# The KPZ equation and the directed landscape

Xuan Wu  Department of Mathematics, University of Chicago, Eckhart Hall 5734 S
University Ave Chicago IL, 60637<EMAIL_ADDRESS>
###### Abstract.

This paper studies the convergence of the narrow wedge solutions of the KPZ
equation to the Airy sheet and the direct landscape. Our main approach is
inspired by the polymer free energy interpretation of the KPZ equation. We
derive a simple inequality relating Busemann functions and quantiles of the
polymer measure (Lemma 4.6) and this is one of the main ingredients for
proving the convergence. Another main ingredient is Proposition 3.5, an
identity for the geometric RSK correspondence.

##  1\. Introduction

###  1.1. Kardar-Parisi-Zhang equation

This paper investigates the large-time asymptotics for solutions to the
Kardar-Parisi-Zhang (KPZ) equation. The KPZ equation was introduced in 1986 by
Kardar, Parisi and Zhang [KPZ86] as a model for random interface growth. In
one spatial dimension (sometimes also called (1+1)-dimension to emphasize that
time is one dimension too), it describes the evolution of a function
$\mathcal{H}(t,y)$ recording the height of an interface at time $t$ above
position $y$. The KPZ equation is written formally as a stochastic partial
differential equation (SPDE),

(1.1)
$\partial_{t}\mathcal{H}=\frac{1}{2}\partial_{y}^{2}\mathcal{H}+\frac{1}{2}(\partial_{y}\mathcal{H})^{2}+\mathscr{\dot{W}},$

where $\mathscr{\dot{W}}$ is a space-time white noise on $\mathbb{R}^{2}$ (for
mathematical background or literature review, see [Cor12, QS15] for instance).
The KPZ equation (1.1) is a canonical member of the associated KPZ
universality class. Its large-time asymptotics is believed to be governed by a
universal Markov process called the KPZ fixed point. We will discuss
progresses along this direction and we refer readers to [MQR21] for the
construction of the KPZ fixed point.

The KPZ equation is related to the stochastic heat equation (SHE) with
multiplicative noise through the Hopf–Cole transformation. Denote
$\mathcal{Z}(t,y)$ as the solution to the following SHE,

(1.2)
$\partial_{t}\mathcal{Z}=\frac{1}{2}\partial_{y}^{2}\mathcal{Z}+\mathcal{Z}\mathscr{\dot{W}}.$

The Hopf-Cole solution to the KPZ equation (1.1) is defined by taking
$\mathcal{H}(t,y)=\log\mathcal{Z}(t,y).$ It was first proved in [Mue91] that
$\mathcal{Z}(t,y)$ is almost surely strictly positive (with positive initial
conditions), which justifies the validity of the transform. The fundamental
solutions to the SHE (1.2) are of great importance. For fixed
$(s,x)\in\mathbb{R}^{2}$, we denote by $\mathcal{Z}(s,x;t,y)$ the solution to
(1.2) on $t>s$, $y\in\mathbb{R}$ with the delta initial condition
$\mathcal{Z}(s,x;s,y)=\delta(y-x)$. We take a logarithm and define the narrow
wedge solutions to (1.1):

(1.3) $\displaystyle\mathcal{H}(s,x;t,y)\coloneqq\log\mathcal{Z}(s,x;t,y).$

Let $\mathbb{R}^{4}_{+}\coloneqq\\{(s,x,t,y)\in\mathbb{R}^{4}\,|\,s<t\\}$. It
is recently proved in [Alb+22] that $\\{\mathcal{Z}(s,x;t,y),\
(s,x;t,y)\in\mathbb{R}^{4}_{+}\\}$ can be coupled in one probability space to
form a process on $\mathbb{R}_{+}^{4}$ with many desired features. In the
following theorem, we collect some of the results in [Alb+22]. We formulate
them in terms of narrow wedge solutions which are more suitable for our
purpose.

######  Theorem 1.1.

There exists a coupling of narrow wedge solutions $\\{\mathcal{H}(s,x;t,y),\
(s,x;t,y)\in\mathbb{R}^{4}_{+}\\}$ with the following properties.

  1. (1)

$\mathcal{H}(s,x;t,y)$ is a random continuous function on $\mathbb{R}^{4}_{+}$
.

  2. (2)

Almost surely for all $(s,x,t,y)\in\mathbb{R}^{4}_{+}$ and $r\in(s,t)$, it
holds that

(1.4)
$\displaystyle\exp\big{(}\mathcal{H}(s,x;t,y)\big{)}=\int_{-\infty}^{\infty}\exp\big{(}\mathcal{H}(s,x;r,z)+\mathcal{H}(r,z;t,y)\big{)}\,dz.$

  3. (3)

For any fixed $(r,z)\in\mathbb{R}^{2}$, as
$C(\mathbb{R}^{4}_{+},\mathbb{R})$-valued random variables,
$\mathcal{H}(s+r,x+u;t+r,y+u)\overset{d}{=}\mathcal{H}(s,x;t,y).$ Also,
$\mathcal{H}(-t,y;-s,x)\overset{d}{=}\mathcal{H}(s,x;t,y).$

  4. (4)

Fix finitely many disjoint open intervals $\\{(s_{j},t_{j})\\}_{j=1}^{m}$. The
random functions $\\{\mathcal{H}(s_{j},\cdot;t_{j},\cdot)\\}_{j=1}^{m}$ are
independent.

For $t=T>s=0$ fixed, the marginal $\mathcal{H}(0,x;T,y)$, viewed as a random
continuous function on $\mathbb{R}^{2}$, is called a KPZ sheet. We denote it
by

(1.5) $\mathcal{H}^{T}(x,y)\coloneqq\mathcal{H}(0,x;T,y).$

###  1.2. The Airy line ensemble, Airy sheets and the directed landscape

In this subsection, we introduce several central objects in the KPZ
universality class: the Airy line ensemble, Airy sheets and the directed
landscape.

######  Definition 1.2.

The stationary Airy line ensemble
$\tilde{\mathcal{A}}=\\{\tilde{\mathcal{A}}_{1}>\tilde{\mathcal{A}}_{2}>\cdots\\}$
is countable many random functions indexed by $\mathbb{N}$. The law of
$\tilde{\mathcal{A}}$ is uniquely determined by its determinantal structure.
More precisely, for any finite set
$I=\\{u_{1},\cdots,u_{k}\\}\subset\mathbb{R}$, the point process on
$I\times\mathbb{R}$ given by
$\\{(s,\tilde{\mathcal{A}}_{i}(s)):i\in\mathbb{N},s\in I\\}$ is a
determinantal point process with kernel given by

$K(s_{1},x_{1};s_{2},x_{2})=\left\\{\begin{array}[]{cc}\int_{0}^{\infty}e^{-z(s_{1}-s_{2})}\textup{Ai}(x_{1}+z)\textup{Ai}(x_{2}+z)dz&\textrm{if}\quad
s_{1}\geq s_{2},\\\
-\int_{-\infty}^{0}e^{-z(s_{1}-s_{2})}\textup{Ai}(x_{1}+z)\textup{Ai}(x_{2}+z)dz&\textrm{if}\quad
s_{1}<s_{2},\end{array}\right.$

where Ai is the Airy function. The (parabolic) Airy line ensemble
${\mathcal{A}}=\\{{\mathcal{A}}_{1}>{\mathcal{A}}_{2}>\dots\\}$ is defined by

(1.6) ${\mathcal{A}}_{i}(x):=\tilde{\mathcal{A}}_{i}(x)-x^{2}.$

The finite-dimensional marginal of the stationary Airy line ensemble was
introduced by Prähofer and Spohn [PS02] in which it was called the “multi-line
Airy process.” Later, Corwin and Hammond [CH14] showed that $\mathcal{A}$ can
be realized as a random continuous function on $\mathbb{N}\times\mathbb{R}$
through the Brownian Gibbs property. The first indexed random function,
$\tilde{\mathcal{A}}_{1}$, is of particular importance and is known as the
$\textup{Airy}_{2}$ process.

In the monumental work [DOV18], Dauvergne, Ortmann and Virág constructed Airy
sheets and the directed landscape via the Airy line ensemble. The directed
landscape can be viewed as “fundamental solutions” to the KPZ fixed point and
Airy sheets are fixed time marginals of the directed landscape. We follow the
presentation in [DOV18] and define Airy sheets and the directed landscape
through their characterization properties. We remark that it was proved in
[DOV18] that those properties uniquely determine the laws of Airy sheets and
the directed landscape respectively.

######  Definition 1.3.

The Airy sheet $\mathcal{S}(x,y)$ is a $C(\mathbb{R}^{2},\mathbb{R})$-valued
random variable which can be coupled with the Airy line ensemble $\mathcal{A}$
with the following properties.

  1. (1)

$\mathcal{S}(\cdot+t,\cdot+t)$ has the same distribution as
$\mathcal{S}(\cdot,\cdot).$

  2. (2)

$S(0,\cdot)=\mathcal{A}_{1}(\cdot)$.

  3. (3)

Almost surely for all $x>0$ and $y_{1},y_{2}$ in $\mathbb{R}$, we have

(1.7)
$\begin{split}\lim_{k\to\infty}\mathcal{A}[(-2^{-1/2}k^{1/2}x^{-1/2},k)\xrightarrow{\infty}(y_{2},1)]-\mathcal{A}[(-2^{-1/2}k^{1/2}x^{-1/2},k)\xrightarrow{\infty}&(y_{1},1)]\\\
&=\mathcal{S}(x,y_{2})-\mathcal{S}(x,y_{1}).\end{split}$

Here $\mathcal{A}[(x,k)\xrightarrow{\infty}(y,1)]$ is the last passage time
from $(x,k)$ to $(y,1)$ on the Airy line ensemble. We refer readers to (2.5)
in Section 2 for the detailed definition. For any $s>0$,
$s\mathcal{S}(s^{-2}x,s^{-2}y)$ is called an Airy sheet of scale $s$.

######  Definition 1.4.

The directed landscape $\mathcal{L}(s,x;t,y)$ is a
$C(\mathbb{R}^{4}_{+},\mathbb{R})$-valued random variable with the following
properties.

  1. (1)

Given $s<t$, $\mathcal{L}(s,\cdot;t,\cdot)$ is distributed as an Airy sheet of
scale $(t-s)^{1/3}$.

  2. (2)

For any finite disjoint open intervals $\\{(s_{j},t_{j})\\}_{j=1}^{m}$, the
random functions $\\{\mathcal{L}(s_{j},\cdot;t_{j},\cdot)\\}_{j=1}^{m}$ are
independent.

  3. (3)

Almost surely for all $s<r<t$ and $x,y\in\mathbb{R}$, it holds that

$\displaystyle\mathcal{L}(s,x;t,y)=\max_{z\in\mathbb{R}}\big{(}\mathcal{L}(s,x;r,z)+\mathcal{L}(r,z;t,y)\big{)}.$

As a direct consequence of Definition 1.4, we have the following description
on the marginal law of $\mathcal{L}$ when the time variables are restricted on
a finite set.

######  Corollary 1.5.

Fix a finite set $\\{t_{1}<t_{2}<\dots<t_{m}\\}$. Then the restriction of the
directed landscape, $\\{\mathcal{L}(t_{i},\cdot;t_{j},\cdot)\\}$ is uniquely
characterized by the following properties.

  1. (1)

For all $1\leq i<j\leq m$, $\mathcal{L}(t_{i},\cdot;t_{j},\cdot)$ is
distributed as an Airy sheet of scale $(t_{j}-t_{i})^{1/3}$.

  2. (2)

$\\{\mathcal{L}(t_{i},\cdot;t_{i+1},\cdot)\\}_{i=1}^{m-1}$ are independent.

  3. (3)

Almost surely for all $x,y\in\mathbb{R}$ and $1\leq i<k<j\leq m$,

$\displaystyle\mathcal{L}(t_{i},x;t_{k},y)=\max_{z\in\mathbb{R}}\left(\mathcal{L}(t_{i},x;t_{j},z)+\mathcal{L}(t_{j},z;t_{k},y)\right).$

###  1.3. Main results

In this subsection, we perform the $1:2:3$ scaling to $\mathcal{H}(s,x;t,y)$
and state our main results concerning the large-time asymptotics. For $T>0$,
define

(1.8)
$\displaystyle\mathfrak{H}^{T}(s,x;t,y):=2^{1/3}T^{-1/3}\mathcal{H}(Ts,2^{1/3}T^{2/3}x;Tt,2^{1/3}T^{2/3}y)+(t-s)2^{1/3}T^{2/3}/24.$

For $t=1$ and $s=0$ fixed, we call the marginal $\mathfrak{H}^{T}(0,x;1,y)$
the scaled KPZ sheet and denote it by

(1.9) $\mathfrak{H}^{T}(x,y)\coloneqq\mathfrak{H}^{T}(0,x;1,y).$

Note that from (1.5), $\mathfrak{H}^{T}$ can be expressed in terms of the KPZ
sheet $\mathcal{H}^{T}$ as

(1.10) $\mathfrak{H}^{T}(x,y)\coloneqq
2^{1/3}T^{-1/3}\mathcal{H}^{T}(2^{1/3}T^{2/3}x,2^{1/3}T^{2/3}y)+{2^{1/3}T^{2/3}}/{24}.$

It is conjectured that the KPZ fixed point describes the large-time behavior
of solutions to the KPZ equation (1.1). In [ACQ11], Amir, Corwin and Quastel
gave strong evidence for this conjecture and proved that
$\mathfrak{H}^{T}(0,0;1,0)$ converges to the Tracy-Widom law. Equivalently,
using the notation introduced above, they showed that
$\mathfrak{H}^{T}(0,0;1,0)$ converges in distribution to
$\mathcal{L}(0,0;1,0)$. Recently, a breakthrough was made by two groups,
Quastel-Sarkar [QS23] and Virág [Vir20]. The authors independently proved that
$\mathfrak{H}^{T}(0,0;1,y)$, as a random function on $\mathbb{R}$, converges
in distribution to $\mathcal{L}(0,0;1,y)$. We remark that in [QS23] and
[Vir20], the convergence can be shown for a large class of initial data.

In this paper, we establish the convergence of $\mathfrak{H}^{T}(s,x;t,y)$ to
$\mathcal{L}(s,x;t,y)$ as a four-parameter process and this is the content of
Theorem 1.6. Before stating the theorem, we note that for a topological space
$\mathcal{T}$, we always equip $C(\mathcal{T},\mathbb{R})$, the collection of
continuous functions on $\mathcal{T}$, with the topology of uniform
convergence on compact subsets.

######  Theorem 1.6.

Fix a finite set $\Lambda=\\{t_{1}<t_{2}<\dots<t_{m}\\}$,
$\\{\mathfrak{H}^{T}(t_{i},x;t_{j},y)\\}$ converges in distribution to the
directed landscape $\\{\mathcal{L}((t_{i},x);(t_{j},y))\\}$ as $T$ goes to
infinity. Here $\mathfrak{H}^{T}(t_{i},x;t_{j},y)$ and
$\mathcal{L}(t_{i},x;t_{j},y)$ are viewed as
$C(\Lambda^{2}_{+}\times\mathbb{R}^{2},\mathbb{R})$-valued random variables
with $\Lambda_{+}^{2}=\\{(s,t)\in\Lambda^{2}\,|\,s<t\\}$.

As a crucial middle step, we show the convergence of the scaled KPZ sheet to
the Airy sheet.

######  Theorem 1.7.

The scaled KPZ sheet $\mathfrak{H}^{T}(x,y)$ converges in distribution to the
Airy sheet $\mathcal{S}(x,y)$ as $T$ goes to infinity. Here
$\mathfrak{H}^{T}(x,y)$ and $\mathcal{S}(x,y)$ are viewed as
$C(\mathbb{R}^{2},\mathbb{R})$-valued random variables .

