# Diffusive scaling limit of the Busemann process in Last Passage Percolation

Ofer Busani University of Bristol, School of Mathematics, Fry Building,
Woodland Rd., Bristol BS8 1UG, UK. O. Busani was supported by EPSRC’s
EP/R021449/1 Standard Grant. E-mail<EMAIL_ADDRESS>
###### Abstract

In exponential last passage percolation, we consider the rescaled Busemann
process $x\mapsto N^{-1/3}B^{\rho}_{0,[xN^{2/3}]e_{1}}\,\,(x\in\mathbb{R})$,
as a process parametrized by the scaled density
$\rho=1/2+\frac{\mu}{4}N^{-1/3}$, and taking values in $C(\mathbb{R})$. We
show that these processes, as $N\rightarrow\infty$, have a càdlàg scaling
limit $G=(G_{\mu})_{\mu\in\mathbb{R}}$, parametrized by $\mu$ and taking
values in $C(\mathbb{R})$. The limiting process $G$, which can be thought of
as the Busemann process under the KPZ scaling, can be described as an ensemble
of ”sticky” lines of Brownian regularity. We believe $G$ is the universal
scaling limit of Busemann processes in the KPZ universality class. Our proof
provides insight into this limiting behaviour by highlighting a connection
between the joint distribution of Busemann functions obtained by Fan and
Seppäläinen in [17], and a sorting algorithm of random walks introduced by
O’Connell and Yor in [32].

###### Contents

  1. 1 Introduction
    1. 1.1 Notation
    2. 1.2 Main results
    3. 1.3 Acknowledgements
  2. 2 Some results on queues in stationarity
    1. 2.1 Preliminaries on Queues
    2. 2.2 Stationary measures for queues in tandem
    3. 2.3 The construction $\mathfrak{D}^{(n)}$
    4. 2.4 The operator $\mathfrak{D}$ as melonization of random walks
  3. 3 The Busemann Process
    1. 3.1 Definition and main properties
    2. 3.2 Upper bound on the number of jumps of the Busemann process
  4. 4 Convergence to the Stationary Horizon
    1. 4.1 The space $D(\mathbb{R},\mathcal{S})$
    2. 4.2 Finite dimensional distributions of the stationary horizon
      1. 4.2.1 The map $\Phi$
      2. 4.2.2 From $\Phi^{k}$ to $\mathcal{D}^{(k)}$
      3. 4.2.3 Controlling the probability of the events and finite dimensional convergence
    3. 4.3 Putting everything together
  5. A Appendix
    1. A.1 Queues
    2. A.2 Results on Random walks
    3. A.3 Topological results
    4. A.4 Proofs of some computations appearing in the text

##  1 Introduction

In last passage percolation (LPP), i.i.d weights are assigned to each vertex
of $\mathbb{Z}^{2}$. If $x,y\in\mathbb{Z}^{2}$ such that $y$ is above and to
the right of $x$, then one defines the last passage time $L(x,y)$ as the
maximum total weight collected by an up-right path on the lattice, starting at
$x$ and terminating at $y$. A path on which the total weight is attained is
called a geodesic from $x$ to $y$. Lattice LPP belongs to a large family of
LPP models, which can be viewed as random directed metric spaces [15], which
seem to share the same limiting behaviour of the fluctuations of their
observables. Since the seminal work of Rost [39] where the first order of
$L(\bm{0},y)$ was established for $y$ large in exponential LPP, much progress
has been made in the study of these models [1, 36, 29, 14, 27].

One of the aspects of LPPs and random metric-like models that has been studied
in the past three decades is that of infinite geodesics. An infinite geodesic
is an up-right path on the random environment, such that its restriction to
between any two points on it is a geodesic. Questions such as the existence
and uniqueness of infinite geodesics as well as other finer results have been
studied in first passage percolation (FPP) by C. Newman and co-authors [25,
26, 30, 28], while results of that flavour for various LPP models can be found
in [20, 33, 6, 7, 3, 27, 41, 42, 43, 37]. One of the main tools of studying
infinite geodesics in random (directed) metrics models is the Busemann
function - a random stationary real valued function defined on
$\mathbb{Z}^{2}\times\mathbb{Z}^{2}$ which satisfies the cocycle property and
a so-called Burke’s property. Originating in hyperbolic geometry, the Busemann
functions were introduced first to FPP by Newman [30] and Hoffman [24] and
later to LPP in [21]. They underlie many of the techniques used to study
infinite geodesics [3, 41] and have deep connections to the set of exceptional
directions at which uniqueness of the geodesics fail [27]. They are also
important in the study of point-to-point geodesics [2], their coalescence [12]
and the regularity of the passage time profile around a point [33, 34, 2, 12,
35]. Busemann fucntions are usually associated with a direction on the first
quadrant of the lattice – for $\rho\in(0,1)$ one defines the Busemann function
$B^{\rho}$ by essentially setting $B^{\rho}(x,y)$ to be the difference in
passage time between two geodesics starting from $x$ and $y$ and going in a
direction that is parametrised by $\rho\in(0,1)$. $B^{\rho}$ turns out to hold
much (and in some models all) the information about infinite geodesics
emanating from every point of the lattice, in the direction associated with
$\rho$. It is possible, to construct Busemann functions of different intensity
$\rho$ on one probability space to obtain the Busemann process
$\bm{B}=\\{B^{\rho}\\}_{\rho\in(0,1)}$. The process $\bm{B}$ turns out to be
very rich, and in the case of Exponential LPP, it was shown to hold all the
information on the exceptional directions of non uniqueness of infinite
geodesics [27]. For more on the magic of Busemann functions in the context of
LPP see [40, 38].

The random picture of passage times and geodesics has an interesting scaling
limit achieved through an incremental progress over the past three decades [1,
36, 29] (to mention a few), culminating in showing that, under the (1:2:3)-KPZ
scaling, the microscopic metric-like space of LPP converges to an object
dubbed the Directed Landscape(DL) [14]. The DL is a continuous function
$\mathcal{L}:\mathbb{R}^{2}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ holding
much of the information on the limiting fluctuations of LPP models in the KPZ
class. The prelimiting geodesics, in turn, scale to continuous functions on
the plane - the geodesics of the DL. While much of the work in recent years
have been to understand the scaling limit of the LPP picture on a compact set
in the DL, the question of the scaling behavior of infinite geodesics to that
of those in the DL is interesting, and first steps were taken very recently in
[37] where the existence of infinite geodesics and Busemann functions was
proven in the DL.

In that context, it would be interesting to see what is the Busemann process
in the DL [13, Remark 1.3] i.e. does the limit

$\displaystyle
B_{DL}^{z}(x,0;y,t)=\lim_{r\rightarrow\infty}\mathcal{L}(rz,-r;x,0)-\mathcal{L}(rz,-r;y,t)\qquad\forall
z,x,y,t\in\mathbb{R},$ (1.1)

exist and what are its properties. Note that the index $z$ in (1.1) plays the
role of the direction while $(x,s)$ and $(y,t)$ are points on the plane. The
existence of $B_{DL}^{z}$ for every $z\in\mathbb{R}$ was proven very recently
in [37, Section 3.8]. Following the same progress as in that of the lattice
LPP, it would be interesting to see what is the distribution of the Busemann
process $z\mapsto B_{DL}^{z}$, its properties, and its connection to infinite
geodesics in the DL. From the co-cycle property of the Busemann functions, the
study of $B_{DL}$ can be broken down into two steps: a) study the Busemann
process on one time horizon, i.e. study the process $H_{DL}^{z}$ defined by
$z\mapsto B_{DL}^{z}(0,0;x,0)$, b) push the process $H_{DL}^{z}$ through
$\mathcal{L}$ to obtain $B_{DL}$ on different time horizons i.e.

$\displaystyle
B_{DL}^{z}(0,0;x,t)=\sup_{y}H^{z}_{DL}(y)\mathcal{L}(y,0;x,t)\qquad
z\in\mathbb{R},t>0.$ (1.2)

From the stationarity of the Busemann process $\bm{B}$ on the lattice and the
(1:2:3)-KPZ limit, it is expected that $H_{DL}$ should emerge as the diffusive
scaling limit of

$\displaystyle H^{\rho}_{x}=B^{1/2+\rho}_{(0,0),xe_{1}}\qquad
x\in\mathbb{R},\rho\in(-1/2,1/2),$

that is, we expect the following sequence

$\displaystyle
G^{N}_{\rho}(x):=N^{-1/3}\big{[}H^{\frac{\rho}{4}N^{-1/3}\rho(z)}_{xN^{2/3}}-2xN^{2/3}\big{]},$
(1.3)

to converge in some sense to an object parameterized by the intensity $\rho$
and taking values in $C(\mathbb{R})$. This limit should, perhaps under some
reparametrization, be equal in distribution to $H_{DL}$, and so, should be
universal in the KPZ class.

The main motivation behind this work is the study of $B_{DL}$ and the
geodesics associated to it in the DL. This work establishes the limit $G$ of
the sequence in (1.3), which we call the Stationary Horizon (SH), and studies
its properties.

Only a few months after the first draft of this paper, Seppäläinen and
Sorensen obtained, among other results, the joint distribution of the unscaled
Busemann process in Brownian LPP (BLPP), i.e. when the set of directions by
which the process is parameterized is macroscopic, as opposed to our work
where we scale the directions around a specific macroscopic direction.
Interestingly, although not a proven result yet, it seems that the (unscaled)
Busemann process in BLPP is essentially the SH. From that perspective, the
results in [42, 43] would complement the results in this work. One possible
explanation for the SH appearance as Busemann process of the BLPP without
scaling the direction, is that the BLPP can be thought of as a lattice LPP
after a diffusive scaling. It is then not surprising that the Busemann process
in BLPP has a scaling invariance property in that direction.

Regarding our proof and techniques, our starting point will be the results of
Fan and Seppäläinen in [17] where the finite dimensional distribution of the
Busemann process was obtained. Our main technique is a ”melonization”
representation of the map presented in [17], which, as it turns out, is
closely related to the RSK picture as non stationary queues developed in [31,
9] by O’Connell and co-authors. Loosely speaking, melonization of a finite
sequence of random walks (RWs) is a sorting process that results in a sequence
of ordered RWs and is closely related to the dynamics of queues in tandem. The
reason for the term melonization is that by Greene’s Theorem the sum of the
height of the $k$ top sorted RWs has the same distribution as that of the
maximal passage weight collected by $k$ non-intersecting paths, which arguably
looks like a melon. Even though we do not study a possible connection between
our ”stationary RSK” and $k$-paths maximal weight in this paper, we shall
still refer to our sorting process as melonization, to point out the relation
to previous work in the standard RSK model. Our representation does not seem
to be completely new, however, our derivation of it is new, and its
interpretation as melonization of random walks is proved important in this
work, and appears to be novel in context of two sided RWs. We discuss this at
length in Subsection 2.4.

This melonization dynamics is behind one of the most salient features of the
SH - although $G$ is two dimensional as function of space $x$ and its drift
$\mu$, it can be represented as an ensemble of ”sticky” lines (Figure 1(b)).
Alternatively, we can (and will) think of $G$ as a right continuous with left
limits (rcll) monotone process which is a jump process when restricted to any
compact interval (Figure 1(a)). This stickiness phenomenon is in line with
similar prelimiting and limiting behavior of LPP models studied in the
literature, for the Difference Profile [5, 8, 19], the Busemann process in
lattice LPP [27, 17] and the Shock Measures [13, 37, 15]. This in turn shows,
that the rcll microscopic behaviour of the Busemann function carries over to
its diffusive limit. It also implies that infinite geodesics associated with a
continuum of directions, emanating from a single point in the DL form a tree
(see 1(b)). We defer this, and the connection between the SH and the infinite
geodesics in the DL to a future work.

###  1.1 Notation

We denote by $\mathbb{N}=\\{1,2,...\\}$ the set of natural numbers. For
$\lambda>0$, we write $X\sim\rm{Exp}(\lambda)$ if $X$ is a random variable
s.t. $\mathbb{P}(X>t)=e^{-\lambda t}$ for $t>0$. For $\rho_{1},\rho_{2}>0$ and
two independent random variables $X^{1}\sim\mathrm{Exp}(\rho_{1})$,
$X^{2}\sim\mathrm{Exp}(\rho_{2})$ we write
$X\sim\mathrm{Exp}(\rho^{2},\rho^{1})$, if $X\sim X^{2}-X^{1}$. For
$x,y\in\mathbb{R}\cup\\{\infty,-\infty\\}$, we denote by $\llbracket
x,y\rrbracket$ the set $\mathbb{Z}\cap[x,y]$. $e_{1}$ and $e_{2}$ stand for
$(1,0)$ and $(0,1)$ respectively.

(a) Dense crossings of geodesics

(b) Sparse crossings of geodesics

Figure 1.1: Two possible behaviours of infinite geodesics starting from
$(N,N)$. From the point $(N,N)$ we consider infinite geodesics going in the
direction $(1-\alpha,1+\alpha)$ where $\alpha=O(N^{-1/3})$. These geodesics
will cross the $x$-axis within $O(N^{2/3})$ distance of the origin. Different
colours represent different intensities. Figure 1(a)) The set of crossing
points of the geodesics with the $x$-axis is ”dense” in $\mathbb{R}$ i.e.
within any interval of size $\Theta(N^{2/3})$ one can find a crossing with
high probability. Our work seems show that this picture is false. Figure 1(b))
The set of crossing points of the geodesics with the $x$-axis on an interval
of size $O(N^{2/3})$ is finite. Our work implies that this picture is the
correct one.

###  1.2 Main results

Let $\mathcal{S}=C(\mathbb{R})$ be the set of continuous functions on the real
line equipped with the topology of uniform convergence on compact intervals,
and let $D(\mathbb{R},\mathcal{S})$ be the space of $\mathcal{S}$-valued rcll
functions on the real line, equipped with the Skorohord topology (see
Subsection 4.1 for the exact definition). Let
$\bm{B}=\\{B^{\rho}\\}_{\rho\in(0,1)}$ be the Busemann process as defined in
Section 3. From $\bm{B}$ we construct a sequence of processes
$G^{N}=\\{G^{N}_{\mu}\\}_{\mu\in\mathbb{R}}$ in $D(\mathbb{R},\mathcal{S})$ as
follows. First define $F^{N}_{\cdot}\in D(\mathbb{R},\mathcal{S})$ through

$\displaystyle\text{$F^{N}_{\mu}$ is the linear interpolation of the discrete
function
$\\{B^{1/2-\frac{\mu}{4}N^{-1/3}}_{0,ie_{1}}\\}_{i\in\mathbb{Z}}$},\quad\forall\mu\in\mathbb{R}.$

Then

$\displaystyle
G^{N}_{\mu}(x)=N^{-1/3}[F^{N}_{\mu}(N^{2/3}x)-2N^{2/3}x],\quad\forall\mu\in\mathbb{R}.$

A rigorous definition of $G^{N}$ is given in (4.6). One should think of
$G^{N}$ as a process, indexed by $\mathbb{R}$, and whose values are two-sided
random walks. To see where the parametrization $\mu\mapsto
1/2-\frac{\mu}{4}N^{-1/3}$ for $G^{N}$ comes from, note first that from the
properties of the Busemann process (Section 3),
$B_{\cdot}^{1/2-\frac{\mu}{4}N^{-1/3}}-2\cdot\,$ is a discrete RW with mean
$\mu N^{-1/3}+O(N^{-2/3})$. This in turn means that $G^{N}_{\mu}$ is a RW with
drift $\mu+o(1)$. In other words, the process $G^{N}$ is parametrized by the
drift of the random walk $G^{N}_{\cdot}$. We are now ready to state our
results.

######  Theorem 1.1.

There exists a process $G\in D(\mathbb{R},\mathcal{S})$ such that the
following limit holds in distribution in $D(\mathbb{R},\mathcal{S})$

$\displaystyle G^{N}\Rightarrow G\qquad\text{as $N\rightarrow\infty$}.$

We call the limiting process $G$ the Stationary Horizon. Here are some of its
properties.

######  Theorem 1.2.

The Stationary Horizon $G$ satisfies the following:

  1. 1.

The one-dimensional distribution of $G$ is that of a Brownian motion with a
drift, i.e.

$\displaystyle G_{\mu}\sim W^{\mu}\quad\mu\in\mathbb{R},$ (1.4)

where $W^{\mu}$ is a doubly-infinite Brownian motion with drift $\mu$ and
diffusivity $2$, conditioned to vanish at the origin.

  2. 2.

For every $k\in\mathbb{N}$, there exists a function
$\Phi^{k}:C(\mathbb{R})^{k}\rightarrow C(\mathbb{R})^{k}$ such that for every
$\mu_{1}<...<\mu_{k}$ in $\mathbb{R}$

$\displaystyle(G_{\mu_{1}},...,G_{\mu_{k}})\sim\Phi^{k}(W^{\mu_{1}},...,W^{\mu_{k}})$

where $W^{\mu_{1}},...,W^{\mu_{k}}$ are independent two-sided Brownian motions
with diffusivity $2$ and drift $\mu_{1},...,\mu_{k}$ respectively, conditioned
to vanish at the origin.

  3. 3.

$G$ is scale-invariant with exponents $(1,-2,1)$, i.e. for $c>0$

$\displaystyle cG_{c\cdot}(c^{-2}\cdot)\sim G.$ (1.5)

  4. 4.

$G$ is stochastically continuous, i.e.

$\displaystyle\mathbb{P}(G_{\mu}-G_{\mu-}=0)=1\quad\forall\mu\in\mathbb{R}.$

  5. 5.

Let $x_{0}>0$ and consider the process $G^{x_{0}}\in
D\big{(}\mathbb{R},C([-x_{0},x_{0}])\big{)}$ defined through

$\displaystyle
G^{x_{0}}_{\mu}(x)=G_{\mu}(x)\quad\forall\mu\in\mathbb{R},x\in[-x_{0},x_{0}].$
(1.6)

In words, $G^{x_{0}}$ retains only the information of $G$ restricted to
$[-x_{0},x_{0}]$ (see Figure 4.1). Then $G^{x_{0}}$ is a pure jump process in
$D\big{(}\mathbb{R},C([-x_{0},x_{0}])\big{)}$, i.e. its increments are the sum
of its jumps in $D\big{(}\mathbb{R},C([-x_{0},x_{0}])\big{)}$. Moreover,
$G^{x_{0}}$ has finitely many jumps on any compact interval
$[-\mu_{0},\mu_{0}]$, and we have the bound

$\displaystyle\mathbb{P}\big{(}\\#\\{\text{$\mu\in[-\mu_{0},\mu_{0}]$:
$G^{x_{0}}$ has a jump at $\mu$ }\\}\geq M\big{)}\leq CM^{-1}\qquad M>0.$
(1.7)

for some constant $C>0$ depending on $x_{0}$ and $\mu_{0}$.

  6. 6.

For any real numbers $x_{1}\leq x_{2}$, the process

$\displaystyle\mu\mapsto G_{\mu}(x_{2})-G_{\mu}(x_{1})$

is a non-decreasing jump process in $D(\mathbb{R},\mathbb{R})$, i.e. a càdlàg
process such that for any $\mu_{1}\leq\mu_{2}$

$\displaystyle G_{\mu_{2}}(x_{2})-G_{\mu_{2}}(x_{1})\geq
G_{\mu_{1}}(x_{2})-G_{\mu_{1}}(x_{1}).$ (1.8)

Figure 4.1 is an illustration of a typical realization of $G$ on a compact
rectangle on the $\mu-x$ plane. Theorem 1.2 shows that some of the properties
of the microscopic Busemann process carry over in the limit as can be seen by
the similarity between properties 4–6 of the SH and the fact that $\rho\mapsto
B^{\rho}_{\bm{0},e_{1}}$ is a Poisson process [17].

######  Remark 1.3.

We believe the upper bound (1.7), on the number of jumps of the SH on a
compact set in $x-\mu$ is far from optimal, and we expect it to be
exponential.

######  Remark 1.4.

While we do not provide a proof, we do not expect $G$ to be Markovian.

Let us now try to give a sketch of the proof of the above result while
pointing out the main challenges in it. In general, in the Skorohord topology
on the real line, a sequence of random variables $X^{N}$ converges to some $X$
if (see A.3)

  1. 1.

The sequence $X^{N}$ is tight,

  2. 2.

$X^{N}$ converges in the sense of finite dimensional distributions.

Proving Item 1 consists mostly of showing that the processes in the sequence
$\\{G^{N}\\}_{N\in\mathbb{N}}$ do not have too many jumps on a compact
rectangle on the $\mu-x$ plane, uniformly in $N$. In Section 2, we develop the
technical tools to tackle this problem. We show that the map $\mathcal{D}$
developed in [17] can be obtained by the top lines of sorted random walks.
These tools are then used in Section 3.2 to obtain the necessary intermediate
result. For Item 2, it is natural to try to use the results on the finite
dimensional distributions of the Busemann process obtained in [17] and the
continuous mapping theorem, to show that the limit exists. Alas, the queueing
mapping $\Phi^{k}$ is not continuous in general. We work around this
difficulty in Section 4.2 where we condition on an event with high probability
on which the map is continuous.

###  1.3 Acknowledgements

The author would like to thank Márton Balázs for reading early parts of the
manuscripts and helpful comments, Patrik Ferrari and Offer Kella for helpful
discussions. The author also thanks Timo Seppäläinen and Evan Sorensen for
helpul comments.

##  2 Some results on queues in stationarity

In this section we develop the tools that will be used later to show that the
prelimiting sequence $\\{G^{N}\\}$ is tight. The main technical result of this
section is Proposition 2.11. We conclude this section with discussing our
results in the context of melonizing RWs, which although are not stated as a
theorem, underlie the main ideas in the proofs of Section 3.

###  2.1 Preliminaries on Queues

Let us recall the basic setup of queuing theory. Let
$\mathcal{X}_{+}=(\mathbb{R}_{+})^{\mathbb{Z}}$. Let
$\mathbf{a}=\\{a_{i}\\}_{i\in\mathbb{Z}}$ and
$\mathbf{s}=\\{s_{i}\\}_{i\in\mathbb{Z}}$ be the sequences in
$\mathcal{X}_{+}$ denoting the inter-arrival times and service times in a
queue respectively. We assume

$\displaystyle\lim_{m\rightarrow-\infty}\sum_{i=m}^{0}(s_{i-1}-a_{i})=-\infty.$
(2.1)

Given two sequences $\mathbf{a},\mathbf{s}\in\mathcal{X}_{+}$ satisfying
(2.1), the waiting time customer $i$ must wait in the queue before being
served is

$\displaystyle w_{i}=\sup_{j\leq
i}\Big{(}\sum_{k=j}^{i}s_{k-1}-a_{k}\Big{)}^{+},$

and we define the waiting time mapping
$W:\mathcal{X}_{+}^{2}\rightarrow\mathcal{X}_{+}$

$\displaystyle\\{w_{i}\\}_{i\in\mathbb{Z}}=W(\mathbf{s},\mathbf{a}).$ (2.2)

######  Remark 2.1.

Although the maps used in this paper are not defined on all the prescribed
domain (only on those points satisfying (2.1)), we shall abuse notations and
write $W:\mathcal{X}_{+}^{2}\rightarrow\mathcal{X}_{+}$.

