$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(s+1)\geq k$, and there exists
$(x,s)\in\operatorname{NU}^{\xi{\scaleobj{0.8}{\boxempty}}}$ such that
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(s+1)\leq-k$. Next, assume, by
way of contradiction, that the set
$\\{x\in\mathbb{R}:(x,0)\in\operatorname{NU}^{\xi{\scaleobj{0.8}{\boxempty}}}\\}$
has an upper bound $b$. Then, by the monotonicity of Theorem 4.5.5(iv), for
all $x\in\mathbb{R}$ with
$(x,s)\in\operatorname{NU}^{\xi{\scaleobj{0.8}{\boxempty}}}$,
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(s+1)\leq
g_{(b,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(s+1)$. But this contradicts the
fact we showed that
$\\{g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(s+1):x\in\mathbb{R}\\}$ is
not bounded above. Hence, there exists a sequence $y_{n}\to\infty$ such that
$(y_{n},s)\in\operatorname{NU}^{\xi{\scaleobj{0.8}{\boxempty}}}$ for all $n$.
By a similar argument, there exists a sequence $x_{n}\to-\infty$ such that
$(x_{n},s)\in\operatorname{NU}^{\xi{\scaleobj{0.8}{\boxempty}}}$ for all $n$.

Now, for arbitrary $s\in\mathbb{R}$, pick a rational number $T>s$. Pick
$(z,T)\in\operatorname{NU}^{\xi{\scaleobj{0.8}{\boxempty}}}$, and let

$\displaystyle
x_{1}=\sup\\{x\in\mathbb{R}:g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(T)\leq
z\\},\qquad\text{and}\qquad
x_{2}=\inf\\{x\in\mathbb{R}:g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(T)\geq
z\\}.$

By the limits in Equation (4.45) of Theorem 4.5.5(v), $x_{1}$ and $x_{2}$ lie
in $\mathbb{R}$.

We first show that $x_{2}\leq x_{1}$. If not, then choose $x\in(x_{1},x_{2})$.
Then,
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(T)<z<g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(T)$,
contradicting the meaning of L and R. Hence $x_{2}\leq x_{1}$. For any
$x>x_{2}$, $g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(T)\geq z$, and by the
limit in Equation (4.43) of Theorem 4.5.5(v),
$g_{(x_{2},s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(T)\geq z$ as well. By an
analogous argument, for $x<x_{1}$,
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(T)\leq z$, and the inequality
$g_{(x_{1},s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(T)\leq z$ holds by the same
argument. Hence, for $x\in[x_{2},x_{1}]$,

$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(T)\leq z,\qquad\text{and}\qquad
g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(T)\geq z.$

Then, by the monotonicity of Theorem 4.5.5(iv), for $t\geq T$,

$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(t)\leq
g_{(z,T)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(t)\leq
g_{(z,T)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(t)\leq
g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(t).$ (4.54)

By assumption that
$(z,T)\in\operatorname{NU}^{\xi{\scaleobj{0.8}{\boxempty}}}$, there exists
$t>T$ such that the middle inequality in (4.54) is strict, so
$(x,s)\in\operatorname{NU}^{\xi{\scaleobj{0.8}{\boxempty}}}$. Furthermore, by
assumption, the set $\\{z\in\mathbb{R}:(z,T)\in\operatorname{NU}\\}$ has
neither an upper or lower bound. Then, by the $t=T$ case of (4.54) and a
similar argument as for the $s=0$ case, the set
$\\{x\in\mathbb{R}:(x,s)\in\operatorname{NU}\\}$ also has neither an upper nor
lower bound.

We lastly show countability of the sets. By (4.42), it suffices to show that
for each $\xi\in\mathbb{Q}$ and $s\in\mathbb{R}$,
$\operatorname{NU}^{\xi}\cap\,\mathcal{H}_{s}$ is countable. The proof is that
of Theorem 3.4, Item 3 in [126], adapted to all horizontal lines
simultaneously. For each $(x,s)\in\operatorname{NU}^{\xi}$, there exists $t>s$
such that $g_{(x,s)}^{\xi,L}(t)<g_{(x,s)}^{\xi,R}(t)$. By continuity of
geodesics, the space between the two geodesics contains an open subset of
$\mathbb{R}^{2}$. By the monotonicity of Theorem 4.5.5(iv), for $x<y$,
$g_{(x,s)}^{\xi,R}(t)\leq g_{(y,s)}^{\xi,L}(t)$ for all $t\geq s$. Hence, for
$x<y$, with $(x,s),(y,s)\in\operatorname{NU}^{\xi}$, the associated open sets
in $\mathbb{R}^{2}$ are disjoint, and
$\operatorname{NU}^{\xi}\cap\,\mathcal{H}_{s}$ is at most countably infinite.
∎

###  4.6 Coalescence and the global geometry of geodesics

We can now describe the global structure of the semi-infinite geodesics,
beginning with coalescence.

######  Theorem 4.6.1.

On a single event of full probability, the following hold across all
directions $\xi\in\mathbb{R}$ and signs
${\scaleobj{0.9}{\boxempty}}\in\\{-,+\\}$.

  1. (i)

For all $p,q\in\mathbb{R}^{2}$, if $g_{1}$ and $g_{2}$ are
$\xi{\scaleobj{0.8}{\boxempty}}$ Busemann geodesics from $p$ and $q$,
respectively, then $g_{1}$ and $g_{2}$ coalesce. If the first point of
intersection of the two geodesics is not $p$ or $q$, then the first point of
intersection is the coalescence point of the two geodesics.

  2. (ii)

Let $g_{1}$ and $g_{2}$ be two distinct $\xi{\scaleobj{0.8}{\boxempty}}$
Busemann geodesics from an initial point
$(x,s)\in\operatorname{NU}^{\xi{\scaleobj{0.8}{\boxempty}}}$. Then, the set
$\\{t>s:g_{1}(t)\neq g_{2}(t)\\}$ is a bounded open interval. That is, after
the geodesics split, they coalesce exactly when they meet again.

  3. (iii)

For each compact set $K\subseteq\mathbb{R}^{2}$, there exists a random
$T=T(K,\xi,{\scaleobj{0.9}{\boxempty}})<\infty$ such that for any two
$\xi{\scaleobj{0.8}{\boxempty}}$ geodesics $g_{1}$ and $g_{2}$ whose starting
points lie in $K$, $g_{1}(t)=g_{2}(t)$ for all $t\geq T$. That is, there is a
time level $T$ after which all semi-infinite geodesics started from points in
$K$ have coalesced into a single path.

######  Remark 4.6.2.

Theorem 1 of [26] implies the following refinements of the results in this
section. In Theorem 4.6.1(ii), $\\{t>s:g_{1}(t)\neq g_{2}(t)\\}=(s,r)$ for
some $r\in(s,\infty)$. Under Condition (i) of Theorem 4.6.3 below, the entire
collection of semi-infinite geodesics in direction $\xi$ is a tree.

The following gives a full classification of the directions in which geodesics
coalesce. We refer the reader to Theorems 4.6.8 and 4.6.9 below for the
connection between coalescence and the regularity of the Busemann process.

######  Theorem 4.6.3.

On a single event of probability one, the following are equivalent.

  1. (i)

$\xi\notin\Xi$.

  2. (ii)

$g_{p}^{\xi-,S}=g_{p}^{\xi+,S}$ for all $p\in\mathbb{R}^{2}$ and
$S\in\\{L,R\\}$.

  3. (iii)

All semi-infinite geodesics in direction $\xi$ coalesce (whether Busemann
geodesics or not).

  4. (iv)

For all $p\in\mathbb{R}^{2}\setminus\operatorname{NU}$, there is a unique
geodesic starting from $p$ with direction $\xi$.

  5. (v)

There is a unique $\xi$-directed semi-infinite geodesic from some
$p\in\mathbb{R}^{2}$.

  6. (vi)

There exists $p\in\mathbb{R}^{2}$ such that $g_{p}^{\xi-,L}=g_{p}^{\xi+,L}$.

  7. (vii)

There exists $p\in\mathbb{R}^{2}$ such that $g_{p}^{\xi-,R}=g_{p}^{\xi+,R}$

Under these equivalent conditions, the following also holds.

  1. (viii)

From any $p\in\mathbb{R}^{2}$, all semi-infinite geodesics in direction $\xi$
are Busemann geodesics.

######  Remark 4.6.4.

The equivalence (i)$\Leftrightarrow$(vi) implies that $\forall\xi\in\Xi$ and
$p\in\mathbb{R}^{2}$, geodesics $g_{p}^{\xi-,L}$ and $g_{p}^{\xi+,L}$ are
distinct. The same is true when $L$ is replaced with $R$. Since
$g_{p}^{\xi-,L}$ and $g_{p}^{\xi+,L}$ are both leftmost geodesics between any
two of their points (Theorem 4.4.9(iv)) then if $\xi\in\Xi$, these two
geodesics must separate at some time $t\geq s$, and they cannot ever come back
together. For each $\xi\in\Xi$, there are two coalescing families of
geodesics, namely the $\xi-$ and $\xi+$ geodesics. (See again Figure 4.1.) In
particular, whenever $\xi\in\Xi$, $s\in\mathbb{R}$, and $x<y$,
$g_{(x,s)}^{\xi+,L}(t)>g_{(y,s)}^{\xi-,R}(t)$ for sufficiently large $t$, as
alluded to in Remark 4.5.6.

####  4.6.1 Proofs

In each of these theorems, the full-probability event is $\Omega_{2}$ (4.28).
We start by proving some lemmas that allow us to prove Theorem 4.6.1. The
proof of Theorem 4.6.3 comes at the very end of this subsection. Section 4.7.1
proves Theorem 4.2.1 as well as lingering results from Section 4.4.

######  Lemma 4.6.5.

Let $\omega\in\Omega_{1}$, $s\in\mathbb{R}$ and $x<y\in\mathbb{R}$. Assume,
for some $\alpha<\xi$ and
${\scaleobj{0.9}{\boxempty}}_{1},{\scaleobj{0.9}{\boxempty}}_{2}\in\\{-,+\\}$,
that
$W_{\alpha{\scaleobj{0.8}{\boxempty}}_{1}}(y,s;x,s)=W_{\xi{\scaleobj{0.8}{\boxempty}}_{2}}(y,s;x,s)$.
We also allow $\alpha=\xi$ if ${\scaleobj{0.9}{\boxempty}}_{1}=-$ and
${\scaleobj{0.9}{\boxempty}}_{2}=+$. If $t>s$ and
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},R}(t)\leq
g_{(y,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},L}(t)$, then for all
$u\in[s,t]$,

$g_{(x,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},R}(u)=g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},R}(u)\qquad\text{and}\qquad
g_{(y,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},L}(u)=g_{(y,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},L}(u).$
(4.55)

###### Proof.

By assumption, whenever $w<z$ and $t\in\mathbb{R}$, Theorem 4.4.1(iii) gives

$W_{\alpha{\scaleobj{0.8}{\boxempty}}_{1}}(z,t;w,t)\leq
W_{\xi{\scaleobj{0.8}{\boxempty}}_{2}}(z,t;w,t).$ (4.56)

For the rest of the proof, we suppress the
${\scaleobj{0.9}{\boxempty}}_{1},{\scaleobj{0.9}{\boxempty}}_{2}$ notation. By
Theorem 4.4.1(ii),(iv),

$\displaystyle W_{\xi}(y,s;x,s)$
$\displaystyle=W_{\xi}(y,s;0,t)-W_{\xi}(x,s;0,t)$
$\displaystyle=\sup_{z\in\mathbb{R}}\\{\mathcal{L}(y,s;z,t)+W_{\xi}(z,t;0,t)\\}-\sup_{z\in\mathbb{R}}\\{\mathcal{L}(x,s;z,t)+W_{\xi}(z,t;0,t)\\},$
(4.57)

and the same with $\xi$ replaced by $\alpha$. Recall that
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(t)$ and
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(t)$ are, respectively, the
leftmost and rightmost maximizers of
$\mathcal{L}(x,s;z,t)+W_{\xi{\scaleobj{0.8}{\boxempty}}}(z,t;0,t)$ over
$z\in\mathbb{R}$. Understanding that these quantities depend on $s$ and $t$,
we use the shorthand notation
$g_{x}^{\xi,R}=g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{1},R}(t)$, and
similarly with the other quantities. Then, we have

$\displaystyle\mathcal{L}(x,s;g_{x}^{\xi,R},t)+W_{\xi}(g_{x}^{\xi,R},t;0,t)$
$\displaystyle-(\mathcal{L}(x,s;g_{x}^{\xi,R},t)+W_{\alpha}(g_{x}^{\xi,R},t;0,t))$
$\displaystyle\geq\sup_{z\in\mathbb{R}}\\{\mathcal{L}(x,s;z,t)+W_{\xi}(z,t;0,t)\\}$
$\displaystyle-\sup_{z\in\mathbb{R}}\\{\mathcal{L}(x,s;z,t)+W_{\alpha}(z,t;0,t)\\}$
(4.58)
$\displaystyle=\sup_{z\in\mathbb{R}}\\{\mathcal{L}(y,s;z,t)+W_{\xi}(z,t;0,t)\\}$
$\displaystyle-\sup_{z\in\mathbb{R}}\\{\mathcal{L}(y,s;z,t)+W_{\alpha}(z,t;0,t)\\}$
$\displaystyle\geq\mathcal{L}(y,s;g_{y}^{\alpha,L},t)+W_{\xi}(g_{y}^{\alpha,L},t;0,t)$
$\displaystyle-(\mathcal{L}(y,s;g_{y}^{\alpha,L},t)+W_{\alpha}(g_{y}^{\alpha,L},t;0,t)),$
(4.59)

where the middle equality came from the assumption that
$W_{\xi}(y,s;x,s)=W_{\alpha}(y,s;x,s)$ and Equation (4.57) applied to both
$\xi$ and $\alpha$. Rearranging the first and last lines yields

$W_{\xi}(g_{y}^{\alpha,L},t;g_{x}^{\xi,R},t)\leq
W_{\alpha}(g_{y}^{\alpha,L},t;g_{x}^{\xi,R},t).$

However, the assumption $g_{x}^{\xi,R}\leq g_{y}^{\alpha,L}$ combined with
(4.56) implies that this inequality is an equality. Hence, inequalities (4.58)
and (4.59) are also equalities. From the equality (4.58),

$\mathcal{L}(x,s;g_{x}^{\xi,R},t)+W_{\alpha}(g_{x}^{\xi,R},t;0,t)=\sup_{z\in\mathbb{R}}\\{\mathcal{L}(x,s;z,t)+W_{\alpha}(z,t;0,t)\\},$

so $z=g_{x}^{\xi,R}$ is a maximizer of
$\mathcal{L}(x,s;z,t)+W_{\alpha}(z,t;0,t)$. By definition, $g_{x}^{\alpha,R}$
is the rightmost maximizer, and by geodesic ordering (Theorem 4.5.5(i)),
$g_{x}^{\xi,R}\geq g_{x}^{\alpha,R}$, so $g_{x}^{\xi,R}=g_{x}^{\alpha,R}$. An
analogous argument applied to (4.59) implies $g_{y}^{\alpha,L}=g_{y}^{\xi,L}$.
We have shown that

$g_{(x,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},R}(t)=g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},R}(t),\qquad\text{and}\qquad
g_{(y,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},L}(t)=g_{(y,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},L}(t).$

Since $g_{(x,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},R}$ and
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},R}$ are both the rightmost
geodesics between any two of their points and similarly with the leftmost
geodesics from $(y,s)$ (Theorem 4.4.9(iv)), Equation (4.55) holds for all
$u\in[s,t]$, as desired. ∎

######  Lemma 4.6.6.

Let $\omega\in\Omega_{2}$, $s\in\mathbb{R}$, and $x<y$. If, for some
$\alpha<\xi$ and
${\scaleobj{0.9}{\boxempty}}_{1},{\scaleobj{0.9}{\boxempty}}_{2}\in\\{-,+\\}$
we have that
$W_{\alpha{\scaleobj{0.8}{\boxempty}}_{1}}(y,s;x,s)=W_{\xi{\scaleobj{0.8}{\boxempty}}_{2}}(y,s;x,s)$,
then $g_{(x,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},R}$ coalesces with
$g_{(y,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},L}$,
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},R}$ coalesces with
$g_{(y,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},L}$, and the coalescence points
of the two pairs of geodesics are the same.

###### Proof.

By Theorem 4.4.9(iii),
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},R}(t)/t\to\xi$ while
$g_{(y,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},L}(t)/t\to\alpha$ as
$t\to\infty$. By this and continuity of geodesics, there exists a minimal time
$t>s$ such that
$z:=g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},R}(t)=g_{(y,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},L}(t).$
By Lemma 4.6.5,

$g_{(x,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},R}(u)=g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},R}(u)\qquad\text{and}\qquad
g_{(y,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},L}(u)=g_{(y,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},L}(u)\qquad\text{for
all }u\in[s,t].$

Since $t$ was chosen to be minimal, Theorem 4.5.5(iv) implies that the pair
$g_{(x,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},R}$,
$g_{(y,s)}^{\alpha{\scaleobj{0.8}{\boxempty}}_{1},L}$ and the pair
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},R}$,
$g_{(y,s)}^{\xi{\scaleobj{0.8}{\boxempty}}_{2},L}$ both coalesce at $(z,t)$. ∎

###### Proof of Theorem 4.6.1.

Item (i) (Coalescence): Let $g_{1}$ and $g_{2}$ be
$\xi{\scaleobj{0.9}{\boxempty}}$ Busemann
geodesics from $(x,s)$ and $(y,t)$, respectively, and take $s\leq t$ without
loss of generality. Let $a=(g_{1}(t)\wedge y)-1$ and $b=(g_{1}(t)\vee y)+1$.
By Theorem 4.5.5(iv), for all $u\geq t$,

$g_{(a,t)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(u)\leq g_{1}(u)\wedge
g_{2}(u)\leq g_{1}(u)\vee g_{2}(u)\leq
g_{(b,t)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(u).$ (4.60)

By Theorem 4.4.1(v), there exists $\alpha$, sufficiently close to $\xi$, (from
the left for ${\scaleobj{0.9}{\boxempty}}=-$ and from the right for
${\scaleobj{0.9}{\boxempty}}=+$) such that
$W_{\xi{\scaleobj{0.8}{\boxempty}}}(b,t;a,t)=W_{\alpha{\scaleobj{0.8}{\boxempty}}}(b,t;a,t)$.
By Lemma 4.6.6, $g_{(a,t)}^{\xi{\scaleobj{0.8}{\boxempty}},R}$ coalesces with
$g_{(b,t)}^{\xi{\scaleobj{0.8}{\boxempty}},L}$. Then, for $u$ large enough,
all inequalities in (4.60) are equalities, and $g_{1}$ and $g_{2}$ coalesce.

If the first point of intersection is not $(y,t)$, then $g_{1}(t)\neq y$, and
the coalescence point of $g_{1}$ and $g_{2}$ is the first point of
intersection by Theorem 4.5.5(iv).

Item (ii) (Geodesics coalesce when they meet): Let
$(x,s)\in\operatorname{NU}^{\xi{\scaleobj{0.8}{\boxempty}}}$, and let $g_{1}$
and $g_{2}$ be two distinct $\xi{\scaleobj{0.8}{\boxempty}}$ Busemann
geodesics from $(x,s)$. The set $\text{GNEQ}:=\\{t>s:g_{1}(t)\neq g_{2}(t)\\}$
is therefore nonempty and infinite by continuity of $g_{1}$ and $g_{2}$.
Assume, by way of contradiction, that GNEQ is not an open interval. By
continuity of geodesics, GNEQ cannot be a closed or half-closed interval, so
GNEQ is not path connected. Thus, there exists $t_{1}<t_{2}<t_{3}$ so that

$g_{1}(t_{1})\neq g_{2}(t_{1}),\quad g_{1}(t_{2})=g_{2}(t_{2}),\quad\text{
and}\quad g_{1}(t_{3})\neq g_{2}(t_{3}).$

The geodesics $g_{1}|_{[t_{1},\infty)}$ and $g_{2}|_{[t_{1},\infty)}$ started
from $(g_{1}(t_{1}),t_{1})$ and $(g_{2}(t_{1}),t_{1})$, respectively, are both
Busemann geodesics by their construction in Theorem 4.4.9. Since the geodesics
$g_{1}|_{[t_{1},\infty)}$ and $g_{2}|_{[t_{1},\infty)}$ start at different
spatial locations (namely $g_{1}(t_{1})$ and $g_{2}(t_{1})$) along the same
time level $t_{1}$, they cannot intersect at either of their starting points.
By Item (i), the two geodesics $g_{1}|_{[t_{1},\infty)}$ and
$g_{2}|_{[t_{1},\infty)}$ must coalesce, and the first point of intersection
is the coalescence point. Since $g_{1}(t_{2})=g_{2}(t_{2})$, this implies that
$g_{1}(t)=g_{2}(t)$ for all $t>t_{2}$, a contradiction to the existence of
$t_{3}$.

Item (iii) (Uniformity of coalescence): Let $\xi\in\mathbb{R}$,
${\scaleobj{0.9}{\boxempty}}\in\\{-,+\\}$, and let the compact set $K$ be
given. Let $S$ be the smallest integer greater than $\max\\{s:(x,s)\in K\\}$.
Set

$A:=\inf\\{g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(S):(x,s)\in
K\\},\qquad\text{and}\qquad
B:=\sup\\{g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(S):(x,s)\in K\\}.$

By Lemma 4.4.13, $-\infty<A\leq B<\infty$. Then, by Theorem 4.5.5(iv),
whenever $g$ is a $\xi{\scaleobj{0.8}{\boxempty}}$ geodesic starting from
$(x,s)\in K$,

$g_{(A,S)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(t)\leq g(t)\leq
g_{(B,S)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(t)\qquad\text{for all }t\geq S.$

To complete the proof, let $T$ be the time at which
$g_{(A,S)}^{\xi{\scaleobj{0.8}{\boxempty}},L}$ and
$g_{(B,S)}^{\xi{\scaleobj{0.8}{\boxempty}},R}$ coalesce, which is guaranteed
to be finite by Item (i). ∎

For two initial points on a horizontal level, as $\xi$ varies, a constant
Busemann process corresponds to a constant coalescence point of the geodesics.
The non-uniqueness of geodesics requires us to be careful about the choice of
left and right geodesic.

######  Definition 4.6.7.

For $s\in\mathbb{R}$ and $x<y$, let
$\mathbf{z}^{\xi{\scaleobj{0.8}{\boxempty}}}(y,s;x,s)$ be the coalescence
point of $g_{(y,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}$ and
$g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}$.

######  Theorem 4.6.8.

On a single event of probability one, for all reals $\alpha<\beta$, $s$, and
$x<y$, the following are equivalent.

  1. (i)

$W_{\alpha+}(y,s;x,s)=W_{\beta-}(y,s;x,s)$.