Our approach for Theorem 1.7 and Theorem 1.6 adopts the interpretation that
narrow wedge solutions are free energies for the continuum directed random
polymers (CDRP). We derive an identity of polymer free energy under the
geometric RSK correspondence, Proposition 3.5, which is a main ingredient of
our argument. We consider Busemann functions on the KPZ line ensemble under
the 1:2:3 KPZ scaling. Another key to our argument is an inequality relating
Busemann functions and quantiles of the polymer measure, see Lemma 4.6. These
allow us to establish the characterizing property (1.7) for all subsequential
limits of the scaled KPZ sheet through the Busemann functions. We note that
Lemma 4.6 holds deterministically and can be applied to other polymer models
such as the log-gamma polymer [Sep].

###  1.4. KPZ line ensembles

Motivated by recent developments on solvable directed polymer models,
O’Connell and Warren [OW16] defined multi-layer extension of the SHE. More
precisely, they defined a hierarchy of partition functions
$\mathcal{Z}_{i}(t,y)$,

(1.11)
$\mathcal{Z}_{i}(t,y)=p(t,y)^{i}\sum_{k=0}^{\infty}\int_{\Delta_{k}(t)}\int_{\mathbb{R}^{k}}R_{k}^{(i)}\left((t_{1},x_{1}),\cdots,(t_{k},x_{k})\right)\mathscr{\dot{W}}(dt_{1}dx_{1})\cdots\mathscr{\dot{W}}(dt_{k}dx_{k}),$

where $p(t,y)=(2\pi y)^{-1/2}e^{-y^{2}/(2t)}$,
$\Delta_{k}(t)=\\{0<t_{1}<\cdots<t_{k}<t\\}$, and $R_{k}^{(i)}$ is the
$k$-point correlation function for a collection of $i$ non-intersecting
Brownian bridges each of which starts at $0$ at time $0$ and ends at $y$ at
time $t$. For notational simplicity, set $\mathcal{Z}_{0}(t,y)\equiv 1$.

######  Definition 1.8.

For $t>0$ fixed, the KPZ line ensemble is a continuous $\mathbb{N}$-indexed
line ensemble $\mathcal{X}^{t}=\\{\mathcal{X}^{t}_{i}\\}_{i\in\mathbb{N}}$ on
$\mathbb{R}$ given by

(1.12)
$\mathcal{X}^{t}_{i}(y):=\log\left(\frac{\mathcal{Z}_{i}(t,y)}{\mathcal{Z}_{i-1}(t,y)}\right),$

with convention $\mathcal{Z}_{0}\equiv 1$. Note that
$\mathcal{X}^{t}_{1}(\cdot)$ equals $\mathcal{H}(0,0;t,\cdot)$, the time $t$
spatial process of the solution to the KPZ equation (1.1).

The KPZ line ensemble also arises as the scaling limits of O’Connell-Yor semi-
discrete polymers [OY01]. We will discuss this convergence in more detail in
Section 4. We note that in [CN17], the convergence to the KPZ line ensemble is
proved for a wide range of polymer models.

The nature of our proof of Theorem 1.7 leads us to the following conjecture
which relates the KPZ sheet and the Busemann functions on the KPZ line
ensemble. It can be viewed as a counterpart of (1.7).

######  Conjecture 1.9.

Fix $T>0$. There exists a coupling of the KPZ line ensemble
$\mathcal{X}^{T}(y)$ and the KPZ sheet $\mathcal{H}^{T}(x,y)$ such that the
following holds.

  1. (1)

$\mathcal{H}^{T}(0,\cdot)=\mathcal{X}^{T}_{1}(\cdot)$.

  2. (2)

For all $x>0$ and $y_{1},y_{2}$ in $\mathbb{R}$, we have

(1.13)
$\lim_{k\to\infty}\bigg{(}\mathcal{X}^{T}[(-kT/x,k)\to(y_{2},1)]-\mathcal{X}^{T}[(-kT/x,k)\to(y_{1},1)]\bigg{)}=\mathcal{H}^{T}(x,y_{2})-\mathcal{H}^{T}(x,y_{1}).$

Here $\mathcal{X}^{T}[(x,k)\to(y,1)]$ stands for the polymer free energy from
$(x,k)$ to $(y,1)$ on $\mathcal{X}^{T}$. We refer readers to (2.5) in Section
2 for the detailed definition.

Even though we do not have a proof for Conjecture 1.9, we are able to reduce
it to certain path regularity for the KPZ line ensemble.

######  Theorem 1.10.

Suppose for any $\varepsilon>0$ and $x>0$, it holds that

(1.14)
$\displaystyle\sum_{k=1}^{\infty}\mathbb{P}\bigg{(}\left|\mathcal{X}^{t}[(0,k+1)\to(x,1)]-k\log
x+\log k!\right|>\varepsilon k\bigg{)}<\infty.$

Then Conjecture 1.9 holds true.

###  1.5. Outline

Section 2 contains the definition of semi-discrete polymer and some basic
deterministic properties. In Section 3, we derive a crucial identity related
to the geometric RSK correspondence, (3.3) in Proposition 3.5. In Section 4,
we introduce the O’Connell-Yor polymer and its scaled versions. We prove
Theorems 1.6, 1.7 in Section 5 and Theorem 1.10 in Section 6. Appendix A
contains proofs for results in Section 2.

###  1.6. Notation

We would like to explain some notation here. The natural numbers are defined
to be $\mathbb{N}=\\{1,2,...\\}$ and $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$.
The positive rational numbers are denoted by $\mathbb{Q}^{+}$. For
$m\leq\ell\in\mathbb{N}_{0}$, we write $\llbracket m,\ell\rrbracket$ for
$\\{m,m+1,m+2,\dots,\ell\\}$. We use a special font $\mathsf{t}$ to denote a
sequence of positive numbers $\\{T_{1}<T_{2}<\dots\\}$ which goes to infinity.
We also denote by $\mathsf{n}$ a sequence of positive integers
$\\{n_{1}<n_{2}<\dots\\}$ which goes to infinity.

For a topological space $\mathcal{T}$, we equip $C(\mathcal{T},\mathbb{R})$,
the collection of continuous functions on $\mathcal{T}$, with the topology of
uniform convergence on compact subsets. A family of
$C(\mathcal{T},\mathbb{R})$-valued random variables converges in distribution
if the corresponding family of measures on $C(\mathcal{T},\mathbb{R})$
converges weakly.

###  1.7. Acknowledge

The author thanks Alan Hammond for the discussion related to the temporal
correlation of the KPZ equation. The author thanks Ivan Corwin for advice on a
draft of this paper.

##  2\. Semi-Discrete Polymer

In this section, we introduce semi-discrete polymers with a deterministic
environment and record some basic properties. The proofs for those properties
can be found in Appendix A.

A semi-discrete environment is given by finitely or countably many continuous
functions defined on an interval.

######  Definition 2.1.

For $n\in\mathbb{N}$ and an interval $I\subset\mathbb{R}$, we define

$\displaystyle C^{n}(I)\coloneqq\left\\{(f_{1},f_{2},\dots,f_{n})\,|\,f_{i}\in
C(I,\mathbb{R})\ \textup{for}\ i\in\llbracket 1,n\rrbracket\right\\}.$

For the special cases $I=(0,T)$ or $I=[0,T)$ for some $0<T$, we denote

$\displaystyle C^{n}(T)\coloneqq C^{n}((0,T))\ \textup{and}\
\overline{C}^{n}(T)\coloneqq C^{n}([0,T)).$

We define the up/right paths connecting two points as follows.

######  Definition 2.2.

For real numbers $x<y$ and positive integers $\ell\geq m$, we denote by
$\mathcal{Q}[(x,\ell)\to(y,m)]$ the collection of non-increasing càdlàg
functions $\pi:[x,y]\to\mathbb{N}$ with $\pi(x)\leq\ell$ and $\pi(y)=m$. We
refer to a member of $\mathcal{Q}[(x,\ell)\to(y,m)]$ as a path from $(x,\ell)$
to $(y,m)$.

There is an injective map from $\mathcal{Q}[(x,\ell)\to(y,m)]$ to
$\mathbb{R}^{\ell-m}$, $\pi\mapsto(t_{\ell},\dots,t_{m+1})$, given by

(2.1) $\displaystyle t_{j}=\inf\\{t\in[x,y]\,|\,\pi(t)\leq j-1\\}\
\textup{for}\ j\in\llbracket m+1,\ell\rrbracket.$

It is convenient to set

(2.2) $\displaystyle t_{\ell+1}=x,\ t_{m}=y.$

In particular, it holds that

(2.3) $\displaystyle\pi(t)=j\ \textup{for}\ t\in(t_{j+1},t_{j})\ \textup{and}\
j\in\llbracket m,\ell\rrbracket.$

The image of $\mathcal{Q}[(x,\ell)\to(y,m)]$ is the closed convex subset
$\\{x\leq t_{\ell}\leq t_{\ell-1}\leq\dots\leq t_{m+1}\leq y\\}.$ We often
abuse the notation and view $\mathcal{Q}[(x,\ell)\to(y,m)]$ as a subset of
$\mathbb{R}^{\ell-m}$.

For $f\in C^{n}([a,b])$ and $\pi\in\mathcal{Q}[(x,\ell)\to(y,m)]$ with
$(x,\ell),(y,m)\in[a,b]\times\llbracket 1,n\rrbracket$, define

(2.4) $\displaystyle
f(\pi)\coloneqq\sum_{j=m}^{\ell}f_{j}(t_{j})-f_{j}(t_{j+1}),$

where $t_{j}$ are given by (2.1) and (2.2). Let $d\pi$ be the Lebesgue measure
on $\mathcal{Q}[(x,\ell)\to(y,m)]$. For $\beta>0$, the $\beta$-free energy
from $(x,\ell)$ to $(y,m)$ is defined by

(2.5) $\displaystyle
f[(x,\ell)\xrightarrow{\beta}(y,m)]\coloneqq\beta^{-1}\log\int_{\mathcal{Q}[(x,\ell)\to(y,m)]}\exp\left(\beta{f(\pi)}\right)\,d\pi.$

We also allow $\beta=\infty$ and set

$\displaystyle
f[(x,\ell)\xrightarrow{\infty}(y,m)]=\max_{\pi\in\mathcal{Q}[(x,\ell)\to(y,m)]}f(\pi).$

For $\beta=1$, we denote $f[(x,\ell)\xrightarrow{1}(y,m)]$ by
$f[(x,\ell)\to(y,m)]$.

The following lemma follows directly from the definition above.

######  Lemma 2.3.

For any $k\in\llbracket m,\ell-1\rrbracket$, we have

$\displaystyle\exp\left(f[(x,\ell)\to(y,m)]\right)=\int_{x}^{y}\exp\left(f[(x,\ell)\to(z,k+1)]+f[(z,k)\to(y,m)]\right)\,dz.$

######  Lemma 2.4.

Fix $n\geq\ell\geq m\geq 1$, $a\leq x_{1}\leq x_{2}<b$ and $f\in
C^{n}([a,b])$. Then the function
$f[(x_{2},\ell)\to(y,m)]-f[(x_{1},\ell)\to(y,m)]$ is monotone non-decreasing
for $y\in(x_{2},b]$.

######  Lemma 2.5.

Fix $n\geq\ell\geq m\geq 1$, $a\leq x<y_{1}\leq y_{2}\leq b$ and $f\in
C^{n}([a,b])$. Then

$\displaystyle f[(x,\ell)\to(y_{1},m)]\leq
f[(x,\ell)\to(y_{2},m)]-f_{m}(y_{1})+f_{m}(y_{2}).$

######  Lemma 2.6.

Fix constants $a_{1},a_{2}>0$, $a_{3},a_{4}\in\mathbb{R}$ and
$\\{a_{5,i}\\}_{i\in\mathbb{N}}$. For $g$ defined by
$g_{i}(x)=a_{1}f_{i}(a_{2}x+a_{3})+a_{4}x+a_{5,i}$, it holds that

$\displaystyle g[(x,\ell)\xrightarrow{\beta}(y,k)]=a_{1}\cdot
f[(a_{2}x+a_{3},\ell)\xrightarrow{a_{1}\beta}(a_{2}y+a_{3},k)]+a_{4}(y-x)-\beta^{-1}(\ell-k)\log
a_{2}.$

Next, we consider multiple paths that do not cross each other. Let $\pi_{1}$
and $\pi_{2}$ be two paths which belong to
$\mathcal{Q}[(x_{1},\ell_{1})\to(y_{1},m_{1})]$ and
$\mathcal{Q}[(x_{2},\ell_{2})\to(y_{2},m_{2})]$ respectively. We write
$\pi_{1}\prec\pi_{2}$ if $\pi_{1}(t)<\pi_{2}(t)$ for all
$t\in(x_{1},y_{1})\cap(x_{2},y_{2})$. In this case, we say $\pi_{1}$ and
$\pi_{2}$ are non-intersecting. The next lemma shows that non-intersecting
paths form a closed convex set.

######  Lemma 2.7.

For $i\in\\{1,2\\}$, let $(x_{i},\ell_{i})$ and $(y_{i},m_{i})$ be pairs with
$x_{i}<y_{i}$ and $\ell_{i}\geq m_{i}$. Further assume $x_{1}\leq x_{2}$ and
$y_{1}\leq y_{2}$. Then the collection of $(\pi_{1},\pi_{2})$ in
$\mathcal{Q}[(x_{1},\ell_{1})\to(y_{1},m_{1})]\times\mathcal{Q}[(x_{2},\ell_{2})\to(y_{2},m_{2})]$
with $\pi_{1}\prec\pi_{2}$ is a closed convex subset in
$\mathbb{R}^{\ell_{1}-m_{1}}\times\mathbb{R}^{\ell_{2}-m_{2}}$.

A pair of sequences in $\mathbb{R}\times\mathbb{N}$ which can be connected by
non-intersecting paths is called an endpoint pair. Its definition is given
below.

######  Definition 2.8.

Fix $k\in\mathbb{N}$. Let $U=\\{(x_{i},\ell_{i})\\}_{i\in\llbracket
1,k\rrbracket}\ \textup{and}\ V=\\{(y_{i},m_{i})\\}_{i\in\llbracket
1,k\rrbracket}$ be two sequences in $\mathbb{R}\times\mathbb{N}$ with
$x_{i}<y_{i}$ and $\ell_{i}\geq m_{i}$ for all $i$. We denote by
$\mathcal{Q}[U\to V]$ the collection of paths $\pi=(\pi_{1},\dots,\pi_{k})$ in
$\prod_{i=1}^{k}\mathcal{Q}[(x_{i},\ell_{i})\to(y_{i},m_{i})]$ that satisfy
$\pi_{1}\prec\pi_{2}\prec\dots\prec\pi_{k}$. We call $(U,V)$ an endpoint pair
if $\mathcal{Q}[U\to V]$ is non-empty and $x_{i}\leq x_{i+1}$, $y_{j}\leq
y_{j+1}$ for $i\in\llbracket 1,k-1\rrbracket$. We may call $(U,V)$ a
$k$-endpoint pair to emphasize that there are $k$ pairs of endpoints.

