# The Directed Landscape is a Black Noise

Zoe Himwich  and  Shalin Parekh  Zoe Himwich: Department of Mathematics,
Columbia University, 2990 Broadway, New York, NY 10027, USA
<EMAIL_ADDRESS>Shalin Parekh: Department of Mathematics, University
of Maryland, College Park, 4176 Campus Drive, College Park, MD 20742 USA
<EMAIL_ADDRESS>
###### Abstract.

We show that the directed landscape is a black noise in the sense of Tsirelson
and Vershik. As a corollary, we show that for any microscopic system in which
the height profile converges in law to the directed landscape, the driving
noise is asymptotically independent of the height profile. This decoupling
result provides one answer to the question of what happens to the driving
noise in the limit under the KPZ scaling, and illustrates a type of noise
sensitivity for systems in the KPZ universality class. Such decoupling and
sensitivity phenomena are not present in the intermediate-disorder or weak-
asymmetry regime, thus illustrating a contrast from the weak KPZ scaling
regime. Along the way, we prove a strong mixing property for the directed
landscape on a bounded time interval under spatial shifts, with a mixing rate
$\alpha(N)\leq Ce^{-dN^{3}}$ for some $C,d>0$.

###### Contents

  1. 1 Introduction
  2. 2 Reducing the black noise problem to an estimate on the Airy sheet
  3. 3 A strong mixing property for the landscape under spatial shifts
  4. 4 Useful estimates on Airy sheets and argmaxes
  5. 5 Estimates on 3-dimensional Bessel processes
  6. 6 Proof of the main theorem
  7. 7 Independence from Gaussian white noise

##  1\. Introduction

###  1.1. Preface

Over the past several decades, mathematicians and physicists who study
interacting particle systems, integrable systems, and the Kardar-Parisi-Zhang
(KPZ) equation have discovered connections between the large-scale behavior of
a wide-ranging class of models. It is known that the height functions of
asymmetric exclusion processes and stochastic vertex models exhibit universal
scaling behavior in $1+1$ dimensions. A common feature in these models is the
appearance of $1:2:3$ scaling exponents and Tracy-Widom limit laws. Similar
results are expected for the Eden model and last passage percolation models
with arbitrary weight distributions, as well as Burgers-type stochastic PDEs
and ballistic deposition models (e.g. Tetris). This phenomenon is called KPZ
universality [Cor12, Qua11].

In recent years, there has been a great deal of work involved in constructing
and understanding the universal limit objects in the KPZ universality class.
Matetski, Quastel, and Remenik [MQR21] constructed the KPZ fixed point, the
invariant limiting Markov process (conjecturally) of all models in the KPZ
universality class. Dauvergne, Ortmann and Virág [DOV22] then constructed the
directed landscape, a generalization which encodes the natural joint couplings
of all KPZ fixed points started from all possible initial data simultaneously.
The directed landscape is a four-parameter field, denoted in this paper as
$\mathcal{L}_{s,t}(x,y)$, see 1.8.

It is a longstanding project of mathematical physicists to have a better
understanding of this phenomenon of universality. It is conjectured that any
model for which it is possible to associate a height function $h(t,x)$ with
the basic properties of local dynamics, a smoothing mechanism, lateral growth,
and space-time forcing with rapid correlation decay, will converge to the KPZ
fixed point under $1:2:3$ scaling, $\epsilon
h(\epsilon^{-3}t,\epsilon^{-2}x)-C_{\epsilon}t$. More generally, the height
mechanisms started simultaneously from several distinct initial data should
converge to KPZ fixed points coupled through the directed landscape. See the
strong and weak KPZ universality conjectures as discussed in the introduction
of [MQR21, HQ18]. Models which satisfy these conditions, and which have been
proved or conjectured to have the KPZ fixed point and directed landscape as a
scaling limit, are said to belong to the KPZ universality class.

The directed landscape (1.8), which is the main object in this paper, was
constructed by Dauvergne, Ortmann, and Virág [DOV22] as the scaling limit of
the last passage metric between points on a Brownian line ensemble (Brownian
LPP). It is a random four-parameter continuous function
$\mathcal{L}:\mathbb{R}^{4}_{\uparrow}\to\mathbb{R}$, with
$\mathbb{R}^{4}_{\uparrow}:=\\{(x,s;y,t)\in\mathbb{R}^{4}|s<t\\},$ which may
be viewed as a metric between space-time points $(s,x)$ and $(t,y)$ that
satisfies a reverse triangle inequality,
$\mathcal{L}_{s,r}(x,z)\geq\mathcal{L}_{s,t}(x,y)+\mathcal{L}_{t,r}(y,z).$
More precisely the following “metric composition” law holds:

$\displaystyle\mathcal{L}_{s,r}(x,z)$
$\displaystyle=\max_{y\in\mathbb{R}}\left(\mathcal{L}_{s,t}(x,y)+\mathcal{L}_{t,r}(y,z)\right),$
$\displaystyle\text{for all }(x,z)\in\mathbb{R}^{2},\;\;s<t<r.$ (1.1)

A series of recent works have demonstrated that the directed landscape is the
scaling limit of many well-known models with simple microscopic dynamics.
Works such as [DOV22, DNV23, DV21] demonstrated that it is the limit of the
Brownian, Poissonian, exponential, and geometric last passage percolation, and
Poisson and Seppäläinen-Johansson line ensembles. A work by Wu [Wu23] combined
with a result of Quastel and Sarkar [QS20] demonstrates that the directed
landscape is the scaling limit of the KPZ equation started from all possible
initial data but coupled through the same realization of the driving noise.
Most recently, a work by Aggarwal, Corwin, and Hegde [ACH24] demonstrated a
similar convergence for ASEP and stochastic six vertex models. The KPZ fixed
point of Matetski, Quastel, and Remenik [MQR21] is a marginal of the directed
landscape by the variational formula
$h_{t}(y)=\sup_{x\in\mathbb{R}}\left(h_{0}(x)+\mathcal{L}_{0,t}(x,y)\right)$,
see [NQR20].

There are still many basic properties of the directed landscape that have yet
to be explored. A natural question to ask is whether it has some simpler
Gaussian structure.

######  Question 1.1.

Is the directed landscape the adapted strong solution of a white-noise driven
SPDE?

This article was motivated by an interest in better understanding what happens
to the driving noise of the KPZ equation under the directed landscape limit.
Another natural question in this vein is the following.

######  Question 1.2.

What happens to the microscopic driving noise of systems like last passage
percolation models, asymmetric exclusion processes, and the KPZ equation as we
take the limit to the directed landscape? In particular, is any memory of this
microscopic noise retained by the limiting directed landscape?

This paper aims to address both of the preceding questions. We study the
directed landscape as an abstract noise in the sense of Tsirelson and Vershik
[TV98], demonstrating that it is a black noise (1.10). Using this property, we
show that the directed landscape is independent of any white noise field
defined on the same filtered probability space (1.11). These results have a
few corollaries. We conclude that the height function associated to any model
which converges to the directed landscape will asymptotically decouple from
the random environment (1.12), which answers 1.2. We can also answer 1.1 in
the negative (1.13).

###  1.2. Background and Results

####  1.2.1. Black Noise

Black noise was first introduced by Tsirelson and Vershik [TV98], and
subsequently explored at length in a survey by Tsirelson [Tsi04]. Building
towards the definition of black noise, we begin by introducing the concept of
a noise.

######  Definition 1.3 (Definition 3d(1), [Tsi04]).

A noise, denoted in this paper as a 4-tuple
$(\Omega,(\mathcal{F}_{s,t})_{s<t},\mathbb{P},(\theta_{h})_{h})$, is a
probability space $(\Omega,\mathcal{F},\mathbb{P})$ with sub-$\sigma$-fields
$\mathcal{F}_{s,t}\subset\mathcal{F}$ ($-\infty<s<t<\infty$) and measurable
maps $\theta_{h}:\Omega\to\Omega$ ($h\in\mathbb{R}$) such that the following
properties are satisfied.

  1. (1)

$\mathcal{F}_{s,t}$ and $\mathcal{F}_{t,u}$ are independent under the
probability measure $P$ if $s<t<u$.

  2. (2)

For all $r<s<t$, $\mathcal{F}_{s,t}$ and $\mathcal{F}_{r,s}$ together generate
$\mathcal{F}_{r,t}$, i.e.,
$\mathcal{F}_{r,s}\vee\mathcal{F}_{s,t}=\mathcal{F}_{r,t}$.

  3. (3)

$\mathcal{F}$ is generated by the union of all $\mathcal{F}_{s,t}.$

  4. (4)

If $A\in\mathcal{F}_{s,t}$ then $\theta_{h}(A)\in\mathcal{F}_{s+h,t+h}$, and
$P(\theta_{h}(A))=P(A).$

  5. (5)

$\theta_{h}\theta_{k}=\theta_{h+k}$ for all $h,k\in\mathbb{R},$ and
$\theta_{0}=\mathrm{Id}$.

This definition of noise uses the language of stochastic flows, in the sense
of Tsirelson [Tsi04]. We can relate this to other, possibly more familiar,
definitions of noise. For example, if $B$ is a one-dimensional two-sided
Brownian motion, the $\sigma$-algebra $\mathcal{F}_{s,t}$ would be generated
by random variables of the form $B(u)-B(s)$ for $u\in[s,t]$, and
$\theta_{h}B(u):=B(u+h)-B(h)$. More generally a Gaussian space-time white
noise on $\mathbb{R}\times\mathbb{R}^{d}$, as discussed in Walsh’s book on
SPDEs [Wal86], associates a Gaussian random variable to every Borel set, and
defines a structure of a noise in a similar and straightforward fashion. These
are both examples of white noise.

We can now give a definition of black noise. There are several equivalent
definitions; Tsirelson [Tsi04] uses the notion of a stable $\sigma$-algebra to
define white noise and black noise. The definition we use in this paper is
more direct.

######  Definition 1.4.

Let
$(\Omega,(\mathcal{F}_{s,t})_{s<t},\mathbb{P},(\theta_{h})_{h\in\mathbb{R}})$
be a noise. A random variable $F:\Omega\to\mathbb{R}$ is called linear if

$\displaystyle\mathbb{E}\left[F|\mathcal{F}_{s,t}\right]+\mathbb{E}\left[F|\mathcal{F}_{t,u}\right]=\mathbb{E}\left[F|\mathcal{F}_{s,u}\right]\;\;\;\;\;\;\;\;\;\mathrm{for\;\;all\;\;\;\;}s<t<u.$

We say that
$(\Omega,(\mathcal{F}_{s,t})_{s<t},\mathbb{P},(\theta_{h})_{h\in\mathbb{R}})$
is a white noise if $\mathcal{F}$ is generated by linear random variables.
Likewise, we say that
$(\Omega,(\mathcal{F}_{s,t})_{s<t},\mathbb{P},(\theta_{h})_{h\in\mathbb{R}})$
is a black noise if $0$ is the only linear random variable.

In general, white noise consists of Gaussian and Poissonian components,
whereas black noise can be much more complex and admits no simple description.
White noise and black noise do not form a dichotomy: there are noises that
satisfy 1.3 which have both a “white noise part” and a “black noise part,” see
[Tsi04, Section 6].

####  1.2.2. The KPZ Universality Class and the Directed Landscape

In this section, we define the directed landscape and several other important
objects in the KPZ universality class. We begin by discussing the Airy process
and Airy line ensemble.

######  Definition 1.5.

The Airy Line Ensemble is a collection of random functions
$\left\\{\mathcal{A}_{i}\right\\}_{i\in\mathbb{N}}$ with law determined by the
Airy kernel (1.2). For any finite set $I\subset\mathbb{R}$, we define the
point process $\\{(x,\mathcal{A}_{i}(x))|x\in I\\}$ on $I\times\mathbb{R}$ as
the determinantal point process [HKPV09, Section 4] with kernel
$K(x_{1},t_{1};x_{2},t_{2})$ given by the extended Airy kernel

$\displaystyle K(x_{1},t_{1};x_{2},t_{2})$
$\displaystyle:=\begin{cases}\int_{0}^{\infty}e^{-r(t_{1}-t_{2})}\mathrm{Ai}(x_{1}+r)\mathrm{Ai}(x_{2}+r)dr&t_{1}\geq
t_{2}\\\
-\int_{-\infty}^{0}e^{-r(t_{1}-t_{2})}\mathrm{Ai}(x_{1}+r)\mathrm{Ai}(x_{2}+r)dr&t_{1}<t_{2}\end{cases}$
(1.2)

In this formula, $\mathrm{Ai}(\cdot)$ is the Airy function,

$\displaystyle\textrm{Ai}(x)$
$\displaystyle=\frac{1}{\pi}\lim_{b\to\infty}\int_{0}^{b}\cos{\left(\frac{t^{3}}{3}+xt\right)}dt$

The parabolic Airy line ensemble is defined by
$\mathcal{P}_{i}(x)=\mathcal{A}_{i}(x)-x^{2}$. We commonly refer to the top
curve $\mathcal{P}_{1}(\cdot)$ as the parabolic Airy2 process.

The parabolic Airy line ensemble can also be defined as the unique line
ensemble with a Brownian Gibbs resampling property and with the parabolic
Airy2 process as its top line. Existence of such a line ensemble was shown by
Corwin and Hammond [CH14] and uniqueness was shown by Dimitrov and Matetski
[DM21].

For a metric space $X$, we let $C(X)$ denote the space of continuous functions
$X\to\mathbb{R}$, equipped with the topology of uniform convergence on
compacts. For the choices of $X$ used in this paper, the space $C(X)$ is
always separable and metrizable, hence Polish. We can now introduce the Airy
sheet, a two-variable random function that can be viewed as the universal
limit of the last passage times for pairs of points of models in the KPZ
class.

######  Definition 1.6.

The Airy sheet is a random continuous function
$\mathcal{S}:\mathbb{R}^{2}\to\mathbb{R}$, i.e., a $C(\mathbb{R}^{2})$-valued
random variable, such that the following properties hold:

  1. (1)

$\mathcal{S}(\cdot,\cdot)$ has the same law as $\mathcal{S}(\cdot+t,\cdot+t)$
for all $t\in\mathbb{R}$, as $C(\mathbb{R}^{2})$-valued random variables.

  2. (2)

$\mathcal{S}$ can be coupled with a parabolic Airy line ensemble so that
$\mathcal{S}(0,\cdot)=\mathcal{P}_{1}(\cdot)$ and for all
$(x,y,z)\in\mathbb{R}^{+}\times\mathbb{R}^{2}$,

$\displaystyle\lim_{k\to\infty}\mathcal{P}\left[\left(-\sqrt{\frac{k}{2x}},k\right)\to\left(z,1\right)\right]-\mathcal{P}\left[\left(-\sqrt{\frac{k}{2x}},k\right)\to\left(y,1\right)\right]=\mathcal{S}(x,z)-\mathcal{S}(x,y),$

where $\mathcal{P}\left[\left(a,k\right)\to\left(b,1\right)\right]$ is the
last passage time between points $(a,k)$ and $(b,1)$ on the Airy line
ensemble,

$\displaystyle\mathcal{P}[(a,k)\to(b,1)]:=\sup_{a=t_{k}<...<t_{0}=b}\sum_{i=1}^{k}\left(\mathcal{P}_{i}(t_{i-1})-\mathcal{P}_{i}(t_{i})\right).$

Dauvergne, Ortmann, and Virág [DOV22] proved that the Airy sheet exists and
that it is unique in law, in fact, it was later shown to be a deterministic
functional of the Airy line ensemble [DV21]. With these definitions in hand,
we can discuss the directed landscape. As explained in [DV21, Example 1.6],
the directed landscape can be interpreted as a random directed spacetime
metric of negative sign. We can think of the directed landscape as a universal
limit of the metric defined by last passage times on a last passage model. We
begin by defining the canonical probability space for the directed landscape.

######  Definition 1.7.

We define the canonical probability space of the directed landscape
$(\Omega,\mathcal{F^{\mathcal{L}}})$. Let
$\mathbb{R}^{4}_{\uparrow}:=\\{(x,s;y,t)\in\mathbb{R}^{4}|s<t\\}$. Define the
underlying probability space $\Omega=C(\mathbb{R}^{4}_{\uparrow})$, equipped
with its Borel sets from the topology of uniform convergence on compact sets.
The canonical process on this space is the identity map
$\mathcal{L}:C(\mathbb{R}^{4}_{\uparrow})\to C(\mathbb{R}^{4}_{\uparrow})$.
The evaluation map at a point $(x,s,y,t)$ applied to $\mathcal{L}$ will be
denoted in this paper as $\mathcal{L}_{s,t}(x,y)$. In this notation, we can
define $\mathcal{F^{\mathcal{L}}}$ as the $\sigma$-algebra generated by

$\mathcal{F}^{\mathcal{L}}_{s,t}:=\sigma\left(\\{\mathcal{L}_{a,b}(x,y):s\leq
a<b\leq
t,(x,y)\in\mathbb{R}^{2}\\}\right),\;\;\;\;\;\;\;\;-\infty<s<t<\infty.$

######  Definition 1.8.

The directed landscape is a four-parameter field of random variables
$\mathcal{L}_{s,t}(x,y)$, indexed by $-\infty<s<t<\infty$ and
$x,y\in\mathbb{R}$, such that

  1. (1)

$\mathcal{L}_{s,t}$ and $\mathcal{L}_{t,u}$ are independent for $s<t<u$.

  2. (2)

$\mathcal{L}_{s,t}\circ\mathcal{L}_{t,u}=\mathcal{L}_{s,u}$ for all $s<t<u$
where the “metric composition” is defined for two functions
$f,g:\mathbb{R}^{2}\to\mathbb{R}$ as

$(f\circ g)(x,z)=\sup_{y\in\mathbb{R}}f(x,y)+g(y,z).$

  3. (3)

For all $s<t$, $\mathcal{L}_{s,t}$ has the same law as
$R_{t-s}\mathcal{L}_{0,1}$ where
$R_{\epsilon}f(x,y):=\epsilon^{1/3}f(\epsilon^{-2/3}x,\epsilon^{-2/3}y).$

  4. (4)

$\mathcal{L}_{0,1}$ is distributed as the Airy Sheet.

As an immediate consequence of this definition, we conclude that the directed
landscape defines a noise.

######  Lemma 1.9.

The directed landscape defines a noise
$(\Omega,\left(\mathcal{F}^{\mathcal{L}}_{s,t}\right)_{s<t},\mathbb{P},\left(\theta_{h}\right)_{h})$
in the sense of 1.3.

###### Proof.

Let $\Omega$ and $\mathcal{F}^{\mathcal{L}}_{s,t}$ be as in 1.7. Define the
measurable maps $\theta_{h}:\Omega\to\Omega$ by
$(\theta_{h}\mathcal{L})_{s,t}(x,y):=\mathcal{L}_{s+h,t+h}(x,y)$. Properties
$(1)$ and $(2)$ of 1.8 verify properties $(1)$ and $(2)$ of 1.3, respectively.
Properties $(4)$ and $(5)$ of 1.3 are satisfied directly by the definition of
the map $\theta_{h}$. Finally, we note that
$\mathcal{F}^{\mathcal{L}}=\bigvee_{-\infty<s<t<\infty}\mathcal{F}^{\mathcal{L}}_{s,t}$
by definition. ∎

We can now state our main theorem.

######  Theorem 1.10.

The noise associated to the directed landscape is a black noise in the sense
of 1.4.

There are a few equivalent conditions for demonstrating that a noise is a
black noise (we use the one stated in 2.3). These conditions usually involve a
variance bound which resembles a noise sensitivity condition with respect to
all functions in $L^{2}(\Omega)$. Noise sensitivity, which was studied in the
context of Boolean functions in foundational works such as [KKL88, BKS99], and
then for its applications to physics and computer science in works such as
[ST99, SS10, MOO10], is closely related to black noise. In fact, another way
to understand the concept of black noise is to consider it as a system with
respect to which all observables (functionals) are noise sensitive. It would
be interesting to explore these noise sensitivity properties further and in a
more precise fashion. Ganguly and Hammond [GH20], for instance, study the
asymptotic noise sensitivity of Brownian last passage, using the notion of the
overlap of perturbed geodesics to measure noise sensitivity.

While we do not study more specific noise sensitivity phenomena in this paper,
we do obtain a few corollaries of 1.10. The next result, 1.11, demonstrates
the independence of the directed landscape from any white noise defined on the
same probability space. For the purpose of this text, a Gaussian white noise
on an open subset $U\subset\mathbb{R}^{d}$ is a family of jointly Gaussian
random variables $\\{\xi(f)\\}_{f\in L^{2}(U)}$ all defined on the same
probability space, satisfying $\mathbb{E}[\xi(f)\xi(g)]=(f,g)_{L^{2}(U)}.$ We
often write $\xi(f):=\int_{U}f(x)\xi(dx)$ or similar. A Gaussian space-time
white noise is just a special case $U=\mathbb{R}^{2},$ where the first
coordinate is interpreted as “time” and the second as “space.”