  2. (ii)

$\mathbf{z}^{\alpha+}(y,s;x,s)=\mathbf{z}^{\beta-}(y,s;x,s)$.

  3. (iii)

There exist $t>s$ and $z\in\mathbb{R}$ such that there are paths
$g_{1}:[s,t]\to\mathbb{R}$ (connecting $(x,s)$ and $(z,t)$) and
$g_{2}:[s,t]\to\mathbb{R}$ (connecting $(y,s)$ to $(z,t)$) such that for all
$\xi\in(\alpha,\beta)$, ${\scaleobj{0.9}{\boxempty}}\in\\{-,+\\}$, and
$u\in[s,t)$,

$\displaystyle g_{1}(u)$
$\displaystyle=g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},R}(u)=g_{(x,s)}^{\alpha+,R}(u)=g_{(x,s)}^{\beta-,R}(u)$
(4.61)
$\displaystyle<g_{2}(u)=g_{(y,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(u)=g_{(y,s)}^{\alpha+,L}(u)=g_{(y,s)}^{\beta-,L}(u).$

###### Proof.

(i)$\Rightarrow$(ii) follows from Lemma 4.6.6.

(ii)$\Rightarrow$(i): Assume
$(z,t):=\mathbf{z}^{\alpha+}(y,s;x,s)=\mathbf{z}^{\beta-}(y,s;x,s)$.
By additivity (Theorem 4.4.1(ii)) and Theorem 4.4.9(iii),

$\displaystyle W_{\alpha+}(y,s;x,s)$
$\displaystyle=W_{\alpha+}(y,s;z,t)-W_{\alpha+}(x,s;z,t)$
$\displaystyle=\mathcal{L}(y,s;z,t)-\mathcal{L}(x,s;z,t)$
$\displaystyle=W_{\beta-}(y,s;z,t)-W_{\beta-}(x,s;z,t)=W_{\beta-}(y,s;x,s).$

(ii)$\Rightarrow$(iii): Let $(z,t)$ be as in the proof of
(ii)$\Rightarrow$(i). By Theorem 4.4.9(iv), the restriction of
$g_{(x,s)}^{\alpha+,R}$ and $g_{(x,s)}^{\beta-,R}$ to the domain $[s,t]$ are
both rightmost geodesics between $(x,s)$ and $(z,t)$, and therefore they agree
on this restricted domain. Similarly, $g_{(y,s)}^{\alpha+,L}$ and
$g_{(y,s)}^{\beta-,L}$ agree on the domain $[s,t]$. By the monotonicity of
Theorem 4.5.5(i), and since $(z,t)$ is the common coalescence point, (4.61)
holds for $u\in[s,t)$, as desired.

(iii)$\Rightarrow$(ii) is immediate. ∎

######  Theorem 4.6.9.

On a single event of probability one, for all reals $s,\xi\in\mathbb{R}$, and
$x<y$, the following are equivalent.

  1. (i)

$W_{\xi-}(y,s;x,s)=W_{\xi+}(y,s;x,s).$

  2. (ii)

$\mathbf{z}^{\xi-}(y,s;x,s)=\mathbf{z}^{\xi+}(y,s;x,s)$.

  3. (iii)

$g_{(x,s)}^{\xi-,R}(t)=g_{(y,s)}^{\xi+,L}(t)$ for some $t>s$, i.e., the paths
$g_{(x,s)}^{\xi-,R}$ and $g_{(y,s)}^{\xi+,L}$ intersect.

######  Remark 4.6.10.

In Item (iii), if $\xi\in\Xi$, then despite intersecting, the geodesics
$g_{(x,s)}^{\xi-,R}$ and $g_{(y,s)}^{\xi+,L}$ cannot coalesce. This follows
from Theorem 4.6.3, which gives a full classification of the directions in
which all semi-infinite geodesics coalesce.

###### Proof of Theorem 4.6.9.

(i)$\Rightarrow$(ii): If $W_{\xi-}(y,s;x,s)=W_{\xi+}(y,s;x,s)$, then
Theorem 4.4.1(v) implies that for some $\alpha<\xi<\beta$,
$W_{\alpha+}(y,s;x,s)=W_{\beta-}(y,s;x,s)$. Then, we apply
(i)$\Rightarrow$(iii) of Theorem 4.6.8 to conclude that for some $t>s$ and
$z\in\mathbb{R}$,

$g_{(x,s)}^{\xi-,R}(u)=g_{(x,s)}^{\xi+,R}(u)<g_{(y,s)}^{\xi-,L}(u)=g_{(y,s)}^{\xi+,L}(u),\qquad\text{for
}u\in[s,t),$

whereas for $u=t$, all terms above equal some common value $z$. Therefore,
$(z,t)=\mathbf{z}^{\xi-}(y,s;x,s)=\mathbf{z}^{\xi+}(y,s;x,s)$.

(ii)$\Rightarrow$(i): Similarly as in the proof of (ii)$\Rightarrow$(i) of
Theorem 4.6.8, if
$(z,t)=\mathbf{z}^{\xi-}(y,s;x,s)=\mathbf{z}^{\xi+}(y,s;x,s)$, then
$W_{\xi-}(y,s;x,s)=\mathcal{L}(y,s;z,t)-\mathcal{L}(x,s;z,t)=W_{\xi+}(y,s;x,s).$

(ii)$\Rightarrow$(iii): Assume
$(z,t)=\mathbf{z}^{\xi-}(y,s;x,s)=\mathbf{z}^{\xi+}(y,s;x,s)$. Then,
$g_{(x,s)}^{\xi-,R}(t)=z=g_{(y,s)}^{\xi+,L}(t)$.

(iii)$\Rightarrow$(ii): Assume that
$g_{(x,s)}^{\xi-,R}(t)=g_{(y,s)}^{\xi+,L}(t)$ for some $t>s$. Let $t$ be the
minimal such time, and let $(z,t)$ be the point where the geodesics first
intersect. By Theorem 4.5.5, Items (i) and (iv), for $u>s$,

$g_{(x,s)}^{\xi-,R}(u)\leq g_{(x,s)}^{\xi+,R}(u)\wedge
g_{(y,s)}^{\xi-,L}(u)\leq g_{(x,s)}^{\xi+,R}(u)\vee g_{(y,s)}^{\xi-,L}(u)\leq
g_{(y,s)}^{\xi+,L}(u).$ (4.62)

In particular, when $u=t$, all inequalities in (4.62) are equalities. Further,
since $g_{(x,s)}^{\xi-,R}$,$g_{(x,s)}^{\xi+,R}$ are rightmost geodesics
between $(x,s)$ and $(z,t)$ (Theorem 4.4.9(iv)),
$g_{(x,s)}^{\xi-,R}(u)=g_{(x,s)}^{\xi+,R}(u)$ for $u\in[s,t]$. Similarly,
$g_{(y,s)}^{\xi-,L}(u)=g_{(y,s)}^{\xi+,L}(u)$ for $u\in[s,t]$. Since $t$ was
chosen minimally for $g_{(x,s)}^{\xi-,R}(t)=g_{(y,s)}^{\xi+,L}(t)$, we have
$(z,t)=\mathbf{z}^{\xi-}(y,s;x,s)=\mathbf{z}^{\xi+}(y,s;x,s)$. ∎

###### Proof of Theorem 4.6.3 (Classification of directions).

(i)$\Rightarrow$(ii): If $\xi\notin\Xi$, then $W_{\xi-}=W_{\xi+}$, so (ii)
follows by the construction of the Busemann geodesics from the Busemann
functions.

(ii)$\Rightarrow$(iii): Since a geodesic in direction $\xi$ from $(x,s)$ must
pass through each horizontal level $t>s$, it is sufficient to show that, for
$s\in\mathbb{R}$ and $x<y$, whenever $g_{1}$ is a semi-infinite geodesic from
$(x,s)$ in direction $\xi$ and $g_{2}$ is a semi-infinite geodesic from
$(y,s)$ in direction $\xi$, $g_{1}$ and $g_{2}$ coalesce. Assuming (ii) and
using Theorem 4.5.7(i), for all $t>s$,

$g_{(x,s)}^{\xi+,L}(t)=g_{(x,s)}^{\xi-,L}(t)\leq g_{1}(t)\wedge g_{2}(t)\leq
g_{1}(t)\vee g_{2}(t)\leq g_{(y,s)}^{\xi+,R}(t).$

By Theorem 4.6.1(i), $g_{(x,s)}^{\xi+,L}$ and $g_{(y,s)}^{\xi+,R}$ coalesce,
so all inequalities above are equalities for large $t$, and $g_{1}$ and
$g_{2}$ coalesce.

(iii)$\Rightarrow$(i): We prove the contrapositive. If $\xi\in\Xi$, then by
Theorem 4.4.1(iii)-(iv),
$W_{\xi-}(y,s;x,s)<W_{\xi+}(y,s;x,s)$ for some $x<y$ and $s\in\mathbb{R}$. By
(i)$\Leftrightarrow$(iii) of Theorem 4.6.9,
$g_{(x,s)}^{\xi-,R}(t)<g_{(y,s)}^{\xi+,L}(t)$ for all $t>s$. In particular,
$g_{(x,s)}^{\xi-,R}$ and $g_{(y,s)}^{\xi+,L}$ do not coalesce.

(ii)$\Rightarrow$(iv): By definition of $\operatorname{NU}$, whenever
$p\notin\operatorname{NU}$,
$g_{p}^{\xi{\scaleobj{0.8}{\boxempty}},L}=g_{p}^{\xi{\scaleobj{0.8}{\boxempty}},R}$
for $\xi\in\mathbb{R}$ and ${\scaleobj{0.9}{\boxempty}}\in\\{-,+\\}$. Hence,
assuming $p\notin\operatorname{NU}$ and $g_{p}^{\xi-,R}=g_{p}^{\xi+,R}$, we
also have $g_{p}^{\xi-,L}=g_{p}^{\xi+,R}$, so there is a unique geodesic from
$p$ in direction $\xi$ by Theorem 4.5.7(i).

(iv)$\Rightarrow$(v): By Theorem 4.5.2(i), on the event $\Omega_{2}$,
$\operatorname{NU}$ contains no points of $\mathbb{Q}^{2}$, and therefore,
$\operatorname{NU}$ is not all of $\mathbb{R}^{2}$.

(v)$\Rightarrow$(vi) and (v)$\Rightarrow$(vii) are direct consequences of
Theorem 4.5.7(i): If there is a unique semi-infinite geodesic in direction
$\xi$ from a point $p\in\mathbb{R}^{2}$, then
$g_{p}^{\xi-,L}=g_{p}^{\xi+,L}=g_{p}^{\xi-,R}=g_{p}^{\xi+,R}$.

(vi)$\Rightarrow$(ii): Let $p$ be a point from which
$g_{p}^{\xi-,L}=g_{p}^{\xi+,L}$, and call this common geodesic $g$. Let $q$ be
an arbitrary point in $\mathbb{R}^{2}$. By Theorem 4.6.1(i),
$g_{q}^{\xi-,L},g_{q}^{\xi+,L},g_{q}^{\xi-,R}$, and $g_{q}^{\xi+,R}$ each
coalesce with $g$, so $g_{q}^{\xi-,L}$ and $g_{q}^{\xi+,L}$ coalesce. Since
both geodesics are the leftmost geodesics between their points by Theorem
4.4.9(iv), they must be the same. Similarly, $g_{q}^{\xi-,R}=g_{q}^{\xi+,R}$.

(vii)$\Rightarrow$(ii): follows by the same proof.

Item (viii): Let $\xi\in\mathbb{R}\setminus\Xi$, and let $g$ be a semi-
infinite geodesic in direction $\xi$, starting from a point
$(x,s)\in\mathbb{R}^{2}$. By Lemma 4.4.14 and Theorem 4.4.9(ii), it is
sufficient to show that for sufficiently large $t$,

$\mathcal{L}(x,s;g(t),t)=W_{\xi}(x,s;g(t),t).$ (4.63)

(we dropped the $\pm$ distinction since $W_{\xi-}=W_{\xi+}$). By Item (iii),
$g$ coalesces with $g_{(x,s)}^{\xi,R}$. Then, for sufficiently large $t$,
$g(t)=g_{(x,s)}^{\xi,R}(t)$ and by Theorem 4.4.9(iii), (4.63) holds. ∎

###  4.7 Remaining proofs from Section 4.4 and the proofs of the main theorems

In this section, we complete the unfinished business of the chapter. In
Section 4.7.1, we start by proving Items (vi)–(viii) of Theorem 4.4.1 and
mixing in Theorem 4.4.3(ii). Then, we prove the first main theorem of the
chapter, namely Theorem 4.2.1. In Section 4.7.2, we prove Theorems 4.4.5(ii)
and 4.2.6, along with some necessary lemmas.

####  4.7.1 Proof of Theorem 4.2.1 and related results

###### Proof of Items (vi)–(viii) of Theorem 4.4.1 and mixing in Theorem
4.4.3(ii).

We
continue to work on $\Omega_{2}$.

Item (vi) of Theorem 4.4.1 (Busemann limits I): By Theorem 4.6.3(viii), if
$\xi\notin\Xi$, all $\xi$-directed semi-infinite geodesics are Busemann
geodesics, and they all coalesce. By Theorem 4.6.1(iii), there exists a level
$T$ such that all geodesics from points starting in the compact set $K$ have
coalesced by time $T$. Let $(Z,T)$ denote the location of the point of the
common geodesics at time $T$. Let $r_{t}=(z_{t},u_{t})_{t\in\mathbb{R}_{\geq
0}}$ be any net with $u_{t}\to\infty$ and $z_{t}/u_{t}\to\xi$. By Theorem
4.5.7(ii), for all sufficiently large $t$ and $p\in K$, all geodesics from $p$
to $r_{t}$ pass through $(Z,T)$. Then, for $p,q\in K$,

$\mathcal{L}(p;r_{t})-\mathcal{L}(q;r_{t})=\mathcal{L}(p;Z,T)+\mathcal{L}(Z,T;r_{t})-(\mathcal{L}(q;Z,T)+\mathcal{L}(Z,T;r_{t})).$

By Theorems 4.4.9(ii)(ii)(b) and 4.4.1(ii), the right-hand side is equal to

$W_{\xi}(p;Z,T)-W_{\xi}(q;Z,T)=W_{\xi}(p;q).$

Item (vii) of Theorem 4.4.1 (Busemann limits II): By Theorem 4.4.5(iii), on
the event $\Omega_{2}$, $\Xi$ contains no rational directions. Then, for
arbitrary $\xi\in\mathbb{R}$, $s\in\mathbb{R}$, $x<y\in\mathbb{R}$,
$\alpha,\beta\in\mathbb{Q}$ with $\alpha<\xi<\beta$, and a net $(z_{r},u_{r})$
with $u_{r}\to\infty$ and $z_{r}/u_{r}\to\xi$, for sufficiently large $r$,
$\alpha u_{r}<z_{r}<\beta u_{r}$. Theorem 4.4.1(vi) gives the existence of the
limits in the first and last lines below, while the monotonicity of Lemma
B.1.5(i) justifies the first and last inequalities:

$\displaystyle W_{\alpha}(y,s;x,s)$
$\displaystyle=\lim_{r\to\infty}\mathcal{L}(y,s;\alpha
u_{r},u_{r})-\mathcal{L}(x,s;\alpha u_{r},u_{r})$
$\displaystyle\leq\liminf_{r\to\infty}\mathcal{L}(y,s;z_{r},u_{r})-\mathcal{L}(x,s;z_{r},u_{r})$
$\displaystyle\leq\limsup_{r\to\infty}\mathcal{L}(y,s;z_{r},u_{r})-\mathcal{L}(x,s;z_{r},u_{r})$
$\displaystyle\leq\lim_{r\to\infty}\mathcal{L}(y,s;\beta
u_{r},u_{r})-\mathcal{L}(x,s;\beta u_{r},u_{r})=W_{\beta}(y,s;x,s).$

Sending $\mathbb{Q}\ni\alpha\nearrow\xi$ and $\mathbb{Q}\ni\beta\searrow\xi$
and using Item (v) completes the proof.

Item (viii) of Theorem 4.4.1 (Global attractiveness): We follow a similar
proof to the attractiveness in Theorem 3.2.2. Let $\xi\notin\Xi$ and assume
$\mathfrak{h}\in\operatorname{UC}$ is a function satisfying the drift
condition (3.7). Recall that we define

$h_{s,t}(x)=\sup_{z\in\mathbb{R}}\\{\mathcal{L}(x,s;z,t)+\mathfrak{h}(z)\\}.$
(4.64)

For $a>0$ and $s<t$, Theorems 4.4.1(v) and 4.4.5(iii) allows us to choose
$\varepsilon=\varepsilon(\xi)>0$ small enough so that $\xi\pm
2\varepsilon\in\mathbb{Q}$ (and thus $\xi\pm 2\varepsilon\notin\Xi$), and so
for all $x\in[-a,a]$,

$W_{\xi\pm 2\varepsilon}(x,s;0,s)=W_{\xi}(x,s;0,s).$ (4.65)

By Theorem 4.6.1(iii), there exists a random $T=T(a,\xi\pm\varepsilon)$ such
that all $\xi-2\varepsilon$ Busemann geodesics have coalesced by time $T$ and
all $\xi+2\varepsilon$ Busemann geodesics have coalesced by time $T$. For
$t>T$, let $g^{\xi\pm 2\varepsilon}(t)$ be locations of these two common
geodesics at time $t$. By Theorem 4.4.9(ii)(ii)(d), $g^{\xi\pm
2\varepsilon}(t)/t\to\xi\pm 2\varepsilon$. By Equation (3.21) in Lemma 3.5.1
applied to the temporally reflected version of $\mathcal{L}$, there exists
$t_{0}(a,\varepsilon(\xi),s)$ so that for $t>t_{0}$, whenever $x\in[-a,a]$ and
$z$ is a maximizer in (4.64),
$g^{\xi-2\varepsilon}(t)<z<g^{\xi+2\varepsilon}(t).$ Then, by Lemma
B.1.5(iii), for such large $t$,

$W_{\xi-2\varepsilon}(x,s;0,s)\leq h_{s,t}(x)-h_{s,t}(0)\leq
W_{\xi+2\varepsilon}(x,s;0,s),$

while for $-a\leq x\leq 0$, the equalities reverse. Combined with (4.65), this
completes the proof.

Item (ii) of Theorem 4.4.3 (Mixing): This proof follows a similar idea as that
in Lemma 7.5 of [11], and the key is that, within a compact set, the Busemann
functions are equal to differences of the directed landscape for large enough
$t$. Then, we use Lemma B.1.4, which states that, as a projection of
$\\{\mathcal{L},W\\}$, the directed landscape $\mathcal{L}$ is mixing under
the transformation $T_{z;a,b}$. Set $r_{z}=(az,bz)$. By a standard
$\pi-\lambda$ argument, it suffices to show that for
$\xi_{1},\ldots\xi_{k}\in\mathbb{R}$ (ignoring the sign
${\scaleobj{0.9}{\boxempty}}$ since $\xi_{i}\notin\Xi$ a.s.), all compact sets
$K:=K_{1}\times
K_{2}^{k}\subseteq\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}\times(\mathbb{R}^{4})^{k}$,
and all Borel sets $A,B\in C(K,\mathbb{R})$,

$\displaystyle\lim_{z\to\infty}\mathbb{P}\Bigl{(}\\{\mathcal{L},W_{\xi_{1:k}}\\}\big{|}_{K}\in
A,\\{T_{z;a,b}\mathcal{L},T_{z;a,b}W_{\xi_{1:k}}\\}\big{|}_{K}\in B\Bigr{)}$
$\displaystyle\qquad\qquad=\mathbb{P}\bigl{(}\\{\mathcal{L},W_{\xi_{1:k}}\\}\big{|}_{K}\in
A\bigr{)}\mathbb{P}\bigl{(}\\{\mathcal{L},W_{\xi_{1:k}}\\}\big{|}_{K}\in
B\bigr{)},$

where we use the shorthand notation

$\\{\mathcal{L},W_{\xi_{1:k}}\\}\big{|}_{K}:=\\{\mathcal{L}(v),W_{\xi_{i}}(p;q):1\leq
i\leq k,(v,p,q)\in K\\},$

and $T_{z;a,b}$ acts on $\mathcal{L}$ and $W$ as projections of
$\\{\mathcal{L},W\\}$. By Theorem 4.4.1(vi), we may choose $t>0$ sufficiently
large so that

$\mathbb{P}(W_{\xi_{i}}(p;q)=\mathcal{L}(p;(t\xi,t))-\mathcal{L}(q;(t\xi,t))\;\forall(p,q)\in
K_{2},1\leq i\leq k)\geq 1-\varepsilon.$ (4.66)

By stationarity of the process under space-time shifts, we also have that for
such large $t$ and all $z\in\mathbb{R}$,

$\mathbb{P}(T_{z;a,b}W_{\xi_{i}}(p;q)=T_{z;a,b}[\mathcal{L}(p;(t\xi,t))-\mathcal{L}(q;(t\xi,t))]\;\forall(p,q)\in
K_{2},1\leq i\leq k)\geq 1-\varepsilon$ (4.67)

Let $C_{z,t}$ be the intersection of the events in (4.66) over $1\leq i\leq k$
with the event (4.67). Then for large enough $t$, $\mathbb{P}(C_{z,t})\geq
1-2\varepsilon$, and