Let $(U,V)$ be a $k$-endpoint pair and $f\in C^{n}([a,b])$ with
$U,V\subset[a,b]\times\llbracket 1,n\rrbracket$. For
$\pi=(\pi_{1},\dots,\pi_{k})\in\mathcal{Q}[U\to V]$, we define

$\displaystyle f(\pi)\coloneqq\sum_{i=1}^{k}f(\pi_{i}),$

where $f(\pi_{i})$ are given in (2.4). In view of Lemma 2.7, $\mathcal{Q}[U\to
V]$ can be identified as a closed convex set in a Euclidean space. Let
$p\in\mathbb{N}_{0}$ be the smallest integer such that $\mathcal{Q}[U\to V]$
is contained in a $p$-dimensional subspace and let $d\pi$ be the
$p$-dimensional Hausdorff measure on $\mathcal{Q}[U\to V]$. We define

(2.6) $\displaystyle f[U\to V]\coloneqq\log\int_{\mathcal{Q}[U\to
V]}\exp\left({f(\pi)}\right)\,d\pi.$

The following reversing map will be used in Section 3. For
$f\in\overline{C}^{n}(T)$ and $z\in(0,T)$, we define
$R_{z}f\in\overline{C}^{n}([0,z])$ by

(2.7) $\displaystyle(R_{z}f)_{i}(t):=-f_{n+1-i,}(z-t).$

Let $U=\\{(x_{i},\ell_{i})\\}_{i\in\llbracket 1,k\rrbracket}\ \textup{and}\
V=\\{(y_{i},m_{i})\\}_{i\in\llbracket 1,k\rrbracket}$ be an endpoint pair with
$U,V\subset[0,z]\times\llbracket 1,n\rrbracket$. Let

$\displaystyle\widetilde{V}\coloneqq$
$\displaystyle\\{(z-x_{k+1-i},n+1-\ell_{k+1-i})\\}_{i\in\llbracket
1,k\rrbracket},$ $\displaystyle\widetilde{U}\coloneqq$
$\displaystyle\\{(z-y_{k+1-i},n+1-m_{k+1-i})\\}_{i\in\llbracket
1,k\rrbracket}.$

######  Lemma 2.9.

Under the setting above, it holds that

$\displaystyle f[U\to V]=(R_{z}f)[\widetilde{U}\to\widetilde{V}].$

It is convenient to introduce certain special sequences in
$\mathbb{R}\times\mathbb{N}$. We use $(x,\ell)^{k}$ to denote the sequence
$\\{\underbrace{(x,\ell),(x,\ell),\dots,(x,\ell)}_{k\ \textup{terms}}\\}.$ For
$1\leq k\leq n$, we set

(2.8) $\begin{split}&V_{k}(x)\coloneqq\\{(x,1),(x,2),\dots,(x,k)\\},\\\
&V^{\prime}_{k}(x)\coloneqq\\{(x,2),(x,3),\dots,(x,k+1)\\},\\\
&U_{n,k}(x)\coloneqq\\{(x,n-k+1),(x,n-k+2),\dots,(x,n)\\}.\end{split}$

Moreover, we denote by $\mathcal{V}_{n,k}(x)$ the collection of
$\\{(x,\ell_{1}),(x,\ell_{2}),\dots,(x,\ell_{k})\\}$ that satisfies
$1\leq\ell_{1}<\ell_{2}<\dots<\ell_{k}\leq n$.

For paths in $\mathcal{Q}[(x,n)^{k}\to V]$, because of the non-intersecting
requirement, the starting points need to pile up. Therefore, they belong to
$\mathcal{Q}[U_{n,k}(x)\to V]$. This is the content of the next lemma.

######  Lemma 2.10.

Fix $n\geq 2$, $1\leq k\leq n-1$, $0<x<y<z<T$ and $f\in{C}^{n}(T)$. Then the
following identities hold.

(2.9) $f[(x,n)^{k+1}\to(y,1)^{k+1}]=f[U_{n,k+1}(x)\to V_{k+1}(y)].$ (2.10)
$f[\\{(x,n)^{k},(y,n)\\}\to(z,1)^{k+1}]=f[\\{U_{n,k}(x),(y,n)\\}\to
V_{k+1}(z)].$ (2.11)
$f[(x,n)^{k+1}\to\\{(y,1),(z,1)^{k}\\}]=f[U_{n,k+1}(x)\to\\{(y,1),V_{k}(z)\\}].$

Lastly, we introduce the down/right paths which are analogous to the up/right
paths.

######  Definition 2.11.

For real numbers $x<y$ and positive integers $m\leq\ell$, we use the notation
$\mathcal{Q}[(x,m)\searrow(y,\ell)]$ to denote the collection of non-
decreasing càdlàg functions $\rho:[x,y]\to\mathbb{N}$ with $\rho(x)\geq m$ and
$\rho(y)=\ell$.

There is an injective map from $\mathcal{Q}[(x,m)\searrow(y,\ell)]$ to
$\mathbb{R}^{\ell-m}$ given by

(2.12) $\displaystyle t_{j}=\inf\\{t\in[x,y]\,|\,\rho(t)\geq j+1\\}\
\textup{for}\ j\in\llbracket m,\ell-1\rrbracket.$

The image of $\mathcal{Q}[(x,m)\searrow(y,\ell)]$ is a closed convex subset
and we often view $\mathcal{Q}[(x,m)\searrow(y,\ell)]$ as the subset of
$\mathbb{R}^{\ell-m}$.

For $f\in C^{n}([a,b])$ and $\rho\in\mathcal{Q}[(x,m)\searrow(y,\ell)]$ with
$(x,m),(y,\ell)\in[a,b]\times\llbracket 1,n\rrbracket$, we define

$\displaystyle f(\rho)\coloneqq\sum_{j=m}^{\ell}f_{j}(t_{j})-f_{j}(t_{j-1}),$

where $t_{j},\ j\in\llbracket m,\ell-1\rrbracket$ are given by (2.12) and
$t_{m-1}=x,\ t_{\ell}=y$. Let $d\rho$ be the Lebesgue measure on
$\mathcal{Q}[(x,\ell)\to(y,m)]$. We define

(2.13) $\displaystyle
f[(x,\ell)\searrow(y,m)]\coloneqq-\log\int_{\mathcal{Q}[(x,\ell)\searrow(y,m)]}\exp\left(-{f(\rho)}\right)\,d\rho.$

We finish this section with the lemma below which shows
$f[V^{\prime}_{k-1}(x)\to V_{k-1}(y)]$ and $f[(x,1)\searrow(y,k)]$ supplement
each other. Here $V^{\prime}_{k-1}(x)$ and $V_{k-1}(y)$ are given in (2.8).

######  Lemma 2.12.

Fix $n\geq 2$, $1\leq k\leq n-1$, $0<x<y<T$ and $f\in C^{n}(T)$. Then it holds
that

$\displaystyle f[V^{\prime}_{k}(x)\to
V_{k}(y)]+f[(x,1)\searrow(y,k+1)]=f[V_{k+1}(x)\to V_{k+1}(y)].$

##  3\. geometric RSK correspondence

In this section we define a geometric variant of the RSK correspondence
introduced in [OCo12]. The main goal of this section is to derive the identity
(3.3) in Proposition 3.5. This identity describes the polymer energy for an
environment under the geometric RSK and plays a crucial rule in convergence of
the scaled KPZ sheet.

Fix $n\geq 2$, $1\leq i\leq n-1$ and $f\in C^{n}(T)$. Suppose
$\int_{0}^{t}\exp(f_{i+1}(s)-f_{i}(s))ds$ is well-defined as an improper
integral. Define $\mathcal{T}_{i}f\in C^{n}(T)$ by

(3.1) $\displaystyle(\mathcal{T}_{i}f)(t)\coloneqq
f(t)+\left(\log\int_{0}^{t}\exp(f_{i+1}(s)-f_{i}(s))ds\right)(e_{i}-e_{i+1}).$

For $1\leq r\leq n-1$, define

$\displaystyle\mathcal{K}_{r}f\coloneqq\mathcal{T}_{r}\mathcal{T}_{r+1}\cdots\mathcal{T}_{n-1}f.$

######  Definition 3.1.

Given $f\in\overline{C}^{n}(T)$, we define $\mathcal{W}f\in C^{n}(T)$ by

(3.2)
$\displaystyle\mathcal{W}f\coloneqq\mathcal{K}_{n-1}\mathcal{K}_{n-2}\cdots\mathcal{K}_{1}f.$

The following Greene’s theorem was proved in [OCo12].

######  Proposition 3.2.

Fix $n\geq 2$ and $f\in\overline{C}^{n}(T)$. Recall that $U_{n,k}(0)$ and
$V_{k}(t)$ are given in (2.8). Then it holds for all $t\in(0,T)$ and $1\leq
k\leq n$ that

$\displaystyle\sum_{i=1}^{k}(\mathcal{W}f)_{i}(t)=f[U_{n,k}(0)\to V_{k}(t)].$

The following invariance of the free energy was proved in [Cor21, Theorem
3.4].

######  Proposition 3.3.

Fix $n\geq 2$, $f\in\overline{C}^{n}(T)$ and an endpoint pair
$U=\\{(x_{i},n)\\}_{i\in\llbracket 1,k\rrbracket}$ and
$V=\\{(y_{i},1)\\}_{i\in\llbracket 1,k\rrbracket}$ with
$U,V\subset(0,T)\times\llbracket 1,n\rrbracket$. Then it holds that

$\displaystyle f[U\to V]=\left(\mathcal{W}f\right)[U\to V].$

######  Remark 3.4.

In [Cor21], Proposition 3.3 was proved under the condition
$x_{1}<x_{2}<\dots<x_{n}$ and $y_{1}<y_{2}<\dots<y_{n}$. This condition can be
removed through approximation.

The next proposition relates $(\mathcal{W}f)[(x,n)\to(z,k)]$ and
$(\mathcal{W}f)_{k}(z).$

######  Proposition 3.5.

Fix $n\geq 2$, $1\leq k\leq n-1$, $0<T$ and $f\in\overline{C}^{n}(T)$. For any
$0<x<z<T$, it holds that

(3.3)
$\displaystyle(\mathcal{W}f)[(x,n)\to(z,k+1)]+(\mathcal{W}R_{z}f)[(z-x,1)\searrow(z,k+1)]=(\mathcal{W}f)_{k+1}(z).$

The rest of this section is devoted to proving Proposition 3.5. We start with
a direct consequence of Lemma 2.10 and Proposition 3.3.

######  Corollary 3.6.

Fix $n\geq 2$, $1\leq k\leq n-1$, $0<x<y<z<T$ and $f\in\overline{C}^{n}(T)$.
The following identities hold.

$\displaystyle f[U_{n,k+1}(x)\to V_{k+1}(y)]$
$\displaystyle=(\mathcal{W}f)[U_{n,k+1}(x)\to V_{k+1}(y)],$ $\displaystyle
f[\\{U_{n,k}(x),(y,n)\\}\to V_{k+1}(z)]$
$\displaystyle=(\mathcal{W}f)[\\{U_{n,k}(x),(y,n)\\}\to V_{k+1}(z)],$
$\displaystyle f[U_{n,k+1}(x)\to\\{(y,1),V_{k}(z)\\}]$
$\displaystyle=(\mathcal{W}f)[U_{n,k+1}(x)\to\\{(y,1),V_{k}(z)\\}].$

######  Lemma 3.7.

Fix $n\geq 2$, $1\leq k\leq n-1$, $0<x<y<T$ and $f\in\overline{C}^{n}(T)$. It
holds that

$\displaystyle(\mathcal{W}f)[(x,n)\to(y,k+1)]+f[U_{n,k}(0)\to
V_{k}(y)]=f[\\{U_{n,k}(0),(x,n)\\}\to V_{k+1}(y)].$

###### Proof.

Let $g=\mathcal{W}f$. Because of the natural measure-preserving injection from
$\mathcal{Q}[\\{U_{n,k}(\varepsilon),(x,n)\\}\to V_{k+1}(y)]$ to
$\mathcal{Q}[U_{n,k}(\varepsilon)\to
V_{k}(y)]\times\mathcal{Q}[(x,n)\to(y,k+1)]$, we have

$\displaystyle g[\\{U_{n,k}(\varepsilon),(x,n)\\}\to V_{k+1}(y)]\leq
g[U_{n,k}(\varepsilon)\to V_{k}(y)]+g[(x,n)\to(y,k+1)].$

Take $\varepsilon$ go to zero and apply Corollary 3.6, we get

$\displaystyle f[\\{U_{n,k}(0),(x,n)\\}\to V_{k+1}(y)]\leq f[U_{n,k}(0)\to
V_{k}(y)]+g[(x,n)\to(y,k+1)].$

Because of the natural measure-preserving injection from

$\mathcal{Q}[U_{n,k}(\varepsilon)\to V_{k}(x)]\times\mathcal{Q}[V_{k}(x)\to
V_{k}(y)]\times\mathcal{Q}[(x,n)\to(y,k+1)]$

to $\mathcal{Q}[\\{U_{n,k}(\varepsilon),(x,n)\\}\to V_{k+1}(y)]$, we have

$\displaystyle g[\\{U_{n,k}(\varepsilon),(x,n)\\}\to V_{k+1}(y)]\geq
g[U_{n,k}(\varepsilon)\to V_{k}(x)]+g[V_{k}(x)\to
V_{k}(y)]+g[(x,n)\to(y,k+1)].$

Take $\varepsilon$ go to zero and apply Corollary 3.6 and Proposition 3.2, we
get

$\displaystyle f[\\{U_{n,k}(0),(x,n)\\}\to V_{k+1}(y)]\geq f[U_{n,k}(0)\to
V_{k}(y)]+g[(x,n)\to(y,k+1)].$

∎

######  Lemma 3.8.

Fix $n\geq 2$, $1\leq k\leq n$, $0<x<T$ and $f\in\overline{C}^{n}(T)$. Recall
that $U_{n,k}(\varepsilon)$ and $V_{k}(x)$ are given in (2.8). Then for any
$V\in\mathcal{V}_{n,k}(x)$ with $V\neq V_{k}(x)$, it holds that

$\displaystyle\lim_{\varepsilon\to
0}\exp\bigg{(}(\mathcal{W}f)[U_{n,k}(\varepsilon)\to V]\bigg{)}=0.$

###### Proof.

Let $g=\mathcal{W}f$ and take $y\in(x,T)$. By separating
$\mathcal{Q}[U_{n,k}(\varepsilon)\to V_{k}(y)]$ according to $\pi(x)$, we have

$\displaystyle\exp\bigg{(}g[U_{n,k}(\varepsilon)\to
V_{k}(y)]\bigg{)}=\sum_{V\in\mathcal{V}_{n,k}(x)}\exp\bigg{(}g[U_{n,k}(\varepsilon)\to
V]+g[V\to V_{k}(y)]\bigg{)}.$

Therefore,

$\displaystyle\sum_{V\in\mathcal{V}_{k}(x),V\neq
V_{k}(x)}\exp\bigg{(}g[U_{n,k}(\varepsilon)\to V]+g[V\to V_{k}(y)]\bigg{)}$
$\displaystyle=$ $\displaystyle\exp\bigg{(}g[U_{n,k}(\varepsilon)\to
V_{k}(y)]\bigg{)}-\exp\bigg{(}g[U_{n,k}(\varepsilon)\to
V_{k}(x)]+g[V_{k}(x)\to V_{k}(y)]\bigg{)}.$

From Corollary 3.6, the limit of the right hand side equals

$\displaystyle\exp\bigg{(}f[U_{n,k}(0)\to
V_{k}(y)]\bigg{)}-\exp\bigg{(}f[U_{n,k}(0)\to V_{k}(x)]+g[V_{k}(x)\to
V_{k}(y)]\bigg{)}.$

From Proposition 3.2, the above vanishes. Therefore for any
$V\in\mathcal{V}_{n,k}(0)$ with $V\neq V_{k}(x)$, we have

$\lim_{\varepsilon\to 0}\exp\big{(}g[U_{n,k}(\varepsilon)\to V]\big{)}=0.$

∎

######  Lemma 3.9.

Fix $n\geq 2$, $1\leq k\leq n-1$, $0<x<y<T$ and $f\in\overline{C}^{n}(T)$.
Then

$\displaystyle f[U_{n,k+1}(0)\to\\{(x,1),V_{k}(y)\\}]=f[U_{n,k+1}(0)\to
V_{k+1}(y)]-(\mathcal{W}f)[(x,1)\searrow(y,k+1)].$

