models with boundary conditions $\sigma$ and $\tau\circ\sigma$, respectively,
are the same. These two inferences complete the proof by recalling the
definition (3.1) of the colored $q$-Boson measure. ∎

######  Remark 3.3.

Unlike the colored S6V model, in the colored $q$-Boson model one cannot merge
colors with color zero (no arrows). This is a consequence of the
$\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{u}$ weights not being stochastic. It
can be seen by the fact that the analog of (3.2) for merging with $0$ (i.e.,
$\tau^{-1}({0})$ contains a non-zero integer) does not hold.

####  3.1.5. Gibbs property for $q$-Boson models

Next we discuss two Gibbs properties, one each for the colored and uncolored
$q$-Boson models, that are consequences of the above description, and which
will be used in later arguments. By Gibbs property, we mean a spatial Markov
property: informally, for a domain $\Lambda$, the conditional distribution of
the configuration strictly inside $\Lambda$, given the configuration on the
boundary of $\Lambda$ as well as outside $\Lambda$, is explicit and depends
only on the configuration on the boundary of $\Lambda$.

We start with the uncolored Gibbs property. We may write the arrow
configuration at a vertex $v$ as $(A_{v},i_{v};B_{v},j_{v})$ where
$i_{v},j_{v}\in\\{0,1\\}$ (presence of arrow) and
$A_{v},B_{v}\in\mathbb{Z}_{\geq 0}$ (number of arrows). Because of the form of
the weight of a configuration as a product of vertex weights, that this model
enjoys a Gibbs property is immediate; the dependency of the conditional
distribution on the configuration on the boundary of the domain is imposed via
the requirement of consistency and arrow conservation. To state the Gibbs
property more precisely, fix a rectangle
$\Lambda:=\llbracket-k,-\ell\rrbracket\times\llbracket
a,b\rrbracket\subseteq\mathbb{Z}_{\leq 0}\times\llbracket 1,N+M\rrbracket$,
let $\Lambda^{c}=\mathbb{Z}_{\leq 0}\times\llbracket
1,N+M\rrbracket\setminus\Lambda$, and let
$\smash{\mathcal{F}^{\mathrm{uB}}_{\Lambda^{c}}}$ (“uB” short for uncolored
$q$-Boson) be the $\sigma$-algebra (where $\sigma(\bm{\cdot})$ for
$\bm{\cdot}$ some collection of random variables means the $\sigma$-algebra
generated by that collection)

$\displaystyle\mathcal{F}^{\mathrm{uB}}_{\Lambda^{c}}:=\sigma\left(\left\\{j_{v}:v\in\Lambda^{c}\right\\}\right);$
(3.3)

note that, by consistency, for every $v$ in the left boundary of $\Lambda$, it
holds that $i_{v}$ is $\mathcal{F}^{\mathrm{uB}}_{\Lambda^{c}}$-measurable,
and by consistency along with arrow conservation, for every $v$ in the bottom
boundary of $\Lambda$, it holds that $A_{v}$ is
$\mathcal{F}^{\mathrm{uB}}_{\Lambda^{c}}$-measurable. This implies that, for
$v$ the top left corner of $\Lambda$, $B_{v}$ is
$\mathcal{F}^{\mathrm{uB}}_{\Lambda^{c}}$-measurable and, iteratively, that
the same holds for every $v$ in the top boundary of $\Lambda$. Continuing
reasoning in this way, we see that $B_{v}$ is
$\mathcal{F}^{\mathrm{uB}}_{\Lambda^{c}}$-measurable for every $v$ in
$\mathbb{Z}_{\leq 0}\times\llbracket
1,N+M\rrbracket\setminus(\llbracket-k,-\ell+1\rrbracket\times\llbracket
a,b-1\rrbracket)$. See the left panel of Figure 11.

$\Gamma_{k}$ Figure 11. Left: The setup for the Gibbs property in the
uncolored $q$-Boson model. Outgoing arrow counts outside the region $\Lambda$
outlined in blue are conditioned upon, and arrows have been displayed in this
depiction if the arrow count is non-zero and has been conditioned on. Arrow
counts entering the blue region are conditioned on due to consistency, but
those exiting it are not. Right: The setup for the Gibbs property in the
colored $q$-Boson model. Horizontally outgoing colored arrows counts are
conditioned on strictly to the left of the blue dashed line (which extends to
the ends of the domain in both directions), while only total outgoing arrow
counts (depicted in black) are conditioned on to the right (including on
$\Gamma_{k}$).

For the statement of the Gibbs property, we adopt the notation
$\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,v}$ for the vertex weights in the
model:
$\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,v}=\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{1}$
for $v\in\mathbb{Z}_{\leq 0}\times\llbracket 1,N\rrbracket$ and
$\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,v}=\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z}$
for $v\in\mathbb{Z}_{\leq 0}\times\llbracket N+1,N+M\rrbracket$.

######  Lemma 3.4 (Uncolored $q$-Boson Gibbs property).

Let $\Lambda=\llbracket-k,-\ell\rrbracket\times\llbracket a,b\rrbracket$ and
$\smash{\mathcal{F}^{\mathrm{uB}}_{\Lambda^{c}}}$ be as in (3.3). Under the
uncolored $q$-Boson measure, the conditional distribution of
$((A_{v},i_{v};B_{v},j_{v}):v\in\Lambda)$ given
$\smash{\mathcal{F}^{\mathrm{uB}}_{\Lambda^{c}}}$ is supported on tuples
$((\smash{A^{\prime}_{v},i^{\prime}_{v};B^{\prime}_{v},j^{\prime}_{v}}):v\in\Lambda)$
such that arrow conservation and consistency are satisfied inside $\Lambda$
and with respect to the boundary conditions; call the (random) set of such
tuples $\mathrm{Supp}^{\mathrm{uB}}$. Further, the conditional probability
that
$((A_{v},i_{v};B_{v},j_{v}):v\in\Lambda)=((A^{\prime}_{v},i^{\prime}_{v};B^{\prime}_{v},j^{\prime}_{v}):v\in\Lambda)$,
for any fixed (i.e., deterministic) selection of the righthand side, is
proportional to
$\prod_{v\in\Lambda}{\scalebox{1.05}{\mbox{$\mathtt{L}$}}}_{z,v}(A^{\prime}_{v},i^{\prime}_{v};B^{\prime}_{v},j^{\prime}_{v})\mathbbm{1}_{(A^{\prime}_{v},i^{\prime}_{v};B^{\prime}_{v},j^{\prime}_{v})_{v\in\Lambda}\in\mathrm{Supp}^{\mathrm{uB}}}$.

###### Proof.

This follows from the form of the uncolored $q$-Boson measure as a product of
vertex weights. ∎

Next we turn to a Gibbs property for the colored $q$-Boson model. Informally,
it explains how to “reveal” colored arrow counts, conditional on total arrow
counts. Fix $k\in\mathbb{N}$. Let
$\Lambda_{k}=\scalebox{0.97}{$\Lparen$}{-}\infty,-k\rrbracket\times\llbracket
1,N+M\rrbracket$ and $\Gamma_{k}=\\{-k\\}\times\llbracket 1,N+M\rrbracket$,
and let $\smash{\mathcal{F}^{\mathrm{cB}}_{k}}$ (“cB” short for colored
$q$-Boson) be the $\sigma$-algebra

$\smash{\mathcal{F}^{\mathrm{cB}}_{k}}:=\sigma\left(\bigl{\\{}\mathbbm{1}_{i_{v}\geq
1}:v\in\mathbb{Z}_{\leq 0}\times\llbracket
1,N+M\rrbracket\bigr{\\}}\cup\bigl{\\{}i_{v}:v\in\Lambda_{k}\bigr{\\}}\right).$
(3.4)

For $\bm{A}\in\mathbb{Z}_{\geq 0}^{N}$, let $|\bm{A}|:=\sum_{i=1}^{N}A_{i}$.
Note that by colored arrow conservation and consistency, the collection of
random variables $\\{(|\bm{A}_{v}|;|\bm{B}_{v}|,\mathbbm{1}_{j_{v}\geq
1}):v\in\mathbb{Z}_{\leq 0}\times\llbracket 1,N+M\rrbracket\\}$ and
$\\{(\bm{A}_{v};\bm{B}_{v},j_{v}):v\in\Lambda_{k+1}\\}$ is
$\mathcal{F}^{\mathrm{cB}}_{k}$-measurable. Thus, in words,
$\smash{\mathcal{F}^{\mathrm{cB}}_{k}}$ contains the information of the
incoming colored arrow counts at $\Gamma_{k}$, incoming and outgoing total
arrow counts of every vertex in the domain, and incoming and outgoing colored
arrow counts of every vertex in $\Lambda_{k+1}$. See the right panel of Figure
11.

The following is the Gibbs property when conditioning on
$\mathcal{F}^{\mathrm{cB}}_{k}$; because of the heterogeneous nature of the
random variables being conditioned (a mixture of colored and total arrow
counts), its proof is slightly more complicated than that of Lemma 3.4. Recall
the definition of $\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,v}$ from before
Lemma 3.4.

######  Lemma 3.5 (Colored $q$-Boson Gibbs property).

Fix $k\in\mathbb{N}$ and $\sigma:\llbracket 1,N\rrbracket\to\llbracket
1,N\rrbracket$, and let $\Gamma_{k}$ and
$\smash{\mathcal{F}^{\mathrm{cB}}_{k}}$ be as in (3.4). Under the colored
$q$-Boson measure with boundary condition $\sigma$, the conditional
distribution of $((\bm{A}_{v};\bm{B}_{v},j_{v}):v\in\Gamma_{k})$ given
$\smash{\mathcal{F}^{\mathrm{cB}}_{k}}$ is supported on tuples
$((\smash{\bm{A}^{\prime}_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v}}):v\in\Gamma_{k})$
such that (i) $\mathbbm{1}_{j^{\prime}_{v}\geq 1}=\mathbbm{1}_{j_{v}\geq 1}$
for all $v\in\Gamma_{k}$ and (ii) colored arrow conservation and consistency
are satisfied. Call the (random) set of all such tuples
$\mathrm{Supp}^{\mathrm{cB}}$. Further, the conditional probability that
$((\bm{A}_{v};\bm{B}_{v},j_{v}):v\in\Gamma_{k})=((\bm{A}^{\prime}_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v}):v\in\Gamma_{k})$,
for any fixed (i.e., deterministic) selection of the righthand side, is
proportional to
$\prod_{v\in\Gamma_{k}}\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,v}(\bm{A}^{\prime}_{v},i_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v})\mathbbm{1}_{(\bm{A}^{\prime}_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v})_{v\in\Gamma_{k}}\in\mathrm{Supp}^{\mathrm{cB}}}$.

###### Proof.

Let $\Lambda_{k}^{c}:=\mathbb{Z}_{\leq 0}\times\llbracket
1,N+M\rrbracket\setminus\Lambda_{k}=\llbracket-k+1,0\rrbracket\times\llbracket
1,N+M\rrbracket$. Let
$\bm{\omega}=((\bm{A}^{\prime}_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v}):v\in\Gamma_{k})$
and let $\mathrm{Val}_{k}(\bm{\omega})$ (for “valid”) be the collection of
configurations
$(\bm{A}^{\prime}_{v},i^{\prime}_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v})_{v\in\Lambda_{k}^{c}}$
which satisfy colored arrow conservation and consistency as well as the
following:

  * $\bullet$

$i^{\prime}_{(-k+1,x)}=j^{\prime}_{(-k,x)}$ for $x\in\llbracket
1,N+M\rrbracket$ (consistency of incoming arrows to $\Gamma_{k-1}$ with
outgoing arrows from $\Gamma_{k}$ as given by $\bm{\omega}$),

  * $\bullet$

$\bm{B}^{\prime}_{(0,N+M)}=(\\#\sigma^{-1}(\\{j\\}))_{j=1}^{N}$ (all arrows
exit vertically from the rightmost column), and

  * $\bullet$

$\mathbbm{1}_{i^{\prime}_{v}\geq 1}=\mathbbm{1}_{i_{v}\geq 1}$,
$\mathbbm{1}_{j^{\prime}_{v}\geq 1}=\mathbbm{1}_{j_{v}\geq 1}$,
$|\bm{A}^{\prime}_{v}|=|\bm{A}_{v}|$, and $|\bm{B}^{\prime}_{v}|=|\bm{B}_{v}|$
for all $v\in\Lambda_{k}^{c}$ (respecting the total arrow counts conditioned
on).