######  Theorem 1.11.

Consider a noise
$(\Omega,(\mathcal{F}_{s,t})_{s<t},\mathbb{P},(\theta_{h})_{h})$. Let
$\mathcal{L}$ and $\xi$ respectively be a copy of a directed landscape and a
Gaussian space-time white noise, defined on this same probability space, which
together generate the entire $\sigma$-algebra
$\mathcal{F}:=\bigvee_{s<t}\mathcal{F}_{s,t}$, and which are both adapted to
the filtration $(\mathcal{F}_{s,t})_{s<t}$, in the sense that

  * •

$\int_{\mathbb{R}^{2}}\phi(s,y)\xi(ds,dy)$ is $\mathcal{F}_{u,t}$-measurable
if $\phi$ is supported on $[u,t]\times\mathbb{R}.$

  * •

$\mathcal{L}_{a,b}(x,y)$ is $\mathcal{F}_{s,t}$ measurable for all $s\leq
a<b\leq t$ and $x,y\in\mathbb{R}.$

Then $\mathcal{L}$ and $\xi$ are independent under $\mathbb{P}$, and all
linear random variables (1.4) are measurable with respect to $\xi$.

The directed landscape is not unique in this respect: the proof will show that
one could replace $\mathcal{L}$ by any space-time field that is known to be a
black noise. This theorem may be surprising, since it illustrates a contrast
with many prelimiting models in the KPZ class, in which case one often has
$\mathcal{L}_{N}=F_{N}(\xi_{N})$ for some adapted path functionals $F_{N}$ of
the discrete driving noise $\xi_{N}$. As we take the limit $N\to\infty$, 1.11
shows that this dependence no longer holds. In fact, the two fields
$\mathcal{L}_{N}$ and $\xi_{N}$ become independent. The following corollary
illustrates this phenomenon in the context of the KPZ equation.

######  Corollary 1.12 (The environment decouples from the height process when
scaling to the landscape).

Let $\xi$ be a Gaussian space-time white noise on
$\mathbb{R}_{+}\times\mathbb{R}$, and let $H^{\varepsilon}(t,x)$ denote the
Hopf-Cole solution of the KPZ equation
$\partial_{t}H^{\varepsilon}=\frac{1}{4}(\partial_{x}H^{\varepsilon})^{2}+\frac{\varepsilon}{4}\partial_{x}^{2}H^{\varepsilon}+\varepsilon^{1/2}\xi+\frac{1}{12}$,
started from any deterministic sequence of upper semicontinuous initial data
$H^{\varepsilon}(0,x)$ satisfying $H^{\varepsilon}(0,x)\leq C(1+|x|)$ for some
$C>0$, and converging as $\varepsilon\to 0$ to some upper semicontinuous
profile $H_{0}(x)$ in the sense of local hypograph convergence [MQR21, Section
3.1]. Consider any joint limit point $(\mathfrak{h},\xi)$ as $\varepsilon\to
0$ of the pair $(H^{\varepsilon},\xi)$ in the topology of
$C((0,\infty)\times\mathbb{R})\times\mathcal{S}^{\prime}(\mathbb{R}^{2}).$
Then $\mathfrak{h}$ is distributed as the KPZ fixed point started from
$H_{0}(x)$, independent of the space-time white noise $\xi.$

Before proving the 1.12, we note that joint limit points do exist because each
coordinate is tight, which implies that the pair is tight. This theorem
characterizes any joint limit point uniquely, and therefore it is a limit
theorem for the pair $(H^{\varepsilon},\xi)$. The local hypograph convergence
for the sequence of initial data is a weaker assumption than uniform
convergence on compacts if all profiles are continuous.

###### Proof.

We define a four-parameter field $\mathscr{H}^{\varepsilon}_{s,t}(x,y)$ for
each $\varepsilon>0$ such that
$(t,y)\mapsto\mathscr{H}^{\varepsilon}_{s,t}(x,y)$ is the Hopf-Cole solution
of the same equation solved by $H^{\varepsilon}$, with the same realization of
the noise $\xi$, started at time $s$ from initial condition given by a narrow
wedge at position $x$ (meaning that the exponential is started from a Dirac
mass centered at location $x$). The main results of [QS20, Wu23] imply that
this four-parameter field $\mathscr{H}^{\varepsilon}$ converges in law (in the
topology of $C(\mathbb{R}^{4}_{\uparrow})$) to the directed landscape.

1.11 implies that any joint limit point of $(\mathscr{H}^{\varepsilon},\xi)$
must be of the form $(\mathcal{L},\xi)$ where $\mathcal{L}$ is a directed
landscape independent of $\xi$. Indeed, one can define the $\sigma$-algebras
$\mathcal{F}_{s,t}$ to be precisely the ones generated by random variables of
the two forms given by the two bullet points in 1.11. The noise property
needed to apply 1.11 holds in that limit simply because it holds trivially in
the prelimit and any limit point will still respect this property.

Now we consider any joint limit point $(\mathfrak{h},\mathcal{L},\xi)$ of the
triple $(H^{\varepsilon},\mathscr{H}^{\varepsilon},\xi)$. On one hand, we
already know that $\mathcal{L}$ and $\xi$ must be independent under this limit
point. On the other hand, the results of [NQR20, Wu23] imply that
$\mathfrak{h}(t,y)=\sup_{x\in\mathbb{R}}H_{0}(x)+\mathcal{L}_{0,t}(x,y).$
Therefore $\mathfrak{h}$ is $\mathcal{L}$-measurable, which implies that it is
independent of $\xi.$ Projecting onto the first and third marginals
$\mathfrak{h}$ and $\xi$ gives the result. ∎

While 1.12 was formulated purely in terms of the KPZ equation, the result of
1.11 implies something much stronger: in any system for which there a height
process which converges to the directed landscape, the underlying field of
environment variables necessarily decouples from the height process under that
limit. In the case of ASEP, the environment consists of the Poisson clocks
that result in executed jumps for the system. In the case of last passage
percolation or directed polymers, the underlying environment consists of the
independent, identically distributed weights through which the last passage
paths travel. 1.11 implies that for any system that converges to the directed
landscape, the environment and the height function asymptotically decouple as
they respectively scale to the white noise and to the landscape. Such
decoupling and sensitivity phenomena are not present in the intermediate-
disorder or weak-asymmetry regime.

This phenomenon illustrates a contrast between the intermediate-disorder
regime and the strong-disorder regime for $(1+1)$-dimensional systems in the
KPZ universality class. For example, in scaling limits where particle systems,
SPDEs, or directed polymers are known to converge to the KPZ equation [BG97,
AKQ14, DT16, DGP17, CST18, GJ14, HQ18, Yan23] as opposed to the KPZ fixed
point, it is straightforward to show that the field of environment variables
and the height process converge jointly in law to the driving noise of the
limiting KPZ equation. This is always the case in the presence of intermediate
disorder or weak asymmetry scaling of models in the KPZ university class.
There are a few intuitive reasons why the same phenomenon does not happen when
scaling to the landscape, i.e., in the strong disorder or fixed asymmetry
regime. At the level of last passage percolation or directed polymers, the
height process only sees the largest variables in the environment field,
whereas most of the smaller weights are ignored. The height process thus
ignores “most” of the environment under the strong disorder scaling, with only
a few special exceptions. In contrast, all of the noise variables contribute
to some extent for the intermediate disorder scaling, as individual large
contributions have a less dramatic effect on the overall height process.

We discuss another, perhaps more precise, conjecture about why the strong
disorder scaling is so different from the intermediate disorder regime with
regard to retaining the memory of the microscopic environment. As before, we
work through the lens of polymers (for the purpose of this discussion, they
may be discrete or continuous as the reader prefers). Polymers admit a chaos
expansion in terms of multilinear polynomials of the underlying environment
field. Distinct terms in this expansion are always orthogonal to one another
in $L^{2}$ of the probability space associated to the underlying environment.
Under intermediate disorder scaling, it is well understood from the work of
Alberts, Khanin, and Quastel [AKQ14] that every individual term of the
prelimiting chaos expansion contributes meaningfully to the scaling limit. In
fact, they prove that the KPZ equation limit appears by proving term-by-term
convergence of the polynomial chaos expansion of the polymer partition
function to that of the exponential of the KPZ equation. In contrast, in the
strong disorder regime, we conjecture that most of the $L^{2}$-mass of the
terms of the chaos expansion of the height process escapes into the tails of
the chaos expansion. And in fact, we conjecture that this “mass escape”
phenomenon will not just occur for the height process itself, but also for
every bounded measurable functional of the height process. The “mass escape”
phenomenon is another way to formulate noise sensitivity [Gar11, Proposition
2.2].

As another corollary of 1.11, we show that the directed landscape is not an
SPDE driven by space-time Gaussian white noise (in fact, any Gaussian noise
that is white in time would not be sufficient to generate the landscape). To
formalize the notion of being an SPDE, we note that any probabilistic strong
solution of an SPDE defines an adapted path functional on the probability
space of the white noise. We therefore show that no such adapted functional
can exist.

######  Corollary 1.13 (The directed landscape is not an SPDE).

Let $\xi$ be a Gaussian space-time white noise on $\mathbb{R}^{2},$ defined on
some probability space $(\Omega,\mathcal{F}^{\xi},\mathbb{P})$. For $s<t$ let
$\mathcal{F}_{s,t}^{\xi}$ denote the $\sigma$-algebra generated by the random
variables $\int_{\mathbb{R}^{2}}\phi(s,y)\xi(ds,dy)$ as $\phi$ ranges over all
smooth functions of support contained in $[s,t]\times\mathbb{R}.$ Then there
does not exist any random variable $\mathcal{L}$ taking values in
$C(\mathbb{R}^{4}_{\uparrow})$, defined on the same probability space
$\Omega$, satisfying the following properties:

  * •

$\mathcal{L}$ is distributed as the directed landscape.

  * •

$\mathcal{L}_{a,b}(x,y)$ is $\mathcal{F}^{\xi}_{s,t}$-measurable whenever
$s\leq a<b\leq t$ and $x,y\in\mathbb{R}.$

The proof is immediate from 1.11, as $\mathcal{L}$ cannot be simultaneously
independent of $\xi$ and adapted to its filtration: this would imply
$\mathcal{L}$ is independent of itself and thus deterministic.

Next, we discuss a few open questions which are related to the content of this
paper. The two-dimensional critical stochastic heat flow is a continuum model
recently constructed by [CSZ23]. It is a flow of random measures on
$\mathbb{R}^{2}$, written as $u_{s,t}(\vec{x},d\vec{y})$ indexed by
$-\infty<s<t<\infty$ and $\vec{x},\vec{y}\in\mathbb{R}^{2}$. We can think of
it as the random field obtained from the universal scaling limit of partition
functions of intermediate-disorder directed polymers in a space-time random
environment in spatial dimension $d=2$. This is another example of a random
field which is a noise in the sense of 1.3, and it exhibits certain properties
which strongly suggest that it will not be a white noise [CSZ23].

######  Conjecture 1.14.

The two-dimensional critical stochastic heat flow constructed in [CSZ23] is a
black noise.

By the remark after 1.11, if this conjecture is true, it would be possible to
prove decoupling theorems analogous to 1.12 for two-dimensional polymers. Our
next conjecture is about the directed landscape being a noise in its space
variables as well as its time variables. To formulate this precisely, one
needs to define the notion of being a two-dimensional noise. The following is
adapted from the definition of a $d$-dimensional noise stated in Tsirelson
[Tsi04], or in Ellis and Feldheim [EF16].

######  Definition 1.15.

A two-dimensional noise is a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ with a collection of sub-$\sigma$-algebras
$\mathcal{F}_{\vec{x},\vec{y}}\subset\mathcal{F}$ associated to all open two-
dimensional rectangles in $\mathbb{R}^{2}$, and a collection of measurable
maps $\theta_{\vec{h}}:\Omega\to\Omega$ indexed by $\vec{h}\in\mathbb{R}^{2}$,
such that the following properties hold.

  1. (1)

$\mathcal{F}_{\vec{x}_{1},\vec{y}_{1}}$ and
$\mathcal{F}_{\vec{x}_{2},\vec{y}_{2}}$ are independent under $\mathbb{P}$
when $R_{1}:=(x_{1}^{(1)},x_{1}^{(2)})\times(y_{1}^{(1)},y_{1}^{(2)})$ and
$R_{2}:=(x_{2}^{(1)},x_{2}^{(2)})\times(y_{2}^{(1)},y_{2}^{(2)})$ satisfy
$R_{1}\cap R_{2}=\emptyset$.

  2. (2)

$\mathcal{F}_{\vec{x}_{1},\vec{y}_{1}}\vee\mathcal{F}_{\vec{x}_{2},\vec{y}_{2}}=\mathcal{F}_{\vec{x}_{3},\vec{y}_{3}}$
whenever the rectangles $R_{1}$ and $R_{2}$ satisfy $R_{1}\cap
R_{2}=\emptyset$ and $\overline{R_{1}\cup R_{2}}=\overline{R_{3}}.$

  3. (3)

$\mathcal{F}$ is generated by the union of all of the
$\mathcal{F}_{\vec{x},\vec{y}}$.

  4. (4)

If $A\in\mathcal{F}_{\vec{x},\vec{y}}$, then
$\theta_{\vec{h}}(A)\in\mathcal{F}_{\vec{x}+(h_{1},h_{1}),\vec{y}+(h_{2},h_{2})}$
and $\mathbb{P}(\theta_{\vec{h}}(A))=\mathbb{P}(A)$.

  5. (5)

$\theta_{\vec{h}_{1}}\theta_{\vec{h}_{2}}=\theta_{\vec{h}_{1}+\vec{h}_{2}}$
for all $\vec{h}_{1},\vec{h}_{2}\in\mathbb{R}^{2}$, and
$\theta_{\vec{0}}=\text{Id}.$

We say this two-dimensional noise is a black noise when the only linear random
variable is $0$, i.e., any $F\in L^{2}(\Omega)$ satisfying
$\mathbb{E}[F|\mathcal{F}_{R_{1}\cup
R_{2}}]=\mathbb{E}[F|\mathcal{F}_{R_{1}}]+\mathbb{E}[F|\mathcal{F}_{R_{2}}]$
for all rectangles $R_{1},R_{2}$ as in Item (2), implies that $F=0$.

There are two well-known examples of two-dimensional black noises arising from
simple probabilistic systems. The first is the scaling limit of critical
planar percolation, which was conjectured to be black noise by Tsirelson
[Tsi04] and later proved by Schramm and Smirnov [SS11]. The second is the
Brownian web, which was proved to be a black noise by Ellis and Feldheim
[EF16]. In the context of this definition, we can state the following
conjecture, due to Bálint Virág (in conversation). Define the
$\sigma$-algebras $\mathcal{F}_{(s,t),(x,y)}$ to be the ones generated by the
random variables $\int_{a}^{b}d\mathcal{L}\circ\pi$ (see [DOV22, Eq. (5)]) as
one varies over all $s\leq a<b\leq t$ and all continuous deterministic paths
$\pi:[a,b]\to[x,y].$ Then one has the following conjecture.

######  Conjecture 1.16.

The directed landscape is a two-dimensional black noise.

The difficulty of the two-dimensional problem is showing that the noise
property holds in the first place, in particular Item (2) in 1.15. If true,
this would address a problem posed by Ellis and Feldheim [EF16]. Their work
demonstrates that the Brownian web is an example of a two-dimensional black
noise, and they ask for further relevant examples of two-dimensional black
noise arising from simple probabilistic systems.

There are several key properties of the directed landscape that are crucial to
the proof that it is a black noise, and which distinguish it from many
prelimiting models in the KPZ universality class that are inherently white
noise.

  1. (1)

Scale invariance: Obtaining a rescaled version of the same space-time process
at every temporal scale is a crucial ingredient to many parts of the proof.
For example, in 3.3 we get a spatial mixing rate for the directed landscape of
order $e^{-dk^{3}}$ on a time interval $[0,1]$. This mixing rate immediately
translates to a mixing rate of $e^{-d\epsilon^{-2}k^{3}}$ for on a smaller
time interval $[0,\epsilon]$.

  2. (2)

Zero Temperature: Many of our estimates would be weakened or fail for positive
temperature analogues like the KPZ equation, as the maximum would get replaced
by some convolution operation which is not as local in some sense.

  3. (3)

Local Brownianity: Even for many zero-temperature conjectural prelimits (such
as inviscid stochastic Burgers equations driven by white noise) the local
Brownianity of the spatial process fails. Improved spatial regularity of the
solution actually hurts many of the estimates one needs to prove the black
noise property, as the better regularity “widens” the set of $x$-values where
the spatial height process is close to the argmax. Controlling the size of the
latter set of $x$-values is really the heart of the proof, as quantified in
6.4.

###  1.3. Notation

Throughout the paper, we use the font $\mathfrak{B},\mathfrak{C}$ as a way to
denote random constants. The letters $C,d$ are used, respectively, to denote
deterministic constants that may get larger or smaller from line to line when
performing an estimate. Typically, such $C,d$ will be (respectively) a
multiplying constant and an exponential decay rate of some quantity. For a
metric space $X$ we will let $C(X)$ denote the space of continuous functions
$X\to\mathbb{R}$, equipped with the topology of uniform convergence on
compacts. We often deal with random variables defined on the underlying
probability space $C(\mathbb{R})$ or $C(\mathbb{R}^{2})$ or
$C(\mathbb{R}^{4}_{\uparrow})$, and these random variables should be
understood as being real-valued Borel-measurable functions with respect to the
topology of uniform convergence on compact sets (which is always separable and
metrizable, hence Polish).

###  1.4. Structure of the Paper

In Section 2, we reduce the problem of demonstrating that the directed
landscape is a black noise to demonstrating a specific variance estimate about
the Airy sheet (2.6). Section 3 through Section 6 are dedicated to obtaining
this estimate, which relies on a mixing property of the directed landscape
(Section 3), as well as estimates on argmaxes of Airy sheets (Section 4) and
estimates on Bessel processes (Section 5). Then in Section 6, we conclude that
the directed landscape is a black noise. Finally in Section 7, we prove 1.11
from 1.10.

###  1.5. Acknowledgements

SP thanks Duncan Dauvergne, Jeremy Quastel, and Bálint Virág for discussions
and context about this problem during a visit to Toronto. SP and ZH thank Ivan
Corwin and Yu Gu for comments on an earlier draft of the paper. ZH was
supported by the Fernholz Foundation’s Minerva Fellowship Program and Ivan
Corwin’s grant, NSF DMS-1811143. ZH thanks E. Himwich and W. Himwich for
discussions about black noise.

##  2\. Reducing the black noise problem to an estimate on the Airy sheet

In this section, we reduce the problem of showing that the directed landscape
is a black noise to a specific variance bound for the Airy sheet.

######  Lemma 2.1.

Let $(\Omega,\mathcal{G},\mathbb{P})$ be a standard probability space, and
suppose that $\mathcal{G}$ is generated by some countably infinite collection
of random variables $\\{X_{n}\\}_{n=1}^{\infty}$ taking values in some Polish
space $\mathcal{X}$. Consider any bounded measurable $F:\Omega\to\mathbb{R}$.
Then for each $m\in\mathbb{N}$ there exists a Borel measurable map
$\Psi_{m}:\mathcal{X}^{m}\to\mathbb{R}$ such that
$\sup_{x_{1},...,x_{m}\in\mathcal{X}}|\Psi_{m}(x_{1},...,x_{m})|\leq\|F\|_{L^{\infty}(\Omega)}$,
and moreover the random variables $F_{m}:=\Psi_{m}(X_{1},...,X_{m})$ satisfy
$F=\lim_{m\to\infty}F_{m}$ $\mathbb{P}$-almost surely as well as
$\lim_{m\to\infty}\|F_{m}-F\|_{L^{p}(\Omega)}=0$ for $1\leq p<\infty$.

###### Proof.

We define $\mathcal{G}_{m}:=\sigma(X_{1},...,X_{m})$ and
$F_{m}:=\mathbb{E}[F|\mathcal{G}_{m}].$ Using approximation by
$\mathcal{G}_{m}$-measurable simple functions, we see that $F_{m}$ are of the
form $\Psi_{m}(X_{1},...,X_{m})$ for some Borel measurable
$\Psi_{m}:\mathcal{X}^{m}\to\mathbb{R}$. If needed, we replace $\Psi_{m}$ by
$(-\|F\|_{L^{\infty}(\Omega)})\vee\Psi_{m}\wedge\|F\|_{L^{\infty}(\Omega)}$,
and this still satisfies the same relation $F_{m}=\Psi_{m}(X_{1},...,X_{m})$
on a set of measure $1$. By Doob’s martingale convergence theorem, $F_{m}\to
F$ almost surely and in every $L^{p}(\Omega)$. ∎

In his survey, Tsirelson [Tsi04] gives a few examples in which it is possible
to prove something is a black noise, and provides a necessary and sufficient
condition for a given noise to be black. We quote the following result from
that survey, see [Tsi04, Proposition 7a(3) Item (b)].