###### Proof.

Let $g=\mathcal{W}f$. From Corollary 3.6,
$f[U_{n,k+1}(0)\to\\{(x,1),V_{k}(y)\\}]$ equals

$\lim_{\varepsilon\to 0}g[U_{n,k+1}(\varepsilon)\to\\{(x,1),V_{k}(y)\\}].$

From Lemma 3.8, the above becomes

$\lim_{\varepsilon\to 0}g[U_{n,k+1}(\varepsilon)\to
V_{k+1}(x)]+g[V^{\prime}_{k}(x)\to V_{k}(y)].$

From Corollary 3.6 and Lemma 2.12, the above equals

$f[U_{n,k+1}(0)\to V_{k+1}(x)]+g[V_{k+1}(x)\to
V_{k+1}(y)]-g[(x,1)\searrow(y,k+1)].$

In view of Proposition 3.2, the above becomes $f[U_{n,k+1}(0)\to
V_{k+1}(y)]-g[(x,1)\searrow(y,k+1)]$. ∎

###### Proof of Proposition 3.5.

From Lemma 3.7, $(\mathcal{W}f)[(x,n)\to(z,k+1)]$ equals

$\displaystyle f[\\{U_{n,k}(0),(x,n)\\}\to V_{k+1}(z)]-f[U_{n,k}(0)\to
V_{k}(z)].$

From Lemma 2.9, the above equals

$\displaystyle(R_{z}f)[U_{n,k+1}(0)\to\\{(z-x,1),V_{k}(z)\\}]-f[U_{n,k}(0)\to
V_{k}(z)].$

From Lemma 3.9, the above equals

$\displaystyle(R_{z}f)[U_{n,k+1}(0)\to
V_{k+1}(z)]-(\mathcal{W}R_{z}f)[(z-x,1)\searrow(z,k+1)]-f[U_{n,k}(0)\to
V_{k}(z)].$

Applying Lemma 2.9 and Proposition 3.2, it becomes

$\displaystyle(\mathcal{W}f)_{k+1}(z)-(\mathcal{W}R_{z}f)[(z-x,1)\searrow(z,k+1)].$

∎

##  4\. O’Connell-Yor polymer model

Let $B_{1},B_{2},\dots$ be i.i.d. standard two-sided Brownian motions. For
$n\in\mathbb{N}$, let $B^{n}=(B_{1},B_{2},\dots,B_{n})$ restricted on
$[0,\infty)$ and define
$Y^{n}=(Y^{n}_{1},\dots,Y^{n}_{n})\coloneqq\mathcal{W}B^{n}$. It was shown by
O’Connell [OCo12] that $Y^{n}$ is a diffusion process in $\mathbb{R}^{n}$ with
an infinitesimal generator related to the quantum Toda lattice and
$\mathfrak{gl}_{n}$-Whittaker functions. Corwin and Hammond [CH16] derived the
$\mathbf{H}$-Brownian Gibbs property for $Y^{n}$ and used this property to
construct the KPZ line ensemble $\mathcal{H}^{T}$ as a scaling limit of
$Y^{n}$. We discuss this convergence in more detail below.

For $n,i\in\mathbb{N}$ and $T>0$, let

$\displaystyle C_{1}(T,n)\coloneqq$ $\displaystyle n^{1/2}T^{-1/2}+2^{-1},\
C_{3,i}(T)\coloneqq-(i-1)\log T+\log(i-1)!$ $\displaystyle
C_{2}(T,n)\coloneqq$ $\displaystyle
n+2^{-1}n^{1/2}T^{1/2}-2^{-1}(n-1)\log(n^{1/2}T^{-1/2}).$

Define
$\mathcal{X}^{T,n}=\\{\mathcal{X}^{T,n}_{1},\mathcal{X}^{T,n}_{2},\dots,\mathcal{X}^{T,n}_{n}\\}$
on $(-T^{1/2}n^{1/2},\infty)$ by

(4.1) $\mathcal{X}^{T,n}_{i}(x)\coloneqq
Y^{n}_{i}(n^{1/2}T^{1/2}+x)-C_{1}(T,n)x-C_{2}(T,n)-C_{3,i}(T),\ i\in\llbracket
1,n\rrbracket$

From Proposition 3.2,

$\displaystyle\mathcal{X}^{T,n}_{1}(x)=B^{n}[(0,n)\to(n^{1/2}T^{1/2}+x,1)]-C_{1}(T,n)x-C_{2}(T,n).$

For $T>0$, $n\geq 1$, $x\in\mathbb{R}$ and $y>-n^{1/2}T^{1/2}+x$, define

(4.2) $\displaystyle\mathcal{H}^{T,n}(x,y)\coloneqq
B^{n}[(x,n)\to(n^{1/2}T^{1/2}+y,1)]-C_{1}(T,n)(y-x)-C_{2}(T,n).$

Because $B_{i}(x+y)-B_{i}(x)\overset{d}{=}B_{i}(y)$, $\mathcal{H}^{T,n}(x,y)$
has the same distribution as $\mathcal{X}^{T,n}_{1}(y-x)$. In view of
Proposition 3.3, for $x>0$, we may rewrite $\mathcal{H}^{T,n}(x,y)$ as polymer
free energy on $Y^{n}$.

(4.3) $\displaystyle\mathcal{H}^{T,n}(x,y)\coloneqq
Y^{n}[(x,n)\to(n^{1/2}T^{1/2}+y,1)]-C_{1}(T,n)(y-x)-C_{2}(T,n).$

Combining [CH16, Theorem 3.10] and [Nic21, Theorem 1.2], we have the following
convergence result.

######  Theorem 4.1.

Fix $T>0$. When $n$ goes to infinity, $\mathcal{X}^{T,n}$ converges in
distribution to the KPZ line ensemble $\mathcal{X}^{T}$. Here
$\mathcal{X}^{T,n}$ and $\mathcal{X}^{T}$ are viewed as
$C(\mathbb{N}\times\mathbb{R},\mathbb{R})$-valued random variables.

Moreover, the random continuous function $\mathcal{H}^{T,n}$ also converges to
$\mathcal{H}^{T}$.

######  Proposition 4.2.

Fix $T>0$. When $n$ goes to infinity, the finite-dimensional marginal of
$\mathcal{H}^{T,n}$ converges in distribution to the finite-dimensional
marginal of the KPZ sheet $\mathcal{H}^{T}$.

###### Proof.

This is essentially proved in [Nic21]. In [Nic21, Theorem 1.2], the author
proved the finite-dimensional convergence for $\mathcal{H}^{T,n}(0,\cdot)$ and
the same argument applies for $\mathcal{H}^{T,n}(\cdot,\cdot)$. See [AKQ,
Section 6.2] for a similar result for discrete polymers.∎

For $1\leq k\leq n-1$, $T>0$, $x>0$ and $y>z>-n^{1/2}T^{1/2}+x$, we define

$\displaystyle F^{T,n}_{k}(x,z)\coloneqq$ $\displaystyle
Y^{n}[(x,n)\to(n^{1/2}T^{1/2}+z,k+1)]-Y^{n}_{k+1}(n^{1/2}T^{1/2}+z)+C_{1}(T,n)x,$

and

$\displaystyle G^{T,n}_{k}(z,y)\coloneqq$ $\displaystyle
Y^{n}[(n^{1/2}T^{1/2}+z,k)\to(n^{1/2}T^{1/2}+y,1)]+Y^{n}_{k+1}(n^{1/2}+z)-C_{1}(T,n)y-C_{2}(T,n).$

From Lemma 2.3 and (4.3), we have

(4.4)
$\exp\left(\mathcal{H}^{T,n}(x,y)\right)=\int_{-n^{1/2}T^{1/2}+x}^{y}\exp\left(F^{T,n}_{k}(x,z)+G^{T,n}_{k}(z,y)\right)\,dz.$

We note that from Lemma 2.6 and (4.1), $F^{T,n}_{k}(x,z)$ and
$G^{T,n}_{k}(z,y)$ can be expressed in terms of $\mathcal{X}^{T,n}$ as

$\begin{split}F^{T,n}_{k}(x,z)=\mathcal{X}^{T,n}[(-n^{1/2}T^{1/2}x,n)\to(z,k+1)]-&\mathcal{X}^{T,n}_{k+1}(z)+(n-1)\log(n^{1/2}T^{-1/2})-C_{3,k+1}.\end{split}$

(4.5) $\displaystyle
G^{T,n}_{k}(z,y)=\mathcal{X}^{T,n}[(z,k)\to(y,1)]+\mathcal{X}^{T,n}_{k+1}(z)+C_{3,k+1}.$

The next lemma concerns the distributional limit of $F^{T,n}_{k}(x,z)$ when
$n$ goes to infinity.

######  Lemma 4.3.

Fix $T>0$, $k\geq 1$, $x>0$ and $z\in\mathbb{R}$. Then when $n$ goes to
infinity, $F^{T,n}_{k}(z)$ converges in distribution to
$\mathcal{X}^{T}[(0,k+1)\to(x,1)]+T^{-1}zx$.

###### Proof.

From Proposition 3.5, and $Y^{n}=\mathcal{W}B^{n}$, $F^{T,n}_{k}(z)$ equals

$\displaystyle-(\mathcal{W}R_{n^{1/2}T^{1/2}+z}B^{n})[(n^{1/2}T^{1/2}+z-x,1)\searrow(n^{1/2}+z,k+1)]+C_{1}(T,n)x.$

Because $\mathcal{W}R_{n^{1/2}T^{1/2}+z}B^{n}\overset{d}{=}Y^{n}$, the above
has the same distribution as

$\displaystyle-Y^{n}[(n^{1/2}T^{1/2}+z-x,1)\searrow(n^{1/2}T^{1/2}+z,k+1)]+C_{1}(T,n)x.$

From (4.1) and Lemma 2.6, the above equals
$-\mathcal{X}^{T,n}[(z-x,1)\searrow(z,k+1)]$. By Theorem 4.1 it converges in
distribution to $-\mathcal{X}^{T}[(z-x,1)\searrow(z,k+1)]$. Because
$\mathcal{X}^{T}(y)$ has the same distribution as $\mathcal{X}^{T}(-y)$,

$-\mathcal{X}^{T}[(z-x,1)\searrow(z,k+1)]\overset{d}{=}\mathcal{X}^{T}[(-z,k+1)\to(-z+x,1)].$

From the stationarity of $\mathcal{X}^{T}(y)+2^{-1}T^{-1}y^{2}$,
$\mathcal{X}^{T}(-z+y)\overset{d}{=}\mathcal{X}(y)+T^{-1}zy-2^{-1}T^{-1}z^{2}$.
Therefore,

$\displaystyle\mathcal{X}^{T}[(-z,k+1)\to(-z+x,1)]\overset{d}{=}\mathcal{X}^{T}[(0,k+1)\to(x,1)]+T^{-1}zx.$

∎

In the rest of the section, we drive a relation between Busemann functions and
quantiles of the polymer measure. In particular, Lemma 4.6, (4.10) and (4.11)
will be used in Sections 5 and 6. Those are deterministic properties and do
not rely on the laws of $\mathcal{X}^{T,n}$ or $\mathcal{H}^{T,n}$.

We start with a simple consequence of Lemma 2.4.

######  Lemma 4.4.

Fix $T>0$, $n\geq 2$, $1\leq k\leq n-1$, $0<x_{1}\leq x_{2}$ and
$-n^{1/2}T^{1/2}<y_{1}\leq y_{2}$. Then
$F^{T,n}_{k}(x_{2},z)-F^{T,n}_{k}(x_{1},z)$ is monotone non-decreasing in
$z\in(-n^{1/2}T^{1/2}+x_{2},\infty)$ and
$G^{T,n}_{k}(z,y_{2})-G^{T,n}_{k}(z,y_{1})$ is monotone non-decreasing in
$z\in(-n^{1/2}T^{1/2},y_{1})$.

We define the random probability measure on $\mathbb{R}$ which corresponds to
(4.4). It is the marginal of the polymer measure.

######  Definition 4.5.

Fix $T>0$, $n\geq 2$, $1\leq k\leq n-1$, $x>0$ and $y>-n^{1/2}T^{1/2}+x$. We
denote by $d\mu^{T,n}_{k,x,y}(z)$ the random probability measure with the
density

$\exp\left(-\mathcal{H}^{T,n}(x,y)+F^{T,n}_{k}(x,z)+G^{T,n}_{k}(z,y)\right)\mathbbm{1}(-n^{1/2}T^{1/2}+x<z<y).$

We also set

$A^{T,n}_{k}(x,y;z)\coloneqq\mu^{T,n}_{k,x,y}([z,\infty)),\
B^{T,n}_{k}(x,y;z)\coloneqq\mu^{T,n}_{k,x,y}((-\infty,z]).$

######  Lemma 4.6.

Fix $T>0$, $n\geq 2$, $1\leq k\leq n-1$, $x_{2}\geq x_{1}>0$ and
$y>-n^{1/2}T^{1/2}+x_{2}$. Then for $z\in(-n^{1/2}T^{1/2}+x_{2},y)$, we have

(4.6) $\displaystyle F^{T,n}_{k}(x_{2},z)-F^{T,n}_{k}(x_{1},z)\leq$
$\displaystyle\mathcal{H}^{T,n}(x_{2},y)-\mathcal{H}^{T,n}(x_{1},y)-\log
A^{T,n}_{k}(x_{1},y;z),$

and

(4.7) $\displaystyle F^{T,n}_{k}(x_{2},z)-F^{T,n}_{k}(x_{1},z)\geq$
$\displaystyle\mathcal{H}^{T,n}(x_{2},y)-\mathcal{H}^{T,n}(x_{1},y)+\log
B^{T,n}_{k}(x_{2},y;z).$

###### Proof.

We start with (4.6).

$\displaystyle\exp\left(\mathcal{H}^{T,n}(x_{2},y)-\mathcal{H}^{T,n}(x_{1},y)\right)=$
$\displaystyle\int_{-n^{1/2}T^{1/2}+x_{2}}^{y}\exp\left(F^{T,n}_{k}(x_{2},z^{\prime})-F^{T,n}_{k}(x_{1},z^{\prime})\right)d\mu^{T,n}_{k,x_{1},y}(z^{\prime})$
$\displaystyle\geq$
$\displaystyle\int_{z}^{y}\exp\left(F^{T,n}_{k}(x_{2},z^{\prime})-F^{T,n}_{k}(x_{1},z^{\prime})\right)d\mu^{T,n}_{k,x_{1},y}(z^{\prime})$
$\displaystyle\geq$
$\displaystyle\exp\left(F^{T,n}_{k}(x_{2},z)-F^{T,n}_{k}(x_{1},z)\right)A^{T,n}_{k}(x_{1},y;z).$

We used Lemma 4.4 in the last inequality. Then (4.6) follows. For (4.7), we
derive similarly,

$\displaystyle\exp\left(\mathcal{H}^{T,n}(x_{1},y)-\mathcal{H}^{T,n}(x_{2},y)\right)\geq$
$\displaystyle\int_{-n^{1/2}T^{1/2}+x_{2}}^{y}\exp\left(F^{T,n}_{k}(x_{1},z^{\prime})-F^{T,n}_{k}(x_{2},z^{\prime})\right)d\mu^{T,n}_{k,x_{2},y}(z^{\prime})$
$\displaystyle\geq$
$\displaystyle\int_{-n^{1/2}T^{1/2}+x_{2}}^{z}\exp\left(F^{T,n}_{k}(x_{1},z^{\prime})-F^{T,n}_{k}(x_{2},z^{\prime})\right)d\mu^{T,n}_{k,x_{2},y}(z^{\prime})$
$\displaystyle\geq$
$\displaystyle\exp\left(F^{T,n}_{k}(x_{1},z)-F^{T,n}_{k}(x_{2},z)\right)B^{T,n}_{k}(x_{2},y;z).$

We again used Lemma 4.4 in the last inequality. Hence (4.7) follows. ∎

The lemma below is analogous to Lemma 4.6 and we omit the proof.