It is immediate from the definition of the colored $q$-Boson measure that the
$\mathcal{F}^{\mathrm{cB}}_{k}$-conditional probability of the event
$\\{((\bm{A}_{v};\bm{B}_{v},j_{v}):v\in\Gamma_{k})=\bm{\omega}\\}$ is
proportional to

$\displaystyle\prod_{v\in\Gamma_{k}}\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,v}(\bm{A}^{\prime}_{v},i_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v})\cdot\left(\sum_{(\bm{A}^{\prime}_{v},i^{\prime}_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v})_{v\in\Lambda_{k}^{c}}\in\mathrm{Val}_{k}(\bm{\omega})}\prod_{w\in\Lambda_{k}^{c}}\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,w}(\bm{A}^{\prime}_{w},i^{\prime}_{w};\bm{B}^{\prime}_{w},j^{\prime}_{w})\right).$
(3.5)

To absorb the factor in parentheses into the normalization constant, we must
show that it is $\smash{\mathcal{F}^{\mathrm{cB}}_{k}}$-measurable. The lemma
will therefore be implied by the following claim: that factor’s value depends
only on $((|\bm{A}_{v}|,\mathbbm{1}_{i_{v}\geq
1};|\bm{B}_{v}|,\mathbbm{1}_{j_{v}\geq 1}):v\in\Lambda_{k}^{c})$, which has
been conditioned upon (it is determined by $(\mathbbm{1}_{i_{v}\geq
1}:v\in\Lambda_{k}^{c})$ by consistency and arrow conservation). This claim in
turn follows by applying (3.2) (with $\tau(k)=1$ for all $k\in\llbracket
1,N\rrbracket$ and $\tau(0)=0$) iteratively for each factor in the second
product in (3.5) (i.e., for each vertex in $\Lambda_{k}^{c}$); notice that the
sum in (3.5) is over $\mathrm{Val}_{k}(\bm{\omega})$, which fixes the total
arrow counts at each vertex according to
$((|\bm{A}_{v}|,\mathbbm{1}_{i_{v}\geq 1};|\bm{B}_{v}|,\mathbbm{1}_{j_{v}\geq
1}):v\in\Lambda_{k}^{c})$. This yields that the parenthetical quantity in
(3.5) equals the partition function for the uncolored $q$-Boson measure on
configurations
$(\tilde{A}_{v},\tilde{i}_{v};\tilde{B}_{v},\tilde{j}_{v})_{v\in\Lambda_{k}^{c}}$
(where $A_{v},B_{v}\in\mathbb{Z}_{\geq 0}$ and $i_{v},j_{v}\in\\{0,1\\}$
represent total arrow counts, as in the beginning of Section 3.1.5)
constrained to satisfy the uncolored analogs of the three bullet points above,
namely, $\tilde{i}_{(-k+1,x)}=\mathbbm{1}_{j_{(-k,x)}\geq 1}$ for
$x\in\llbracket 1,N+M\rrbracket$; $\tilde{B}_{(0,N+M)}=N$; and
$\tilde{i}_{v}=\mathbbm{1}_{i_{v}\geq 1},\tilde{j}_{v}=\mathbbm{1}_{j_{v}\geq
1}$, $\tilde{A}_{v}=|\bm{A}_{v}|$, and $\tilde{B}_{v}=|\bm{B}_{v}|$ for all
$v\in\Lambda_{k}^{c}$ (this actually includes the first point since
$\mathbbm{1}_{j_{(-k,x)}\geq 1}=\mathbbm{1}_{i_{(-k+1,x)}\geq 1}$). This
verifies the claim and completes the proof. ∎

###  3.2. Colored Hall-Littlewood line ensemble

Next we define a rewriting of a configuration sampled from the colored
$q$-Boson measure in a more probabilistically appealing form, namely, the
colored line ensemble. First we specify what we mean by a discrete line
ensemble.

######  Definition 3.6 (Discrete line ensemble).

Fix a (possibly infinite) interval $\Lambda\subseteq\mathbb{Z}$. A _$\Lambda$
-indexed discrete line ensemble_ $\bm{L}=(L_{1},L_{2},\ldots)$ is a random
variable defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$
taking values in the space of functions $\mathbb{N}\times\Lambda\to\mathbb{Z}$
(here the first argument is written as a subscript, i.e.,
$L_{i}(\bm{\cdot}):=\bm{L}(i,\bm{\cdot})$), endowed with the discrete
topology, such that: (i) $L_{i}(y+1)-L_{i}(y)\in\\{0,-1\\}$ for any
$i\in\mathbb{N}$ and $y$ such that $y,y+1\in\Lambda$, and (ii) $L_{i}(y)\geq
L_{i+1}(y)$ for any $y\in\Lambda$ and $i\in\mathbb{N}$.

Note that we take discrete line ensembles, and all the line ensembles we work
with in this paper, to have an infinite number of curves. This differs
slightly from some previous studies focused on tightness of line ensembles
(e.g., [CH14, CH16, CD18]).

We will refer to the set of vertices $\\{-k\\}\times\llbracket
1,N+M\rrbracket$ in the domain $\mathbb{Z}_{\leq 0}\times\llbracket
1,N+M\rrbracket$ of the colored $q$-Boson model as the $k$th column. The
colored line ensemble will be given in terms of colored arrow counts in
columns at a finite distance from the rightmost (zeroth) column. Recall that,
in the notation from Section 3.1.1, for $y\in\llbracket 1,N+M\rrbracket$, and
$k\in\mathbb{N}$, $j_{(-k,y)}$ is the color of the arrow exiting horizontally
from vertex $(-k,y)$; and $j_{(-k,y)}=0$ if there is no such arrow.

######  Definition 3.7 (Colored and uncolored Hall-Littlewood line ensembles).

Let $\sigma:\llbracket 1,N\rrbracket\to\llbracket 1,N\rrbracket$ and let
$\\{(\bm{A}_{v},i_{v};\bm{B}_{v},j_{v}):v\in\mathbb{Z}_{\leq
0}\times\llbracket 1,N+M\rrbracket\\}$ be a configuration sampled from the
colored $q$-Boson measure with boundary condition $\sigma$ (from Section
3.1.2). For each $k\in\llbracket 1,N\rrbracket$, we define the discrete line
ensemble
$\bm{L}^{\mathrm{cHL},\smash{(k)}}=(L^{\mathrm{cHL},\smash{(k)}}_{1},L^{\mathrm{cHL},\smash{(k)}}_{2},\ldots):\mathbb{N}\times\llbracket
0,N+M\rrbracket\to\mathbb{Z}$ by letting $\smash{L^{\mathrm{cHL},(k)}_{i}(y)}$
be the number of arrows of color at least $k$ which exit horizontally in the
$i$th column strictly above row $y$, i.e,

$L^{\mathrm{cHL},(k)}_{i}(y):=\\#\bigl{\\{}y^{\prime}>y:j_{(-i,y^{\prime})}\geq
k\bigr{\\}}.$ (3.6)

The object
$\bm{L}^{\mathrm{cHL}}=(\bm{L}^{\mathrm{cHL},(1)},\bm{L}^{\mathrm{cHL},(2)},\ldots,\bm{L}^{\mathrm{cHL},(N)})$
is called the _colored Hall-Littlewood line ensemble_ with boundary condition
$\sigma$ (see Figure 3). The line ensemble $\bm{L}^{\mathrm{cHL},(1)}$ is
called the _uncolored Hall-Littlewood line ensemble_.

######  Remark 3.8.

The terminology comes from the fact that, if one interprets the colored Hall-
Littlewood line ensemble as random sequences of compositions (see [AB24,
Definition 4.1]), the resulting process is the same as the colored Hall-
Littlewood process introduced in [BW22, Section 1.6] (though we do not need or
prove this).

We call $\bm{L}^{\mathrm{cHL},(1)}$ the uncolored Hall-Littlewood line
ensemble as it is determined purely by total arrow counts in the colored
$q$-Boson model. As in the uncolored $q$-Boson model, there is no boundary
condition to be specified for the uncolored Hall-Littlewood line ensemble.
Analogous to the colored case, the uncolored Hall-Littlewood line ensemble is
associated to the (uncolored) Hall-Littlewood process (see e.g., [BC14,
Dim18]), a collection of measures on sequences of partitions that is specified
by a choice of specializations and a down-right path. The specializations (in
some cases) can be thought of as our choices of spectral parameters, and the
down-right path as the choice of boundary conditions of the uncolored
$q$-Boson model (allowing arrows to horizontally enter only at rows with
spectral parameter $1$, but which may be different than the bottom $N$ rows as
we have fixed here; see also [AB24, Remark 3.11]). The choice of boundary
condition and spectral parameters we have fixed (and the only one considered
in this article) is equivalent to the “ascending Hall-Littlewood process” with
a homogeneous choice of specializations.

Note that we do not include $\sigma$ in the notation for the colored Hall-
Littlewood line ensembles, but we will specify $\sigma$ in the relevant
statements. Next we record some simple consequences of Definition 3.7.

######  Lemma 3.9.

Fix $\sigma:\llbracket 1,N\rrbracket\to\llbracket 1,N\rrbracket$, and for each
$k\in\llbracket 1,N\rrbracket$, let
$\bm{L}^{\mathrm{cHL},(k)}=(L^{\mathrm{cHL},(k)}_{1},L^{\mathrm{cHL},(k)}_{2}$,
$\ldots)$ be defined as in (3.6) with boundary condition $\sigma$. It holds
deterministically that, for any $i\in\mathbb{N}$ and $y\in\llbracket
0,N+M-1\rrbracket$,

(i) $L^{\mathrm{cHL},(k)}_{i}(y)\geq L^{\mathrm{cHL},(k+1)}_{i}(y)$, (ii)
$L^{\mathrm{cHL},(k)}_{i}(y)\geq L^{\mathrm{cHL},(k)}_{i+1}(y)$, and (iii)
$L^{\mathrm{cHL},(k)}_{i}(y+1)-L^{\mathrm{cHL},(k)}_{i}(y)=-\mathbbm{1}_{j_{(-i,y+1)}\geq
k}\in\\{0,-1\\}$. (3.7)

###### Proof.

The first and third points are immediate from the definition. The second
relies on the fact that arrows can only move up and to the right, so that any
arrow exiting the $(i+1)$st column at a row higher than $y$ must exit the
$i$th column at a row higher than $y$. ∎

We next record a statement of color merging for the colored Hall-Littlewood
line ensemble.

######  Lemma 3.10 (Color merging for $\bm{L}^{\mathrm{cHL}}$).

Fix $\sigma:\llbracket 1,N\rrbracket\to\llbracket 1,N\rrbracket$ and
$k\in\llbracket 1,N\rrbracket$, and let $\tau:\llbracket
1,N\rrbracket\to\llbracket 1,k\rrbracket$ be surjective and non-decreasing.
Let $\bm{L}^{\mathrm{cHL},\sigma}$ and $\bm{L}^{\mathrm{cHL},\tau\circ\sigma}$
be the colored Hall-Littlewood line ensembles with boundary conditions
$\sigma$ and $\tau\circ\sigma$ as in Definition 3.7, respectively. For
$j\in\llbracket 1,k\rrbracket$, let $t_{j}=\min\tau^{-1}(\\{j\\})$. Then

$\displaystyle(\bm{L}^{\mathrm{cHL},\sigma,(t_{j})})_{j\in\llbracket
1,k\rrbracket}\stackrel{{\scriptstyle
d}}{{=}}(\bm{L}^{\mathrm{cHL},\tau\circ\sigma,(j)})_{j\in\llbracket
1,k\rrbracket}.$

In particular, defining $\tau$ by $\tau(j)=1$ for all $j\in\llbracket
1,N\rrbracket$, it follows that $\bm{L}^{\mathrm{cHL},\sigma,(1)}$ under any
boundary condition $\sigma$ has the same distribution as
$\bm{L}^{\mathrm{cHL},\sigma^{\prime},(1)}$ with $\sigma^{\prime}(j)=1$ for
all $j\in\llbracket 1,N\rrbracket$, i.e., the colored line ensemble associated
with the uncolored $q$-Boson model.

###### Proof.

This is an immediate consequence of combining Definition 3.7 with the color
merging statement Lemma 3.2. ∎

####  3.2.1. Restricting the domain

We need to connect the top curve of $\smash{\bm{L}^{\mathrm{cHL},(j)}}$ for
different $j$ with the height function of the colored S6V model (we focus on
S6V since we will ultimately deal with ASEP via a limit to it from the former;
see Section 5). To do so, we must first modify the setup of the colored S6V
model slightly. Recall from (2.17) that the prelimiting analog
$\mathcal{S}^{\mathrm{S6V},\varepsilon}$ of $\mathcal{S}$ concerns counts of
the number of arrows above a varying vertex on the vertical line
$x=\lfloor\varepsilon^{-1}\rfloor$, and that arrows horizontally enter the
system from the left side at coordinates $(1,i)$ for
$i\in\llbracket-N,N\rrbracket$. In the packed boundary condition, the arrow
entering horizontally at $(1,i)$ has color $i$, and thus all arrow colors
entering at or below the $x$-axis are not positive. In contrast, note that the
colored Hall-Littlewood line ensemble we defined are associated with only
positive colors. (We needed to allow arrows of negative color in the colored
S6V model, as $\mathcal{S}^{\mathrm{S6V},\varepsilon}$ needs to allow negative
values in the first argument.)

We first record a consequence of color merging (of the colored S6V model) that
says that, when considering the height function for colors $1$ or higher, we
may restrict the domain to $\mathbb{Z}_{\geq 1}\times\llbracket 1,N\rrbracket$
without changing the distribution of the colored height function. This will
address the discrepancy pointed out in the previous paragraph.