######  Proposition 2.2.

Let $(\Omega,(\mathcal{F}_{s,t})_{s<t},\mathbb{P},\theta_{h})$ be a noise in
the sense of 1.3. The space $M$ of $\mathcal{F}_{0,1}$-measurable linear
random variables (1.4) is a closed linear subspace of $L^{2}(\Omega)$, and
moreover the orthogonal projection of any $F\in L^{2}(\Omega)$ onto $M$ is
given by

$\lim_{n\to\infty}\sum_{k=1}^{2^{n}}\big{(}\mathbb{E}\left[F\big{|}\mathcal{F}_{(k-1)2^{-n},k2^{-n}}\right]-\mathbb{E}[F]\big{)},$

where the limit is understood in $L^{2}(\Omega).$

######  Corollary 2.3.

A noise $(\Omega,(\mathcal{F}_{s,t})_{s<t},\mathbb{P},\theta_{h})$ is a black
noise if and only if the following condition holds on a dense linear subspace
$D\subset L^{2}(\Omega)$:

$\sum_{k=1}^{2^{n}}\big{(}\mathbb{E}\left[F\big{|}\mathcal{F}_{(k-1)2^{-n},k2^{-n}}\right]-\mathbb{E}[F]\big{)}\stackrel{{\scriptstyle
L^{2}(\Omega)}}{{\to}}0,\;\;\;\;\;\;\;\;\mathrm{for\;\;all\;\;\;}F\in D,$

or equivalently,

$\displaystyle\lim_{n\to\infty}\sum_{k=1}^{2^{n}}\mathrm{Var}\left(\mathbb{E}\left[F\big{|}\mathcal{F}_{(k-1)2^{-n},k2^{-n}}\right]\right)=0,\;\;\;\;\;\;\;\;\mathrm{for\;\;all\;\;\;}F\in
D.$ (2.1)

We recall the definition of the Directed Landscape and the associated noise
(1.8) and use $\Omega$ to denote the underlying probability space of
$\mathcal{L}$. There is a particular class of functions that are most useful
to us in the context of the directed landscape. We define $D\subset
L^{2}(\Omega)$ to be those functionals of the landscape that are finite linear
combinations of “polynomial functions” of the form

$F=\prod_{j=1}^{m}\prod_{i=1}^{n_{j}}\mathcal{L}_{s_{j},t_{j}}(x_{ij},y_{ij}),$
(2.2)

where $(s_{j},t_{j})$ are disjoint intervals of $\mathbb{R}$ whose endpoints
$s_{j},t_{j}$ are both dyadic numbers (of the form $k2^{-N}$ with
$k\in\mathbb{Z}$ and $N\in\mathbb{N}),$ furthermore
$x_{ij},y_{ij}\in\mathbb{R}$ are all distinct points in $\mathbb{R}$, and
$m,n_{j}\in\mathbb{N}$.

######  Lemma 2.4.

The set $D$ is a dense linear subspace of $L^{2}(\Omega)$ with $\Omega$ as in
1.7.

###### Proof.

The space $L^{2}(\gamma_{1})\otimes\cdots\otimes L^{2}(\gamma_{m})$ is
isometrically isomorphic as a Hilbert space to
$L^{2}(\gamma_{1}\otimes\cdots\otimes\gamma_{m})$ for any Borel probability
measures $\gamma_{i}$ on a Polish space, under the canonical isomorphism
$(F_{1}\otimes\cdots\otimes F_{m})(x_{1},...,x_{m})\cong F_{1}(x_{1})\cdots
F_{m}(x_{m})$. Therefore, in order to show that functions of the form (2.2)
are dense in $L^{2}(\Omega)$, it suffices to show that any function of the
form
$\prod_{j=1}^{m}\varphi_{j}(\mathcal{L}_{s_{j},t_{j}}(x_{1},y_{1}),...,\mathcal{L}_{s_{j},t_{j}}(x_{n},y_{n}))$
is in the $L^{2}$-closure of $D$, where $m,n\in\mathbb{N}$ and $\varphi_{j}$
are bounded measurable functions on $\mathbb{R}^{n}$. Indeed, functionals of
the latter form are dense in $L^{2}(\Omega)$ by the following argument: by 2.1
one can take $n\uparrow\infty$ to approximate any functional of the form
$\prod_{j=1}^{m}\Phi_{j}(\mathcal{L}_{s_{j},t_{j}})$ where $m\in\mathbb{N}$
and $\Phi_{j}:C(\mathbb{R}^{2})\to\mathbb{R}$ are bounded measurable, and
$\mathcal{L}_{s,t}:\mathbb{R}^{2}\to\mathbb{R}$ denotes the random function
given by $(x,y)\mapsto\mathcal{L}_{s,t}(x,y)$. Then in turn by taking linear
combinations and using the tensorization property of $L^{2}$ one can
approximate any functional of the form
$\Psi(\mathcal{L}_{s_{1},t_{1}},...,\mathcal{L}_{s_{m},t_{m}})$ where
$m\in\mathbb{N}$ and $\Psi:C(\mathbb{R}^{2})^{m}\to\mathbb{R}$ is bounded
measurable, and finally by 2.1 one can take $m\uparrow\infty$ and the latter
types of functionals can then approximate any general functional
$F(\mathcal{L}).$

We therefore focus on showing that any function of the form
$\prod_{j=1}^{m}\varphi_{j}(\mathcal{L}_{s_{j},t_{j}}(x_{1},y_{1}),...,\mathcal{L}_{s_{j},t_{j}}(x_{n},y_{n}))$
is in the $L^{2}(\Omega)$-closure of $D$, where $m,n\in\mathbb{N}$ and
$\varphi_{j}$ are bounded measurable functions on $\mathbb{R}^{n}$. By
independence of the different terms in the product, it suffices to prove that,
for $m=1$, such functions may be approximated in $L^{2}(\Omega)$ by functions
of the form (2.2) with $m=1$. In turn, it suffices to show that polynomial
functions on $\mathbb{R}^{n}$ are dense in $L^{2}(\mu)$ where $\mu$ is the
probability measure on $\mathbb{R}^{n}$ given by the joint law of
$(\mathcal{L}_{s,t}(x_{1},y_{1}),...,\mathcal{L}_{s,t}(x_{n},y_{n}))$ for some
fixed interval $(s,t)$ and fixed $\vec{x},\vec{y}\in\mathbb{R}^{n}$. Since the
marginals here are deterministically shifted Tracy-Widom GUE laws which have a
super-exponential tail decay, we conclude that $\mu$ must have a finite moment
generating function everywhere. Therefore, if $f\in L^{2}(\mu)$ was orthogonal
to all polynomials, then by taking the Taylor expansion of the exponential and
interchanging the infinite sum with the integral over $\mathbb{R}^{n}$, we
would have that
$\int_{\mathbb{R}^{n}}f(x)e^{i\sum_{1}^{n}\lambda_{j}x_{j}}\mu(dx)=0$ for all
$\lambda=(\lambda_{1},...,\lambda_{n})\in\mathbb{R}^{n}$. This interchange can
be justified by Fubini, using the super-exponential tail decay on $\mu$. By
inverting the Fourier transform (which is a linear isomorphism on
$\mathcal{S}^{\prime}(\mathbb{R}^{n}))$, this implies that the measure
$f\;d\mu\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ must be the zero measure,
i.e., $f=0$ $\mu$-almost everywhere. ∎

Due to the independence of the $\sigma$-algebras associated to disjoint
intervals (property $(1)$ in 1.8), we can ignore the product over $j$ in
(2.2), and thus without loss of generality, we may assume that $m=1$ in (2.2).
We label the single interval by $(s_{1},t_{1})=(s,t)$. By shifting and
rescaling the landscape, we can assume without loss of generality that
$(s,t)=(0,1).$ Therefore, we have reduced the black noise problem to
demonstrating that

$\displaystyle\lim_{n\to\infty}\sum_{k=1}^{2^{n}}\mathrm{Var}\left(\mathbb{E}\left[\prod_{i=1}^{n}\mathcal{L}_{0,1}(x_{j},y_{j})\bigg{|}\mathcal{F}^{\mathcal{L}}_{(k-1)2^{-n},k2^{-n}}\right]\right)=0,$
$\displaystyle\text{for all }\vec{x},\vec{y}\in\mathbb{R}^{n},\text{ and
}n\in\mathbb{N}.$ (2.3)

The entire problem is now formulated in terms of the Airy sheet
$\mathcal{S}:=\mathcal{L}_{0,1}$. In fact, we can simplify things further.

######  Proposition 2.5.

In order to show that the directed landscape is a black noise, it suffices to
show that for any $\eta\in(0,1/2)$ and $n\in\mathbb{N}$ and
$\vec{x},\vec{y}\in\mathbb{R}^{n}$,

$\lim_{n\to\infty}\sum_{\eta 2^{n}\leq
k\leq(1-\eta)2^{n}}\mathrm{Var}\left(\mathbb{E}\left[\prod_{i=1}^{n}\mathcal{L}_{0,1}(x_{j},y_{j})\bigg{|}\mathcal{F}^{\mathcal{L}}_{(k-1)2^{-n},k2^{-n}}\right]\right)=0,$
(2.4)

###### Proof.

Let $F:=\prod_{i=1}^{n}\mathcal{L}_{0,1}(x_{j},y_{j}),$ and let $G$ denote the
linear random variable obtained by applying the orthogonal projection of $F$
onto the closed linear subspace of $L^{2}(\Omega)$ consisting of linear
$\mathcal{F}^{\mathcal{L}}_{0,1}$ measurable random variables.

By applying 2.2, we see from (2.4) that $G$ must be measurable with respect to
$\mathcal{F}^{\mathcal{L}}_{0,\eta}\vee\mathcal{F}^{\mathcal{L}}_{1-\eta,1}$,
in other words, all dependency on $\mathcal{F}^{\mathcal{L}}_{\eta,1-\eta}$
vanishes. This is true for all $\eta\in(0,1/2)$. Therefore, $G$ is measurable
with respect to
$\mathcal{H}:=\bigcap_{\eta\in(0,1/2)}\mathcal{F}^{\mathcal{L}}_{0,\eta}\vee\mathcal{F}^{\mathcal{L}}_{1-\eta,1}.$
We claim that $\mathcal{H}$ is a trivial $\sigma$-algebra, only containing
sets of measure $0$ or $1$. It is clear that $\mathcal{H}$ is contained in
$\mathcal{F}^{\mathcal{L}}_{0,1}.$ However, $\mathcal{H}$ is also independent
of the $\sigma$-algebra generated by
$\bigcup_{\eta\in(0,1/2)}\mathcal{F}^{\mathcal{L}}_{\eta,1-\eta}.$ We claim
that the $\sigma$-algebra generated by
$\bigcup_{\eta\in(0,1/2)}\mathcal{F}^{\mathcal{L}}_{\eta,1-\eta}$ is all of
$\mathcal{F}^{\mathcal{L}}_{0,1}.$ To see this, note by the continuity of the
directed landscape in all parameters that e.g. the random variable
$\mathcal{L}_{0,1}(x,y)$ can be written as the almost sure limit as $\eta\to
0$ of $\mathcal{L}_{\eta,1-\eta}(x,y).$ Thus
$\mathcal{H}\subset\mathcal{F}^{\mathcal{L}}_{0,1}$ is independent of
$\mathcal{F}^{\mathcal{L}}_{0,1}$, proving that it is a trivial
$\sigma$-algebra.

Since the random variable $G$ is $\mathcal{H}$-measurable, we conclude that it
is almost surely constant. But $G$ is an $L^{2}$-limit of the sequence of
random variables as in 2.2, and therefore (2.3) holds. ∎

The following corollary is an immediate consequence.

######  Corollary 2.6.

Fix some $\varrho>0$. In order to show that the directed landscape is a black
noise, it suffices to show that for any $\eta\in(0,1/2)$, $n\in\mathbb{N}$ and
$\vec{x},\vec{y}\in\mathbb{R}^{n}$, there exists some
$C=C(n,\eta,\vec{x},\vec{y})>0$ such that for all $a,b\in[\eta,1-\eta]$,

$\mathrm{Var}\left(\mathbb{E}\left[\prod_{i=1}^{n}\mathcal{L}_{0,1}(x_{j},y_{j})\bigg{|}\mathcal{F}^{\mathcal{L}}_{a,b}\right]\right)\leq
C(b-a)^{1+\varrho},$ (2.5)

The rest of the paper will focus on proving (2.5), for a value of $\varrho$
that is slightly smaller than $\frac{1}{15}$. This bound may not be sharp.

##  3\. A strong mixing property for the landscape under spatial shifts

In this section, we prove a strong mixing property for the directed landscape
under spatial shifts (3.5) that is crucial to the arguments in Section 4.
First, we state a theorem from [DOV22] that is used in the proof.

######  Theorem 3.1 (Theorem 1.7 [DOV22]).

Fix $u=(x,t;y,s)\in\mathbb{R}^{4}_{\uparrow}$. There is almost surely a unique
directed geodesic $\Pi_{u}$ from $(x,s)$ to $(y,t)$, i.e., a path
$\pi:[s,t]\to\mathbb{R}$ such that $\pi(s)=x,\pi(t)=y$ and for all
$k\in\mathbb{N}$ and all $s=s_{0}<...<s_{k}=t$,

$\mathcal{L}_{s,t}(x,y)=\sum_{j=1}^{k}\mathcal{L}_{t_{k-1},t_{k}}(\pi(t_{k-1}),\pi(t_{k})).$

Its distribution only depends on $u$ through scaling: as random continuous
functions from $[0,1]\to\mathbb{R}$,

$\displaystyle\Pi_{(x,t;y,s)}(s+(t-s)r)\overset{d}{=}\Pi_{(0,0;0,1)}(r)+x+(y-x)r.$

Moreover, for $u=(0,0;0,1)$, there is a random constant $\mathfrak{C}$ such
that, for all $s>0$ with $t,t+s\in[0,1]$,

$\displaystyle\big{|}\Pi_{u}(t+s)-\Pi_{u}(t)\big{|}\leq\mathfrak{C}s^{\frac{2}{3}}\log^{\frac{1}{3}}{\left(\frac{2}{s}\right)}.$

The random constant satisfies
$\mathbb{E}\big{[}a^{\mathfrak{C}^{3}}\big{]}<\infty$ for some $a>1$.

With 3.1 in hand, we prove that there exists a probability space on which we
can couple three copies of the directed landscape in such a way that two of
them are independent, and the third agrees with the other two sufficiently far
from the origin.

######  Lemma 3.2.

There exist three copies of the directed landscape
$\mathcal{L}^{i}=(\mathcal{L}^{i}_{s,t})_{0\leq s<t\leq 1}$ with $i=0,1,2$,
all coupled onto the same probability space
$(\Omega,\mathcal{F},\mathbb{P}_{\mathrm{triple}})$ so that:

  1. (1)

$\mathcal{L}^{1},\mathcal{L}^{2}$ are independent under
$\mathbb{P}_{\mathrm{triple}}$.

  2. (2)

There exists a positive random variable $X:\Omega\to(0,\infty)$ such that
$\mathcal{L}_{s,t}^{0}(x,y)=\mathcal{L}_{s,t}^{1}(x,y)$ for all $x,y>X$ and
$0\leq s<t\leq 1$, and furthermore
$\mathcal{L}_{s,t}^{0}(x,y)=\mathcal{L}_{s,t}^{2}(x,y)$ for all $x,y<-X$ and
$0\leq s<t\leq 1$.

  3. (3)

$\mathbb{P}_{\mathrm{triple}}(X>u)\leq Ce^{-du^{3}}$ where $C,d>0$ do not
depend on $u>0$.

The proof is inspired by [Dau24, Proposition 2.6], which proves a coupling
result for copies of the directed landscape via approximation by a last
passage model.

###### Proof of 3.2.

We consider two fields of independent, identically distributed exponential
random variables $\omega^{i}=\\{\omega^{i}_{t,x}\\}_{t\geq 0,x\in\mathbb{Z}}$
for $i=1,2$, independent of one another. We define a third field of
independent, identically distributed variables $\omega^{0}_{t,x}$ by
$\omega^{0}_{t,x}:=\omega^{1}_{t,x}$ if $x\geq 0$ and
$\omega^{0}_{t,x}:=\omega^{2}_{t,x}$ if $x<0$.

For $i=0,1,2$ and $x,y\in\mathbb{Z}$ and $0\leq m<n\in\mathbb{Z}$, we consider
the fields of last passage times

$L_{m,n}^{i}(x,y):=\max\\{W^{i}(\pi):\pi:\\{m,...,n\\}\to\mathbb{Z};\pi(m)=x,\pi(n)=y\\}$

where $\pi$ denotes a nearest neighbor path $\pi:\\{m,...,n\\}\to\mathbb{Z}$
and its length is defined by

$W^{i}(\pi):=\sum_{i=m}^{n}\omega^{i}_{i,\pi(i)}.$

For $x,y\in\varepsilon^{2/3}\mathbb{Z}$ and $s,t\in\varepsilon\mathbb{Z}$, we
then define a rescaled last passage time

$\mathcal{L}^{i,\varepsilon}_{s,t}(x,y):=\varepsilon^{1/3}\big{(}L_{\varepsilon^{-1}s,\varepsilon^{-1}t}^{i}(\varepsilon^{-2/3}x,\varepsilon^{-2/3}y)-\varepsilon^{-1}(t-s)\big{)}.$

[DV21, Theorem 1.20] shows that the marginal laws of each
$\mathcal{L}^{i,\varepsilon}_{s,t}(x,y)$ individually converge in law as
$\varepsilon\to 0$ to the directed landscape in the uniform-on-compact
topology. Here, the compact set can be taken to be any compact subset of
$\\{(s,t,x,y):0\leq s<t\leq 1,(x,y)\in\mathbb{R}^{2}\\},$ see 1.7.

Furthermore, if $\pi_{\mathrm{opt}}^{i,m,n,x,y}$ denotes the maximizing path,
satisfying $L^{i}_{m,n}(x,y)=W^{i}(\pi_{\mathrm{opt}}^{i,m,n,x,y})$, then
[DV21, Theorem 8.7] implies that the paths

$\Pi^{i,s,t,x,y;\varepsilon}(r):=\varepsilon^{2/3}\pi_{\mathrm{opt}}^{i,\varepsilon^{-1}s,\varepsilon^{-1}t,\varepsilon^{-2/3}x,\varepsilon^{-2/3}y}(\varepsilon^{-1}r),$

converge in law as $\varepsilon\to 0$, jointly in all variables $(s,t,x,y,r)$,
with respect to the uniform-on-compact topology on $\\{(s,t,x,y,r):0\leq
s<t\leq 1,s\leq r\leq t,(x,y)\in\mathbb{Z}^{2}\\}$, to geodesic paths
$\Pi^{i,s,t,x,y}(r)$ of the corresponding directed landscapes
$\mathcal{L}^{i}$, for $i=0,1,2$. This convergence occurs jointly with the
height profiles $\mathcal{L}^{i,\varepsilon}_{s,t}(x,y)$.

Now let $(\mathcal{L}^{0},\mathcal{L}^{1},\mathcal{L}^{2})$ be any joint limit
point of
$(\mathcal{L}^{0,\varepsilon},\mathcal{L}^{1,\varepsilon},\mathcal{L}^{2,\varepsilon})$
as $\varepsilon\to 0.$ We define the random variable

$X:=\inf\\{x\in\mathbb{N}:\inf_{r\in[0,1]}\Pi^{0,0,1,x,x}(r)>0,\sup_{r\in[0,1]}\Pi^{0,0,1,-x,-x}(r)<0\\},$

which tracks the smallest positive integer value of $x$ for which both of the
directed geodesic paths from $x\to x$ and from $-x\to-x$ do not cross $0$ in
$\mathcal{L}^{0}$. We verify that $\mathcal{L}^{1},\mathcal{L}^{2}$ are
independent and that $\mathcal{L}_{s,t}^{0}(x,y)=\mathcal{L}_{s,t}^{1}(x,y)$
for all $x,y>X$ and $0\leq s<t\leq 1$ by noting that the analogous facts hold
trivially in the prelimit. Furthermore,
$\mathcal{L}_{s,t}^{0}(x,y)=\mathcal{L}_{s,t}^{2}(x,y)$ for all $x,y<-X$ and
$0\leq s<t\leq 1$, again because the analogous fact holds in the prelimit.