Define the queuing mapping
$D(\cdot,\cdot):\mathcal{X}_{+}^{2}\rightarrow\mathcal{X}_{+}$ through

$\displaystyle\\{d_{i}\\}_{i\in\mathbb{Z}}$
$\displaystyle=D(\mathbf{s},\mathbf{a})$ (2.3) $\displaystyle d_{i}=(w_{i-1}$
$\displaystyle+s_{i-1}-a_{i})^{-}+s_{i}.$ (2.4)

The third map is the so-called sojourn mapping
$S_{oj}(\cdot,\cdot):\mathcal{X}_{+}^{2}\rightarrow\mathcal{X}_{+}$, which
stands for the time a customer spends in the queue once arriving it, and is
given by

$\displaystyle\\{t_{i}\\}_{i\in\mathbb{Z}}$
$\displaystyle=S_{oj}(\mathbf{s},\mathbf{a})$ (2.5) $\displaystyle t_{i}$
$\displaystyle=w_{i}+s_{i}.$

The fourth map $R(\cdot,\cdot):\mathcal{X}_{+}^{2}\rightarrow\mathcal{X}_{+}$
is given by

$\displaystyle\\{r_{i}\\}_{i\in\mathbb{Z}}$
$\displaystyle=R(\mathbf{s},\mathbf{a})$ (2.6) $\displaystyle
r_{i}=a_{i}\wedge t_{i-1}$ $\displaystyle=a_{i}-(w_{i-1}+s_{i-1}-a_{i})^{-}.$
(2.7)

Lastly, define the idle time mapping to be
$Er(\cdot,\cdot):\mathcal{X}^{2}\rightarrow\mathcal{X}$, accounting for the
time the sever is idle, defined through

$\displaystyle\\{e_{i}\\}_{i\in\mathbb{Z}}$
$\displaystyle=Er(\mathbf{s},\mathbf{a})$ (2.8) $\displaystyle e_{i}=(w_{i-1}$
$\displaystyle+s_{i-1}-a_{i})^{-}.$

Although the queueing mappings defined in (2.2)–(2.6) are usually defined in
the literature on $\mathcal{X}_{+}$, it is not hard to see that they are well
defined on the larger space $\mathcal{X}=\mathbb{R}^{\mathbb{Z}}$ as long as
(2.1) holds. From here on, we shall think of the queuing mappings as maps from
$\mathcal{X}^{2}$ to $\mathcal{X}$. Let
$\bm{\rho}=\\{\rho_{i}\\}_{i\in\mathbb{N}}\in\mathbb{R}_{+}^{\mathbb{N}}$ be a
strictly decreasing sequence of positive numbers. Let
$\\{I^{i}\\}_{i\in\mathbb{N}}$ be a sequence of independent r.v.s in
$\mathcal{X}$ i.e. $I^{i}=\\{I^{i}_{k}\\}_{k\in\mathbb{Z}}\in\mathcal{X}$ is a
random sequence of numbers. We further assume that
$I^{i}=\\{I^{i}_{k}\\}_{k\in\mathbb{Z}}$ is a sequence of i.i.d. r.v.s with
marginal distribution $I^{i}_{0}\sim\text{Exp}(\rho_{i})$. Let
$\overline{I}=(I^{1},I^{2},...)\in\mathcal{X}^{\mathbb{N}}$, we then write
$\overline{I}\sim\nu^{\bm{\rho}}$, i.e. $\nu^{\bm{\rho}}$ denotes the product
measure of i.i.d sequences of Exponential random variables with intensities
$\bm{\rho}$.

###  2.2 Stationary measures for queues in tandem

In this subsection we introduce the construction of the multi-species
stationary distribution of queues introduced in [17]. Fix $n\in\mathbb{N}$.
Let $\bm{\rho}^{n}=(\rho_{1},...,\rho_{n})\in\mathbb{R}_{+}^{n}$. Define the
state spaces

$\displaystyle\mathcal{Y}^{n}$
$\displaystyle:=\Big{\\{}(I^{1},...,I^{n})\in\mathcal{X}^{n}:\lim_{m\rightarrow\infty}m^{-1}\sum_{i=-m}^{0}I^{k}_{i}<\lim_{m\rightarrow\infty}m^{-1}\sum_{i=-m}^{0}I^{k+1}_{i}\quad\text{for
$1\leq k\leq n-1$}\Big{\\}}$ $\displaystyle\mathcal{Y}_{\bm{\rho}^{n}}$
$\displaystyle:=\Big{\\{}(I^{1},...,I^{n})\in\mathcal{X}^{n}:\lim_{m\rightarrow\infty}m^{-1}\sum_{i=-m}^{0}I^{k}_{i}=(\rho_{k})^{-1}\quad\text{for
$1\leq k\leq n$}\Big{\\}}.$

Suppose $\bm{\rho}^{n}=(\rho_{1},...,\rho_{n})\in\mathbb{R}_{+}^{n}$ is a
strictly decreasing vector, i.e. $\rho_{1}>...>\rho_{n}$. Let
$\rho_{1}<\rho_{0}$, and let $I^{0}\in\mathcal{X}$ such that

$\displaystyle\lim_{m\rightarrow\infty}m^{-1}\sum_{i=-m}^{0}I^{0}_{i}=(\rho_{0})^{-1}.$

We define the mapping $\Phi:\mathcal{Y}_{\bm{\rho}^{n}}\rightarrow\mathcal{X}$
associated with $I^{0}$ through

$\displaystyle\Phi\big{(}(I^{1},...,I^{n})\big{)}=(D(I^{0},I^{1}),...,D(I^{0},I^{n}))\qquad(I^{1},...,I^{n})\in\mathcal{Y}_{\bm{\rho}^{n}}.$

Now assume that $I^{0}$ is an i.i.d. sequence with
$I^{0}_{0}\sim\text{Exp}(\rho_{0})$ and that
$\overline{I}^{n}=(I^{1},...,I^{n})\in\mathcal{Y}_{\bm{\rho}^{n}}$ is random
and independent of $I^{0}$. A natural question in the context of queues in
tandem is under what distribution $\mu^{\bm{\rho}_{n}}$ of $\overline{I}^{n}$
the mapping $\Phi$ is stationary, i.e.

$\displaystyle\Phi\overline{I}^{n}\sim\mu^{\bm{\rho}_{n}},$

or, abusing notation, in terms of distributions

$\displaystyle\Phi\mu^{\bm{\rho}_{n}}=\mu^{\bm{\rho}_{n}}.$ (2.9)

This question was answered in [17] by Fan and Seppäläinen. In order to
describe their construction of the measure $\mu^{\bm{\rho}_{n}}$, we introduce
some notation. First extend the queueing operator $D$ to
$D^{(k)}:\mathcal{Y}^{k}\rightarrow\mathcal{X}$ through

$\displaystyle D^{(1)}(I^{1})$ $\displaystyle=I^{1}$ $\displaystyle
D^{(2)}(I^{1},I^{2})$ $\displaystyle=D(I^{1},D^{(1)}(I^{2}))$ $\displaystyle
D^{(3)}(I^{1},I^{2},I^{3})$ $\displaystyle=D(I^{1},D^{(2)}(I^{2},I^{3}))$
$\displaystyle\vdots$ $\displaystyle\text{ and in general
}D^{(k)}(I^{1},...,I^{k})$
$\displaystyle=D(I^{1},D^{(k-1)}(I^{2},...,I^{k}))\qquad k\geq 2.$ (2.10)

Note that for $1\leq k<n$

$\displaystyle
D^{(n)}(I^{1},...,I^{n})=D^{(k+1)}\Big{(}I^{1},...,I^{k},D^{(n-k)}(I^{k+1},...,I^{n})\Big{)}.$
(2.11)

Next define the map
$\mathcal{D}^{(n)}:\mathcal{Y}^{n}\rightarrow\mathcal{Y}^{n}$ given by

$\displaystyle[\mathcal{D}^{(n)}(\overline{I})]_{i}=D^{(i)}(I^{1},...,I^{i})\qquad
1\leq i\leq n,$ (2.12)

where we used the notation $[\bm{v}]_{i}=v_{i}$ for any vector $\bm{v}$ of
size $n$ such that $1\leq i\leq n$. The following result from [17, Theorem
5.5] is the foundation upon which we build our results.

######  Theorem 2.2.

[17] There exists a unique measure $\mu^{\bm{\rho}_{n}}$ satisfying (2.9), and
that is defined as the push-forward distribution with respect to the measure
$\nu^{\bm{\rho}_{n}}$ and $\mathcal{D}^{(n)}$, i.e.

$\displaystyle\mu^{\bm{\rho}_{n}}=\nu^{\bm{\rho}_{n}}\circ(\mathcal{D}^{(n)})^{-1}.$
(2.13)

From here to the end of the section we introduce notation and obtain some
auxilary results that will be used in the next subsection. For
$i,j\in\mathbb{N}$ and $m\in\mathbb{N}\cup\\{\infty\\}$ such that $m\geq j\geq
i$, define $\Pi_{[i,j]}$ to be the projection onto the i’th to j’th components
i.e.

$\displaystyle\Pi_{[i,j]}(\overline{I})=(I^{i},...,I^{j}),\quad\text{ for
}\overline{I}\in\mathcal{X}^{m}.$

We will abbreviate $\Pi_{i}:=\Pi_{[i,i]}$. Let
$\bm{\rho}=(\rho_{1},\rho_{2},...)\in\mathbb{R}_{+}^{\mathbb{N}}$ be a
decreasing vector i.e. $\rho_{i}>\rho_{i+1}$ for $i\in\mathbb{N}$. Define

$\displaystyle\mathcal{Z}:=\Big{\\{}(I^{1},I^{2},...)\in\mathcal{X}^{\mathbb{N}}:\text{
the following limit exists
}\lim_{m\rightarrow\infty}m^{-1}\sum_{k=-m}^{0}I^{i}_{k}\quad\text{for all
$i\in\mathbb{N}$}\Big{\\}}.$

Define the operator $\Gamma:\mathcal{Z}\rightarrow\mathbb{R}^{\mathbb{N}}$
through

$\displaystyle[\Gamma(\overline{I})]_{i}=\Bigg{[}\lim_{m\rightarrow\infty}m^{-1}\sum_{k=-m}^{0}I^{i}_{k}\Bigg{]}^{-1},\qquad\overline{I}\in\mathcal{Z},i\in\mathbb{N}.$

In words, the operator $\Gamma$ accepts as input a vector of sequences and
returns the vector of their densities. Next define, for each $i\in\mathbb{N}$,
the operator $\sigma_{i}:\mathcal{Z}\rightarrow\mathcal{Z}$ through

$\displaystyle(\sigma_{i}\overline{I})_{k}=\begin{cases}\overline{I}_{k}&\text{$k\notin\\{i,i+1\\}$
or $[\Gamma(\overline{I})]_{i}\leq[\Gamma(\overline{I})]_{i+1}$}\\\
D(\overline{I}_{i},\overline{I}_{i+1})&\text{$k=i$ and
$[\Gamma(\overline{I})]_{i}>[\Gamma(\overline{I})]_{i+1}$}\\\
R(\overline{I}_{i},\overline{I}_{i+1})&\text{$k=i+1$ and
$[\Gamma(\overline{I})]_{i}>[\Gamma(\overline{I})]_{i+1}$}.\end{cases}$ (2.14)

To see that $\sigma_{i}$ is well defined and maps into $\mathcal{Z}$, note
that for $i$ such that
$[\Gamma(\overline{I})]_{i}>[\Gamma(\overline{I})]_{i+1}$

$\displaystyle[\Gamma(\sigma_{i}\overline{I})]_{k}=\begin{cases}[\Gamma(\overline{I})]_{k}&k\notin\\{i,i+1\\}\\\
[\Gamma(\overline{I})]_{i+1}&k=i\\\
[\Gamma(\overline{I})]_{i}&k=i+1.\end{cases}$

We shall also need the following operator: for
$m\in\mathbb{N}\cup\\{\infty\\}$ and $1\leq i\leq m$, let
$\sigma^{*}_{i}:\mathbb{R}_{+}^{m}\rightarrow\mathbb{R}_{+}^{m}$

$\displaystyle[\sigma^{*}_{i}\bm{\rho}]_{l}=\begin{cases}\rho_{l}&\text{$l\notin\\{i,i+1\\}$
or $\rho_{i}<\rho_{i+1}$}\\\ \rho_{i+1}&\text{$l=i$ and
$\rho_{i}>\rho_{i+1}$}\\\ \rho_{i}&\text{$l=i+1$ and
$\rho_{i}>\rho_{i+1}$}.\end{cases}$

In words, the operator $\sigma^{*}_{i}$ takes a vector of length $m$ and
switches places between the $i$’th and $i+1$’th elements iff they are
reversely ordered i.e. $\rho_{i}>\rho_{i+1}$. Note that for
$\overline{I}\in\mathcal{X}^{m}$ we have

$\displaystyle\Gamma(\sigma_{i}\overline{I})=\sigma^{*}_{i}(\Gamma(\overline{I})).$
(2.15)

We say the operations $\sigma_{i}\overline{I}$ or
$\sigma^{*}_{i}\Gamma(\overline{I})$ are active if
$[\Gamma(\overline{I})]_{i}>[\Gamma(\overline{I})]_{i+1}$, otherwise we say
they are repressed. Note that the operation $\sigma_{i}\overline{I}$
($\sigma^{*}_{i}\Gamma(\overline{I})$) is active if and only if
$\sigma_{i}\overline{I}\neq\overline{I}$
($\sigma^{*}_{i}\Gamma(\overline{I})\neq\Gamma(\overline{I})$). The following
result establishes the effect of the operation $\sigma_{i}$ on the
distribution $\nu_{\bm{\rho}}$.

######  Lemma 2.3.

Suppose $\overline{I}\sim\nu_{\bm{\rho}}$. Fix $i\in\mathbb{N}$ and consider
$\sigma_{i}$ defined in (2.14). Then

$\displaystyle\sigma_{i}\overline{I}\sim\nu^{\sigma^{*}_{i}\bm{\rho}}.$ (2.16)

###### Proof.

From the law of large numbers applied to each element of $\overline{I}$, with
probability $1$ $\Gamma(\overline{I})=\bm{\rho}$. This means there are two
cases to check - a) $[\Gamma(\overline{I})]_{i}<[\Gamma(\overline{I})]_{i+1}$
a.s.- here nothing happens with probability $1$ and so (2.16) holds. (b)
$[\Gamma(\overline{I})]_{i}>[\Gamma(\overline{I})]_{i+1}$ a.s. - here with
probability $1$, the application of $\sigma_{i}$ results in

$\displaystyle\sigma_{i}\overline{I}=\big{(}\Pi_{[1,i-1]}(\overline{I}),D(I^{i},I^{i+1}),R(I^{i},I^{i+1}),\Pi_{[i+2,\infty)}(\overline{I})\big{)}.$

By hypothesis $\overline{I}\sim\nu_{\bm{\rho}}$, the result now follows from
Lemma A.4 and (2.15). ∎

In what follows, if $\overline{I}=(I^{1},...,I^{n})\in\mathcal{X}^{n}$, we
denote $D^{(n)}(\overline{I})=D^{(n)}(I^{1},...,I^{n})$. Next, define

$\displaystyle\sigma_{[i,j]}\overline{I}=\sigma_{i}\sigma_{i+1}\cdots\sigma_{j}\overline{I}.$
(2.17)

For $c\in\mathbb{R}$, we let $\mathbf{c}\in\mathcal{X}$ denote the constant
sequence. i.e. $\mathbf{c}_{i}=c$ for all $i\in\mathbb{Z}$. We denote
$\mathbf{c}^{k}=(\underbrace{\mathbf{c},...,\mathbf{c}}_{\text{k times}})$.
The following result will be used later in the paper, it shows that
translation of the input of the function $\mathcal{D}^{(n)}$ results in
translation of the output.

######  Lemma 2.4.

For every $n\in\mathbb{N}$ and $c\in\mathbb{R}$

$\displaystyle\mathcal{D}^{(n)}(\overline{I})-\mathbf{c}^{n}=\mathcal{D}^{(n)}(\overline{I}-\mathbf{c}^{n}).$
(2.18)

###### Proof.

From the definition (2.12), it is enough to show that

$\displaystyle
D^{(i)}(I^{1},...,I^{i})-\mathbf{c}=D^{(i)}(I^{1}-\mathbf{c},...,I^{i}-\mathbf{c})\quad
1\leq i\leq n.$ (2.19)

Clearly (2.19) holds for $i=1$, the proof for $i\geq 2$ will follow by
induction on $i$. For the base case $i=2$, it is not hard to see that for all
$k\in\mathbb{Z}$

$\displaystyle[D(I^{1},I^{2})]_{k}-c=d_{k}-c$
$\displaystyle=\big{(}W(I^{1}-\mathbf{c},I^{2}-\mathbf{c})_{k-1}+(I^{1}_{k-1}-c)-(I^{2}_{k}-c)\big{)}^{-}+I^{1}_{k}-c=[D(I^{1}-\mathbf{c},I^{2}-\mathbf{c})]_{k}.$

Assume the hypothesis holds up to $i-1$, using (2.2)

$\displaystyle
D^{(i)}(I^{1},...,I^{i})-\mathbf{c}=D\big{(}I^{1},D^{(i-1)}(I^{2},...,I^{i})\big{)}-\mathbf{c}$
$\displaystyle=D\big{(}I^{1}-\mathbf{c},D^{(i-1)}(I^{2},...,I^{i})-\mathbf{c}\big{)}=D\big{(}I^{1}-\mathbf{c},D^{(i-1)}(I^{2}-\mathbf{c},...,I^{i}-\mathbf{c})\big{)}$
$\displaystyle=D^{(i)}(I^{1}-\mathbf{c},...,I^{i}-\mathbf{c}).$

We have thus proven (2.19) and so (2.18). ∎

###  2.3 The construction $\mathfrak{D}^{(n)}$

In this subsection we obtain the main technical tool that underlies the
development of much of the main results in this paper. Recall from Theorem 2.2
that in order to obtain the distribution $\mu^{\bm{\rho}_{n}}$, one can simply
do the following - start with $\overline{I}\sim\nu^{\bm{\rho}_{n}}$ and apply
to it the map $\mathcal{D}^{(n)}$. In what comes next, we construct the
distribution $\mu^{\bm{\rho}_{n}}$ differently. Although the new construction
is arguably more convoluted than $\mathcal{D}^{(n)}$, it presents an important
feature that facilitates much of the results to come. Let us try to explain
here the advantage of the new construction $\mathfrak{D}^{(n)}$, which equals
$\mathcal{D}^{(n)}$. The map $\mathfrak{D}^{(n)}$ takes as input
$\overline{I}\sim\nu^{\bm{\rho}_{n}}$ and spits out
$(f^{1},...,f^{n})\in\mathcal{X}^{n}$. The construction $\mathfrak{D}^{(n)}$
is such that for each $2\leq i\leq n$, there exists $u^{i}\in\mathcal{X}$ such
that $f^{i}=D(f^{i-1},u^{i})$. The upshot, and one of the main results of this
section (Proposition 2.11), is that the r.v.s (recall (2.5))
$\\{S_{oj}(f^{i-1},u^{i})_{m}\\}_{2\leq i\leq n}$ are mutually independent,
for any $m\in\mathbb{Z}$. This property is important since when the difference
between $\rho_{i-1}$ and $\rho_{i}$ is small, $S_{oj}(f^{i-1},u^{i})_{m}$ gets
large which then implies that $f_{i}=f_{i-1}$ on a fixed compact interval with
high probability. We will then use the results in this section to show that,
on a fixed compact interval, there are only few $i$’s for which $f^{i}\neq
f^{i+1}$, which is key to show relative compactness in the Skorohord space. We
note here that although the map $\mathfrak{D}$ is equivalent to the map
constructed in [17, Section 6.2], its derivation as a sequential sorting of
RWs, is new and plays an important role in the proof of Lemma 3.3 (see the
last part of Subsection 2.4).

Let $\bm{\rho}=(\rho_{1},\rho_{2},...)$ be a decreasing sequence and let
$\overline{I}=(I^{1},I^{2},...)\sim\nu^{\bm{\rho}}$. Set $f^{1}=I^{1}$ and for
$i=2$

$\displaystyle\overline{I}^{2}$ $\displaystyle=\overline{I}$ $\displaystyle
f^{2}$ $\displaystyle=D^{(2)}(\Pi_{[1,2]}\overline{I}^{2})$ $\displaystyle
u^{2}=I^{2}$ $\displaystyle\qquad v^{2}=I^{1}.$

For $i\geq 3$

$\displaystyle\overline{I}^{i}$
$\displaystyle=\sigma_{[1,i-2]}\overline{I}^{i-1}$ (2.20) $\displaystyle
f^{i}$ $\displaystyle=D^{(i)}(\Pi_{[1,i]}\overline{I}^{i})$ $\displaystyle
u^{i}=D^{(i-1)}(\Pi_{[2,i]}\overline{I}^{i})$ $\displaystyle\qquad
v^{i}=\Pi_{1}\overline{I}^{i}.$ (2.21)

Note that from (2.11) with $k=1$ it follows that

$\displaystyle f^{i}=D(v^{i},u^{i}).$ (2.22)

If $\bm{\rho}_{n}=(\rho_{1},...,\rho_{n})$ and
$\overline{I}^{(n)}\sim\nu^{\bm{\rho}_{n}}$, we define the map
$\mathfrak{D}^{(n)}:\mathcal{X}^{n}\rightarrow\mathcal{X}^{n}$ by

$\displaystyle\mathfrak{D}^{(n)}(\overline{I}^{(n)})=(f^{1},...,f^{n}).$
(2.23)

The following is a simple yet important observation.