######  Lemma 3.11.

Fix $N\in\mathbb{N}$ and let
$\sigma:\llbracket-N,N\rrbracket\to\scalebox{0.97}{$\Lparen$}{-}\infty,N\rrbracket\cup\\{-\infty\\}$,
$\sigma_{\geq 1}:\llbracket 1,N\rrbracket\to\llbracket 1,N\rrbracket$ be such
that $\sigma(k)=\sigma_{\geq 1}(k)\geq 1$ for $k\in\llbracket 1,N\rrbracket$
and $\sigma(k)\leq 0$ for $k\in\llbracket-N,0\rrbracket$. Consider the colored
S6V model on the domains $\mathbb{Z}_{\geq 1}\times\llbracket-N,\infty\Rparen$
and $\mathbb{Z}_{\geq 1}\times\llbracket 1,\infty\Rparen$ with boundary
conditions $\sigma$ and $\sigma_{\geq 1}$, respectively (i.e., an arrow of
color $\sigma(k)$ or $\sigma_{\geq 1}(k)$ enters horizontally at $(1,k)$ for
$k\in\llbracket-N,N\rrbracket$ and $\llbracket 1,N\rrbracket$, respectively),
and let $h^{\mathrm{S6V}}$ and $h^{\mathrm{S6V}}_{\geq 1}$ be their respective
colored height functions as defined in (2.17). Then
$(h^{\mathrm{S6V}}(x,0;y,t))_{x,y,t\in\mathbb{N}}\stackrel{{\scriptstyle\smash{d}}}{{=}}(h^{\mathrm{S6V}}_{\geq
1}(x,0;y,t))_{x,y,t\in\mathbb{N}}$.

###### Proof.

This is an immediate consequence of color merging for colored S6V (Remark
2.18): in the colored S6V model on $\mathbb{Z}_{\geq
1}\times\llbracket-N,\infty\Rparen$ with boundary condition $\sigma$ in the
statement, we merge the colors
$\scalebox{0.97}{$\Lparen$}{-}\infty,0\rrbracket\cup\\{-\infty\\}$ with
$-\infty$ and thus obtain the colored S6V model on $\mathbb{Z}_{\geq
1}\times\llbracket 1,\infty\Rparen$ with boundary condition $\sigma_{\geq 1}$.
∎

As noted, the convergence of the S6V and ASEP sheets to $\mathcal{S}$ is a
statement about the colored height functions at a fixed time, so to prove
Theorems 2.8 and 2.20 we can restrict our attention to the colored height
function at time $M$ for fixed $M$, i.e.,
$h^{\mathrm{S6V}}(\bm{\cdot},0;\bm{\cdot},M)$. Note that we switched the
notation for the time argument from $t$ to $M$; this is so as to match the
notation of the colored $q$-Boson model as well as [AB24].

####  3.2.2. Relation to colored S6V

With these preliminaries, we may state the crucial link between the colored
Hall-Littlewood line ensemble and the height functions in colored S6V. Its
proof, originally due to [AB24], relies on the Yang-Baxter equation of colored
S6V and the colored $q$-Boson model. Because our arguments fundamentally rely
on this result, we give a self-contained proof in Appendix A.

######  Proposition 3.12 ([AB24, Theorem 4.7]).

Fix $N,M\in\mathbb{N}$, $\sigma:\llbracket 1,N\rrbracket\to\llbracket
1,N\rrbracket$, $q\in[0,1)$, and $z\in(0,1)$. Let $h^{\mathrm{S6V}}$ be the
height function associated with the colored S6V on $\mathbb{Z}_{\geq
1}\times\llbracket 1,\infty\Rparen$ as in (2.17) and
$\bm{L}^{\mathrm{cHL},(j)}$ be as in Definition 3.7 (with the associated
colored $q$-Boson model on domain $\mathbb{Z}_{\leq 0}\times\llbracket
1,N+M\rrbracket$), both with boundary condition $\sigma$. We have the equality
in distribution of $(h^{\mathrm{S6V}}(k,0;y,M))_{y\in\llbracket
1,N-1\rrbracket,k\in\llbracket 1,N\rrbracket}$ and
$(L^{\mathrm{cHL},\smash{(k)}}_{1}(y))_{y\in\llbracket
1,N-1\rrbracket,k\in\llbracket 1,N\rrbracket}$.

Next we move to describing a Gibbs property of the colored Hall-Littlewood
line ensemble.

####  3.2.3. Colored Hall-Littlewood Gibbs property

From the definition of the colored Hall-Littlewood line ensemble in terms of
the colored $q$-Boson model, and the Gibbs property of the latter recorded in
Lemma 3.5, we obtain a Gibbs property for the former. We restrict ourselves to
stating the case where the system contains only two colors, i.e.,
$\sigma:\llbracket 1,N\rrbracket\to\llbracket 1,2\rrbracket$, as this will
turn out to suffice for our purposes (by performing a color merging later, see
Section 4), though more general Gibbs properties involving more colors also
hold. In this case, $\bm{L}^{\mathrm{cHL},(1)}$ and
$\bm{L}^{\mathrm{cHL},(2)}$ determine the colored arrow configurations at all
vertices in the system via Definition 3.7. Let
$\smash{\mathcal{F}^{\mathrm{cHL}}_{k}}$ (“cHL” short for colored Hall-
Littlewood) be the $\sigma$-algebra

$\displaystyle\smash{\mathcal{F}^{\mathrm{cHL}}_{k}}=\sigma\left(\bigl{\\{}\smash{L^{\mathrm{cHL},(1)}_{i}}(x),\smash{L^{\mathrm{cHL},(2)}_{j}}(x):i\in\mathbb{N},j\in\llbracket
k+1,\infty\Rparen,x\in\llbracket 0,N+M\rrbracket\bigr{\\}}\right);$ (3.8)

in words, we condition on all curves of the line ensemble
$\bm{L}^{\mathrm{cHL},(1)}$, and the curves indexed $k+1$ or larger of
$\bm{L}^{\mathrm{cHL},(2)}$. We now state the Gibbs property of the colored
Hall-Littlewood line ensemble that we will need. Recall the notation
$\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,v}$ from before Lemma 3.4.

######  Lemma 3.13 (Colored Hall-Littlewood Gibbs property).

Under the colored Hall-Littlewood line ensemble measure (with boundary
condition $\sigma:\llbracket 1,N\rrbracket\to\llbracket 1,2\rrbracket$), the
conditional distribution of $\smash{L^{\mathrm{cHL},(2)}_{k}}(\bm{\cdot})$
given $\smash{\mathcal{F}^{\mathrm{cHL}}_{k}}$ is supported on the collection
of Bernoulli paths $L^{\prime}:\llbracket 0,N+M\rrbracket\to\mathbb{Z}$ such
that (i) $\smash{L^{\mathrm{cHL},(1)}_{k}}(\bm{\cdot})-L^{\prime}(\bm{\cdot})$
is a Bernoulli path and (ii) $L^{\prime}(\bm{\cdot})\geq
L^{\mathrm{cHL},(2)}_{k+1}(\bm{\cdot})$ with $L^{\prime}(N+M)=0$ and
$L^{\prime}(0)=L^{\smash{\mathrm{cHL},(2)}}_{k+1}(0)$; call the (random) set
of such Bernoulli paths $\mathrm{Supp}^{\mathrm{cHL}}$.

For such a Bernoulli path $L^{\prime}$, let
$(\bm{A}^{\prime}_{v},i_{v};\bm{B}^{\prime}_{v},j_{v}^{\prime})_{v\in\Gamma_{k}}$
be the associated tuple determined by $L^{\prime}(\bm{\cdot})$,
$L^{\mathrm{cHL},\smash{(1)}}_{k}(\bm{\cdot})$,
$L^{\mathrm{cHL},\smash{(1)}}_{k+1}(\bm{\cdot})$, and
$L^{\mathrm{cHL},\smash{(2)}}_{k+1}(\bm{\cdot})$ on the event that
$L^{\mathrm{cHL},\smash{(2)}}_{k}(\bm{\cdot})=L^{\prime}(\bm{\cdot})$ (we
write $i_{v}$ and not $i_{v}^{\prime}$ as it has no dependence on
$L^{\prime}(\bm{\cdot})$). The condition
$L^{\prime}(\bm{\cdot})\in\mathrm{Supp}^{\mathrm{cHL}}$ is equivalent to
$(\bm{A}^{\prime}_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v})_{v\in\Gamma_{k}}\in\mathrm{Supp}^{\mathrm{cB}}$
(the latter as in Lemma 3.5). Further, the conditional probability that
$L^{\mathrm{cHL},(2)}_{k}(\bm{\cdot})=L^{\prime}(\bm{\cdot})$ , for any fixed
(i.e., deterministic) selection of the righthand side, is proportional to
$\prod_{v\in\Gamma_{k}}\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,v}(\bm{A}^{\prime}_{v},i_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v})\mathbbm{1}_{(\bm{A}^{\prime}_{v};\bm{B}^{\prime}_{v},j^{\prime}_{v})_{v\in\Gamma_{k}}\in\mathrm{Supp}^{\mathrm{cB}}}$.

###### Proof.

The lemma follows from Lemma 3.5 and Definition 3.7 relating the color Hall-
Littlewood line ensemble and the arrow configurations of the colored $q$-Boson
model; note that (i) and (ii) in the hypotheses of Lemma 3.13 correspond to
(i) and (ii) in Lemma 3.5 (for (i), using that
$L^{\smash{\mathrm{cHL},(1)}}_{k}(\bm{\cdot})-\smash{L^{\mathrm{cHL},(2)}_{k}}(\bm{\cdot})$
decreases by $1$ from $x$ to $x+1$ exactly at those $x$ such that an arrow of
color $1$ exits horizontally from $(-k,x)$, by Lemma 3.9 (iii)). ∎

There is also a Gibbs property for the uncolored Hall-Littlewood line
ensemble, i.e., $\bm{L}^{\mathrm{cHL},(1)}$, obtained by translating Lemma
3.4. We will discuss this in more detail in Section 3.3 next.

###  3.3. Convergence of line ensembles

In this section we discuss convergence results for a class of discrete line
ensembles possessing a certain Gibbs property (that of the uncolored Hall-
Littlewood line ensemble). The statement we give for the limit of line
ensembles is under a general framework which we introduce now. Because of the
general setup, these line ensembles are not necessarily related to the
$q$-Boson vertex model; having said that, as mentioned, the example the
framework will be applied to in this paper is the uncolored Hall-Littlewood
line ensemble $\bm{L}^{\mathrm{cHL},(1)}$.

####  3.3.1. Hall-Littlewood Gibbs property

The line ensembles we consider will possess the _Hall-Littlewood Gibbs
property_ , which describes the interactions that curves in the ensemble have
with each other. We start by precisely specifying the data we condition on in
the Gibbs property.

######  Definition 3.14.

Let $\bm{L}:\mathbb{N}\times\Lambda\to\mathbb{Z}$ be a discrete line ensemble
(recall Definition 3.6). Let $\llbracket j,k\rrbracket\subset\mathbb{N}$ and
$\llbracket a,b\rrbracket\subset\Lambda$ be intervals. We define
$\mathcal{F}_{\mathrm{ext}}(\llbracket j,k\rrbracket,\llbracket
a,b\rrbracket,\bm{L})$ to be the $\sigma$-algebra generated by the collection
of random variables
$\\{L_{i}(x):(i,x)\in\mathbb{N}\times\Lambda\setminus(\llbracket
j,k\rrbracket\times\llbracket a+1,b-1\rrbracket)\\}$.

When the line ensemble under discussion is clear, we will drop $\bm{L}$ from
the notation. Similarly when $j=1$, we will replace $\llbracket 1,k\rrbracket$
by simply $k$. Thus $\mathcal{F}_{\mathrm{ext}}(k,\llbracket a,b\rrbracket)$
would denote $\mathcal{F}_{\mathrm{ext}}(\llbracket 1,k\rrbracket,\llbracket
a,b\rrbracket,\bm{L})$.

Next, recall the definition of a Bernoulli path from Definition 2.9. For
$a,b,x,y\in\mathbb{Z}$ with $a<b$, $y\leq x\leq y+b-a$, we define a _Bernoulli
random walk bridge_ $B:\llbracket a,b\rrbracket\to\mathbb{Z}$ from $(a,x)$ to
$(b,y)$ to be a uniformly selected random path from the set of all Bernoulli
paths $\gamma:\llbracket a,b\rrbracket\to\mathbb{Z}$ with $\gamma(a)=x$ and
$\gamma(b)=y$ (assuming this set is non-empty).

######  Definition 3.15 (Weight factor).

Let $\llbracket a,b\rrbracket\subseteq\mathbb{Z}$ and fix $q\in[0,1)$. Let
$\bm{\gamma}=(\gamma_{1},\ldots,\gamma_{k})$, $f$, and $g$ be a collection of
Bernoulli paths ($f$ and $g$ should be thought of as upper and lower boundary
curves respectively), all defined on $\llbracket a,b\rrbracket$, with
$\gamma_{i}(x)\geq\gamma_{i+1}(x)$ for $i\in\llbracket 1,k-1\rrbracket$,
$\gamma_{1}(x)\leq f(x)$, and $\gamma_{k}(x)\geq g(x)$ for all $x\in\llbracket
a,b\rrbracket$. We define the weight factor $W(\bm{\gamma},f,g,\llbracket
a,b\rrbracket)$ by

$\displaystyle W(\bm{\gamma},f,g,\llbracket
a,b\rrbracket):=\prod_{i=0}^{k}\prod_{x=a+1}^{b}\left(1-q^{\Delta_{i}(x-1)}\mathbbm{1}_{\Delta_{i}(x)=\Delta_{i}(x-1)-1}\right),$
(3.9)

where $\Delta_{i}(x)=\gamma_{i}(x)-\gamma_{i+1}(x)$ and $\gamma_{0}=f$ and
$\gamma_{k+1}=g$ by convention. For curves which do not satisfy the ordering
conditions mentioned above, we define $W(\bm{\gamma},f,g,\llbracket
a,b\rrbracket)=0$; observe that (3.9) equals 0 if at any location the
separation between consecutive curves drops from $0$ to $-1$. See also Figure
12. When the interval under consideration is obvious from the context we will
drop $\llbracket a,b\rrbracket$ from the notation. Moreover, if $f=\infty$, we
drop it from the notation; if in addition $g=-\infty$, then we drop it from
the notation as well.