######  Lemma 4.7.

Fix $T>0$, $n\geq 2$, $1\leq k\leq n-1$, $x>0$ and $y_{2}\geq
y_{1}>-n^{1/2}T^{1/2}+x$. Then for $z\in(-n^{1/2}T^{1/2}+x,y_{1})$, we have

(4.8) $\displaystyle G^{T,n}_{k}(z,y_{2})-G^{T,n}_{k}(z,y_{1})\leq$
$\displaystyle\mathcal{H}^{T,n}(x,y_{2})-\mathcal{H}^{T,n}(x,y_{1})-\log
A^{T,n}_{k}(x,y_{1},z),$

and

(4.9) $\displaystyle G^{T,n}_{k}(z,y_{2})-G^{T,n}_{k}(z,y_{1})\geq$
$\displaystyle\mathcal{H}^{T,n}(x,y_{2})-\mathcal{H}^{T,n}(x,y_{1})+\log
B^{T,n}_{k}(x,y_{2},z).$

The following two inequalities will be applied in Section 5 and Section 6.
Combining (4.5), (4.8), (4.9) and $A^{T,n}_{k}(x,y;z)+B^{T,n}_{k}(x,y;z)=1$,
we have

(4.10)
$\begin{split}\mathcal{X}^{T,n}[(z,k)\to(y_{2},1)]-\mathcal{X}^{T,n}[(z,k)\to(y_{1},1)]-\mathcal{H}^{T,n}&(x,y_{2})+\mathcal{H}^{T,n}(x,y_{1})\\\
&\leq-\log\left(1-B^{T,n}_{k}(x,y_{1},z)\right),\end{split}$

and

(4.11)
$\begin{split}\mathcal{X}^{T,n}[(z,k)\to(y_{2},1)]-\mathcal{X}^{T,n}[(z,k)\to(y_{1},1)]-\mathcal{H}^{T,n}(x&,y_{2})+\mathcal{H}^{T,n}(x,y_{1})\\\
&\geq\log\left(1-A^{T,n}_{k}(x,y_{2},z)\right).\end{split}$

##  5\. Proof of Theorems 1.6 and 1.7

We present the proofs for Theorems 1.6 and 1.7 in this section. We begin with
the scaled KPZ line ensemble and record some convergence results.

Recall that in (1.10), the scaled KPZ sheet is given by

$\displaystyle\mathfrak{H}^{T}(x,y)\coloneqq
2^{1/3}T^{-1/3}\mathcal{H}^{T}(2^{1/3}T^{2/3}x,2^{1/3}T^{2/3}y)+{2^{1/3}T^{2/3}}/{24}.$

We accordingly define the scaled KPZ line ensemble
$\mathfrak{X}^{T}=\\{\mathfrak{X}^{T}_{1},\mathfrak{X}^{T}_{2},\dots\\}$ as

$\mathfrak{X}^{T}_{i}(x)\coloneqq
2^{1/3}T^{-1/3}\mathcal{X}^{T}(2^{1/3}T^{2/3}x)+{2^{1/3}T^{2/3}}/{24}.$

The pre-limit versions are given by

(5.1)
$\begin{split}\mathfrak{X}^{T,n}_{i}(x)\coloneqq&2^{1/3}T^{-1/3}\mathcal{X}^{T,n}(2^{1/3}T^{2/3}x)+{2^{1/3}T^{2/3}}/{24},\\\
\mathfrak{H}^{T,n}(x,y)\coloneqq&2^{1/3}T^{-1/3}\mathcal{H}^{T,n}(2^{1/3}T^{2/3}x,2^{1/3}{T}^{2/3}y)+2^{1/3}{T^{2/3}}/{24}.\end{split}$

The convergence of the scaled KPZ line ensemble to the Airy line ensemble is a
consequence of a series of works [QS23, Vir20, DM21, Wu22].

######  Theorem 5.1.

The scaled KPZ line ensemble $\mathfrak{X}^{T}(x)$ converges in distribution
to the Airy line ensemble $\mathcal{A}(x)$ when $T$ goes to infinity. Here
$\mathfrak{X}^{T}$ and $\mathcal{A}$ are considered as
$C(\mathbb{N}\times\mathbb{R},\mathbb{R})$-valued random variables.

Next, we record the tightness of the scaled KPZ sheet.

######  Proposition 5.2.

When $T$ goes to infinity, the scaled KPZ sheet $\mathfrak{H}^{T}(x,y)$ is
tight in $C(\mathbb{R}^{2},\mathbb{R})$.

###### Proof.

It suffices to prove that for all $b>0$, $\mathfrak{H}^{T}|_{[-b,b]^{2}}$ is
tight in $C([-b,b]^{2},\mathbb{R})$. The tightness of
$\mathfrak{H}^{T}(0,0)=\mathfrak{H}^{T}(0,0;1,0)$ follows direct from its
convergence [ACQ11]. It remains to obtain the modulus of continuity. From
[CGH21, Theorem 1.3], there exist $D_{1}>0$ depending on and $b$ such that for
all $T\geq 1$, $d\in(0,1]$, $K\geq 0$ and $x,x+d\in[-2b,2b]$, we have

(5.2)
$\displaystyle\mathbb{P}\left(|\mathfrak{X}_{1}^{T}(x+d)-|\mathfrak{X}_{1}^{T}(x)|>Kd^{1/2}\right)\leq
D_{1}e^{-D_{1}^{-1}K^{3/2}}.$

Note that
$\mathfrak{H}^{T}(\cdot,y)\overset{d}{=}\mathfrak{X}^{T}_{1}(y-\cdot)$ and
$\mathfrak{H}^{T}(x,\cdot)\overset{d}{=}\mathfrak{X}^{T}_{1}(\cdot-x)$.
Therefore, for any $(x,y)\in[-b,b]^{2}$,

$\displaystyle\mathbb{P}\left(|\mathfrak{H}^{T}(x+d,y)-\mathfrak{H}^{T}(x,y)|>Kd^{1/2}\right)\leq
D_{1}e^{-D_{1}^{-1}K^{3/2}},$

provided $x+d\in[-b,b]$. Similarly, if $y+d\in[-b,b]$, then

$\displaystyle\mathbb{P}\left(|\mathfrak{H}^{T}(x,y+d)-\mathfrak{H}^{T}(x,y)|>Kd^{1/2}\right)\leq
D_{1}e^{-D_{1}^{-1}K^{3/2}},$

From [DV21, Lemma 3.3], there exists a random constant $C^{T,n}$ such that
almost surely for all $(x,y),(x^{\prime},y^{\prime})\in[-b,b]^{2}$ with
$|x-x^{\prime}|,|y-y^{\prime}|\leq 1$, we have

$\displaystyle|\mathfrak{H}^{T}(x,y)-\mathfrak{H}^{T}(x^{\prime},y^{\prime})|\leq
C^{T,n}\left(|x-x^{\prime}|^{1/2}\log^{2/3}(2/|x-x^{\prime}|)+|y-y^{\prime}|^{1/2}\log^{2/3}(2/|y-y^{\prime}|)\right).$

Moreover, $\mathbb{P}(C^{T,n}>K)<D_{2}e^{-D_{2}^{-1}K^{3/2}}$ for some
constant $D_{2}$ depending only on $T$ and $b$. By the Kolmogorov-Chentsov
criterion (see Corollary 14.9 in [Kal97]) this implies the tightness of
$\mathfrak{H}^{T}|_{[-b,b]^{2}}$. ∎

The next proposition shows that any subsequential limit of $\mathfrak{H}^{T}$
and the Airy line ensemble can be coupled together with desired properties.

######  Proposition 5.3.

Let $\mathfrak{H}$ be a distributional limit of $\mathfrak{H}^{T}$ along some
sequence. Then there exists a coupling of $\mathfrak{H}$ and the Airy line
ensemble $\mathcal{A}$ such that the following holds.

  1. (1)

$\mathfrak{H}(0,\cdot)=\mathcal{A}_{1}(\cdot)$.

  2. (2)

Almost surely for all $x>0$ and $y_{1},y_{2}$ in $\mathbb{R}$, we have

(5.3)
$\begin{split}\lim_{k\to\infty}\mathcal{A}[(-2^{-1/2}k^{1/2}x^{-1/2},k)\xrightarrow{\infty}(y_{2},1)]-\mathcal{A}[(-2^{-1/2}k^{1/2}x^{-1/2},k)&\xrightarrow{\infty}(y_{1},1)]\\\
&=\mathfrak{H}(x,y_{2})-\mathfrak{H}(x,y_{1}).\end{split}$

###### Proof of Theorem 1.7.

Let $\mathfrak{H}$ be a distributional limit of $\mathfrak{H}^{T}$ along some
sequence. From Proposition 5.3, (5.3) holds. Because of (3) in Theorem 1.1,
$\mathfrak{H}(\cdot+t,\cdot+t)$ has the same distribution as
$\mathfrak{H}(\cdot,\cdot)$. From Definition 1.3, $\mathfrak{H}$ has the same
law as the Airy sheet. As a result, $\mathfrak{H}^{T}$ converges to the Airy
sheet in distribution. ∎

It remains to prove Proposition 5.3. We begin with the rescaled version of
Lemma 4.6, (4.10) and (4.11). Define

(5.4) $\mathfrak{R}^{T,n}_{k}(x,z)\coloneqq
2^{1/3}T^{-1/3}F^{T,n}_{k}(2^{1/3}T^{2/3}x,2^{1/3}T^{2/3}z)-2^{3/2}k^{1/2}x^{1/2}-2zx.$
(5.5)
$\begin{split}\mathfrak{A}^{T,n}_{k}(x,y;z)\coloneqq&A^{T,n}_{k}(2^{1/3}T^{2/3}x,2^{1/3}T^{2/3}y;2^{1/3}T^{2/3}\bar{z}),\\\
\mathfrak{B}^{T,n}_{k}(x,y;z)\coloneqq&B^{T,n}_{k}(2^{1/3}T^{2/3}x,2^{1/3}T^{2/3}y;2^{1/3}T^{2/3}\bar{z}).\end{split}$

######  Lemma 5.4.

Fix $T\geq 2$, $n\geq 2$, and $1\leq k\leq n-1$. Then for all $x,\bar{x}>0$,
and $y>\max\\{-2^{-1/3}n^{1/2}T^{-1/6}+x,-2^{-1/3}n^{1/2}T^{-1/6}+\bar{x}\\}$,
the following statements hold. Let $\bar{z}=-2^{-1/2}k^{1/2}\bar{x}^{-1/2}$.
If $\bar{x}\geq x$, then

(5.6)
$\begin{split}\log\mathfrak{A}^{T,n}_{k}(x,y;\bar{z})+2^{1/2}k^{1/2}\bar{x}^{1/2}&\left(1-\bar{x}^{-1/2}x^{1/2}\right)^{2}\\\
&\leq\mathfrak{H}^{T,n}(\bar{x},y)-\mathfrak{H}^{T,n}(x,y)-\mathfrak{R}^{T,n}_{k}(\bar{x},\bar{z})+\mathfrak{R}^{T,n}_{k}(x,\bar{z}).\end{split}$

If $\bar{x}\leq x$, then

(5.7)
$\begin{split}\log\mathfrak{B}^{T,n}_{k}(x,y;\bar{z})+2^{1/2}k^{1/2}\bar{x}^{1/2}&\left(1-\bar{x}^{-1/2}x^{1/2}\right)^{2}\\\
&\leq\mathfrak{H}^{T,n}(\bar{x},y)-\mathfrak{H}^{T,n}(x,y)-\mathfrak{R}^{T,n}_{k}(\bar{x},\bar{z})+\mathfrak{R}^{T,n}_{k}(x,\bar{z}).\end{split}$

###### Proof.

First, we consider the case $\bar{x}\geq x$. From (4.6), (5.1) and (5.5), we
have

$\displaystyle
2^{-1/3}T^{1/3}\big{(}\mathfrak{H}^{T,n}(\bar{x},y)-\mathfrak{H}^{T,n}(x,y)\big{)}$
$\displaystyle-\log\mathfrak{A}^{T,n}_{k}(x,y;z)$ $\displaystyle\geq
F^{T,n}_{k}(2^{1/3}T^{2/3}\bar{x},2^{1/3}T^{2/3}\bar{z})-F^{T,n}_{k}(2^{1/3}T^{2/3}x,2^{1/3}T^{2/3}\bar{z}).$

From (5.4), the right hand side equals

$\displaystyle
2^{-1/3}T^{1/3}\bigg{(}2^{1/2}k^{1/2}\bar{x}^{1/2}\left(1-\bar{x}^{-1/2}x^{1/2}\right)^{2}+\mathfrak{R}^{T,n}_{k}(\bar{x},\bar{z})-\mathfrak{R}^{T,n}_{k}(x,\bar{z})\bigg{)}.$

Together with $T\geq 2$, (5.6) follows by rearranging terms. The proof of
(5.7) is analogous. ∎

######  Lemma 5.5.

Fix $T\geq 2$, $n\geq 2$, $1\leq k\leq n-1$, $x>0$, and $y_{2}\geq y_{1}\geq
z>-2^{-1/3}n^{1/2}T^{-1/6}+x$ . Then it holds that

(5.8)
$\begin{split}\mathfrak{H}^{T,n}(x,y_{2})-\mathfrak{H}^{T,n}(x,y_{1})-\mathfrak{X}^{T,n}[(z,k)\xrightarrow{(T/2)^{1/3}}(y_{2},1)]+\mathfrak{X}^{T,n}[(z,k)&\xrightarrow{(T/2)^{1/3}}(y_{1},1)]\\\
\geq&\log\big{(}1-\mathfrak{B}^{T,n}_{k}(x,y_{1};z)\big{)}.\end{split}$

Similarly, it holds that

(5.9)
$\begin{split}\mathfrak{H}^{T,n}(x,y_{2})-\mathfrak{H}^{T,n}(x,y_{1})-\mathfrak{X}^{T,n}[(z,k)\xrightarrow{(T/2)^{1/3}}(y_{2},1)]+\mathfrak{X}^{T,n}[(z&,k)\xrightarrow{(T/2)^{1/3}}(y_{1},1)]\\\
\leq&-\log\big{(}1-\mathfrak{A}^{T,n}_{k}(x,y_{2};z)\big{)}.\end{split}$

###### Proof.

From (5.1) and Lemma 2.6,
$\mathfrak{X}^{T,n}[(z,k)\xrightarrow{(T/2)^{1/3}}(y_{2},1)]-\mathfrak{X}^{T,n}[(z,k)\xrightarrow{(T/2)^{1/3}}(y_{1},1)]$
equals

$\displaystyle
2^{1/3}T^{-1/3}\left(\mathcal{X}^{T,n}[(2^{1/3}T^{2/3}z,k){\longrightarrow}(2^{1/3}T^{2/3}y_{2},1)]-\mathcal{X}^{T,n}[(2^{1/3}T^{2/3}z,k){\longrightarrow}(2^{1/3}T^{2/3}y_{1},1)]\right).$

From (4.10), it is bounded from above by

$\displaystyle
2^{1/3}T^{-1/3}\bigg{(}\mathcal{H}^{T,n}(2^{1/3}T^{2/3}x,2^{1/3}T^{2/3}y_{2})-$
$\displaystyle\mathcal{H}^{T,n}(2^{1/3}T^{2/3}x,2^{1/3}T^{2/3}y_{1})$
$\displaystyle-\log
A^{T,n}_{k}(2^{1/3}T^{2/3}x,2^{1/3}T^{2/3}y_{1};2^{1/3}T^{2/3}z)\bigg{)}.$

From (5.1), the above equals

$\displaystyle\mathfrak{H}^{T,n}(x,y_{2})-\mathfrak{H}^{T,n}(x,y_{1})-2^{1/3}T^{-1/3}\log\left(1-\mathfrak{B}^{T,n}_{k}(x,y_{1};z)\right).$

Together with $T\geq 2$, (5.8) follows by rearranging terms. The proof for
(5.9) is similar. ∎

The next proposition provides the coupling which allows us to prove
Proposition 5.3.