######  Lemma 2.5.

Fix $i\geq 4$ and define $X^{k}=\sigma_{[k+1,i-2]}\overline{I}^{i-1}$ for any
$0\leq k\leq i-3$ and $X^{i-2}=\overline{I}^{i-1}$. Then for every $0\leq
k\leq i-2$ the operation $\sigma_{k}X^{k}$ is active.

###### Proof.

First note that from (2.20), $\overline{I}^{i-1}$ is obtained from
$\overline{I}$ by applying $\sigma$ operations affecting only the first to
$i-2$ elements. As $\Pi_{[1,n]}\overline{I}\in\mathcal{Y}^{n}$ for every
$n\in\mathbb{N}$, it follows that

$\displaystyle[\Gamma(\overline{I}^{i-1})]_{j}>[\Gamma(\overline{I}^{i-1})]_{i-1}\quad\text{
for $1\leq j<i-1$},$

and therefore that

$\displaystyle\text{$\sigma_{i-2}\overline{I}^{i-1}$ is active and
$[\Gamma(\sigma_{i-2}\overline{I}^{i-1})]_{j}>[\Gamma(\sigma_{i-2}\overline{I}^{i-1})]_{i-2}\quad\text{
for $1\leq j<i-2$}$}.$ (2.24)

Continuing by induction on $k$ with (2.24) as our base case we conclude the
result. ∎

The next lemma shows that $\mathfrak{D}^{(n)}$ is indeed a representation of
$\mathcal{D}^{(n)}$.

######  Lemma 2.6.

Fix $n\in\mathbb{N}$. Then

$\displaystyle\mathcal{D}^{(n)}(\Pi_{[1,n]}\overline{I})=\mathfrak{D}^{(n)}(\Pi_{[1,n]}\overline{I})$

###### Proof.

For $i\in[n]$ we must show that

$\displaystyle[\mathcal{D}^{(n)}(\Pi_{[1,n]}\overline{I})]_{i}=f^{i},$

where $f^{i}$ was defined in (2.20). By definition

$\displaystyle[\mathcal{D}^{(n)}(\Pi_{[1,n]}\overline{I})]_{i}=D^{(i)}(\Pi_{[1,i]}\overline{I})\quad
1\leq i\leq n.$

Note that $\overline{I}^{i}$ is obtained from $\overline{I}$ by a sequence of
$\sigma$ operations affecting only the first to $i-1$’th elements of the
sequence $\overline{I}$ i.e.

$\displaystyle\overline{I}^{i}_{j}=\overline{I}_{j}\qquad\forall i\leq j.$

we see that

$\displaystyle
f^{i}=D^{(i)}(\Pi_{[1,i]}\overline{I}^{i})\stackrel{{\scriptstyle\eqref{si3}}}{{=}}D^{(i)}(\Pi_{[1,i]}\overline{I})\stackrel{{\scriptstyle\eqref{md}}}{{=}}[\mathcal{D}^{(n)}(\Pi_{[1,n]}\overline{I})]_{i}.$

∎

Recall $v^{i}$ and $u^{i}$ from (2.21). The following identities will come
useful in the sequel.

######  Lemma 2.7.

For $i\geq 3$

$\displaystyle
v^{i-1}=\Pi_{1}\Big{(}\sigma_{[2,i-2]}\overline{I}^{i-1}\Big{)}$ (2.25)
$\displaystyle
u^{i-1}=\Pi_{2}\Big{(}\sigma_{[2,i-2]}\overline{I}^{i-1}\Big{)}$ (2.26)

and

$\displaystyle D(v^{i-1},u^{i-1})=\Pi_{1}\Big{(}\overline{I}^{i}\Big{)}$
(2.27) $\displaystyle
R(v^{i-1},u^{i-1})=\Pi_{2}\Big{(}\overline{I}^{i}\Big{)}.$ (2.28)

Moreover,

$\displaystyle v^{i}=D(v^{i-1},u^{i-1})=f^{i-1}\quad\text{ for $i\geq 2$}.$
(2.29)

and

$\displaystyle f^{i}=D(f^{i-1},u^{i}).$ (2.30)

###### Proof.

We begin with showing (2.25), which is equivalent, from the definition of
$v^{i}$ in (2.21), to showing that

$\displaystyle\Pi_{1}\overline{I}^{i-1}=\Pi_{1}\Big{(}\sigma_{[2,i-2]}\overline{I}^{i-1}\Big{)}.$
(2.31)

As the operation $\sigma_{[2,i-2]}$ does not affect the first element of
$\overline{I}^{i-1}$, (2.31) follows. Next we show (2.26). Define

$\displaystyle h_{k}=\Pi_{k}\sigma_{[k,i-2]}\overline{I}^{i-1}\qquad
k\in\\{2,...,i-2\\}.$

Using Lemma 2.5, the following recursive relation holds for $1\leq k\leq i-3$

$\displaystyle h_{k}$
$\displaystyle=\Pi_{k}\sigma_{[k,i-2]}\overline{I}^{i-1}=\Pi_{k}\sigma_{k}\sigma_{[k+1,i-2]}\overline{I}^{i-1}$
(2.32)
$\displaystyle=\Pi_{1}\big{(}D(\overline{I}^{i-1}_{k},\Pi_{k+1}\sigma_{[k+1,i-2]}\overline{I}^{i-1}),R(\overline{I}^{i-1}_{k},\Pi_{k+1}\sigma_{[k+1,i-2]}\overline{I}^{i-1})\big{)}$
$\displaystyle=D(\overline{I}^{i-1}_{k},\Pi_{k+1}\sigma_{[k+1,i-2]}\overline{I}^{i-1})=D(\overline{I}^{i-1}_{k},h_{k+1})$

with the boundary condition

$\displaystyle
h_{i-2}=D\big{(}\overline{I}^{i-1}_{i-2},\overline{I}^{i-1}_{i-1}\big{)}.$
(2.33)

It is not hard to see that from (2.32)–(2.33) we have

$\displaystyle\Pi_{2}\big{(}\sigma_{[2,i-2]}\overline{I}^{i-1}\big{)}=h_{2}=D^{(i-2)}\big{(}\overline{I}^{i-1}_{2},...,\overline{I}^{i-1}_{i-1}\big{)}=D^{(i-2)}(\Pi_{[2,i-1]}\overline{I}^{i-1})=u^{i-1},$

thus proving (2.26). Finally we prove (2.27)–(2.28).

$\displaystyle\Pi_{1}\big{(}\overline{I}^{i}\big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{Ii}}}{{=}}\Pi_{1}\big{(}\sigma_{1}\sigma_{[2,i-2]}\overline{I}^{i-1}\big{)}=D\Big{(}\Pi_{1}\Big{(}\sigma_{[2,i-2]}\overline{I}^{i-1}\Big{)},\Pi_{2}\Big{(}\sigma_{[2,i-2]}\overline{I}^{i-1}\Big{)}\Big{)}=D\big{(}v^{i-1},u^{i-1}\big{)}$
$\displaystyle\Pi_{2}\big{(}\overline{I}^{i}\big{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{Ii}}}{{=}}\Pi_{2}\big{(}\sigma_{1}\sigma_{[2,i-2]}\overline{I}^{i-1}\big{)}=R\Big{(}\Pi_{1}\Big{(}\sigma_{[2,i-2]}\overline{I}^{i-1}\Big{)},\Pi_{2}\Big{(}\sigma_{[2,i-2]}\overline{I}^{i-1}\Big{)}\Big{)}=R\big{(}v^{i-1},u^{i-1}\big{)},$

where in both lines, the second equality follows from the definition of
$\sigma_{i}$, while the third equality follows from (2.25)–(2.26). From
(2.21), (2.22) and (2.27) we conclude (2.29). Finally, (2.30) follows by
plugging (2.29) in (2.22). ∎

Let $\bm{\rho}\in\mathbb{R}_{+}^{\mathbb{N}}$ be a decreasing vector. In what
comes next we use the construction above with
$\overline{I}\sim\nu^{\bm{\rho}}$. Recall $u^{i}$ and $v^{i}$ from (2.21) and
define $J^{i}=S_{oj}(v^{i},u^{i})\in\mathcal{X}$ for $i\geq 2$.

######  Lemma 2.8.

Fix $i\geq 2$ and $x\in\mathbb{Z}$. Then $\\{J^{j}_{x}\\}_{j\geq i+1}$ is
independent of $J^{i}_{x}=S_{oj}(v^{i},u^{i})_{x}$.

###### Proof.

We first claim that it is enough to show that

$\displaystyle\\{[\overline{I}^{j}_{z}]_{l}\\}_{j\geq i+1,l\leq
x,z\in\mathbb{N}}\text{ is independent of }J^{i}_{x}.$ (2.34)

To see why this is true, first note that from (2.5), for every $j\geq 2$ and
$x\in\mathbb{Z}$, $J^{j}_{x}$ is a function of
$\\{u^{j}_{l},v^{j}_{l}\\}_{l\leq x}$. Moreover, for $j\geq i+1$, by (2.21)
$\\{u^{j}_{l},v^{j}_{l}\\}_{l\leq x}$ is a function of
$\\{[\overline{I}^{j}_{z}]_{l}\\}_{j\geq i+1,l\leq x,z\in\mathbb{N}}$ and so
(2.34) implies the result. Next we claim that (2.34) can be further weakened
to showing that

$\displaystyle\\{[\overline{I}^{i+1}_{z}]_{l}\\}_{l\leq
x,z\in\mathbb{N}}\text{ is independent of }J^{i}_{x}.$ (2.35)

To see why (2.35) implies (2.34), first note that the random variables
$\\{[\overline{I}^{j}_{z}]_{l}\\}_{j\geq i+1,l\leq x,z\in\mathbb{Z}}$ can be
obtained from $\overline{I}$ by applying the $\sigma$ operations. Next observe
that the operator $\sigma$ does not ’look into the future’ i.e. for every
$x\in\mathbb{Z}$

$\displaystyle\Big{\\{}\Pi_{1}\big{(}\sigma(I^{1},I^{2})\big{)}_{i},\Pi_{2}\big{(}\sigma(I^{1},I^{2})\big{)}_{i}\Big{\\}}_{i\leq
x}=\Big{\\{}D(I^{1},I^{2})_{i},R(I^{1},I^{2})_{i}\Big{\\}}_{i\leq x}$
$\displaystyle\text{ is measureable with respect to the sigma algebra
generated by }\\{I^{1}_{i},I^{2}_{i}\\}_{i\leq x}.$

This implies that $\\{[\overline{I}^{j}_{z}]_{l}\\}_{j\geq i+1,l\leq
x,z\in\mathbb{Z}}$ is measurable with respect to the sigma-algebra generated
by $\\{[\overline{I}^{i+1}_{z}]_{l}\\}_{l\leq x,z\in\mathbb{Z}}$ which shows
that (2.35) implies (2.34).

We now move on to show (2.35). Let
$\overline{I}^{\prime}=\sigma_{[2,i-1]}\overline{I}^{i}$. From (2.20)
$\overline{I}^{i+1}=\sigma_{1}\sigma_{[2,i-1]}\overline{I}^{i}=\sigma_{1}\overline{I}^{\prime}$.
From Lemma 2.3 we see that

$\displaystyle\Pi_{[3,\infty]}\overline{I}^{\prime}\text{ is independent of
}\Pi_{[1,2]}\overline{I}^{\prime}.$

From (2.25)–(2.26)

$\displaystyle\Pi_{[1,2]}\overline{I}^{\prime}=\big{(}v^{i},u^{i}\big{)}.$
(2.36)

It follows that

$\displaystyle\Pi_{[3,\infty]}\overline{I}^{\prime}\text{ is independent of
}\big{(}D(v^{i},u^{i}),R(v^{i},u^{i}),S_{oj}(v^{i},u^{i})\big{)}.$ (2.37)

From (2.37) and Lemma A.4 (i)

$\displaystyle\\{[D(v^{i},u^{i})]_{j}\\}_{j\leq
x},\\{[R(v^{i},u^{i})]_{j}\\}_{j\leq
x},\\{\Pi_{[3,\infty]}\overline{I}^{\prime}\\}_{j\leq x}\,\,\text{ are
independent of }\,\,J^{i}_{x}=S_{oj}(v^{i},u^{i})_{x}.$ (2.38)

It follows that

$\displaystyle\overline{I}^{i+1}$
$\displaystyle=\sigma_{1}\overline{I}^{\prime}=\sigma_{1}\Big{(}\Pi_{[1,2]}\overline{I}^{\prime},\Pi_{[3,\infty]}\overline{I}^{\prime}\Big{)}=\Big{(}\sigma_{1}\Pi_{[1,2]}\overline{I}^{\prime},\Pi_{[3,\infty]}\overline{I}^{\prime}\Big{)}$
(2.39)
$\displaystyle\stackrel{{\scriptstyle\eqref{id}}}{{=}}\Big{(}\sigma_{1}\big{(}v^{i},u^{i}\big{)},\Pi_{[3,\infty]}\overline{I}^{\prime}\Big{)}=\Big{(}D\big{(}v^{i},u^{i}\big{)},R\big{(}v^{i},u^{i}\big{)},\Pi_{[3,\infty]}\overline{I}^{\prime}\Big{)}.$

From (2.38) and (2.39) follows (2.35). The proof is now complete. ∎

The following result is the main feature of the construction
$\mathfrak{D}^{n}$.

######  Corollary 2.9.

Fix $x\in\mathbb{Z}$. Then $\\{J^{j}_{x}\\}_{j\geq 1}$ are independent.

###### Proof.

This follows by induction and Lemma 2.8. ∎

Next we would like to determine the distribution of $J^{i}_{x}$. In order to
do that, we first have to determine the distributions of $v^{i}$ and $u^{i}$.

######  Lemma 2.10.

Fix $i\in\mathbb{N}$. Then $v^{i}$ and $u^{i}$ are independent i.i.d.
sequences with the following marginals

$\displaystyle v^{i}_{0}\sim\mathrm{Exp}(\rho_{i-1})$ (2.40) $\displaystyle
u^{i}_{0}\sim\mathrm{Exp}(\rho_{i}).$ (2.41)

###### Proof.

By Lemma 2.3 and (2.25)–(2.26), both $v^{i}$ and $u^{i}$ are independent
sequences of Exponential random variables. It is left to verify the
intensities in (2.40)–(2.41). Note that from (2.20), the $\sigma$ operators
affect only the first to $i-1$ elements of $\overline{I}$ to obtain
$\overline{I}^{i}$. This implies that

$\displaystyle\Pi_{i}\overline{I}^{i}=[\overline{I}]_{i}.$

and therefore that

$\displaystyle[\Gamma(\overline{I}^{i})]_{k}>[\Gamma(\overline{I}^{i})]_{i}\quad
1\leq k<i.$

It follows that

$\displaystyle[\Gamma(v^{i})]_{1}\stackrel{{\scriptstyle\eqref{xieta}}}{{=}}[\Gamma(\overline{I}^{i})]_{1}=[\Gamma(\sigma_{[1,i-2]}\overline{I}^{i-1})]_{1}\stackrel{{\scriptstyle\eqref{sigmai}}}{{=}}[\sigma^{*}_{[1,i-2]}(\Gamma(\overline{I}^{i-1}))]_{1}=[\Gamma(\overline{I}^{i-1})]_{i-1}=\rho_{i-1},$

concluding (2.40). Finally, (2.41) follows from

$\displaystyle\Gamma(u^{i})\stackrel{{\scriptstyle\eqref{xieta}}}{{=}}\Gamma(D^{(i-1)}(\Pi_{[2,i]}\overline{I}^{i}))=\Gamma([\overline{I}]_{i})=\rho_{i}.$

∎

The following proposition is the main result of this section.

######  Proposition 2.11.

Let $\bm{\rho}\in\mathbb{R}_{+}^{\mathbb{N}}$ be a decreasing vector. Assume
the construction above and fix $x\in\mathbb{Z}$. The random variables
$\\{J^{i}_{x}\\}_{i\in\mathbb{N}}$ are independent and

$\displaystyle J^{i}_{x}\sim\mathrm{Exp}(\rho_{i-1}-\rho_{i}).$ (2.42)

###### Proof.

Independence of $\\{J^{i}_{x}\\}_{i\in\mathbb{N}}$ is a consequence of
Corollary 2.9. (2.42) follows from Lemma A.1 and Lemma 2.10. ∎

Figure 2.1: The transformation
$(I^{1},I^{2})\mapsto\big{(}R(I^{1},I^{2}),D(I^{1},I^{2})\big{)}$ restricted
to $\mathbb{N}$ as melonization of two random walks. To simplify notation we
annotate the curves by their sequence of differences rather than their true
value i.e. we write $I^{1}$ instead of $S^{0,\cdot}(I^{1})$. (1) The input of
the melonization is two infinite sequences $I^{1}$ and $I^{2}$ from which we
obtain $S_{oj}(I^{1},I^{2})$. (2) The curve $I^{1}$ is raised by
$S_{oj}(I^{1},I^{2})$. (3) We construct an upper (blue) and lower (red) curves
(4) We lower the upper curve by $S_{oj}(I^{1},I^{2})$ to obtain
$D(I^{1},I^{2})$ (blue). The algorithm holds up to error of $O(1)$ which
disappears upon scaling. Figure 2.2: Illustration of the construction of
$\mathfrak{D}$ as melonization of random walks. From the top, at each of the
three lines of the figure, we construct and element of the vector
$\mathfrak{D}(I^{1},I^{2},I^{3})=\Big{(}I^{2},D(I^{1},I^{2}),D\big{(}D(I^{1},I^{2}),D(R(I^{1},I^{2}),I^{3})\big{)}\Big{)}$.
At each line, the black curve represents a new RW that is independent of all
other RW introduced in earlier stages of the algorithm. Curves with the same
colour are ordered. In (1) we introduce the first RW, $I^{1}$, which is also
the first element of $\mathfrak{D}$. In (2)-(3) we construct the second
element of the $\mathfrak{D}$ as follows. We start with the output of (1), to
which we add the RW $I^{2}$. We then order the two RWs to obtain (3). The top
line, $D(I^{1},I^{2})$ is the second element of $\mathfrak{D}$. In (4)–(6) we
construct the third element of $\mathfrak{D}$ in a similar fashion. Use the
line ensemble from $(3)$ as an input and add $I^{3}$ to it. We then start
sorting the curves from bottom to top, $I^{3}$ is first sorted with the bottom
line in (5) yielding the two blue curves, the top line of which, is then
sorted with respect to the red line to give the ordered ensemble in (6). The
top line of this ensemble is the third element of $\mathfrak{D}$.

###  2.4 The operator $\mathfrak{D}$ as melonization of random walks

In this subsection we discuss in more detail the connection of the map
$\mathfrak{D}$ from Subsection 2.3 to the non-intersecting-RWs-RSK
correspondence of O’Connell and Yor from [32].

#### Two types of melonization

The connection between queues and LPP goes back to the works of Szczotka and
Kelly [44], and Glynn and Whitt [22]. In [32], O’Connel and Yor showed that a
sequence of independent Brownian motions (Bm) conditioned not to intersect can
be represented as a sequence of queues in tandem. Let us now sketch their
idea. Let $\overline{X}=(X^{1},X^{2},...,X^{N})$ be $N\geq 1$ independent Bms
on $[0,\infty)$. Let

$\displaystyle E(f,g)(t)=\max_{s\in[0,t]}g(s)-f(s),$

and define

$\displaystyle\alpha^{U}(f,g)(t)$ $\displaystyle=f(t)+E(f,g)(t)$ (2.43)
$\displaystyle\alpha^{D}(f,g)(t)$ $\displaystyle=g(t)-E(f,g)(t).$ (2.44)

We then define

$\displaystyle\alpha_{i}(\overline{X})=\big{(}X_{1},...,\alpha^{U}(X_{i},X_{i+1}),\alpha^{D}(X_{i},X_{i+1}),...,X_{N}\big{)}.$

Next we define

$\displaystyle\hat{\alpha}_{i}=\alpha_{1}\cdots\alpha_{i-1}\alpha_{i}\quad
1\leq i\leq N-1.$

Finally let

$\displaystyle\mathcal{W}(\overline{X})=\hat{\alpha}_{N-1}\hat{\alpha}_{N-2}\cdots\hat{\alpha}_{1}\overline{X}.$

It then holds that $\mathcal{W}$ has the same distribution as that of
$\overline{X}$ conditioned on $\\{X_{1}(t)\geq X_{2}(t)\geq...\geq
X_{N}(t)\\}\quad\forall t\geq 0$. Confirming to the literature, we refer to
$\mathcal{W}$ as the packed melon of $\overline{X}$, where we added the word
packed to distinguish it from a similar model we shall shortly discuss. We
shall often refer to the algorithm behind the melon as melonization or packed
melonization. From this, and the connection between LPP and queueing theory,
follows the connection between the extremal values of LPP and the eigenvalues
of the GUE random matrices [4, 23]. Indeed, by Greene’s Theorem and the
results of O’Connell in [31], the sum of the top $k$ values in
$\mathcal{W}(\overline{X})$ at time $t=1$ equals the maximal passage time of
$k$ non-intersecting paths on $\overline{X}$, starting from $X^{N}$ at time
$0$ and terminating at time $1$ on $X^{1}$. These paths, looking like the
stripes of a watermelon, are where the name melon comes from. This picture is
expected to hold more generally, as was shown by Biane, Bougerol and O’Connell
in [31, 9] where the connection between non-intersecting RWs and the RSK
algorithm was studied, and in fact recently used by Dauvergne and Virag in
[15] to prove the convergence of various models to the Directed Landscape.