Sometimes, to emphasize whether we are focusing on interaction with the upper
curve $f$, between the curves of $\bm{\gamma}$, or with the lower curve $g$,
we will write

$\displaystyle W(\bm{\gamma},f,g)=W_{\mathrm{up}}(\gamma_{1},f)\cdot
W_{\mathrm{int}}(\bm{\gamma})\cdot W_{\mathrm{low}}(\gamma_{k},g),$

where, in (3.9), $W_{\mathrm{up}}(\gamma_{1},f)$ is the factor corresponding
to $i=0$, $W_{\mathrm{int}}(\bm{\gamma})$ is the product corresponding to
$i=1,\ldots,k-1$, and $W_{\mathrm{low}}(\gamma_{k},g)$ to the factor with
$i=k$.

The following is the Gibbs property that the line ensembles in the general
framework we are setting up will possess; it originally appeared and was
studied in [CD18].

######  Definition 3.16.

(Hall-Littlewood Gibbs property) We say a discrete line ensemble
$\bm{L}:\mathbb{N}\times\Lambda\to\mathbb{Z}$ satisfies the (uncolored) _Hall-
Littlewood Gibbs property_ (with parameter $q\in[0,1)$) if the following holds
for any $\llbracket j,k\rrbracket\subset\mathbb{N}$ and $\llbracket
a,b\rrbracket\subset\Lambda$. Conditionally on
$\mathcal{F}=\mathcal{F}_{\mathrm{ext}}(\llbracket j,k\rrbracket,\llbracket
a,b\rrbracket,\bm{L})$, the distribution of $\bm{L}|_{\llbracket
j,k\rrbracket\times\llbracket a,b\rrbracket}$ is given by a collection of
$k-j+1$ Bernoulli random walk bridges $\bm{B}=(B_{j},\ldots,B_{k})$, where
$B_{i}(a)=L_{i}(a)$ and $B_{i}(b)=L_{i}(b)$ for each $i\in\llbracket
j,k\rrbracket$, reweighted by the Radon-Nikodym derivative
$W(\bm{B},L_{j-1},L_{k+1})/\mathbb
E_{\mathcal{F}}[W(\bm{B},L_{j-1},L_{k+1})]$, where $L_{0}\equiv\infty$ by
convention.

Equivalently, for any bounded function $F:\mathbb{Z}^{\llbracket
j,k\rrbracket\times\llbracket a,b\rrbracket}\to\mathbb{R}$,

$\displaystyle\mathbb E_{\mathcal{F}}\left[F(L_{j}|_{\llbracket
a,b\rrbracket},\ldots,L_{k}|_{\llbracket a,b\rrbracket})\right]=\frac{\mathbb
E_{\mathcal{F}}\left[F(B_{j},\ldots,B_{k})W(\bm{B},L_{j-1},L_{k+1})\right]}{\mathbb
E_{\mathcal{F}}\left[W(\bm{B},L_{j-1},L_{k+1})\right]}$

(here $\mathbb E_{\mathcal{F}}[\bm{\cdot}]$ is shorthand for
$\mathbb{E}[\bm{\cdot}\mid\mathcal{F}]$, and we will also use analogous
notation $\mathbb P_{\mathcal{F}}(\bm{\cdot})$ for
$\mathbb{P}(\bm{\cdot}\mid\mathcal{F})$). We will sometimes write $Z=\mathbb
E_{\mathcal{F}}[W(\bm{B},L_{j-1},L_{k+1})]$, but this will always be made
explicit.

As above, we will usually refer to this property as the Hall-Littlewood Gibbs
property (dropping the “uncolored” adjective); this will not cause confusion
as we will always explicitly say “colored” when referring to its colored
analog.

This Gibbs property is satisfied by $\smash{\bm{L}^{\mathrm{cHL},(1)}}$, but
not in general by $\bm{L}^{\mathrm{cHL},(j)}$ for $j\geq 2$.

######  Proposition 3.17.

$\bm{L}^{\mathrm{cHL},(1)}|_{\mathbb{N}\times\llbracket
0,N-1\rrbracket}=(L^{\mathrm{cHL},(1)}_{1}|_{\llbracket
0,N-1\rrbracket},L^{\mathrm{cHL},(1)}_{2}|_{\llbracket
0,N-1\rrbracket},\ldots)$ has the Hall-Littlewood Gibbs property for any
boundary condition $\sigma:\llbracket 1,N\rrbracket\to\llbracket
1,N\rrbracket$.

###### Proof.

This is a direct consequence of Lemma 3.4 after translating to the line
ensemble notation from Definition 3.7, using also from Lemma 3.10 that
$\bm{L}^{\mathrm{cHL},(1)}$ is in distribution the line ensemble associated to
the uncolored $q$-Boson model. ∎

A derivation of Proposition 3.17 is also contained in [AB24, Proposition 5.1],
and an alternate derivation is present in [CD18, Proposition 3.9] after
matching $\bm{L}^{\mathrm{cHL},(1)}$ with the (uncolored) Hall-Littlewood
process.

$\Delta_{0}(x+1)=2$$\Delta_{1}(x)=1$ Figure 12. An example of the Hall-
Littlewood weight factor with a single curve $\gamma$. The top and bottom
curves (in black) are $f$ and $g$, respectively, and are fixed, and the blue
curve is $\gamma$. In calculating the weight factor $W$, the separation of
$\gamma$ from $f$ and $g$ as captured by $\Delta_{i}(\bm{\cdot})$ is used. In
this example, $W(\gamma,f,g)=(1-q^{2})\cdot(1-q)^{2}$: the separation
$\Delta_{0}(\bm{\cdot})$ between $f$ and $\gamma$ reduces only at the
penultimate step, from $2$ to $1$, and the separation $\Delta_{1}(\bm{\cdot})$
between $\gamma$ and $g$ reduces twice, each time from $1$ to $0$.

####  3.3.2. Heuristic behavior of the Gibbs property

For intuition, the Gibbs property should be thought of as a resampling
property, as in Figure 15 ahead: it says that the following resampling
procedure leaves the distribution of the line ensemble unchanged. First, with
$\llbracket j,k\rrbracket\subseteq\mathbb{N}$ and $\llbracket
a,b\rrbracket\subseteq\Lambda$ fixed, we erase $\bm{L}$ on $\llbracket
j,k\rrbracket\times\llbracket a+1,b-1\rrbracket$ while retaining the
information of its values on $\mathbb{N}\times\Lambda\setminus(\llbracket
j,k\rrbracket\times\llbracket a+1,b-1\rrbracket)$. Then we attempt to replace
$L_{j},\ldots,L_{k}$ on $\llbracket a,b\rrbracket$ with a tuple $\bm{B}$ of
$k-j+1$ independent Bernoulli random walk bridges with endpoints of $B_{i}$
equaling $L_{i}(a)$ and $L_{i}(b)$ at $a$ and $b$, respectively: the attempt
succeeds with probability $W(\bm{B},L_{j-1},L_{k+1})$ as given in Definition
3.15, independent of everything else; note that if any individual curve of
$\bm{B}$ crosses $L_{j-1}$, $L_{k+1}$, or any other curve of $\bm{B}$, then
$W(\bm{B},L_{j-1},L_{k+1})=0$ and the attempt certainly fails. On the event of
$\bm{B}$ successfully replacing $L_{j},\ldots,L_{k}$ on $\llbracket
a,b\rrbracket$, the former’s law is exactly that of $k$ independent Bernoulli
random walk bridges under the reweighting by the Radon-Nikodym derivative
$W(\bm{B},L_{j-1},L_{k+1})/\mathbb
E_{\mathcal{F}}[W(\bm{B},L_{j-1},L_{k+1})]$. The basic idea underlying many of
our arguments is that behavior which is unlikely for such reweighted Bernoulli
random walk bridges must also be unlikely for the original line ensemble.

The above is really a perspective on Gibbs properties of line ensembles in
general, and now we discuss the particular features of the Hall-Littlewood
Gibbs property that constitute a major challenge and necessitate the
development of new techniques, compared to previous studies of line ensembles
with different Gibbs properties. In our setup, we will consider sequences of
line ensembles indexed by a parameter $N$ (which should not be confused with
the $N$ parameter of the colored S6V model or the colored $q$-Boson model in
Section 3.1). Our goal in Section 3.3 is to give a result of convergence of
these line ensembles, rescaled by the KPZ exponents, to the parabolic Airy
line ensemble. This means that we scale the spatial argument of the line
ensemble by $N^{2/3}$ and the value of the lines by $N^{1/3}$ (both up to
constants) in order to obtain a tight, or unit-order, sequence.

At short range, the resampling property penalizes curves coming closer
together, as encoded by the indicator function in (3.9). This is a somewhat
complicated interaction as it is not always active, and, more importantly,
does not satisfy a certain convenient stochastic monotonicity property. Before
discussing these aspects, let us first consider how the interaction behaves on
the larger scale on which our limits will be taken. This means evaluating the
weight function (3.9) when $\llbracket a,b\rrbracket$ is an interval of size
of order $N^{2/3}$ and $\Delta_{i}(x)$ is of order $N^{1/3}$: then $W$ is
$\smash{(1-q^{\Theta(N^{1/3})})^{\Theta(kN^{2/3})}}$, i.e., essentially 1 when
$q\in[0,1)$ is fixed. This assumes the separation between consecutive curves
is of order $N^{1/3}$ throughout the interval; if instead at some location
consecutive curves cross, the weight factor becomes zero as before. Thus at
large $N$ the weight factor is expected to behave like the indicator function
of the curves not intersecting, mimicking the Brownian Gibbs property (see
Definition 6.4) for the parabolic Airy line ensemble (the putative limit of
our discrete one).

####  3.3.3. Lack of stochastic monotonicity

Actually establishing this separation (that the above heuristic relies on) is
non-trivial and will constitute the bulk of the proof of our convergence
results. Some challenges are exactly the earlier mentioned complicated nature
of the interaction at short range, from which we must argue we can escape to a
larger scale, and that the Hall-Littlewood Gibbs property does not possess a
stochastic monotonicity property present in other Gibbs properties previously
studied. Stochastic monotonicity means that if one considers the law of the
curves of the line ensemble inside $\llbracket j,k\rrbracket\times\llbracket
a,b\rrbracket$ conditional on the boundary data, and then considers the same
law with the boundary points increased and/or the upper and lower curves
increased, the second law is stochastically larger than the first (i.e., there
is a coupling of the two laws under which each curve in the second collection,
with higher boundary data, is pointwise larger than the corresponding curve in
the first collection). The Hall-Littlewood Gibbs property fails to satisfy
this because $W$ from (3.9) is a monotone function of the discrete derivative
of the distance between consecutive curves and not of the distance itself. See
Figure 13 for an example demonstrating the lack of stochastic monotonicity.
The violations of stochastic monotonicity can be fairly severe and it is also
possible to find counterexamples to versions of stochastic monotonicity with
more conditions imposed, e.g., comparing only boundary conditions which share
the same endpoints.

$(0,0)$$(0,-k)$$W(B,g)=1-q^{k+1}$$\mathbb{P}(B)=\frac{1-q^{k+1}}{2-q^{k+1}}$$(0,0)$$(0,-k)$$W(B,g)=1$$\mathbb{P}(B)=\frac{1}{2-q^{k+1}}$
Figure 13. An example of the failure of stochastic monotonicity for the Hall-
Littlewood Gibbs property. Here $g$ is the lower boundary curve drawn in black
(indexed by its starting location $(0,-k)$) and $B$ is the random curve drawn
in blue (whose endpoints are fixed). We see that the probability of the higher
possibility for $B$ equals $\smash{\frac{1-q^{k+1}}{2-q^{k+1}}}$, which is
increasing in $k$, while stochastic monotonicity requires it to be non-
increasing.

The approach to line ensemble tightness we develop relies instead on a form of
weak monotonicity for a single curve (a minor refinement of a notion
introduced in [CD18]); see Section 8. By this we mean a comparison of
partition functions (up to a, possibly large, constant) of certain pairs of
laws corresponding to a single curve under different boundary conditions,
which translates to a comparison (up to the same constant) of one-point
probabilities for a single curve. In particular, it does not hold for
collections of more than one curve, and does not imply the existence of a
coupling of the two laws such that all points on the pair of curves are
ordered. Using this weak monotonicity, we iteratively establish uniform
separation of order $N^{1/3}$ between curves further and further down in the
line ensemble, simultaneously establishing control on how far down such curves
can be. We will discuss and develop this more later (see Section 9); here we
next introduce the general framework under which our arguments operate.

####  3.3.4. Assumptions on line ensembles and their tightness

In this section we formulate some assumptions on general line ensembles under
which we will prove tightness. Before stating them, we specify the space we
will work on and some conventions about the objects.