######  Proposition 5.6.

Fix a sequence $\mathsf{t}_{0}$. Then there exists a sequence $\mathsf{n}$, a
subsequence $\mathsf{t}\subset\mathsf{t}_{0}$ and a coupling of
$\\{\mathfrak{X}^{T,n},\mathfrak{H}^{T,n}\\}_{(T,n)\in\mathsf{t}\times\mathsf{n}}$,
$\\{\mathfrak{X}^{T},\mathfrak{H}^{T}\\}_{T\in\mathsf{t}}$ and $\mathcal{A}$
such that the following statements hold.

First, fix any $T\in\mathsf{t}_{0}$. Almost surely $\mathfrak{X}^{T,n}$
converges to $\mathfrak{X}^{T}$ in $C(\mathbb{N}\times\mathbb{R},\mathbb{R})$,
$\mathfrak{H}^{T,n}(x,y)$ converges to $\mathfrak{H}^{T}(x,y)$ for all
$x,y\in\mathbb{Q}$, and
$\mathfrak{R}^{T,n}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$ converges for all
$k\geq 1$ and $x,\bar{x}\in\mathbb{Q}^{+}$. We denote the limits by
$\mathfrak{R}^{T}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$.

Second, almost surely $\mathfrak{X}^{T}$ converges to $\mathcal{A}$ in
$C(\mathbb{N}\times\mathbb{R},\mathbb{R})$, $\mathfrak{H}^{T}(x,y)$ converges
in $C(\mathbb{R}^{2},\mathbb{R})$, and
$\mathfrak{R}^{T}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$ converges for all
$k\geq 1$ and $x,\bar{x}\in\mathbb{Q}^{+}$. We denote the limits by
$\mathfrak{H}$ and $\mathfrak{R}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$
respectively.

Lastly, $\mathfrak{H}(0,\cdot)=\mathcal{A}_{1}(\cdot)$. For all
$x,\bar{x}\in\mathbb{Q}^{+}$, it holds almost surely

(5.10)
$\lim_{k\to\infty}|k^{-1/2}\mathfrak{R}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})|=0.$

###### Proof.

Fix $T\in\mathsf{t}_{0}$ and an arbitrary sequence $\mathsf{n}_{0}$. From
Theorem 4.1, $\\{\mathfrak{X}^{T,n}\\}_{n\in\mathsf{n}_{0}}$ is tight in
$C(\mathbb{N}\times\mathbb{R},\mathbb{R})$. From Proposition 4.2, the finite-
dimensional distribution of $\\{\mathfrak{H}^{T,n}\\}_{n\in\mathsf{n}_{0}}$ is
tight. From (5.4) and Lemma 4.3, we have the convergence in distribution of
$\mathfrak{R}^{T,n}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$ to

$\mathfrak{X}^{T}[(0,k+1)\xrightarrow{(T/2)^{1/3}}(\bar{x},1)]-2^{3/2}k^{1/2}\bar{x}^{1/2}+2^{1/3}T^{-1/3}k\log(2^{1/3}T^{2/3}).$

By the Skorokhod’s representation theorem [Bil99, Theorem 6.7], we may find a
subsequence $\mathsf{n}^{\prime}\subset\mathsf{n}_{0}$ and a coupling of
$\\{\mathfrak{X}^{T,n},\mathfrak{H}^{T,n}\\}_{n\in\mathsf{n}^{\prime}}$ such
that along $\mathsf{n}^{\prime}$, $\mathfrak{X}^{T,n}$,
$\mathfrak{H}^{T,n}(x,y)$ and
$\mathfrak{R}^{T,n}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$ converge almost
surely. We note that the convergences of the latter two hold at rational
points. From Theorem 4.1, the limit of $\mathfrak{X}^{T,n}$ is distributed as
the scaled KPZ line ensemble and we denote it by $\mathfrak{X}^{T}$. From
Proposition 4.2, we may augment the probability space to accommodate a scaled
KPZ sheet $\mathfrak{H}^{T}$ such that $\mathfrak{H}^{T,n}(x,y)$ converges
almost surely to $\mathfrak{H}^{T}(x,y)$ for all $x,y\in\mathbb{Q}$. We note
that since $\mathfrak{H}^{T,n}(0,\cdot)=\mathfrak{X}^{T,n}_{1}(\cdot)$, we may
further require $\mathfrak{H}^{T}(0,\cdot)=\mathfrak{X}^{T}_{1}(\cdot)$. The
limits of $\mathfrak{R}^{T,n}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$ are
denoted by $\mathfrak{R}^{T}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$ .
Moreover,

$\displaystyle\mathfrak{R}^{T}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})\overset{d}{=}\mathfrak{X}^{T}[(0,k+1)\xrightarrow{(T/2)^{1/3}}(\bar{x},1)]-2^{3/2}k^{1/2}\bar{x}^{1/2}+2^{1/3}T^{-1/3}k\log(2^{1/3}T^{2/3}).$

By a diagonal argument, we can find a sequence $\mathsf{n}$ and couplings of
$\\{\mathfrak{X}^{T,n},\mathfrak{H}^{T,n}\\}_{n\in\mathsf{n}}$,
$\mathfrak{X}^{T}$ and $\mathfrak{H}^{T}$ for each $T\in\mathsf{t}_{0}$ such
that along $\mathsf{n}$, the convergences in the previous paragraph hold. From
now on we fix such a sequence $\mathsf{n}$. From Theorem 5.1,
$\\{\mathfrak{X}^{T}\\}_{T\in\mathsf{t}_{0}}$ is tight in
$C(\mathbb{N}\times\mathbb{R},\mathbb{R})$. From Proposition 5.2,
$\\{\mathfrak{H}^{T}\\}_{T\in\mathsf{t}_{0}}$ is tight in
$C(\mathbb{R}^{2},\mathbb{R})$. Similarly,
$\\{\mathfrak{R}^{T}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})\\}_{T\in\mathsf{t}_{0}}$
is tight. By the Skorokhod’s representation theorem, we can find a subsequence
$\mathsf{t}\subset\mathsf{t}_{0}$ and a coupling such that along $\mathsf{t}$,
$\mathfrak{X}^{T}$, $\mathfrak{H}^{T}$ and
$\mathfrak{R}^{T}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$ converge almost
surely. From Theorem 5.1, the limit of $\mathfrak{X}^{T}$ is distributed as an
Airy line ensemble and we denote it by $\mathcal{A}$. Denote by $\mathfrak{H}$
and $\mathfrak{R}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$ the limits of
$\mathfrak{H}^{T}$ and
$\mathfrak{R}^{T}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})$ respectively. From
$\mathfrak{H}^{T}(0,\cdot)=\mathfrak{X}^{T}_{1}(\cdot)$, we have
$\mathfrak{H}(0,\cdot)=\mathcal{A}_{1}(\cdot)$. Moreover,

$\displaystyle\mathfrak{R}_{k}(\bar{x},-2^{-1/2}k^{1/2}x^{-1/2})\overset{d}{=}\mathcal{A}[(0,k+1)\xrightarrow{\infty}(\bar{x},1)]-2^{3/2}k^{1/2}\bar{x}^{1/2}.$

From [DOV18, Theorem 6.3], for all $\varepsilon>0$,

$\displaystyle\sum_{k=1}^{\infty}\mathbb{P}\left(|\mathfrak{R}_{k}(\bar{x},-k^{1/2}x^{-1/2})|>\varepsilon
k^{1/2}\right)<\infty.$

Then (5.10) follows from the Borel-Cantelli lemma. ∎

###### Proof of Proposition 5.3.

Let $\mathfrak{H}$ be the distributional limit of $\mathfrak{H}^{T}$ along
some sequence $\mathsf{t}_{0}$. From Proposition 5.6, we can find a sequence
$\mathsf{n}$, a subsequence $\mathsf{t}\subset\mathsf{t}_{0}$ and a coupling
of
$\\{\mathfrak{X}^{T,n},\mathfrak{H}^{T,n}\\}_{(T,n)\in\mathsf{t}\times\mathsf{n}}$,
$\mathfrak{H}$ and the Airy line ensemble $\mathcal{A}$ such that the
assertions in Proposition 5.6 holds. In particular,
$\mathfrak{H}(0,\cdot)=\mathcal{A}_{1}(\cdot)$. From Definition 1.3, we may
further augment the probability space to accommodate an Airy sheet
$\mathcal{S}$ such that on an event with probability one,

(5.11)
$\begin{split}\lim_{k\to\infty}\mathcal{A}[(-2^{-1/2}k^{1/2}x^{-1/2},k)\xrightarrow{\infty}(y_{2},1)]-\mathcal{A}[(-2^{-1/2}k^{1/2}x^{-1/2},k)&\xrightarrow{\infty}(y_{1},1)]\\\
&=\mathcal{S}(x,y_{2})-\mathcal{S}(x,y_{1}),\end{split}$

for all $x>0$ and $y_{1},y_{2}$ in $\mathbb{R}$. From now one, we fix an event
$\Omega_{0}$ such that for each element in $\Omega_{0}$, all assertions in
Proposition 5.6 and (5.11) hold. Our goal is to prove that when this event
$\Omega_{0}$ occurs,

(5.12)
$\displaystyle\mathfrak{H}(x,y_{2})-\mathfrak{H}(x,y_{1})=\mathcal{S}(x,y_{2})-\mathcal{S}(x,y_{1}),$

for all $x>0$ and $y_{1},y_{2}$ in $\mathbb{R}$.

Fix $x_{-}<x_{0}$ in $\mathbb{Q}^{+}$ and $y_{1}\leq y_{2}$ in $\mathbb{Q}$.
We want to show that

(5.13)
$\mathfrak{H}(x_{0},y_{2})-\mathfrak{H}(x_{0},y_{1})\geq\mathcal{S}(x_{-},y_{2})-\mathcal{S}(x_{-},y_{1}).$

Let $z_{k}=-2^{-1/2}k^{1/2}x_{-}^{-1/2}$. From (5.8), we have

(5.14)
$\begin{split}\mathfrak{H}^{T,n}(x_{0},y_{2})-\mathfrak{H}^{T,n}(x_{0},y_{1})-\mathfrak{X}^{T,n}[(z_{k},k)\xrightarrow{(T/2)^{1/3}}(y_{2},1)]+\mathfrak{X}^{T,n}&[(z_{k},k)\xrightarrow{(T/2)^{1/3}}(y_{1},1)]\\\
&\geq\log\big{(}1-\mathfrak{B}^{T,n}_{k}(x_{0},y_{1};z_{k})\big{)}.\end{split}$

From our arrangement,

$\displaystyle\lim_{k\to\infty}\lim_{\begin{subarray}{c}T\in\mathsf{t}\\\
T\to\infty\end{subarray}}\lim_{\begin{subarray}{c}n\in\mathsf{n}\\\
n\to\infty\end{subarray}}\bigg{(}\textup{LHS\ of\
\eqref{equ:Airy_middle}}\bigg{)}=\mathfrak{H}(x_{0},y_{2})-\mathfrak{H}(x_{0},y_{1})-\mathcal{S}(x_{-},y_{2})+\mathcal{S}(x_{-},y_{1}).$

Therefore, to prove (5.13), it suffices to show

(5.15)
$\displaystyle\liminf_{k\to\infty}\liminf_{\begin{subarray}{c}T\in\mathsf{t}\\\
T\to\infty\end{subarray}}\liminf_{\begin{subarray}{c}n\in\mathsf{n}\\\
n\to\infty\end{subarray}}\left(\log\mathfrak{B}^{T,n}_{k}(x_{0},y_{1};z_{k})\right)=-\infty.$

Applying (5.7) with $x=x_{0}$ and $\bar{x}=x_{-}$,
$\log\mathfrak{B}^{T,n}_{k}(x_{0},y_{1};z_{k})$ is bounded from above by

$\displaystyle-2^{1/2}k^{1/2}x_{-}^{1/2}\left(1-x_{-}^{-1/2}x_{0}^{1/2}\right)^{2}+\mathfrak{H}^{T,n}(x_{-},y_{1})-\mathfrak{H}^{T,n}(x_{0},y_{1})-\mathfrak{R}^{T,n}_{k}(x_{-},z_{k})+\mathfrak{R}^{T,n}_{k}(x_{0},z_{k}).$

Because of (5.10), the above goes to $-\infty$. Therefore (5.15) holds. A
similar argument yields

(5.16)
$\mathfrak{H}(x_{0},y_{2})-\mathfrak{H}(x_{0},y_{1})\leq\mathcal{S}(x_{+},y_{2})-\mathcal{S}(x_{+},y_{1}),$

for all $x_{0}<x_{+}$ in $\mathbb{Q}^{+}$ and $y_{1}\leq y_{2}$ in
$\mathbb{Q}$. As a result, (5.12) holds for all $x\in\mathbb{Q}^{+}$ and
$y_{1},y_{2}\in\mathbb{Q}$. By the continuity, (5.12) holds for all $x>0$ and
$y_{1},y_{2}\in\mathbb{R}$.

∎

In the rest of the section, we prove Theorem 1.6, which is actually a simple
consequence of Theorem 1.7. For $T>0$, recall that

$\displaystyle\mathfrak{H}^{T}(s,x;t,y)=2^{1/3}T^{-1/3}\mathcal{H}(Ts,2^{1/3}T^{2/3}x;Tt,2^{1/3}T^{2/3}y)+(t-s)2^{1/3}T^{2/3}/24.$

From (3) in Theorem 1.1 and (1.9), for fixed $s<t$ it holds that

(5.17)
$\displaystyle\mathfrak{H}^{T}(s,x;t,y)\overset{d}{=}(t-s)^{1/3}\mathfrak{H}^{(t-s)T}((t-s)^{-2/3}x,(t-s)^{-2/3}y).$

The both sides of (5.17) are viewed as $C(\mathbb{R}^{2},\mathbb{R})$-valued
random variables. The linearity (1.4) can be rewritten as

(5.18)
$\begin{split}\mathfrak{H}^{T}(s,x;t,y)=2^{1/3}T^{-1/3}\log\int_{-\infty}^{\infty}\exp\big{[}2^{-1/3}T^{1/3}\big{(}\mathfrak{H}^{T}(s,x;&\tau,z)+\mathfrak{H}^{T}(\tau,z;t,y)\big{)}\big{]}dz\\\
&+2^{1/3}T^{-1/3}\log(2^{1/3}T^{2/3}).\end{split}$

###### Proof of Theorem 1.6.

Fix a finite set $\\{t_{1}<t_{2}<\dots<t_{m}\\}$. From (5.17) and Theorem 1.7,
$\\{\mathfrak{H}^{T}(t_{i},\cdot;t_{j},\cdot)\\}_{i<j}$ is tight. Denote by
$\\{\mathfrak{H}(t_{i},\cdot;t_{j},\cdot)\\}_{i<j}$ a subsequential limit. By
the Skorokhod’s representation theorem [Bil99, Theorem 6.7], we may take a
coupling such that $\mathfrak{H}^{T}(t_{i},\cdot;t_{j},\cdot)$ converges to
$\mathfrak{H}(t_{i},\cdot;t_{j},\cdot)$ almost surely in
$C(\mathbb{R}^{2},\mathbb{R})$. From (4) in Theorem 1.1,
$\\{\mathfrak{H}(t_{i},\cdot;t_{i+1},\cdot)\\}_{i=1}^{m-1}$ are independent.
Moreover, from (5.17) and Theorem 1.7, $\mathfrak{H}(t_{i},\cdot;t_{j},\cdot)$
is distributed as an Airy sheet of scale $(t_{j}-t_{i})^{1/3}$. In view of
Corollary 1.5, it remains to prove that for any $t_{i}<t_{j}<t_{k}$, it holds
almost surely

(5.19)
$\displaystyle\mathfrak{H}(t_{i},x;t_{k},y)=\max_{z\in\mathbb{R}}\left(\mathfrak{H}(t_{i},x;t_{j},z)+\mathfrak{H}(t_{j},z;t_{k},y)\right).$

From [DOV18, Proposition 9.2], the right hand side of (5.19) is well-defined
as a random variable on $C(\mathbb{R}^{2},\mathbb{R})$. Moreover, it is
distributed as an Airy sheet of scale $(t_{k}-t_{i})^{1/3}$. Therefore, it
suffices to show that almost surely for all $x,y\in\mathbb{R}$,

(5.20)
$\displaystyle\mathfrak{H}(t_{i},x;t_{k},y)\geq\max_{z\in\mathbb{R}}\left(\mathfrak{H}(t_{i},x;t_{j},z)+\mathfrak{H}(t_{j},z;t_{k},y)\right).$

Let $\Omega_{0}$ be the event on which the following holds. First,
$\mathfrak{H}^{T}(t_{i},\cdot;t_{j},\cdot)$ converges to
$\mathfrak{H}(t_{i},\cdot;t_{j},\cdot)$ in $C(\mathbb{R}^{2},\mathbb{R})$ for
all $t_{i}<t_{j}$. Second, the right hand side of (5.19) defines a continuous
function in $x$ and $y$. We will show that (5.20) holds on $\Omega_{0}$.