Recall the sequences $\mathbf{a},\mathbf{s},\bm{d}$ and $\bm{r}$ defined in
Subsection 2.1, and let us now try to see how our description of the map
$\mathfrak{D}$ is related to the melon of RWs. Define

$\displaystyle A_{n}=\sum_{i=1}^{n}a_{i},\quad S_{n}=\sum_{i=1}^{n}s_{i}\quad
D_{n}=\sum_{i=1}^{n}d_{i}\quad\text{and}\quad
R_{n}=\sum_{i=1}^{n}r_{i}\quad\text{for $n\geq 1$}.$

Recall the sequence $\\{e_{i}\\}_{i\in\mathbb{Z}}=Er(\mathbf{s},\mathbf{a})$
from (2.8). From (A.5)

$\displaystyle E_{n}:=$
$\displaystyle\sum_{i=1}^{n}e_{i}=\sum_{i=1}^{n}(w_{i-1}+s_{i-1}-a_{i})^{-}=\Big{(}\inf_{1\leq
i\leq n-1}J_{0}+S_{i-1}-A_{i-1}-a_{i}\Big{)}^{-}$ $\displaystyle=$
$\displaystyle\Big{(}\sup_{1\leq i\leq
n-1}A_{i-1}-S_{i-1}+a_{i}-s_{0}-w_{0}\Big{)}^{+}$

From (2.4) and (2.7) we conclude that

$\displaystyle D_{n}$ $\displaystyle=S_{n}+E_{n}$ (2.45) $\displaystyle R_{n}$
$\displaystyle=A_{n}-E_{n}.$ (2.46)

Now assume there exists a scaling $\stackrel{{\scriptstyle N}}{{\rightarrow}}$
such that

$\displaystyle A\stackrel{{\scriptstyle N}}{{\rightarrow}}f_{A},\quad
S\stackrel{{\scriptstyle N}}{{\rightarrow}}f_{S},\quad
E\stackrel{{\scriptstyle N}}{{\rightarrow}}f_{E},\quad
w_{0}\stackrel{{\scriptstyle N}}{{\rightarrow}}W,\quad
D\stackrel{{\scriptstyle N}}{{\rightarrow}}f_{D},\quad\text{and
}R\stackrel{{\scriptstyle N}}{{\rightarrow}}f_{R}\quad\text{ as
$N\rightarrow\infty$},$

where $f_{A},f_{S},f_{E},f_{D},f_{R}\in C[0,\infty)$ vanish at $0$ and
$W\in\mathbb{R}_{+}$. Assume further that under $\stackrel{{\scriptstyle
N}}{{\rightarrow}}$ the contribution of $|a_{i}|$ and $|s_{i}|$ is negligible.
Then, putting aside some (serious) technical issues, it follows that

$\displaystyle f_{D}$ $\displaystyle=f_{S}+E(f_{S}+W_{0},f_{A})^{+}$ (2.47)
$\displaystyle f_{R}$ $\displaystyle=f_{A}-E(f_{S}+W_{0},f_{A})^{+}.$ (2.48)

Consider Figure 2.1 for an illustration of the sorting mechanism for 2
functions. Comparing (2.43)–(2.44) with (2.47) –(2.48), we see that the
difference between the 2-functions melonization
$(f,g)\mapsto(\alpha_{1},\alpha_{2})$, used in [32], and
$(f_{S},f_{A})\mapsto(f_{D},f_{R})$ – the one we are using in this paper,
comes from the waiting time $W_{0}$. Alternatively, the difference in the two
melonizations comes from the fact that the sequences $\mathbf{s}$ and
$\mathbf{a}$ are bi-infinite rather than just infinite in one direction. To
distinguish the $(f_{S},f_{A})\mapsto(f_{D},f_{R})$ defined through (2.47)
–(2.48) from the packed melon, we shall refer to the former as the stationary
melon(ization).

#### From melonziation to $\mathfrak{D}$

Now that we have some intuition to the basic stationary melonization operation
$(\mathbf{s},\mathbf{a})\mapsto\big{(}D(\mathbf{s},\mathbf{a}),R(\mathbf{s},\mathbf{a})\big{)}$
as a geometric picture of sorting two functions, let us explain how we obtain
the mapping $\mathfrak{D}$ from a sequence of such operations. Let
$\overline{J}\sim\nu_{\bm{\rho}}$ for some decreasing vector
$\bm{\rho}=(\rho_{1},...,\rho_{k})$. Recall the operator $\sigma_{i}$ from
(2.14) and $\sigma_{[i,j]}$ from (2.17) and define

$\displaystyle\mathcal{W}^{(1)}(\overline{J})$ $\displaystyle=J^{1}$ (2.49)
$\displaystyle\mathcal{W}^{(i)}(\overline{J})$
$\displaystyle=\sigma_{[1,i-1]}\Big{(}\mathcal{W}^{(i-1)}(\overline{J}),J^{i}\Big{)}\quad\text{for
$2\leq i\leq k$}.$ (2.50)

Note that $\mathcal{W}^{(i)}$ is a vector of size $i$. In words, we receive
the vector $\overline{J}$ as input and we sort its elements in the following
way; having sorted the elements $J^{1}$ through $J^{i-1}$ to obtain
$\mathcal{W}^{(i-1)}(\overline{J})$, we sort the new (and independent) element
$J^{i}$ from the bottom ($\mathcal{W}^{(i-1)}_{i-1}$) up to the top line
($\mathcal{W}^{{(i-1)}}_{1}$) to obtain $\mathcal{W}^{i}$ (see Figure 2.2).
Finally we define

$\displaystyle\mathcal{W}^{\text{top}}(\overline{J})=(\mathcal{W}^{(1)}_{1},...,\mathcal{W}^{(k)}_{1}),$

the vector registering the top most line in each of the iterations
$\\{\mathcal{W}^{(i)}\\}_{i=1}^{k}$. We now claim that

$\displaystyle\mathfrak{D}^{(k)}(\overline{J})=\mathcal{W}^{\text{top}}(\overline{J}).$
(2.51)

To see that, recall the infinite vector of sequences $\overline{I}^{i}$ from
(2.20), and note that

$\displaystyle\Pi_{[1,i-1]}\overline{I}^{i}=\mathcal{W}^{(i-1)}(\Pi_{[1,k]}\overline{I})\quad
i\geq 2,$

and that

$\displaystyle
f^{i}=D^{(i)}\big{(}\Pi_{[1,i]}\overline{I}^{i}\big{)}=D^{(i)}\big{(}\Pi_{[1,i-1]}\overline{I}^{i},I^{i}\big{)}=D^{(i)}\big{(}\mathcal{W}^{(i-1)}(\Pi_{[1,i-1]}\overline{I}),I^{i}\big{)}.$

Setting the $k$ first elements of $\overline{I}$ to be $\overline{J}$

$\displaystyle
f^{i}=D^{(i)}\big{(}\mathcal{W}^{(i-1)}(\overline{J}),J^{i}\big{)}\quad 1\leq
i\leq k.$ (2.52)

Next note that the operator $D^{(i)}$ applies the operator $D$ successively,
hence sorting the sequence $J^{i}$ through $\mathcal{W}^{(i-1)}(\overline{J})$
from bottom to top. We have thus showed that

$\displaystyle
D^{(i)}\big{(}\mathcal{W}^{(i-1)}(\overline{J}),J^{i}\big{)}=\mathcal{W}^{(i)}_{1}\quad\text{for
$1\leq i\leq k$}.$ (2.53)

Plugging (2.53) in (2.52) and using (2.23) implies (2.51).

#### The effect of large $W_{0}$

From Burke’s Theorem [11], it follows that no matter how large the size the
vector $\overline{J}$ ($k$) is, $\mathcal{W}^{(k)}_{1}$ cannot be
stochastically larger than $J^{k}$. This is in contrast to the packed melon
where $\mathcal{W}_{1}\sim\sqrt{k}$, i.e. the top line of the packed melon
grows like square root of the number of lines in the melon. The fact that the
system does not ”blow up” is of course expected from a stationary system. What
is not so straightforward is the fact that the system is kept ”in check” by
sticking successive lines to one another. This phenomena can be attributed to
the role of $W_{0}$ in the stationary melonization. If the sequence $J^{k}$ is
kept fixed while the number of sequences ($k$) goes to infinity, then from
condition (2.1) it follows that the different elements in $\overline{J}$ must
be close, which one could then expect would lead to large waiting times in the
stationary melonziation. A large waiting time $W_{0}$ in the operation
$\sigma_{i}$ for some $1\leq i\leq k$ can be thought of as a potential well
that prevents the $\sigma_{i}$ from being carried out. Indeed, setting $W_{0}$
large enough in (2.47)–(2.48) would imply that $f_{D}=f_{S}$ and
$f_{R}=f_{A}$, and therefore that
$\sigma_{i}(f_{S},f_{A})=(f_{D},f_{R})=(f_{S},f_{A})$. Rendering many of the
$\sigma$ operations in (2.2) inactive results in many of the elements of
$\mathcal{W}^{\text{top}}(\overline{J})$ (and therefore the elements of
$\mathfrak{D}^{(k)}(\overline{J})$) being equal. This stickiness feature of
the Busemann process has been observed in [17] where the authors showed that
when varying the density $\rho\in(0,1)$, $\rho\mapsto B^{\rho}_{0,e_{1}}$ is a
Poisson process, a phenomena that can be attributed to the coalescence of
geodesics. Our results show that this feature of the Busemann process carries
over to its behaviour around a prescribed density when scaled diffusively.

#### Relevant literature

Our representation of $\mathfrak{D}^{(n)}$, developed in Subsection 2.3, is
not new and is equivalent to that in [17, Section 6.2] (we thank Timo
Sepäläinen for pointing this out). However, the derivation of the
representation in [17] is different and constructed using two triangular
arrays rather than one. In particular, the connection to the packed melon did
not appear in [17] and seems to be new. This observation plays a crucial role
in one of the key results in this paper - Lemma 3.3, by controlling the top
line of the stationary melon by that of the packed one (see the discussion
therein). Moreover, although we did not pursue such a result, we believe this
new picture of the map $\mathcal{D}^{(k)}$ as sorting random walks can be used
to improve (1.7) significantly (although not optimally).

The fact that large $W_{0}$ results in constant values of the Busemann process
around a fixed density $\rho$ was first used in [2] to obtain strong local
stationary results of the point to point geodesics. In [12], the authors used
similar ideas to obtain the exact exponent for the probability that the point
to point geodesic does not coalesce with the stationary one. In this sense,
this work is a natural progression as we consider the constant times of the
Busemann process across a continuum of densities around a fixed density.

##  3 The Busemann Process

In this section we recall the definition and main properties of the Busesmann
process, and obtain the necessary results to be used later in the paper. The
Busemann Process is a random process parametrized by $0<\rho<1$ and taking
values in the space of real valued functions defined on
$\mathbb{Z}^{2}\times\mathbb{Z}^{2}$. More specifically, for a fixed
$0<\rho<1$, the process $B^{\cdot}$ attains the value
$B^{\rho}\in\mathbb{R}^{\mathbb{Z}^{2}\times\mathbb{Z}^{2}}$ which is called
the Busemann function (associated with the density $\rho$). This function has
proven to be an important tool in studying infinite geodesics in LPP on the
lattice (where the ideas extend to positive temperature polymers on the
lattice). For example, $B^{\rho}$ allows one to explicitly construct an
infinite geodesic starting from any point on the lattice in the direction
associated with the density $\rho$.

This section is divided into two parts, in Section 3.1 we provide the
definition of the Busemann process and give its main properties. Section 3.2
shows that under the diffusive scaling, the Busemman process does not take to
many values on a compact interval. This will later imply the modulus of
continuity condition (Condition 2 of Theorem A.15) for the prelimiting
sequence $G^{N}$.

###  3.1 Definition and main properties

Below we recall the main properties of the Busemann process.

######  Theorem 3.1.

[40, Theorem 4.2] For each $0<\rho<1$ there exists a function
$B^{\rho}_{\cdot,\cdot}(\omega):\mathbb{Z}^{2}\times\mathbb{Z}^{2}\rightarrow\mathbb{R}$
and a family of random weights $\\{\hat{\omega}_{x}\\}_{x\in\mathbb{Z}^{2}}$
such that the following properties hold:

  1. 1.

for any $x,y,z\in\mathbb{Z}^{2}$

$\displaystyle B^{\rho}_{x,y}+B^{\rho}_{y,z}=B^{\rho}_{x,z}.$ (3.1)

  2. 2.

$B^{\rho}$ is stationary with marginal distribution

$\displaystyle B_{x,x+e_{1}}\sim\mathrm{Exp}(\rho)\quad
B_{x,x+e_{2}}\sim\mathrm{Exp}(1-\rho)$

  3. 3.

For any down-right path $\mathcal{Y}=(y_{k})_{k\in\mathbb{Z}}$ in
$\mathbb{Z}^{2}$, the random variables

$\displaystyle\\{\hat{\omega}_{z}\\}_{z\in\mathcal{G}_{-}},\quad\\{t(y_{k-1},y_{k}):k\in\mathbb{Z}\\}\quad\text{and}\quad\\{\omega_{z}\\}_{z\in\mathcal{G}_{+}}$
(3.2)

are mutually independent, where

$\displaystyle t(e)=\begin{cases}B^{\rho}_{x-e_{1},x}&e=(x-e_{1},x)\\\
B^{\rho}_{x-e_{2},x}&e=(x-e_{2},x)\end{cases}$

and where

$\displaystyle\mathcal{G}_{+}=\mathcal{Y}+\mathbb{Z}^{2}_{>0},\quad\text{and}\quad\mathcal{G}_{-}=\mathcal{Y}+\mathbb{Z}^{2}_{<0}.$

  4. 4.

the following identities hold

$\displaystyle\hat{\omega}_{x-e_{1}-e_{2}}$
$\displaystyle=B^{\rho}_{x-e_{1}-e_{2},x-e_{2}}\wedge
B^{\rho}_{x-e_{1}-e_{2},x-e_{1}}$ $\displaystyle B^{\rho}_{x-e_{1},x}$
$\displaystyle=\omega_{x}+(B^{\rho}_{x-e_{1}-e_{2},x-e_{2}}-B^{\rho}_{x-e_{1}-e_{2},x-e_{1}})^{-}$
$\displaystyle B^{\rho}_{x-e_{2},x}$
$\displaystyle=\omega_{x}+(B^{\rho}_{x-e_{1}-e_{2},x-e_{1}}-B^{\rho}_{x-e_{1}-e_{2},x-e_{2}})^{-}$

  5. 5.

There exists a single event $\Omega_{2}$ of full probability such that for
every $\omega\in\Omega$, all $x\in\mathbb{Z}^{2}$ and all $\rho_{1}<\rho_{2}$

$\displaystyle B^{\rho_{2}}_{x,x+e_{1}}(\omega)\leq
B^{\rho_{1}}_{x,x+e_{1}}(\omega),\quad B^{\rho_{1}}_{x,x+e_{2}}(\omega)\leq
B^{\rho_{2}}_{x,x+e_{2}}(\omega),$ (3.3)

and for each $x,y\in\mathbb{Z}^{2}$, the function $\rho\rightarrow
B^{\rho}_{x,y}(\omega)$ is right-continuous with left limits.

  6. 6.

For each fixed $0<\rho<1$ there exists $\Omega_{2}^{(\rho)}$ of full
probability such that for every $\omega\in\Omega_{2}^{(\rho)}$ and any
sequence $v_{n}\in\mathbb{Z}^{2}$ such that $|v_{n}|_{1}\rightarrow\infty$ and

$\displaystyle\lim_{n\rightarrow\infty}\frac{v_{n}}{|v_{n}|_{1}}=\Bigg{(}\frac{\rho^{2}}{(1-\rho)^{2}+\rho^{2}},\frac{(1-\rho)^{2}}{(1-\rho)^{2}+\rho^{2}}\Bigg{)}$

we have the following limit

$\displaystyle
B^{\rho}_{x,y}(\omega)=\lim_{n\rightarrow\infty}[G_{x,v_{n}}(\omega)-G_{y,v_{n}}(\omega)].$

In this work we shall be primarily interested in the distribution of the
Busemann process restricted on a horizontal line on the lattice. Hence, for a
fixed $0<\rho<1$ define the sequence $I^{\rho}$ through

$\displaystyle I^{\rho}_{i}=B^{\rho}_{(i-1)e_{1},ie_{1}}\quad i\in\mathbb{Z}.$
(3.4)

The following is from [17]. It relates the results in Section 2 with the
Busemann functions. Recall the measure $\mu^{\bm{\rho}^{n}}$ from Theorem 2.2.

######  Theorem 3.2.

[17, Theorem 3.2] Let $\bm{\rho}^{n}=(\rho_{1},..,\rho_{n})$ be a decreasing
vector with elements in $(0,1)$. Then

$\displaystyle(I^{\rho_{1}},...,I^{\rho_{n}})\sim\mu^{\bm{\rho}^{n}}.$

###  3.2 Upper bound on the number of jumps of the Busemann process

Recall $\\{I^{\rho}\\}_{\rho\in(0,1)}$ from (3.4). The main result of this
subsection is Proposition 3.5 – on an interval of size $O(N^{2/3})$, the
process $\mu\mapsto I^{\mu,N}:=I^{1/2-\frac{\mu}{4}N^{-1/3}}$ does not see
much change i.e. with high probability it will have only a finite number of
’jumps’. It is in this subsection that we use the results of Subsection 2.3 as
input.

Fix $\rho^{*}>0$ and let $M\in\mathbb{N}$. Define the sets

$\displaystyle\mathcal{E}(M)$
$\displaystyle=[-\rho^{*},\rho^{*}]\cap\\{i2^{-M}\\}_{i\in\mathbb{Z}}$
$\displaystyle\mathcal{E}_{ind}(M)$ $\displaystyle=2^{M}\mathcal{E}.$

Define the set of intensities

$\displaystyle\rho_{i}=1/2-i2^{-M}N^{-1/3}\quad i\in\mathcal{E}_{ind}.$

The vector $\bm{\rho}:\mathcal{E}_{ind}\rightarrow\mathbb{R}_{+}$, defined
through $\bm{\rho}_{i}=\rho_{i}$ is decreasing. Let
$\overline{I}=\\{I^{i}\\}_{i\in\mathcal{E}_{ind}}\in\mathcal{X}^{\mathcal{E}_{ind}}$
be a vector of independent sequences with exponential distribution with
intensities $\bm{\rho}$ i.e. $\overline{I}\sim\nu^{\bm{\rho}}$. We can then
define

$\displaystyle\overline{f}^{M,N}=(f^{1},...f^{|\mathcal{E}_{ind}|})=\mathcal{D}^{(|\mathcal{E}_{ind}|)}(\overline{I})=\mathfrak{D}^{(|\mathcal{E}_{ind}|)}(\overline{I}),$
(3.5)

where the map $\mathcal{D}^{(n)}$ was defined in (2.12). In the following, we
omit the superscript and simply write $\overline{f}$. In particular,

$\displaystyle f^{j}\geq f^{i}\quad\text{for $j,i\in\mathcal{E}_{ind}$ such
that $j\geq i$}.$

In what follows we shall make use of the following set

$\displaystyle\mathcal{E}^{-i}(M)=\\{j\in\mathcal{E}_{ind}:j\leq\sup\\{\mathcal{E}_{ind}\\}-i\\}\quad\text{for
$1\leq i\leq|\mathcal{E}_{ind}|-1$}.$

In words, the set $\mathcal{E}^{-i}(M)$ takes all indices in
$\mathcal{E}_{ind}$ except the $i$ largest ones. Let $x_{0}>0$ and define
$x_{0}^{N}=x_{0}N^{2/3}$. Define the following events

$\displaystyle B^{i,N}=\\{f^{i}_{j}\neq f^{i+1}_{j}\text{ for some
$j\in[-x_{0}^{N},x_{0}^{N}]\cap\mathbb{Z}$}\\}\quad\forall
i\in\mathcal{E}^{-1},N\in\mathbb{N}.$ (3.6)

Define

$\displaystyle
F_{B^{i}}:=1_{B^{i,N}}+1_{B^{i+1,N}}+1_{B^{i+2,N}}\quad\text{for all
$i\in\mathcal{E}^{-3}$}.$

In words, the function $F_{B^{i}}$ counts the number of consecutive changes in
the four elements of the vector $\overline{f}$, sitting between index $i$ to
$i+3$. The following two maps will be useful. For any $k,l\in\mathbb{Z}$ such
that $k\leq l$ we define

$\displaystyle S^{k,l}(I)=\sum_{i=k}^{l}I_{i}.$ (3.7)

For $m\in\mathbb{Z}$, define the map
$\psi^{m}:\mathbb{R}_{+}\times\mathcal{X}\times\mathcal{X}\rightarrow\mathcal{X}^{[m,\infty)}$
through

$\displaystyle\psi^{m}(J,I^{1},I^{2})_{i}=\inf_{m\leq j\leq
i}(J+S^{m,j-1}(I^{1}-I^{2})-I^{2}_{j})^{-}.$ (3.8)

The following result shows that it is not typical to see three consecutive
changes in the vector $\overline{f}$ when restricted to
$[-x_{0}^{N},x_{0}^{N}]$.

######  Lemma 3.3.

Let $\rho^{*}\geq 1$,$x_{0}\geq 1$, $(32)^{3}\vee 64(\rho^{*})^{3}<N$ and
$\frac{1}{2}\log_{2}\big{(}x_{0}\big{)}+4<M$. Then for any
$i\in\mathcal{E}_{ind}^{-3}$

$\displaystyle\mathbb{P}\big{(}F_{B^{i}}\geq 2\big{)}\leq 2^{20}x_{0}2^{-2M}.$
(3.9)

###### Proof.

Fix $i\in\mathcal{E}_{ind}^{-3}$. Let
$\overline{I}=(I^{1},I^{2},I^{3},I^{4})\sim\nu_{\bm{\rho}_{4}}$ where
$\bm{\rho}_{4}=(\rho_{i},\rho_{i+1},\rho_{i+2},\rho_{i+3})$. Let
$(\hat{f}^{1},\hat{f}^{2},\hat{f}^{3},\hat{f}^{4})=\mathcal{D}^{(4)}(\overline{I})=\mathfrak{D}^{(4)}(\overline{I})$.
Define

$\displaystyle\hat{B}^{N,l}=\\{\hat{f}^{l}_{j}\neq\hat{f}^{l+1}_{j}\text{ for
some $j\in[-x_{0}^{N},x_{0}^{N}]\cap\mathbb{Z}$}\\}\quad l\in\\{1,2,3\\}.$

From Theorem 3.2 $(\hat{B}^{N,1},\hat{B}^{N,2},\hat{B}^{N,3})$ has the same
distribution as $(B^{N,i},B^{N,i+1},B^{N,i+2})$ from (3.6), so it is enough to
show that

$\displaystyle\mathbb{P}\big{(}F_{\hat{B}}\geq 2\big{)}\leq
2^{20}x_{0}2^{-2M},$ (3.10)

where

$\displaystyle
F_{\hat{B}}:=1_{\hat{B}^{N,1}}+1_{\hat{B}^{N,2}}+1_{\hat{B}^{N,3}}.$

From (2.29)–(2.30), (2.4) and (2.8),

$\displaystyle\hat{f}^{l}=D(v^{l},u^{l})=D(\hat{f}^{l-1},u^{l})=\hat{f}^{l-1}+Er(\hat{f}^{l-1},u^{l})\quad
l\in\\{2,3,4\\}.$ (3.11)

From (3.11) and the definition of $\hat{B}^{N,l}$, it follows that

$\displaystyle\hat{B}^{N,l}=\\{S^{-x_{0}^{N},x_{0}^{N}}(Er(\hat{f}^{l},u^{l+1}))>0\\}\stackrel{{\scriptstyle\eqref{se}}}{{=}}\\{\psi^{-x_{0}^{N}}(J^{l},\hat{f}^{l},u^{l+1})_{x_{0}^{N}}>0\\}\quad\forall
l\in\\{1,2,3\\},$ (3.12)

where

$\displaystyle J^{l}=S_{oj}(\hat{f}^{l},u^{l+1})_{x_{0}^{N}-1}.$

What’s ahead?
Let us take a pause here to explain the main idea of the rest of the proof,
which otherwise might be lost on the reader for the trees (or displays). A
quick inspection of the map $\psi$ from (3.8) and equation (3.12), shows that
the probability of the event $\hat{B}^{N,l}$ can be bounded from above by the
probability that the random walk

$\displaystyle\mathbb{S}^{l}:=S^{-x_{0},\cdot}(u^{l+1},\hat{f}^{l})$

does not go above the value $J^{l}>0$ \- in other words, in order for the
event $\hat{B}^{N,l}$ to happen, the random walk $\mathbb{S}^{l}$ has to go
over a potential wall of size $J^{l}$. The statistics of
$\\{J^{l}\\}_{l=1}^{3}$ are known to us from Proposition 2.11 \- these are
i.i.d. exponential random variables with intensity $2^{-M}N^{-1/3}$. This
property is crucial, indeed, in the diffusive scaling the picture translates
to three Brownian motions of drifts $\mu_{l}=\mu_{0}+l2^{-M+2}$ for some
$\mu_{0}$ depending on the index $i$ from (3.9), each trying to cross a random
barrier of order $2^{M}$. When $M$ is large, from the independence, we might
expect that the probability of two events in $\\{\hat{B}^{N,l}\\}_{l=1}^{3}$
to occur will be of order ${2^{-2M}}$. This argument could be valid if it were
not for the fact that the random walks $\\{\mathbb{S}^{l}\\}_{l=1}^{3}$ are
not independent of the barriers $\\{J^{l}\\}_{l=1}^{3}$, nor are the random
walks themselves independent.