Our sequence of discrete line ensembles will be denoted by
$\bm{L}^{N}=(L_{1}^{N},L_{2}^{N},\ldots)$. Here $N$ is an indexing parameter
which has no relation to the parameter in the colored S6V or colored $q$-Boson
models. Though discrete valued, we will regard $\bm{L}^{N}$ as an element of
$\mathcal{C}(\mathbb{N}\times\mathbb{R},\mathbb{R})$, by linearly
interpolating $L^{N}_{i}(x)$ for $x\in[j,j+1]$ for each $i\in\mathbb{N}$ and
$j\in\mathbb{Z}$; here $\mathbb{N}\times\mathbb{R}$ is given the product
topology of the discrete topology on $\mathbb{N}$ and the usual Euclidean
topology on $\mathbb{R}$. We endow the space
$\mathcal{C}(\mathbb{N}\times\mathbb{R},\mathbb{R})$ with the topology of
uniform convergence on compact sets, i.e.,
$\smash{\bm{f}^{(n)}},\bm{f}\in\mathcal{C}(\mathbb{N}\times\mathbb{R},\mathbb{R})$
satisfy $\smash{\bm{f}^{(n)}}\to\bm{f}$ if for any compact set
$K\subseteq\mathbb{R}$ and $i\in\mathbb{N}$ fixed, $\smash{f^{(n)}_{i}}(x)\to
f_{i}(x)$ uniformly over $x\in K$.

Further, we allow a sequence $g_{N}$ going to $\infty$ with $N\to\infty$ such
that the values of $L^{N}_{i}(x)$ are not specified for $|x|\geq
g_{N}N^{2/3}$, as these values do not affect the existence or value of the
limits we consider (which are determined by uniform convergence on compact
sets on the $N^{2/3}$ spatial scale); in particular, $L^{N}_{i}(x)$ may be
defined for such values in any way such that $\bm{L}^{N}$ is an element of
$\mathcal{C}(\mathbb{N}\times\mathbb{R},\mathbb{R})$, without regard for
satisfying the assumptions below.

Now we may turn to the precise assumptions we assume on our line ensembles
$\bm{L}^{N}=(L^{N}_{1},L^{N}_{2},\ldots)$.

######  Assumption 1 (Hall-Littlewood Gibbs property).

$\bm{L}^{N}$ has the Hall-Littlewood Gibbs property with parameter $q\in[0,1)$
fixed (Definition 3.16).

######  Assumption 2 (One-point tightness of top curve around parabola).

There exist $p\in(-1,0)$ and $\lambda>0$ such that, for any $\delta>0$ there
exists $M=M(\delta)$ such that

$\displaystyle\sup_{x\in\mathbb{R}}\,\limsup_{N\to\infty}\,\mathbb{P}\left(|L^{N}_{1}(xN^{2/3})-pxN^{2/3}+\lambda
x^{2}N^{1/3}|>MN^{1/3}\right)\leq\delta.$

Equivalently, for any $\delta>0$ there exists $M$ such that for all
$x\in\mathbb{R}$ there exists $N_{0}$ such that, for $N>N_{0}$,

$\displaystyle\mathbb{P}\left(|L^{N}_{1}(xN^{2/3})-pxN^{2/3}+\lambda
x^{2}N^{1/3}|>MN^{1/3}\right)\leq\delta.$ (3.10)

By our convention on the definition of $L^{N}_{i}(x)$ for $|x|\geq
g_{N}N^{2/3}$, Assumption 1 is only needed for
$\bm{L}^{N}|_{\mathbb{N}\times[-g_{N}N^{2/3},g_{N}N^{2/3}]}$. We will not
explicitly mention this point or the general point about the definition of
$L^{N}_{i}(x)$ for $|x|\geq g_{N}N^{2/3}$ further.

We note that Assumption 2 is satisfied if it is known that the union over
$x\in\mathbb{R}$ of the set of subsequential weak limits of
$\\{N^{-1/3}(L^{N}_{1}(xN^{2/3})-pxN^{2/3}+\lambda
x^{2}N^{1/3})\\}_{N=1}^{\infty}$ is a tight collection (for instance, as will
be the case for our models of interest, if it is a single distribution, e.g.,
a constant multiple of the GUE Tracy-Widom distribution).

We will need to rescale our line ensembles to state our results. For
$p\in(-1,0)$, $\lambda>0$, and $N\in\mathbb{N}$, we introduce a scaling
operator
$T_{p,\lambda,N}:\mathcal{C}(\mathbb{N}\times\mathbb{R},\mathbb{R})\to\mathcal{C}(\mathbb{N}\times\mathbb{R},\mathbb{R})$
defined by the following for
$\bm{f}\in\mathcal{C}(\mathbb{N}\times\mathbb{R},\mathbb{R})$,
$i\in\mathbb{N}$, and $x\in\mathbb{R}$:

$\displaystyle\bigl{(}T_{p,\lambda,N}(\bm{f})\bigr{)}_{i}(x):=\sigma^{-1}N^{-1/3}\left(f_{i}(\beta
xN^{2/3})-p\beta xN^{2/3}\right)$ (3.11)

where $\beta$ and $\sigma$ are such that

$\displaystyle\sigma^{-1}\beta^{2}\lambda=1\quad\text{and}\quad\sigma^{-2}\beta
p(1-p)=2.$ (3.12)

For $\bm{L}^{N}$ satisfying Assumption 2, and with $p$ and $\lambda$ as there,
define $\bm{\mathfrak{L}}^{N}:\mathbb{N}\times\mathbb{R}\to\mathbb{R}$ by

$\displaystyle\bm{\mathfrak{L}}^{N}:=T_{p,\lambda,N}(\bm{L}^{N}).$ (3.13)

The equations (3.12) are the analogs of the conditions (2.9). The first
condition will ensure that $\smash{\mathfrak{L}^{N}_{1}(x)}$ decays like
$-x^{2}$; the second will ensure that the large $N$ limit of
$\bm{\mathfrak{L}}^{N}$ has the Brownian Gibbs property (Definition 6.4) with
the correct diffusion coefficient of 2.

Now we may state the tightness result we prove, which is a key result needed
to establish our main results, Theorems 2.8 and 2.20; the proof will be given
in Section 6.4.

######  Theorem 3.18 (Tightness of line ensembles).

Let $\bm{L}^{N}:\mathbb{N}\times\mathbb{R}\to\mathbb{R}$ be a line ensemble
satisfying Assumptions 1 and 2. Then
$\\{\bm{\mathfrak{L}}^{N}\\}_{N=1}^{\infty}$ is tight in the space
$\mathcal{C}(\mathbb{N}\times\mathbb{R},\mathbb{R})$, under the topology of
uniform convergence on compact sets.

The following is a third assumption which will be needed to ensure that there
is a unique limiting line ensemble, and that it is the parabolic Airy line
ensemble $\bm{\mathcal{P}}$ from Definition 2.1.

######  Assumption 3 (GUE Tracy-Widom convergence).

There exists $x\in\mathbb{R}$ such that
$\mathfrak{L}^{N}_{1}(x)+x^{2}\xrightarrow{\smash{d}}\mathrm{GUE}\text{-}\mathrm{TW}$
(the GUE Tracy-Widom distribution, as defined at the end of Definition 2.1) as
$N\to\infty$.

The following gives the convergence to $\bm{\mathcal{P}}$ of the rescaled line
ensemble $\bm{\mathfrak{L}}^{N}$. It will be proved in Section 6 as a quick
consequence of the tightness from Theorem 3.18, together with the recent
characterization result for Brownian Gibbsian line ensembles from [AH23a].

######  Theorem 3.19.

Suppose $\bm{L}^{N}$ satisfies Assumptions 1 and 2. Then any subsequential
limit of $\\{\bm{\mathfrak{L}}^{N}\\}_{N=1}^{\infty}$ is
$\bm{\mathcal{P}}+\mathfrak{c}$, where $\mathfrak{c}\in\mathbb{R}$ is a random
variable independent of $\bm{\mathcal{P}}$ (but which may depend on the
subsequence taken), under the topology of uniform convergence on compact sets.
If additionally Assumption 3 is satisfied, then $\mathfrak{c}=0$, i.e,
$\bm{\mathfrak{L}}^{N}\xrightarrow{\smash{d}}\bm{\mathcal{P}}$.

###  3.4. Last passage percolation in the colored Hall-Littlewood line
ensemble

We now turn to stating our results on the relation between
$\bm{L}^{\mathrm{cHL},(1)}$ and
$(\bm{L}^{\mathrm{cHL},(2)},\bm{L}^{\mathrm{cHL},(3)},\ldots)$ (under general
boundary condition $\sigma$) as defined in (3.6). In particular, we give a
precise statement of the approximate representation of
$\smash{L^{\mathrm{cHL},(j)}_{1}}$ as a last passage percolation (LPP) problem
in $\bm{L}^{\mathrm{cHL},(1)}$, which in fact holds exactly when $q=0$.

Recall the definition of $\smash{\bm{L}^{\mathrm{cHL},(j)}}$ from (3.6) and
the notation for LPP problems from (2.2); this notation is for an LPP problem
in an environment given by a family of continuous functions, and for this
purpose as well as in the next result, we regard $\bm{L}^{\mathrm{cHL},(1)}$
as a collection of continuous functions on the real interval $[0,N+M]$ by
linear interpolation, as in the setup introduced in Section 3.3.4. We prove
the next result in Section 4.

######  Theorem 3.20.

Fix $N,M\in\mathbb{N}$, any boundary condition $\sigma:\llbracket
1,N\rrbracket\to\llbracket 1,N\rrbracket$ and $q\in[0,1)$. Let
$(\bm{L}^{\mathrm{cHL},(1)},\ldots,\bm{L}^{\mathrm{cHL},(N)})$ be the
associated colored Hall-Littlewood line ensemble as given by Definition 3.7.
There exist positive constants $c$ (absolute), and $K=K(q)$ such that for any
$j,k\geq 1$ and $m\geq K\log N$,

$\displaystyle\mathbb{P}\left(\sup_{y\in[0,N+M]}\left|L^{\mathrm{cHL},(j)}_{1}(y)-\sup_{z\leq
y}\left(L^{\mathrm{cHL},(j)}_{k+1}(z)+\bm{L}^{\mathrm{cHL},(1)}[(z,k)\to(y,1)]\right)\right|\geq
km\right)\leq kq^{cm^{2}}.$ (3.14)

When $q=0$, $L^{\mathrm{cHL},(j)}_{1}(y)=\sup_{z\leq
y}(L^{\mathrm{cHL},(j)}_{k+1}(z)+\bm{L}^{\mathrm{cHL},(1)}[(z,k)\to(y,1)])$
for all $y\in[0,N+M]$, $j\in\llbracket 1,N\rrbracket$, and $k\in\mathbb{N}$.

Recall the definition of the Airy sheet from Definition 2.3, the definition of
$\mathcal{S}^{\mathrm{S6V},\varepsilon}$ from (2.20), and the relation of
$\smash{L^{\mathrm{cHL},(j)}_{1}}$ to the S6V height function
$h^{\mathrm{S6V}}$ from Proposition 3.12. At this point in our discussion, it
should not be hard to believe that, given the representation of
$\smash{L^{\mathrm{cHL},(j)}_{1}}$ in Theorem 3.20 and the convergence of
$\bm{L}^{\mathrm{cHL},(1)}$ to $\bm{\mathcal{P}}$ from Theorem 3.19, the
convergence of $\mathcal{S}^{\mathrm{S6V},\varepsilon}$ to $\mathcal{S}$ will
follow (and similarly the convergence of
$\mathcal{S}^{\mathrm{ASEP},\varepsilon}$ which, as mentioned in the beginning
of Section 3, will be proved by a limit to ASEP from S6V). Indeed, given the
results we have stated in this section, the argument to establish this is at
this point well-known, following presentations in [DOV22] and [DV21a]; as a
result we give the proof of Theorems 2.20 and 2.8 in Section 7.1 after
relegating some of the standard details to Appendix C.

##  4\. Approximate LPP representation

In this section we prove Theorem 3.20. Recall that we allow a general boundary
condition encoded by a function $\sigma:\llbracket 1,N\rrbracket\to\llbracket
1,N\rrbracket$, where $\sigma(k)$ is the color of the arrow horizontally
entering the domain $\mathbb{Z}_{\leq 0}\times\llbracket 1,N+M\rrbracket$ of
the colored $q$-Boson model at $(1,k)$.

###  4.1. Pitman transform and the $q=0$ case

For two functions $f,g:\mathbb{Z}_{\geq 0}\to\mathbb{R}$, we define the
_Pitman transform_ $\mathrm{PT}(f,g):\mathbb{Z}_{\geq 0}\to\mathbb{R}$ by

$\displaystyle\mathrm{PT}(f,g)(y):=f(y)+\max_{0\leq y^{\prime}\leq
y}\left(g(y^{\prime})-f(y^{\prime})\right).$ (4.1)

The Pitman transform should be thought of as an LPP problem across two curves,
with the bottom one equal to $g$ and the top equal to $f$ (this interpretation
assumes $g(0)=0$, as will be the case in our context). Indeed, we will
ultimately access the LPP value across $k$ lines by iterated applications of
$\mathrm{PT}$.

We start by stating and proving a statement in the case of $q=0$, in order to
make the source of the Pitman transform in the colored Hall-Littlewood line
ensemble more transparent; we will subsequently turn to obtaining a similar
statement in the case of general $q\in[0,1)$ in Section 4.2.