It remains to verify the tail bound in Item (3) of the lemma statement. The
moment bound of 3.1 implies that for any fixed $x\in\mathbb{R}$ the
probability of any geodesic path from $x\to x$ varying more than a distance
$u>0$ from its starting point decays like $e^{-du^{3}}$ for some $d>0$. This
directly implies Item $(3)$ through a union bound. ∎

We use 3.2 to prove a strong mixing property for the landscape under diagonal
shifts.

######  Proposition 3.3.

The directed landscape $\mathcal{L}=(\mathcal{L}_{s,t})_{0\leq s<t\leq 1}$ is
$\alpha$-mixing under spatial shifts, with mixing rate $\alpha(k)\leq
Ce^{-dk^{3}}$ for some universal constants $C,d>0$. More precisely, we have
the following stronger estimate. For $-\infty\leq a<b\leq\infty$ let
$\mathcal{G}_{a,b}$ denote the $\sigma$-algebra generated by the random
variables $\mathcal{L}_{s,t}(x,y)$ where $0\leq s<t\leq 1$ and $x,y\in(a,b)$.
Then

$|\operatorname{Cov}(F,G)|\leq
Ce^{-dk^{3}}\mathbb{E}[F^{4}]^{1/4}\mathbb{E}[G^{4}]^{1/4},$

where $C,d>0$ are independent of $k$ and $F,G\in L^{4}(\Omega)$ that are
respectively measurable with respect to $\mathcal{G}_{-\infty,0}$ and
$\mathcal{G}_{k,\infty}.$

The $\alpha$-mixing property is a very strong form of mixing that is uniform
over all observables (see the survey by Bradley [Bra05] for further
explanation). Roughly speaking, the result says that under translation of the
landscaped by $k$ spatial units, the decay of correlation is upper bounded by
$e^{-dk^{3}}$, where $d>0$ is bounded away from zero uniformly over all
observables and $k>0$. Moreover, the remarkably fast cubic-exponential rate of
decay means that we can treat the field as virtually independent on well-
separated scales.

###### Proof.

By translation invariance, it suffices to prove the claim with the assumption
“$F$ measurable with respect to $\mathcal{G}_{-\infty,0}$” replaced by the
statement “$F$ measurable with respect to $\mathcal{G}_{-\infty,-k}$” (this
replacement is just for notational convenience). Therefore, we will consider
$F,G$ that are respectively measurable with respect to
$\mathcal{G}_{-\infty,-k}$ and $\mathcal{G}_{k,\infty}.$

We assume that $\mathcal{L}^{0},\mathcal{L}^{1},\mathcal{L}^{2}$ are the same
couplings and $X$ is the same random variable as in 3.2. We will use
$\mathbb{E}_{\mathrm{triple}}[\cdot]$ to denote the expectation with respect
to the measure on the coupled space from 3.2. Consider the measurable
functions $F,G$ as above and write
$F_{i}:=F(\mathcal{L}^{i}),G_{i}:=G(\mathcal{L}^{i})$ for $i=0,1,2$. Then by
independence,
$\mathbb{E}_{\text{triple}}[F_{2}G_{1}]=\mathbb{E}_{\text{triple}}[F_{2}]\mathbb{E}_{\text{triple}}[G_{1}],$
thus

$\displaystyle|\operatorname{Cov}(F,G)|$
$\displaystyle=|\mathbb{E}_{\text{triple}}[F_{0}G_{0}-F_{2}G_{1}]|$
$\displaystyle\leq|\mathbb{E}_{\text{triple}}[(F_{0}-F_{2}+F_{2})(G_{0}-G_{1}+G_{1})-F_{2}G_{1}]|$
$\displaystyle=|\mathbb{E}_{\text{triple}}[(F_{0}-F_{2})(G_{0}-G_{1})]+\mathbb{E}_{\text{triple}}[(F_{0}-F_{2})G_{1}]+\mathbb{E}_{\text{triple}}[F_{2}(G_{0}-G_{1})]|$
$\displaystyle\leq\mathbb{E}_{\text{triple}}[(F_{0}-F_{2})^{2}]^{1/2}\mathbb{E}_{\text{triple}}[(G_{0}-G_{1})^{2}]^{1/2}+\mathbb{E}_{\text{triple}}[(F_{0}-F_{2})^{2}]^{1/2}\mathbb{E}_{\text{triple}}[G_{1}^{2}]^{1/2}$
$\displaystyle\;\;\;\;\;\;+\mathbb{E}_{\text{triple}}[F_{2}^{2}]^{1/2}\mathbb{E}_{\text{triple}}[(G_{0}-G_{1})^{2}]^{1/2}.$

Now recall that $F,G$ are, respectively, measurable with respect to
$\mathcal{G}_{-\infty,-k}$ and $\mathcal{G}_{k,\infty}$. Therefore,
$F_{0}=F_{2}$ and $G_{0}=G_{1}$ on the event that $X\leq k$. By 3.2 we thus
have that

$\displaystyle\mathbb{E}_{\text{triple}}[(F_{0}-F_{2})^{2}]^{1/2}$
$\displaystyle=\mathbb{E}_{\text{triple}}[(F_{0}-F_{2})^{2}1_{\\{X>k\\}}]^{1/2}$
$\displaystyle\leq
2\mathbb{E}[F^{4}]^{1/4}\mathbb{P}_{\text{triple}}(X>k)^{1/2}\leq
2\mathbb{E}[F^{4}]^{1/4}Ce^{-\frac{1}{2}dk^{3}}.$

We used Cauchy-Schwarz and $(F_{0}-F_{2})^{2}\leq 2(F_{0}^{2}+F_{2}^{2})$ in
the first inequality. Likewise, one may prove that
$\mathbb{E}_{\text{triple}}[(G_{0}-G_{1})^{2}]^{1/2}\leq
2\mathbb{E}[G^{4}]^{1/4}Ce^{-\frac{1}{2}dk^{3}}$. On the other hand, it is
clear from Jensen that
$\mathbb{E}_{\text{triple}}[F_{2}^{2}]^{1/2}\leq\mathbb{E}_{\text{triple}}[F_{2}^{4}]^{1/4}=\mathbb{E}[F^{4}]^{1/4}$
and likewise
$\mathbb{E}_{\text{triple}}[G_{1}^{2}]^{1/2}\leq\mathbb{E}_{\text{triple}}[G_{1}^{4}]^{1/4}=\mathbb{E}[G^{4}]^{1/4}$.
∎

The choice of $L^{4}$ is not optimal in this proof, and applying Hölder’s
inequality rather than Cauchy-Schwarz could improve the assumptions to $F,G\in
L^{p}(\Omega)$ with a bound
$\mathbb{E}[|F|^{p}]^{1/p}\mathbb{E}[|G|^{p}]^{1/p}$ for any $p>2$. It is
natural to ask what the strongest or optimal form of mixing is for spatial
shifts in the landscape, such as $\beta$-mixing or $\phi$-mixing, but we will
not pursue this question. 3.2 and 3.3 provide a flavor of how to prove mixing
estimates, but we actually need a stronger version of these results.

######  Theorem 3.4.

Fix $T>0$, and consider any compact disjoint intervals
$I_{1},J_{1},...,I_{N},J_{N}\subset\mathbb{R}$. Define the $\sigma$-fields

$\mathcal{F}_{I}:=\sigma(\\{\mathcal{L}_{s,t}(x,y):0\leq s<t\leq T,x\text{ and
}y\text{ are in the same interval }I_{r}\text{ for some }r\leq N\\}$
$\mathcal{F}_{J}:=\sigma(\\{\mathcal{L}_{s,t}(x,y):0\leq s<t\leq T,x\text{ and
}y\text{ are in the same interval }J_{r}\text{ for some }r\leq N\\}.$

Define the positive real number

$D:=\inf\bigg{\\{}|x-y|:x\in\bigcup_{1}^{N}I_{r},\;\;y\in\bigcup_{1}^{N}J_{r}\bigg{\\}}.$

Then we have the bound

$|\operatorname{Cov}(F,G)|\leq
CNe^{-dT^{-2}D^{3}}\mathbb{E}[F^{4}]^{1/4}\mathbb{E}[G^{4}]^{1/4}.$

Here $C,d>0$ are universal constants that are independent of
$N,T,\\{I_{r},J_{r}\\}_{r=1}^{N},D$ and $F,G\in L^{4}(\Omega)$ that are
respectively measurable with respect to $\mathcal{F}_{I}$ and
$\mathcal{F}_{J}.$

We remark that not all $I_{r}$ need to be strictly to the left of all $J_{r}$
or vice versa: they may very well be mixed amongst each other in a nontrivial
way.

###### Proof.

Without loss of generality, as in Section 2, we may assume that $T=1$. By
scale invariance of the directed landscape 1.8 (4), all of the intervals could
be replaced by their $T^{-2/3}$-rescalings. This changes the value of $D$ to
$T^{-2/3}D$, which respects the estimate. In the case that $T=1$, the proof
uses an explicit coupling construction that is very similar to the proofs of
3.2 and 3.3. We now sketch the details of this coupling construction.

Without loss of generality, we assume that the intervals are ordered
$I_{1},J_{1},I_{2},J_{2},...,I_{N},J_{N}$ so that the supremum of $I_{1}$ is
less than or equal to the infimum of $J_{1}$, the supremum of $J_{1}$ is less
than the infimum of $I_{2}$, and so on. This does not lose any generality
because we can always combine consecutive instances of the $I_{r}$ or $J_{r}$
without changing the value of $D$.

We relabel $I_{1},J_{1},I_{2},J_{2},...,I_{N},J_{N}$ as $U_{1},...,U_{2N}$ and
define $a_{r}$ to be the midpoint of the segment in the complement of
$\bigcup_{i\leq 2N}U_{i}$ that is between $U_{r}$ and $U_{r+1}$. We define a
fattening of $I_{r}$ and $J_{r},$ by setting $\bar{I}_{r}:=(a_{2r-1},a_{2r})$
and $\bar{J}_{r}:=(a_{2r},a_{2r+1})$ for $r=1,...,N$. In this notation, we say
that $a_{2N+1}:=\infty$. We also define $b_{r},c_{r}$ to be those points in
the segment in the complement of $\bigcup_{i\leq 2N}U_{i}$ that is between
$U_{r}$ and $U_{r+1}$ that are (respectively) exactly one-third and two-thirds
of the way across this segment.

If $L^{1},L^{2}\in C(\mathbb{R}_{\uparrow}^{4})$ and $I\subset\mathbb{R}$,
then we will say that $L^{1},L^{2}$ agree on $I\times[0,T]$ if
$L^{1}_{a,b}(x,y)=L^{2}_{a,b}(x,y)$ for all $0\leq a<b\leq T$ and all $a,b\in
I.$ Just as we did in 3.2, we construct three copies
$\mathcal{L}^{0},\mathcal{L}^{1},\mathcal{L}^{2}$ of the directed landscape,
so that $\mathcal{L}^{1},\mathcal{L}^{2}$ are independent, and furthermore,
with high probability, $\mathcal{L}^{0}$ agrees with $\mathcal{L}^{1}$ on
$I_{r}\times[0,T]$ and $\mathcal{L}^{0}$ agrees with $\mathcal{L}^{2}$ on
$J_{r}\times[0,T]$ for all $1\leq r\leq N$. We have the following construction
based on the same argument as 3.2.

Consider fields of independent, identically distributed exponential random
variables $\omega^{i}=\\{\omega^{i}(t,x)\\}_{t\geq 0,x\in\mathbb{Z}}$ for
$i=1,2$, independent of one another. Define a third field of independent,
identically distributed variables
$\omega_{\varepsilon}^{0}=\\{\omega_{\varepsilon}^{0}(t,x)\\}_{t\geq
0,x\in\mathbb{Z}}$ which agrees with $\omega^{1}$ on $\mathbb{Z}_{\geq
0}\times\bigcup((\varepsilon^{-2/3}\bar{I}_{r})\cap\mathbb{Z})$ and agrees
with $\omega^{2}$ on $\mathbb{Z}_{\geq
0}\times\bigcup((\varepsilon^{-2/3}\bar{J}_{r})\cap\mathbb{Z})$. Just as we
did in 3.2, we take a joint limit point of the last passage times for all
three fields (together with all of the geodesic paths associated to the finite
collection of endpoints given by $b_{1},c_{1},...,b_{N},c_{N}$) as
$\varepsilon\to 0$. We will obtain three copies of the directed landscape
$\mathcal{L}^{0},\mathcal{L}^{1},\mathcal{L}^{2}$ coupled onto the same space
(together with the respective geodesics).

Figure 1.  A visual depiction of the “geodesic separation” argument in the
proof of 3.4. We view the horizontal axis as a spatial axis, and likewise view
the vertical axis as a temporal one. If all of the prelimiting geodesic paths
from $b_{r}$ to $b_{r}$ and from $c_{r}$ to $c_{r}$ do not intersect the
dotted lines or the endpoints of the intervals, then the non-crossing property
of the geodesics in the prelimiting model implies that all last passage values
between two space-time points inside the left shaded region will agree with
those determined by $\omega^{1}$, while all last passage values between two
space-time points inside the right shaded region will agree with those
determined by $\omega^{2}$.

Consider the event $E$ in which all of the $4N$ geodesic paths from the points
$b_{r}$ to $b_{r}$ and $c_{r}$ to $c_{r}$ in $\mathcal{L}^{0}$ do not cross
any of the values $a_{r}$, and also do not cross the endpoints of the
intervals $I_{r},J_{r}$ (which in particular implies that they do not
intersect). By the same argument as in 3.2, the values
$\mathcal{L}^{0}_{s,t}(x,y)$ will agree with either of $\mathcal{L}^{1}$ or
$\mathcal{L}^{2}$ as long as $x,y$ lie strictly in between any given
consecutive geodesics $\Pi_{r}$ and $\Pi_{r+1}$. See Fig. 1 for a geometric
depiction of this event.

We apply a union bound over all $4N$ of the geodesics, together with an
application of 3.1, to show that the event $E$ occurs with probability greater
than $1-C(4N)e^{-dD^{3}}$, where $C,d$ do not depend on $N,D$ or the choice of
intervals. Consequently, by repeating the exact argument as that in 3.3
(replacing the event $\\{X>k\\}$ appearing there with the event $E$ here) we
obtain the required bound. ∎

Specializing 3.4 to the case $T=1$ and projecting the result to the two-time
marginal $(s,t)=(0,1)$, we obtain the following corollary about strong mixing
of the Airy sheet under diagonal shifts.

######  Corollary 3.5.

Consider the Airy sheet $\mathcal{S}=\mathcal{L}_{0,1}$. Fix $T>0$, and
consider any compact disjoint intervals
$I_{1},...,I_{N},J_{1},...,J_{N}\subset\mathbb{R}^{2}$. Define the
$\sigma$-fields

$\displaystyle\mathcal{G}_{I}$
$\displaystyle:=\sigma(\\{\mathcal{S}(x,y):(x,y)\in I_{r}\times I_{r}\text{
for some }r\leq N\\}),$ $\displaystyle\mathcal{G}_{J}$
$\displaystyle:=\sigma(\\{\mathcal{S}(x,y):(x,y)\in J_{r}\times J_{r}\text{
for some }r\leq N\\}).$

Define

$D:=\inf\bigg{\\{}|x-y|:x\in\bigcup_{1}^{N}I_{r},\;\;y\in\bigcup_{1}^{N}J_{r}\bigg{\\}}.$

Then we have the bound

$|\operatorname{Cov}(F,G)|\leq
CNe^{-dD^{3}}\mathbb{E}[F^{4}]^{1/4}\mathbb{E}[G^{4}]^{1/4}.$

Here $C,d>0$ are universal constants that are independent of
$N,\\{I_{r},J_{r}\\}_{r=1}^{N},D$ and $F,G\in L^{4}(\Omega)$ that are
respectively measurable with respect to $\mathcal{G}_{I}$ and
$\mathcal{G}_{J}.$

##  4\. Useful estimates on Airy sheets and argmaxes

We begin this section by establishing some useful notation. For $f,g\in
C(\mathbb{R})$ and $L\in C(\mathbb{R}^{2})$ we define

$f\circ L\circ g:=\sup_{x,y\in\mathbb{R}}f(x)+L(x,y)+g(y).$

In this section, we prove a bound (4.8) on the covariance of
$f\circ\mathcal{L}_{0,\epsilon}\circ g$ and
$u\circ\mathcal{L}_{0,\epsilon}\circ v$ for deterministic functions $f,g,u,v$
all lying within a certain restricted domain. This estimate, and the
additional estimates on the covariance of products of functions of this nature
(4.9 and 4.10), are critical inputs to the proof of the estimate (2.5). Most
of this section is devoted to setting up bounds which are used to prove 4.8,
4.9, and 4.10. The following lemma from [DSV22] is useful in our arguments.

######  Lemma 4.1 (Lemma 5.3, [DSV22]).

There exists a deterministic $c>0$ and a random $\mathfrak{C}>0$ so that

$|\mathcal{S}(x,y)+(x-y)^{2}|\leq\mathfrak{C}+c\log^{2/3}(2+|x|+|y|).$

Furthermore $\mathbb{E}[a^{\mathfrak{C}^{3/2}}]<\infty$ for some $a>1.$

We use this lemma to prove a helpful property of the point $(x,y)$ which
maximizes $f\circ\mathcal{L}_{0,\epsilon}\circ g$.

######  Proposition 4.2.

Let $f,g:\mathbb{R}\to\mathbb{R}$ be deterministic continuous functions
satisfying $-2x^{2}-B\leq f(x),g(x)\leq-\frac{1}{2}x^{2}+B$ for some $B>0.$ We
define the random variable

$\displaystyle(X_{\epsilon},Y_{\epsilon}):=\mathrm{argmax}_{(x,y)\in\mathbb{R}^{2}}\;\;f(x)+\mathcal{L}_{0,\epsilon}(x,y)+g(y).$

We define $z_{\max}:=\mathrm{argmax}_{z\in\mathbb{R}}f(z)+g(z),$ and assume
that this point is unique. Then for $M^{2}>1+100B$,

$\mathbb{P}(X_{\epsilon}^{2}+Y_{\epsilon}^{2}>M^{2})\leq
Ce^{-d\epsilon^{-1/2}M^{3}}$

where $C,d>0$ are universal constants not depending on $f,g,M,\epsilon,B$.

###### Proof.

By the scale invariance of the directed landscape, we can replace
$\mathcal{L}_{0,\epsilon}(x,y)$ by the process

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

In the notation of 4.1 we have

$|\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)+\epsilon^{-1}(x-y)^{2}|\leq\mathfrak{C}\epsilon^{1/3}+c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|x|+|y|)),$

where $c$ is deterministic and $\mathfrak{C}$ is a non-negative random
variable satisfying $\mathbb{P}(\mathfrak{C}>u)\leq Ce^{-du^{3/2}}$ for $u\geq
0$. From now on, we use the notation
$\mathcal{S}^{\epsilon}(x,y):=\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)$.
In particular,

$\displaystyle\begin{split}(f+g)(z_{\max})-\mathfrak{C}\epsilon^{1/3}-c\epsilon^{1/3}\log^{2/3}(2+2\epsilon^{-2/3}|z_{\max}|)&\leq
f(z_{\max})+\mathcal{S}^{\epsilon}(z_{\max},z_{\max})+g(z_{\max})\\\ &\leq
f(X_{\epsilon})+\mathcal{S}^{\epsilon}(X_{\epsilon},Y_{\epsilon})+g(Y_{\epsilon})\end{split}$
(4.1)

By the assumptions of the proposition, on the event
$X_{\epsilon}^{2}+Y_{\epsilon}^{2}>M^{2}$ we have that

$f(X_{\epsilon})+g(Y_{\epsilon})\leq
2B-\frac{1}{2}(X_{\epsilon}^{2}+Y_{\epsilon}^{2})\leq 2B-\frac{1}{2}M^{2}.$

However, 4.1 also implies that

$\mathcal{S}^{\epsilon}(X_{\epsilon},Y_{\epsilon})\leq\mathfrak{C}\epsilon^{1/3}+c\epsilon^{1/3}\log^{2/3}(\epsilon^{-2/3}(2+|X_{\epsilon}|+|Y_{\epsilon}|)).$

From the conditions imposed on $f,g$ by the statement of the proposition, it
is clear that $z_{\max}\in[-2\sqrt{B},2\sqrt{B}]$. Combining all of this with
(4.1) we obtain the following bound on $\mathfrak{C}$:

$\displaystyle 2\mathfrak{C}\epsilon^{1/3}$
$\displaystyle\geq(f+g)(z_{\max})-2B+\frac{1}{2}M^{2}+c\epsilon^{1/3}\bigg{(}-\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))-\log^{2/3}(2+2\epsilon^{-2/3}|z_{\max}|)\bigg{)}$
$\displaystyle\geq-
z_{\max}^{2}-4B+\frac{1}{2}M^{2}+c\epsilon^{1/3}\bigg{(}-\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))-\log^{2/3}(2+2\epsilon^{-2/3}|z_{\max}|)\bigg{)}$
$\displaystyle\geq-8B+\frac{1}{2}M^{2}+c\epsilon^{1/3}\bigg{(}-\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))-\log^{2/3}(2+2\epsilon^{-2/3}B)\bigg{)}.$

The deterministic constant $c$ may grow from line to line. Therefore, for
sufficiently small $\epsilon>0$ (below some threshold that is deterministic
and independent of $B\geq 2$),
$c\epsilon^{1/3}\log^{2/3}(2+2\epsilon^{-2/3}B)\leq B$.