Our strategy to work around this difficulty is two fold. First, we decouple
the random walks from the barriers by showing that they can be controlled by
random walks coming from empty queues starting at $-x_{0}$, i.e. random walks
coming from packed-melonization (see the discussion of packed melonization vs
stationary one in Section 2.4), this technical bit is proved in Section A.1.
Let us denote by $\tilde{f}\in\mathcal{X}^{4}$ this packed-melon and let

$\displaystyle\mathbb{S}^{l,\text{packed}}:=S^{-x_{0},\cdot}(\tilde{u}^{l+1},\tilde{f}^{l})$
(3.13)

be the new random walks. The next step is to find one random walk that bounds
all $\\{\mathbb{S}^{l,\text{packed}}\\}_{l=1}^{3}$ uniformly in $l$. Indeed,
this will relieve us of trying to figure out the statistics of
$\\{\mathbb{S}^{l,\text{packed}}\\}_{l=1}^{3}$ which seem to be quite
complicated. The first thing to note is that as $\tilde{f}$ is increasing,
replacing $\tilde{f}^{l}$ in (3.13) with $\tilde{f}^{1}$ will result in a
random walk that dominates $\mathbb{S}^{l}$ for any $l\in\\{1,2,3\\}$.
Similarly, the top line of the melon, $\hat{f}^{4}$, dominate all other lines
that appear in the process of the packed-melonization of $\tilde{f}$, in
particular $\\{\tilde{u}^{l+1}\\}_{l=1}^{3}$. To conclude, we can dominate the
random walk in (3.13) by
$\mathbb{S}^{\text{top}}:=S^{-x_{0},\cdot}(\tilde{f}^{4},\tilde{f}^{1})$. As
$\tilde{f}^{1}$ and $\tilde{f}^{4}$ are not independent, their joint
statistics is not trivial, and we use a sub-optimal bound on
$\mathbb{S}^{\text{top}}$ developed in Lemma A.8.

From the proof, it seems that it should be possible to prove that the
probability of seeing $k$ consecutive changes in the vector $\overline{f}$
from (3.5) should decay like $2^{-Mk}$. We believe that these ideas could help
to improve the bound in (1.7), but may not be enough to obtain the optimal
bound which we expect to be exponential in $M$. Indeed, the fact that the size
of $\mathbb{S}^{\text{top}}$ grows with $M$ i.e.
$\mathbb{S}^{\text{top}}\approx\tilde{f}^{M}\approx 2\sqrt{M}$, could be in
the way of obtaining a sharp bound on the left hand side of (1.7) using our
strategy.

Let us now get back to the proof. Next we control
$\\{u^{l}\\}_{l\in\\{2,3,4\\}}$. From the definition of $u^{l}$ in (2.21)

$\displaystyle u^{2}$ $\displaystyle=I^{2}$ $\displaystyle u^{3}$
$\displaystyle=D(R(I^{1},I^{2}),I^{3})$ $\displaystyle u^{4}$
$\displaystyle=D^{(3)}\Big{(}R\big{(}D(I^{1},I^{2}),D(R(I^{1},I^{2}),I^{3})\big{)},R\big{(}R(I^{1},I^{2}),I^{3}\big{)},I^{4}\Big{)}.$

In the next display we use the map $D_{m,0}$ from (A.7) and (A.9). For any
$m\in\mathbb{Z}$

$\displaystyle\hat{u}^{2}:=\Pi_{[m,\infty)}(u^{2})$
$\displaystyle=\Pi_{[m,\infty)}(I^{2})$ (3.14)
$\displaystyle\hat{u}^{3}:=\Pi_{[m,\infty)}(u^{3})$ $\displaystyle\leq
D_{m,0}(R(I^{1},I^{2}),I^{3})$
$\displaystyle\hat{u}^{4}:=\Pi_{[m,\infty)}(u^{4})$ $\displaystyle\leq
D^{(3)}_{m,0}\Big{(}R\big{(}D(I^{1},I^{2}),D(R(I^{1},I^{2}),I^{3})\big{)},R\big{(}R(I^{1},I^{2}),I^{3}\big{)},I^{4}\Big{)}.$

In what follows, we shall use the function $\Phi_{m,n}$ defined in (A.11). As
$\hat{f}^{1}\leq\hat{f}^{l}$ for all $l\in\\{2,3,4\\}$,

$\displaystyle\hat{C}^{N,l}$
$\displaystyle:=\\{J^{l-1}<\Phi_{-x_{0}^{N},x_{0}^{N}}(\hat{u}^{l}-\hat{f}^{1})\\}\qquad\forall
l\in\\{2,3,4\\}$ (3.15)
$\displaystyle\supseteq\\{\psi^{-x_{0}^{N}}(J^{l-1},\hat{f}^{l-1},\hat{u}^{l})_{x_{0}^{N}}>0\\}=\hat{B}^{N,l}.$

By definition $\hat{f}^{1}=I^{1}$ so

$\displaystyle\Phi_{-x_{0}^{N},x_{0}^{N}}(\hat{u}^{l}-\hat{f}^{1})$
$\displaystyle=\Phi_{-x_{0}^{N},x_{0}^{N}}(\hat{u}^{l}-I^{1}).$

In the next display we use the map $D_{m,0}$ from (A.7), Lemma A.7 and that by
definition $R(I^{1},I^{2})\leq I^{1},I^{2}$ for $I^{1},I^{2}\in\mathcal{X}$.
From (3.14), for $m\leq j$

$\displaystyle S^{m,j}(\hat{u}^{3})$
$\displaystyle\stackrel{{\scriptstyle\eqref{ode4}}}{{\leq}}S^{m,j}\big{(}D_{m,0}(I^{2},I^{3})\big{)}$
(3.16) $\displaystyle S^{m,j}(\hat{u}^{4})$
$\displaystyle\stackrel{{\scriptstyle\eqref{ode4}}}{{\leq}}S^{m,j}\Bigg{(}D^{(3)}_{m,0}\Big{(}D(R(I^{1},I^{2}),I^{3}),I^{3},I^{4}\Big{)}\Bigg{)}$
(3.17)
$\displaystyle\stackrel{{\scriptstyle\eqref{ode3}}}{{\leq}}S^{m,j}\Bigg{(}D^{(3)}_{m,0}\Big{(}D_{m,0}(R(I^{1},I^{2}),I^{3}),I^{3},I^{4}\Big{)}\Bigg{)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{ode4}}}{{\leq}}S^{m,j}\Bigg{(}D^{(3)}_{m,0}\Big{(}D_{m,0}(I^{2},I^{3}),I^{3},I^{4}\Big{)}\Bigg{)}.$

From (3.16)-(3.17) with $m=-x_{0}^{N}$ and $i=x_{0}^{N}$

$\displaystyle\Phi_{-x_{0}^{N},x_{0}^{N}}(\hat{u}^{l}-I^{1})\leq\begin{cases}\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{2}-I^{1})&l=2\\\
\Phi_{-x_{0}^{N},x_{0}^{N}}\big{(}D_{m,0}(I^{2},I^{3})-I^{1}\big{)}&l=3\\\
\Phi_{-x_{0}^{N},x_{0}^{N}}\Big{(}D^{(3)}_{m,0}\Big{(}D_{m,0}(I^{2},I^{3}),I^{3},I^{4}\Big{)}-I^{1}\Big{)}&l=4.\end{cases}$
(3.18)

Using (3.18) in Lemma A.8 with $m=-x_{0}^{N}$ and $i=x_{0}^{N}$

$\displaystyle\Phi_{-x_{0}^{N},x_{0}^{N}}(\hat{u}^{l}-I^{1})\leq\begin{cases}\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{2}-I^{1})&l=2\\\
\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{2}-I^{1})+\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{3}-I^{2})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3})&l=3\\\
\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{2}-I^{1})+2\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{3}-I^{2})+\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{3})&l=4\\\
+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{4})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3}+I^{4}),\end{cases}$

where the computation for case $l=4$ is as follows

$\displaystyle\Phi_{-x_{0}^{N},x_{0}^{N}}(\hat{u}^{4}-I^{1})$
$\displaystyle\leq\Phi_{-x_{0}^{N},x_{0}^{N}}(D_{-x_{0}^{N},0}(I^{2},I^{3})-I^{1})+\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{3}-D_{-x_{0}^{N},0}(I^{2},I^{3}))+\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{3})$
$\displaystyle+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{4})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3}+I^{4})$
$\displaystyle\stackrel{{\scriptstyle\eqref{bs}}}{{\leq}}\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{2}-I^{1})+\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{3}-I^{2})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3})+\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{3}-I^{2})$
$\displaystyle+\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{3})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{4})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3}+I^{4})$
$\displaystyle=\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{2}-I^{1})+2\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{3}-I^{2})+\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{3})$
$\displaystyle+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{4})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3}+I^{4}),$

where we used Lemma A.8 twice and the fact that $D(I^{2},I^{3})\geq I^{2}$. We
see that for all $l\in\\{2,3,4\\}$

$\displaystyle\Phi_{-x_{0}^{N},x_{0}^{N}}(\hat{u}^{l}-I^{1})\leq\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{2}-I^{1})+2\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{3}-I^{2})+\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{3})$
$\displaystyle+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{4})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3}+I^{4}):=X_{max}.$

We have

$\displaystyle\mathbb{P}\big{(}X_{max}>t\big{)}$ (3.19)
$\displaystyle\leq\mathbb{P}(\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{2}-I^{1})+2\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{3}-I^{2})+\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{3})>t/2)$
$\displaystyle+\mathbb{P}(\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{4})+\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3}+I^{4})>t/2)$
$\displaystyle\leq 4\mathbb{P}(\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{1})>t/8)$
$\displaystyle+\mathbb{P}(2\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{3})+2\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{4})>t/2)$
$\displaystyle\leq 4\mathbb{P}(\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{1})>t/8)$
$\displaystyle+4\mathbb{P}(\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{4})>t/8),$

where we used the fact that $\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{1})$
($I^{4}$) stochastically dominates all the terms in the second line(fifth
line) of (3.19). As $I^{4}-I^{1}\sim\mathrm{Exp}(\rho_{i}+3\cdot
2^{-M}N^{-1/3},\rho_{i})$ for some $\rho_{i}\in\mathcal{E}$, by Lemma A.9,

$\displaystyle\mathbb{P}(\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{1})>t/8)$ (3.20)
$\displaystyle\leq\Bigg{(}1+\frac{(2x_{0}^{N}+1)^{-1}}{(\rho_{i}-\sqrt{(3\cdot
2^{-M}N^{-1/3})^{2}+4(2x_{0}^{N}+1)^{-1}})\rho_{i}}\Bigg{)}^{2x_{0}^{N}+1}e^{-\frac{\sqrt{(3\cdot
2^{-M}N^{-1/3})^{2}+4(2x_{0}^{N}+1)^{-1}}-3\cdot 2^{-M}N^{-1/3}}{2}{t/8}}$
$\displaystyle\leq\Bigg{(}1+\frac{(2x_{0}^{N}+1)^{-1}}{[1/2-\rho^{*}N^{-1/3}-\sqrt{(3\cdot
2^{-M}N^{-1/3})^{2}+4(2x_{0}^{N}+1)^{-1}}](1/2-\rho^{*}N^{-1/3})}\Bigg{)}^{2x_{0}^{N}+1}$
$\displaystyle\times e^{-\frac{\sqrt{(3\cdot
2^{-M}N^{-1/3})^{2}+4(2x_{0}^{N}+1)^{-1}}-3\cdot 2^{-M}N^{-1/3}}{2}{t/8}}$

Setting $\rho^{*}\geq 1$, $x_{0}\geq 1$, $(32)^{3}\vee 64(\rho^{*})^{3}<N$ we
have the following bounds

$\displaystyle 1/2-\rho^{*}N^{-1/3}>1/4,\quad\sqrt{(3\cdot
2^{-M}N^{-1/3})^{2}+4(2x_{0}^{N}+1)^{-1}}<1/8\quad\text{and}\quad 3\cdot
2^{-M}N^{-1/3}<\frac{1}{16}.$ (3.21)

Setting $\frac{1}{2}\log_{2}\big{(}x_{0}\big{)}+4<M$ gives

$\displaystyle\frac{\sqrt{(3\cdot
2^{-M}N^{-1/3})^{2}+4(2x_{0}^{N}+1)^{-1}}-3\cdot
2^{-M}N^{-1/3}}{2}\geq\frac{1}{4}(2x_{0}+1)^{-1/2}N^{-1/3}$ (3.22)

Using (3.21) and (3.22) in (3.20) leads to

$\displaystyle\mathbb{P}(\Phi_{-x_{0}^{N},x_{0}^{N}}(I^{4}-I^{1})>t/8)$ (3.23)
$\displaystyle\leq\big{(}1+(32)^{-1}(2x_{0}^{N}+1)^{-1}\big{)}^{2x_{0}^{N}+1}e^{-\frac{1}{32}(2x_{0}+1)^{-1/2}{N^{-1/3}t}}\leq
e^{-\frac{1}{32}(2x_{0}+1)^{-1/2}{N^{-1/3}t}+1},$

where we used $(1+\frac{x}{n})^{n}\leq e^{x}$. Using union bound and (3.21)

$\displaystyle\mathbb{P}(\mathcal{M}_{-x_{0}^{N},x_{0}^{N}}(I^{4})>t/8)\leq(2x_{0}^{N}+1)e^{-(1/2-\rho^{*}N^{-1/3})t/8}\leq(2x_{0}^{N}+1)e^{-t/32}.$
(3.24)

Plugging (3.23) and (3.24) into (3.19),

$\displaystyle\mathbb{P}\big{(}X_{max}>t\big{)}\leq
4e^{-\frac{1}{32}(2x_{0}+1)^{-1/2}{N^{-1/3}t}+1}+4(2x_{0}^{N}+1)e^{-t/32}.$
(3.25)

Recall the set $\hat{C}^{N,l}$ from (3.15). Denote

$\displaystyle
F_{\hat{C}}:=1_{\hat{C}^{l,N}}+1_{\hat{C}^{l+1,N}}+1_{\hat{C}^{l+2,N}}.$

It follows that

$\displaystyle\mathbb{P}(F_{\hat{C}^{i}}\geq 2)\leq$ (3.26)
$\displaystyle\mathbb{P}(\hat{C}^{N,i+2}\cap\hat{C}^{N,i+3})+\mathbb{P}(\hat{C}^{N,i+2}\cap\hat{C}^{N,i+4})+\mathbb{P}(\hat{C}^{N,i+3}\cap\hat{C}^{N,i+4})$
$\displaystyle\leq 4\mathbb{P}\big{(}\max\\{J^{2},J^{3}\\}\leq
X_{max}\big{)}.$

By Proposition 2.11, $\\{J^{l}\\}_{l\in\\{2,3,4\\}}$ are i.i.d. r.v.s and

$\displaystyle
J^{l}\sim\textrm{Exp}(\rho_{i-1}-\rho_{i})=\textrm{Exp}(2^{-M}N^{-1/3})\quad\text{for
$l\in\\{1,2,3\\}$}.$

Using the joint independence of $J^{2},J^{3}$ and $X_{max}$

$\displaystyle\mathbb{P}\big{(}\max\\{J^{2},J^{3}\\}\leq
X_{max}\big{)}=2\int_{0}^{\infty}2^{-M}e^{-2^{-M}t}(1-e^{-2^{-M}t})\mathbb{P}(X_{max}>tN^{1/3})dt.$
(3.27)

Plugging (3.25) in (3.27)

$\displaystyle\mathbb{P}\big{(}\max\\{J^{2},J^{3}\\}\leq X_{max}\big{)}\leq
2^{18}x_{0}2^{-2M},$ (3.28)

where the full computation can be found in (A.4). Plugging (3.28) in (3.26)
gives

$\displaystyle\mathbb{P}(F_{\hat{C}^{i}}\geq 2)\leq 2^{20}x_{0}2^{-2M}$ (3.29)

From (3.15) $\mathbb{P}(F_{\hat{B}^{i}}\geq
2)\leq\mathbb{P}(F_{\hat{C}^{i}}\geq 2)$, so that (3.29) implies (3.10). The
proof is complete. ∎

######  Corollary 3.4.

Let $\rho^{*}\geq 1$, $x_{0}\geq 1$, $(32)^{3}\vee 64(\rho^{*})^{3}<N$ and
$\frac{1}{2}\log_{2}\big{(}x_{0}\big{)}+4<M$.

$\displaystyle\mathbb{P}\big{(}F_{B^{i}}\geq 2\text{ for some
$i\in\mathcal{E}^{-3}$}\big{)}\leq 2^{20}x_{0}2^{-M}.$ (3.30)

###### Proof.

As $|\mathcal{E}^{-2}|=2^{M}-3$, using union bound and Lemma 3.3 we obtain the
result. ∎

Recall $I^{\rho}$ from (3.4). Let $x_{0}>0$ and $N_{0}>(\rho^{*})^{3}$. For
$N>N_{0}$ define the family of functions
$\\{F^{N,x_{0}}_{\mu}\\}_{\mu\in[-\rho^{*},\rho^{*}]}$ through

$\displaystyle
F^{N,x_{0}}_{\mu}(i)=I^{1/2-\frac{\mu}{4}N^{-1/3}}_{i}\quad\forall
i\in\llbracket-x_{0}^{N},x_{0}^{N}\rrbracket$ (3.31)

When the interval is clear from the context we shall omit the superscript
$x_{0}$. We call $\rho_{0}\in(-\rho^{*},\rho^{*})$ an epoch of $F^{N}$ if for
some $i\in[-x_{0}^{N},x_{0}^{N}]$, $F^{N}_{\cdot}(i)$ is not constant on any
open interval containing $\rho_{0}$. Our next result shows that with high
probability the epochs of $F^{N}_{\rho}$ are not too close. Let $\delta>0$ and
define

$\displaystyle\mathcal{A}^{\delta,N,\rho^{*},x_{0}}:=\\{\text{there exist
epochs $\rho_{0},\rho_{1}\in[-\rho^{*},\rho^{*}]$ of $F^{N}$ such that
$|\rho_{1}-\rho_{0}|\leq\delta$}\\}.$ (3.32)

######  Proposition 3.5.

For $\rho^{*}\geq 1$, $x_{0}\geq 1$, $(32)^{3}\vee 64(\rho^{*})^{3}<N$ and
$\delta<2^{-4}x_{0}^{-1/2}$

$\displaystyle\mathbb{P}\big{(}\mathcal{A}^{\delta,N,\rho^{*},x_{0}}\big{)}\leq
2^{20}x_{0}\delta.$

###### Proof.

Suppose there exist two epochs $\rho_{0},\rho_{1}\in[-\rho^{*},\rho^{*}]$ of
$F^{N}$ such that $|\rho_{1}-\rho_{0}|=\delta<1$. Let $M\in\mathbb{N}$ be such
that

$\displaystyle 2^{-M}<\delta\leq 2^{-M+1}.$ (3.33)

Then there exists $\rho_{e}\in\mathcal{E}(M)$ and two possible cases (without
loss of generality we assume $\rho_{0}<\rho_{1}$)

  1. Case 1:

The two epochs belong to two adjacent intervals of size $2^{-M}$ i.e.

$\displaystyle\rho_{0}\in[\rho_{e},\rho_{e}+2^{-M})\quad\text{and}\quad\rho_{1}\in[\rho_{e}+2^{-M},\rho_{e}+2^{-M+1}).$
(3.34)

  2. Case 2:

$\rho_{0}$ and $\rho_{1}$ belong to the leftmost and rightmost intervals of
three consecutive intervals of size $2^{-M}$ i.e.