In this section we will mainly restrict to the case where $\sigma:\llbracket
1,N\rrbracket\to\llbracket 1,2\rrbracket$, i.e., arrows in the colored
$q$-Boson model are only of color $1$ or $2$, as this will suffice for the
proof of Theorem 3.20 (see the end of this section) by color merging. We drop
the superscript of “cHL” when writing the colored Hall-Littlewood line
ensemble; for instance we will write $\smash{L^{(j)}_{i}}$ for
$\smash{L^{\mathrm{cHL},(j)}_{i}}$. We will also refer to arrows of color 2 as
$2$-arrows, and analogously $1$-arrows for arrows of color $1$. In particular,
recalling the relation of the colored Hall-Littlewood line ensemble and the
colored $q$-Boson model from Definition 3.7, $\smash{L^{(2)}_{k}}$ corresponds
to counts of 2-arrows in the $k$th column (i.e., $\\{-k\\}\times\llbracket
1,N+M\rrbracket$) of the colored $q$-Boson model, while $\smash{L^{(1)}_{k}}$
corresponds to combined arrow counts of $1$\- and $2$-arrows in the same
column.

######  Lemma 4.1.

Let $q=0$. For any boundary condition $\sigma:\llbracket
1,N\rrbracket\to\llbracket 1,2\rrbracket$ and any $k\in\mathbb{N}$,

$L^{(2)}_{k}=\mathrm{PT}\left(L^{(1)}_{k},L^{(2)}_{k+1}\right).$

It turns out that a lower bound actually holds deterministically for all
$q\in[0,1)$. We state and prove it first before giving the proof of Lemma 4.1
that equality holds when $q=0$.

######  Lemma 4.2.

For any $q\in[0,1)$, boundary condition $\sigma:\llbracket
1,N\rrbracket\to\llbracket 1,2\rrbracket$, and $k\in\mathbb{N}$,

$L^{(2)}_{k}\geq\mathrm{PT}\left(L^{(1)}_{k},L^{(2)}_{k+1}\right).$

###### Proof.

Fix $y$. Recalling the definition (4.1) of the Pitman transform, we must show
that

$\displaystyle L^{(2)}_{k}(y)\geq L^{(1)}_{k}(y)+\max_{y^{\prime}\leq
y}\left(L^{(2)}_{k+1}(y^{\prime})-L^{(1)}_{k}(y^{\prime})\right).$

Let us write $O_{k}(y)=L^{(1)}_{k}(y)-L^{(2)}_{k}(y)$, so that $O_{k}(y)$ is
the number of $1$-arrows exiting the $k$th column in the colored $q$-Boson
model at row strictly higher than $y$. Then, from the previous display, what
we have to show is equivalent to

$\displaystyle\min_{y^{\prime}\leq
y}\left(O_{k}(y^{\prime})+L^{(2)}_{k}(y^{\prime})-L^{(2)}_{k+1}(y^{\prime})\right)\geq
O_{k}(y).$ (4.2)

Now, recall from (3.7) that $L^{(2)}_{k}(y^{\prime})\geq
L^{(2)}_{k+1}(y^{\prime})$ for all $y^{\prime}$ and observe that $O_{k}(y)$ is
non-increasing in $y$ by its definition as the count of $1$-arrows (analogous
to the fact that $\smash{L^{(j)}_{k}(y+1)\leq L^{(j)}_{k}(y)}$ noted in the
third part of (3.7)). This yields that the lefthand side of the previous
display is at least the righthand side. ∎

###### Proof of Lemma 4.1.

We continue with the notation $O_{k}(y)=L^{(1)}_{k}(y)-L^{(2)}_{k}(y)$ from
the previous proof for the count of 1-arrows exiting the $k$th column at row
strictly higher than $y$. By Lemma 4.2, we must show (4.2) holds with
equality, i.e., for some $y^{\prime}\leq y$,

$O_{k}(y^{\prime})+L^{(2)}_{k}(y^{\prime})-L^{(2)}_{k+1}(y^{\prime})=O_{k}(y).$
(4.3)

First suppose that $O_{k}(y^{\prime})=O_{k}(y)$ for all $0\leq y^{\prime}\leq
y$, which is the case that no $1$-arrows exit horizontally from any such
vertex $(-k,y^{\prime})$. We observe that
$\smash{L^{(2)}_{k}(0)=L^{(2)}_{k+1}(0)}$, since both quantities are simply
the total number of $2$-arrows in the system, and we have established (4.3) at
$y^{\prime}=0$.

Next we assume that there exists at least one vertex $(-k,y^{\prime})$ with
$y^{\prime}\leq y$ from which a 1-arrow exits horizontally. Let $y_{0}$ be the
largest such $y^{\prime}$; equivalently, $y_{0}$ is the largest
$y^{\prime}\leq y$ such that $O_{k}(y^{\prime})=O_{k}(y^{\prime}-1)-1$.
Observe that then $O_{k}(y_{0})=O_{k}(y)$ since $O_{k}$ is non-increasing for
each $k$.

We claim that then $L^{(2)}_{k}(y_{0})=L^{(2)}_{k+1}(y_{0})$, which would
establish (4.3). Indeed, we know from the weights in the $q=0$ case from
Figure 10 that since a $1$-arrow exits horizontally from $(-k,y_{0})$, there
is no $2$-arrow entering that vertex from the left or bottom. The arrows
vertically entering $(-k,y_{0})$ are precisely the arrows that have exited
$(-(k+1),y^{\prime})$ horizontally for some $y^{\prime}<y_{0}$ and which have
not exited the $k$th column strictly below $y_{0}$. Thus all 2-arrows in the
$(k+1)$st column that have horizontally exited that column strictly below row
$y_{0}$ have also exited the $k$th column strictly below row $y_{0}$, and, as
we already noted, a $2$-arrow does not exit horizontally from $(-k,y_{0})$ or
$(-(k+1),y_{0})$. In other words,
$L^{(2)}_{k}(0)-L^{(2)}_{k}(y_{0})=L^{(2)}_{k+1}(0)-L^{(2)}_{k+1}(y_{0})$,
which implies our claim, since $L^{(2)}_{k}(0)=L^{(2)}_{k+1}(0)$. ∎

As we saw in Lemma 4.2,
$L^{(2)}_{k}(y)\geq\mathrm{PT}(L^{(1)}_{k},L^{(2)}_{k+1})(y)$
deterministically. The next statement gives a bound on how much the deviation
can be in the case of general $q\in[0,1)$ and is the basic ingredient of the
proof of Theorem 3.20.

######  Proposition 4.3.

There exist positive constants $c$ and $K=K(q)$ such that, for any
$N\in\mathbb{N}$, boundary condition $\sigma:\llbracket
1,N\rrbracket\to\llbracket 1,2\rrbracket$, $k\in\mathbb{N}$, $q\in[0,1)$, and
$m\geq K\log N$,

$\displaystyle\mathbb{P}\left(\max_{y\in\llbracket
0,N+M\rrbracket}\left(L^{(2)}_{k}(y)-\mathrm{PT}(L^{(1)}_{k},L^{(2)}_{k+1})(y)\right)\geq
m\right)\leq q^{cm^{2}}.$

Before turning to the proof of this statement in Section 4.2, we use it to
establish Theorem 3.20.

###### Proof of Theorem 3.20.

Recall that we have dropped the superscript $\mathrm{cHL}$ from the notation
of the colored line ensembles. Next, it suffices to specialize to the case of
$j=2$. Indeed, the original statement will follow from this special case by
Lemma 3.10 (color merging), i.e., the fact that the joint distribution of
$(\bm{L}^{(1)},\bm{L}^{(j)})$ associated to boundary condition $\sigma$ is the
same as that of $(\bm{L}^{(1)},\bm{L}^{(2)})$ associated to boundary condition
$\tau\circ\sigma$, where $\tau(i)=1$ if $i\in\llbracket 1,j-1\rrbracket$ and
$\tau(i)=2$ if $i\in\llbracket j,N\rrbracket$. In other words, we merge the
colors in $\llbracket 1,j-1\rrbracket$ to be $1$ and those in $\llbracket
j,N\rrbracket$ to be $2$.

Inductively define $\mathrm{PT}^{(1)}=\mathrm{PT}$ and for $k\geq 2$

$\displaystyle\mathrm{PT}^{(k)}\left((L^{(1)}_{i})_{i=1}^{k},L^{(2)}_{k+1}\right):=\mathrm{PT}^{(k-1)}\left((L^{(1)}_{i})_{i=1}^{k-1},\mathrm{PT}\left(L^{(1)}_{k},L^{(2)}_{k+1}\right)\right);$
(4.4)

It is easy to show that

$\displaystyle\mathrm{PT}^{(k)}\left((L^{(1)}_{i})_{i=1}^{k},L^{(2)}_{k+1}\right)=\max_{z\leq
y}\left(L^{(2)}_{k+1}(z)+\bm{L}^{(1)}[(z,k)\to(y,1)]\right).$ (4.5)

With this we see that

$\displaystyle\mathbb{P}\left(\max_{y\in\llbracket
0,N+M\rrbracket}\left|L^{(2)}_{1}(y)-\max_{z\leq
y}\left(L^{(2)}_{k+1}(z)+\bm{L}^{(1)}[(z,k)\to(y,1)]\right)\right|\geq
km\right)$ $\displaystyle\leq\mathbb{P}\left(\max_{y\in\llbracket
0,N+M\rrbracket}\left|L^{(2)}_{1}(y)-\mathrm{PT}(L^{(1)}_{1},L^{(2)}_{2})(y)\right|\geq
m\right)$ (4.6)
$\displaystyle\quad+\sum_{j=1}^{k-1}\mathbb{P}\left(\max_{y\in\llbracket
0,N+M\rrbracket}\left|\mathrm{PT}^{(j)}\left((L^{(1)}_{i})_{i=1}^{j},L^{(2)}_{j+1}\right)(y)-\mathrm{PT}^{(j+1)}\left((L^{(1)}_{i})_{i=1}^{j+1},L^{(2)}_{j+2}\right)(y)\right|\geq
m\right).$

In the $q=0$ case, Lemma 4.1 says that the previous display is $0$ for any
$m>0$. For the case of $q\in(0,1)$, we observe that if $h$ is a function with
$\max|h|\leq\varepsilon$ for some $\varepsilon>0$, then it follows from the
representation (4.5) of $\mathrm{PT}^{(k)}$ and the triangle inequality that,
for any $f_{1},\ldots,f_{k}$ and $g$,

$\displaystyle|\mathrm{PT}^{(k)}((f_{i})_{i=1}^{k},g)-\mathrm{PT}^{(k)}((f)_{i=1}^{k},g+h)|\leq
2\varepsilon.$ (4.7)

From Lemma 4.2 and Proposition 4.3 it holds that, for any $j\geq 0$ and $m\geq
K\log N$,

$\displaystyle\mathbb{P}\left(\max_{y\in\llbracket
0,N+M\rrbracket}\left|L^{(2)}_{j+1}(y)-\mathrm{PT}(L^{(1)}_{j+1},L^{(2)}_{j+2})(y)\right|\geq\tfrac{1}{2}m\right)\leq
q^{cm^{2}}.$

Combining this with (4.7) and the definition (4.4) of $\mathrm{PT}^{(j)}$
yields that the $j$th summand in (4.6) is upper bounded by $q^{cm^{2}}$ which,
on tracing back, completes the proof. ∎

The rest of the section is devoted to proving Proposition 4.3.

###  4.2. The case of general $q$

As we saw in the proof of Lemma 4.1, the Pitman transform representation was
essentially a consequence of the fact that, when $q=0$, the colored $q$-Boson
vertex weights force $2$-arrows to exit horizontally before $1$-arrows. In
$q>0$, $1$-arrows can exit before $2$-arrows, but will be penalized in weight
by a factor of $q^{m}$, where $m$ is the number of $2$-arrows vertically
entering at the vertex where the $1$-arrow is exiting (see Figure 10). At a
high level, the heuristic behind the proof of Proposition 4.3 is as follows:
on the event that the deviation of the height function in the $q>0$ case from
the Pitman transform expression (as it would be in the $q=0$ case) is $m$, we
can find order $m$ many different locations where a 1-arrow horizontally exits
before a $2$-arrow, and where there are order $m$ many $2$-arrows “in queue”,
i.e., vertically entering; this will lead to the $\smash{q^{cm^{2}}}$
probability bound for this event.

The precise proof of Proposition 4.3 requires introducing some notation. Let
$U=N+M$ (the number of rows in the colored $q$-Boson model). Fix
$\bm{v}=(v_{1},\ldots,v_{U})\in\\{0,1,2\\}^{U}$ and
$\bm{x}=(x_{1},\ldots,x_{U})\in\\{0,1\\}^{U}$. Here $v_{y}$ represents the
color of the arrow horizontally entering at the $y$th row and $x_{y}$
represents the number (of non-zero color) arrows horizontally exiting at the
$y$th row, both counted from the bottom (see Figure 14).

For a $\bm{v}\in\\{0,1,2\\}^{U}$, $j\in\\{1,2\\}$ and $y\in\llbracket
1,U\rrbracket$, let the height function (which we call $\mathrm{Ht}$ for
short) of the $j$th color be defined by

$\mathrm{Ht}_{j}(\bm{v},y):=\\#\Bigl{\\{}y^{\prime}>y:v_{y^{\prime}}=j\Bigr{\\}}=\sum_{y^{\prime}=y+1}^{U}\mathbbm{1}_{v_{y^{\prime}}=j},$

i.e., the number of arrows of color $j$ strictly above row $y$ in $\bm{v}$.