Under the additional assumption that
$X_{\epsilon}^{2}+Y_{\epsilon}^{2}\leq(2M)^{2}$, we find that
$c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))\leq\frac{1}{4}M^{2}$
for sufficiently small $\epsilon>0$ (below some threshold that is
deterministic and independent of $M\geq 2$). Therefore, the last expression
admits a lower bound by $-9B+\frac{1}{4}M^{2}.$ We have shown that

$\mathbb{P}(M^{2}\leq
X_{\epsilon}^{2}+Y_{\epsilon}^{2}\leq(2M)^{2})\leq\mathbb{P}(2\mathfrak{C}\epsilon^{1/3}\geq-9B+\frac{1}{4}M^{2}),$

for $\epsilon\leq\epsilon_{0}$ with $\epsilon_{0}$ deterministic and
independent of $B,M.$ Summing over dyadically sized concentric annuli, we
obtain

$\displaystyle\mathbb{P}(X_{\epsilon}^{2}+Y_{\epsilon}^{2}>M^{2})$
$\displaystyle=\sum_{k=0}^{\infty}\mathbb{P}(4^{k}M^{2}<X_{\epsilon}^{2}+Y_{\epsilon}^{2}\leq
4^{k+1}M^{2})$
$\displaystyle\leq\sum_{k=0}^{\infty}\mathbb{P}(2\mathfrak{C}\epsilon^{1/3}\geq-9B+4^{k-1}M^{2})$
$\displaystyle\leq\sum_{k=0}^{\infty}Ce^{-d\epsilon^{-1/2}(4^{k-1}M^{2}-9B)^{3/2}}$
$\displaystyle\leq\sum_{k=0}^{\infty}Ce^{-d\epsilon^{-1/2}8^{k-2}(M^{2}-36B)^{3/2}}.$

In the last line we used the bound $9B\leq 9\cdot 4^{k}B$. We can obtain a
factor of $8^{-k}$ in front of the exponential terms, through the bounds
$e^{-u}\leq 2u^{-1}e^{-u/2}\leq 2u^{-1}e^{-u/8^{k}}$. This allows us to bound
the infinite sum by a constant and obtain a bound only in terms of the first
summand. In summary, the final line of the equation above can be bounded above
by a quantity of the form

$C\epsilon^{1/2}(M^{2}-36B)^{-3/2}e^{-d\epsilon^{-1/2}(M^{2}-36B)^{3/2}},$

where the value of $d$ can be reduced if desired. If $M^{2}$ is larger than
$1+100B$, then $(M^{2}-36B)^{-3/2}$ can be bounded above by $1$. Furthermore,
$M^{2}>1+100B>72B$ implies that $M^{2}-36B>\frac{1}{2}M^{2},$ and therefore
$e^{-d\epsilon^{-1/2}(M^{2}-36B)^{3/2}}\leq e^{-d\epsilon^{-1/2}M^{3}},$ where
once again, we can make $c$ larger and $d$ smaller if desired. ∎

We use 4.2 to obtain the following bound on $|X_{\epsilon}-Y_{\epsilon}|$.

######  Proposition 4.3.

Let $f,g:\mathbb{R}\to\mathbb{R}$ be deterministic continuous functions
satisfying $-2x^{2}-B\leq f(x),g(x)\leq-\frac{1}{2}x^{2}+B$ for some $B>0.$
Define the random variable

$\displaystyle(X_{\epsilon},Y_{\epsilon}):=\mathrm{argmax}_{(x,y)\in\mathbb{R}^{2}}\;\;f(x)+\mathcal{L}_{0,\epsilon}(x,y)+g(y).$

We define $z_{\max}:=\mathrm{argmax}_{z\in\mathbb{R}}f(z)+g(z),$ and assume
that this point is unique. Then

$\mathbb{P}(|X_{\epsilon}-Y_{\epsilon}|>\epsilon^{\frac{1}{2}-\delta})\leq
Ce^{-d\epsilon^{-1/2}(\epsilon^{-2\delta}-108B)^{3/2}}+Ce^{-d\log^{6/5}(1/\epsilon)},$

where $C,d>0$ are universal constants independent of $f,g,B,\epsilon\in(0,1].$

Intuitively, the bound $\epsilon^{\frac{1}{2}-\delta}$ should not be optimal,
as one would expect based on the scaling properties of the landscape that the
typical value of $|X_{\epsilon}-Y_{\epsilon}|$ should be of order
$\epsilon^{2/3}$. Without additional regularity assumptions on $f$ or $g$,
this is simply not true. However, with additional Hölder regularity
assumptions we will later show that this is indeed true, see 4.5 below.

###### Proof.

As in previous arguments, we replace $\mathcal{L}_{0,\epsilon}(x,y)$ by
$\epsilon^{1/3}\mathcal{L}_{0,1}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)=\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)$.
From 4.1, we have the bound

$|\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)+\epsilon^{-1}(x-y)^{2}|\leq\mathfrak{C}\epsilon^{1/3}+c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|x|+|y|)),$

where $c$ is deterministic and $\mathfrak{C}$ is a non-negative random
variable satisfying $\mathbb{P}(\mathfrak{C}>u)\leq Ce^{-du^{3/2}}$ for $u\geq
0$. We estimate the probability of the event
$K_{\epsilon}:=\\{|X_{\epsilon}-Y_{\epsilon}|>\epsilon^{\frac{1}{2}-\delta}\\}$.
On one hand, on the event $K_{\epsilon}$,
$\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}>\epsilon^{-2\delta}$. From the
conditions on $f,g$ imposed by the statement of the proposition, it is also
clear that $|z_{\max}|\leq\sqrt{2}B.$ Therefore, on $K_{\epsilon}$ we find
that

$\displaystyle-\epsilon^{-2\delta}$
$\displaystyle+2B\geq-\epsilon^{-2\delta}+\sup_{x}f(x)+\sup_{y}g(y)$
$\displaystyle\geq
f(X_{\epsilon})-\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}+g(Y_{\epsilon})$
$\displaystyle\geq\big{[}f(X_{\epsilon})+\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}X_{\epsilon},\epsilon^{-2/3}Y_{\epsilon})+g(Y_{\epsilon})\big{]}-\mathfrak{C}\epsilon^{1/3}-c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))$
$\displaystyle\geq\big{[}f(z_{\max})+\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}z_{\max},\epsilon^{-2/3}z_{\max})+g(z_{\max})\big{]}-\mathfrak{C}\epsilon^{1/3}-c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))$
$\displaystyle\geq
f(z_{\max})+g(z_{\max})-\bigg{(}\mathfrak{C}\epsilon^{1/3}+c\epsilon^{1/3}\log^{2/3}(2+2\epsilon^{-2/3}|z_{\max}|)\bigg{)}$
$\displaystyle\;\;\;\;\;-\mathfrak{C}\epsilon^{1/3}-c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))$
$\displaystyle\geq(-B-\frac{1}{2}z_{\max}^{2})+(-B-\frac{1}{2}z_{\max}^{2})-2\mathfrak{C}\epsilon^{\frac{1}{3}}$
$\displaystyle\;\;\;\;\;-c\epsilon^{1/3}\bigg{(}\log^{2/3}(2+2\epsilon^{2/3}|z_{\max}|)+\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))\bigg{)}$
$\displaystyle\geq-4B-2\mathfrak{C}\epsilon^{1/3}-c\epsilon^{1/3}\bigg{(}\log^{2/3}(2+2\epsilon^{-2/3}B)+\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))\bigg{)}.$

Since $c$ is deterministic, for sufficiently small deterministic $\epsilon>0$
(below some threshold independent of $B\geq 2$ say) we have an elementary
bound $c\epsilon^{1/3}\log^{2/3}(2+2\epsilon^{-2/3}B)\leq 2B$. We define the
event

$H_{\epsilon}:=\\{|X_{\epsilon}|+|Y_{\epsilon}|\leq
1+\log^{2/5}(1/\epsilon)+100B\\}.$

By 4.2 we find that $1-\mathbb{P}(H_{\epsilon})\leq
Ce^{-d\log^{6/5}(1/\epsilon)}.$ On the event $H_{\epsilon}$ we likewise have
that
$c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))\leq
100B$, again assuming that $\epsilon$ is below some sufficiently small
positive deterministic threshold. On $K_{\epsilon}\cap H_{\epsilon}$ we find
that $2\mathfrak{C}\epsilon^{1/3}\geq\epsilon^{-2\delta}-108B.$ Therefore, we
obtain

$\displaystyle\mathbb{P}(K_{\epsilon})$
$\displaystyle\leq\mathbb{P}(H_{\epsilon}\cap
K_{\epsilon})+\mathbb{P}(H_{\epsilon}^{c})$
$\displaystyle\leq\mathbb{P}(2\mathfrak{C}\epsilon^{1/3}\geq\epsilon^{-2\delta}-108B)+Ce^{-d\log^{6/5}(1/\epsilon)}$
$\displaystyle\leq
Ce^{-d\epsilon^{-1/2}(\epsilon^{-2\delta}-108B)^{3/2}}+Ce^{-d\log^{6/5}(1/\epsilon)}.$

This completes the proof. ∎

We introduce several quantities which will be important to our argument.

######  Definition 4.4.

Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions decaying
parabolically at infinity, that is, $-Ax^{2}-B\leq f(x),g(x)\leq-Cx^{2}-D$ for
some $A>C>0$ and $B,D\in\mathbb{R}.$ Assume that $f,g$ achieve their maximum
values at unique input values $z_{\max}^{f},z_{\max}^{g}$. For $\epsilon,M>0$
and $\alpha\in(0,1)$, we define

$\displaystyle\mathscr{S}(f;\epsilon)$
$\displaystyle:=\\{z\in\mathbb{R}:f(z)\geq f(z_{\max}^{f})-\epsilon\\}$ (4.2)
$\displaystyle\Gamma(f,g;\epsilon)$
$\displaystyle:=\inf\\{|z-z^{\prime}|:|f(z)-f(z_{\max}^{f})|<\epsilon,\;\;|g(z^{\prime})-g(z_{\max}^{g})|<\epsilon\\}.$
(4.3) $\displaystyle\mathfrak{Hol}(g,\alpha;M)$
$\displaystyle:=\sup_{\begin{subarray}{c}s,t\in[-M,M]\\\ s\neq
t\end{subarray}}\frac{|g(t)-g(s)|}{|t-s|^{\alpha}}.$

We define $N_{\epsilon}(f)$ to be the smallest number of intervals of length
$\epsilon^{2}$ required to cover $\mathscr{S}(f;\epsilon).$

######  Proposition 4.5.

Let $f,g:\mathbb{R}\to\mathbb{R}$ be deterministic functions satisfying
$-2x^{2}-B\leq f(x),g(x)\leq-\frac{1}{2}x^{2}+B$ for some $B\geq 1.$ Define
the random variable

$\displaystyle(X_{\epsilon},Y_{\epsilon}):=\mathrm{argmax}_{(x,y)\in\mathbb{R}^{2}}\;\;f(x)+\mathcal{L}_{0,\epsilon}(x,y)+g(y).$

We also denote $z_{\max}:=\mathrm{argmax}_{z\in\mathbb{R}}f(z)+g(z),$ which we
assume is unique. Fix $\delta\in(0,1/3]$. Then

$\displaystyle\mathbb{P}\big{(}|X_{\epsilon}-Y_{\epsilon}|>\epsilon^{\frac{2}{3}-\delta}\big{)}$
$\displaystyle\leq Ce^{-d\log^{6/5}(1/\epsilon)}+1_{\\{B\geq
d\epsilon^{-1/3}\\}}+1_{\big{\\{}\mathfrak{Hol}\big{(}g,\;\frac{1}{2}(1-\frac{3}{10}\delta),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}>\epsilon^{-\delta/10}\big{\\}}}$

where $C,d>0$ may depend on $\delta$ but are independent of $f,g,B$ and
$\epsilon\in(0,1]$.

Here and in later propositions, we do not optimize the constants such as
$3/10,2/5,6/5,100$. The current values suffice for arguments in Section 6.

###### Proof.

By scale invariance of the directed landscape we can replace
$\mathcal{L}_{0,\epsilon}(x,y)$ by
$\epsilon^{1/3}\mathcal{L}_{0,1}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)=\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)$,
where $\mathcal{S}$ is the Airy sheet as usual. In the notation of 4.1 we have

$|\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)+\epsilon^{-1}(x-y)^{2}|\leq\mathfrak{C}\epsilon^{1/3}+c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|x|+|y|)),$

where $c$ is deterministic and $\mathfrak{C}$ is a non-negative random
variable satisfying $\mathbb{P}(\mathfrak{C}>u)\leq Ce^{-du^{3/2}}$ for $u\geq
0$. We choose $v:=\frac{3\delta}{10-6\delta}>\frac{3}{10}\delta$ and define an
inductive sequence of exponents $\kappa_{n}$ for $n=0,1,2,3,...$ as follows:

$\kappa_{0}=\frac{1}{3}\text{ and
}\kappa_{n+1}:=\frac{1-v}{2}+\frac{1}{2}\bigg{(}\frac{1}{2}-v\bigg{)}\kappa_{n}.$

As $n\to\infty$, $\kappa_{n}\uparrow\frac{2}{3}-\frac{1}{2}\delta$. This
convergence necessarily occurs exponentially quickly by standard iteration
theory for linear recurrences. For $n=0,1,2,3,...$ define the events

$K^{(n)}_{\epsilon}:=\\{|X_{\epsilon}-Y_{\epsilon}|\leq\epsilon^{\kappa_{n}}\\}.$

We will now prove that there exist universal constants $C,d>0$ so for all
$n\geq 1$ and $\epsilon\in(0,1]$,

$\mathbb{P}(K^{(n-1)}_{\epsilon}\backslash K_{\epsilon}^{(n)})\leq 1_{\\{B\geq
d\epsilon^{-1}\\}}+1_{\\{\mathfrak{Hol}(g;\frac{1}{2}-\frac{v}{2};1+100B+\log^{2/5}(1/\epsilon))>\epsilon^{-\delta\kappa_{n-1}/2}\\}}+Ce^{-d\log^{6/5}(1/\epsilon)}.$
(4.4)

To prove this, notice

$\displaystyle f(X_{\epsilon})$
$\displaystyle-\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}+g(Y_{\epsilon})$
$\displaystyle\geq
f(X_{\epsilon})+\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}X_{\epsilon},\epsilon^{-2/3}Y_{\epsilon})+g(Y_{\epsilon})-\mathfrak{C}\epsilon^{1/3}-c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))$
$\displaystyle\geq
f(z_{\max})+\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}z_{\max},\epsilon^{-2/3}z_{\max})+g(z_{\max})-\mathfrak{C}\epsilon^{1/3}-c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))$
$\displaystyle\geq
f(z_{\max})+g(z_{\max})-\bigg{(}\mathfrak{C}\epsilon^{1/3}+c\epsilon^{1/3}\log^{2/3}(2+2\epsilon^{-2/3}|z_{\max}|)\bigg{)}$
$\displaystyle\;\;\;\;\;\;-\mathfrak{C}\epsilon^{1/3}-c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))$

From the conditions on $f,g$ imposed by the statement of the proposition, it
follows that $|z_{\max}|\leq\sqrt{2}B.$ Since $c$ is deterministic, for
sufficiently small deterministic $\epsilon>0$, which does not depend on the
choice of $B\geq 1$, we have an elementary bound
$c\epsilon^{1/3}\log^{2/3}(2+2\epsilon^{-2/3}B)\leq
C\epsilon^{\frac{1}{3}-\frac{2}{3}v}B^{v}.$ This follows from the fact that
$\log^{2/3}(2+u)\leq Cu^{v}$ for all $u\geq 1$ where $C=C(v)$ does not depend
on $u.$ We define $H_{\epsilon}$ to be the event

$H_{\epsilon}:=\\{|X_{\epsilon}|+|Y_{\epsilon}|\leq
1+\log^{2/5}(1/\epsilon)+100B\\}.$

By 4.2, we conclude $1-\mathbb{P}(H_{\epsilon})\leq
Ce^{-d\log^{6/5}(1/\epsilon)}.$ Likewise, on the event $H_{\epsilon}$ we have
$c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))\leq
C\epsilon^{\frac{1}{3}-\frac{2}{3}v}B^{v}$. Here we have assumed again that
$\epsilon$ is below some sufficiently small positive deterministic threshold.
Consequently, on the event $H_{\epsilon}$, we obtain the following bound

$f(X_{\epsilon})-\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}+g(Y_{\epsilon})\geq
f(z_{\max})+g(z_{\max})-2\mathfrak{C}\epsilon^{1/3}-C\epsilon^{\frac{1}{3}-\frac{2}{3}v}B^{v}.$
(4.5)

On the event $K^{(n-1)}_{\epsilon}\cap H_{\epsilon}$,
$|g(X_{\epsilon})-g(Y_{\epsilon})|\leq
C\epsilon^{-v\kappa_{n-1}/2}|X_{\epsilon}-Y_{\epsilon}|^{\frac{1}{2}-\frac{v}{2}}\leq\epsilon^{(\frac{1}{2}-v)\kappa_{n-1}}$
as long as we have
$\mathfrak{Hol}(g;\frac{1}{2}-\frac{v}{2};1+100B+\log^{2/5}(1/\epsilon))\leq\epsilon^{-v\kappa_{n-1}/2}.$
When we combine this bound with (4.5), on the event $H_{\epsilon}\cap
K^{(n-1)}_{\epsilon}$, we obtain the bound

$\displaystyle
f(z_{\max})-\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}+g(z_{\max})$
$\displaystyle\geq
f(X_{\epsilon})-\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}+g(X_{\epsilon})$
$\displaystyle\geq
f(X_{\epsilon})-\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}+g(Y_{\epsilon})-|g(X_{\epsilon})-g(Y_{\epsilon})|$
$\displaystyle\geq
f(z_{\max})+g(z_{\max})-2\mathfrak{C}\epsilon^{1/3}-C\epsilon^{\frac{1}{3}-\frac{2}{3}v}B^{v}-\epsilon^{(\frac{1}{2}-v)\kappa_{n-1}}$
$\displaystyle\geq
f(z_{\max})+g(z_{\max})-2\mathfrak{C}\epsilon^{1/3}-C\epsilon^{(\frac{1}{2}-v)\kappa_{n-1}}B^{v}.$

In the final line, we use the fact that $B\geq 1$ and that
$(\frac{1}{2}-v)\kappa_{n}$ is always less than $\frac{1}{3}-\frac{2}{3}v.$ As
long as
$\mathfrak{Hol}(g;\frac{1}{2}-\frac{v}{2};1+100B+\log^{2/5}(1/\epsilon))\leq\epsilon^{-v\kappa_{n-1}/2}$,
we have by the recursive definition of the $\kappa_{n}$ that
$1+(\frac{1}{2}-v)\kappa_{n-1}=2\kappa_{n}+v$ and therefore the above
expression immediately gives

$(X_{\epsilon}-Y_{\epsilon})^{2}\leq
2\mathfrak{C}\epsilon^{4/3}+C\epsilon^{1+(\frac{1}{2}-v)\kappa_{n-1}}B^{v}=2\mathfrak{C}\epsilon^{4/3}+C\epsilon^{2\kappa_{n}+v}B^{v},$

for some large enough deterministic constant $C=C(v)$ that does not depend on
$B,f,g,\epsilon.$ The right-hand side of the previous expression is less than
$\epsilon^{2\kappa_{n}}$ as long as
$\mathfrak{C}\leq\frac{1}{4}\epsilon^{-2v}$ (because
$2\kappa_{n}\leq\frac{4}{3}-2v$ for all $n)$ and
$B\leq\frac{1}{2}\epsilon^{-1}C^{-\frac{1}{v}},$ which proves (4.4) once we
use the facts that $\mathbb{P}(H_{\epsilon}^{c})\leq
Ce^{-d\log^{6/5}(1/\epsilon)}$ and
$\mathbb{P}(\mathfrak{C}>\frac{1}{2}\epsilon^{-2v})\leq
Ce^{-d\epsilon^{-3v}}\leq Ce^{-d\log^{6/5}(1/\epsilon)}$.