$\displaystyle\Bigl{|}\mathbb{P}\Bigl{(}\\{\mathcal{L},W_{\xi_{1:k}}\\}|_{K}\in
A,\\{T_{z;a,b}\mathcal{L},T_{z;a,b}W_{\xi_{1:k}}\\}|_{K}\in B\Bigr{)}$
$\displaystyle\qquad-\mathbb{P}\Bigl{(}\\{\mathcal{L},W_{\xi_{1:k}}\\}|_{K}\in
A\Bigr{)}\mathbb{P}\Bigl{(}\\{\mathcal{L},W_{\xi_{1:k}}\\}|_{K}\in
B\Bigr{)}\Bigr{|}$
$\displaystyle\leq\Bigl{|}\mathbb{P}\Bigl{(}\\{\mathcal{L},W_{\xi_{1:k}}\\}|_{K}\in
A,\\{T_{z;a,b}\mathcal{L},T_{z;a,b}W_{\xi_{1:k}}\\}|_{K}\in B,C_{z,t}\Bigr{)}$
$\displaystyle\qquad-\mathbb{P}\Bigl{(}\\{\mathcal{L},W_{\xi_{1:k}}\\}|_{K}\in
A,C_{z,t}\Bigr{)}\mathbb{P}\Bigl{(}\\{\mathcal{L},W_{\xi_{1:k}}\\}|_{K}\in
B,C_{z,t}\Bigr{)}\Bigr{|}+C\varepsilon$
$\displaystyle=\Bigl{|}\mathbb{P}\Bigl{(}\\{\mathcal{L}(v),\mathcal{L}(p;(t\xi_{1:k},t))-\mathcal{L}(q;(t\xi_{1:k},t))\\}|_{K}\in
A,$
$\displaystyle\qquad\qquad\\{T_{z;a,b}\mathcal{L}(v),T_{z;a,b}[\mathcal{L}(p;(t\xi_{1:k},t))-\mathcal{L}(q;(t\xi_{1:k},t))]\\}|_{K}\in
B,C_{z,t}\Bigr{)}$
$\displaystyle\qquad-\mathbb{P}\Bigl{(}\\{\mathcal{L}(v),\mathcal{L}(p;(t\xi_{1:k},t))-\mathcal{L}(q;(t\xi_{1:k},t))\\}|_{K}\in
A,C_{z,t}\Bigr{)}$
$\displaystyle\qquad\qquad\times\mathbb{P}\Bigl{(}\\{\mathcal{L}(v),\mathcal{L}(p;(t\xi_{1:k},t))-\mathcal{L}(q;(t\xi_{1:k},t))\\}|_{K}\in
B,C_{z,t}\Bigr{)}\Bigr{|}+C\varepsilon$
$\displaystyle\leq\Bigl{|}\mathbb{P}\Bigl{(}\\{\mathcal{L}(v),\mathcal{L}(p;(t\xi_{1:k},t))-\mathcal{L}(q;(t\xi_{1:k},t))\\}|_{K}\in
A,$
$\displaystyle\qquad\qquad\\{T_{z;a,b}\mathcal{L}(v),T_{z;a,b}[\mathcal{L}(p;(t\xi_{1:k},t))-\mathcal{L}(q;(t\xi_{1:k},t))]\\}|_{K}\in
B\Bigr{)}$
$\displaystyle\qquad-\mathbb{P}\Bigl{(}\\{\mathcal{L}(v),\mathcal{L}(p;(t\xi_{1:k},t))-\mathcal{L}(q;(t\xi_{1:k},t))\\}|_{K}\in
A\Bigr{)}$
$\displaystyle\qquad\qquad\times\mathbb{P}\Bigl{(}\\{\mathcal{L}(v),\mathcal{L}(p;(t\xi_{1:k},t))-\mathcal{L}(q;(t\xi_{1:k},t))\\}|_{K}\in
B\Bigr{)}\Bigr{|}+C^{\prime}\varepsilon,$

where the constants $C,C^{\prime}$ came as the cost of adding and removing the
high probability event $C_{z,t}$. The proof is complete by sending
$z\to\infty$ and using the mixing of $\mathcal{L}$ under the shift $T_{z;a,b}$
(Lemma B.1.4). ∎

###### Proof of Theorem 4.2.1.

Item (i) (All geodesics have a direction): First, we show that, on
$\Omega_{2}$, if $g$ is a semi-infinite geodesic starting from $(x,s)$, then

$-\infty<\liminf_{t\to\infty}t^{-1}{g(t)}\leq\limsup_{t\to\infty}t^{-1}{g(t)}<\infty.$
(4.68)

We show the rightmost inequality, the leftmost being analogous. Assume, by way
of contradiction, that $\limsup_{t\to\infty}g(t)/t=\infty$. By the
directedness of Theorem 4.4.9(iii), $\forall\xi\in\mathbb{R}$ there exists an
infinite sequence $t_{i}\to\infty$ such that
$g(t_{i})>g_{(x,s)}^{\xi+,L}(t_{i})$ for all $i$. Since $g_{(x,s)}^{\xi+,L}$
is the leftmost geodesic between any two of its points (Theorem 4.4.9(iv)), we
must have $g(t)\geq g_{(x,s)}^{\xi+,L}(t)$ $\forall\xi\in\mathbb{R}$ and
$t\in\mathbb{R}$. By Theorem 4.5.5(iii), $g(t)=\infty$ $\forall t>s$, a
contradiction.

Having established (4.68), assume by way of contradiction that

$\liminf_{t\to\infty}t^{-1}{g(t)}<\limsup_{t\to\infty}t^{-1}{g(t)}.$

Choose some $\xi$ strictly between the two values above. By the directedness
of Theorem 4.4.9(iii), there exists a sequence $t_{i}\to\infty$ such that
$g_{(x,s)}^{\xi+,R}(t_{i})<g(t_{i})$ for $i$ even and
$g_{(x,s)}^{\xi+,R}(t_{i})>g(t_{i})$ for $i$ odd. This cannot occur since
$g_{(x,s)}^{\xi+,R}$ is the rightmost geodesic between any two of its points.

By Theorem 4.4.9(iii), for each $\xi\in\mathbb{R}$ and
$(x,s)\in\mathbb{R}^{2}$, $g_{(x,s)}^{\xi+,R}$, for example, is a semi-
infinite geodesic from $(x,s)$ in direction $\xi$, justifying the claim that
there is at least one semi-infinite geodesic from each point and in every
direction.

Item (ii) (Coalescence): The first statement follows from the equivalences
(i)$\Leftrightarrow$(iii)$\Leftrightarrow$(iv) of Theorem 4.6.3. By Theorem
4.5.2(i), $\mathbb{P}(p\in\operatorname{NU})=0$ $\forall p\in\mathbb{R}^{2}$.
This and Tonelli’s theorem imply that the expected Lebesgue measure of the set
$\operatorname{NU}$ is

$\mathbb{E}\int_{(x,s)\in\mathbb{R}^{2}}\mathbf{1}(p\in\operatorname{NU})\,dx\,ds=\int_{(x,s)\in\mathbb{R}^{2}}\mathbb{P}(p\in\operatorname{NU})\,dx\,ds=0,$

so $\operatorname{NU}$ almost surely has planar Lebesgue measure zero.

Item (iii) (Non-uniqueness in exceptional directions): This follows from
Remark 4.6.4. ∎

####  4.7.2 Proof of Theorem 4.2.6 and related results.

Recall the functions $f_{s,\xi}(x)=W_{\xi+}(x,s;0,s)-W_{\xi-}(x,s;0,s)$
defined in (4.8) and the sets $\mathfrak{S}_{s,\xi}$ from (4.1):

$\displaystyle\mathfrak{S}_{s,\xi}$
$\displaystyle:=\\{x\in\mathbb{R}:\exists\text{ {disjoint}}\text{ }\text{semi-
infinite geodesics from }(x,s)\text{ in direction }\xi\\}$ (4.69)
$\displaystyle\mathfrak{S}$ $\displaystyle:=\bigcup_{s\hskip 0.9pt\in\hskip
0.9pt\mathbb{R},\,\xi\hskip 0.9pt\in\hskip
0.9pt\Xi}\mathfrak{S}_{s,\xi}\times\\{s\\}.$

######  Theorem 4.7.1.

On a single event of full probability, the function $f_{s,\xi}$ is
nondecreasing simultaneously for all $s\in\mathbb{R}$ and $\xi\in\Xi$. Denote
the set of local variation of $f_{s,\xi}$ by

$\mathcal{D}_{s,\xi}=\\{x\in\mathbb{R}:f_{s,\xi}(x-\varepsilon)<f_{s,\xi}(x+\varepsilon)\;\forall\varepsilon>0\\}.$
(4.70)

Then, on a single event of full probability, simultaneously for each
$s\in\mathbb{R}$ and $\xi\in\Xi$,

$\mathcal{D}_{s,\xi}=\mathfrak{S}_{s,\xi}^{L}\cup\mathfrak{S}_{s,\xi}^{R}\;\subseteq\;\mathfrak{S}_{s,\xi},$
(4.71)

where for $S\in\\{L,R\\}$,

$\mathfrak{S}_{s,\xi}^{S}:=\\{x\in\mathbb{R}:g_{(x,s)}^{\xi-,S}\text{ and
}g_{(x,s)}^{\xi+,S}\text{ are disjoint}\\}.$ (4.72)

Furthermore, $(\mathfrak{S}_{s,\xi}\setminus\mathcal{D}_{s,\xi})\times\\{s\\}$
is contained in the countable set $\operatorname{NU}^{\xi-}\cap\hskip
0.9pt\operatorname{NU}^{\xi+}\cap\,\mathcal{H}_{s}$.

######  Remark 4.7.2.

Presently, we do not know if $\mathcal{D}_{s,\xi}$ equals
$\mathfrak{S}_{s,\xi}$. Since
$\operatorname{NU}^{\xi-}\cap\operatorname{NU}^{\xi+}\subseteq\operatorname{NU}$
and $\operatorname{NU}\cap\,\mathcal{H}_{s}$ is at most countable (Theorem
4.5.2(ii)), $\mathfrak{S}_{s,\xi}$ and $\mathcal{D}_{s,\xi}$ have the same
Hausdorff dimension for all $s\in\mathbb{R}$ and $\xi\in\Xi$.

######  Remark 4.7.3.

In contrast with $\mathfrak{S}$ in (4.69), the sets $\mathfrak{S}^{S}$ are
concerned only with leftmost ($S=L$) and rightmost ($S=R$) Busemann geodesics.
In BLPP, the analogues of $\mathfrak{S}^{L}$ and $\mathfrak{S}^{R}$ are both
equal to the set of initial points from which some geodesic travels initially
vertically (Theorems 2.10 and 4.30 in [135]). Furthermore, in BLPP, the
analogue of this set contains $\operatorname{NU}$. We do not presently know
whether either is true in DL.

###### Proof of Theorem 4.7.1.

The full-probability event is $\Omega_{2}$ in (4.28). The monotonicity of the
function $f_{s,\xi}$ follows from (4.37). We now prove that
$\mathcal{D}_{s,\xi}=\mathfrak{S}_{s,\xi}^{L}\cup\mathfrak{S}_{s,\xi}^{R}$.
Assume that $y\notin\mathcal{D}_{s,\xi}$. Then, there exist $a<y<b$ such that
$f_{s,\xi}$ is constant on $[a,b]$. Hence, for $a\leq x<y$,

$W_{\xi+}(x,s;0,s)-W_{\xi-}(x,s;0,s)=W_{\xi+}(y,s;0,s)-W_{\xi-}(y,s;0,s),$

and by additivity (Theorem 4.4.1(ii)), $W_{\xi-}(y,s;x,s)=W_{\xi+}(y,s;x,s)$.
Choose $t>s$ sufficiently small so that
$g_{(x,s)}^{\xi+,R}(t)<g_{(y,s)}^{\xi-,L}(t)$. By Lemma 4.6.5,
$g_{(y,s)}^{\xi-,L}(u)=g_{(y,s)}^{\xi+,L}(u)$ for $u\in[s,t]$. By a symmetric
argument, instead choosing a point $x>y$, $g_{(y,s)}^{\xi-,R}$ and
$g_{(y,s)}^{\xi+,R}$ agree near the starting point $(y,s)$. Hence,
$y\notin\mathfrak{S}_{s,\xi}^{L}\cup\mathfrak{S}_{s,\xi}^{R}$.

Next, assume that $y\in\mathcal{D}_{s,\xi}$. Then, for all $x<y<z$,

$W_{\xi+}(x,s;0,s)-W_{\xi-}(x,s;0,s)<W_{\xi+}(z,s;0,s)-W_{\xi-}(z,s;0,s),$

Since the difference is a monotone function, either (i)
$W_{\xi-}(y,s;x,s)<W_{\xi+}(y,s;x,s)$ for all $x<y$ or (ii)
$W_{\xi-}(z,s;y,s)<W_{\xi+}(z,s;y,s)$ for all $z>y.$

We show that $g_{(y,s)}^{\xi-,L}$ and $g_{(y,s)}^{\xi+,L}$ are disjoint in the
first case. A symmetric proof shows that $g_{(y,s)}^{\xi-,R}$ and
$g_{(y,s)}^{\xi+,R}$ are disjoint in the second case. So assume
$W_{\xi-}(y,s;x,s)<W_{\xi+}(y,s;x,s)$ for all $x<y$. Sending $x\nearrow y$,
$g_{(x,s)}^{\xi-,R}$ converges to $g_{(y,s)}^{\xi-,L}$ by Theorem 4.5.5(v).
Assume, by way of contradiction, that
$g_{(y,s)}^{\xi-,L}(u)=g_{(y,s)}^{\xi+,L}(u)$ for some $u>s$. This implies
then $g_{(y,s)}^{\xi-,L}(t)=g_{(y,s)}^{\xi+,L}(t)$ for all $t\in[s,u]$ since
both paths are the leftmost geodesic between any two of their points (Theorem
4.4.9(iv)). For $t\geq s$, the convergence $g_{(x,s)}^{\xi-,R}(t)\to
g_{(y,s)}^{\xi-,L}(t)$ is monotone by Theorem 4.5.5(iv). Since geodesics are
continuous paths, Dini’s theorem implies that, as $x\nearrow y$,
$g_{(x,s)}^{\xi-,R}(t)$ converges to
$g_{(y,s)}^{\xi-,L}(t)=g_{(y,s)}^{\xi+,L}(t)$ uniformly in $t\in[s,u]$. Lemma
B.2.1 implies that, for sufficiently close $x<y$, $g_{(x,s)}^{\xi-,R}$ and
$g_{(y,s)}^{\xi+,L}$ are not disjoint. This contradicts
(i)$\Leftrightarrow$(iii) of Theorem 4.6.9 since we assumed
$W_{\xi-}(y,s;x,s)<W_{\xi+}(y,s;x,s)$ for all $x<y$.

Lastly, we show that
$(\mathfrak{S}_{s,\xi}\setminus\mathcal{D}_{s,\xi})\times\\{s\\}\subseteq\operatorname{NU}^{\xi-}\cap\operatorname{NU}^{\xi+}\cap\,\mathcal{H}_{s}$.
Let $x\in\mathfrak{S}_{s,\xi}\setminus\mathcal{D}_{s,\xi}$. By Theorem
4.5.7(i), $g_{(x,s)}^{\xi-,L}$ is the leftmost $\xi$-directed geodesic from
$(x,s)$, and $g_{(x,s)}^{\xi+,R}$ is the rightmost. Since
$x\in\mathfrak{S}_{s,\xi}$, these two geodesics must be disjoint. Since
$x\notin\mathcal{D}_{s,\xi}$, $g_{(x,s)}^{\xi-,L}$ and $g_{(x,s)}^{\xi+,L}$
are not disjoint, and $g_{(x,s)}^{\xi-,R}$ and $g_{(x,s)}^{\xi+,R}$ are not
disjoint. Since the leftmost/rightmost semi-infinite geodesics are
leftmost/rightmost geodesics between their points (Theorem 4.4.9(iv)), there
exists $\varepsilon>0$ such that for $t\in(s,s+\varepsilon)$,

$g_{(x,s)}^{\xi-,L}(t)=g_{(x,s)}^{\xi+,L}(t)<g_{(x,s)}^{\xi-,R}(t)=g_{(x,s)}^{\xi+,R}(t),$

and therefore,
$(x,s)\in\operatorname{NU}^{\xi-}\cap\operatorname{NU}^{\xi+}\cap\,\mathcal{H}_{s}$.
Recall this set is at most countable by Theorem 4.5.2(ii). ∎

######  Lemma 4.7.4.

Given $\omega\in\Omega_{2}$ and
$(x,s;y,u)\in\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}$, let
$g:[s,u]\to\mathbb{R}$ be the leftmost (resp. rightmost) geodesic between
$(x,s)$ and $(y,u)$. Then, $(g(t),t)\in\mathfrak{S}^{L}$ ${\rm(}\text{resp.\
}\mathfrak{S}^{R}{\rm)}$ for some $t\in[s,u)$. Furthermore, among the
directions $\xi$ for which $g_{(x,s)}^{\xi-,L}$ and $g_{(x,s)}^{\xi+,L}$
separate at some $t\in[s,u)$, there is a unique direction $\widehat{\xi}$ such
that

$g_{(x,s)}^{\widehat{\xi}-,L}(u)\leq y<g_{(x,s)}^{\widehat{\xi}+,L}(u).$

The same holds with $L$ replaced by $R$ and the strict and weak inequalities
swapped.

###### Proof.

We prove the statement for leftmost geodesics. The proof for rightmost
geodesics is analogous. Set

$\widehat{\xi}:=\sup\\{\xi\in\mathbb{R}:g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(u)\leq
y\\}=\inf\\{\xi\in\mathbb{R}:g_{(x,s)}^{\xi{\scaleobj{0.8}{\boxempty}},L}(u)>y\\}.$
(4.73)

The monotonicity of Theorem 4.5.5(i) guarantees that the second equality
holds, and that the definition is independent of the choice of
${\scaleobj{0.9}{\boxempty}}\in\\{-,+\\}$. Theorem 4.5.5(iii) guarantees that
$\widehat{\xi}\in\mathbb{R}$. By definition of $\widehat{\xi}$ and the
monotonicity of Theorem 4.5.5(i),
$g_{(x,s)}^{\alpha{\scaleobj{0.8}{\boxempty}},L}(u)\leq
y=g(u)<g_{(x,s)}^{\beta{\scaleobj{0.8}{\boxempty}},L}(u)$ whenever
$\alpha<\widehat{\xi}<\beta$ and ${\scaleobj{0.9}{\boxempty}}\in\\{-,+\\}$.
But by Theorem 4.5.5(ii), the $\beta{\scaleobj{0.9}{\boxempty}}$ and
$\widehat{\xi}+$ geodesics agree locally when $\beta$ is close enough to
$\widehat{\xi}$. We can conclude that

$g_{(x,s)}^{\widehat{\xi}-,L}(u)\leq y=g(u)<g_{(x,s)}^{\widehat{\xi}+,L}(u).$
(4.74)

Since all three are leftmost geodesics (recall Theorem 4.4.9(iv)),

$g_{(x,s)}^{\widehat{\xi}-,L}(t)\leq g(t)\leq
g_{(x,s)}^{\widehat{\xi}+,L}(t)\qquad\text{for }t\in[s,u].$ (4.75)

By (4.74) the paths $g_{(x,s)}^{\widehat{\xi}-,L}$ and
$g_{(x,s)}^{\widehat{\xi}+,L}$ must separate at some time $t\in[s,u)$.
Furthermore, once $g_{(x,s)}^{\widehat{\xi}-,L}$ splits from
$g_{(x,s)}^{\widehat{\xi}+,L}$ at a point $(z_{1},t_{1})$, the geodesics must
stay apart. Otherwise, they would meet again at a point $(z_{2},t_{2})$, and
Theorem 4.4.9(iv) implies that both paths are the leftmost geodesic between
$(z_{1},t_{1})$ and $(z_{2},t_{2})$. See Figure 4.6. Set
$\hat{t}=\inf\\{t>s:g_{(x,s)}^{\widehat{\xi}-,L}(t)<g_{(x,s)}^{\widehat{\xi}+,L}(t)\\}$.
Then, $g_{(x,s)}^{\widehat{\xi}-,L}(t)<g_{(x,s)}^{\widehat{\xi}+,L}(t)$ for
all $t>\hat{t}$. By (4.75) and continuity of geodesics,
$g_{(x,s)}^{\widehat{\xi}-,L}(t)=g(t)=g_{(x,s)}^{\widehat{\xi}+,L}(t)$ for
$t\in[s,\hat{t}\hskip 0.7pt]$, and so $(g(\hat{t}),\hat{t}\hskip
0.7pt)\in\mathfrak{S}^{L}$. ∎

Figure 4.6: The black/thin path is the path $g$. The red/thick paths are the
semi-infinite geodesics $g_{(x,s)}^{\widehat{\xi}-,L}$ and
$g_{(x,s)}^{\widehat{\xi}+,L}$ after they split from $g$. Once the red paths
split, they cannot return, or else there would be two leftmost geodesics from
$(g(\hat{t}),\hat{t})$ to the point where they come back together.

It remains to prove Theorems 4.4.5(ii) and 4.2.6. Recall the definition of the
function from (4.8): $f_{s,\xi}(x)=W_{\xi+}(x,s;0,s)-W_{\xi-}(x,s;0,s).$

Let $\Omega_{3}$ be the subset of $\Omega_{2}$ on which the following holds:
for each $T\in\mathbb{Z}$, (4.76) $\displaystyle\text{whenever
$\xi\in\mathbb{R}$ is such that $f_{T,\,\xi}\neq 0$,
then}\lim_{x\to\pm\infty}f_{T,\,\xi}(x)=\pm\infty.$

By Theorem 4.4.3(iii) and Corollary 2.6.8, $\mathbb{P}(\Omega_{3})=1$.

###### Proof of Theorem 4.4.5(ii).

We work on the full-probability event $\Omega_{3}$. The statement (4.9) to be
proved is $\xi\in\Xi\iff\forall
s\in\mathbb{R}:\lim_{x\to\pm\infty}f_{s,\xi}(x)=\pm\infty$. If, for any $s$,
$f_{s,\xi}\to\pm\infty$ as $x\to\pm\infty$, then $W_{\xi-}(x,s;0,s)\neq
W_{\xi+}(x,s;0,s)$ for $|x|$ sufficiently large, and $\xi\in\Xi$. It remains
to prove the converse statement. From (4.39),

$\Xi=\bigcup_{T\in\mathbb{Z}}\\{\xi\in\mathbb{R}:W_{\xi-}(x,T;0,T)\neq
W_{\xi+}(x,T;0,T)\text{ for some }x\in\mathbb{R}\\}.$

To finish the proof of (4.9), by definition of $\Omega_{3}$, it suffices to
show these two statements:

  1. (i)

If $f_{s,\xi}\neq 0$ for some $s,\xi\in\mathbb{R}$ then $f_{T,\,\xi}\neq 0$
for all $T>s$.

  2. (ii)

For $T\in\mathbb{Z},\xi\in\mathbb{R}$, if $f_{T,\,\xi}\neq 0$, then for all
$s<T$, $\lim_{x\to\pm\infty}f_{s,\xi}(x)=\pm\infty$.