$\displaystyle\rho_{0}\in[\rho_{e},\rho_{e}+2^{-M})\quad\text{and}\quad\rho_{1}\in[\rho_{e}+2^{-M+1},\rho_{e}+2^{-M+2}).$
(3.35)

Form (3.3), if $\rho_{0}$ and $\rho_{1}$ are epochs of $F^{N}$ satisfying
either (3.34) or (3.35), then

  1. Case 1:

(3.34) implies

$\displaystyle I^{\rho_{e},N}<I^{\rho_{e}+2^{-M},N}<I^{\rho_{e}+2^{-M+1},N}.$
(3.36)

  2. Case 2:

(3.35) implies

$\displaystyle I^{\rho_{e},N}<I^{\rho_{e}+2^{-M},N}<I^{\rho_{e}+2^{-M+2},N}.$
(3.37)

The inequalities in (3.36)–(3.37) should be understood element-wise, where
strict inequality $<$ means that the vector to its right has at least one
element that is strictly larger than the corresponding element of the vector
to the left of it. Denote

$\displaystyle\mathcal{B}^{M,N}:=\\{\text{there exists epochs
$\rho_{0},\rho_{1}\in[-\rho^{*},\rho^{*}]$ of $F^{N}$ such that
$2^{-M}<|\rho_{1}-\rho_{0}|\leq 2^{-M+1}$}\\}$
$\displaystyle\mathcal{C}^{M,N}:=\\{\text{$\exists\rho_{e}\in\mathcal{E}$ such
that either \eqref{xe2} or \eqref{xe3} hold}\\}.$

We have thus shown that

$\displaystyle\mathcal{B}^{M,N}\subseteq\mathcal{C}^{M,N}.$

It follows from Corollary 3.4 that

$\displaystyle\mathbb{P}(\mathcal{C}^{M,N})\leq 2^{20}x_{0}2^{-M}.$ (3.38)

Note that

$\displaystyle\mathcal{D}^{M,N}\subseteq\bigcup_{M\leq
i}\mathcal{B}^{i,N}\subseteq\bigcup_{M\leq i}\mathcal{C}^{i,N},$

where

$\displaystyle\mathcal{D}^{M,N}:=\\{\text{there exists epochs
$\rho_{0},\rho_{1}\in[-\rho^{*},\rho^{*}]$ of $F^{N}$ such that
$|\rho_{1}-\rho_{0}|\leq 2^{-M+1}$}\\}$

Using (3.38) and union bound,

$\displaystyle\mathbb{P}(\mathcal{D}^{M,N})\leq\sum_{i=M}^{\infty}\mathbb{P}(\mathcal{C}^{i,N})\leq
2^{20}x_{0}2^{-M+1}.$ (3.39)

Finally, using (3.33) in (3.39)

$\displaystyle\mathbb{P}\big{(}\text{$\exists$ epochs
$\rho_{0},\rho_{1}\in[-\rho^{*},\rho^{*}]$ of $F^{N}$ such that
$|\rho_{1}-\rho_{0}|\leq\delta$}\big{)}\leq\mathbb{P}(\mathcal{D}^{M,N})$
$\displaystyle\leq 2^{21}x_{0}\delta.$

∎

##  4 Convergence to the Stationary Horizon

In this section we prove the main results of this work, i.e. we show that the
Busemann process on a horizontal line, converges to a non-trivial limit under
diffusive scaling - Theorem 1.1 and Theorem 1.2. The sense in which the
convergence takes place will be defined rigorously in Subsection 4.1, here we
only try to give a general idea of our intentions. Recall
$\\{I^{\mu}\\}_{\mu\in\mathbb{R}}$ from (3.4), a family of sequences sampling
the value of the Busemann process on the horizontal edges that belong to the
$x$-axis. As we have an eye to use $\\{I^{\mu}\\}_{\mu\in\mathbb{R}}$ as
initial conditions for our LPP model in future work, a problem to which the
relevant scaling is the $1:2:3$-KPZ scaling, the scaling for the initial
condition is the diffusive one - $1:2$. For a fixed $\mu\in\mathbb{R}$ we
interpolate between the points of $I^{\mu,N}=I^{1/2-\frac{\mu}{4}N^{-1/3}}$
and scale diffusively to obtain a continuous function $G^{N}_{\mu}$ on the
real line. One should think of $G^{N}$ as a process that takes values in the
space of continuous functions and is indexed by its drift $\mu\in\mathbb{R}$.
The goal of this section is to show that $G^{N}$ converges in some sense to
some non-trivial limit $G$ which we refer to as the Stationary Horizon. The
processes $G^{N}$ are rcll, inherited from Property (5) of the Busemann
process, hence the appropriate and perhaps natural sense of convergence is
that of the Skorohod topology.

This section contains four subsections, intended to establish the two
necessary and sufficient conditions for convergence in the Skorohord topology
(see Theorem A.15). Section 4.1 introduces the prelimiting sequence
$\\{G^{N}\\}_{N\in\mathbb{N}}$, the space in which it lives and its associated
topology. Here we also prove Condition 2 of Theorem A.15. Section 4.2 shows
that the finite dimensional distributions of $G^{N}$ converge weakly, and so
verifying Condition 1 of Theorem A.15. Finally, Section 4.3 combines the
results of the preceding Sections to complete the proof of Theorem 1.1 and
Theorem 1.2.

###  4.1 The space $D(\mathbb{R},\mathcal{S})$

This subsection is dedicated to defining the processes $G^{N}$, the space
$\mathcal{S}$ where they take values, and the space
$D(\mathbb{R},\mathcal{S})$ in which they ’live’. We will also prove some of
the properties of these processes in this subsection. Here we also prove one
of the main result of this section – Proposition 4.3, essentially tightness
for the sequence $G^{N}$

Let $I\subset\mathbb{R}$ be an interval and let $C(I,\mathbb{R})$ be the set
of real valued continuous functions on $I$. We will use the abbreviation
$C(I)$. For $n\in\mathbb{N}$ define the pseudo-metric $d_{n}$ on
$C(\mathbb{R},\mathbb{R})$

$\displaystyle d_{n}(f,g)=\sup_{x\in[-n,n]}|f(x)-g(x)|.$

Let $\mathcal{S}=C(\mathbb{R},\mathbb{R})$ be the set of continuous functions
on the real line equipped with the following metric

$\displaystyle
d(f,g)=\sum_{i=1}^{\infty}2^{-i}\frac{d_{i}(f,g)}{1+d_{i}(f,g)}.$

The space $\mathcal{S}$ is a complete separable metric space (Lemma A.13).
Note that since for any $f,g\in\mathcal{S}$ it holds that $d_{i}(f,g)\leq
d_{i+1}(f,g)$, we have the following bound which we will make use of often in
the sequel

$\displaystyle d(f,g)\leq d_{n}(f,g)+2^{-n}\quad\forall n\in\mathbb{N}.$ (4.1)

Let $I=\\{I_{i}\\}_{i\in\mathbb{Z}}\in\mathcal{X}$. Define the linear map
$\Psi:\mathcal{X}\rightarrow\mathcal{S}$ through

$\displaystyle\Psi(I)(t)=S^{I}_{\lfloor t\rfloor}+(t-\lfloor
t\rfloor)(S^{I}_{\lfloor t\rfloor+1}-S^{I}_{\lfloor t\rfloor})\quad
t\in\mathbb{R}$ (4.2)

where

$\displaystyle S^{I}_{i}=\begin{cases}S^{1,i}(I)&0<i\\\ 0&i=0\\\
-S^{i+1,0}(I)&i<0,\end{cases}$ (4.3)

and where the map $S^{k,l}$ was defined in (3.7). In words, the map $\Psi$
constructs a function $\Psi(I)$ in the following way; it considers its input,
say $I$, as the increments of the piecewise linear function $\Psi(I)$ on the
integers i.e..

$\displaystyle\Psi(I)(t)-\Psi(I)(t-1)=I_{t}\quad t\in\mathbb{Z}.$ (4.4)

As such continuous functions are only unique up to an additive constant,
$\Psi(I)$ is chosen such that it vanishes at the origin. Display (4.4) implies
that for integers $s\leq t$

$\displaystyle\Psi(I)(t)-\Psi(I)(s)=\sum_{s+1}^{t}I_{i}.$ (4.5)

Recall the sequences $I^{\rho}$ from (3.4). Let
$G^{N}:\mathbb{R}\rightarrow\mathcal{S}$ be defined through

$\displaystyle
G^{N}_{\mu}(x)=\begin{cases}N^{-1/3}\Psi(I^{1/2-\frac{\mu}{4}N^{-1/3}}-\bm{2})(xN^{2/3})&-N^{1/3}\leq\mu<N^{1/3}\\\
N^{-1/3}\Psi(I^{1/4}-\bm{2})(xN^{2/3})&\qquad N^{1/3}\leq\mu\\\
N^{-1/3}\Psi(I^{3/4}-\bm{2})(xN^{2/3})&\qquad\mu<-N^{1/3}\end{cases}$ (4.6)

For fixed $N$, we will consider $G^{N}$ as a function $\mu\mapsto\mathcal{S}$.
For $x_{0}>0$, we let $F^{N,x_{0}}:\mathbb{R}\rightarrow C[-x_{0},x_{0}]$ be
the restriction of $G^{N}$ to $[-x_{0},x_{0}]$ i.e.

$\displaystyle F^{N,x_{0}}(x)=G^{N}(x)\quad x\in[-x_{0},x_{0}].$

Recall that $D(\mathbb{R},\mathcal{S})$ is the space of rcll functions from
$\mathbb{R}$ to $\mathcal{S}$. The next lemma shows that
$\\{G^{N}\\}_{N\in\mathbb{N}}$ is a sequence of elements in
$D(\mathbb{R},\mathcal{S})$.

######  Lemma 4.1.

$G^{N}\in D(\mathbb{R},\mathcal{S})$ for every $N\in\mathbb{N}$.

###### Proof.

Clearly $G^{N}_{\mu}\in\mathcal{S}$ for every $\mu\in\mathbb{R}$, so we must
show that for every $N\in\mathbb{N}$ and $\mu\in\mathbb{R}$

$\displaystyle\lim_{h\downarrow 0}d(G^{N}_{\mu},G^{N}_{\mu+h})=0,$ (4.7)

and the following limit exists in $\mathcal{S}$

$\displaystyle\lim_{h\uparrow 0}G^{N}_{\mu+h}.$ (4.8)

Note that from (4.6), we need only show (4.7)–(4.8) for
$-N^{1/3}\leq\mu<N^{1/3}$. From the rcll property (5) of the Busemann process,

$\displaystyle\lim_{h\downarrow
0}d_{n}(F^{N,n}_{\mu},F^{N,n}_{\mu+h})=0\qquad-N^{1/3}\leq\mu<N^{1/3},n\in\mathbb{N}.$
(4.9)

Now, fix $\mu\in[-N^{1/3},N^{1/3})$ and let $M\in\mathbb{N}$. From (4.9),
there exists $h>0$ small enough such that

$\displaystyle
d(G^{N}_{\mu},G^{N}_{\mu+h})<\sum_{i=1}^{M}2^{-i}\frac{d_{i}(F^{N,i}_{\mu},F^{N,i}_{\mu+h})}{1+d_{i}(F^{N,i}_{\mu},F^{N,i}_{\mu+h})}+2^{-M-1}<2^{-M}.$
(4.10)

As (4.10) holds for every $M\in\mathbb{N}$, (4.7) follows. Next we show (4.8).
Once again, from the rcll property (5) of the Busemann process, the following
limit exists in $(d_{n},C([-n,n],\mathbb{R}))$ for every $n\in\mathbb{N}$

$\displaystyle\hat{F}^{N,n}_{\mu-}:=\lim_{h\uparrow 0}F^{N,n}_{\mu+h}.$ (4.11)

For $n\in\mathbb{N}$, construct the function

$\displaystyle\tilde{F}^{N,n}_{\mu-}(x)=\begin{cases}\hat{F}^{N,n}_{\mu-}(x)&x\in[-n,n]\\\
\hat{F}^{N,n}_{\mu-}(-n)&x<-n\\\ \hat{F}^{N,n}_{\mu-}(n)&x>n.\end{cases}$

It is not hard to see that $\tilde{F}^{N,n}_{\mu-}\in\mathcal{S}$ and that
$\\{\tilde{F}^{N,n}_{\mu-}\\}_{n\in\mathbb{N}}$ is a Cauchy sequence in
$\mathcal{S}$, so that there exists $G^{N}_{\mu-}\in\mathcal{S}$ such that

$\displaystyle d(\tilde{F}^{N,n}_{\mu-},G^{N}_{\mu-})\rightarrow 0\qquad
n\rightarrow\infty.$ (4.12)

For $M\in\mathbb{N}$ and $h>0$

$\displaystyle
d(G^{N}_{\mu-h},G^{N}_{\mu-})\leq\sum_{i=1}^{M}2^{-i}\frac{d_{i}(F^{N,i}_{\mu-h},G^{N}_{\mu-})}{1+d_{i}(F^{N,i}_{\mu-h},G^{N}_{\mu-})}+2^{-M-1}$
$\displaystyle\leq\sum_{i=1}^{M}2^{-i}\frac{d_{i}(\tilde{F}^{N,n}_{\mu-},G^{N}_{\mu-})+d_{i}(F^{N,i}_{\mu-h},\tilde{F}^{N,n}_{\mu-})}{1+d_{i}(F^{N,i}_{\mu-h},G^{N}_{\mu-})}+2^{-M-1}.$

From the construction of $\tilde{F}^{N,n}_{\mu-}$ and (4.12), for every
$i\in\mathbb{N}$ and $n>i$, $d_{i}(\hat{F}^{N,n}_{\mu-},G^{N}_{\mu-})=0$.
This, with (4.11) implies that for $h$ small enough

$\displaystyle d(G^{N}_{\mu-h},G^{N}_{\mu-})\leq 2^{-M}.$ (4.13)

As (4.13) holds for every $M\in\mathbb{N}$

$\displaystyle\lim_{h\uparrow 0}G^{N}_{\mu+h}=G^{N}_{\mu-},$

which proves (4.8). ∎

Next we would like to show that $G^{N}$ is stochastically continuous. We
recall the definition of stochastic continuity. For $f\in
D(\mathbb{R},\mathcal{S})$ we denote $\triangle f(\mu)=f(\mu)-f(\mu-)$. Note
that $\triangle f(\mu)\neq 0$ if and only if $f$ has a jump at $\mu$. A
process in $D(\mathbb{R},\mathcal{S})$ is stochastically continuous if for
every $\mu\in\mathbb{R}$ $\mathbb{P}(\triangle f(\mu)=0)=1$. Our next result
shows that $G^{N}$ is stochastically continuous. In fact we prove something
stronger.

######  Lemma 4.2.

Fix $\mu\in\mathbb{R}$ and $0<\epsilon<1$. Then for $0<\delta<1$ and $N\geq
2(5+|\mu|)^{3}$

$\displaystyle\mathbb{P}\Big{(}\sup_{s,t\in(\mu-\delta,\mu]}d(G^{N}_{t},G^{N}_{s})>\epsilon\Big{)}\leq(2^{40}\epsilon^{-20}+1)\delta.$

###### Proof.

Without loss of generality, we assume $\mu\in[-N^{1/3}+1,N^{1/3})$, the
extension to $\mathbb{R}$ is straightforward as $G^{N}$ is constant outside
$[-N^{1/3}+1,N^{1/3})$. From (4.1)

$\displaystyle\Big{\\{}\sup_{s,t\in(\mu-\delta,\mu]}d(G^{N}_{t},G^{N}_{s})>\epsilon\Big{\\}}\subseteq\Big{\\{}\sup_{s,t\in(\mu-\delta,\mu]}d_{n}(G^{N}_{t},G^{N}_{s})>\epsilon-2^{-n}\Big{\\}}$
(4.14)

Recall (3.4) and observe that

$\displaystyle\Big{\\{}\sup_{s,t\in(\mu-\delta,\mu]}d_{n}(G^{N}_{t},G^{N}_{s})>0\Big{\\}}$
$\displaystyle\subseteq\big{\\{}d_{n}(G^{N}_{\mu},G^{N}_{\mu-\delta})>0\big{\\}}$
$\displaystyle=\big{\\{}I^{1/2-\frac{\mu}{4}N^{-1/3}}_{i}\neq
I^{1/2-\frac{\mu-\delta}{4}N^{-1/3}}_{i},\quad\text{ for some }-nN^{2/3}-1\leq
i\leq nN^{2/3}+1\big{\\}},$ (4.15)

where in the second line we used the definition of $G^{N}$ and (3.3) to
conclude that $d_{n}(G^{N}_{t},G^{N}_{s})\leq
d_{n}(G^{N}_{\mu},G^{N}_{\mu-\delta})$ for any $s,t\in(\mu-\delta,\mu]$. Next
use [2][Lemma 5.8] (see also the discussion on p.19 therein) with
$\alpha=1/2-\frac{\mu-\delta}{4}N^{-1/3}$, $\beta=1/2-\frac{\mu}{4}N^{-1/3}$,
$\theta=N^{-1/3}$ and $N\geq 2(5+|\mu|)^{3}$ to conclude that

$\displaystyle\mathbb{P}(\text{the event in
\eqref{eq15}})\leq\delta(1+e^{10n}).$ (4.16)

For the details on the derivation of (4.16) see (A.4). For
$n=\lceil\log_{2}(2\epsilon^{-1})\rceil$, using (4.14)

$\displaystyle\mathbb{P}\Big{(}\sup_{s,t\in(\mu-\delta,\mu]}d(G^{N}_{t},G^{N}_{s})>\epsilon\Big{)}\leq\mathbb{P}\Big{(}\sup_{s,t\in(\mu-\delta,\mu]}d_{n}(G^{N}_{t},G^{N}_{s})>\epsilon/2\Big{)}.$
(4.17)

Using the bound (4.16) in (4.15) and then in (4.17), that
$n\leq\log_{2}(2\epsilon^{-1})+1$ and that $\log_{2}(e)<2$ gives the bound

$\displaystyle\mathbb{P}\Big{(}\sup_{s,t\in(\mu-\delta,\mu]}d(G^{N}_{t},G^{N}_{s})>\epsilon\Big{)}\leq\big{(}1+2^{40}\epsilon^{-20}\big{)}\delta.$

∎

Next we would like to show that condition 2 of Theorem A.15 holds for the
sequence $G^{N}$. Recall the modulus continuity function $\omega$ for rcll
functions from A.30.

######  Proposition 4.3.

Fix $\mu^{*}>0$. For every $\epsilon>0$

$\displaystyle\lim_{\delta\rightarrow
0}\limsup_{N\rightarrow\infty}\mathbb{P}\Big{(}\omega(G^{N},\mu^{*},\delta)>\epsilon\Big{)}=0.$
(4.18)

###### Proof.

Recall (A.29)–(A.30). For $m\in\mathbb{N}$ and $a<b$ we define

$\displaystyle\theta^{n}_{X}[a,b)=\sup_{s,t\in[a,b)}d_{n}\big{(}X(t),X(s)\big{)}\qquad
X\in D(\mathbb{R},\mathcal{S}),$

and

$\displaystyle\omega^{n}(X,t,\delta)=\inf\\{\max_{1\leq i\leq
m}\theta^{n}_{X}[t_{i-1},t_{i})$ $\displaystyle:\exists m\geq
1,-t=t_{0}<t_{1}<...<t_{m}=t$ (4.19) $\displaystyle\text{such that
$t_{i}-t_{i-1}>\delta$ for all $i\leq m$}\\}.$

In words, the difference between $\omega^{n}$ and $\omega$ is that the former
considers only the values of $X$ restricted to the interval $[-n,n]$. From
(4.1) it follows that

$\displaystyle\omega(X,\mu^{*},\delta)\leq\omega^{n}(X,\mu^{*},\delta)+2^{-n}.$

Setting $n=\lceil\log_{2}\epsilon^{-1}\rceil+1$, from the last display it
follows that in order to show (4.18) it is enough to show that

$\displaystyle\lim_{\delta\rightarrow
0}\limsup_{N\rightarrow\infty}\mathbb{P}\Big{(}\omega^{n}(G^{N},\mu^{*},\delta)>\frac{\epsilon}{2}\Big{)}=0.$
(4.20)

Recall the event $\mathcal{A}^{\delta,N,\mu^{*},n}$ from (3.32). Note that

$\displaystyle\Big{(}\mathcal{A}^{2\delta,N,\mu^{*},n}\Big{)}^{c}\subseteq\\{\omega^{n}(G^{N},\mu^{*},\delta)=0\\}.$

From Proposition 3.5, for $(32)^{3}\vee 64(\rho^{*})^{3}<N$ and
$\delta<2^{-6}n^{-1/2}$

$\displaystyle\mathbb{P}\Big{(}\omega^{n}(G^{N},\mu^{*},\delta)>\frac{\epsilon}{2}\Big{)}\leq\mathbb{P}\big{(}\omega^{n}(G^{N},\mu^{*},\delta)>0\big{)}\leq\mathbb{P}\big{(}\mathcal{A}^{2\delta,N,\mu^{*},n}\big{)}\leq
2^{18}n\delta,$

which implies (4.20), which in turn implies that (4.18) indeed holds. ∎

Fix $x_{0}>0$ and define the sequence of functions
$\\{G^{x_{0},N}\\}_{N\in\mathbb{N}}$ in $D(\mathbb{R},C([-x_{0},x_{0}]))$
through

$\displaystyle
G^{x_{0},N}_{\mu}(x)=G^{N}_{\mu}(x)\qquad\mu\in\mathbb{R},x\in[-x_{0},x_{0}].$
(4.21)

In words, $G^{x_{0},N}$ retains the information of $G^{N}$ restricted to the
interval $[-x_{0},x_{0}]$. As before, we call $\mu\in\mathbb{R}$ an epoch of
$G^{x_{0},N}$, if $G^{x_{0},N}$ is non-constant on any open interval
containing $\mu$. The next result says that the process $G^{N}$, when
restricted to $[-x_{0},x_{0}]$, attains finitely many values on
$[-\mu_{0},\mu_{0}]$.

######  Proposition 4.4.

Fix $x_{0}\geq 1$, $\mu_{0}\geq 1$. For $N,M\in\mathbb{N}$ , define

$\displaystyle\mathcal{B}^{M,N,\mu_{0},x_{0}}$ $\displaystyle=\\{\text{the
process $G^{x_{0},N}$ has no less than $M$ epochs in
$[-\mu_{0},\mu_{0}]$}\\}.$ (4.22)

Then, for $M>1+2\mu_{0}x_{0}^{1/2}2^{4}$ and $N>(32)^{3}\vee 64\mu_{0}^{3}$

$\displaystyle\mathbb{P}(\mathcal{B}^{M,N,\mu_{0},x_{0}})\leq
2^{21}\frac{x_{0}\mu_{0}}{M-1}.$ (4.23)

In particular, with probability one and for any $N\geq 1$, the process
$G^{x_{0},N}$ attains finitely many values on $[-\mu_{0},\mu_{0}]$.