Fix $t_{i}<t_{j}$ and $x,y\in\mathbb{R}$. Denote by $Z_{j}(t_{i},x;t_{k},y)$
the collection of maximum points of
$\mathfrak{H}(t_{i},x;t_{j},z)+\mathfrak{H}(t_{j},z;t_{k},y)$. Note that when
$\Omega_{0}$ occurs, $Z_{j}(t_{i},x;t_{k},y)\neq\varnothing$. For $M>0$,
consider the event
$\Omega_{0}\cap\\{Z_{j}(t_{i},x;t_{k},y)\cap[-M,M]\neq\varnothing\\}$. When
such an event occurs, we have

$\displaystyle\max_{z\in\mathbb{R}}\left(\mathfrak{H}(t_{i},x;t_{j},z)+\mathfrak{H}(t_{j},z;t_{k},y)\right)$
$\displaystyle=$
$\displaystyle\max_{z\in[-M,M]}\left(\mathfrak{H}(t_{i},x;t_{j},z)+\mathfrak{H}(t_{j},z;t_{k},y)\right)$
$\displaystyle=$
$\displaystyle\lim_{T\to\infty}2^{1/3}T^{-1/3}\log\int_{-M}^{M}\exp\big{[}2^{-1/3}T^{1/3}\big{(}\mathfrak{H}^{T}(t_{i},x;t_{j},z)+\mathfrak{H}^{T}(t_{j},z;t_{k},y)\big{)}\big{]}\,dz$
$\displaystyle\leq$
$\displaystyle\lim_{T\to\infty}\mathfrak{H}^{T}(t_{i},x;t_{k},y)-2^{1/3}T^{-1/3}\log(2^{1/3}T^{2/3})=\mathfrak{H}(t_{i},x;t_{k},y).$

This implies (5.20) and finishes the proof. ∎

##  6\. Proof of Theorem 1.10

In this section, we prove Theorem 1.10. That is, we show that Conjecture 1.9
is true provided (1.14) holds.

We begin by giving upper bounds for $A^{T,n}_{k}(x,y;z)$ and
$B^{T,n}_{k}(x,y;z)$ (see Definition 4.5). We need the following elementary
inequality. Fix $\varepsilon>0$ and $k\geq 1$. There exists a constant
$D=D(\varepsilon)>0$ such that for all $T>0$ and
$x,\bar{x}\in[\varepsilon,\varepsilon^{-1}]$, we have

(6.1) $\displaystyle k\log\bar{x}+T^{-1}\bar{z}\bar{x}-k\log
x-T^{-1}\bar{z}x\geq D^{-1}k|x-\bar{x}|^{2},$

where $\bar{z}=-kT/\bar{x}$. Define

(6.2) $\displaystyle R^{T,n}_{k}(x,z)\coloneqq F^{T,n}_{k}(x,z)-k\log
x-T^{-1}zx+\log k!.$

######  Lemma 6.1.

Fix $T,\varepsilon>0$, $n\geq 2$, and $1\leq k\leq n-1$. There exists
$D=D(\varepsilon)>0$ such that for all
$x,\bar{x}\in[\varepsilon,\varepsilon^{-1}]$, and
$y>\max\\{-n^{1/2}T^{1/2}+x,-n^{1/2}T^{1/2}-\bar{x}\\}$, the following
statements hold. Let $\bar{z}=-kT/\bar{x}$. If $\bar{x}\geq x$, then

(6.3) $\begin{split}\log
A^{T,n}_{k}(x,y;\bar{z})\leq&-D^{-1}k|x-\bar{x}|^{2}+\mathcal{H}^{T,n}(\bar{x},y)-\mathcal{H}^{T,n}(x,y)-R^{T,n}_{k}(\bar{x},\bar{z})+R^{T,n}_{k}(x,\bar{z}).\end{split}$

If $\bar{x}\leq x$, then

(6.4) $\begin{split}\log
B^{T,n}_{k}(x,y;\bar{z})\leq&-D^{-1}k|x-\bar{x}|^{2}+\mathcal{H}^{T,n}(\bar{x},y)-\mathcal{H}^{T,n}(x,y)-R^{T,n}_{k}(\bar{x},\bar{z})+R^{T,n}_{k}(x,\bar{z}).\end{split}$

###### Proof.

First, we consider the case $\bar{x}\geq x$. From (4.6), we have

$\displaystyle\mathcal{H}^{T,n}(\bar{x},y)-\mathcal{H}^{T,n}(x,y)-\log
A^{T,n}_{k}(x,y;\bar{z})\geq$ $\displaystyle
F^{T,n}_{k}(\bar{x},\bar{z})-F^{T,n}_{k}(x,\bar{z}).$

Using (6.2), the right hand side of the above equals

$\displaystyle(k\log\bar{x}+T^{-1}\bar{z}\bar{x})-(k\log
x+T^{-1}\bar{z}x)+R^{T,n}_{k}(\bar{x},\bar{z})-R^{T,n}_{k}(x,\bar{z}).$

Using (6.1), it is bounded from below by

$\displaystyle
D^{-1}k|x-\bar{x}|^{2}+R^{T,n}_{k}(\bar{x},\bar{z})-R^{T,n}_{k}(x,\bar{z}).$

Hence (6.3) follows by rearranging terms.

Next, we consider the case $\bar{x}\leq x$. From (4.7), we have

$\displaystyle\mathcal{H}^{T,n}(x,y)-\mathcal{H}^{T,n}(\bar{x},y)+\log
B^{T,n}_{k}(x,y;\bar{z})\leq
F^{T,n}_{k}(x,\bar{z})-F^{T,n}_{k}(\bar{x},\bar{z}).$

Using (6.2), the right hand side of the above equals

$\displaystyle(k\log
x+T^{-1}\bar{z}x)-(k\log\bar{x}+T^{-1}\bar{z}\bar{x})+R^{T,n}_{k}(x,\bar{z})-R^{T,n}_{k}(\bar{x},\bar{z}).$

Using (6.1), it is bounded from above by

$\displaystyle-D^{-1}k|x-\bar{x}|^{2}+R^{T,n}_{k}(x,\bar{z})-R^{T,n}_{k}(\bar{x},\bar{z}).$

Rearranging the terms gives (6.4). ∎

The next proposition provides us the coupling to prove Theorem 1.10.

######  Proposition 6.2.

Fix $T>0$. There exists a sequence $\mathsf{n}$, and a coupling of
$\\{\mathcal{X}^{T,n},\mathcal{H}^{T,n}\\}_{n\in\mathsf{n}}$,
$\mathcal{X}^{T}$ and $\mathcal{H}^{T}$ such that the following statements
hold.

Almost surely, $\mathcal{X}^{T,n}$ converges to $\mathcal{X}^{T}$ in
$C(\mathbb{N}\times\mathbb{R},\mathbb{R})$, $\mathcal{H}^{T,n}(x,y)$ converges
to $\mathcal{H}^{T}(x,y)$ for all $x,y\in\mathbb{Q}$ and
${R}^{T,n}_{k}(x,-Tk/\bar{x})$ converge for all $k\geq 1$ and
$x,\bar{x}\in\mathbb{Q}^{+}$. The limits of ${R}^{T,n}_{k}(x,-Tk/\bar{x})$ are
denoted by ${R}^{T}_{k}(x,-Tk/\bar{x})$. It holds that
$\mathcal{H}^{T}(0,\cdot)=\mathcal{X}^{T}_{1}(\cdot)$.

Moreover, suppose (1.14) holds. Then for all $x,\bar{x}\in\mathbb{Q}^{+}$, it
holds almost surely

(6.5) $\lim_{k\to\infty}|R^{T}_{k}(x,-Tk/\bar{x})/k|=0.$

###### Proof.

From Theorem 4.1, $\\{\mathcal{X}^{T,n}\\}_{n\in\mathbb{N}}$ is tight in
$C(\mathbb{N}\times\mathbb{R},\mathbb{R})$. From Proposition 4.2, the finite-
dimensional distribution of $\mathcal{H}^{T,n}$ is tight. From Lemma 4.3,
${R}^{T,n}_{k}(x,-Tk/\bar{x})$ has the distributional limit

$\mathcal{X}^{T}[(0,k+1)\to(x,1)]-k\log x+\log k!.$

By the Skorokhod’s representation theorem [Bil99, Theorem 6.7], we may find a
sequence $\mathsf{n}$ and a coupling of
$\\{\mathcal{X}^{T,n},\mathcal{H}^{T,n}\\}_{n\in\mathsf{n}}$ such that along
$\mathsf{n}$, $\mathcal{X}^{T,n}$, $\mathcal{H}^{T,n}(x,y)$ and
${R}^{T,n}_{k}(x,-Tk/\bar{x})$ converge almost surely. We note that the
convergences of the latter two hold at rational points. From Theorem 4.1, the
limit of $\mathcal{X}^{T,n}$ is distributed as the KPZ line ensemble and we
denote it by $\mathcal{X}^{T}$. From Proposition 4.2, we may augment the
probability space to accommodate the KPZ sheet $\mathcal{H}^{T}$ such that
$\mathcal{H}^{T,n}(x,y)$ converges to $\mathcal{H}^{T}(x,y)$ for all
$x,y\in\mathbb{Q}$. From
$\mathcal{H}^{T,n}(0,\cdot)=\mathcal{X}^{T,n}_{1}(\cdot)$, we may further
require $\mathcal{H}^{T}(0,\cdot)=\mathcal{X}^{T}_{1}(\cdot)$. Denote the
limits of ${R}^{T,n}_{k}(x,-Tk/\bar{x})$ by ${R}^{T}_{k}(x,-Tk/\bar{x})$. From
Proposition 4.3,

$\displaystyle
R^{T}_{k}(x,-Tk/\bar{x})\overset{d}{=}\mathcal{X}^{T}[(0,k+1)\to(x,1)]-k\log
x+\log k!.$

Suppose (1.14) holds. This implies for all $\varepsilon>0$,

$\displaystyle\sum_{k=1}^{\infty}\mathbb{P}\left(|R^{T}_{k}(x,-Tk/\bar{x})|>\varepsilon
k\right)<\infty.$

Then (6.5) follows from the Borel-Cantelli lemma. ∎

###### Proof of Theorem 1.10.

Throughout this proof, we assume (1.14) is valid. Fix $T>0$. From Proposition
6.2, we can find a sequence $\mathsf{n}$ and a coupling of
$\\{\mathcal{X}^{T,n},\mathcal{H}^{T,n}\\}_{n\in\mathsf{n}}$ with the
following property. There exists an event $\Omega_{0}$ with probability one on
which the statements below hold.

  1. (1)

$\mathcal{X}^{T,n}$ converges to the KPZ line ensemble $\mathcal{X}^{T}$ in
$C(\mathbb{N}\times\mathbb{R},\mathbb{R})$.

  2. (2)

$\mathcal{H}^{T,n}(x,y)$ converges to the KPZ sheet $\mathcal{H}^{T}(x,y)$ for
all $x,y\in\mathbb{Q}$.

  3. (3)

$R^{T,n}_{k}(x,-Tk/\bar{x})$ converges to $R^{T}_{k}(x,-Tk/\bar{x})$ for all
$x,\bar{x}\in\mathbb{Q}^{+}$ and $k\in\mathbb{N}$.

  4. (4)

(6.5) holds.

Our goal is to show that (1.13) holds on $\Omega_{0}$.

Fix arbitrary $x_{0}<x_{+}$ in $\mathbb{Q}^{+}$ and $y_{1}\leq y_{2}$ in
$\mathbb{Q}$. We claim that

(6.6)
$\begin{split}\limsup_{k\to\infty}\bigg{(}\mathcal{X}^{T}[(-Tk/x_{0},k)\to(y_{2},1)]-\mathcal{X}^{T}[(-Tk/x_{0},k)&\to(y_{1},1)]\bigg{)}\\\
&\leq\mathcal{H}^{T}(x_{+},y_{2})-\mathcal{H}^{T}(x_{+},y_{1}).\end{split}$

Let $z_{k}=-kT/x_{0}$. From (4.10), we have

$\begin{split}\mathcal{X}^{T,n}[(z_{k},k)\to(y_{2},1)]-\mathcal{X}^{T,n}[(z_{k},k)\to(y_{1},1)]-\mathcal{H}^{T,n}&(x_{+},y_{2})+\mathcal{H}^{T,n}(x_{+},y_{1})\\\
&\leq-\log\left(1-B^{T,n}_{k}(x_{+},y_{1},z_{k})\right).\end{split}$

Let $n$ and $k$ go to infinity, we have

$\displaystyle\limsup_{k\to\infty}\bigg{(}\mathcal{X}^{T}[(z_{k},k)\to(y_{2},1)]-\mathcal{X}^{T}[(z_{k},k)\to$
$\displaystyle(y_{1},1)]\bigg{)}-\mathcal{H}^{T}(x_{+},y_{2})+\mathcal{H}^{T}(x_{+},y_{1})$
$\displaystyle\leq$
$\displaystyle-\log\bigg{(}1-\limsup_{k\to\infty}\limsup_{\begin{subarray}{c}n\in\mathsf{n}\\\
n\to\infty\end{subarray}}B^{T,n}_{k}(x_{+},y_{1};z_{k})\bigg{)}.$

To obtain (6.6), it suffice to show that the limit of
$B^{T,n}_{k}(x_{+},y_{1};z_{k})$ is zero. Equivalently,

(6.7) $\limsup_{k\to\infty}\limsup_{\begin{subarray}{c}n\in\mathsf{n}\\\
n\to\infty\end{subarray}}\log B^{T,n}_{k}(x_{+},y_{1};z_{k})=-\infty.$

Applying (6.4) with $\bar{x}=x_{0}$ and $x=x_{+}$, we have

$\displaystyle\log B^{T,n}_{k}(x_{+},y_{1};z_{k})\leq$
$\displaystyle-D^{-1}k|x_{+}-x_{0}|^{2}+\mathcal{H}^{T,n}(x_{0},y_{1})-\mathcal{H}^{T,n}(x_{+},y_{1})-R^{T,n}_{k}(x_{0},z_{k})+R^{T,n}_{k}(x_{+},z_{k}).$

Because of (6.5), the limit of the right hand side is $-\infty$. Therefore we
proved (6.7) and (6.6). For any $x_{-}<x_{0}$ in $\mathbb{Q}^{+}$, a similar
argument yields

(6.8)
$\begin{split}\liminf_{k\to\infty}\bigg{(}\mathcal{X}^{T}[(-Tk/x_{0},k)\to(y_{2},1)]-\mathcal{X}^{T}[(-Tk/x_{0},k)&\to(y_{1},1)]\bigg{)}\\\
&\geq\mathcal{H}^{T}(x_{-},y_{2})-\mathcal{H}^{T}(x_{-},y_{1}).\end{split}$

Combining (6.6) and (6.8), we obtain (1.13) for $x,y_{1},y_{2}\in\mathbb{Q}$.