Part (i) follows from the equality below. By (4.38), for $s<T$,

$\displaystyle W_{\xi{\scaleobj{0.8}{\boxempty}}}(x,s;0,s)$
$\displaystyle=\sup_{z\in\mathbb{R}}\\{\mathcal{L}(x,s;z,T)+W_{\xi{\scaleobj{0.8}{\boxempty}}}(z,T;0,T)\\}$
(4.77)
$\displaystyle\qquad\qquad-\sup_{z\in\mathbb{R}}\\{\mathcal{L}(0,s;z,T)+W_{\xi{\scaleobj{0.8}{\boxempty}}}(z,T;0,T)\\}.$

To prove (ii), we show the limits as $x\to+\infty$, and the limits as
$x\to-\infty$ follow analogously. Let $T\in\mathbb{Z},\xi\in\mathbb{R}$ be
such that $f_{T,\,\xi}\neq 0$, and let $R>0$. By definition of the event
$\Omega_{3}$, we may choose $Z>0$ sufficiently large so that $\inf_{z\geq
Z}\\{f_{T,\,\xi}(z)\\}\geq R$. Then, by Equation (4.45) of Theorem 4.5.5(v),
for all sufficiently large $x$ and ${\scaleobj{0.9}{\boxempty}}\in\\{-,+\\}$,

$\sup_{z\in\mathbb{R}}\\{\mathcal{L}(x,s;z,T)+W_{\xi{\scaleobj{0.8}{\boxempty}}}(z,T;0,T)\\}=\sup_{z\geq
Z}\\{\mathcal{L}(x,s;z,T)+W_{\xi{\scaleobj{0.8}{\boxempty}}}(z,T;0,T)\\}.$

Let

$A:=\sup_{z\in\mathbb{R}}\\{\mathcal{L}(x,s;z,T)+W_{\xi+}(z,T;0,T)\\}-\sup_{z\in\mathbb{R}}\\{\mathcal{L}(x,s;z,T)+W_{\xi-}(z,T;0,T)\\},$

and note that this does not depend on $x$. Then, by (4.77),

$\displaystyle-f_{s,\xi}(x)$ $\displaystyle=\sup_{z\geq
Z}\\{\mathcal{L}(x,s;z,T)+W_{\xi-}(z,T;0,T)\\}$
$\displaystyle\qquad\qquad-\sup_{z\geq
Z}\\{\mathcal{L}(x,s;z,T)+W_{\xi+}(z,T;0,T)\\}+A$
$\displaystyle\leq\sup_{z\geq Z}\\{W_{\xi-}(z,T;0,T)-W_{\xi+}(z,T;0,T)\\}+A$
$\displaystyle=-\inf_{z\geq Z}\\{f_{T,\,\xi}(z)\\}+A\leq-R+A,$

so that $f_{s,\xi}(x)\geq R-A$. Since $A$ is constant in $x$ and $R$ is
arbitrary, the desired result follows.

Note that (4.9) immediately proves (4.10) in the case $x=0$. The general case
follows from additivity of the Busemann functions (Theorem 4.4.1(ii)) and
(4.9). ∎

###### Proof of Theorem 4.2.6.

Item (i) ($\mathfrak{S}$ is dense): Work on the full-probability event
$\Omega_{2}$. Since
$\mathfrak{S}\supseteq\mathfrak{S}^{L}\cup\mathfrak{S}^{R}$, it suffices to
show that for $(x,s)\in\mathbb{R}^{2}$ there is a sequence
$(y_{n},t_{n})\in\mathfrak{S}^{L}$ converging to $(x,s)$. Let $g$ be the
leftmost geodesic from $(x,s)$ to $(x,s+1)$. Then $\forall n\geq 1$,
$g|_{[s,s+n^{-1}]}$ is the leftmost geodesic from $(x,s)$ to $(x,s+n^{-1})$.
By Lemma 4.7.4, $\forall n\in\mathbb{Z}_{>0}$
$\exists(x_{n},t_{n})\in\mathfrak{S}^{L}$ such that $x_{n}=g(t_{n})$ and
$s\leq t_{n}\leq s+n^{-1}$. The proof is complete by continuity of geodesics.

Item (ii) ($\mathbb{P}(p\in\mathfrak{S})=0$ for all $p\in\mathbb{R}^{2}$): If
there exist disjoint semi-infinite geodesics from $(x,s)$, then for each level
$t>s$, there exist disjoint geodesics from $(x,s)$ to some points
$(y_{1},t),(y_{2},t)$. For each fixed $(x,s)$, with probability one, this
occurs for no such points by [23, Remark 1.12].

Item (iii) (Hausdorff dimension of $\mathfrak{S}_{s,\xi})$: Since $s$ is
fixed, it suffices to take $s=0$. By Theorem 4.4.3(iii),
$\\{W_{\xi+}({\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}},0;0,0)\\}\overset{d}{=}G^{\sqrt{2}}$,
and by Theorem 4.4.5(ii), $\xi\in\Xi$ if and only if $f_{0,\xi}\neq 0$.
Therefore, Corollary 2.6.9 implies that, with probability one,
$\dim_{H}(\mathcal{D}_{0,\xi})=\frac{1}{2}$ for all $\xi\in\Xi$. Then, as
noted in Remark 4.7.2,
$\mathbb{P}(\dim_{H}(\mathfrak{S}_{0,\xi})=\frac{1}{2}\;\forall\xi\in\Xi)=1$.

Item (iv) ($\mathfrak{S}_{s,\xi}$ is nonempty and unbounded for all $s$): By
Theorem 4.4.5(ii), on the event $\Omega_{3}$, whenever $\xi\in\Xi$, for all
$s\in\mathbb{R}$, $f_{s,\xi}(x)\to\pm\infty$ as $x\to\pm\infty$. Since
$f_{s,\xi}$ is continuous (Theorem 4.4.1(i)), the set $\mathcal{D}_{s,\xi}$ is
unbounded in both directions. The proof is complete since
$\mathcal{D}_{s,\xi}\subseteq\mathfrak{S}_{s,\xi}$ by definition. ∎

##  Chapter 5 Scaling limit of the TASEP speed process

###  5.1 Introduction

####  5.1.1 The totally asymmetric simple exclusion process

In this final chapter, we show how the SH appears as a scaling limit for
coupled initial data for a particle system know as the totally asymmetric
simple exclusion process (TASEP). In the simplest TASEP dynamics each site of
$\mathbb{Z}$ contains either a particle or a hole. Each site has an
independent rate 1 Poisson clock. If at time $t$ the clock rings at site
$x\in\mathbb{Z}$ the following happens. If there is a particle at site $x$ and
no particle at site $x+1$ then the particle at site $x$ jumps to site $x+1$,
while the other sites remain unchanged. If there is no particle at site $x$ or
there is a particle at site $x+1$ then the jump is suppressed. In other words,
a particle can jump to the right only if the target site has no particle at
the time of the jump attempt. This is the exclusion rule. TASEP is a Markov
process on the compact state space $\\{0,1\\}^{\mathbb{Z}}$. Generic elements
of $\\{0,1\\}^{\mathbb{Z}}$, or particle configurations, are denoted by
$\eta=\\{\eta(x)\\}_{x\in\mathbb{Z}}$, where $\eta(x)=1$ means that site $x$
is occupied by a particle and $\eta(x)=0$ that site $x$ is occupied by a hole,
in other words, is empty.

For each density $\rho\in[0,1]$, the i.i.d. Bernoulli distribution
$\nu^{\rho}$ on $\\{0,1\\}^{\mathbb{Z}}$ with density $\rho$ is the unique
translation-invariant extremal stationary distribution of particle density
$\rho$ under the TASEP dynamics.

There is a natural way to couple multiple TASEPs from different initial
conditions but with the same driving dynamics. Let
$\\{\mathcal{N}_{x}:x\in\mathbb{Z}\\}$ be a $\mathbb{Z}$-indexed collection of
independent rate $1$ Poisson processes on $\mathbb{R}$. The clock at location
$x$ rings at the times that correspond to points in $\mathcal{N}_{x}$. One can
then take two densities $0<\rho^{1}<\rho^{2}<1$ and ask whether there exists a
coupling measure $\pi^{\rho_{1},\rho_{2}}$ on
$\\{0,1\\}^{\mathbb{Z}}\times\\{0,1\\}^{\mathbb{Z}}$ with Bernoulli marginals
$\nu^{\rho_{1}}$ and $\nu^{\rho_{2}}$ that is stationary under the joint TASEP
dynamics and ordered. In other words, the twin requirements are that if
initially $(\eta^{1},\eta^{2})\sim\pi^{\rho_{1},\rho_{2}}$, then
$(\eta^{1}_{t},\eta^{2}_{t})\sim\pi^{\rho_{1},\rho_{2}}$ at all subsequent
times $t\geq 0$, and $\eta^{1}(x)\leq\eta^{2}(x)$ for all $x\in\mathbb{Z}$
with $\pi^{\rho_{1},\rho_{2}}$-probability one. Such a two-component
stationary distribution exists and is unique [103].

One reason for the interest in stationary measures of more than one density
comes from the connection between the TASEP dynamics on $k$ coupled profiles
in the state space $(\\{0,1\\}^{\mathbb{Z}})^{k}$ and the TASEP dynamics on
particles with classes in $\llbracket 1,k\rrbracket=\\{1,\ldots,k\\}$, called
multiclass or multitype dynamics. In the $k$-type dynamics, each particle has
a class in $\llbracket 1,k\rrbracket$ that remains the same for all time. A
particle jumps to the right, upon the ring of a Poisson clock, only if there
is either a hole or a particle of lower class (higher label) to the right. If
this happens, the lower class particle moves left. The state space of $k$-type
dynamics is $\\{1,\ldots,k,\infty\\}^{\mathbb{Z}}$, with generic
configurations denoted again by $\eta=\\{\eta(x)\\}_{x\in\mathbb{Z}}$. A value
$\eta(x)=i\in\llbracket 1,k\rrbracket$ means that site $x$ is occupied by a
particle of class $i$, and $\eta(x)=\infty$ means that site $x$ is empty,
equivalently, occupied by a hole. Denoting a hole by $\infty$ is convenient
now because holes can be equivalently viewed as particles of the absolute
lowest class. For $k=1$ the multitype dynamics is the same as basic TASEP.

The next question is whether we can couple all the invariant multiclass
distributions so that the resulting construction is still invariant under
TASEP dynamics. This was achieved by [2]: such couplings can be realized by
applying projections to an object they constructed and named the TASEP speed
process. We describe briefly the construction. To start, each site
$i\in\mathbb{Z}$ is occupied by a particle of class $i$. This creates the
initial profile $\eta_{0}\in\mathbb{Z}^{\mathbb{Z}}$ such that
$\eta_{0}(i)=i$. Let $\eta_{t}$ evolve under TASEP dynamics, now interpreted
so that a particle switches places with the particle to its right only if the
particle to the right is of lower class, that is, has a higher label. Note
that now each site is always occupied by a particle of some integer label. The
limit from [111] implies that each particle has a well-defined limiting speed:
if $X_{t}(i)$ denotes the time-$t$ position of the particle initially at site
$i$, then the following random limit exists almost surely:

$U_{i}=\lim_{t\to\infty}t^{-1}X_{t}(i).$ (5.1)

The process $\\{U_{i}\\}_{i\in\mathbb{Z}}$ is the TASEP speed process. It is a
random element of the space $[-1,1]^{\mathbb{Z}}$.

######  Theorem 5.1.1 ([2, Theorem 1.5]).

The TASEP speed process $\\{U_{i}\\}_{i\in\mathbb{Z}}$ is the unique invariant
distribution of TASEP that is ergodic under translations of the lattice
$\mathbb{Z}$ and such that each $U_{i}$ is uniformly distributed on $[-1,1]$.

In the context of the theorem above, the TASEP state
$\eta=\\{\eta(i)\\}_{i\in\mathbb{Z}}$ is a real-valued sequence, but the
meaning of the dynamics is the same as before. Namely, at each pair
$\\{i,i+1\\}$ of nearest-neighbor sites, at the rings of a rate one
exponential clock, the variables $\eta(i)$ and $\eta(i+1)$ are swapped if
$\eta(i)<\eta(i+1)$, otherwise left unchanged.

A key point is that the TASEP speed process projects to multitype stationary
distributions.

######  Theorem 5.1.2 ([63, Theorem 2.1], [2, Theorem 2.1]).

Let $k\in\mathbb{N}$ be the number of
classes. Let $\bar{\rho}=(\rho_{1},\ldots,\rho_{k})\in(0,1)^{k}$ be a
parameter vector such that $\sum_{i=1}^{k}\rho_{i}\leq 1$. Then there is a
translation-invariant stationary distribution $\widecheck{\pi}^{\bar{\rho}}$
for the $k$-type TASEP which is unique under the conditions (i) and (ii), and
also under the conditions (i) and (ii’) below:

(i)
$\widecheck{\pi}^{\bar{\rho}}\\{\eta\in\\{1,\ldots,k,\infty\\}^{\mathbb{Z}}:\eta(x)=j\\}=\rho_{j}$
for each site $x\in\mathbb{Z}$ and class $j\in\llbracket 1,k\rrbracket$;

(ii) under $\widecheck{\pi}^{\bar{\rho}}$, for each $\ell\in\llbracket
1,k\rrbracket$, the distribution of the sequence
$\\{\mathbf{1}[\eta(x)\leq\ell]\\}_{x\in\mathbb{Z}}$ of indicators is the
i.i.d. Bernoulli measure $\nu^{\sum_{j=1}^{\ell}\rho_{j}}$ of intensity
$\sum_{j=1}^{\ell}\rho_{j}$;

(ii’) $\widecheck{\pi}^{\bar{\rho}}$ is ergodic under the translation of the
lattice $\mathbb{Z}$.

Furthermore, $\widecheck{\pi}^{\bar{\rho}}$ is extreme among translation-
invariant stationary measures of the $k$-type dynamics with jumps to the
right.

Theorem 5.1.2 is not stated exactly in this form in either reference. It can
be proved with the techniques of Section VIII.3 of Liggett [104].

######  Lemma 5.1.3 ([2, Corollary 5.4]).

Let $F:[-1,1]\rightarrow\\{1,\dotsc,k,\infty\\}$ be a nondecreasing function
and $\lambda_{j}=\frac{1}{2}{\rm Leb}\big{(}F^{-1}(j)\big{)}$, i.e., one-half
the Lebesgue measure of the interval mapped to the value
$j\in\\{1,\dotsc,k,\infty\\}$. Then the distribution of the
$\\{1,\dotsc,k,\infty\\}$-valued sequence $\\{F(U_{i})\\}_{i\in\mathbb{Z}}$ is
the stationary measure $\widecheck{\pi}^{(\lambda_{1},\dotsc,\lambda_{k})}$
described in Theorem 5.1.2 for the $k$-type TASEP with jumps to the right.

For example, the case $k=1$ of Lemma 5.1.3 tells us that to produce a particle
configuration with Bernoulli distribution $\nu^{\rho}$ from the TASEP speed
process, assign a particle to each site $x$ such that $U_{x}\leq 2\rho-1$.
Lemma 5.1.3 follows readily from Theorems 5.1.1 and 5.1.2 because the
nondecreasing projection $F$ commutes with the pathwise dynamics.

######  Remark 5.1.4 (Direction of the jumps).

Throughout this section, jumps in TASEP go to the right. Later in Section 5.2,
we use the convention from [63] whereby TASEP jumps proceed left. This is
convenient because then discrete time in the queuing setting agrees with the
order on $\mathbb{Z}$. Notationally, $\widecheck{\pi}^{\bar{\rho}}$ denotes
the multiclass stationary measure under rightward jumps, as in Theorem 5.1.2
and Lemma 5.1.3 above, while $\pi^{\bar{\rho}}$ will denote the stationary
measure under leftward jumps. These measures are simply reflections of each
other (see Theorem 5.2.1).

####  5.1.2 Scaling limit of the speed process

The space $\\{0,1\\}^{\mathbb{Z}}$ of TASEP particle configurations $\eta$ can
be mapped bijectively onto the space of continuous interfaces
$f:\mathbb{R}\to\mathbb{R}$ such that $f(0)=0$, $|f(x)-f(x+1)|=1$ for all
$x\in\mathbb{Z}$, and $f(x)$ interpolates linearly between integer points.
Define $\mathcal{P}:\\{0,1\\}^{\mathbb{Z}}\rightarrow C(\mathbb{R})$ by
stipulating that on integers $i$ the image function $\mathcal{P}[\eta]$ is
given by

$\mathcal{P}[\eta](i)=\begin{cases}\sum_{j=0}^{i-1}(2\eta(j)-1),&i\in\mathbb{N}\\\
0,&i=0\\\ -\sum_{j={i}}^{-1}(2\eta(j)-1),&i\in-\mathbb{N}\end{cases}$ (5.2a)
and then extend $\mathcal{P}[\eta]$ to the reals by linear interpolation:
$\text{for}\quad
x\in\mathbb{R}\setminus\mathbb{Z},\quad\mathcal{P}[\eta](x)=(\lceil{x}\rceil-x)\mathcal{P}[\eta](\lfloor{x}\rfloor)+(x-\lfloor{x}\rfloor)\mathcal{P}[\eta](\lceil{x}\rceil).$
(5.2b)

TASEP can therefore be thought of as dynamics on continuous interfaces
$f:\mathbb{R}\to\mathbb{R}$ such that for all $x\in\mathbb{Z}$,
$f(x)\in\mathbb{Z}$ and $f(x\pm 1)\in\\{f(x)-1,f(x)+1\\}$. When a particle at
location $x$ lies immediately to the left of a hole at location $x+1$, the
interface has a local maximum at location $x+1$. When the particle changes
places with the hole, the local maximum becomes a local minimum.

Let $U=\\{U_{j}\\}_{j\in\mathbb{Z}}$ be the TASEP speed process and for
$s\in\mathbb{R}$,
$\mathbf{1}_{U<s}=\\{\mathbf{1}_{U_{j}<s}\\}_{j\in\mathbb{Z}}$ a shorthand for
the $\\{0,1\\}$-valued sequence of indicators. For each value of the centering
$v\in(-1,1)$ and a scaling parameter $N\in\mathbb{N}$, use the mapping (5.2)
to define from the speed process a $C(\mathbb{R})$-valued process indexed by
$\xi\in\mathbb{R}$:

$H^{N}_{\xi}(x)=H^{\sigma,v,N}_{\xi}(x)=N^{-1/2}\,\mathcal{P}\big{[}\mathbf{1}_{U\leq
v+\xi(1-v^{2})N^{-1/2}}\big{]}\Bigl{(}\frac{\sigma^{2}x}{1-v^{2}}N\Bigr{)}-\frac{\sigma^{2}vx}{1-v^{2}}N^{1/2},\qquad\xi,x\in\mathbb{R}.$
(5.3)

The main theorem of this chapter is the process-level weak limit of
$H^{\sigma,v,N}=\\{H^{\sigma,v,N}_{\xi}\\}_{\xi\in\mathbb{R}}$. The path space
of $\xi\mapsto H^{\sigma,v,N}_{\xi}$ is the Skorokhod space
$D(\mathbb{R},C(\mathbb{R}))$ of $C(\mathbb{R})$-valued cadlag paths on
$\mathbb{R}$, with its usual Polish topology. This topology is discussed in
more detail in [34].

######  Theorem 5.1.5.

Let $\sigma>0$ and $G^{\sigma}$ be the stationary horizon. Then, for each
$v\in(-1,1)$, as $N\rightarrow\infty$, the distributional limit
$H^{\sigma,v,N}\Rightarrow G^{\sigma}$ holds on the path space
$D(\mathbb{R},C(\mathbb{R}))$.

Theorem 5.1.5 is taken from the author’s joint work with Busani and
Seppäläinen [34]. In that work, we focus on the case $\sigma=\sqrt{2}$, but
the extension to general $\sigma$ follows the same proof (or alternatively
from the scaling relations of Theorem 2.3.2(ii)). As is typical, the proof
splits into two main steps: (i) weak convergence of finite-dimensional
distributions of $H^{N}$ to the limiting object and (ii) tightness of
$\\{H^{N}\\}_{N\in\mathbb{N}}$ on $D(\mathbb{R},C(\mathbb{R}))$. Both parts
use the Ferrari-Martin queuing representation of the multitype stationary
measures. The first part shows that, in the limit, the queuing representation
recovers the queuing structure that defines the SH [63]. In this dissertation,
we only show the finite-dimensional convergence (Proposition 5.2.5), and refer
the reader to [34] for the proof of tightness.

###  5.2 Finite-dimensional convergence

####  5.2.1 The space $D(\mathbb{R},C(\mathbb{R}))$

We observe here why the path $\xi\mapsto H^{N}_{\xi}$ defined in (5.3) lies in
$D(\mathbb{R},C(\mathbb{R}))$. Restriction of $x\mapsto H^{N}_{\xi}(x)$ to a
bounded interval $[-x_{0},x_{0}]$ is denoted by
$H^{N,x_{0}}_{\xi}=H^{N}_{\xi}|_{[-x_{0},x_{0}]}$. Then note that for
$\xi<\rho$, $H^{N,x_{0}}_{\xi}\neq H^{N,x_{0}}_{\rho}$ if and only if
$U_{j}\in(v+\xi(1-v^{2})N^{-1/2},v+(1-v^{2})\rho N^{-1/2}]$ for some
$j\in\llbracket\,\lfloor{-\,\frac{\sigma^{2}x_{0}}{1-v^{2}}N}\rfloor,\lceil{\frac{\sigma^{2}x_{0}}{1-v^{2}}N}\rceil\,\rrbracket$.
Since this range of indices is finite, for each $\xi\in\mathbb{R}$ and
$x_{0}>0$ there exists $\varepsilon>0$ such that
$H^{N,x_{0}}_{\xi}=H^{N,x_{0}}_{\rho}$ for $\rho\in[\xi,\xi+\varepsilon]$ and
$H^{N,x_{0}}_{\rho}=H^{N,x_{0}}_{\sigma}$ for
$\rho,\sigma\in[\xi-\varepsilon,\xi)$.

####  5.2.2 Ferrari-Martin representation of multiclass measures

This section describes the queuing construction of stationary multiclass
measures from [63]. We use the convention of [63] that TASEP particles jump to
the left rather than to the right, because this choice leads to the more
natural queuing set-up where time flows on $\mathbb{Z}$ from left to right.
This switch is then accounted for when we apply the results of this section.

##### Queues with a single customer stream

Let $\mathcal{U}_{1}:=\\{1,\infty\\}^{\mathbb{Z}}$ be the space of
configurations of particles on $\mathbb{Z}$ with the following interpretation:
a configuration $\boldsymbol{x}=\\{x(j)\\}_{j\in\mathbb{Z}}\in\mathcal{U}_{1}$
has a particle at time $j\in\mathbb{Z}$ if $x(j)=1$, otherwise
$\boldsymbol{x}$ has a hole at time $j\in\mathbb{Z}$. Let
$\boldsymbol{a},\boldsymbol{s}\in\mathcal{U}_{1}$. Think of $\boldsymbol{a}$
as arrivals of customers to a queue, and of $\boldsymbol{s}$ as the available
services in the queue. For $i\leq j\in\mathbb{Z}$ let $a^{\leq 1}[i,j]$ be the
number of customers, that is, the number of $1$’s in $\boldsymbol{a}$, that
arrive to the queue during time interval $[i,j]$. Similarly, let $s[i,j]$ be
the number of services available during time interval $[i,j]$. The queue
length at time $i$ is then given by

$Q_{i}=\sup_{j:j\leq i}\big{(}a^{\leq 1}[j,i]-s[j,i]\hskip 0.9pt\big{)}^{+}.$
(5.4)

In principle this makes sense for arbitrary sequences $\boldsymbol{a}$ and
$\boldsymbol{s}$ if one allows infinite queue lengths $Q_{i}=\infty$. However,
in our treatment $\boldsymbol{a}$ and $\boldsymbol{s}$ are always such that
queue lengths are finite. We will not repeat this point in the sequel.