###### Proof.

From definition (4.6), it is clear, that for $\mu_{0}^{3}<N$ the epochs of
$G^{x_{0},N}$ are the same as that of $F^{N,x_{0}}$ from (3.31). More
precisely, for $M>0$, define the event

$\displaystyle\mathcal{C}^{M,N,\mu_{0},x_{0}}$ $\displaystyle=\\{\text{the
process $F^{N,x_{0}}$ has no less than $M$ epochs in
$[-\mu_{0},\mu_{0}]$}\\},$

then for $\mu_{0}^{3}<N$

$\displaystyle\mathcal{B}^{M,N,\mu_{0},x_{0}}=\mathcal{C}^{M,N,\mu_{0},x_{0}}.$

This implies that

$\displaystyle\mathcal{B}^{M,N,\mu_{0},x_{0}}=\mathcal{C}^{M,N,\mu_{0},x_{0}}\subseteq\mathcal{A}^{\frac{2\mu_{0}}{M-1},N,\mu_{0},x_{0}}.$

To see why the last display holds, note that if there are $M\geq 2$ epochs of
the process $F^{N,x_{0}}$ in an interval of size $2\mu_{0}$, there must be two
epochs within distance of $\frac{2\mu_{0}}{M-1}$ from each other. From
Proposition 3.5 it follows that for $M>1+2\mu_{0}x_{0}^{1/2}2^{4}$ and
$N>(32)^{3}\vee 64\mu_{0}^{3}$

$\displaystyle\mathbb{P}(\mathcal{B}^{M,N,\mu_{0},x_{0}})=\mathbb{P}(\mathcal{B}^{\lceil
M\rceil,N,\mu_{0},x_{0}})\leq 2^{20}x_{0}\frac{2\mu_{0}}{M-1},$

which implies the result. ∎

###  4.2 Finite dimensional distributions of the stationary horizon

This subsection is mostly devoted to showing finite dimensional convergence of
the sequence $\\{G^{N}\\}$, which is necessary in order to show convergence of
$\\{G^{N}\\}$ in $D(\mathbb{R},\mathcal{S})$. The convergence of $G^{N}$ in
the finite dimensional sense will be achieved in three steps;

  1. 1.

In Section 4.2.1 we introduce the map
$\Phi^{k}:\mathcal{S}^{k}\rightarrow\mathcal{S}^{k}$, which is the analogue of
the map $\mathcal{D}^{(k)}$ for elements in $\mathcal{S}^{k}$; just as in the
prelimit the map $\mathcal{D}^{(k)}$ constructs the joint distribution of the
Busemann process from independent i.i.d. random walks, $\Phi^{k}$ constructs
the joint distribution of the SH from independent Brownian motions. Although
the map $\Phi^{k}$ is not continuous on $\mathcal{S}^{k}$, we introduce a
”good” set $A_{\text{good}}\subset\mathcal{S}^{k}$ on which $\Phi^{k}$ is
almost Lipschitz.

  2. 2.

In Section 4.2.2, we show that on a ”good” set
$B_{\text{good}}\subset\mathcal{X}^{k}$ one can approximate the map
$\Phi^{k}\Psi$ with the map $\Psi\mathcal{D}^{k}$, where
$\Psi:\mathcal{X}^{k}\rightarrow\mathcal{S}^{k}$ is an interpolation operator.

  3. 3.

Section 4.2.3 shows that the ”good” events $A_{\text{good}}$ and
$B_{\text{good}}$ are of high probability and combines the results form the
previous steps to show that a scaled version of $\Psi\mathcal{D}^{k}$, from
which the finite dimensional distribution of $G^{N}$ can be obtained,
converges in distribution.

####  4.2.1 The map $\Phi$

Consider the following map $\Phi:C(\mathbb{R})\times C(\mathbb{R})\rightarrow
C(\mathbb{R})$ defined through

$\displaystyle\Phi(f,g)=\begin{cases}f(t)+\Big{[}W_{0}(f-g)+\inf_{0\leq s\leq
t}f(s)-g(s)\Big{]}^{-}&t\geq 0\\\ f(t)-\Big{[}W_{t}(f-g)+\inf_{t<s\leq
0}f(s)-f(t)-[g(s)-g(t)]\Big{]}^{-}&t<0\end{cases}$ (4.24)

where

$\displaystyle W_{t}(f)=\sup_{s\leq t}\big{[}f(t)-f(s)\big{]}.$ (4.25)

One can think of the map $\Phi$ as a continuous version of the map $D$ acting
on continuous functions rather than sequences, where the first argument can be
thought of as the service times while the second as the inter-arrival times.

######  Remark 4.5.

It is not hard to show that the map $\Phi$ in (4.24) is equivalent to the one
in [32, Eq. (13)]. In particular, if $X^{1}$ and $X^{2}$ are two bi-infinite
Brownian motions conditioned to vanish at the origin, with drift
$\mu_{1}<\mu_{2}$ respectively, then by Burke’s property

$\displaystyle\Phi(X^{1},X^{2})\sim X^{2}.$

This suggests that, in the limit, we should expect that the joint distribution
of the SH, $(G_{\mu_{1}},G_{\mu_{2}})$, equal in distribution to
$\big{(}X^{1},\Phi(X^{1},X^{2})\big{)}$. This will indeed be proven in
Proposition 4.19.

Let $P_{t}(f)$ be the position in time where $W_{t}$ attains its maximum.

$\displaystyle P_{t}(f)=\underset{s\leq
t}{\mathrm{argmax\,\,}}\big{[}f(t)-f(s)\big{]}.$ (4.26)

Define the set

$\displaystyle A^{N}=\\{(f,g):P_{-N}(f-g)\in[-2N,-N]\\}.$

When $f,g\in\mathcal{S}$ are random, we denote

$\displaystyle A^{N}(f,g)=\\{(f,g)\in A^{N}\\}.$ (4.27)

######  Remark 4.6.

The function $\Phi$ is not continuous from
$\mathcal{S}\times\mathcal{S}\rightarrow\mathcal{S}$. To see that, consider
the sequence of functions $(f^{n},g^{n})\in\mathcal{S}^{2}$ defined by

$\displaystyle f^{n}(x)$
$\displaystyle=\begin{cases}2n(x+n+1)-2n&x\in[-n-1,-n]\\\
-2n(x+n+1)-2n&x\in[-n-2,-n-1]\\\ 0&\text{otherwise}\end{cases}$ (4.28)
$\displaystyle g^{n}$ $\displaystyle=Id.$ (4.29)

where $Id(x)=x$. Clearly, $(f^{n},g^{n})\rightarrow(0,Id)$ in
$\mathcal{S}^{2}$. However, as $W_{-m}(f^{n}-g^{n})=n$ for every $n\geq m\geq
0$ it is not hard to verify that $\Phi(f^{n},g^{n})\rightarrow 0$ in
$\mathcal{S}$ while $\Phi(0,Id)=Id$. Hence the map $\Phi$ is not continuous.
The cause of the discontinuity comes form the fact that
$P_{-1}(f-g)\rightarrow\infty$ as $n\rightarrow\infty$. When
$f,g\in\mathcal{S}$ are considered as the input of the map $\Phi$, the event
$A^{N}(f,g)$ will play an important role throughout this section. Conditioning
on this event, we essentially require the behaviour of $\Phi(f,g)$ on
$[-N,\infty)$ to be independent of $(-\infty,-2N]$ for every $n$, ruling out
the existence of a sequence such as in (4.28). We shall refer to this property
as locality. Throughout the this section we shall introduce sets and events
which possess the locality property.

Next we need to extend the (pseudo) metrics $d$ and $d_{N}$ to vectors in
$\mathcal{S}$. Let $k\in\mathbb{N}$ and
$\overline{f},\overline{g}\in\mathcal{S}^{k}$. Define

$\displaystyle d^{(k)}(\overline{f},\overline{g})$ $\displaystyle=\max_{1\leq
i\leq k}d(f_{i},g_{i})$ $\displaystyle d^{(k)}_{N}(\overline{f},\overline{g})$
$\displaystyle=\max_{1\leq i\leq k}d_{N}(f_{i},g_{i}).$

The following result shows that on $A^{N}$, the map $\Phi$ is almost
continuous.

######  Lemma 4.7.

Fix $N\in\mathbb{N}$. Then

$\displaystyle d(\Phi(f_{1},g_{1}),\Phi(f_{2},g_{2}))\leq
2^{2N+4}d^{(2)}\big{(}(f_{1},g_{1}),(f_{2},g_{2})\big{)}+2^{-N}\quad\forall(f_{1},g_{1}),(f_{2},g_{2})\in
A^{N}.$

###### Proof.

First we note that for $(f_{1},g_{1}),(f_{2},g_{2})\in A^{N}$

$\displaystyle d_{k}\big{(}\Phi(f_{1},g_{1}),\Phi(f_{2},g_{2})\big{)}\leq
7d^{(2)}_{2N}\big{(}(f_{1},g_{1}),(f_{2},g_{2})\big{)}\quad\forall 1\leq k\leq
N.$ (4.30)

To see that, note that on $A^{N}\subseteq\mathcal{S}\times\mathcal{S}$ the
values of the function $\Phi$ on the interval $[-N,N]$ depend only on the
values of the input functions in the interval $[-2N,2N]$, and that the number
of occurrences of the input functions in the map $\Phi$ in (4.24) is $7$. It
follows that

$\displaystyle
d\big{(}\Phi(f_{1},g_{1}),\Phi(f_{2},g_{2})\big{)}\leq\sum_{i=1}^{N}2^{-i}\frac{d_{i}\big{(}\Phi(f_{1},g_{1}),\Phi(f_{2},g_{2})\big{)}}{1+d_{i}\big{(}\Phi(f_{1},g_{1}),\Phi(f_{2},g_{2})\big{)}}$
(4.31)
$\displaystyle+\sum_{i=N+1}^{\infty}2^{-i}\frac{d_{i}\big{(}\Phi(f_{1},g_{1}),\Phi(f_{2},g_{2})\big{)}}{1+d_{i}\big{(}\Phi(f_{1},g_{1}),\Phi(f_{2},g_{2})\big{)}}\leq
2^{-N}+7\frac{d^{(2)}_{2N}\big{(}(f_{1},g_{1}),(f_{2},g_{2})\big{)}}{1+d^{(2)}_{2N}\big{(}(f_{1},g_{1}),(f_{2},g_{2})\big{)}}.$

For $k\in\mathbb{N}$

$\displaystyle d^{(2)}\big{(}(f_{1},g_{1}),(f_{2},g_{2})\big{)}\geq
2^{-k}\max\Big{\\{}\frac{d_{k}(f_{1},g_{1})}{1+d_{k}(f_{1},g_{1})},\frac{d_{k}(f_{2},g_{2})}{1+d_{k}(f_{2},g_{2})}\Big{\\}}=2^{-k}\frac{d^{(2)}_{k}\big{(}(f_{1},g_{1}),(f_{2},g_{2})\big{)}}{1+d^{(2)}_{k}\big{(}(f_{1},g_{1}),(f_{2},g_{2})\big{)}},$
(4.32)

where in the equality we used that $\frac{x}{1+x}$ is an increasing function.
Using (4.32) in the last term of (4.31) with $k=2N$ implies the result. ∎

Next we would like to find an analogue of $\mathcal{D}^{(k)}$ to elements in
$\mathcal{S}^{k}$ for some $k\in\mathbb{N}$. Define the map
$\Phi^{k}:\mathcal{S}^{k}\rightarrow\mathcal{S}^{k}$ as follows

  1. 1.

$\Phi^{1}(f_{1})=f_{1}$

  2. 2.

$\Phi^{2}(f_{1},f_{2})=[f_{1},\Phi(f_{1},f_{2})]$

  3. 3.

$\displaystyle\Phi^{k}(f_{1},...,f_{k})=[f_{1},\Phi\big{(}f_{1},[\Phi^{k-1}(f_{2},...,f_{k})]_{1}\big{)},...,\Phi\big{(}f_{1},[\Phi^{k-1}(f_{2},...,f_{k})]_{k-1}\big{)}]\quad
k\geq 3.$ (4.33)

It is helpful to compare the map $\Phi^{k}$ to the map $\mathcal{D}^{(k)}$
form (2.12), as well as the construction in (4.33) with that in (2.2). For
$1\leq i\leq k$ we denote the $i$’th element of $\Phi^{k}$ by $\Phi_{i}^{k}$.
For $h\in\mathcal{S}$, define $\bm{h}^{k}$ to be the element in
$\mathcal{S}^{k}$ whose every coordinate is the function $h$. Analogously to
Lemma 2.4 we have the following

######  Lemma 4.8.

Let $k\geq 1$, $c>0$ and $h\in\mathcal{S}$. For
$\overline{f}\in\mathcal{S}^{k}$

$\displaystyle\Phi^{k}\big{(}c(\overline{f}-\bm{h}^{k})\big{)}=c(\Phi^{k}(\overline{f})-\bm{h}^{k})$
(4.34)

###### Proof.

For $k=1$ the result is trivial. For $k\geq 2$ continue by induction. The base
case, $k=2$, follows easily from considering the map $\Phi$ in (4.24) i.e.

$\displaystyle\Phi(c(f-h),c(g-h))=c\Phi(f,g)-ch.$ (4.35)

For the general step $k\geq 3$, assume the hypothesis holds for $k-1$

$\displaystyle\Phi^{k}(c(f_{1}-h),...,c(f_{k}-h))$
$\displaystyle\stackrel{{\scriptstyle\eqref{it}}}{{=}}\Big{[}c(f_{1}-h),\Phi\Big{(}c(f_{1}-h),[\Phi^{k-1}(c(f_{2}-h),...,c(f_{k}-h))]_{1}\Big{)}$
$\displaystyle,...,\Phi\Big{(}c(f_{1}-h),[\Phi^{k-1}(c(f_{2}-h),...,c(f_{k}-h))]_{k-1}\Big{)}\Big{]}$
$\displaystyle=\Big{[}c(f_{1}-h),\Phi\Big{(}c(f_{1}-h),c([\Phi^{k-1}(f_{2},...,f_{k})]_{1}-h)\Big{)},...,\Phi\Big{(}c(f_{1}-h),[c(\Phi^{k-1}(f_{2},...,f_{k})]_{k-1}-h)\Big{)}\Big{]}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq14}}}{{=}}[cf_{1}-ch,c\Phi\Big{(}f_{1},[\Phi^{k-1}(f_{2},...,f_{k})]_{1}\Big{)}-ch,...,c\Phi\Big{(}f_{1},[\Phi^{k-1}(f_{2},...,f_{k})]_{k-1}\Big{)}-ch]$
$\displaystyle=c\Phi^{k}(f_{1},...,f_{k})-c\bm{h}^{k},$

where in the second equality we used the induction hypothesis for $k-1$. ∎

Our next result is an analogue of (4.7) for $\Phi^{k}$, for its statement we
shall need to introduce an analogue to the event $A^{N}(\cdot,\cdot)$ from
(4.27). For $R\in\mathbb{N}$, define the sequence of numbers

$\displaystyle a_{1}^{R}$ $\displaystyle=R+1$ (4.36) $\displaystyle
a_{j+1}^{R}$ $\displaystyle=4a_{j}^{R}+5\quad j\in\mathbb{N}.$

For random vectors $\overline{f},\overline{g}\in\mathcal{S}^{k}$, $1\leq j\leq
k-1$ and $1\leq i\leq k-j$, define the events

$\displaystyle A^{\overline{f},R}_{j,i}$
$\displaystyle=\big{\\{}P_{-a_{j}^{R}}\big{(}f_{j},\Phi_{i}^{k-j}(f_{j+1},...,f_{k})\big{)}\in[-2a_{j}^{R},-a_{j}^{R}]\big{\\}}$
(4.37) $\displaystyle A^{\overline{g},R}_{j,i}$
$\displaystyle=\big{\\{}P_{-a_{j}^{R}}\big{(}g_{j},\Phi_{i}^{k-j}(g_{j+1},...,g_{k})\big{)}\in[-2a_{j}^{R}+,-a_{j}^{R}]\big{\\}}$
$\displaystyle A_{R}^{\overline{f},\overline{g}}$
$\displaystyle=\bigcap_{\begin{subarray}{c}1\leq j\leq k-1\\\ 1\leq i\leq
j\end{subarray}}[A^{\overline{f},R}_{j,i}\cap A^{\overline{g},R}_{j,i}].$

The event $A_{R}^{\overline{f},\overline{g}}$ assures that at each of the
steps in (4.33), constructing $\Phi^{k}$ from either $\overline{f}$ or
$\overline{g}$, the locality property holds. We are now ready to state the
main result of this part. It shows, that the map $\Phi^{k}$ can be thought of
as (almost) continuous when conditioning on a certain event.

######  Lemma 4.9.

Fix $k\in\mathbb{N}$ and let $\overline{f},\overline{g}\in\mathcal{S}^{k}$ be
random vectors. Fix $R\in\mathbb{N}$. Then on the event
$A_{R}^{\overline{f},\overline{g}}$ it holds that

$\displaystyle
d^{(k-j+1)}\big{(}\Phi^{k-j+1}(f_{j},...,f_{k}),\Phi^{k-j+1}(g_{j},...,g_{k})\big{)}$
(4.38) $\displaystyle\leq
2^{k4^{k+3}R}d^{(k-j+1)}\big{(}(f_{j},...,f_{k}),(g_{j},...,g_{k})\big{)}+2^{-R}\quad
1\leq j\leq k-1.$

###### Proof.

For $k=1$, the result trivially holds. We prove the result for $k\geq 2$. The
proof uses induction on $1\leq j\leq k-1$, with base case $j=k-2$. More
precisely, we would like to show that if the following holds

$\displaystyle
d^{(k-j)}\big{(}\Phi^{k-j}(f_{j+1},...,f_{k}),\Phi^{k-j}(g_{j+1},...,g_{k})\big{)}$
(4.39) $\displaystyle\leq
2^{F_{j+1}}d^{(k-j)}\big{(}(f_{j+1},...,f_{k}),(g_{j+1},...,g_{k})\big{)}+\sum_{i=j+1}^{k-1}2^{G_{{j+1},i}}$
(4.40)

where

$\displaystyle F_{j+1}$ $\displaystyle=2\sum_{i=j+1}^{k-1}a^{R}_{i}+4(k-j-1)$
(4.41) $\displaystyle G_{{j+1},i}$
$\displaystyle=2\sum_{l=j+1}^{i-1}a_{l}^{R}+4(i-j-1)-a_{i}^{R}\quad j+1\leq
i\leq k-1.$

The functions above satisfy

$\displaystyle F_{j}$ $\displaystyle=F_{j+1}+2a_{j}^{R}+4$ (4.42)
$\displaystyle G_{j,i}$ $\displaystyle=G_{j+1,i}+2a_{j}^{R}+4\quad j+1\leq
i\leq k-1$ $\displaystyle G_{j,j}$ $\displaystyle=-a_{j}^{R}$

then, on the event

$\displaystyle\bigcap_{1\leq i\leq k-j}[A^{\overline{f},R}_{j,i}\cap
A^{\overline{g},R}_{j,i}]$ (4.43)

it holds that

$\displaystyle
d^{(k-j+1)}\big{(}\Phi^{k-j+1}(f_{j},...,f_{k}),\Phi^{k-j+1}(g_{j},...,g_{k})\big{)}$
(4.44) $\displaystyle\leq
2^{F_{j}}d^{(k-j+1)}\big{(}(f_{j},...,f_{k}),(g_{j},...,g_{k})\big{)}+\sum_{i=j}^{k-1}2^{G_{j,i}}.$

The base case $j=k-2$ was proved in Lemma 4.7, indeed, setting $j=k-2$ in
(4.41)

$\displaystyle F_{k-1}=2a_{k-1}^{R}+4\quad\quad G_{k-1,k-1}=-a_{k-1}^{R}.$

using Lemma 4.7, on $A^{\overline{f},R}_{k-1,1}\cap
A^{\overline{g},R}_{k-1,1}$

$\displaystyle
d^{(2)}\big{(}\Phi^{2}(f_{k-1},f_{k}),\Phi^{2}(g_{k-1},g_{k})\big{)}\leq
2^{a_{k-1}^{R}+4}d^{(2)}\big{(}(f_{k-1},f_{k}),(g_{k-1},g_{k})\big{)}+2^{-a_{k-1}^{R}},$
(4.45)

where we used that
$d\big{(}\Phi^{2}_{1}(f_{k-1},f_{k}),\Phi^{2}_{1}(g_{k-1},g_{k})\big{)}=d\big{(}f_{k-1},g_{k-1}\big{)}$
is smaller or equal to the r.h.s. of (4.45). For the general step, $j<k-2$,
note that (4.39) implies that

$\displaystyle
d\big{(}\Phi_{i}^{k-j}(f_{j+1},...,f_{k}),\Phi_{i}^{k-j}(g_{j+1},...,g_{k})\big{)}$
$\displaystyle<\eqref{ub48}\quad\forall 1\leq i\leq k-j.$

Moreover, clearly

$\displaystyle d(f_{j},g_{j})\leq
2^{F_{j+1}}d(f_{j},g_{j})+\sum_{i=j+1}^{k-1}2^{G_{{j+1},i}}.$

We then conclude from Lemma 4.7 that on the event in (4.43)

$\displaystyle
d\big{(}\Phi_{i}^{k-j+1}(f_{j},...,f_{k}),\Phi_{i}^{k-j+1}(g_{j},...,g_{k})\big{)}\qquad\qquad\forall
2\leq i\leq k-j+1$ (4.46)
$\displaystyle=d\big{(}\Phi\big{(}f_{j},\Phi_{i-1}^{k-j}(f_{j+1},...,f_{k})\big{)},\Phi\big{(}g_{j},\Phi_{i-1}^{k-j}(g_{j+1},...,g_{k})\big{)}\big{)}$
$\displaystyle\leq
2^{2a_{j}^{R}+4}\Big{[}2^{F_{j+1}}d^{(k-j+1)}\big{(}(f_{j},...,f_{k}),(g_{j},...,g_{k})\big{)}+\sum_{i=j+1}^{k-1}2^{G_{{j+1},i}}\Big{]}+2^{-a_{j}^{R}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{FG2}}}{{=}}2^{F_{j}}d^{(k-j+1)}\big{(}(f_{j},...,f_{k}),(g_{j},...,g_{k})\big{)}+\sum_{i=j}^{k-1}2^{G_{{j},i}}.$
(4.47)

Next we would like to bound the second term in (4.47).