We say that $\bm{v}$ and $\bm{x}$ are _compatible_ if they can arise as the
incoming color counts and outgoing total arrow counts at a column,
respectively, while satisfying arrow conservation; more precisely, if and only
if
$\sum_{j=1}^{2}(\mathrm{Ht}_{j}(\bm{v},0)-\mathrm{Ht}_{j}(\bm{v},y))\geq\sum_{y^{\prime}=1}^{y}x_{y^{\prime}}$
for each $y$ and equality holds at $y=U$.

For given $\bm{v}\in\\{0,1,2\\}^{U}$ and $\bm{x}\in\\{0,1\\}^{U}$ which are
compatible, define $\mathrm{Val}(\bm{v},\bm{x})$ (short for “valid”)

$\mathrm{Val}(\bm{v},\bm{x}):=\left\\{\bm{w}\in\\{0,1,2\\}^{U}:\parbox{239.00298pt}{\centering$w_{y}=0\iff
x_{y}=0$, $\mathrm{Ht}_{j}(\bm{w},0)=\mathrm{Ht}_{j}(\bm{v},0),$\\\
$\mathrm{Ht}_{j}(\bm{w},y)\geq\mathrm{Ht}_{j}(\bm{v},y)\text{ for }j=1,2,\
y\in\llbracket 0,U-1\rrbracket$\@add@centering}\right\\},$

to be the set of valid assignments of outgoing colors at a column with
incoming color counts given by $\bm{v}$ and outgoing arrow counts given by
$\bm{x}$ (see Figure 14); here, valid informally means that colored arrow
conservation holds.

$v_{1}$$x_{1}$$v_{2}$$x_{2}$$v_{3}$$x_{3}$$v_{4}$$x_{4}$$1$$0$$0$$0$$2$$1$$0$$1$$\bm{v}$$\bm{x}$$1$$0$$0$$0$$2$$2$$0$$1$$\bm{v}$$\bm{w}$
Figure 14. A depiction of $\bm{v}\in\\{0,1,2\\}^{U}$, $\bm{x}\in\\{0,1\\}^{U}$
(left panel), an example where they are compatible (middle panel), and an
example of a $\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})$ (right panel). Here the
$1$ and $0$ in black on the right side of the middle panel respectively
indicate the presence or absence of an arrow of any color, while the colored
numbers on the left side of the middle panel and both sides of the right panel
indicate the color of the arrow.

For $\bm{v}\in\\{0,1,2\\}^{U}$ and $\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})$,
define $\mathrm{VertCt}_{j}$ (short for “vertical count”) by

$\begin{split}\mathrm{VertCt}_{j}(\bm{v},\bm{w},y)&:=\bigl{(}\mathrm{Ht}_{j}(\bm{v},0)-\mathrm{Ht}_{j}(\bm{v},y)\bigr{)}-\bigl{(}\mathrm{Ht}_{j}(\bm{w},0)-\mathrm{Ht}_{j}(\bm{w},y)\bigr{)}\\\
&=\mathrm{Ht}_{j}(\bm{w},y)-\mathrm{Ht}_{j}(\bm{v},y)=\sum_{y^{\prime}=1}^{y}\mathbbm{1}_{v_{y^{\prime}}=j}-\mathbbm{1}_{w_{y^{\prime}}=j}\end{split}$
(4.8)

be the number of arrows of color $j$ vertically exiting the vertex at height
$y$; the second equality is since
$\mathrm{Ht}_{j}(\bm{v},0)=\mathrm{Ht}_{j}(\bm{w},0)$ is the total number of
$j$-arrows in the system (equal since $\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})$).
Note that $\mathrm{VertCt}_{j}(\bm{v},\bm{w},y)$ is a function only of
$v_{1},\ldots,v_{y}$ and $w_{1},\ldots,w_{y}$.

We next define a specific assignment
$\bm{w}^{*}\in\mathrm{Val}(\bm{v},\bm{x})$ for any given compatible pair
$(\bm{v},\bm{x})$, which is the deterministic (and unique) assignment of
colors to the outgoing arrows that would occur in the $q=0$ colored $q$-Boson
model. It is defined as follows: we specify $w^{*}_{y}$ iteratively starting
with $y=1$ and continuing in increasing order by:

  * $\bullet$

$w^{*}_{y}=2$ if
$\mathbbm{1}_{v_{y}=2}+\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{*},y-1)\geq 1$
(i.e., if there is at least one $2$-arrow that can exit the vertex at height
$y$) and $x_{y}=1$; note that the condition is well-defined since
$\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{*},y-1)$ does not depend on
$w^{*}_{y^{\prime}}$ for any $y^{\prime}\geq y$;

  * $\bullet$

$w^{*}_{y}=1$ if
$\mathbbm{1}_{v_{y}=2}+\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{*},y-1)=0$ and
$\mathbbm{1}_{v_{y}=1}+\mathrm{VertCt}_{1}(\bm{v},\bm{w}^{*},y-1)\geq 1$
(i.e., if there are no 2-arrows and at least one $1$-arrow that can exit the
vertex at height $y$) and $x_{y}=1$;

  * $\bullet$

$w^{*}_{y}=0$ if $x_{y}=0$.

The fact that $(\bm{v},\bm{x})$ are compatible ensures that exactly one of the
above three conditions holds for each $y$.

Fix $\bm{v}\in\\{0,1,2\\}^{U}$ and $\bm{x}\in\\{0,1\\}^{U}$ which are
compatible, let $\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})$, and let $\bm{w}^{*}$
be as defined above. We define the Pitman error $\mathrm{PE}(\bm{w})$ of
$\bm{w}$ by

$\displaystyle\mathrm{PE}(\bm{w}):=\max_{y\leq
U}\bigl{(}\mathrm{Ht}_{2}(\bm{w},y)-\mathrm{Ht}_{2}(\bm{w}^{*},y)\bigr{)},$

i.e., it is the maximal height difference of $2$-arrows between $\bm{w}$ and
$\bm{w}^{*}$ over all rows. Observe that
$\mathrm{Ht}_{2}(\bm{w},y)\geq\mathrm{Ht}_{2}(\bm{w}^{*},y)$ for each $y\leq
U$. Indeed, as we said, it is easy to show that $\bm{w}^{*}$ is the
configuration that would arise in the $q=0$ setting, so by Lemma 4.1,
$\mathrm{Ht}_{2}(\bm{w}^{*},\bm{\cdot})$ may be regarded as the Pitman
transform of the incoming color counts $\bm{v}$ and the outgoing arrow counts
$\bm{x}$; then Lemma 4.2 implies that
$\mathrm{Ht}_{2}(\bm{w},y)\geq\mathrm{Ht}_{2}(\bm{w}^{*},y)$. Alternatively,
since $\bm{w}^{*}$ always releases $2$-arrows as soon as it is possible,
$\mathrm{Ht}_{2}(\bm{w}^{*},y)$ is minimal over all
$\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})$ for any $y$, which again verifies that
$\mathrm{Ht}_{2}(\bm{w},y)\geq\mathrm{Ht}_{2}(\bm{w}^{*},y)$.

We need some notation for the weight of configurations. For a compatible pair
$(\bm{v},\bm{x})$ and $\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})$, let
$\bm{\mathrm{VertCt}}(\bm{v},\bm{w},y)=(\mathrm{VertCt}_{1}(\bm{v},\bm{w},y),\mathrm{VertCt}_{2}(\bm{v},\bm{w},y))$.
Further let
$\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,y}=\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{1}$
if $y\in\llbracket 1,N\rrbracket$ and
$\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,y}=\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z}$
(recall that $z\in(0,1)$ is the fixed spectral parameter) if $y\in\llbracket
N+1,N+M\rrbracket$, where $\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{u}$ is read
from Figure 10. Now define

$\mathrm{Wgt}_{y}(\bm{v},\bm{w}):=\scalebox{1.05}{\mbox{$\mathtt{L}$}}_{z,y}\bigl{(}\bm{\mathrm{VertCt}}(\bm{v},\bm{w},y-1),v_{y};\bm{\mathrm{VertCt}}(\bm{v},\bm{w},y),w_{y}\bigr{)}$
(4.9)

to be the vertex weight at $y$, with incoming and outgoing horizontal arrows
of $v_{y}$ and $w_{y}$, respectively, and incoming and outgoing vertical
arrows of $\bm{\mathrm{VertCt}}(\bm{v},\bm{w},y-1)$ and
$\bm{\mathrm{VertCt}}(\bm{v},\bm{w},y)$, respectively. Next, define the weight
of the configuration determined by $\bm{v}$ and $\bm{w}$ by

$\displaystyle\mathrm{Wgt}(\bm{v},\bm{w}):=\prod_{y=1}^{U}\mathrm{Wgt}_{y}(\bm{v},\bm{w}).$
(4.10)

The next lemma is a deterministic statement about the weights of
configurations $w$ with high values of $\mathrm{PE}(\bm{w})$.

######  Lemma 4.4.

Fix $N,M\in\mathbb{N}$. There exist positive absolute constants $c$ and $C$
and $K=K(q)$, such that the following holds for any spectral parameter
$z\in(0,1)$ (which may depend on $N,M$). Fix a compatible pair
$(\bm{v},\bm{x})\in\\{0,1,2\\}^{U}\times\\{0,1\\}^{U}$. There exists a mapping
$f:\mathrm{Val}(\bm{v},\bm{x})\to\mathrm{Val}(\bm{v},\bm{x})$ such that, if
$\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})$ with $m:=\mathrm{PE}(\bm{w})$ and
$\mathrm{Wgt}(\bm{v},\bm{w})>0$, then
$\mathrm{Wgt}(\bm{v},\bm{w})/\mathrm{Wgt}(\bm{v},f(\bm{w}))\leq Cq^{cm^{2}}$
whenever $m>K\log N$. Further, for any $\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})$
with $m:=\mathrm{PE}(\bm{w})$ and $m\geq K\log N$,
$\\#f^{-1}(\\{\bm{w}\\})\leq N^{2m}$.

This statement should be interpreted as giving a bound on the conditional
probability of a given configuration exhibiting
$\max_{y}\smash{L^{(2)}_{k}(y)-\mathrm{PT}(L^{(1)}_{k},L^{(2)}_{k+1})(y)}\geq
m$ given $\smash{L^{(1)}_{k}}$ and $\smash{L^{(2)}_{k+1}}$ (which will
determine $\bm{v}$ and $\bm{x}$). Then Proposition 4.3 will follow by
averaging over the conditioned data; we give this proof now before proving
Lemma 4.4 in Section 4.3.

###### Proof of Proposition 4.3.

Recall from (3.8) that $\mathcal{F}^{\mathrm{cHL}}_{k}$ is the
$\sigma$-algebra generated by $\bm{L}^{(1)}$, $L^{(2)}_{k+1},L^{(2)}_{k+2}$,
$\ldots$. We condition on $\mathcal{F}^{\mathrm{cHL}}_{k}$ and use the tower
property of conditional expectations to write

$\displaystyle\mathbb{P}\left(\max_{y\in\llbracket
0,N+M\rrbracket}L^{(2)}_{k}(y)-\mathrm{PT}(L^{(1)}_{k},L^{(2)}_{k+1})(y)\geq
m\right)$ $\displaystyle=\mathbb{E}\left[\mathbb{P}\left(\max_{y\in\llbracket
0,N+M\rrbracket}L^{(2)}_{k}(y)-\mathrm{PT}(L^{(1)}_{k},L^{(2)}_{k+1})(y)\geq
m\ \Big{|}\ \mathcal{F}^{\mathrm{cHL}}_{k}\right)\right].$

Let $\bm{v}\in\\{0,1,2\\}^{U}$ and $\bm{x}\in\\{0,1\\}^{U}$ be given by

$\displaystyle
v_{y}=\left(L^{(2)}_{k+1}(y-1)-L^{(2)}_{k+1}(y)\right)+\left(L^{(1)}_{k+1}(y-1)-L^{(1)}_{k+1}(y)\right)\quad\text{and}\quad
x_{y}=L^{(1)}_{k}(y-1)-L^{(1)}_{k}(y);$

in words, they are the horizontally incoming colors and the horizontally
outgoing arrow counts at column $k$. They are compatible. Let
$A(\bm{v},\bm{x},m^{\prime})$ be the set of
$\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})$ such that
$\mathrm{PE}(\bm{w})=m^{\prime}$ and $\mathrm{Wgt}(\bm{v},\bm{w})>0$. Now, it
is immediate from the Gibbs property of the colored Hall-Littlewood line
ensemble, Lemma 3.13 (and can also be seen directly from the Gibbs property of
the colored $q$-Boson model, Lemma 3.5), that

$\begin{split}&\mathbb{P}\left(\max_{y\in\llbracket
0,N+M\rrbracket}L^{(2)}_{k}(y)-\mathrm{PT}(L^{(1)}_{k},L^{(2)}_{k+1})(y)\geq
m\ \Big{|}\ \mathcal{F}^{\mathrm{cHL}}_{k}\right)\\\
&=\frac{\sum_{m^{\prime}=m}^{\infty}\sum_{\bm{w}\in
A(\bm{v},\bm{x},m^{\prime})}\mathrm{Wgt}(\bm{v},\bm{w})}{\sum_{w\in\mathrm{Val}(\bm{v},\bm{x})}\mathrm{Wgt}(\bm{v},\bm{w})}.\end{split}$
(4.11)