The departures from the queue come from the mapping
$\boldsymbol{d}=D_{d}(\boldsymbol{a},\boldsymbol{s}):\mathcal{U}_{1}\times\mathcal{U}_{1}\rightarrow\mathcal{U}_{1}$,
given by

$d(i)=\begin{cases}1&\text{$s(i)=1$ and either $Q_{i-1}>0$ or $a(i)=1$},\\\
\infty&\text{otherwise}.\end{cases}$ (5.5)

In other words, a customer leaves the queue at time $i$ (and $d(i)=1$) if
there is a service at time $i$ and either the queue is not empty or a customer
just arrived at time $i$. The sequence
$\boldsymbol{u}:=U_{d}(\boldsymbol{a},\boldsymbol{s})$ of unused services is
given by a mapping
$U_{d}:\mathcal{U}_{1}\times\mathcal{U}_{1}\rightarrow\mathcal{U}_{1}$ defined
by

$u(j)=\begin{cases}1&\text{if $s(j)=1$, $Q_{j-1}=0$, and $a(j)=\infty$},\\\
\infty&\text{otherwise}.\end{cases}$ (5.6)

Last, we define the map
$R_{d}:\mathcal{U}_{1}\times\mathcal{U}_{1}\rightarrow\mathcal{U}_{1}$ as
$\boldsymbol{r}=R_{d}(\boldsymbol{a},\boldsymbol{s})$ with

$r(j)=\begin{cases}1&\text{ if either $a(j)=1$ or $u(j)=1$},\\\ \infty&\text{
if $a(j)=u(j)=\infty$}.\end{cases}.$ (5.7)

Extend the departure operator $D_{d}$ to queues in tandem. Let
$D_{d}^{1}(\boldsymbol{x})=\boldsymbol{x}$ be the identity, and for $n\geq 2$,

$\displaystyle D_{d}^{(2)}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})$
$\displaystyle=D_{d}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})$ (5.8)
$\displaystyle
D_{d}^{3}(\boldsymbol{x}_{1},\boldsymbol{x}_{2},\boldsymbol{x}_{3})$
$\displaystyle=D_{d}\big{(}D_{d}^{(2)}(\boldsymbol{x}_{1},\boldsymbol{x}_{2}),\boldsymbol{x}_{3}\big{)}$
$\displaystyle\ \;\vdots$ $\displaystyle
D_{d}^{(n)}(\boldsymbol{x}_{1},\boldsymbol{x}_{2},\dotsc,\boldsymbol{x}_{n})$
$\displaystyle=D_{d}\big{(}D_{d}^{(n-1)}(\boldsymbol{x}_{1},\boldsymbol{x}_{2},\dotsc,\boldsymbol{x}_{n-1}),\boldsymbol{x}_{n}\big{)}.$

We use the notation $D_{d}$ (with the subscript $d$) to denote discrete, in
comparison to the operators on continuous functions in (2.3),(2.5).

##### Queues with priorities

Now consider queues with customers of different classes. For $m\in\mathbb{N}$,
let
$\mathcal{U}_{m}:=\\{1,2,\dotsc,m,\infty\\}^{\mathbb{Z}}$ be the space of
configurations of particles on $\mathbb{Z}$ with classes in $\llbracket
1,m\rrbracket=\\{1,2,\dotsc,m\\}$. A lower label indicates higher class and,
as before, the value $\infty$ signifies an empty time slot. To illustrate the
notation for an arrival sequence $\boldsymbol{a}\in\mathcal{U}_{m}$, the value
$a(j)=k\in\llbracket 1,m\rrbracket$ means that a customer of class $k$ arrives
at time $j\in\mathbb{Z}$, while $a(j)=\infty$ means no arrival at time $j$.
Define

$a^{\leq k}[j]=\begin{cases}1&\text{if $a(j)\leq k$},\\\ 0&\text{if
$a(j)>k$}.\end{cases}$

Consistently with earlier definitions, $a^{\leq k}[i,j]=\sum_{l=i}^{j}a^{\leq
k}[l]$ is the number of customers in classes $\llbracket 1,k\rrbracket$ that
arrive to the queue in the time interval $[i,j]$. Let
$\boldsymbol{s}\in\mathcal{U}_{1}$ be the sequence of available services. The
number of customers in classes $\llbracket 1,k\rrbracket$ in the queue at time
$i$ is then

$Q^{\leq k}_{i}(\boldsymbol{a},\boldsymbol{s})=\sup_{j:j\leq i}\big{(}a^{\leq
k}[j,i]-s[j,i]\big{)}^{+},\qquad i\in\mathbb{Z}.$

The multiclass departure map
$\boldsymbol{d}=F_{m}(\boldsymbol{a},\boldsymbol{s}):\mathcal{U}_{m}\times\mathcal{U}_{1}\rightarrow\mathcal{U}_{m+1}$
is defined so that customers of higher class (lower label) are served first.
These are the rules:

$\begin{cases}d(i)\leq k&\text{for $k\in\llbracket 1,m\rrbracket$ if $s(i)=1$
and either $Q^{\leq k}_{i-1}>0$ or $a(i)\leq k$},\\\ d(i)=m+1&\text{if
$s(i)=1$, $Q^{\leq m}_{i-1}=0$, and $a(i)=\infty$},\\\ d(i)=\infty&\text{if
}s(i)=\infty.\end{cases}$ (5.9)

The map $F_{m}$ works as follows. The queue is fed with arrivals
$\boldsymbol{a}\in\mathcal{U}_{m}$ of customers in classes $1$ to $m$. Suppose
a service is available at time $i\in\mathbb{Z}$ ($s(i)=1$). Then the customer
of the highest class (lowest label in $\llbracket 1,m\rrbracket$) in the queue
at time $i$, or just arrived at time $i$, is served at time $i$, and its label
becomes the value of $d(i)$. If no customer arrived at time $i$
($a(i)=\infty$) and the queue is empty ($Q^{\leq m}_{i-1}=0$), then the unused
service $s(i)=1$ is converted into a departing customer of class $m+1$:
$d(i)=m+1$. If there is no service available at time $i\in\mathbb{Z}$
($s(i)=\infty$), then no customer leaves at time $i$ and $d(i)=\infty$.

In particular, for $m=1$, the output
$\boldsymbol{d}=F_{1}(\boldsymbol{a},\boldsymbol{s})$ satisfies

$d(i)=\begin{cases}1,&[D_{d}(\boldsymbol{a},\boldsymbol{s})]_{i}=1\\\
2,&[U_{d}(\boldsymbol{a},\boldsymbol{s})]_{i}=1\\\
\infty,&s(i)=\infty.\end{cases}$ (5.10)

For $n\in\mathbb{N}$ define the space
$\mathcal{X}_{n}=\mathcal{U}_{1}^{n}=\\{1,\infty\\}^{\mathbb{Z}\times\\{1,\dotsc,n\\}}$
of $n$-tuples of sequences. Let
$\bar{\lambda}=(\lambda_{1},\dotsc,\lambda_{n})\in(0,1)^{n}$ be a parameter
vector such that $\sum_{r=1}^{n}\lambda_{r}\leq 1$. Define the product measure
$\nu^{\bar{\lambda}}$ on $\mathcal{X}_{n}$ so that if
$\bar{\boldsymbol{x}}=(\boldsymbol{x}_{1},\dotsc,\boldsymbol{x}_{n})\sim\nu^{\bar{\lambda}}$
then the sequences $\boldsymbol{x}_{k}$ are independent and each
$\boldsymbol{x}_{k}$ has the i.i.d. product Bernoulli distribution with
intensity $\sum_{i=1}^{k}\lambda_{i}$. From this input we define a new process
$\bar{\boldsymbol{v}}=(\boldsymbol{v}_{1},\dotsc,\boldsymbol{v}_{n})$ such
that each $\boldsymbol{v}_{m}\in\mathcal{U}_{m}$ by the iterative formulas

$\displaystyle\boldsymbol{v}_{1}$
$\displaystyle=\boldsymbol{x}_{1}\qquad\text{and}$ (5.11)
$\displaystyle\boldsymbol{v}_{m}$
$\displaystyle=F_{m-1}(\boldsymbol{v}_{m-1},\boldsymbol{x}_{m})\qquad\text{for
}\ m=2,\dotsc,n.$

We denote this map by
$\bar{\boldsymbol{v}}=\mathcal{V}(\bar{\boldsymbol{x}})=(\mathcal{V}_{1}(\bar{\boldsymbol{x}}),\dotsc,\mathcal{V}_{n}(\bar{\boldsymbol{x}}))$.
For a vector $\bar{\lambda}=(\lambda_{1},\dotsc,\lambda_{n})$ and
$\bar{\boldsymbol{x}}\sim\nu^{\bar{\lambda}}$, define the distribution
$\pi^{\bar{\lambda}}$ as the image of $\nu^{\bar{\lambda}}$ under this map:

$\pi^{\bar{\lambda}}=\nu^{\bar{\lambda}}\circ\mathcal{V}_{n}^{-1}\ \iff\
\mathcal{V}_{n}(\bar{\boldsymbol{x}})\sim\pi^{\bar{\lambda}}.$ (5.12)

######  Theorem 5.2.1.

[63, Theorem 2.1] For each $m\in\llbracket 1,n\rrbracket$, the distribution of
$\boldsymbol{v}_{m}$ under $\pi^{\bar{\lambda}}$ is the unique translation-
ergodic stationary distribution of the $m$-type TASEP on $\mathbb{Z}$ with
leftward jumps and with density $\lambda_{r}$ of particles of class
$r\in\llbracket 1,m\rrbracket$. The distribution of the reversed configuration
$\\{\boldsymbol{v}_{m}(-i)\\}_{i\in\mathbb{Z}}$ is the unique distribution
$\widecheck{\pi}^{\bar{\lambda}}$ described in Theorem 5.1.2, in other words,
the unique translation-ergodic stationary distribution of the $m$-type TASEP
on $\mathbb{Z}$ with rightward jumps, with density $\lambda_{r}$ of particles
of class $r\in\llbracket 1,m\rrbracket$.

######  Remark 5.2.2.

The statement about the TASEP with rightward jumps is not included in [63],
but its proof is straightforward: reflecting the index does not change the
density of the particles, so the values $\lambda_{r}$ are preserved. Consider
an $m$-type TASEP with left jumps $\\{\eta_{t}\\}_{t\geq 0}$ defined by the
Poisson clocks $\\{\mathcal{N}_{i}\\}_{i\in\mathbb{Z}}$ and started from
initial profile $\eta_{0}\sim\boldsymbol{v}_{m}$. Let
$\\{\widecheck{\eta}_{t}\\}_{t\geq 0}$ be TASEP with right jumps defined by
the Poisson clocks $\\{\mathcal{N}_{-i}\\}_{i\in\mathbb{Z}}$ and started from
initial profile
$\\{\widecheck{\eta}_{0}(i)\\}_{i\in\mathbb{Z}}:=\\{\eta_{0}(-i)\\}_{i\in\mathbb{Z}}$,
which has distribution $\\{\boldsymbol{v}_{m}(-i)\\}_{i\in\mathbb{Z}}$. Then,
in the process $\eta_{t}$, a particle jumps from site $i$ to site $i-1$
exactly when a particle in the process $\widecheck{\eta}_{t}$ jumps from site
$-i$ to site $-i+1$. By the invariance of $\eta_{0}$ under TASEP with left
jumps,

$\\{\widecheck{\eta}_{t}(i)\\}_{i\in\mathbb{Z}}=\\{\eta_{t}(-i)\\}_{i\in\mathbb{Z}}\overset{d}{=}\\{\eta_{0}(-i)\\}_{i\in\mathbb{Z}}=\\{\widecheck{\eta}_{0}(i)\\}_{i\in\mathbb{Z}},$

so $\\{\boldsymbol{v}_{m}(-i)\\}_{i\in\mathbb{Z}}$ is the invariant measure
for TASEP with right jumps and densities $\lambda_{r}$.

For $\boldsymbol{x}\in\mathcal{U}_{n}$, define

$\operatorname{Cls}^{[i,j]}_{m}(\boldsymbol{x})=\\#\\{l\in[i,j]:{x}(l)\leq
m\\},\qquad m\in\llbracket 1,n\rrbracket.$

$\operatorname{Cls}^{[i,j]}_{m}(\boldsymbol{x})$ records the number of
customers in classes $\llbracket 1,m\rrbracket$ during time interval $[i,j]$
in the sequence $\boldsymbol{x}$. Note that a customer of class $m$ appears in
$[i,j]$ if and only if
$\operatorname{Cls}^{[i,j]}_{m}(\boldsymbol{x})>\operatorname{Cls}^{[i,j]}_{m-1}(\boldsymbol{x})$,
with the convention $\operatorname{Cls}_{0}\equiv 0$. The key technical lemma
is that the iteration in (5.11) can be represented by tandem queues.

######  Lemma 5.2.3.

Let $n\in\mathbb{N}$ and
$\bar{\boldsymbol{x}}=(\boldsymbol{x}_{1},\dotsc,\boldsymbol{x}_{n})\in\mathcal{X}_{n}$.
Let $\boldsymbol{v}_{n}=\mathcal{V}_{n}(\bar{\boldsymbol{x}})$, where
$\mathcal{V}_{n}$ is given in (5.11). Define

$\displaystyle\boldsymbol{d}^{n,i}:=D_{d}^{(n-i+1)}(\boldsymbol{x}_{i},\boldsymbol{x}_{i+1},\dotsc,\boldsymbol{x}_{n})\quad\text{for
}\
i=1,\dotsc,n-1,\quad\text{and}\quad\boldsymbol{d}^{n,n}:=\boldsymbol{x}_{n}.$

Then for all time intervals $[i,j]$,

$\big{(}\operatorname{Cls}^{[i,j]}_{1}(\boldsymbol{v}_{n}),\dotsc,\operatorname{Cls}^{[i,j]}_{n}(\boldsymbol{v}_{n})\big{)}=\big{(}d^{n,1}[i,j],\dotsc,d^{n,n}[i,j]\big{)}.$
(5.13)

###### Proof.

The proof goes by induction on $n$, with base case $n=2$. From (5.10),
$\boldsymbol{d}^{2,1}=D_{d}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})$ registers
the first class departures out of the queue
$F_{1}(\boldsymbol{x}_{1},\boldsymbol{x}_{2})$ while
$\boldsymbol{d}^{2,2}=\boldsymbol{x}_{2}$ is the combined number of first and
second class customers coming out of the queue. The case $n=2$ of (5.13) has
been verified.

Assume (5.13) holds for some $n=k\geq 2$. This means that for each
$m\in\llbracket 1,k\rrbracket$, $\boldsymbol{d}^{k,m}$ registers the customers
in classes $\llbracket 1,m\rrbracket$ in $\boldsymbol{v}_{k}$. In the next
step, $\boldsymbol{v}_{k+1}=F_{k}(\boldsymbol{v}_{k},\boldsymbol{x}_{k+1})$,
and

$\big{(}\boldsymbol{d}^{k+1,1},\dotsc,\boldsymbol{d}^{k+1,k+1}\big{)}=\big{(}D_{d}(\boldsymbol{d}^{k,1},\boldsymbol{x}_{k+1}),\dotsc,D_{d}(\boldsymbol{d}^{k,k},\boldsymbol{x}_{k+1}),\boldsymbol{x}_{k+1}\big{)}.$

Since the same service process $\boldsymbol{x}_{k+1}$ acts in both queuing
maps, the outputs match in the sense that for each $m\leq k$,
$\boldsymbol{d}^{k+1,m}=D_{d}(\boldsymbol{d}^{k,m},\boldsymbol{x}_{k+1})$
registers the customers in classes $\llbracket 1,m\rrbracket$ in
$\boldsymbol{v}_{k+1}$. Under
$F_{k}(\boldsymbol{v}_{k},\boldsymbol{x}_{k+1})$, unused services become
departures of class $k+1$. Hence, every service event of
$\boldsymbol{x}_{k+1}$ becomes a departure of some class in $\llbracket
1,k+1\rrbracket$. This verifies the equality
$\operatorname{Cls}^{[i,j]}_{k+1}(\boldsymbol{v}_{k+1})={x}_{k+1}[i,j]=d^{k+1,k+1}[i,j]$
of the last coordinate. Thereby, the validity of (5.13) has been extended from
$k$ to $k+1$. ∎

####  5.2.3 Convergence of queues

This section shows the finite-dimensional weak convergence of the TASEP speed
process, using the representation of stationary distributions in terms of
queuing mappings. To do this, we derive a convenient representation for the
random walk defined by the departure mapping $D_{d}$ (5.14). Consistently with
the count notation $x[i,j]$ introduced above for
$\boldsymbol{x}\in\mathcal{U}_{1}$, abbreviate
$x[i]=x[i,i]=\mathbf{1}_{x(i)=1}$. With this convention, configurations
$\boldsymbol{x}$ can also be thought of as members of the sequence space
$\\{0,1\\}^{\mathbb{Z}}$. Recall the operation $\mathcal{P}$ from (5.2).

######  Lemma 5.2.4.

For $i\in\mathbb{Z}$,

$\mathcal{P}[D_{d}(\boldsymbol{a},\boldsymbol{s})](i)=\mathcal{P}[\boldsymbol{s}](i)+\sup_{-\infty<j\leq
0}[\mathcal{P}[\boldsymbol{s}](j)-\mathcal{P}[\boldsymbol{a}](j)]-\sup_{-\infty<j\leq
i}[\mathcal{P}[\boldsymbol{s}](j)-\mathcal{P}[\boldsymbol{a}](j)].$ (5.14)

###### Proof.

Recall the definition of $D$ from (5.5). Observe that

$D_{d}(\boldsymbol{a},\boldsymbol{s})[j,i]=Q_{j-1}-Q_{i}+a[j,i],$ (5.15)

because any arrival that cannot be accounted for in $Q_{i}$ must have left by
time $i$. Use also the empty interval convention $x[i+1,i]=0$. Then, from
(5.4), we can equivalently write

$Q_{i}=\sup_{j:j\leq i+1}\bigl{(}a^{\leq 1}[j,i]-s[j,i]\bigr{)}.$ (5.16)

Now, observe that for $\boldsymbol{x}\in\mathcal{U}_{1}$,

$2x[j,i]-(i-j+1)=\sum_{k=j}^{i}(2x[k]-1)=\mathcal{P}[\boldsymbol{x}](i+1)-\mathcal{P}[\boldsymbol{x}](j)$
(5.17)

Combining (5.15)–(5.17) and the definition $\mathcal{P}[\boldsymbol{x}](0)=0$,
gives for $i>0$,

$\displaystyle\mathcal{P}[D_{d}(\boldsymbol{a},\boldsymbol{s})](i)$
$\displaystyle=\sum_{k=0}^{i-1}(2D_{d}(\boldsymbol{a},\boldsymbol{s})[k]-1)=2D_{d}(\boldsymbol{a},\boldsymbol{s})[0,i-1]-i$
$\displaystyle=2a[0,i-1]-i+2Q_{-1}-2Q_{i-1}$
$\displaystyle\overset{\eqref{xincs}}{=}\mathcal{P}[\boldsymbol{a}](i)-\mathcal{P}[\boldsymbol{a}](0)+2\sup_{-\infty<j\leq
0}[a^{\leq 1}[j,-1]-s[j,-1]]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-2\sup_{-\infty<j\leq
i}[a^{\leq 1}[j,i-1]-s[j,i-1]]$
$\displaystyle=\mathcal{P}[\boldsymbol{a}](i)+\sup_{-\infty<j\leq 0}[2a^{\leq
1}[j,-1]+j-(2s[j,-1]+j)]$ $\displaystyle\qquad\qquad-\sup_{-\infty<j\leq
i}[2a^{\leq 1}[j,i-1]-(i-j)-(2s[j,i-1]-(i-j))]$
$\displaystyle\overset{\eqref{xincs}}{=}\mathcal{P}[\boldsymbol{a}](i)+\sup_{-\infty<j\leq
0}[\mathcal{P}[\boldsymbol{a}](0)-\mathcal{P}[\boldsymbol{a}](j)-\mathcal{P}[\boldsymbol{s}](0)+\mathcal{P}[\boldsymbol{s}](j)]$
$\displaystyle\qquad\qquad\qquad-\sup_{-\infty<j\leq
i}[\mathcal{P}[\boldsymbol{a}](i)-\mathcal{P}[\boldsymbol{a}](j)-\mathcal{P}[\boldsymbol{s}](i)+\mathcal{P}[\boldsymbol{s}](j)]$
$\displaystyle=\mathcal{P}[\boldsymbol{s}](i)+\sup_{-\infty<j\leq
0}[\mathcal{P}[\boldsymbol{s}](j)-\mathcal{P}[\boldsymbol{a}](j)]-\sup_{-\infty<j\leq
i}[\mathcal{P}[\boldsymbol{s}](j)-\mathcal{P}[\boldsymbol{a}](j)].$

The case $i<0$ follows an analogous proof. ∎

######  Proposition 5.2.5.

Fix the diffusion coeffieient $\sigma>0$ the centering $v\in(-1,1)$. Then the
scaled TASEP speed process $H^{N}$ of (5.3) satisfies the weak convergence
$(H_{\xi_{1}}^{N},\dotsc,H_{\xi_{k}}^{N})$ $\Rightarrow$
$(G^{\sigma}_{\xi_{1}},\dotsc,G^{\sigma}_{\xi_{k}})$ on $C(\mathbb{R})^{k}$
for any finite sequence $(\xi_{1},\dotsc,\xi_{k})\in\mathbb{R}^{k}$.

###### Proof.

Without loss of generality, take $\xi_{1}<\xi_{2}<\cdots<\xi_{k}$. For
$N>|\xi_{1}|^{3}\vee|\xi_{k}|^{3}$, consider the following nondecreasing map
$F:[-1,1]\to\\{1,\ldots,k\\}\cup\\{\infty\\}$:

$\text{For }U\in[-1,1],\;F(U)=\begin{cases}1,&U\leq
v+\xi_{1}(1-v^{2})N^{-1/2}\\\ 2,&v+\xi_{1}(1-v^{2})N^{-1/2}<U\leq
v+\xi_{2}(1-v^{2})N^{-1/2}\\\ \vdots&\\\ k,&v+\xi_{k-1}(1-v^{2})N^{-1/2}<U\leq
v+\xi_{k}(1-v^{2})N^{-1/2}\\\
\infty,&U>v+\xi_{k}(1-v^{2})N^{-1/2}.\end{cases}$

By considering the output of this map as classes, Lemma 5.1.3 implies that
$\\{F(U_{i})\\}_{i\in\mathbb{Z}}$ is distributed as the stationary
distribution for $k$-type TASEP with right jumps and densities

$\bar{\lambda}=\biggl{(}\frac{1+v+\xi_{1}(1-v^{2})N^{-1/2}}{2},\frac{(\xi_{2}-\xi_{1})(1-v^{2})N^{-1/2}}{2},\ldots,\frac{(\xi_{k}-\xi_{k-1})(1-v^{2})N^{-1/2}}{2}\biggr{)}.$

The reflection of Theorem 5.2.1 and translation invariance then imply that
$\\{F(U_{-i-1})\\}_{i\in\mathbb{Z}}$ has the stationary distribution
$\pi^{\bar{\lambda}}$ for TASEP with left jumps. Lemma 5.2.3 implies that

$\displaystyle\bigl{(}\mathbf{1}_{U_{-i-1}\leq
v+\xi_{1}(1-v^{2})N^{-1/2}},\ldots,\mathbf{1}_{U_{-i-1}\leq
v+\xi_{k-1}(1-v^{2})N^{-1/2}}\,,\mathbf{1}_{U_{-i-1}\leq
v+\xi_{k}(1-v^{2})N^{-1/2}}\bigr{)}_{i\in\mathbb{Z}}$ (5.18)
$\displaystyle\qquad\qquad\qquad\overset{d}{=}\bigl{(}D_{d}^{(k)}(\boldsymbol{x}_{1}^{N},\ldots,\boldsymbol{x}_{k}^{N})[i]\,,\ldots,D_{d}^{(2)}(\boldsymbol{x}^{N}_{k-1},\boldsymbol{x}_{k}^{N})[i]\,,\boldsymbol{x}_{k}^{N}[i]\bigr{)}_{i\in\mathbb{Z}},$

where
$(\boldsymbol{x}_{1}^{N},\ldots,\boldsymbol{x}_{k}^{N})\sim\nu^{\bar{\lambda}}$.
Remark 5.2.7 at the end of the section gives an alternative way to justify the
index reversal on the left-hand side above when $v=0$.