We use (4.4) to prove the bound in the proposition. By using 4.3 and the fact
that $\kappa_{0}$ is chosen to be 1/3,

$\mathbb{P}(K^{(0)}_{\epsilon})\leq
Ce^{-d\epsilon^{-1/2}(\epsilon^{-1/3}-108B)^{3/2}}\leq
Ce^{-d\epsilon^{-5/6}}+1_{\\{200B\geq\epsilon^{-1/3}\\}}.$

Using this bound and then repeatedly summing over (4.4) up to a large enough
value of $n$ such that $\kappa_{n}$ reaches the desired threshold will give
the desired bound (this is a finite sum). This is because
$v>\frac{3}{10}\delta$ for $\delta\in(0,1/3]$, as explained earlier. ∎

######  Proposition 4.6.

Let $f,g:\mathbb{R}\to\mathbb{R}$ be deterministic functions satisfying
$-2x^{2}-B\leq f(x),g(x)\leq-\frac{1}{2}x^{2}+B$ for some $B\geq 1.$ Define
the random variable

$\displaystyle(X_{\epsilon},Y_{\epsilon}):=\mathrm{argmax}_{(x,y)\in\mathbb{R}^{2}}\;\;f(x)+\mathcal{L}_{0,\epsilon}(x,y)+g(y).$

We also denote $z_{\max}:=\mathrm{argmax}_{z\in\mathbb{R}}f(z)+g(z)$, which we
assume is unique. Fix $\delta\in(0,1/3]$. Then

$\displaystyle\mathbb{P}\big{(}X_{\epsilon}\notin\mathscr{S}(f+g,\epsilon^{\frac{1}{3}-\delta})\big{)}$
$\displaystyle\leq Ce^{-d\log^{6/5}(1/\epsilon)}+1_{\\{B\geq
d\epsilon^{-1/3}\\}}+1_{\big{\\{}\mathfrak{Hol}\big{(}g,\;\frac{1}{2}(1-\frac{3}{20}\delta),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}>\epsilon^{-\delta/20}\big{\\}}},$
$\displaystyle\mathbb{P}\big{(}Y_{\epsilon}\notin\mathscr{S}(f+g,\epsilon^{\frac{1}{3}-\delta})\big{)}$
$\displaystyle\leq Ce^{-d\log^{6/5}(1/\epsilon)}+1_{\\{B\geq
d\epsilon^{-1/3}\\}}+1_{\big{\\{}\mathfrak{Hol}\big{(}f,\;\frac{1}{2}(1-\frac{3}{20}\delta),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}>\epsilon^{-\delta/20}\big{\\}}}$

where $C,d>0$ may depend on $\delta$ but are independent of $f,g,B$ and
$\epsilon\in(0,1]$.

###### Proof.

The bound on $Y_{\epsilon}$ follows from the bound on $X_{\epsilon}$ by a
symmetry argument. Therefore, we focus on the first bound. By the scale
invariance of the directed landscape, we replace
$\mathcal{L}_{0,\epsilon}(x,y)$ by
$\epsilon^{1/3}\mathcal{L}_{0,1}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)=\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)$.
In the notation of 4.1 we have

$|\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)+\epsilon^{-1}(x-y)^{2}|\leq\mathfrak{C}\epsilon^{1/3}+c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|x|+|y|)),$

where $c$ is deterministic and $\mathfrak{C}$ is a non-negative random
variable satisfying $\mathbb{P}(\mathfrak{C}>u)\leq Ce^{-du^{3/2}}$ for $u\geq
0$. We fix an arbitrary $\delta_{1}>0$ (the value of which will be specified
at a later point). We define the event

$H_{\epsilon}:=\\{|X_{\epsilon}|+|Y_{\epsilon}|\leq
1+\log^{2/5}(1/\epsilon)+100B\\}.$

We recall on $H_{\epsilon}$, from (4.5) for $\epsilon$ below a sufficiently
small positive deterministic threshold

$f(X_{\epsilon})-\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}+g(Y_{\epsilon})\geq
f(z_{\max})+g(z_{\max})-2\mathfrak{C}\epsilon^{1/3}-C\epsilon^{\frac{1}{3}-\frac{2}{3}\delta_{1}}B^{\delta_{1}}.$

We also define the event

$F_{\epsilon}:=\\{|X_{\epsilon}-Y_{\epsilon}|\leq\epsilon^{\frac{2}{3}-\delta_{1}}\\}.$

By 4.5, $F_{\epsilon}$ has probability greater than
$1-Ce^{-d\log^{6/5}(1/\epsilon)}$ as long as $B\leq d\epsilon^{-1/3}$ and

$\mathfrak{Hol}\big{(}g,\frac{1}{2}(1-\delta_{2}),1+100B+\log^{2/5}(1/\epsilon)\big{)}\leq\epsilon^{-\delta_{2}/3},$

where $\delta_{2}:=\frac{3}{10}\delta_{1}$. From the expression above, if
$\mathfrak{Hol}\big{(}g,\frac{1}{2}(1-\delta_{2}),1+100B+\log^{2/5}(1/\epsilon)\big{)}\leq\epsilon^{-\delta_{2}/3}$,
then on the event $F_{\epsilon}\cap H_{\epsilon}$ we have the bound

$\displaystyle f(X_{\epsilon})+g(X_{\epsilon})$
$\displaystyle>f(X_{\epsilon})-\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}+g(X_{\epsilon})$
$\displaystyle\geq
f(X_{\epsilon})-\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}+g(Y_{\epsilon})-|g(Y_{\epsilon})-g(X_{\epsilon})|$
$\displaystyle\geq
f(X_{\epsilon})-\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}+g(Y_{\epsilon})-\epsilon^{-\delta_{2}/3}|X_{\epsilon}-Y_{\epsilon}|^{\frac{1}{2}-\delta_{2}}$
$\displaystyle\geq
f(z_{\max})+g(z_{\max})-2\mathfrak{C}\epsilon^{1/3}-C\epsilon^{\frac{1}{3}-\frac{2}{3}\delta_{1}}B^{\delta_{1}}-\epsilon^{\frac{1}{3}-\delta_{2}-\frac{1}{2}\delta_{1}+\delta_{1}\delta_{2}}.$

With $\delta$ as in the proposition statement, we make a specific choice of
$\delta_{1}$ so that $\frac{2}{3}\delta_{1}<\frac{1}{2}\delta$ and
$\delta_{2}+\frac{1}{2}\delta_{1}-\delta_{1}\delta_{2}<\frac{1}{2}\delta.$ For
precision, take $\delta_{1}:=\frac{1}{2}\delta$. From the previous expression,
it follows that in order to have
$X_{\epsilon}\notin\mathscr{S}(f+g,\epsilon^{\frac{1}{3}-\delta})$, for
$\epsilon<3^{-2/\delta}$ it must be true that
$\mathfrak{C}>\frac{1}{6}\epsilon^{-\delta}$ or
$B>\frac{1}{6}\epsilon^{-\frac{2}{3}}C^{-\frac{1}{\delta_{1}}}.$ Therefore,

$\mathbb{P}(X_{\epsilon}\notin\mathscr{S}(f+g,\epsilon^{\frac{1}{3}-\delta})\leq\mathbb{P}(\mathfrak{C}>\frac{1}{6}\epsilon^{-\delta})+1_{\\{B>\frac{1}{6}\epsilon^{-\frac{2}{3}}C^{-\frac{1}{\delta_{1}}}\\}}+\mathbb{P}(H_{\epsilon}^{c})+\mathbb{P}(F_{\epsilon}^{c}).$

To finish the proof, we apply the tail bounds on $\mathfrak{C}$ and the upper
bounds on the latter two terms, all of which have already been discussed. ∎

Using the preceding bounds, we can prove the following estimate on the
directed landscape.

######  Proposition 4.7.

Let $f,g:\mathbb{R}\to\mathbb{R}$ be deterministic functions satisfying
$-2x^{2}-B\leq f(x),g(x)\leq-\frac{1}{2}x^{2}+B$ for some $B\geq 1.$ Then for
any $\delta>0$ and $p>1$,

$\mathbb{E}[|f\circ\mathcal{L}_{0,\epsilon}\circ g-\max(f+g)|^{p}]^{1/p}\\\
\leq
CB^{2\delta}\epsilon^{\frac{1}{3}-\delta}+CB\bigg{(}Ce^{-d\log^{6/5}(1/\epsilon)}+1_{\\{B\geq
d\epsilon^{-1/3}\\}}+1_{\big{\\{}\mathfrak{Hol}\big{(}g,\;\frac{1}{2}(1-\frac{3}{40}\delta),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}>\epsilon^{-\delta/40}\big{\\}}}\bigg{)}.$

Here $C,d$ may depend on $p$ but are uniform over all $f,g,B,$ and
$\epsilon\in(0,1]$.

###### Proof.

As before, by scale invariance of the directed landscape, we use the
substitution, $\mathcal{L}_{0,\epsilon}(x,y)$ by
$\epsilon^{1/3}\mathcal{L}_{0,1}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)=\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)$.
In the notation of 4.1 we have

$|\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)+\epsilon^{-1}(x-y)^{2}|\leq\mathfrak{C}\epsilon^{1/3}+c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|x|+|y|)),$
(4.6)

where $c$ is deterministic and $\mathfrak{C}$ is a non-negative random
variable satisfying $\mathbb{P}(\mathfrak{C}>u)\leq Ce^{-du^{3/2}}$ for $u\geq
0$. Let $X_{\epsilon},Y_{\epsilon},z_{\max}$ be as in 4.5 and 4.6. We note
that

$\displaystyle|f(X_{\epsilon})$
$\displaystyle+\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y)+g(Y_{\epsilon})-(f(z_{\max})+g(z_{\max}))|$
$\displaystyle\leq|f(X_{\epsilon})+g(X_{\epsilon})-(f(z_{\max})+g(z_{\max}))|+|\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}X_{\epsilon},\epsilon^{-2/3}Y_{\epsilon})+\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}|$
$\displaystyle\;\;\;\;\;+|g(Y_{\epsilon})-g(X_{\epsilon})|+\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}.$

We can now define the event

$J_{\epsilon}:=\\{X_{\epsilon}\in\mathscr{S}(f+g,\epsilon^{\frac{1}{3}-\delta})\\}\cap\\{|X_{\epsilon}-Y_{\epsilon}|\leq\epsilon^{\frac{2}{3}-\delta}\\}\cap\\{|X_{\epsilon}|+|Y_{\epsilon}|\leq
1+100B+\log^{2/5}(1/\epsilon)\\}.$

Furthermore, under the assumption that
$\mathfrak{Hol}(g,\frac{1}{2}-\delta,1+100B+\log^{2/5}(1/\epsilon))<{\epsilon^{-\delta}},$
the last expression is bounded (term-by-term) by

$\epsilon^{\frac{1}{3}-\delta}+\big{(}\mathfrak{C}\epsilon^{1/3}+c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|X_{\epsilon}|+|Y_{\epsilon}|))\big{)}+\epsilon^{-\delta/3}|X_{\epsilon}-Y_{\epsilon}|^{\frac{1}{2}-\delta}+\epsilon^{-1}(X_{\epsilon}-Y_{\epsilon})^{2}\\\
\leq\epsilon^{\frac{1}{3}-\delta}+(\mathfrak{C}\epsilon^{1/3}+\epsilon^{\frac{1}{3}-\frac{2}{3}\delta}B^{\delta})+\epsilon^{(\frac{2}{3}-\delta)(\frac{1}{2}-\delta)}+\epsilon^{\frac{1}{3}-2\delta}.$

For the second term we used the following argument: since $c$ is
deterministic, for sufficiently small deterministic $\epsilon>0$ which does
not depend on the choice of $B\geq 1$, we have an elementary bound
$c\epsilon^{1/3}\log^{2/3}(2+2\epsilon^{-2/3}B)\leq
C\epsilon^{\frac{1}{3}-\frac{2}{3}\delta}B^{\delta}.$ This bound follows
immediately from the fact that $\log^{2/3}(2+u)\leq Cu^{\delta}$ for all
$u\geq 1$, where $C=C(\delta)$ does not depend on $u.$ Note that all powers of
$\epsilon$ in the last expression are at worst $\frac{1}{3}-2\delta$. Since
$\mathfrak{C}$ is the only random term in the last expression, these bounds
imply that

$\mathbb{E}[|f\circ\mathcal{L}_{0,\epsilon}\circ g-\max(f+g)|^{p}\cdot
1_{J_{\epsilon}}]^{1/p}\leq CB^{\delta}\epsilon^{\frac{1}{3}-2\delta}.$

Replacing $\delta$ by $2\delta$ we obtain a bound of the same form as in the
proposition statement. It remains to bound what occurs outside $J_{\epsilon}.$
Using 4.2 and 4.5 and 4.6, and the elementary inequality $(a+b)^{1/2}\leq
a^{1/2}+b^{1/2}$, we obtain

$\mathbb{P}(J_{\epsilon}^{c})^{1/2}\leq
Ce^{-d\log^{6/5}(1/\epsilon)}+1_{\\{B\geq
d\epsilon^{-1/3}\\}}+1_{\big{\\{}\mathfrak{Hol}\big{(}g,\;\frac{1}{2}(1-\frac{3}{20}\delta),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}>\epsilon^{-\delta/20}\big{\\}}}.$
(4.7)

By Cauchy-Schwarz and Minkowski, we furthermore have

$\mathbb{E}[|f\circ\mathcal{L}_{0,\epsilon}\circ g-\max(f+g)|^{p}\cdot
1_{J_{\epsilon}^{c}}]^{1/p}\leq\bigg{(}\mathbb{E}[|f\circ\mathcal{L}_{0,\epsilon}\circ
g|^{2p}]^{1/2p}+\max(f+g)\bigg{)}\cdot\mathbb{P}(J_{\epsilon}^{c})^{1/2}.$

Clearly, $\max(f+g)\leq 2B.$ We will also show that
$\mathbb{E}[|f\circ\mathcal{L}_{0,\epsilon}\circ g|^{2p}]^{1/2p}$ is bounded
above by $B$ times some absolute constant. As before, we write
$\mathcal{L}_{0,\epsilon}$ as
$\mathcal{S}^{\epsilon}(x,y):=\epsilon^{1/3}\mathcal{S}(\epsilon^{-2/3}x,\epsilon^{-2/3}y).$
We apply (4.6) to obtain that

$\displaystyle f(x)+\mathcal{S}^{\epsilon}(x,y)+g(y)$ $\displaystyle\leq
2B-\frac{1}{2}(x^{2}+y^{2})+\mathfrak{C}\epsilon^{1/3}+c\epsilon^{1/3}\log^{2/3}(2+\epsilon^{-2/3}(|x|+|y|))$
$\displaystyle\leq
2B-\frac{1}{2}(x^{2}+y^{2})+\mathfrak{C}\epsilon^{1/3}+2+|x|+|y|,$

where the last bound is valid for sufficiently small $\epsilon>0$ below some
deterministic threshold. Let $m$ denote the maximum value achieved by the
function $(x,y)\mapsto 2+|x|+|y|-\frac{1}{2}(x^{2}+y^{2})$ over all of
$\mathbb{R}^{2}$. Then in order for the last expression to exceed $2B+m+u$, it
would be necessary for the constant $\mathfrak{C}$ to exceed $u$. Therefore,

$\mathbb{P}(f\circ\mathcal{L}_{0,\epsilon}\circ g>2B+m+u)\leq Ce^{-du^{3/2}},$
(4.8)

which implies that $\mathbb{E}[(f\circ\mathcal{L}_{0,\epsilon}\circ
g)_{+}^{2p}]^{1/2p}$ (where $u_{+}:=\max\\{u,0\\}$) is bounded above by $B$
multiplied by an absolute constant. To bound the negative part, we note by
similar arguments that the probability that
$f\circ\mathcal{L}_{0,\epsilon}\circ g<-2B$ yields a large value of
$\mathfrak{C}$. One can show this by considering what happens at
$(z_{\max},z_{\max})$. This yields an analogous bound on
$\mathbb{E}[(f\circ\mathcal{L}_{0,\epsilon}\circ g)_{-}^{2p}]^{1/2p}$ (where
$u_{-}=\max\\{0,-u\\}$). ∎

We now come to the main result of this section, which we obtain by applying
the mixing results of Section 3 with the estimates proved in this section. We
begin by stating a covariance bound that will be crucial to understanding the
mechanisms behind the black noise property.

######  Theorem 4.8.

Consider functions $f,g,u,v$ each lying in between $-B-2x^{2}$ and
$B-\frac{1}{2}x^{2}$. Recall $\Gamma$ and $N_{\epsilon}$ as defined in (4.3).
Then for any $\delta>0$,

$\displaystyle|\operatorname{Cov}(f\circ\mathcal{L}_{0,\epsilon}\circ
g,u\circ\mathcal{L}_{0,\epsilon}\circ v)|$ $\displaystyle\leq
CB^{2\delta}\min\big{\\{}\epsilon^{\frac{2}{3}-\delta},\big{(}N_{\epsilon^{\frac{1}{3}-\delta}}(f+g)+N_{\epsilon^{\frac{1}{3}-\delta}}(u+v)\big{)}e^{-c\epsilon^{-2}\Gamma\big{(}f+g,u+v;\epsilon^{\frac{1}{3}-\delta}\big{)}^{3}}\big{\\}}$
$\displaystyle+CB^{2}\cdot(\mathfrak{E}_{\epsilon}(f;B)+\mathfrak{E}_{\epsilon}(g;B)+\mathfrak{E}_{\epsilon}(u;B)+\mathfrak{E}_{\epsilon}(v;B)).$

Here $C,d$ are universal constants uniform over all $f,g,u,v$, $\epsilon,B$
and

$\mathfrak{E}_{\epsilon}(f,B):=e^{-d\log^{6/5}(1/\epsilon)}+1_{\\{B\geq
d\epsilon^{-1/3}\\}}+1_{\big{\\{}\mathfrak{Hol}\big{(}f,\;\frac{1}{2}(1-\frac{3}{40}\delta),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}>\epsilon^{-\delta/40}\big{\\}}}.$

###### Proof.

By Cauchy-Schwartz,

$\displaystyle|\operatorname{Cov}(f\circ\mathcal{L}_{0,\epsilon}\circ
g,u\circ\mathcal{L}_{0,\epsilon}\circ v)|$
$\displaystyle=\big{|}\operatorname{Cov}\big{(}f\circ\mathcal{L}_{0,\epsilon}\circ
g-\max(f+g),u\circ\mathcal{L}_{0,\epsilon}\circ v-\max(u+v)\big{)}\big{|}$
$\displaystyle\leq\mathbb{E}[(f\circ\mathcal{L}_{0,\epsilon}\circ
g-\max(f+g))^{2}]^{1/2}\mathbb{E}[(u\circ\mathcal{L}_{0,\epsilon}\circ
v-\max(u+v))^{2}]^{1/2}.$

Using 4.7, we get that the last expression is bounded above by
$CB^{2\delta}\epsilon^{\frac{2}{3}-\delta}+CBe^{-d\log^{6/5}(1/\epsilon)}$ as
long as $B\leq d\epsilon^{-1/3}$ and
$\mathfrak{Hol}\big{(}g,\;\frac{1}{2}(1-\frac{3}{40}\delta),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}\leq\epsilon^{-\delta/40}$
and
$\mathfrak{Hol}\big{(}v,\;\frac{1}{2}(1-\frac{3}{40}\delta),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}\leq\epsilon^{-\delta/40}.$

We define the event
$J_{\epsilon}^{f,g}:=\\{X_{\epsilon}^{f,g},Y_{\epsilon}^{f,g}\in\mathscr{S}(f+g,\epsilon^{\frac{1}{3}-\delta})\\}\cap\\{|X_{\epsilon}^{f,g}-Y_{\epsilon}^{f,g}|\leq\epsilon^{\frac{2}{3}-\delta}\\}\cap\\{|X_{\epsilon}^{f,g}|+|Y_{\epsilon}^{f,g}|\leq
1+100B+\log^{2/5}(1/\epsilon)\\}$, where
$X_{\epsilon}^{f,g},Y_{\epsilon}^{f,g}$ are the argmaxes as defined in 4.5 and
4.6 respectively. We define the event $J^{u,v}_{\epsilon}$ in the same way,
but with $f,g$ replaced by $u,v$. Finally, we define
$J_{\epsilon}^{f,g,u,v}:=J^{f,g}_{\epsilon}\cap J^{u,v}_{\epsilon}$. We
condition on the event $J_{\epsilon}^{f,g,u,v}$, and this allows us to use 3.4
to obtain a bound of the form
$\big{(}N_{\epsilon^{\frac{1}{3}-\delta}}(f+g)+N_{\epsilon^{\frac{1}{3}-\delta}}(u+v)\big{)}e^{-c\epsilon^{-2}\Gamma\big{(}f+g,u+v;\epsilon^{\frac{1}{3}-\delta}\big{)}^{3}}.$
Indeed by setting $T:=\epsilon$ in 3.4, we see that the value of $D$ in that
theorem corresponds to $\Gamma(f+g,u+v,\epsilon^{\frac{1}{3}-\delta}),$ and
furthermore
$N_{\epsilon^{\frac{1}{3}-\delta}}(f+g)+N_{\epsilon^{\frac{1}{3}-\delta}}(u+v)$
gives an upper bound for the value of $N$ which appears in that theorem.
Consequently, we apply 3.4 to obtain the bound