$\displaystyle
G_{j,i+1}-G_{j,i}=2\sum_{l=j}^{i}a_{l}^{R}+4(i+1-j)-a_{i+1}^{R}-\big{[}2\sum_{l=j}^{i-1}a_{l}^{R}+4(i-j)-a_{i}^{R}\big{]}=3a_{i}^{R}+4-a_{i+1}^{R}$
$\displaystyle\stackrel{{\scriptstyle\eqref{ak}}}{{=}}-a_{i}-1<-1.$

It follows that

$\displaystyle\sum_{i=j}^{k-1}2^{G_{{j},i}}\leq\sum_{i=0}^{k-1-j}2^{G_{j,j}-i}\leq
2^{G_{j+1,j+1}+1}\stackrel{{\scriptstyle\eqref{FG2}}}{{=}}2^{-a_{j}^{R}+1}\leq
2^{-a_{1}^{R}+1}\stackrel{{\scriptstyle\eqref{ak}}}{{=}}2^{-N}\quad\forall
1\leq j\leq k-1.$ (4.48)

Next we turn to bound

$\displaystyle F_{j}\leq F_{1}=2\sum_{i=1}^{k-1}a_{i}^{R}+4(k-1).$ (4.49)

First from (4.36)

$\displaystyle
a_{j}^{R}=a_{1}^{R}\sum_{i=0}^{j-2}4^{i}+5\sum_{i=0}^{j-3}4^{i}\leq
4^{j}(a_{1}^{R}+5).$ (4.50)

Plugging (4.50) in (4.49)

$\displaystyle F_{j}\leq 2k4^{k}(a_{1}^{R}+5)+4(j-1)=2k4^{k}(R+4)+4(k-1)\leq
2k4^{k}(R+5)\leq k4^{k+3}R,$ (4.51)

where in the penultimate inequality we used that $4^{k}\geq 4k$ for all $k\geq
1$.

(4.48) and (4.51) in (4.44) implies the result. ∎

####  4.2.2 From $\Phi^{k}$ to $\mathcal{D}^{(k)}$

The main results of this part, Lemma 4.11, essentially shows that
$\Psi\circ\mathcal{D}\sim\Phi\circ\Psi$ on a certain event. In words, it shows
that given a $\overline{I}\in\mathcal{X}^{k}$, first applying the map
$\mathcal{D}^{(k)}$ and then using the map $\Psi$ to interpolate the output,
is not much different than first applying $\Psi$ and then $\Phi$.

Recall definition (4.27).

######  Lemma 4.10.

Let $I^{1},I^{2}\in\mathbb{R}^{\mathbb{Z}}$ be random. On
$A^{N}(\Psi(I^{1}),\Psi(I^{2}))$

$\displaystyle
d_{N}\Big{(}\Psi\big{(}D(I^{1},I^{2})\big{)},\Phi\big{(}\Psi(I^{1}),\Psi(I^{2})\big{)}\Big{)}\leq
16\sup_{i\in[-2N-2,2N]}|I^{1}_{i}|+|I^{2}_{i}|$ (4.52)

###### Proof.

Let us start by computing $\Psi(D(I^{1},I^{2}))$. Let
$\tau:\mathcal{X}\rightarrow\mathbb{R}^{\mathbb{Z}}$ be the translation
operator i.e. $[\tau I]_{i}=I_{i-1}$. Recall (4.3), and note that

$\displaystyle
S^{D(I^{1},I^{2})}_{i}=\begin{cases}S^{1,i}(I^{1})+\Big{[}\big{[}\sup_{l\leq
0}S^{l,0}(\tau I^{1}-I^{2})\big{]}^{+}+\inf_{1\leq l\leq i}S^{1,l}(\tau
I^{1}-I^{2})\Big{]}^{-}&0<i\\\ 0&i=0\\\
-S^{i+1,0}(I^{1})-\Big{[}\big{[}\sup_{l\leq i}S^{l,i}(\tau
I^{1}-I^{2})\big{]}^{+}+\inf_{(i+1)\leq l\leq 0}S^{i+1,l}(\tau
I^{1}-I^{2})\Big{]}^{-}&i<0.\end{cases}$

Let $\mathcal{L}:\mathbb{R}^{\mathbb{Z}}\rightarrow C(\mathbb{R})$ be the
interpolation map, i.e.

$\displaystyle\mathcal{L}(J)=J_{\lfloor t\rfloor}+(t-\lfloor
t\rfloor)(J_{\lfloor t\rfloor+1}-J_{\lfloor t\rfloor})\quad t\in\mathbb{R}.$

As $\mathcal{L}$ is a linear map,

$\displaystyle\Psi(D(I^{1},I^{2}))=\mathcal{L}(S^{D(I^{1},I^{2})})=\mathcal{L}(\mathcal{S}^{1})+\mathcal{L}(\mathcal{S}^{2})$
(4.53)

where

$\displaystyle\mathcal{S}^{1}_{i}=\begin{cases}S^{1,i}(I^{1})&0<i\\\ 0&i=0\\\
-S^{i+1,0}(I^{1})&i<0\end{cases}$

and

$\displaystyle\mathcal{S}^{2}_{i}=\begin{cases}\Big{[}\big{[}H^{1}\big{]}^{+}+H^{2}_{i}\Big{]}^{-}&0<i\\\
0&i=0\\\ -\Big{[}\big{[}H^{3}_{i}]^{+}+H^{4}_{i}\Big{]}^{-}&i<0.\end{cases}$

where

$\displaystyle H^{1}$ $\displaystyle:=\sup_{l\leq 0}S^{l,0}(\tau I^{1}-I^{2})$
$\displaystyle H^{2}_{i}$ $\displaystyle:=\inf_{1\leq l\leq i}S^{1,l}(\tau
I^{1}-I^{2})\quad i\in\mathbb{N}$ $\displaystyle H^{3}_{i}$
$\displaystyle:=\sup_{l\leq i}S^{l,i}(\tau I^{1}-I^{2})\quad
i\in\mathbb{Z}_{-}$ $\displaystyle H^{4}_{i}$ $\displaystyle:=\inf_{(i+1)\leq
l\leq 0}S^{i+1,l}(\tau I^{1}-I^{2})i\in\mathbb{Z}_{-}.$

Denote $f=\Psi(I^{1})$ and $g=\Psi(I^{2})$. From the definition of $\Psi$

$\displaystyle\mathcal{L}(\mathcal{S}^{1})=f.$ (4.54)

We would like to approximate $\mathcal{L}(\mathcal{S}^{2})$ by

$\displaystyle
h(t)=\begin{cases}\Big{[}\big{[}h^{1}\big{]}^{+}+h^{2}(t)\Big{]}^{-}&t\geq
0\\\ -\Big{[}\big{[}h^{3}(t)\big{]}^{+}+h^{4}(t)]\Big{]}^{-}&t<0.\end{cases}$

where

$\displaystyle h^{1}$ $\displaystyle:=\sup_{s\leq 0}f(-1)-f(s-2)+g(s-1)$
(4.55) $\displaystyle h^{2}(t)$ $\displaystyle:=\inf_{1\leq s\leq
t}f(s-1)-f(-1)-g(s)\quad t>0$ (4.56) $\displaystyle h^{3}(t)$
$\displaystyle:=\sup_{s\leq t}f(t-1)-f(s-2)-g(t)+g(s-1)\quad t<0$ (4.57)
$\displaystyle h^{4}(t)$ $\displaystyle:=\inf_{t+1<s\leq
0}f(s-1)-f(t-1)-[g(s)-g(t)]\quad t<0.$ (4.58)

From (4.5)

$\displaystyle S^{i,j}(\tau I^{1}-I^{2})=f(j-1)-f(i-2)-[g(j)-g(i-1)].$ (4.59)

As a linear interpolation attains its maxima and minima on the interpolated
points, the $\sup$ and $\inf$ operators commute with $\mathcal{L}$. It follows
from (4.59) that

$\displaystyle h^{i}=\mathcal{L}(H^{i})\quad i\in\\{1,2,3,4\\},$ (4.60)

are therefore that

$\displaystyle\mathcal{L}(\mathcal{S}^{2})(t)=h(t)\quad t\in\mathbb{Z}.$

The difference between $h$ and $\mathcal{L}(\mathcal{S}^{2})$ comes from the
fact that the operator $\mathcal{L}$ does not commute with the functions
$(\cdot)^{+}$ and $(\cdot)^{-}$. Let us now bound this error. Let
$F\in\mathcal{X}$, and define $F^{+}\in\mathbb{R}^{\mathbb{Z}}$ to be
$F^{+}_{i}=(F_{i})^{+}$. We have the following bound

$\displaystyle
d_{[a,b]}(\mathcal{L}(F^{+}),[\mathcal{L}(F)]^{+})\leq\sup_{i\in[-a+1,b]}|F_{i}-F_{i-1}|,$
(4.61)

where we denote by $d_{[a,b]}$ the sup metric on the interval $[a,b]$. To see
why (4.61) should hold, note that the difference between $\mathcal{L}(I^{+})$
and $[\mathcal{L}(I)]^{+})$ appears only between integer points $x$ and $x+1$
such that $I_{x}\cdot I_{x+1}<0$, in which case the difference cannot exceed
$|I_{x}-I_{x+1}|$. Similar result holds for the operator $(\cdot)^{-}$. In the
following, $er_{i}(t)\,\,\,i\in\\{1,2\\}$ is a real continuous function whose
value may change form line to line

$\displaystyle\Big{[}\big{[}h^{3}(t)\big{]}^{+}+h^{4}(t)\Big{]}^{-}$ (4.62)
$\displaystyle\stackrel{{\scriptstyle\eqref{met1}}}{{=}}\Big{[}\mathcal{L}\Big{(}\big{[}H^{3}\big{]}^{+}\Big{)}+er_{1}(t)+\mathcal{L}(H^{4})\Big{]}^{-}$
$\displaystyle=\Big{[}\mathcal{L}\Big{(}\big{[}H^{3}\big{]}^{+}+H^{4}\Big{)}\Big{]}^{-}+er_{1}(t)$
$\displaystyle\stackrel{{\scriptstyle\eqref{met1}}}{{=}}\mathcal{L}\Big{(}\Big{[}\big{[}H^{3}\big{]}^{+}+H^{4}\Big{]}^{-}\Big{)}+er_{1}(t)$

where $er_{1}(t)$ in the last line of (4.62) can be bounded by

$\displaystyle\sup_{t\in[-N,0]}|er_{1}(t)|\leq
3\sup_{i\in[-N+1,0]}|I^{1}_{i-1}|+|I^{2}_{i}|.$ (4.63)

Similarly we show that

$\displaystyle\Big{[}[h^{1}]^{+}+h^{2}(t)\Big{]}^{-}=\mathcal{L}\Big{(}\Big{[}\big{[}H^{1}\big{]}^{+}+H^{2}\Big{]}^{-}\Big{)}+er_{2}(t)$
(4.64)

where $er_{2}(t)$ can be bounded by

$\displaystyle\sup_{t\in[1,N]}|er_{2}(t)|\leq\sup_{i\in[1,N]}|I^{1}_{i-1}|+|I^{2}_{i}|.$
(4.65)

It follows from (4.62)–(4.65) that

$\displaystyle d_{N}(h,\mathcal{L}(\mathcal{S}^{2}))\leq
3\sup_{i\in[-N,N]}|I^{1}_{i-1}|+|I^{2}_{i}|.$

Use the above display in (4.53) and (4.54) to conclude that

$\displaystyle d_{N}(\Psi(D(I^{1},I^{2})),f+h)\leq
3\sup_{i\in[-N,N]}|I^{1}_{i-1}|+|I^{2}_{i}|.$ (4.66)

Our next step towards proving the result is to bound

$\displaystyle d_{N}\Big{(}f+h,\Phi\big{(}f,g\big{)}\Big{)},$

where $\Phi$ is the map defined in (4.24). Comparing the two terms in the
metric of the above display, it is enough to bound

$\displaystyle d_{[0,N]}\Big{(}h,\Big{[}W_{0}(f-g)+\inf_{0\leq s\leq
t}g(s)-f(s)\Big{]}^{-}\Big{)}\leq
5\sup_{i\in[-2N-2,N]}|I^{1}_{i}|+|I^{2}_{i}|$ (4.67) $\displaystyle
d_{[-N,0]}\Big{(}h,-\Big{[}W_{t}(f-g)+\inf_{t<s\leq
0}f(t)-f(s)-[g(t)-g(s)]\Big{]}^{-}\Big{)}\leq
5\sup_{i\in[-2N-2,0]}|I^{1}_{i}|+|I^{2}_{i}|.$

To see that (4.67) holds, recall first that on the event $A^{N}(f,g)$
appearing in the hypothesis, $P_{-N}(f,g),P_{0}(f,g)\in[-2N,0]$ where
$P_{\cdot}$ was defined in (4.26). This implies that

$\displaystyle|W_{0}(f-g)-h^{1}|\leq\sup_{-2N\leq s\leq
0}|f(-1)|+|f(s)-f(s-2)|+|g(s)-g(s-1)|\leq$ (4.68) $\displaystyle=\sup_{-2N\leq
s\leq 0}|I^{1}_{0}|+|I^{1}_{s}|+|I^{1}_{s-1}|+|I^{2}_{s}|\leq 3\sup_{-2N-2\leq
s\leq 0}|I^{1}_{i}|+|I^{2}_{i}|.$

A similar bound can be obtained for the difference between $W_{t}(f-g)$ and
$h^{3}$, while the bounds for the error coming from comparing $h^{2}$ and
$h^{4}$ to their counterparts in $\Phi(f,g)$ give a bound similar to (4.68)
only with a prefactor of $2$. Combining all these bounds together give (4.67).
From (4.67)

$\displaystyle d_{N}\Big{(}f+h,\Phi\big{(}f,g\big{)}\Big{)}\leq
10\sup_{i\in[-2N-2,N]}|I^{1}_{i}|+|I^{2}_{i}|.$ (4.69)

Using (4.66) and (4.69) we obtain (4.52). ∎

The following result is the analogue of Lemma 4.10 for $\Phi^{k}$ and $k\geq
3$. Before we state it we shall need to define the following event which will
assure that the locality property holds in the steps constructing
$\mathcal{D}^{(k)}$. For $k\geq 2$, let $\overline{I}=(I_{1},..,I_{k})$ and

$\displaystyle\mathcal{A}^{\overline{I}}_{N}:=\bigcap_{1\leq l\leq
k-2}\bigcap_{1\leq m\leq
l+1}A^{2^{k-2-l}N}\Big{(}\Psi(I^{k-l-1}),\Psi\big{(}[\mathcal{D}^{(l+1)}(I^{k-l},...,I^{k})]_{m}\big{)}\Big{)}.$
(4.70)

In what follows, if $\overline{I}=(I^{1},...,I^{k})\in\mathcal{X}^{k}$, we
write $\Psi(\overline{I})$ to denote $(\Psi(I^{1}),...,\Psi(I^{k}))$.

######  Lemma 4.11.

Let $k\geq 2$ and let $\overline{I}\in\mathcal{X}^{k}$. For $N\in\mathbb{N}$
and $1\leq l\leq k$, on the set $\mathcal{A}^{\overline{I}}_{N}$

$\displaystyle
d^{(l)}_{2^{k-l}N}\Big{(}\Psi\big{(}\mathcal{D}^{(l)}(I^{k-l+1},...,I^{k})\big{)},\Phi^{l}\big{(}\Psi(I^{k-l+1},...,I^{k})\big{)}\Big{)}$
(4.71) $\displaystyle\leq 16\sum_{i=0}^{l-2}7^{l-2-i}\sup_{|j|\leq
2^{k-1-i}N+2}\Big{[}|I^{k-1-i}_{j}|+\sup_{1\leq q\leq
i+1}|\tilde{I}^{i+1,q}_{j}|\Big{]}$

where for $1\leq m\leq l+1$

$\displaystyle\tilde{I}^{i+1,q}$
$\displaystyle=[\mathcal{D}^{(i+1)}(I^{k-l},...,I^{k})]_{q}$
$\displaystyle\quad 1\leq i\leq k-2$ $\displaystyle\tilde{I}^{i+1,q}$
$\displaystyle=I^{k}$ $\displaystyle\quad i=0.$

###### Proof.

The proof is by induction. Suppose that for $1\leq l\leq k-2$ we have

$\displaystyle
d^{(l+1)}_{2^{k-1-l}N}\Big{(}\Psi\big{(}\mathcal{D}^{(l+1)}(I^{k-l},...,I^{k})\big{)},\Phi^{l+1}\big{(}\Psi(I^{k-l},...,I^{k})\big{)}\Big{)}$
$\displaystyle\leq 16\sum_{i=0}^{l-1}7^{l-1-i}\sup_{|j|\leq
2^{k-1-i}N+2}\Big{[}|I^{k-1-i}_{j}|+\sup_{1\leq q\leq
i+1}|\tilde{I}^{i+1,q}_{j}|\Big{]}$

then using (4.30), for every $1\leq m\leq l+1$

$\displaystyle
d_{2^{k-1-l-1}N}\Bigg{(}\Phi\Big{(}\Psi(I^{k-l-1}),\Psi\big{(}[\mathcal{D}^{(l+1)}(I^{k-l},...,I^{k})]_{m}\big{)}\Big{)},\Phi\big{(}\Psi(I^{k-l-1}),\Phi^{l+1}_{m}\big{(}\Psi(I^{k-l},...,I^{k})\big{)}\big{)}\Bigg{)}$
(4.72) $\displaystyle\leq 7\times 16\sum_{i=0}^{l-1}7^{l-1-i}\sup_{|j|\leq
2^{k-1-i}N+2}\Big{[}|I^{k-1-i}_{j}|+\sup_{1\leq q\leq
i+1}|\tilde{I}^{i+1,q}_{j}|\Big{]}.$

Then by Lemma 4.10, on $\bigcap_{1\leq m\leq
l+1}A^{2^{k-1-l-1}N}\Big{(}\Psi(I^{k-l-1}),\Psi\big{(}[\mathcal{D}^{(l+1)}(I^{k-l},...,I^{k})]_{m}\big{)}\Big{)}$,
for every $1\leq m\leq l+1$

$\displaystyle
d_{2^{k-1-l-1}N}\Bigg{(}\Phi\Big{(}\Psi(I^{k-l-1}),\Psi\big{(}[\mathcal{D}^{(l+1)}(I^{k-l},...,I^{k})]_{m}\big{)}\Big{)},\Psi\big{(}D(I^{k-l-1},[\mathcal{D}^{(l+1)}(I^{k-l-1},...,I^{k})]_{m})\big{)}\Bigg{)}$
(4.73)
$\displaystyle=d_{2^{k-1-l-1}N}\Bigg{(}\Phi\Big{(}\Psi(I^{k-l-1}),\Psi\big{(}[\mathcal{D}^{(l+1)}(I^{k-l},...,I^{k})]_{m}\big{)}\Big{)},\Psi\big{(}[\mathcal{D}^{(l+2)}(I^{k-l-1},...,I^{k})]_{m+1}\big{)}\Bigg{)}$
$\displaystyle\leq 16\sup_{|j|\leq
2^{k-1-l}N+2}\Big{[}|I^{k-1-l}_{j}|+|\tilde{I}^{l+1,m}_{j}|\Big{]}\leq
16\sup_{|j|\leq 2^{k-1-l}N+2}\Big{[}|I^{k-1-l}_{j}|+\sup_{1\leq q\leq
l+1}|\tilde{I}^{l+1,q}_{j}|\Big{]},$

By definition,

$\displaystyle[\mathcal{D}^{(l+2)}(I^{k-l-1},...,I^{k})]_{1}=\Psi(I^{k-l-1})=\Phi_{1}^{l+2}\big{(}\Psi(I^{k-l-1},...,I^{k})\big{)}.$
(4.74)

Combining (4.72)–(4.74)

$\displaystyle
d^{(l+2)}_{2^{k-1-l-1}N}\big{(}\Phi^{l+2}\big{(}\Psi(I^{k-l-1},...,I^{k})\big{)},\Psi(\mathcal{D}^{(l+2)}(I^{k-l-1},..,I^{k}))\big{)}$
(4.75) $\displaystyle\leq 16\sup_{|j|\leq
2^{k-1-l}N+2}\Big{[}|I^{k-1-l}_{j}|+\sup_{1\leq q\leq
l+1}\tilde{I}^{l+1,q}_{j}\Big{]}$ $\displaystyle+7\times
16\sum_{i=0}^{l-1}7^{l-1-i}\sup_{|j|\leq
2^{k-1-i}N+2}\Big{[}|I^{k-1-i}_{j}|+\sup_{1\leq q\leq
i+1}|\tilde{I}^{i+1,q}_{j}|\Big{]}$
$\displaystyle=16\sum_{i=0}^{l}7^{l-i}\sup_{|j|\leq
2^{k-1-i}N+2}\Big{[}|I^{k-1-i}_{j}|+\sup_{1\leq q\leq
i+1}|\tilde{I}^{i+1,q}_{j}|\Big{]}.$

We have thus established the induction step. To finish the proof, we note that
the base case $l=1$ was proven in Lemma 4.10. ∎

####  4.2.3 Controlling the probability of the events and finite dimensional
convergence

The main result in this part, Proposition 4.16, is also the main result of
this subsection. It establishes the finite dimensional convergence of
sequences that are $\mu_{\bm{\rho}^{N}}$ distributed, under the diffusive
scaling. Towards establishing the result, we will obtain control on the
probability of the different ’locality’-events appearing in the previous parts
4.2.1–4.2.2.

For $\tau>0$ define the scaling operator
$\mathcal{L}^{\tau}:\mathcal{S}\rightarrow\mathcal{S}$ through

$\displaystyle\mathcal{L}^{\tau}(f)(t)=f(\tau t).$

If $\overline{f}\in\mathcal{S}^{k}$, we denote
$\overline{f}^{\tau}=\mathcal{L}^{\tau}(\overline{f})=(\mathcal{L}^{\tau}(f_{1}),...,\mathcal{L}^{\tau}(f_{k}))$.
We shall need the following.

######  Lemma 4.12.

For $\tau>0$

$\displaystyle\mathcal{L}^{\tau}\big{(}\Phi^{k}(\overline{f})\big{)}$
$\displaystyle=\Phi^{k}\big{(}\overline{f}^{\tau}\big{)}$ (4.76)
$\displaystyle
d^{(k)}_{N}\big{(}\overline{f}^{\tau},\overline{g}^{\tau}\big{)}$
$\displaystyle=d^{(k)}_{\tau N}\big{(}\overline{f},\overline{g}\big{)}.$
(4.77)

###### Proof.

The proof of (4.76) is by induction on $k$. For $k=2$ the proof follows by
observing that
$\mathcal{L}^{\tau}\big{(}\Phi(f,g)\big{)}=\Phi\big{(}f^{\tau},g^{\tau}\big{)}$
which can be verified directly by considering the effect of applying
$\mathcal{L}^{\tau}$ to the different terms in (4.24). The case $k\geq 3$ is