Before proceeding with the proof, we give a roadmap. First, by definition of
$\mathcal{P}$, if $\widecheck{U}_{i}=U_{-i-1}$, then for $x\in\mathbb{R}$,

$\displaystyle H_{\xi}^{N}(x)$
$\displaystyle=N^{-1/2}\mathcal{P}[\mathbf{1}_{U\leq
v+\xi(1-v^{2})N^{-1/2}}]\Bigl{(}\frac{\sigma^{2}x}{1-v^{2}}N\Bigr{)}-\frac{\sigma^{2}vx}{1-v^{2}}N^{1/2}$
(5.19) $\displaystyle=-N^{-1/2}\mathcal{P}[\mathbf{1}_{\widecheck{U}\leq
v+\xi(1-v^{2})N^{-1/2}}]\Bigl{(}-\frac{\sigma^{2}x}{1-v^{2}}N\Bigr{)}-\frac{\sigma^{2}vx}{1-v^{2}}N^{1/2}.$

Our goal is to show the weak limit

$\displaystyle\biggl{(}-N^{-1/2}\mathcal{P}[\boldsymbol{x}_{k}^{N}]\Bigl{(}\frac{\sigma^{2}\hskip
0.5pt{\hskip 1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt}}{1-v^{2}}N\Bigr{)}+\frac{\sigma^{2}v\hskip 0.5pt{\hskip
1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt}}{1-v^{2}}N^{1/2}\,,$ (5.20)
$\displaystyle\qquad-N^{-1/2}\mathcal{P}[D_{d}(\boldsymbol{x}^{N}_{k-1},\boldsymbol{x}_{k}^{N})]\Bigl{(}\frac{\sigma^{2}\hskip
0.5pt{\hskip 1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt}}{1-v^{2}}N\Bigr{)}+\frac{\sigma^{2}v\hskip 0.5pt{\hskip
1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt}}{1-v^{2}}N^{1/2},\ldots,$
$\displaystyle\qquad\qquad-N^{-1/2}\mathcal{P}[D_{d}^{k}(\boldsymbol{x}_{1}^{N},\ldots,\boldsymbol{x}_{k}^{N})]\Bigl{(}\frac{2\hskip
0.5pt{\hskip 1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt}}{1-v^{2}}N\Bigr{)}+\frac{2v\hskip 0.5pt{\hskip
1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt}}{1-v^{2}}N^{1/2}\biggr{)}$
$\displaystyle\qquad\qquad\Longrightarrow(G^{\sigma}_{-\xi_{k}},\ldots,G^{\sigma}_{-\xi_{1}}).$

Once we show (5.20), from (5.20), (5.18) and (5.19) it follows that
$(H_{\xi_{1}}^{N},\ldots,H_{\xi_{k}}^{N})\Longrightarrow(G^{\sigma}_{-\xi_{1}}(-\hskip
0.5pt{\hskip 1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt}),\ldots,G^{\sigma}_{-\xi_{k}}(-\hskip 0.5pt{\hskip
1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip 1.1pt}))$. This limit has
the same distribution as $(G^{\sigma}_{\xi_{1}},\ldots,G^{\sigma}_{\xi_{k}})$
by Theorem 2.3.2(iv). As mentioned previously, these reflections in the proof
are a consequence of having time flow left to right in the queuing setting.

Now, we prove (5.20). By construction, for $j\in\llbracket 1,k\rrbracket$,
$\\{{x}^{N}_{j}[i]\\}_{i\in\mathbb{Z}}$ is an i.i.d. Bernoulli sequence with
intensity $\sum_{\ell\leq
j}\lambda_{\ell}=\frac{1}{2}{(1+v+\xi_{j}(1-v^{2})N^{-1/2})}$. Hence, for
$j\in\llbracket 1,k\rrbracket$,

$-N^{-1/2}\mathcal{P}[\boldsymbol{x}_{j}^{N}]\Bigl{(}\frac{\sigma^{2}{\hskip
1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt}}{1-v^{2}}N\Bigr{)}+\frac{\sigma^{2}v\hskip 0.5pt{\hskip
1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip 1.1pt}}{1-v^{2}}N^{1/2}$
(5.21)

converges in distribution to a Brownian motion with diffusivity $\sigma$ and
drift $-\sigma^{2}\xi_{j}$. To elevate this to the joint convergence of
(5.20), we utilize the queuing mappings in (5.18) and the transformations
$D^{(k)}$ from (2.5) that construct the SH.

By Skorokhod representation ([54, Thm. 11.7.2], [59, Thm. 3.1.8]), we may
couple $\\{\boldsymbol{x}_{j}^{N}\\}_{j=1,\ldots,k}$ and independent Brownian
motions $\\{Z^{j}\\}_{j\in\llbracket 1,k\rrbracket}$ with diffusivity $\sigma$
and drift $-\sigma^{2}\xi_{j}$ so that, with probability one, for
$j\in\llbracket 1,k\rrbracket$, (5.21) converges uniformly on compact sets to
$Z^{j}$. Let $\mathbb{P}$ be the law of this coupling (To be precise, the
sequences $\boldsymbol{x}_{j}^{N}$ are functions of the converging processes
(5.21), which Skorokhod representation couples with their limiting Brownian
motions).

By Proposition 2.3.1(ii) and definition of $D^{(k)}$ (2.5), for reals
$\xi_{1}<\cdots<\xi_{k}$, the valued marginal
$(G^{\sigma}_{-\xi_{k}},\dotsc,G^{\sigma}_{-\xi_{1}})$ of the SH can be
constructed as follows:

$\displaystyle G^{\sigma}_{-\xi_{k}}$ $\displaystyle=D(Z^{k})=Z^{k},\qquad
G^{\sigma}_{-\xi_{k-1}}=D^{(2)}(Z^{k},Z^{k-1})=D(Z^{k-1},Z^{k}),\ldots$
$\displaystyle G^{\sigma}_{-\xi_{1}}$
$\displaystyle=D^{(k)}(Z^{1},Z^{2},\dotsc,Z^{k})=D(D^{(k-1)}(Z^{1},\ldots,Z^{k-1}),Z^{k}).$

We recall (2.3) that The map $D$ is

$D(Z,B)(y)=B(y)+\sup_{-\infty<x\leq y}\\{Z(x)-B(x)\\}-\sup_{-\infty<x\leq
0}\\{Z(x)-B(x)\\}.$

By a union bound, it suffices to show that, under this coupling, for each
$\varepsilon>0,a>0$, and each $j=0,\ldots,k-1$,

$\displaystyle\limsup_{N\to\infty}\mathbb{P}\biggl{(}\;\sup_{x\in[-a,a]}\Bigl{|}-N^{-1/2}\mathcal{P}[D_{d}^{(j+1)}(\boldsymbol{x}_{k-j}^{N},\ldots,\boldsymbol{x}_{k}^{N})]$
$\displaystyle\Bigl{(}\frac{2x}{1-v^{2}}N\Bigr{)}+\frac{2vx}{1-v^{2}}N^{1/2}$
(5.22)
$\displaystyle-D^{(j+1)}(Z^{k-j},\ldots,Z^{k})(x)\Bigr{|}>\varepsilon\biggr{)}=0,$

where the argument of the discrete operator is understood via linear
interpolation. We prove (5.22) by induction on $j$. The base case $j=0$
follows by the almost sure uniform convergence on compact sets of (5.21) to
$Z^{j}$. Now, assume the statement holds for some $j-1\in\\{0,\ldots,k-2\\}$.
The proof is completed by Lemma 5.2.6 below, once we recall (5.8) that

$D_{d}^{(j+1)}(\boldsymbol{x}_{k-j}^{N},\ldots,\boldsymbol{x}_{k}^{N})=D_{d}(D_{d}^{(j)}(\boldsymbol{x}_{k-j}^{N},\ldots,\boldsymbol{x}_{k-1}^{N}),\boldsymbol{x}_{k}^{N}).\qed$

######  Lemma 5.2.6.

Let $v\in(-1,1)$. For each $N>0$, let $\boldsymbol{a}^{N}$ and
$\boldsymbol{s}^{N}$ be $\\{0,1\\}^{\mathbb{Z}}$-valued i.i.d. sequences such
that the intensity of $\boldsymbol{s}^{N}$ is strictly greater than the
intensity of $\boldsymbol{a}^{N}$. Assume further that these sequences are
coupled together with Brownian motions $Z^{1},Z^{2}$ with diffusivity $\sigma$
and drifts $-\sigma^{2}\xi_{1}>-\sigma^{2}\xi_{2}$ so that, for each
$\varepsilon>0$ and $a>0$,

$\displaystyle\limsup_{N\to\infty}\mathbb{P}\Biggl{(}\sup_{y\in[-a,a]}\Bigl{|}-N^{-1/2}\mathcal{P}[\boldsymbol{a}^{N}]\Bigl{(}\frac{\sigma^{2}y}{1-v^{2}}N\Bigr{)}+\frac{\sigma^{2}vy}{1-v^{2}}N^{1/2}-Z^{1}(y)\Bigr{|}$
(5.23)
$\displaystyle\qquad\qquad\vee\Bigl{|}-N^{-1/2}\mathcal{P}[\boldsymbol{s}^{N}]\Bigl{(}\frac{\sigma^{2}y}{1-v^{2}}N\Bigr{)}+\frac{\sigma^{2}vy}{1-v^{2}}N^{1/2}-Z^{2}(y)\Bigr{|}>\varepsilon\Biggr{)}=0.$

Then, for every $\varepsilon>0$ and $a>0$,

$\limsup_{N\to\infty}\mathbb{P}\Biggl{(}\sup_{y\in[-a,a]}\Bigl{|}-N^{-1/2}\mathcal{P}[D_{d}(\mathbf{a}^{N},\mathbf{s}^{N})]\Bigl{(}\frac{\sigma^{2}y}{1-v^{2}}N\Bigr{)}+\frac{\sigma^{2}vy}{1-v^{2}}N^{1/2}\\\
-D(Z^{1},Z^{2})(y)\Bigr{|}>\varepsilon\Biggr{)}=0.$ (5.24)

###### Proof.

From (5.14), we have, for $\frac{\sigma^{2}Nx}{1-v^{2}}\in\mathbb{Z}$,

$\displaystyle-N^{-1/2}\mathcal{P}[D_{d}(\boldsymbol{a}^{N},\boldsymbol{s}^{N})]\Bigl{(}\frac{\sigma^{2}x}{1-v^{2}}N\Bigr{)}+\frac{\sigma^{2}vx}{1-v^{2}}N^{1/2}$
(5.25)
$\displaystyle=-N^{-1/2}\mathcal{P}[\boldsymbol{s}^{N}]\Bigl{(}\frac{\sigma^{2}x}{1-v^{2}}N\Bigr{)}+\frac{\sigma^{2}vx}{1-v^{2}}N^{1/2}$
$\displaystyle\qquad\qquad+N^{-1/2}\sup_{-\infty<j\leq\sigma^{2}Nx/(1-v^{2})}[\mathcal{P}[\boldsymbol{s}^{N}](j)-\mathcal{P}[\boldsymbol{a}^{N}](j)]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-N^{-1/2}\sup_{-\infty<j\leq
0}[\mathcal{P}[\boldsymbol{s}^{N}](j)-\mathcal{P}[\boldsymbol{a}^{N}](j)]$
$\displaystyle=-N^{-1/2}\mathcal{P}[\boldsymbol{s}^{N}]\Bigl{(}\frac{\sigma^{2}x}{1-v^{2}}N\Bigr{)}+\frac{\sigma^{2}vx}{1-v^{2}}N^{1/2}$
$\displaystyle\qquad\qquad+\sup_{-\infty<j\leq\sigma^{2}Nx/(1-v^{2})}[-N^{-1/2}\mathcal{P}[\boldsymbol{a}^{N}](j)-(-N^{-1/2}\mathcal{P}[\boldsymbol{s}^{N}](j))]$
$\displaystyle\qquad\qquad\qquad-\sup_{-\infty<j\leq
0}[-N^{-1/2}\mathcal{P}[\boldsymbol{a}^{N}](j)-(-N^{-1/2}\mathcal{P}[\boldsymbol{s}^{N}](j))].$

Hence, from (5.25) and the assumed convergence of
$N^{-1/2}\mathcal{P}[\boldsymbol{s}^{N}](\frac{\sigma^{2}N}{1-v^{2}}\hskip
0.5pt{\hskip 1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip 1.1pt})$ in
probability (5.23), to prove (5.24), it suffices to show that, for each $a>0$
and $\varepsilon>0$,

$\limsup_{N\to\infty}\mathbb{P}\biggl{(}\,\sup_{y\in[-a,a]}\Bigl{|}\sup_{-\infty<x\leq[\sigma^{2}Ny/(1-v^{2})]}\bigl{[}-N^{-1/2}\mathcal{P}[\boldsymbol{a}^{N}](x)-(-N^{-1/2}\mathcal{P}[\boldsymbol{s}^{N}](x))\bigr{]}\\\
-\sup_{-\infty<x\leq y}[Z^{1}(x)-Z^{2}(x)]\Bigr{|}>\varepsilon\biggr{)}=0.$
(5.26)

For $x\in\mathbb{R}$, recall that $[x]$ denotes the integer closest to $x$
with $|[x]|\leq|x|$. Note that there is a drift term for both the walks
$\mathcal{P}[\boldsymbol{a}^{N}]$ and $\mathcal{P}[\boldsymbol{s}^{N}]$ that
cancels when they are subtracted. For shorthand, let

$X^{N}(x)=-N^{-1/2}\mathcal{P}[\boldsymbol{a}^{N}](x)-(-N^{-1/2}\mathcal{P}[\boldsymbol{s}^{N}](x))$

For the $a$ in the hypothesis of the lemma and arbitrary $S>a$, let
$E_{N,a,S}$ be the event where the following three conditions all hold:

  1. (i)

$\sup_{-\infty<x\leq[-\sigma^{2}Na/(1-v^{2})]}\bigl{[}X^{N}(x)\bigr{]}=\sup_{[-\sigma^{2}NS/(1-v^{2})]\leq
x\leq[-\sigma^{2}Na/(1-v^{2})]}\bigl{[}X^{N}(x)\bigr{]}.$

  2. (ii)

$\sup_{-\infty<x\leq-a}[Z^{1}(x)-Z^{2}(x)]=\sup_{-S\leq
x\leq-a}[Z^{1}(x)-Z^{2}(x)].$

  3. (iii)

$\sup_{y\in[-a,a]}\Bigl{|}\sup_{[-\sigma^{2}NS/(1-v^{2})]\leq
x\leq[\sigma^{2}Nx/(1-v^{2})]}X^{N}(y)-\sup_{-S\leq x\leq
y}[Z^{1}(x)-Z^{2}(x)]\Bigr{|}\leq\varepsilon.$

For every $S>a$, the event in (5.26) is contained in $E_{N,a,S}^{c}$. By
assumption (5.23) and Lemma A.3.4 (applied to the random walk
$-\mathcal{P}[\boldsymbol{a}^{N}]+\mathcal{P}[\boldsymbol{s}^{N}]$ with
$m=\xi_{2}-\xi_{1}$, $\sigma=\sqrt{2\sigma^{2}}$,
$\varphi(N)=N^{-1/2},\xi(N)=\sigma^{2}N/(1-v^{2})$, and $B=Z^{1}({\hskip
1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt}/(2\sigma^{2}))-Z^{2}({\hskip
1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip 1.1pt}/(2\sigma^{2}))$),
$\lim_{S\to\infty}\limsup_{N\to\infty}\mathbb{P}(E_{N,a,S}^{c})=0$, completing
the proof. ∎

######  Remark 5.2.7.

For $v=0$, one can alternatively arrive at (5.18) by considering the speed
process for TASEP with left jumps. As in (5.1), let $X_{i}(t)$ be the position
of the right-going particle with label $i$ that starts at $X_{i}(0)=i$ and
define the right-going speed process by
$U_{i}=\lim_{t\to\infty}t^{-1}X_{i}(t)$. To flip the space direction, define
left-going particles $\widetilde{X}_{i}(t)=-X_{-i}(t)$ and the corresponding
speed process
$\widetilde{U}_{i}=\lim_{t\to\infty}t^{-1}\widetilde{X}_{i}(t)=-U_{-i}$.
Reversing the lattice direction reversed the priorities of the labels (for the
walks $\widetilde{X}$, lower label means lower priority), so the non-
decreasing projection $F$ to left-going multiclass stationary measures has to
be applied to speeds $-\widetilde{U}_{i}=U_{-i}$.

The distributional equality
$\\{\widetilde{U}_{i}\\}_{i\in\mathbb{Z}}\overset{d}{=}\\{U_{i}\\}_{i\in\mathbb{Z}}$
from [2, Proposition 5.2] implies that both speed processes have the same SH
limit. This can also be verified by replacing $U$ with $\widetilde{U}$ in
(5.3), rearranging, taking the limit, and using the reflection property
(Proposition 2.3.1(iv)) of the SH.

##  Appendix A Auxiliary technical inputs

###  A.1 Maximizers of continuous functions

Recall the definitions of $f(x,y)$ and $f\;{\leq}_{\rm inc}\;g$ from Section
1.5.

######  Lemma A.1.1.

Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions satisfying $f(x)\vee
g(x)\to-\infty$ as $x\to\pm\infty$ and $f\;{\leq}_{\rm inc}\;g$. Let
$x_{f}^{L}$ and $x_{f}^{R}$ be the leftmost and rightmost maximizers of $f$
over $\mathbb{R}$, and similarly defined for $g$. Then, $x_{f}^{L}\leq
x_{g}^{L}$ and $x_{f}^{R}\leq x_{g}^{R}$.

###### Proof.

By the definition of $x_{g}^{R}$ and by the assumption $f\;{\leq}_{\rm
inc}\;g$, for all $x>x_{g}^{R}$

$f(x_{g}^{R},x)\leq g(x_{g}^{R},x)<0.$

Hence, the rightmost maximizer of $f$ must be less than or equal to
$x_{g}^{R}$. We get the statement for leftmost maximizers by considering the
functions $x\mapsto f(-x)$ and $g\mapsto g(-x)$. ∎

######  Lemma A.1.2.

Assume that $f,g:\mathbb{R}\to\mathbb{R}$ satisfy $f\;{\leq}_{\rm inc}\;g$.
Then, for $a\leq x\leq y\leq b$,

$0\leq g(x,y)-f(x,y)\leq g(a,b)-f(a,b).$

###### Proof.

The first inequality follows immediately from the assumption $f\;{\leq}_{\rm
inc}\;g$. The second follows from the inequality

$f(a,b)-f(x,y)=f(a,x)+f(y,b)\leq g(a,x)+g(y,b)=g(a,b)-g(x,y).\qed$

######  Lemma A.1.3.

Let $S\subseteq\mathbb{R}^{n}$, and let $f_{n}:S\rightarrow\mathbb{R}$ be
continuous functions that converge uniformly to $f:S\rightarrow\mathbb{R}$.
Let $c_{n}$ be a maximizer of $f_{n}$ and assume $c_{n}\to c\in S$. Then $c$
is a maximizer of $f$.

###### Proof.

$f_{n}(c_{n})\geq f_{n}(x)$ for all $x\in S$, so it suffices to show that
$f_{n}(c_{n})\rightarrow f(c)$. This follows from the uniform convergence of
$f_{n}$ to $f$, the continuity of $f$, and

$|f_{n}(c_{n})-f(c)|\leq|f_{n}(c_{n})-f(c_{n})|+|f(c_{n})-f(c)|.\qed$

The following is a well-known fact, but is often stated without proof, so we
include full justification for the sake of completeness. For example, it
appears as Equation (1.4) in [124] and Equation (13) in [119].

######  Lemma A.1.4.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that

$\lim_{x\rightarrow\pm\infty}f(x)=\pm\infty.$

Set $F(y)=\sup_{-\infty<x\leq y}f(x)$. Then,

$\inf_{y\leq x<\infty}(2F(x)-f(x))=F(y).$ (A.1)

###### Proof.

The left-hand side of (A.1) is

$\inf_{y\leq x<\infty}(2F(x)-f(x))=\inf_{y\leq
x<\infty}\bigl{(}2\sup_{-\infty<u\leq x}f(u)-f(x)\bigr{)}.$

For all $x\geq y$, $\sup_{\infty<u\leq x}f(u)$ is greater than or equal to
both $f(x)$ and $\sup_{-\infty<u\leq y}f(u)$. Therefore,

$2\sup_{-\infty<u\leq x}f(u)-f(x)\geq\sup_{-\infty<u\leq
y}f(u)+f(x)-f(x)=F(y).$

This establishes one direction of (A.1). To show the other direction, we show
that there exists $x\geq y$ such that

$2F(x)-f(x)=F(y).$

Note that $f(y)\leq F(y)$ recall the assumption
$\lim_{x\rightarrow\infty}f(x)=\infty$. By the intermediate value theorem,
$f(x)=F(y)$ for some $x\geq y$. Let

$x^{\star}=\inf\\{x\geq y:f(x)=F(y)\\}.$

Then, by continuity, $f(x^{\star})=F(x^{\star})=F(y)$, and

$2F(x^{\star})-f(x^{\star})=2F(y)-F(y)=F(y).\qed$

###  A.2 Miscellaneous probabilistic facts

######  Lemma A.2.1.

For a nonnegative random variable $Z$ on $(\Omega,\mathcal{F},\mathbb{P})$ and
$\alpha\in\mathbb{R}$, the following holds

$\int_{\Omega}e^{-\alpha Z}\,d\mathbb{P}=1-\alpha\int_{0}^{\infty}e^{-\alpha
z}\mathbb{P}(Z>z)\,dz.$

###### Proof.

We use Tonelli’s Theorem below.