$\bigg{|}\mathrm{Cov}\bigg{(}\big{(}f\circ\mathcal{L}_{0,\epsilon}\circ
g\big{)}\cdot
1_{J_{\epsilon}^{f,g,u,v}},\big{(}u\circ\mathcal{L}_{0,\epsilon}\circ
v\big{)}\cdot
1_{J_{\epsilon}^{f,g,u,v}}\bigg{)}\bigg{|}\leq\big{(}N_{\epsilon^{\frac{1}{3}-\delta}}(f+g)+N_{\epsilon^{\frac{1}{3}-\delta}}(u+v)\big{)}e^{-c\epsilon^{-2}\Gamma\big{(}f+g,u+v;\epsilon^{\frac{1}{3}-\delta}\big{)}^{3}},$

because on $J_{\epsilon}^{f,g,u,v}$ the maximum on all of $\mathbb{R}^{2}$ in
the definitions of $f\circ\mathcal{L}_{0,\epsilon}\circ
g,u\circ\mathcal{L}_{0,\epsilon}\circ v$ can be replaced by a maximum on some
set of the form which appears in 3.4. It remains to control what happens on
the complement of $J_{\epsilon}^{f,g,u,v}.$ By applying Cauchy-Schwarz twice
we have

$\displaystyle\bigg{|}\mathrm{Cov}$
$\displaystyle\big{(}f\circ\mathcal{L}_{0,\epsilon}\circ
g,u\circ\mathcal{L}_{0,\epsilon}\circ
v\big{)}\bigg{|}\leq\bigg{|}\mathrm{Cov}\bigg{(}\big{(}f\circ\mathcal{L}_{0,\epsilon}\circ
g\big{)}\cdot
1_{J_{\epsilon}^{f,g,u,v}},\big{(}u\circ\mathcal{L}_{0,\epsilon}\circ
v\big{)}\cdot 1_{J_{\epsilon}^{f,g,u,v}}\bigg{)}\bigg{|}$
$\displaystyle\;\;\;\;+2\mathbb{E}[(f\circ\mathcal{L}_{0,\epsilon}\circ
g-\max(f+g))^{4}]^{1/4}\mathbb{E}[(u\circ\mathcal{L}_{0,\epsilon}\circ
v-\max(u+v))^{4}]^{1/4}\cdot(1-\mathbb{P}(J_{\epsilon}^{f,g,u,v}))^{\frac{1}{2}}.$

Using arguments similar to (4.7), it follows that
$(1-\mathbb{P}(J_{\epsilon}^{f,g,u,v}))^{\frac{1}{2}}$ yields an upper bound
consisting of terms of the form appearing in $\mathfrak{E}_{\epsilon}$. The
upper bounds for all other terms have already been discussed. ∎

######  Theorem 4.9.

Consider functions $f,g,u,v,\bar{u},\bar{v}$ each lying in between $-B-2x^{2}$
and $B-\frac{1}{2}x^{2}$. Define
$\Gamma(\epsilon):=\min\\{\Gamma\big{(}f+g,u+v;\epsilon^{\frac{1}{3}-\delta_{1}}\big{)},\Gamma\big{(}f+g,\bar{u}+\bar{v};\epsilon^{\frac{1}{3}-\delta_{1}}\big{)}\\}.$
Then for any $\delta>0$

$\displaystyle\bigg{|}\mathrm{Cov}\bigg{(}f\circ$
$\displaystyle\mathcal{L}_{0,\epsilon}\circ
g\;\;,\;\;(u\circ\mathcal{L}_{0,\epsilon}\circ
v-\max(u+v))((\bar{u}\circ\mathcal{L}_{0,\epsilon}\circ\bar{v}-\max(\bar{u}+\bar{v}))\bigg{)}\bigg{|}$
$\displaystyle\leq
CB^{2\delta}\min\big{\\{}\epsilon^{1-\delta},\big{(}N_{\epsilon^{\frac{1}{3}-\delta}}(f+g)+N_{\epsilon^{\frac{1}{3}-\delta}}(u+v)+N_{\epsilon^{\frac{1}{3}-\delta}}(\bar{u}+\bar{v})\big{)}e^{-c\epsilon^{-2}\Gamma(\epsilon)^{3}}\big{\\}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+CB^{3}\cdot(\mathfrak{E}_{\epsilon}(f;B)+\mathfrak{E}_{\epsilon}(g;B)+\mathfrak{E}_{\epsilon}(u;B)+\mathfrak{E}_{\epsilon}(v;B)+\mathfrak{E}_{\epsilon}(\bar{u};B)+\mathfrak{E}_{\epsilon}(\bar{v};B)).$

Here $C,d$ are universal constants uniform over all $f,g,u,v$, $\epsilon,B$,
and $\mathfrak{E}_{\epsilon}$ was defined in 4.8.

###### Proof.

The proof is very similar to that of the previous theorem. Note, by applying
Cauchy-Schwartz twice, that

$\bigg{|}\mathrm{Cov}\bigg{(}f\circ\mathcal{L}_{0,\epsilon}\circ
g,(u\circ\mathcal{L}_{0,\epsilon}\circ
v-\max(u+v))((\bar{u}\circ\mathcal{L}_{0,\epsilon}\circ\bar{v}-\max(\bar{u}+\bar{v}))\bigg{)}\bigg{|}\\\
\leq\mathbb{E}[(f\circ\mathcal{L}_{0,\epsilon}\circ
g-\max(f+g))^{2}]^{1/2}\mathbb{E}[(u\circ\mathcal{L}_{0,\epsilon}\circ
v-\max(u+v))^{4}]^{1/4}\mathbb{E}[(\bar{u}\circ\mathcal{L}_{0,\epsilon}\circ\bar{v}-\max(\bar{u}+\bar{v}))^{4}]^{1/4}.$

Using 4.7, we get that the last expression is bounded above by
$CB^{2\delta}\epsilon^{1-\delta}+CBe^{-d\log^{6/5}(1/\epsilon)}$ as long as
$B\leq d\epsilon^{-1/3}$ and
$\mathfrak{Hol}\big{(}g,\;\frac{1}{2}(1-\delta_{1}),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}\leq\epsilon^{-\delta_{1}/3}$
and
$\mathfrak{Hol}\big{(}v,\;\frac{1}{2}(1-\delta_{1}),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}\leq\epsilon^{-\delta_{1}/3},$
where $\delta_{1}=\frac{3}{40}\delta.$

Next, we proceed exactly as in the previous theorem. We condition on the event
$J_{\epsilon}^{f,g,u,v,\bar{u},\bar{v}}$, which allows us to use 3.4 to obtain
a bound of the form

$\big{(}N_{\epsilon^{\frac{1}{3}-\delta_{1}}}(f+g)+N_{\epsilon^{\frac{1}{3}-\delta_{1}}}(u+v)+N_{\epsilon^{\frac{1}{3}-\delta_{1}}}(\bar{u}+\bar{v})\big{)}e^{-c\epsilon^{-2}\Gamma(\epsilon)^{3}}.$

Indeed by setting $T:=\epsilon$ in that theorem, the value of $D$ corresponds
to $\Gamma(\epsilon),$ and moreover
$N_{\epsilon^{\frac{1}{3}-\delta_{1}}}(f+g)+N_{\epsilon^{\frac{1}{3}-\delta_{1}}}(u+v)+N_{\epsilon^{\frac{1}{3}-\delta_{1}}}(\bar{u}+\bar{v})$
gives an upper bound for the value of $N$ appearing in that theorem. It
remains to control what occurs on the complement of
$J_{\epsilon}^{f,g,u,v,\bar{u},\bar{v}}$. This can be done completely
analogously to the proof of Theorem 4.8. As we observed there, this will only
yield error terms of the form $\mathfrak{E}_{\epsilon}$. ∎

######  Proposition 4.10.

Consider functions $f_{i},g_{i},u_{j},v_{j}$ ($1\leq i\leq k,1\leq j\leq\ell)$
each lying in between $-B-2x^{2}$ and $B-\frac{1}{2}x^{2}$. Then for any
$\delta_{1}>0$ there exists $\delta_{2}>0$ so that

$\bigg{|}\mathrm{Cov}\bigg{(}\prod_{i=1}^{k}(f_{i}\circ\mathcal{L}_{0,\epsilon}\circ
g_{i}-\max(f_{i}+g_{i})),\prod_{j=1}^{\ell}(u_{j}\circ\mathcal{L}_{0,\epsilon}\circ
v_{j}-\max(u_{j}+v_{j}))\bigg{)}\bigg{|}\\\ \leq
CB^{\delta_{2}}\epsilon^{\frac{1}{3}(k+\ell)-\delta_{1}}+CB^{k+\ell}\cdot(\sum_{i}\mathfrak{E}_{\epsilon}(f_{i};B)+\mathfrak{E}_{\epsilon}(g_{i};B)+\sum_{j}\mathfrak{E}_{\epsilon}(u_{j};B)+\mathfrak{E}_{\epsilon}(v_{j};B)).$

Here $C,d$ are universal constants uniform over all $f_{i},g_{i},u_{i},v_{i}$,
$\epsilon,B$, and $\mathfrak{E}_{\epsilon}$ is as defined in 4.8.

###### Proof.

We apply Cauchy-Schwartz to the covariance, and then apply Hölder’s inequality
to obtain

$\displaystyle\bigg{|}\mathrm{Cov}$
$\displaystyle\bigg{(}\prod_{i=1}^{k}(f_{i}\circ\mathcal{L}_{0,\epsilon}\circ
g_{i}-\max(f_{i}+g_{i})),\prod_{j=1}^{\ell}(u_{j}\circ\mathcal{L}_{0,\epsilon}\circ
v_{j}-\max(u_{j}+v_{j}))\bigg{)}\bigg{|}$
$\displaystyle\leq\mathbb{E}\bigg{[}\prod_{i=1}^{k}(f_{i}\circ\mathcal{L}_{0,\epsilon}\circ
g_{i}-\max(f_{i}+g_{i}))^{2}\bigg{]}^{1/2}\mathbb{E}\bigg{[}\prod_{j=1}^{\ell}(u_{j}\circ\mathcal{L}_{0,\epsilon}\circ
v_{j}-\max(u_{j}+v_{j}))^{2}\bigg{]}^{1/2}$
$\displaystyle\leq\prod_{i=1}^{k}\mathbb{E}\bigg{[}(f_{i}\circ\mathcal{L}_{0,\epsilon}\circ
g_{i}-\max(f_{i}+g_{i}))^{2k}\bigg{]}^{1/(2k)}\prod_{j=1}^{\ell}\mathbb{E}\bigg{[}(u_{j}\circ\mathcal{L}_{0,\epsilon}\circ
v_{j}-\max(u_{j}+v_{j}))^{2k}\bigg{]}^{1/(2k)}.$

We use 4.7 to obtain the claim (note here that unlike the previous two
theorems, one does not need 3.4 as we are simply using the unrefined bound
without leveraging the mixing property, as this will be sufficient later). ∎

######  Remark 4.11.

In all propositions above, the constants $1/2$ and $2$ in the parabolic decay
rates of $f,g$ could have been replaced by $A^{-1}$ and $A$ for arbitrary
$A>1$. This would affect the constants such as $100$ and $108$ as well as the
constants $C,d$. While we cannot control the behavior of these constants as
$A\uparrow\infty$, there is never an issue for a fixed finite $A$, no matter
how large. This will be important later due to the nature of 2.6, because we
will want $f,g$ to be “typical” sample paths of parabolic Airy2 processes of
scales within the interval $[\eta,1]$ for some fixed but very small $\eta>0$.
In this case $A$ can be as large as $C\eta^{-1}$ for some universal $C>0.$

Another remark is that the constant $Ce^{-d\log^{6/5}(1/\epsilon)}$ can be
improved to $Ce^{-d\log^{3M}(1/\epsilon)}$ for any $M>0$, wherever it appears.
However, the cost is that we then need to replace all instances of
$\big{\\{}\mathfrak{Hol}\big{(}f,\;\frac{1}{2}(1-\tilde{\delta}),\;\;1+100B+\log^{2/5}(1/\epsilon)\big{)}>\epsilon^{-\tilde{\delta}/3}\big{\\}}$
by the respective interval
$\big{\\{}\mathfrak{Hol}\big{(}f,\;\frac{1}{2}(1-\tilde{\delta}),\;\;1+100B+\log^{M}(1/\epsilon)\big{)}>\epsilon^{-\tilde{\delta}/3}\big{\\}}$,
where $\tilde{\delta}$ denotes the various instances of
$\frac{3}{10}\delta,\frac{3}{20}\delta,\frac{3}{40}\delta$ and other
convenient choices of multiplying constants that we made in the propositions
above. For the purposes of this paper, the value of $M$ can be taken to be any
value larger than $1/3$, however we chose $M=2/5$ in order to be precise.

##  5\. Estimates on 3-dimensional Bessel processes

######  Lemma 5.1.

Let $R$ be a three-dimensional Bessel process, and let $p\in\mathbb{N}$. Then
there exists $C_{p}>0$ such that we have a uniform bound over all
$s_{1}\leq...\leq s_{p}$ and $\epsilon\in(0,1]$

$\displaystyle\mathbb{P}\bigg{(}\inf_{t\in[s_{i},s_{i}+1]}R(t)\leq\epsilon,\;\forall
1\leq i\leq p\bigg{)}ds_{1}\cdots ds_{p}\leq
C_{p}\epsilon^{p}\prod_{i=0}^{p-1}\min\big{\\{}1,(s_{i+1}-s_{i})^{-3/2}(1+\log_{+}^{3/2}(s_{i+1}-s_{i}))\big{\\}},$

where $s_{0}:=0$.

###### Proof.

We will prove the bound first in the case that $\epsilon=1$. We interpret the
Bessel process as the Euclidean norm of a three-dimensional standard Brownian
motion, that is, $R(t)=|\vec{B}(t)|.$ It is a helpful fact that if
$\vec{B}=(B_{1},B_{2},B_{3})$, then through a union bound and well-known
formulas for the distribution of the supremum of a Brownian motion on $[0,1]$,

$\mathbb{P}(\sup_{t\in[0,1]}|\vec{B}(t)|>a)\leq
3\mathbb{P}(\sup_{t\in[0,1]}B_{1}(t)>a/3)\leq 6e^{-a^{2}/18}.$ (5.1)

It is also helpful to note the following fact

$\mathbb{P}\bigg{(}|\vec{B}(1)|>b,\;\inf_{t\in[0,1]}|\vec{B}(t)|\leq
1\bigg{|}|\vec{B}(0)|=a\bigg{)}\leq Ce^{-\frac{1}{36}(a^{2}+b^{2})},$ (5.2)

where $C>0$ does not depend on $a,b>0.$ This follows from the strong Markov
property. Indeed, define the stopping time $\tau$ as the first time $t$ that
$|\vec{B}(t)|=1$ and the Brownian motion
$\vec{W}(t):=\vec{B}(t+\tau)-\vec{B}(\tau)$. The event
$\\{|\vec{B}(1)|>b,\;\inf_{t\in[0,1]}|\vec{B}(t)|\leq 1\\}$ is contained in
the event $\\{\sup_{t\in[0,1]}|\vec{W}(t)|>b-1\\},$ and the second event is
moreover independent of the $\mathcal{F}^{B}_{\tau}$-measurable event
$\\{\inf_{t\in[0,1]}|\vec{B}(t)|<1\\}=\\{\tau\leq 1\\}.$ Therefore, by
conditioning on the latter event, we see that the left side of (5.2) is
bounded above by

$\mathbb{P}(\sup_{t\in[0,1]}|\vec{W}(t)|>b-1)\cdot\mathbb{P}(\inf_{t\in[0,1]}|\vec{B}(t)|<1|\vec{B}(0)=a).$

By (5.1) this is bounded above by $36e^{-\frac{1}{18}((a-1)^{2}+(b-1)^{2})}.$
Now use $(b-1)^{2}\geq\frac{1}{2}b^{2}-1^{2}$ and
$(a-1)^{2}\geq\frac{1}{2}a^{2}-1^{2}$ and we finally arrive at (5.2) with
$C=36e^{\frac{1}{9}}$.

We return to proving the lemma in the case $\epsilon=1$. Without loss of
generality, we will assume that $s_{i+1}-s_{i}>3$. In order for
$\inf_{t\in[s_{i},s_{i}+1]}R(t)\leq 1$ for all $1\leq i\leq p+1$, at least one
of the following three things must happen for each $i=1,...,p$:

  1. (1)

$\sup_{t\in[s_{i},s_{i}+1]}|B(t)-B(s_{i}+1)|>9\sqrt{\log(s_{i}-s_{i-1})}$.

  2. (2)

$|B(s_{i+1})-B(s_{i}+1)|\leq 18\sqrt{\log(s_{i}-s_{i-1})}.$

  3. (3)

$\sup_{t\in[s_{i+1},s_{i+1}+1]}|B(t)-B(s_{i+1})|>9\sqrt{\log(s_{i}-s_{i-1})}$.

We partition $[s_{1},s_{p})$ into $2p$ disjoint intervals of the form
$[0,s_{1}),[s_{1},s_{1}+1),[s_{1}+1,s_{2}),[s_{2},s_{2}+1),[s_{2}+1,s_{3}),...,[s_{p},s_{p}+1),$
which we denote by $I_{1},J_{1},I_{2},J_{2},I_{3},...,J_{p}$. For a given
realization of the Brownian motion $\vec{B}$, we say that an interval $I_{i}$
or $J_{i}$ is active if one of the conditions above holds on that interval. We
have shown that among any three consecutive sub-intervals in the partition
$I_{1},J_{1},I_{2},J_{2},I_{3},...,J_{p}$, at least one must be active. Some
intervals may furthermore be doubly active in the sense that the conditions
within the above conditions may be satisfied for two distinct values of $i$.
Furthermore, at least one of the first two sub-intervals $I_{1},J_{1}$ must be
active. Consequently, by counting the doubly active intervals twice, there
must be at least $p$ sub-intervals from
$I_{1},J_{1},I_{2},J_{2},I_{3},...,J_{p}$ that are active, each corresponding
to a unique value of $i$ in the list above. Notice that
$e^{-(9\sqrt{\log(s_{i+1}-s_{i})})^{2}/18}\leq 1\wedge(s_{i+1}-s_{i})^{-2}\leq
1\wedge(s_{i+1}-s_{i})^{-3/2}$, and the standard $3$-dimensional Gaussian tail
bound tells us that

$\mathbb{P}\big{(}|B(s_{i+1})-B(s_{i}+1)|\leq
18\sqrt{\log(s_{i+1}-s_{i})}\big{)}\leq\max\\{1,18^{3}\cdot(s_{i+1}-s_{i})^{-3/2}(1+\log^{3/2}_{+}(s_{i+1}-s_{i}))\\}.$
(5.3)

By applying a union bound over all possible choices of $p$-tuples of active
and doubly active sub-intervals from
$I_{1},J_{1},I_{2},J_{2},I_{3},...,J_{p}$, and applying the independence of
increments of the $3$-dimensional Brownian motion, we obtain the bound in the
lemma. This is a consequence of (5.2) for doubly active intervals and (5.1) or
(5.3) for active ones. Each of the bounds for the increments $[s_{i+1},s_{i}]$
will be used exactly once. This proves the claim when $\epsilon=1$.