Next, we show that (1.13) holds for $x\in\mathbb{Q}^{+}$ and
$y_{1},y_{2}\in\mathbb{R}$. Let $y_{1,j}$ and $y_{2,j}$ be a sequence of
rational numbers that converge to $y_{1}$ and $y_{2}$ respectively. We further
require $y_{1,j}\leq y_{1}$ and $y_{2,j}\geq y_{2}$, from Lemma 2.5, we have

$\displaystyle\mathcal{X}^{T}[(-Tk/x,k)\to(y_{2},1)]\leq$
$\displaystyle\mathcal{X}^{T}[(-Tk/x,k)\to(y_{2,j},1)]-\mathcal{X}^{T}_{1}(y_{2,j})+\mathcal{X}^{T}_{1}(y_{2}),$
$\displaystyle\mathcal{X}^{T}[(-Tk/x,k)\to(y_{1},1)]\geq$
$\displaystyle\mathcal{X}^{T}[(-Tk/x,k)\to(y_{1,j},1)]-\mathcal{X}^{T}_{1}(y_{1,j})+\mathcal{X}^{T}_{1}(y_{1}).$

Therefore,

$\begin{split}&\limsup_{k\to\infty}\bigg{(}\mathcal{X}^{T}[(-Tk/x,k)\to(y_{2},1)]-\mathcal{X}^{T}[(-Tk/x,k)\to(y_{1},1)]\bigg{)}\\\
\leq&\mathcal{H}^{T}(x,y_{2,j})-\mathcal{H}^{T}(x,y_{1,j})-\mathcal{X}^{T}_{1}(y_{2,j})+\mathcal{X}^{T}_{1}(y_{2})+\mathcal{X}^{T}_{1}(y_{1,j})-\mathcal{X}^{T}_{1}(y_{1}).\end{split}$

Let $j$ go to infinity, we get

$\begin{split}\limsup_{k\to\infty}\bigg{(}\mathcal{X}^{T}[(-Tk/x,k)\to(y_{2},1)]-\mathcal{X}^{T}[(-Tk/x,k)&\to(y_{1},1)]\bigg{)}\\\
&\leq\mathcal{H}^{T}(x,y_{2})-\mathcal{H}^{T}(x,y_{1}).\end{split}$

The other direction can be proved similarly.

Lastly, we can remove the condition $x\in\mathbb{Q}_{+}$ by noting that from
Lemma 2.4,
$\mathcal{X}^{T}[(-Tk/x,k)\to(y_{2},1)]-\mathcal{X}^{T}[(-Tk/x,k)\to(y_{1},1)]$
is monotone non-decreasing in $x$. ∎

## Appendix A

In the appendix we provide proofs for basic results in Section 2.

###### Proof of Lemma 2.4.

We use an induction argument on $\ell-m$. The assertion holds when $\ell=m$
because $f[(x,m)\to(y,m)]=f_{m}(y)-f_{m}(x)$. From Lemma 2.3, we have

$\displaystyle\exp\big{(}f[(x,\ell)\to(y,m)]\big{)}=\int_{x}^{y}\exp\big{(}f[(x,\ell)\to(z,m+1)]+f_{m}(y)-f_{m}(z)\big{)}\,dz.$

Hence
$\displaystyle\frac{d}{dy}\bigg{(}f[(x_{2},\ell)\to(y,m)]-f[(x_{1},\ell)\to(y,m)]\bigg{)}$
equals

(A.1)
$\begin{split}&\left(\int_{x_{2}}^{y}\exp\big{(}f[(x_{2},\ell)\to(z,m+1)]-f_{m}(z)\big{)}\,dz\right)^{-1}\exp\bigg{(}f[(x_{2},\ell)\to(y,m+1)]-f_{m}(y)\bigg{)}\\\
-&\left(\int_{x_{1}}^{y}\exp\big{(}f[(x_{1},\ell)\to(z,m+1)]-f_{m}(z)\big{)}\,dz\right)^{-1}\exp\bigg{(}f[(x_{1},\ell)\to(y,m+1)]-f_{m}(y)\bigg{)}.\end{split}$

From the induction hypothesis,
$f[(x_{2},\ell)\to(z,m+1)]-f[(x_{1},\ell)\to(z,m+1)]$ is non-decreasing in
$z$. Therefore,

$\displaystyle\int_{x_{2}}^{y}\exp\big{(}f[(x_{2},\ell)\to(z,m+1)]-f_{m}(z)\big{)}\,dz$
$\displaystyle\leq$
$\displaystyle\exp\big{(}f[(x_{2},\ell)\to(y,m+1)]-f[(x_{1},\ell)\to(y,m+1)]\big{)}\times\int_{x_{2}}^{y}\exp\big{(}f[(x_{1},\ell)\to(z,m+1)]-f_{m}(z)\big{)}\,dz$
$\displaystyle\leq$
$\displaystyle\exp\big{(}f[(x_{2},\ell)\to(y,m+1)]-f[(x_{1},\ell)\to(y,m+1)]\big{)}\times\int_{x_{1}}^{y}\exp\big{(}f[(x_{1},\ell)\to(z,m+1)]-f_{m}(z)\big{)}\,dz.$

Apply the above inequality to (A.1), we obtain
$\displaystyle\frac{d}{dy}\bigg{(}f[(x_{2},\ell)\to(y,m)]-f[(x_{1},\ell)\to(y,m)]\bigg{)}\geq
0$. ∎

###### Proof of Lemma 2.5.

Consider the following measure-preserving injection from
$\mathcal{Q}[(x,\ell)\to(y_{1},m)]$ to $Q[(x,\ell)\to(y_{2},m)]$. Given
$\pi\in\mathcal{Q}[(x,\ell)\to(y_{1},m)]$, let

$\displaystyle\bar{\pi}(t)=\left\\{\begin{array}[]{cc}\pi(t),&t\in[x,y_{1}],\\\
m,&t\in(y_{1},y_{2}].\end{array}\right.$

Then the assertion follows $f(\pi)=f(\bar{\pi})-f_{m}(y_{2})+f_{m}(y_{1})$. ∎

###### Proof of Lemma 2.6.

There is a natural map from $\mathcal{Q}[(x,\ell)\to(y,m)]$ to
$\mathcal{Q}[(a_{2}x+a_{3},\ell)\to(a_{2}y+a_{3},m)]$ given by
$\pi(t)\mapsto\pi^{\prime}(t)=\pi(a_{2}^{-1}t-a_{2}^{-1}a_{3})$. Moreover,
$d\pi=a_{2}^{-(\ell-m)}d\pi^{\prime}$. Together with
$g(\pi)=a_{1}f(\pi^{\prime})+a_{4}(y-x)$, we derive

$\displaystyle g[(x,\ell)\xrightarrow{\beta}(y,m)]=$
$\displaystyle\beta^{-1}\log\int_{\mathcal{Q}[(x,\ell)\to(y,m)]}\exp(\beta
g(\pi))d\pi$ $\displaystyle=$
$\displaystyle\beta^{-1}\log\int_{\mathcal{Q}[(a_{2}x+a_{3},\ell)\to(a_{2}y+a_{3},m)]}a_{2}^{-(\ell-m)}\exp(a_{1}\beta
f(\pi^{\prime})+a_{4}\beta(y-x))d\pi^{\prime}$ $\displaystyle=$ $\displaystyle
a_{1}\cdot
f[(a_{2}x+a_{3},\ell)\xrightarrow{a_{1}\beta}(a_{2}y+a_{3},m)]+a_{4}(y-x)-\beta^{-1}(\ell-m)\log
a_{2}.$

∎

###### Proof of Lemma 2.7.

Fix $\pi_{i}\in\mathcal{Q}[(x_{i},\ell_{i})\to(y_{i},m_{i})]$ and let
$(t_{i,j})_{j\in\llbracket m_{i}+1,\ell_{i}\rrbracket}$ be the coordinates of
$\pi_{i}$ under the identification (2.1). We again follow the convention (2.2)
and set $t_{i,\ell_{i}+1}=x_{i}$ and $t_{i,m_{i}}=y_{i}.$ It suffices to show
that $\pi_{1}\prec\pi_{2}$ if and only if for all $j_{1}\in\llbracket
m_{1},\ell_{1}\rrbracket$ and $j_{2}\in\llbracket m_{2},\ell_{2}\rrbracket$
with $j_{1}\geq j_{2}$, it holds that

(A.2) $\displaystyle t_{1,j_{1}}\leq t_{2,j_{2}+1}.$

Suppose $\pi_{1}\prec\pi_{2}$ fails. There exists
$t_{0}\in(x_{1},y_{1})\cap(x_{2},y_{2})$ such that
$\pi_{1}(t_{0})\geq\pi_{2}(t_{0})$. Set $j_{i}=\pi_{i}(t_{0})$. Because
$\pi_{i}$ are càdlàg and integer-valued, there exists $\varepsilon>0$ such
that $\pi_{i}(t)=j_{i}$ for $t\in[t_{0},t_{0}+\varepsilon)$. In view of (2.3),
this implies $t_{1,j_{1}}>t_{2,j_{2}+1}$ and $\eqref{equ:t1t2}$ is violated.

Suppose $t_{1,j_{1}}>t_{2,j_{2}+1}$ for some $j_{1}\in\llbracket
m_{1},\ell_{1}\rrbracket$ and $j_{2}\in\llbracket m_{2},\ell_{2}\rrbracket$
with $j_{1}\geq j_{2}$. Because $x_{1}\leq x_{2}$ and $y_{1}\leq y_{2}$, we
may assume $(t_{1,j_{1}+1},t_{1,j_{1}})$ and $(t_{2,j_{2}+1},t_{2,j_{2}})$ are
non-empty by increasing $j_{1}$ or decreasing $j_{2}$ if necessary. Moreover,
by further increasing $j_{1}$, we may assume
$(t_{1,j_{1}+1},t_{1,j_{1}})\cap(t_{2,j_{2}+1},t_{2,j_{2}})$ is non-empty. In
view of (2.3), this implies there exists $t\in(x_{1},y_{1})\cap(x_{2},y_{2})$
such that $\pi_{1}(t)=j_{1}\geq\pi_{2}(t)=j_{2}$ and hence
$\pi_{1}\prec\pi_{2}$ fails. ∎

###### Proof of Lemma 2.9.

For $i\in\llbracket 1,k\rrbracket$, let
$(\tilde{x}_{i},\tilde{\ell}_{i})=(z-y_{k+1-i},n+1-m_{k+1-i})$ and
$(\tilde{y}_{i},\tilde{m}_{i})=(z-x_{k+1-i},n+1-\ell_{k+1-i})$. There is a
natural measure-preserving bijection between
$\mathcal{Q}[(x_{k+1-i},\ell_{k+1-i})\to(y_{k+1-i},m_{k+1-i})]$ and
$\mathcal{Q}[(\tilde{x}_{i},\tilde{\ell}_{i})\to(\tilde{y}_{i},\tilde{m}_{i})]$.
Given
$\pi_{k+1-i}\in\mathcal{Q}[(x_{k+1-i},\ell_{k+1-i})\to(y_{k+1-i},m_{k+1-i})]$,
set
$\tilde{\pi}_{i}\in\mathcal{Q}[(\tilde{x}_{i},\tilde{\ell}_{i})\to(\tilde{y}_{i},\tilde{m}_{i})]$
as follows. Let $\tilde{\pi}_{i}(\tilde{y}_{i})=\tilde{m}_{i}$ and
$\tilde{\pi}_{i}(t)=n+1-\lim_{t^{\prime}\to t+}\pi_{k+1-i}(z-t)$. It can be
checked that $f(\pi_{k+1-i})=(R_{z}f)(\tilde{\pi}_{i})$ and then
$f(\pi)=(R_{z}f)(\tilde{\pi})$. Hence the assertion follows.

∎

###### Proof of Lemma 2.10.

We give the proof for (2.9). The arguments for (2.10) and (2.11) are
analogous. Consider a map
$\mathbf{G}:\mathcal{Q}[(x,n)^{k+1}\to(y,1)^{k+1}]\to\mathcal{Q}[U_{n,k+1}(x)\to
V_{k+1}(y)]$ given by the following. For
$\pi=(\pi_{1},\dots,\pi_{k+1})\in\mathcal{Q}[(x,n)^{k+1}\to(y,1)^{k+1}]$, we
define $\mathbf{G}(\pi)=\bar{\pi}=(\bar{\pi}_{1},\dots,\bar{\pi}_{k+1})$
through

$\displaystyle\bar{\pi}_{j}(t)=\left\\{\begin{array}[]{cc}\pi_{j}(t),&x\leq
t<y,\\\ j,&t=y.\end{array}\right.$

It can be checked that $\bar{\pi}\in\mathcal{Q}[U_{n,k+1}(x)\to V_{k+1}(y)]$
and $\mathbf{G}$ is a bijection. Moreover, $\mathbf{G}$ is the restriction of
a projection map between Euclidean spaces. This implies $\mathbf{G}$ is
measure-preserving. Together with $f(\pi)=f(\mathbf{G}(\pi))$, (2.9) follows.
∎

###### Proof of Lemma 2.12.

Consider the map from $\mathcal{Q}[V^{\prime}_{k}(x)\to V_{k}(y)]$ to
$\mathcal{Q}[(x,1)\searrow(y,k+1)]$ given by the following. For
$\pi=(\pi_{1},\dots,\pi_{k})\in\mathcal{Q}[V^{\prime}_{k}(x)\to V_{k}(y)]$,
let $\rho$ be defined in the way such that for all $t\in[x,y]$,
$\\{\pi_{1}(t),\pi_{2}(t),\dots,\pi_{k}(t),\rho(t)\\}=\llbracket
1,k+1\rrbracket$. It is straightforward to check that $\rho$ belongs to
$\mathcal{Q}[(x,1)\searrow(y,k+1)]$. Moreover, this map is a measure-
preserving bijection. Together with

$f(\pi)+f(\rho)=\sum_{i=1}^{k+1}f_{i}(y)-f_{i}(x),$

we have

$\displaystyle f[V^{\prime}_{k-1}(x)\to V_{k-1}(y)]$
$\displaystyle=\log\int_{\mathcal{Q}[V^{\prime}_{k}(x)\to
V_{k}(y)]}\exp(f(\pi))\,d\pi$
$\displaystyle=\sum_{i=1}^{k+1}f_{i}(y)-f_{i}(x)+\log\int_{\mathcal{Q}[(x,1)\searrow(y,k+1)]}\exp(-f(\rho))\,d\rho$
$\displaystyle=f[V_{k+1}(x)\to V_{k+1}(y)]-f[(x,1)\searrow(y,k+1)].$

∎

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