$\displaystyle-\alpha\int_{0}^{\infty}e^{-\alpha
z}\mathbb{P}(Z>z)\,dz=-\alpha\int_{0}^{\infty}\int_{\Omega}e^{-\alpha
z}1(Z>z)\,d\mathbb{P}dz$ $\displaystyle=$
$\displaystyle-\alpha\int_{\Omega}\int_{0}^{\infty}e^{-\alpha
z}1(Z>z)\,dzd\mathbb{P}=-\alpha\int_{\Omega}\int_{0}^{Z}e^{-\alpha
z}\,dzd\mathbb{P}$ $\displaystyle=$ $\displaystyle\int_{\Omega}(e^{-\alpha
Z}-1)\,d\mathbb{P}=\int_{\Omega}e^{-\alpha Z}d\mathbb{P}-1.$

Rearranging this equation completes the proof. ∎

The following is a classical result. We provide a proof for the specific
formulation we use.

######  Theorem A.2.2 (The Cameron-Martin-Girsanov Theorem).

Let $X:[0,\infty)\to\mathbb{R}$ be a standard Brownian motion with drift
$\lambda\in\mathbb{R}$ on the space $(\Omega,\mathcal{F},\mathbb{P})$. Then,
for $t>0$ and $\rho\in\mathbb{R}$ consider the measure on
$(\Omega,\mathcal{F})$ defined by

$\widehat{\mathbb{P}}(A)=\mathbb{E}\Bigl{[}\exp\Bigl{(}(\rho-\lambda)X(t)+\frac{\lambda^{2}-\rho^{2}}{2}t\Bigr{)}\mathbf{1}_{A}\Bigr{]}.$

Then, under $\widehat{\mathbb{P}}$, $\\{X(s):0\leq s\leq t\\}$ is a standard
Brownian motion with drift $\rho$.

###### Proof.

The measures $\mathbb{P}$ and $\widehat{\mathbb{P}}$ are mutually absolutely
continuous, so continuity is preserved. It remains to show that, under
$\widehat{\mathbb{P}}$, for $0=t_{0}<t_{1}<\cdots<t_{n-1}<t_{n}=t$,
$X(t_{1}),X(t_{2})-X(t_{1}),\ldots,X(t_{n})-X(t_{n-1})$ are independent
Gaussian random variables with mean $\rho(t_{i}-t_{i-1})$ and variance
$t_{i}-t_{i-1}$. We prove this via moment generating functions: For $1\leq
i\leq n$, set $s_{i}=t_{i}-t_{i-1}$ and $Y_{i}=X(t_{i})-X(t_{i-1})$. Let
$\alpha_{1},\ldots,\alpha_{n}\in\mathbb{R}$. Then, using the fact that
$t=s_{1}+\cdots+s_{n}$ and $X(t)=Y_{1}+\cdots+Y_{n}$,

$\displaystyle\quad\;\widehat{\mathbb{E}}\Bigl{[}\exp\Bigl{(}\sum_{i=1}^{n}\alpha_{i}Y_{i}\Bigr{)}\Bigr{]}=\mathbb{E}\Bigl{[}\exp\Bigl{(}(\rho-\lambda)(Y_{1}+\cdots+Y_{n})+\frac{\lambda^{2}-\rho^{2}}{2}t+\sum_{i=1}^{n}\alpha_{i}Y_{i}\Bigr{)}\Bigr{)}\Bigr{]}$
$\displaystyle=\int_{\mathbb{R}^{n}}\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi
s_{i}}}\exp\bigl{(}-\frac{(y_{i}-\lambda
s_{i})^{2}}{2s_{i}}+(\rho-\lambda)y_{i}+\alpha_{i}y_{i}+\frac{\lambda^{2}-\rho^{2}}{2}s_{i}\bigr{)}\,dy_{i}$
$\displaystyle=\prod_{i=1}^{n}\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi
s_{i}}}\exp\bigl{(}-\frac{(y_{i}-\lambda
s_{i})^{2}}{2s_{i}}+(\rho-\lambda)y_{i}+\alpha_{i}y_{i}+\frac{\lambda^{2}-\rho^{2}}{2}s_{i}\bigr{)}\,dy_{i}$
$\displaystyle=\prod_{i=1}^{n}\exp\bigl{(}\lambda
s_{i}(\rho-\lambda+\alpha_{i})+\frac{s_{i}(\rho-\lambda+\alpha_{i})^{2}}{2}+\frac{\lambda^{2}-\rho^{2}}{2}s_{i}\Bigr{)}=\prod_{i=1}^{n}\exp(\rho\alpha_{i}s_{i}+s_{i}\alpha_{i}^{2}/2).\;\;\qed$

The following is derived from a classical result.

######  Lemma A.2.3.

[29, Equation 1.1.4 (1) on pg 251 ] For a standard Brownian motion $B$ and
$\xi>0$,

$\sup_{-\infty<u\leq 0}\\{\sqrt{2}B(u)+\xi u\\}\sim\operatorname{Exp}(\xi).$

###### Proof.

The statement given in [29] is

$\sup_{0\leq u\leq\infty}\\{B(u)-\xi u\\}\sim\operatorname{Exp}(2\xi).$

To get to the result we use, observe that

$\sup_{-\infty<u\leq 0}\\{\sqrt{2}B(u)+\xi u\\}=\sqrt{2}\sup_{0\leq
u<\infty}\bigl{\\{}B(-u)-\frac{\xi}{\sqrt{2}}u\bigr{\\}}\sim\sqrt{2}\operatorname{Exp}(\sqrt{2}\xi)\sim\operatorname{Exp}(\xi).\qed$

The following is also well-known (see, for example, [29, Equation 1.2.4 on
page 251]; this result is stated for the running infimum, but can be modified
by reflection invariance of Brownian motion). We provide a short proof for the
specific choice of parameterization we use

######  Lemma A.2.4.

Let $B$ be a standard Brownian motion. Then, for all $x\geq 0$,

$\mathbb{P}\bigl{(}\sup_{0\leq u\leq 1}\\{\sqrt{2}B(u)+\xi u\\}\leq
x\bigr{)}=\Phi\Bigl{(}\frac{x-\xi}{\sqrt{2}}\Bigr{)}-e^{\xi
x}\Phi\Bigl{(}\frac{-x-\xi}{\sqrt{2}}\Bigr{)}.$

###### Proof.

We prove that

$\mathbb{P}\bigl{(}\sup_{0\leq u\leq 1}\\{B(u)+\xi u\\}\leq
x\bigr{)}=\Phi(x-\xi)-e^{2\xi x}\Phi(-x-\xi),$ (A.2)

and the statement given follows by replacing $x$ with $x/\sqrt{2}$ and $\xi$
with $\xi/\sqrt{2}$. Let

$W^{\xi}(u)=B(u)+\xi u,\quad M^{\xi}=\sup_{0\leq u\leq 1}\\{W^{\xi}(u)\\},$

and denote $M=M^{0}$ and $W=W^{0}$. A classical fact due to the reflection
principle (see for example, [55, Exercise 7.4.3]), is that $M$ and $W(1)$ have
joint density

$f(m,w)=\frac{2(2m-w)}{\sqrt{2\pi}}e^{-(2m-w)^{2}/2}\mathbf{1}(m\geq
0,-\infty<w\leq m).$

By Theorem A.2.2, $M^{\xi}$ and $W^{\xi}(1)$ have joint density

$f^{\xi}(m,w)=\frac{2(2m-w)}{\sqrt{2\pi}}e^{-(2m-w)^{2}/2+\xi
w-\xi^{2}/2}\mathbf{1}(m>0,-\infty<w<m).$

Hence,

$\displaystyle\mathbb{P}(M^{\xi}\leq x)$
$\displaystyle=\int_{-\infty}^{x}\int_{w}^{x}\frac{2(2m-w)}{\sqrt{2\pi}}e^{-(2m-w)^{2}/2+\xi
w-\xi^{2}/2}dm\,dw$
$\displaystyle=\int_{-\infty}^{x}\int_{w}^{2x-w}\frac{u}{\sqrt{2\pi}}e^{-u^{2}/2+\xi
w-\xi^{2}/2}\,du\,dw$ $\displaystyle=\int_{-\infty}^{x}\frac{e^{\xi
w-\xi^{2}/2}}{\sqrt{2\pi}}\Bigl{[}e^{-w^{2}/2}-e^{-(2x-w)^{2}/2}\Bigr{]}\,dw$
$\displaystyle=\int_{-\infty}^{x}\frac{e^{-(w-\xi)^{2}/2}}{\sqrt{2\pi}}\,dw-e^{2x\xi}\int_{-\infty}^{x}\frac{e^{-(w-(2x+\xi))^{2}/2}}{\sqrt{2\pi}}\,dx,$

and this equals (A.2) after a simple change of variables. ∎

###  A.3 Maximum of random walk

We first state a random walk lemma that comes from p. 519–520 in [128]. See
also Chapter VIII, Section 6 in [4].

######  Lemma A.3.1 ([128], Proposition 6.9.4. See also Chapter VIII, Section
6 in [4]).

Let $\mu_{N}$ be a sequence of strictly positive numbers with $\mu_{N}\to 0$.
Let $\sigma_{N}$ be a sequence satisfying $\sigma_{N}\to 1$. For each $N$, let
$\\{X_{N,i}:i\in\mathbb{Z}\\}$ be a collection of i.i.d. random variables with
mean $\mu_{N}$ and variance $\sigma_{N}^{2}$. Further, suppose that the
sequence $\\{X_{N,0}^{2}:N\geq 1\\}$ is uniformly integrable, meaning that

$\lim_{b\to\infty}\sup_{N\geq
1}\mathbb{E}\bigl{[}X_{N,0}^{2}\mathbf{1}_{|X_{N,0}|>b}\bigr{]}=0.$

Let $B$ be a Brownian motion with diffusion coefficient $1$ and zero drift.
Then, the following convergence in distribution holds:

$\sup_{-\infty<x\leq 0}\mu_{N}S^{N}(\lceil
x\rceil)\underset{N\to\infty}{\Longrightarrow}\sup_{-\infty<x\leq
0}\\{B(x)+x\\}.$

Next, we reformulate the statement slightly to suit our purposes.

######  Lemma A.3.2.

Let $\mu_{N}$ be a sequence of strictly positive numbers with $\mu_{N}\to 0$.
Let $\sigma_{N}$ be a sequence satisfying $\sigma_{N}\to\sigma>0$. Let
$\varphi(N)$ be a sequence satisfying $\mu_{N}/\varphi(N)\to m>0$. For each
$N$, let $\\{X_{N,i}:i\in\mathbb{Z}\\}$ be a collection of i.i.d. random
variables with mean $\mu_{N}$ and variance $\sigma_{N}^{2}$. Further, suppose
that the sequence $\\{X_{N,0}^{2}:N\geq 1\\}$ is uniformly integrable. Let
$S^{N}(m)$ be defined as

$S^{N}(m)=\begin{cases}-\sum_{i=m}^{-1}X_{N,i}&m\leq 0\\\
\sum_{i=0}^{m-1}X_{N,i}&m\geq 0\end{cases}$ (A.3)

with $S^{N}(0)=0$. Let $B$ be a Brownian motion with diffusion coefficient $1$
and zero drift. Then, the following convergence in distribution holds:

$\sup_{-\infty<x\leq
0}\varphi(N)S^{N}(\lceil{x}\rceil)\underset{N\to\infty}{\Longrightarrow}\sup_{-\infty<x\leq
0}\\{\sigma B(x)+mx\\}$ (A.4)

######  Remark A.3.3.

It is immediate that on the left-hand side of (A.4), one can replace $x$ with
$\lceil\xi(N)x\rceil$ for any strictly positive sequence $\xi(N)$.

###### Proof.

The random variables $X_{N,i}/\sigma_{N}$ satisfy the conditions of Lemma
A.3.1 with $\mu_{N}$ replaced by $\mu_{N}/\sigma_{N}$. Then,

$\displaystyle\,\sup_{-\infty<x\leq
0}\varphi(N)S^{N}(x)=\frac{\varphi(N)\sigma_{N}^{2}}{\mu_{N}}\sup_{-\infty<x\leq
0}\frac{\mu_{N}}{\sigma_{N}}\cdot\frac{S^{N}(x)}{\sigma_{N}}$
$\displaystyle=\frac{\varphi(N)\sigma_{N}^{2}}{\mu_{N}}\sup_{-\infty<x\leq
0}\frac{\mu_{N}}{\sigma_{N}}\cdot\frac{S^{N}(\lceil\sigma_{N}^{2}x/\mu_{N}^{2}\rceil)}{\sigma_{N}}\
\overset{\rm Lemma\ref{lem:resnick}}{\Longrightarrow}\
\frac{\sigma^{2}}{m}\sup_{-\infty<x\leq 0}\\{B(x)+x\\}$
$\displaystyle=\frac{\sigma^{2}}{m}\sup_{-\infty<x\leq
0}\Bigl{\\{}B\Bigl{(}\frac{m^{2}}{\sigma^{2}}x\Bigr{)}+\frac{m^{2}}{\sigma^{2}}x\Bigr{\\}}\overset{d}{=}\sup_{-\infty<x\leq
0}\\{\sigma B(x)+mx\\},$

so we may now apply Lemma A.3.1. ∎

For $x\in\mathbb{R}$, recall that $[x]$ denotes the integer closest to $x$
with $|[x]|\leq|x|$.

######  Lemma A.3.4.

Consider the setting of Lemma A.3.2. Let $\xi(N)$ be a sequence satisfying
$\varphi(N)^{2}\xi(N)\to R>0$. Then, for each $S<T\in\mathbb{R}$,

$\displaystyle\,\lim_{N\to\infty}\mathbb{P}\Bigl{[}\,\sup_{-\infty<x\leq[S\xi(N)]}\varphi(N)S^{N}(x)>\sup_{[S\xi(N)]\leq
x\leq[T\xi(N)]}\varphi(N)S^{N}(x)\Bigr{]}$ (A.5)
$\displaystyle\qquad\qquad=\mathbb{P}\bigl{[}\sup_{-\infty<x\leq S}\\{\sigma
B(Rx)+mRx\\}>\sup_{S\leq x\leq T}\\{\sigma B(Rx)+mRx\\}\bigr{]}.$

###### Proof.

Observe that

$\displaystyle\mathbb{P}\Bigl{[}\,\sup_{-\infty<x\leq[S\xi(N)]}\varphi(N)S^{N}(x)>\sup_{[S\xi(N)]\leq
x\leq[T\xi(N)]}\varphi(N)S^{N}(x)\Bigr{]}$
$\displaystyle=\mathbb{P}\Bigl{[}\,\sup_{-\infty<x\leq[S\xi(N)]}\varphi(N)S^{N}([S\xi(N)],x)>\sup_{[S\xi(N)]\leq
x\leq[T\xi(N)]}\varphi(N)S^{N}([S\xi(N)],x)\Bigr{]}.$

Now, note that

$\sup_{-\infty<x\leq[S\xi(N)]}\varphi(N)S^{N}([S\xi(N)],x),\quad\text{and}\quad\sup_{[S\xi(N)]\leq
x\leq[T\xi(N)]}\varphi(N)S^{N}([S\xi(N)],x)$

are independent. By convergence of random walk to Brownian motion with drift
(with respect to the topology of uniform convergence on compact sets) , we get
that

$\sup_{[S\xi(N)]\leq
x\leq[T\xi(N)]}\varphi(N)S^{N}(RS,[S\xi(N)],x)\Longrightarrow\sup_{S\leq x\leq
T}\\{\sigma B(RS,Rx)+R(m-S)x\\}.$

By shift invariance of random walk and Lemma A.3.2,

$\displaystyle\,\sup_{-\infty<x\leq[S\xi(N)]}\varphi(N)S^{N}([S\xi(N)],x)\overset{d}{=}\sup_{-\infty<x\leq
0}\varphi(N)S^{N}(x)$ $\displaystyle\Longrightarrow\sup_{-\infty<x\leq
0}\\{\sigma B(x)+mx\\}=\sup_{-\infty<x\leq 0}\\{\sigma B(Rx)+mRx\\}$
$\displaystyle\overset{d}{=}\sup_{-\infty<x\leq S}\\{\sigma
B(RS,Rx)+m(R-S)x\\}.$

By independence, we have shown the following joint convergence:

$\displaystyle\,\Bigl{(}\sup_{-\infty<x\leq[S\xi(N)]}\varphi(N)S^{N}([S\xi(N)],x),\sup_{[S\xi(N)]\leq
x\leq[T\xi(N)]}\varphi(N)S^{N}([S\xi(N)],x)\Bigr{)}$ (A.6)
$\displaystyle\Longrightarrow\Bigl{(}\sup_{-\infty<x\leq S}\\{\sigma
B(RS,Rx)+m(R-S)x\\},\sup_{S\leq x\leq T}\\{\sigma
B(RS,Rx)+m(R-S)x\\}\Bigr{)}.$

The right-hand side of (A.6) consists of two independent random variables with
continuous distribution. Therefore,

$\displaystyle\quad\lim_{N\to\infty}\mathbb{P}\Bigl{[}\,\sup_{-\infty<x\leq[S\xi(N)]}\varphi(N)S^{N}(x)>\sup_{[S\xi(N)]\leq
x\leq[T\xi(N)]}\varphi(N)S^{N}(x)\Bigr{]}$
$\displaystyle=\lim_{N\to\infty}\mathbb{P}\Bigl{[}\,\sup_{-\infty<x\leq[S\xi(N)]}\varphi(N)S^{N}([S\xi(N)],x)>\sup_{[S\xi(N)]\leq
x\leq[T\xi(N)]}\varphi(N)S^{N}([S\xi(N)],x)\Bigr{]}$
$\displaystyle=\mathbb{P}\bigl{[}\sup_{-\infty<x\leq S}\\{\sigma
B(RS,Rx)+m(R-S)x\\}>\sup_{S\leq x\leq T}\\{\sigma B(Rx)+m(R-S)x\\}\bigr{]}$
$\displaystyle=\mathbb{P}\bigl{[}\sup_{-\infty<x\leq S}\\{\sigma
B(Rx)+mRx\\}>\sup_{S\leq x\leq T}\\{\sigma B(Rx)+mRx\\}\bigr{]},$

with the second equality holding because the event on the right-hand side is a
continuity set for the joint vector on the right in (A.6). ∎

##  Appendix B The directed landscape

###  B.1 Bounds and distributional invariances

The directed landscape satisfies the following symmetries.

######  Lemma B.1.1.

[47, Lemma 10.2],[49, Proposition 1.23] As a random continuous function of
$(x,s;y,t)\in\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}$, the directed
landscape $\mathcal{L}$ satisfies the following distributional symmetries, for
all $r,c\in\mathbb{R}$ and $q>0$.

  1. (i)

(Space-time stationarity)
$\mathcal{L}(x,s;y,t)\overset{d}{=}\mathcal{L}(x+c,s+r;y+c,t+r).$

  2. (ii)

(Skew stationarity)
$\mathcal{L}(x,s;y,t)\overset{d}{=}\mathcal{L}(x+cs,s;y+ct,t)-2c(x-y)+(t-s)c^{2}.$

  3. (iii)

(Spatial and temporal reflections)
$\mathcal{L}(x,s;y,t)\overset{d}{=}\mathcal{L}(-x,s;-y,t)\overset{d}{=}\mathcal{L}(y,-t;x,-s).$

  4. (iv)

(Rescaling)
$\mathcal{L}(x,s;y,t)\overset{d}{=}q\mathcal{L}(q^{-2}x,q^{-3}s;q^{-2}y,q^{-3}t).$

######  Lemma B.1.2.

[47, Corollary 10.7] There exists a
random constant $C$ such that for all
$v=(x,s;y,t)\in\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}$, we have

$\Bigl{|}\mathcal{L}(x,s;y,t)+\frac{(x-y)^{2}}{t-s}\Bigr{|}\leq
C(t-s)^{1/3}\log^{4/3}\Bigl{(}\frac{2(\|v\|+2)}{t-s}\Bigr{)}\log^{2/3}(\|v\|+2),$

where $\|v\|$ is the Euclidean norm.

######  Lemma B.1.3.

[44, Proposition 2.6] For every $i=1,\ldots,k$ and $\varepsilon>0$, let

$K_{i,\varepsilon}=\\{(x,s;y,t)\in\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}:s,t\in[0,\varepsilon],x,y\in[i-1/4,i+1/4]\\}.$
(B.1)

Then, there exists a coupling of $k+1$ copies of the directed landscape
$\mathcal{L}_{0},\mathcal{L}_{1},\ldots,\mathcal{L}_{k}$ so that
$\mathcal{L}_{1},\ldots,\mathcal{L}_{k}$ are independent, and almost surely,
there exists a random $\varepsilon>0$ such that for $1\leq i\leq k$,
$\mathcal{L}_{0}|_{K_{i,\varepsilon}}=\mathcal{L}_{i}|_{K_{i,\varepsilon}}$.

On a measure space $(\Omega,\mathcal{F},\mathbb{P})$, a measure-preserving
transformation $T$ satisfies $T^{-1}E\in\mathcal{F}$ and
$\mathbb{P}(T^{-1}E)=\mathbb{P}(E)$ for all $E\in\mathcal{F}$. Such a
transformation is said to be ergodic if $\mathbb{P}(E)\in\\{0,1\\}$ whenever
$T^{-1}E=E$. The transformation $T$ is said to be mixing if, for all
$A,B\in\mathcal{F}$, $\mathbb{P}(A\cap T^{-k}B)\to\mathbb{P}(A)\mathbb{P}(B)$
as $k\to\infty$. By setting $A=B$, one sees that mixing implies ergodicity.

######  Lemma B.1.4.

For $a,b\in\mathbb{R}$, not both $0$ and $z>0$, consider the shift operator
$T_{z;a,b}$ acting on the directed landscape $\mathcal{L}$ as

$T_{z;a,b}\mathcal{L}(x,s;y,t)=\mathcal{L}(x+az,s+bz;y+az;t+bz),$

where both sides are understood as a process on
$\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}$. Then, $\mathcal{L}$ is mixing
under this transformation. That is, for all Borel subsets $A,B$ of the space
$C(\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4},\mathbb{R})$,

$\mathbb{P}(\mathcal{L}\in A,T_{z;a,b}\mathcal{L}\in
B)\overset{z\to\infty}{\longrightarrow}\mathbb{P}(\mathcal{L}\in
A)\mathbb{P}(\mathcal{L}\in B).$

In words, the directed landscape is mixing (and therefore ergodic) under
space-time shifts in any planar direction.