For arbitrary $\epsilon>0$, we need to explain why we obtain a factor
$\epsilon^{p}$. We begin by explaining this factor in the case that $p=1$. If
we condition on reaching the value $1$ inside the interval $[s_{i},s_{i}+1]$,
the probability of reaching $\epsilon$ inside that interval is bounded by the
probability of reaching $\epsilon$ ever again after time $s_{i}$. Using the
Markov property, and the optional stopping theorem for the martingale $R^{-1}$
(stopping when this martingale hits $\epsilon^{-1})$, we show that the latter
probability is exactly $\epsilon.$ Therefore, for the $p=1$ case we have the
bound

$\displaystyle\mathbb{P}\big{(}\inf_{t\in[s_{i},s_{i}+1]}R(t)<\epsilon\big{)}$
$\displaystyle=\mathbb{P}\bigg{(}\inf_{t\in[s_{i},s_{i}+1]}R(t)<\epsilon\bigg{|}\inf_{t\in[s_{i},s_{i}+1]}R(t)<1\bigg{)}\mathbb{P}\big{(}\inf_{t\in[s_{i},s_{i}+1]}R(t)<1\big{)}$
$\displaystyle\leq\epsilon\;\;\cdot\;\;s_{1}^{-3/2}(1+\log_{+}^{3/2}s_{1}).$

While this illustrates the bound in the case $p=1$, the same argument extends
in a straightforward way to general values of $p$. ∎

######  Proposition 5.2.

Let $R$ be a standard two-sided $3$-dimensional Bessel process. Let
$N_{\epsilon}(R)$ denote the smallest number of intervals of length
$\epsilon^{2}$ needed to cover the random set
$\\{t\in\mathbb{R}:R(t)\leq\epsilon\\}.$ Then
$\sup_{\epsilon\in(0,1]}\mathbb{E}[N_{\epsilon}(R)^{p}]<\infty$ for all $p\geq
1.$

###### Proof.

This random variable is bounded above by

$\epsilon^{-2}\int_{\mathbb{R}}1_{\\{\inf_{t\in[s,s+\epsilon^{2}]}R(t)\leq\epsilon\\}}ds.$

By scale invariance of the Bessel process, we know that the above random
variable has the same law as

$\int_{\mathbb{R}}1_{\\{\inf_{t\in[s,s+1]}R(t)\leq 1\\}}ds.$

Let $p$ be an even integer, and take the $p^{th}$ moment, then apply Fubini to
interchange the integral and expectation. We obtain that

$\displaystyle\mathbb{E}[N_{\epsilon}(R)^{p}]$
$\displaystyle\leq\int_{\mathbb{R}^{p}}\mathbb{P}\bigg{(}\inf_{t\in[s_{i},s_{i+1}]}R(t)\leq
1,\;\forall 1\leq i\leq p\bigg{)}ds_{1}\cdots ds_{p}$
$\displaystyle=2^{p}p!\cdot\int_{\\{0<s_{1}<...<s_{p}\\}}\mathbb{P}\bigg{(}\inf_{t\in[s_{i},s_{i}+1]}R(t)\leq
1,\;\forall 1\leq i\leq p\bigg{)}ds_{1}\cdots ds_{p}$

By 5.1 the integrand is bounded above by

$\displaystyle
C_{p}\prod_{i=0}^{p-1}\min\big{\\{}1,(s_{i+1}-s_{i})^{-3/2}(1+\log_{+}^{3/2}(s_{i+1}-s_{i}))\big{\\}},$

where $s_{0}:=0$. The fact that previous integral is finite follow from this
bound. ∎

######  Definition 5.3.

We define a Bessel process pinned at $t\in\mathbb{R}$ to be the process
$R(\bullet-t)$ where $R$ is a two-sided $3$-dimensional Bessel process,
centered at zero. We will denote by $\mathbb{P}_{\mathrm{Bes3}}^{t_{1},t_{2}}$
to be the law on the canonical space $C(\mathbb{R})^{2},$ of two independent
Bessel processes $R_{1},R_{2}$ pinned at $t_{1},t_{2}$ respectively.

######  Proposition 5.4 (The key estimate).

Fix $\delta\in(0,1/2)$ and $0<\bar{\gamma}<\gamma<1$. Then there exists $C>0$
so that for all $\epsilon\in(0,1]$ we have the estimate

$\displaystyle\sup_{|t_{1}-t_{2}|\geq\epsilon^{-\gamma}}$
$\displaystyle\mathbb{P}_{\mathrm{Bes3}}^{t_{1},t_{2}}\bigg{(}\mathrm{there\;\;exist\;\;}s_{1},s_{2}\in\mathbb{R}\mathrm{\;\;such\;\;that\;\;}R_{1}(s_{1})+R_{2}(s_{2})\leq\epsilon^{-\delta}\mathrm{\;\;and\;\;}|s_{1}-s_{2}|\leq
2\epsilon^{-\bar{\gamma}}\bigg{)}$ $\displaystyle\leq
C\epsilon^{\frac{3}{2}\gamma-\delta(2+\gamma-\bar{\gamma})-\frac{1}{2}\bar{\gamma}}.$

For the black noise result, we will see that this bound is most useful when
$\gamma$ is slightly smaller than $1/3$ and $\delta,\bar{\gamma}$ are very
close to 0. In this case, note that the right side can be made arbitrarily
close to $\epsilon^{1/2}$. In practice, we will only use this estimate with
$\gamma=4/15$ and $\bar{\gamma}=\delta$ close to 0.

###### Proof.

The proof will proceed using a simple union bound. If such values of
$s_{1},s_{2}\in\mathbb{R}$ exist as specified above, then there exists some
$n\in\mathbb{Z}$ such that all of the following simultaneously occur:
$|s_{1}-n|\leq\epsilon^{-\bar{\gamma}}$,
$|s_{s}-n|\leq\epsilon^{-\bar{\gamma}}$,
$R_{1}(s_{1})\leq\epsilon^{-\delta},R_{2}(s_{2})\leq\epsilon^{-\delta}.$
Applying this union bound over all $n\in\mathbb{Z}$, we find that the desired
probability is upper bounded by

$\sum_{n\in\mathbb{Z}}\mathbb{P}_{\mathrm{Bes3}}^{t_{1},t_{2}}\bigg{(}\text{
there exist }s_{1},s_{2}\in\mathbb{R}\text{ such that
}|s_{1}-\epsilon^{-\bar{\gamma}}n|\leq\epsilon^{-\bar{\gamma}},|s_{2}-\epsilon^{-\bar{\gamma}}n|\leq\epsilon^{-\bar{\gamma}},R_{1}(s_{1})\leq\epsilon^{-\delta},R_{2}(s_{2})\leq\epsilon^{-\delta}\bigg{)}.$

Without any loss of generality we can replace $(t_{1},t_{2})$ by
$(0,t_{2}-t_{1})$ and assume that $T:=t_{2}-t_{1}>0,$ thus by assumption
$T>\epsilon^{-\gamma}$. We can estimate the probability in the summand, using
the scaling invariance of the Bessel process to rewrite

$\displaystyle\mathbb{P}_{\mathrm{Bes3}}^{0,T}\bigg{(}\text{ there exist
}s_{1},s_{2}\in\mathbb{R}\text{ such that
}|s_{1}-\epsilon^{-\bar{\gamma}}n|\leq\epsilon^{-\bar{\gamma}},|s_{2}-\epsilon^{-\bar{\gamma}}n|\leq\epsilon^{-\bar{\gamma}},R_{1}(s_{1})\leq\epsilon^{-\delta},R_{2}(s_{2})\leq\epsilon^{-\delta}\bigg{)}$
$\displaystyle=\mathbb{P}_{\mathrm{Bes3}}^{0,\epsilon^{\bar{\gamma}}T}\bigg{(}\text{
there exist }s_{1},s_{2}\in\mathbb{R}\text{ such that }|s_{1}-n|\leq
1,|s_{2}-n|\leq
1,R_{1}(s_{1})\leq\epsilon^{\frac{1}{2}\bar{\gamma}-\delta},R_{2}(s_{2})\leq\epsilon^{\frac{1}{2}\bar{\gamma}-\delta}\bigg{)}$
$\displaystyle=\prod_{i=1}^{2}\mathbb{P}_{\mathrm{Bes3}}^{0,\epsilon^{\bar{\gamma}}T}\bigg{(}\text{
there exists }s_{i}\in[n-1,n+1]\text{ such that
}R_{i}(s_{i})\leq\epsilon^{\frac{1}{2}\bar{\gamma}-\delta}\bigg{)}$
$\displaystyle\leq
C\bigg{[}\epsilon^{\frac{1}{2}\bar{\gamma}-\delta}|n|^{-\frac{3}{2}+\delta}\bigg{]}\bigg{[}\epsilon^{\frac{1}{2}\bar{\gamma}-\delta}|\epsilon^{\bar{\gamma}}T-n|^{-\frac{3}{2}+\delta}\bigg{]}$
$\displaystyle=C\epsilon^{\bar{\gamma}-2\delta}|n|^{-\frac{3}{2}+\delta}|\epsilon^{\bar{\gamma}}T-n|^{-\frac{3}{2}+\delta}.$

In the third line, we used the independence of $R_{1}$ and $R_{2}$, and 5.1
with $p=1$ in the fourth line (we replaced the logarithmic correction by a
weaker bound of size $|n|^{\delta}$ for simplicity). Now notice that uniformly
over $N\in\mathbb{N}$ we have by symmetry

$\sum_{n=1}^{N}\frac{1}{n^{\frac{3}{2}-\delta}(N-n)^{\frac{3}{2}-\delta}}\leq
2\sum_{1\leq n\leq
N/2}\frac{1}{n^{\frac{3}{2}-\delta}(N-n)^{\frac{3}{2}-\delta}}\leq
2\left(\frac{2}{N}\right)^{\frac{3}{2}-\delta}\sum_{1\leq n\leq
N/2}n^{-\frac{3}{2}+\delta}\leq\frac{2^{5/2}\zeta(\tfrac{3}{2}-\delta)}{N^{\frac{3}{2}-\delta}}.$

Using this bound with
$N=\epsilon^{\bar{\gamma}}T\geq\epsilon^{\bar{\gamma}-\gamma},$ one finds that
the desired probability is upper bounded by
$C\epsilon^{\bar{\gamma}-2\delta}(\epsilon^{\bar{\gamma}}T)^{-\frac{3}{2}+\delta}\leq
C\epsilon^{\bar{\gamma}-2\delta}(\epsilon^{\bar{\gamma}-\gamma})^{-\frac{3}{2}+\delta}=C\epsilon^{\frac{1}{2}(3\gamma-\bar{\gamma})-\delta(2+\gamma-\bar{\gamma})}.$
∎

##  6\. Proof of the main theorem

######  Proposition 6.1.

Consider a standard Brownian motion $B$ on $[0,1]$, and let $T$ denote its
unique argmax. Fix $\alpha\in(0,1)$. Define the process
$U:=(B(T)-B(T+t))_{t\in[-\alpha T,\alpha(1-T)]}$. Consider a three-dimensional
Bessel process $R$ independent of $T$ and $B$, and define the process
$V:=(R(t))_{t\in[-\alpha T,\alpha(1-T)]}$. Then $U$ is absolutely continuous
with respect to $V$, and the Radon-Nikodym derivative is deterministically
bounded by $C(1-\alpha)^{-1/2}$ for some universal constant $C>0$.

In the statement of 6.1, we can think of a continuous function on a random
closed interval $I\subset[-1,1]$ as an element of $C[-1,1]$ that is constant
away from $I$. The last statement is sharp: if $\alpha=1$ then the Radon-
Nikodym is only in $L^{3-\delta}$ as opposed to $L^{\infty}$, as discussed in
[DSV22, Theorem 2.15].

###### Proof.

Conditioned on $T$, the process $(B(T)-B(T+t))_{t\in[-T,1-T]}$ is a two-sided
Brownian meander, as discussed in [DSV22, Theorem 2.15]. The left side is a
meander of length $T$ and the right side is a meander of length $1-T$. When we
condition on $T$, these two meanders are independent of one another.
Therefore, it suffices to show that for a Brownian meander $X$ of length $1$,
the Radon-Nikodym derivative $(X(t))_{t\in[0,\alpha]}$ is absolutely
continuous with respect to a Bessel process on the same interval
$(R(t))_{t\in[0,\alpha]}.$

In the proof of [DSV22, Theorem 2.15], the authors observe that if $\alpha=1$,
then the Radon-Nikodym derivative is simply given by $\sqrt{\pi/2}\cdot
R(1)^{-1}.$ Let $\mathcal{F}^{R}_{t}$ denote the filtration of the Bessel
process. Then for general $\alpha\in[0,1]$ the Radon-Nikodym derivative of $X$
with respect to $R$ on the interval $[0,\alpha]$ is simply given by

$\sqrt{\pi/2}\cdot\mathbb{E}[R(1)^{-1}|\mathcal{F}^{R}_{\alpha}]=\sqrt{\pi/2}\cdot
f_{\alpha}(R(\alpha)),$

where the expectation is taken with respect to a $3$-dimensional Bessel
Process and $f_{\alpha}:[0,\infty)\to(0,\infty)$ is the unique function
satisfying

$f_{\alpha}(|\vec{x}|)=(2\pi(1-\alpha))^{-3/2}\int_{\mathbb{R}^{3}}e^{-\frac{|\vec{y}|^{2}}{2(1-\alpha)}}|\vec{x}-\vec{y}|^{-1}d\vec{y}.$

In other words, $\vec{x}\mapsto f_{\alpha}(|\vec{x}|)$ is the radial function
on $\mathbb{R}^{3}$ given by convolving the three-dimensional heat kernel with
the harmonic function $\vec{x}\mapsto|\vec{x}|^{-1}$. The equality above
follows from the Markov property and the fact that the Bessel process $R$ can
be realized as the Euclidean norm of a three-dimensional Brownian motion. A
direct computation shows that $\|f_{\alpha}\|_{L^{\infty}(\mathbb{R}^{3})}\leq
C(1-\alpha)^{-1/2}$ for a universal constant $C.$ ∎

######  Definition 6.2.

An Airy process of scale $s>0$ centered at $y\in\mathbb{R}$ is the process

$\mathcal{A}_{y}^{s}(x):=s^{1/3}\mathcal{P}_{1}(s^{-2/3}(x-y)),\;\;\;\;\;\;x\in\mathbb{R}$

where $\mathcal{P}_{1}$ is the parabolic Airy2 process; the top curve of the
line ensemble in 1.5.

######  Proposition 6.3.

Fix $\eta\in(0,1)$ and let $\mathcal{A}_{x}^{t}$ and
$\tilde{\mathcal{A}_{y}^{s}}$ denote two independent Airy processes of scales
$t,s\in(\eta,1]$ centered at $x,y\in\mathbb{R}$ respectively. We also define
the notation $\mathcal{V}:=\mathcal{A}_{x}^{t}+\tilde{\mathcal{A}_{y}^{s}}$.
Let $I=[a,b]$ be a compact interval. Then
$(\mathcal{V}(t+a)-V(a))_{t\in[0,b-a]}$ is absolutely continuous with respect
to a Brownian motion of diffusion rate $4$. Furthermore, for all $p\geq 1$,
the Radon-Nikodym derivative has a finite $p^{\text{th}}$ moment which is
bounded above by $Ce^{Cp^{2}(a^{2}+b^{2})}$ for some universal constant
$C=C(p,\eta,x,y)>0$ not depending on $p,a,b,s,t$ such that $0\leq b-a\leq 10$.

The choice $b-a\leq 10$ is arbitrary, but we will never use this bound for
intervals larger than length $10$.

###### Proof.

We reduce the claim to the case where $a=-b.$ Since the Airy2 process is
stationary without the addition of the parabola, and since the difference of
two quadratic functions is linear, translating the interval $I$ to be centered
at the origin amounts to adding a linear drift term of the form $mt+b$ to the
process $\mathcal{V}(t)$, for some $m$ whose absolute value can be bounded
above by $C(|a|+|b|)$.

The Airy process on $[-a,a]$ with $|a|\leq 10$ is absolutely continuous with
respect to a standard Brownian motion on that interval, and moreover the
Radon-Nikodym derivative is deterministically bounded due to [Dau23, Corollary
1.3].

Brownian motion with a linear drift of slope $m$ on $[0,b-a]$ is absolutely
continuous with respect to Brownian motion with no drift on that same
interval, with Radon-Nikodym derivative given by $e^{\pm
mB_{b-a}-\frac{m^{2}}{2}(b-a)^{2}}.$ The $p^{\text{th}}$ moment of this random
variable is simply $e^{(p^{2}-p)m^{2}(b-a)^{2}}.$ Combining this with the
result of the previous paragraph finishes the proof. ∎

######  Proposition 6.4.

(The crucial bound) Fix $\eta>0$ and recall $\Gamma$ from (4.3). Let
$\mathcal{A}_{x}^{t},\tilde{\mathcal{A}_{y}^{s}},\bar{\mathcal{A}_{u}^{a}},\hat{\mathcal{A}_{v}^{b}}$
denote four independent Airy processes of scale $t,s,a,b$ respectively,
centered at fixed real numbers $x,y,u,v$ respectively. For any $\delta>0$,
there exists a constant $C=C(\eta,x,y,u,v,\delta)>0$ independent of
$\delta\in(0,1/3)$, such that we have the following bound uniformly over
$\epsilon\in(0,1]$:

$\sup_{\eta\leq s,t,a,b\leq
1}\mathbb{P}\big{(}\Gamma(\mathcal{A}_{x}^{t}+\tilde{\mathcal{A}_{y}^{s}},\bar{\mathcal{A}_{u}^{a}}+\hat{\mathcal{A}_{v}^{b}},\epsilon^{\frac{1}{3}-\delta})\leq\epsilon^{\frac{2}{3}-\delta}\big{)}\leq
C\epsilon^{\frac{2}{5}-4\delta}.$

We are not sure whether $\epsilon^{\frac{2}{5}}$ is the optimal bound, however
we are unable to improve it through the methods of this paper. The crucial
input to proving that the directed landscape is a black noise is the fact that
$\frac{2}{5}-\delta$ is larger than $1/3$ for sufficiently small $\delta$. The
bound in 6.4 is sufficient because it will give us a bound almost as small as
$\epsilon^{2/3}\cdot\epsilon^{2/5}=\epsilon^{16/15}$ in (6.3). This eventually
implies a bound in (2.5) with $\varrho$ smaller than $1/15$.

###### Proof.

In this proof, we leverage the cube-exponential tail decay of the argmaxes of
the Airy processes, and combine this with the results of 6.1 and 6.3 to write
everything in terms of Bessel processes, where we can apply the bound from
5.4. We make the following definitions:

  * •

Denote $\mathcal{V}_{1}:=\mathcal{A}_{x}^{t}+\tilde{\mathcal{A}_{y}^{s}}$ and
$\mathcal{V}_{2}:=\bar{\mathcal{A}_{u}^{a}}+\hat{\mathcal{A}_{v}^{b}}$
henceforth.

  * •

Let $G_{1}$ denote the argmax on $\mathbb{R}$ of $\mathcal{V}_{1}$ and let
$G_{2}$ denote the argmax on $\mathbb{R}$ of $\mathcal{V}_{2}$.

  * •

For $k\in\mathbb{Z}$, let $E^{(1)}_{k}$ denote the event that
$G_{1}\in[k,k+5]$ and let $E^{(2)}_{k}$ denote the event that
$G_{2}\in[k,k+5]$. Note that $E^{(i)}_{k}$ are independent for distinct $i$.

  * •

For $k_{1},k_{2},\ell\in\mathbb{Z}$ let $F_{k_{1},k_{2},\ell}$ denote the
event that $G_{1}\in[k_{1},k_{1}+5]$ , $G_{2}\in[k_{2},k_{2}+5]$, and
furthermore the maximum values of both $\mathcal{V}_{1},\mathcal{V}_{2}$ on
the interval $[\ell,\ell+5]$ are within 1 of their absolute maximum values on
all of $\mathbb{R}.$

  * •

For $i=1,2$ and $k\in\mathbb{Z}$ let $H^{(i)}_{k}$ denote the event that there
exist two points of distance at least $k$ from each other such that the values
of $\mathcal{V}_{i}$ at both points are within 1 of
$\max_{z\in\mathbb{R}}\mathcal{V}_{i}(z).$ Note that $H^{(i)}_{k}$ are
independent for distinct $i$, and

$F_{k_{1},k_{2},\ell}\subset H^{(1)}_{k_{1}-\ell}\cap H^{(2)}_{k_{2}-\ell}.$

  * •

For $k\in\mathbb{Z}$ and $i=1,2$, let $G_{i}^{(k)}$ denote the argmax on
$[k,k+5]$ of $\mathcal{V}_{i}$ and $M_{k}^{(i)}:=\mathcal{V}_{i}(G^{(k)}_{i})$
denote the maximum value of $\mathcal{V}_{i}$ on $[k,k+5]$.

  * •

For $\ell\in\mathbb{Z}$ let $K_{\ell}^{\epsilon}$ denote the event that there
are points
$z_{1}\in[\frac{4}{5}\ell+\frac{1}{5}G_{1}^{(\ell)},\frac{4}{5}(\ell+5)+\frac{1}{5}G_{1}^{(\ell)}]\subset[\ell,\ell+5]$
and
$z_{2}\in[\frac{4}{5}\ell+\frac{1}{5}G_{2}^{(\ell)},\frac{4}{5}(\ell+5)+\frac{1}{5}G_{2}^{(\ell)}]\subset[\ell,\ell+5]$
such that $|z_{1}-z_{2}|\leq\epsilon^{\frac{2}{5}}$, and moreover
$\mathcal{V}_{i}(z_{i})>M^{(i)}_{\ell}-\epsilon^{\frac{1}{3}-\delta}$ for
$i=1,2.$

We use a union bound for the probability that