\delta}}\mathbb{E}(\mathsf{B}_{[a,a+\delta]}|\mathcal{F}_{s})\right]=\mathbb{E}\left[\mathbf{1}_{\mathsf{E}_{a}}\mathbf{1}_{\mathsf{E}_{a+\delta}}\mathbb{P}_{s}(\mathsf{B}_{[a,a+\delta]})\right].$

Due to the monotone coupling, we know
$\mathbb{P}_{s}(\mathsf{B}_{[a,a+\delta]})\leq\mathbb{P}_{\mathbf{free}}(\mathsf{B}_{[a,a+\delta]})$.
Let $\mathfrak{B}$ be a Brownian bridge on $[0,\delta^{\prime}]$ with
$\mathfrak{B}(0)=\mathfrak{B}(\delta^{\prime})=0$. Then, the law of
$\mathfrak{B}(x)+\tilde{\mathfrak{g}}_{t}(a)\frac{\delta^{\prime}-x}{\delta^{\prime}}+\tilde{\mathfrak{g}}_{t}(a+\delta)\frac{x}{\delta^{\prime}}$
is same as $\mathbb{P}_{\mathbf{free}}$. So, we have

$\displaystyle\mathbb{P}(\mathsf{E}_{a}\cap\mathsf{E}_{a+\delta}\cap\mathsf{B}_{[a,a+\delta]})$
$\displaystyle\leq\mathbb{E}\Big{[}\mathbf{1}_{\mathsf{E}_{a}}\mathbf{1}_{\mathsf{E}_{a+\delta}}\mathbb{P}\Big{(}\frac{a^{2}}{2}+\inf_{x\in[0,\delta^{\prime}]}\big{(}\mathfrak{B}(x)+\tilde{\mathfrak{g}}_{t}(a)\frac{\delta^{\prime}-x}{\delta^{\prime}}+\tilde{\mathfrak{g}}_{t}(a+\delta)\frac{x}{\delta^{\prime}}\big{)}\leq-s\Big{)}\Big{]}$