###### Proof.

This key to the proof is Lemma B.1.3, and I thank Duncan Dauvergne for
pointing this out to me. We prove the case $a\neq 0$, and the case $a=0$ and
$b\neq 0$ will be proven separately. Further, since we send $z\to\infty$, it
suffices to show the result for $a=1$, for then the result also holds for
arbitrary $a$ and $b=ab$. With this simplification, we use the shorthand
notation $T_{z;b}=T{z:1,b}$. By Lemma B.1.1(i), $\mathcal{L}$ is stationary
under the shift $T_{z;b}$. By Dynkin’s $\pi$-$\lambda$ theorem, it suffices to
show that for a compact set
$K\subseteq\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}$ and Borel sets
$A\subseteq C(K,\mathbb{R})$ and $B\subseteq C(K,\mathbb{R})$,

$\mathbb{P}(\mathcal{L}|_{K}\in A,(T_{z;b}\mathcal{L})|_{K}\in
B)\overset{z\to\infty}{\longrightarrow}\mathbb{P}(\mathcal{L}|_{K}\in
A)\mathbb{P}(\mathcal{L}|_{K}\in B).$

Further, by temporal stationarity, it suffices to assume that
$\inf\\{s:(x,s;y,t)\in K\\}\geq 0$. Consider the coupling
$\mathcal{L}_{0},\mathcal{L}_{1},\mathcal{L}_{2}$ of Lemma B.1.3 with $k=2$.
Then, using the rescaling and spatial stationarity of Lemma B.1.1,

$\displaystyle\quad\;\mathbb{P}(\mathcal{L}|_{K}\in
A,(T_{z;b}\mathcal{L})|_{K}\in B)$
$\displaystyle=\mathbb{P}(\mathcal{L}_{0}(x,s;y,t)|_{K}\in
A,\mathcal{L}_{0}(x+z,s+bz;y+z,t+bz)|_{K}\in B)$
$\displaystyle=\mathbb{P}(z^{1/2}\mathcal{L}_{0}(z^{-1}x,z^{-3/2}s;z^{-1}y,z^{-3/2}t)|_{K}\in
A,$ $\displaystyle\qquad\qquad
z^{1/2}\mathcal{L}_{0}(z^{-1}x+1,z^{-3/2}s+z^{-1/2}b;z^{-1}y+1,z^{-3/2}t+z^{-1/2}b)|_{K}\in
B)$
$\displaystyle=\mathbb{P}(z^{1/2}\mathcal{L}_{0}(z^{-1}x+1,z^{-3/2}s;z^{-1}y+1,z^{-3/2}t)|_{K}\in
A,$ $\displaystyle\qquad\qquad
z^{1/2}\mathcal{L}_{0}(z^{-1}x+2,z^{-3/2}s+z^{-1/2}b;z^{-1}y+2,z^{-3/2}t+z^{-1/2}b)|_{K}\in
B).$ (B.2)

Specifically, we used the rescaling property with $q=z^{1/2}$ in the second
equality, and in the third equality, we shifted the entire process by $1$ in
the spatial direction. Above, the restrictions $|_{K_{j}}$ mean that
$(x,s;y,t)\in K_{j}$. Since $K$ is compact and we assumed $s\geq 0$ for all
$(x,s;y,t)\in K$, for any $\varepsilon>0$, there exists $Z>0$ such that for
$z>Z$,

$\displaystyle\\{(z^{-1}x+1,z^{-3/2}s;z^{-1}y+1,z^{-3/2}t):(x,s;y,t)\in
K\\}\subseteq K_{1,\varepsilon},\quad\text{and}$ (B.3)
$\displaystyle\\{(z^{-1}x+2,z^{-3/2}s+z^{-1/2}b;z^{-1}y+2,z^{-3/2}t+z^{-1/2}b):(x,s;y,t)\in
K\\}\subseteq K_{2,\varepsilon},$

where $K_{i,\varepsilon}$ are defined in (B.1). Let $C_{z}$ be the event where
both these containments hold for the random $\varepsilon>0$ in Lemma B.1.3,
and let $D_{z}$ be the event in (B.2). Then, $\mathbb{P}(C_{z})\to 1$ as
$z\to+\infty$. Then, for any $\delta>0$, whenever $z$ is sufficiently large so
that $1-\mathbb{P}(C_{z})=\delta>0$,

$\displaystyle\quad\;\Big{|}\mathbb{P}(D_{z})-\mathbb{P}(\mathcal{L}|_{K}\in
A)\mathbb{P}(\mathcal{L}|_{K}\in B)\Big{|}$
$\displaystyle\leq\Big{|}\mathbb{P}(D_{z}\cap
C_{z})-\mathbb{P}(\mathcal{L}|_{K}\in A)\mathbb{P}(\mathcal{L}|_{K}\in
B)\Big{|}+\delta$
$\displaystyle=\Big{|}\mathbb{P}(z^{1/2}\mathcal{L}_{1}(z^{-1}x+1,z^{-3/2}s;z^{-1}y+1,z^{-3/2}t)|_{K}\in
A,$ $\displaystyle\qquad\qquad
z^{1/2}\mathcal{L}_{2}(z^{-1}x+2,z^{-3/2}s+z^{-1/2}b;z^{-1}y+2,z^{-3/2}t+z^{-1/2}b)|_{K}\in
B,C_{z})$ $\displaystyle\qquad\qquad\qquad-\mathbb{P}(\mathcal{L}_{1}|_{K}\in
A)\mathbb{P}(\mathcal{L}_{1}|_{K}\in B)\Big{|}+\delta$
$\displaystyle\leq\Big{|}\mathbb{P}(z^{1/2}\mathcal{L}_{0}(z^{-1}x+1,z^{-3/2}s;z^{-1}y+1,z^{-3/2}t)|_{K}\in
A,$ $\displaystyle\qquad\qquad
z^{1/2}\mathcal{L}_{0}(z^{-1}x+2,z^{-3/2}s+z^{-1/2}b;z^{-1}y+2,z^{-3/2}t+z^{-1/2}b)|_{K}\in
B)$ $\displaystyle\qquad\qquad\qquad-\mathbb{P}(\mathcal{L}_{1}|_{K}\in
A)\mathbb{P}(\mathcal{L}_{2}|_{K}\in B)\Big{|}+2\delta$
$\displaystyle=\Big{|}\mathbb{P}(\mathcal{L}_{1}|_{K}\in
A,\mathcal{L}_{2}|_{K}\in B)-\mathbb{P}(\mathcal{L}_{1}|_{K}\in
A)\mathbb{P}(\mathcal{L}_{2}|_{K}\in B)\Big{|}+2\delta=2\delta,$

completing the proof since $\delta$ is arbitrary. Specifically, in the two
inequalities, we added and removed the event $C_{z}$ at the cost of $\delta$.
In the first equality above, we used the fact that the containments (B.3) hold
on $C_{z}$ and
$\mathcal{L}_{0}|_{K_{i},\varepsilon}=\mathcal{L}_{i}|_{K_{i},\varepsilon}$
for $i=1,2$. In the last line, we reversed the application of the rescaling
and spatial stationarity, then used the independence of $\mathcal{L}_{1}$ and
$\mathcal{L}_{2}$ from Lemma B.1.3.

The statement for the vertical shift operator when $a=0$ is simpler. For a
compact set $K$, the processes $\mathcal{L}|_{K}$ and
$T_{z;0,b}\mathcal{L}|_{K}$ are independent for sufficiently large $b$ by the
independent increments property of the directed landscape, and the desired
result follows. ∎

Recall the definition of the state space $\operatorname{UC}$ (3.5) for the KPZ
fixed point. Recall that the KPZ fixed point
$h_{\mathcal{L}}(t,{\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}};s,\mathfrak{h})$
with initial data $\mathfrak{h}$ at time $s$ can be represented as

$h_{\mathcal{L}}(t,y;s,\mathfrak{h})=\sup_{x\in
R}\\{\mathfrak{h}(x)+\mathcal{L}(x,s;y,t)\\}\qquad\text{for }t>s.$ (B.4)

######  Lemma B.1.5.

[47, 20, 68, 120] Let
$\mathcal{L}:\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}\to\mathbb{R}$ be a
continuous function satisfying the metric composition law (3.3) and such that
maximizers in (3.3) exist. Then,

  1. (i)

Whenever $s<t$, $x_{1}<x_{2}$, $y_{1}<y_{2}$,

$\mathcal{L}(x_{2},s;y_{1},t)-\mathcal{L}(x_{1},s;y_{1},t)\leq\mathcal{L}(x_{2},s;y_{2},t)-\mathcal{L}(x_{1},s;y_{2},t).$

Let $\mathfrak{h}^{1},\mathfrak{h}^{2}\in\operatorname{UC}$, and For $i=1,2$
and $t>s$, define $h_{\mathcal{L}}(t,y;s,\mathfrak{h}^{i})$ by (B.4). Then,
assuming that maximizers in (B.4) exist, the following hold.

  1. (ii)

If $\mathfrak{h}^{1}\;{\leq}_{\rm inc}\;\mathfrak{h}^{2}$, then
$h_{\mathcal{L}}(t,{\hskip
1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt};s,\mathfrak{h}^{1})\;{\leq}_{\rm inc}\;h_{\mathcal{L}}(t,{\hskip
1.1pt\raisebox{1.5pt}{\scaleobj{0.5}{\bullet}}\hskip
1.1pt};s,\mathfrak{h}^{2})$ for all $t>s$.

  2. (iii)

For $t>s$ and $i=1,2$, set

$Z_{\mathcal{L}}(t,y;s,\mathfrak{h}^{i})=\max\operatorname*{arg\,max}_{x\in\mathbb{R}}\\{\mathfrak{h}^{i}(x)+\mathcal{L}(x,s;y,t)\\}.$

Then, if $x<y$ and $Z_{\mathcal{L}}(t,y;s,\mathfrak{h}^{1})\leq
Z_{\mathcal{L}}(t,x;s,\mathfrak{h}^{2})$,

$h_{\mathcal{L}}(t,y;s,\mathfrak{h}^{1})-h_{\mathcal{L}}(t,x;s,\mathfrak{h}^{1})\leq
h_{\mathcal{L}}(t,y;s,\mathfrak{h}^{2})-h_{\mathcal{L}}(t,x;s,\mathfrak{h}^{2}).$

###  B.2 Geodesics in the directed landscape

We start by citing some results from [23] and [48].

######  Lemma B.2.1.

[23, Theorem 1.18] There exists a single event of full probability on which,
for any compact set $K\subseteq\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}$,
there is a random $\varepsilon>0$ such that the following holds. If
$v_{1}=(x,s;y,u)\in K$ and $v_{2}=(z,s;w,u)\in K$ admit geodesics $\gamma_{1}$
and $\gamma_{2}$ satisfying $|\gamma_{1}(t)-\gamma_{2}(t)|\leq\varepsilon$ for
all $t\in[s,u]$, then $\gamma_{1}$ and $\gamma_{2}$ are not disjoint, i.e.,
$\gamma_{1}(t)=\gamma_{2}(t)$ for some $t\in[s,u]$.

Let $g$ be a geodesic from $(x,s)$ to $(y,u)$. Define the graph of this
geodesic as

$\mathcal{G}g:=\\{(g(t),t):t\in[s,u]\\}.$

######  Lemma B.2.2.

[48, Lemma 3.1] The following holds on a single event of full probability. Let
$(p_{n};q_{n})\to(p,q)=(x,s;y,t)\in\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}$,
and let $g_{n}$ be any sequence of geodesics from $p_{n}$ to $q_{n}$. Then,
the sequence of graphs $\mathcal{G}g_{n}$ is precompact in the Hausdorff
metric, and any subsequential limit of $\mathcal{G}g_{n}$ is the graph of a
geodesic from $p$ to $q$.

######  Lemma B.2.3.

[48, Lemma 3.3] The following holds on a single event of full probability. Let
$(p_{n};q_{n})=(x_{n},s_{n};y_{n},u_{n})\in\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}\to(p;q)=(x,s;y,u)\in\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}$,
and let $g_{n}$ be any sequence of geodesics from $p_{n}$ to $q_{n}$. Suppose
that either

  1. (i)

For all $n$, $g_{n}$ is the unique geodesic from $(x_{n},s_{n})$ to
$(y_{n},u_{n})$ and $\mathcal{G}g_{n}\to\mathcal{G}g$ for some geodesic $g$
from $p$ to $q$, or

  2. (ii)

There is a unique geodesic $g$ from $p$ to $q$.

Then, the overlap

$O(g_{n},g):=\\{t\in[s_{n},u_{n}]\cap[s,u]:g_{n}(t)=g(t)\\}$

is an interval for all $n$ whose endpoints converge to $s$ and $u$.

######  Remark B.2.4.

We note that condition (i) is slightly different from that stated in [48].
There, it is assumed instead that
$(x_{n},s_{n};y_{n},u_{n})\in\mathbb{Q}^{4}\cap\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}$
for all $n$. The only use of this requirement in the proof is to ensure that
there is a unique geodesic from $(x_{n},s_{n})$ to $(y_{n},u_{n})$ for all
$n$, so there is no additional justification needed for the statement we use
here.

######  Lemma B.2.5.

On the intersection of the full probability events from Lemmas B.2.2 and
B.2.3, the following holds. For all ordered triples $s<t<u$ and compact sets
$K\subseteq\mathbb{R}$, the set

$\\{g(t):g\text{ is the unique geodesic between }(x,s)\text{ and }(y,u)\text{
for some }x\in K,y\in K\\}$ (B.5)

is finite.

Lemma B.2.5 is known. Its derivation from Lemma B.2.1 and some results of [48]
are shown in [35]. Lemma 3.12 in [70] (posted after the first version of [35])
provides a stronger quantitative statement, but we do not need it for our
purposes. This stronger estimate can be traced back to the work of [21] in
exponential LPP using integrable methods.

###### Proof of Lemma B.2.5.

Assume, without loss of generality, that $K$ is a closed interval $[a,b]$. We
observe that by planarity, all geodesics from $(x,s)$ to $(y,u)$ for $x,y\in
K$ lie between the leftmost geodesic from $(a,s)$ to $(a,u)$ and the rightmost
geodesic from $(b,s)$ to $(b,u)$. Hence, the set (B.5) is contained in a
compact set, and it suffices to show that the set has no limit points.

Assume, to the contrary, that there exists a point
$(\hat{x},s;\hat{y},u)\in(K\times\\{s\\}\times
K\times\\{u\\})\cap\mathbb{R}_{{\scaleobj{0.7}{\uparrow}}}^{4}$ with unique
geodesic $\hat{g}$ such that there exists a sequence of points $x_{n},y_{n}\in
K$ such that for all $n$, the geodesic $g_{n}$ from $(x_{n},s)$ to $(y_{n},t)$
is unique and so that $g_{n}(t)\to\hat{g}(t)$ but $g_{n}(t)\neq\hat{g}(t)$ for
all $n$. By compactness, there exists a convergent subsequence
$(x_{n_{k}},y_{n_{k}})\to(x,y)$. By Lemma B.2.2, there exists a further
subsequence $(x_{n_{k_{\ell}}},y_{n_{k_{\ell}}})$ such that the geodesic
graphs $\mathcal{G}g_{n_{k_{\ell}}}$ converge to the graph of some geodesic
$\mathcal{G}g$ from $(x,s)$ to $(y,u)$ in the Hausdorff metric. Since
$g_{n}(t)\to\hat{g}(t)$, we have $g(t)=\hat{g}(t)$. By Lemma B.2.3, the
overlap $O(g_{n_{k_{\ell}}},g)$ is an interval whose endpoints converge to $s$
and $u$, so $g_{n_{k_{\ell}}}(t)=g(t)=\hat{g}(t)$ for sufficiently large
$\ell$, contradicting the definition of the sequence $g_{n}$. ∎

## Bibliography

  * [1] Tom Alberts, Firas Rassoul-Agha and Mackenzie Simper  “Busemann functions and semi-infinite O’Connell-Yor polymers”  In _Bernoulli_ 26.3, 2020, pp. 1927–1955  URL: https://doi.org/10.3150/19-BEJ1177
  * [2] Gideon Amir, Omer Angel and Benedek Valkó  “The TASEP speed process”  In _Ann. Probab._ 39.4, 2011, pp. 1205–1242  URL: https://doi.org/10.1214/10-AOP561
  * [3] Enrique Daniel Andjel and Maria Eulália Vares  “Hydrodynamic equations for attractive particle systems on ${\bf Z}$”  In _J. Statist. Phys._ 47.1-2, 1987, pp. 265–288  URL: https://doi.org/10.1007/BF01009046
  * [4] Søren Asmussen  “Applied Probability and Queues”  J. WileySons, 1987
  * [5] Antonio Auffinger, Michael Damron and Jack Hanson  “50 years of first-passage percolation” 68, University Lecture Series  American Mathematical Society, Providence, RI, 2017, pp. v+161  URL: https://doi.org/10.1090/ulect/068
  * [6] Jinho Baik, Percy Deift and Kurt Johansson  “On the distribution of the length of the longest increasing subsequence of random permutations”  In _J. Amer. Math. Soc._ 12.4, 1999, pp. 1119–1178  URL: https://doi.org/10.1090/S0894-0347-99-00307-0
  * [7] Yuri Bakhtin  “Burgers equation with random boundary conditions”  In _Proc. Amer. Math. Soc._ 135.7, 2007, pp. 2257–2262  URL: https://doi.org/10.1090/S0002-9939-07-08736-9
  * [8] Yuri Bakhtin  “Ergodic theory of the Burgers equation”  In _Probability and statistical physics in St. Petersburg_ 91, Proc. Sympos. Pure Math.  Amer. Math. Soc., Providence, RI, 2016, pp. 1–49  URL: https://doi.org/10.1090/pspum/091/01535
  * [9] Yuri Bakhtin  “Inviscid Burgers equation with random kick forcing in noncompact setting”  In _Electron. J. Probab._ 21, 2016, pp. Paper No. 3750  URL: https://doi.org/10.1214/16-EJP4413
  * [10] Yuri Bakhtin  “The Burgers equation with Poisson random forcing”  In _Ann. Probab._ 41.4, 2013, pp. 2961–2989  URL: https://doi.org/10.1214/12-AOP747
  * [11] Yuri Bakhtin, Eric Cator and Konstantin Khanin  “Space-time stationary solutions for the Burgers equation”  In _J. Amer. Math. Soc._ 27.1, 2014, pp. 193–238  URL: https://doi.org/10.1090/S0894-0347-2013-00773-0
  * [12] Yuri Bakhtin and Konstantin Khanin  “On global solutions of the random Hamilton-Jacobi equations and the KPZ problem”  In _Nonlinearity_ 31.4, 2018, pp. R93–R121  URL: https://doi.org/10.1088/1361-6544/aa99a6
  * [13] Yuri Bakhtin and Liying Li  “Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation”  In _Comm. Pure Appl. Math._ 72.3, 2019, pp. 536–619  URL: https://doi.org/10.1002/cpa.21779
  * [14] Yuri Bakhtin and Liying Li  “Zero temperature limit for directed polymers and inviscid limit for stationary solutions of stochastic Burgers equation”  In _J. Stat. Phys._ 172.5, 2018, pp. 1358–1397  URL: https://doi.org/10.1007/s10955-018-2104-z
  * [15] M. Balázs, E. Cator and T. Seppäläinen  “Cube root fluctuations for the corner growth model associated to the exclusion process”  In _Electron. J. Probab._ 11, 2006, pp. no. 421094–1132  URL: https://doi.org/10.1214/EJP.v11-366
  * [16] Márton Balázs, Ofer Busani and Timo Seppäläinen  “Local stationarity in exponential last-passage percolation”  In _Probab. Theory Related Fields_ 180.1-2, 2021, pp. 113–162  URL: https://doi.org/10.1007/s00440-021-01035-7
  * [17] Márton Balázs, Ofer Busani and Timo Seppäläinen  “Non-existence of bi-infinite geodesics in the exponential corner growth model”  In _Forum Math. Sigma_ 8, 2020, pp. Paper No. e4634  URL: https://doi.org/10.1017/fms.2020.31
  * [18] Albert-László Barabási  “Roughening of growing surfaces: Kinetic models and continuum theories”  In _Computational Materials Science_ 6.2, 1996, pp. 127–134  URL: https://doi.org/10.1016/0927-0256(96)00026-2
  * [19] Yu. Baryshnikov  “GUEs and queues”  In _Probab. Theory Related Fields_ 119.2, 2001, pp. 256–274  URL: https://doi.org/10.1007/PL00008760
  * [20] Riddhipratim Basu, Shirshendu Ganguly and Alan Hammond  “Fractal geometry of $\rm Airy_{2}$ processes coupled via the Airy sheet”  In _Ann. Probab._ 49.1, 2021, pp. 485–505  URL: https://doi.org/10.1214/20-AOP1444
  * [21] Riddhipratim Basu, Christopher Hoffman and Allan Sly  “Nonexistence of bigeodesics in planar exponential last passage percolation”  In _Comm. Math. Phys._ 389.1, 2022, pp. 1–30  URL: https://doi.org/10.1007/s00220-021-04246-0
  * [22] Riddhipratim Basu, Sourav Sarkar and Allan Sly  “Coalescence of geodesics in exactly solvable models of last passage percolation”  In _J. Math. Phys._ 60.9, 2019, pp. 09330122  URL: https://doi.org/10.1063/1.5093799
  * [23] Erik Bates, Shirshendu Ganguly and Alan Hammond  “Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape”  In _Electron. J. Probab._ 27, 2022, pp. Paper No. 144  URL: https://doi.org/10.1214/21-ejp706
  * [24] Albert Benassi and Jean-Pierre Fouque  “Hydrodynamical limit for the asymmetric simple exclusion process”  In _Ann. Probab._ 15.2, 1987, pp. 546–560  URL: http://links.jstor.org/sici?sici=0091-1798(198704)15:2%3C546:HLFTAS%3E2.0.CO;2-1&origin=MSN
  * [25] Manan Bhatia  “Atypical stars on a directed landscape geodesic”  In _Preprint_ , 2022
  * [26] Manan Bhatia  “Duality in the directed landscape and its applications to fractal geometry”  In _Preprint_ , 2023
  * [27] Manan Bhatia  “Moderate deviation and exit time estimates for stationary last passage percolation”  In _J. Stat. Phys._ 181.4, 2020, pp. 1410–1432  URL: https://doi.org/10.1007/s10955-020-02632-x
  * [28] Patrick Billingsley  “Convergence of probability measures” A Wiley-Interscience Publication, Wiley Series in Probability and Statistics: Probability and Statistics  John Wiley & Sons, Inc., New York, 1999, pp. x+277  URL: https://doi.org/10.1002/9780470316962
  * [29] Andrei N. Borodin and Paavo Salminen  “Handbook of Brownian motion—facts and formulae”, Probability and its Applications  Birkhäuser Verlag, Basel, 2002, pp. xvi+672  URL: https://doi.org/10.1007/978-3-0348-8163-0
  * [30] Paul J. Burke  “The output of a queuing system”  In _Operations Res._ 4, 1956, pp. 699–704 (1957)  URL: https://doi.org/10.1287/opre.4.6.699
  * [31] Ofer Busani  “Diffusive scaling limit of the Busemann process in Last Passage Percolation”  In _Preprint_ , 2021