For each $\bm{w}\in A(\bm{v},\bm{x},m^{\prime})$, let
$f(\bm{w})\in\mathrm{Val}(\bm{v},\bm{x})$ be the configuration obtained from
Lemma 4.4, and recall $\\#f^{-1}(\bm{w})\leq N^{2m^{\prime}}$. Since
$m^{\prime}\geq m\geq K\log N$, we know that
$\mathrm{Wgt}(\bm{v},\bm{w})/\mathrm{Wgt}(\bm{v},f(\bm{w}))$ is upper bounded
by $Cq^{c(m^{\prime})^{2}}$. So we see that, since
$\\#f^{-1}(\\{\bm{w}\\})\leq N^{2m^{\prime}}$ for $\bm{w}\in
A(\bm{v},\bm{x},m^{\prime})$,

$\displaystyle\sum_{\bm{w}\in
A(\bm{v},\bm{x},m^{\prime})}\\!\mathrm{Wgt}(\bm{v},\bm{w})\leq
Cq^{c(m^{\prime})^{2}}\\!\sum_{\bm{w}\in
A(\bm{v},\bm{x},m^{\prime})}\\!\mathrm{Wgt}(\bm{v},f(\bm{w}))\leq
CN^{2m^{\prime}}q^{c(m^{\prime})^{2}}\\!\sum_{\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})}\\!\mathrm{Wgt}(\bm{v},\bm{w}).$

Substituting this into (4.11), and using that
$q^{\frac{1}{2}c(m^{\prime})^{2}}\leq C^{-1}N^{-2m^{\prime}}$ by raising $K$
if necessary (and recalling $m^{\prime}\geq m\geq K\log N$), yields that the
righthand side of that equation is upper bounded by $q^{c^{\prime}m^{2}}$ for
some absolute constant $c>0$. This completes the proof. ∎

###  4.3. Construction of the alternate configuration

###### Proof of Lemma 4.4.

First, to ensure that the $f$ we need to produce is defined for all inputs,
for any $\bm{w}\in\mathrm{Val}(\bm{v},\bm{x})$ such that
$\mathrm{PE}(\bm{w})<K\log N$ (for a $K=K(q)$ to be set later) or
$\mathrm{Wgt}(\bm{v},\bm{w})=0$, we define $f(\bm{w})=\bm{w}$. So in the
remainder of the proof, we may assume that $\bm{w}$ is such that
$\mathrm{PE}(\bm{w})>K\log N$ and $\mathrm{Wgt}(\bm{v},\bm{w})>0$. It will be
useful to keep track of a running Pitman error by row $y$, for which we define

$\displaystyle\mathrm{PE}(\bm{w},y):=\max_{y^{\prime}\leq
y}\left(\mathrm{Ht}_{2}(\bm{w},y^{\prime})-\mathrm{Ht}_{2}(\bm{w}^{*},y^{\prime})\right).$

Let $m^{\prime}\in\llbracket 1,m\rrbracket$. We fix $y=y(m^{\prime})$ to be
the smallest such that $\mathrm{PE}(\bm{w},y)=m^{\prime}$; since
$\mathrm{PE}(\bm{w},0)=0$, $\bm{w}$ is such that
$\mathrm{PE}(\bm{w},U)=\mathrm{PE}(\bm{w})=m$, and
$\mathrm{PE}(\bm{w},y^{\prime})$ increases by at most $1$ when $y^{\prime}$
increases by $1$, such a $y$ exists.

We claim that $w_{y}=1$ and $w_{y}^{*}=2$. We first observe that since
$\mathrm{PE}(\bm{w},y)>\mathrm{PE}(\bm{w},y-1)$ and
$\mathrm{Ht}_{2}(\bm{w},\bm{\cdot})$, $\mathrm{Ht}_{2}(\bm{w}^{*},\bm{\cdot})$
are non-increasing and have increments in $\\{-1,0\\}$, it must hold that
$\mathrm{Ht}_{2}(\bm{w},y)=\mathrm{Ht}_{2}(\bm{w},y-1)$ and
$\mathrm{Ht}_{2}(\bm{w}^{*},y)=\mathrm{Ht}_{2}(\bm{w}^{*},y-1)-1$. This means
that $w^{*}_{y}=2$, which implies $w_{y}=1$ since $w_{y}\in\\{1,2\\}$ (both
$\bm{w}$ and $\bm{w}^{*}$ have the same outgoing arrow counts) and
$\mathrm{Ht}_{2}(\bm{w},y)=\mathrm{Ht}_{2}(\bm{w},y-1)$.

Next we claim that $\mathrm{VertCt}_{2}(\bm{v},\bm{w},y-1)\geq m^{\prime}$. To
see this we first write, using the definition (4.8) of
$\mathrm{VertCt}_{2}(\bm{v},\bm{w},y-1)$ and noting that, by definition of
$y$,
$\mathrm{Ht}_{2}(\bm{w},y-1)-\mathrm{Ht}_{2}(\bm{w}^{*},y-1)=m^{\prime}-1$,

$\displaystyle\mathrm{VertCt}_{2}(\bm{v},\bm{w},y-1)$
$\displaystyle=\mathrm{Ht}_{2}(\bm{w},y-1)-\mathrm{Ht}_{2}(\bm{v},y-1)$
$\displaystyle=\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{*},y-1)+\mathrm{Ht}_{2}(\bm{w},y-1)-\mathrm{Ht}_{2}(\bm{w}^{*},y-1)$
(4.12)
$\displaystyle=\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{*},y-1)+m^{\prime}-1.$

Now it suffices to show that $\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{*},y-1)\geq
1$. We know that $w_{y}^{*}=2$, so $v_{y}=2$ or
$\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{*},y-1)\geq 1$ (or both). But $w_{y}=1$
and the weight of the vertex with a horizontally incoming $2$-arrow and an
outgoing $1$-arrow is zero, by Figure 10. So we have that $v_{y}\neq 2$ (since
we have assumed $\mathrm{Wgt}(\bm{v},\bm{w})>0$), which implies that
$\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{*},y-1)\geq 1$. Returning to (4.12), we
have shown that $\mathrm{VertCt}_{2}(\bm{v},\bm{w},y-1)\geq m^{\prime}$. Thus
we have now established that with $y$ defined as above, it holds that

$\displaystyle
w_{y}=1,\,w^{*}_{y}=2,\,\mathrm{VertCt}_{2}(\bm{v},\bm{w},y-1)\geq
m^{\prime},\text{ and }\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{*},y-1)\geq 1.$
(4.13)

With these preliminaries in place, we turn to exhibiting a particular
configuration $f(\bm{w})$ such that
$\mathrm{Wgt}(\bm{w})/\mathrm{Wgt}(f(\bm{w}))\leq Cq^{cm^{2}}$. We let
$K_{0}:=2(\log q^{-1})^{-1}$ and $K:=2K_{0}$. For $m\geq K\log N$, we will
define a sequence of configurations
$\bm{w}^{(0)},\bm{w}^{(1)},\ldots,\bm{w}^{(R)}\in\mathrm{Val}(\bm{v},\bm{x})$
where $\bm{w}^{(0)}:=\bm{w}$ and $R:=m-K_{0}\log N$ such that $\bm{w}^{(i)}$
has $\mathrm{PE}(\bm{w}^{(i)})\geq m-i$ and

$\displaystyle\frac{\mathrm{Wgt}(\bm{v},\bm{w}^{(i-1)})}{\mathrm{Wgt}(\bm{v},\bm{w}^{(i)})}\leq
q^{m-i-K_{0}\log N-1}(1+q^{K_{0}\log N-1})^{N}.$ (4.14)

We define $f(\bm{w})$ to be $\bm{w}^{(R)}$. Clearly this will suffice to
establish $\mathrm{Wgt}(\bm{w})/\mathrm{Wgt}(f(\bm{w}))\leq Cq^{cm^{2}}$ as
then by multiplying together the inequalities we would obtain that, for
absolute constants $c$ and $C$,

$\displaystyle\frac{\mathrm{Wgt}(\bm{v},\bm{w})}{\mathrm{Wgt}(\bm{v},\bm{w}^{(R)})}\leq
q^{mR-\sum_{i=1}^{R}i-RK_{0}\log N-R}(1+q^{K_{0}\log N-1})^{RN}\leq
Cq^{cm^{2}},$ (4.15)

the final inequality for $m\geq K\log N$. So now we must show (4.14); we will
establish that $\\#f^{-1}(\\{\tilde{\bm{w}}\\})\leq N^{2m}$ for any
$\tilde{\bm{w}}$ with $\mathrm{PE}(\tilde{\bm{w}})=m\geq K\log N$ and
$\mathrm{Wgt}(\bm{v},\bm{w})>0$ after specifying the sequence of
configurations.

Reindexing, to establish (4.14), we must show that if we have a configuration
$\bm{w}$ with $\mathrm{PE}(\bm{w})\geq m^{\prime}\geq K\log N$, then we can
produce a configuration $\bm{w}^{\prime}$ with
$\mathrm{PE}(\bm{w}^{\prime})\geq m^{\prime}-1$ and

$\frac{\mathrm{Wgt}(\bm{v},\bm{w})}{\mathrm{Wgt}(\bm{v},\bm{w}^{\prime})}\leq
q^{m^{\prime}-K_{0}\log N-1}(1+q^{K_{0}\log N-1})^{N}.$ (4.16)

Now, by (4.13), since $\mathrm{PE}(\bm{w})=m$, for any
$m^{\prime}\in\llbracket K\log N,m\rrbracket$ there exists a first vertex
(whose row height we labeled $y$), where a $1$-arrow exits horizontally and
there are at least $m^{\prime}$ 2-arrows entering vertically, i.e., $w_{y}=1$
and $\mathrm{VertCt}_{2}(\bm{v},\bm{w},y-1)\geq m^{\prime}$. Since we have
assumed $m^{\prime}\geq K\log N$ and all $2$-arrows eventually exit, there
exists a value $y^{\prime}>y$ which is the smallest one greater than $y$ such
that $v_{y^{\prime}}\neq 2$, $w_{y^{\prime}}=2$, and
$\mathrm{VertCt}_{2}(\bm{w},y^{\prime}-1)=\lceil\frac{1}{2}K\log
N\rceil=\lceil K_{0}\log N\rceil$. In particular, for all $u\in\llbracket
y,y^{\prime}-1\rrbracket$,

$\mathrm{VertCt}_{2}(\bm{v},\bm{w},u)\geq\tfrac{1}{2}K\log N=K_{0}\log N.$
(4.17)

We define $\bm{w}^{\prime}$ by swapping the exiting arrows at $y$ and
$y^{\prime}$, i.e., by $w^{\prime}_{u}=w_{u}$ for all
$u\not\in\\{y,y^{\prime}\\}$, $w^{\prime}_{y^{\prime}}=w_{y}=1$, and
$w^{\prime}_{y}=w_{y^{\prime}}=2$. Observe that since there are at most $N$
possible locations for such a swap, and $f(\bm{w})$ is defined by a sequence
of $R=m-K_{0}\log N<m$ such swaps, it follows that
$\\#f^{-1}(\\{\tilde{\bm{w}}\\})\leq\binom{N}{2}^{m}<N^{2m}$ for any
$\tilde{\bm{w}}\in\mathrm{Val}(\bm{v},\bm{x})$ with
$\mathrm{PE(}\tilde{\bm{w}})=m$. Next observe that, for $u\in\llbracket
y,y^{\prime}-1\rrbracket$,

$\begin{split}\mathrm{VertCt}_{1}(\bm{v},\bm{w}^{\prime},u)&=\mathrm{VertCt}_{1}(\bm{v},\bm{w},u)+1\
\text{ and }\\\
\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{\prime},u)&=\mathrm{VertCt}_{2}(\bm{v},\bm{w},u)-1.\end{split}$
(4.18)

It is immediate that $\mathrm{PE}(\bm{w}^{\prime})\geq m^{\prime}-1$. To upper
bound $\mathrm{Wgt}(\bm{v},\bm{w})/\mathrm{Wgt}(\bm{v},\bm{w}^{\prime})$, we
note that the vertex configurations at $u$ are identical in $\bm{w}$ and
$\bm{w}^{\prime}$ for $u\not\in\llbracket y,y^{\prime}\rrbracket$, so the
ratio is the ratio of the product of the vertex weights for vertices at $u$
with $u\in\llbracket y,y^{\prime}\rrbracket$, i.e., (recall the definition of
$\mathrm{Wgt}_{u}$ and $\mathrm{Wgt}$ from (4.9) and (4.10) respectively)

$\displaystyle\frac{\mathrm{Wgt}(\bm{v},\bm{w})}{\mathrm{Wgt}(\bm{v},\bm{w}^{\prime})}=\prod_{u=y}^{y^{\prime}}\frac{\mathrm{Wgt}_{u}(\bm{v},\bm{w})}{\mathrm{Wgt}_{u}(\bm{v},\bm{w}^{\prime})}.$

To estimate this, we first note that $w_{u}=w^{\prime}_{u}$ for
$u\in\llbracket y+1,y^{\prime}-1\rrbracket$ and recall that (4.18) holds for
$u\in\llbracket y,y^{\prime}-1\rrbracket$.

Recall the spectral parameter $z\in(0,1)$. For a vertex height $u\in\llbracket
1,U\rrbracket$, let

$\displaystyle\tilde{z}(u):=z\text{ if }u\in\llbracket N+1,N+M\rrbracket\text{
and }\tilde{z}(u):=1\text{ if }u\in\llbracket 1,N\rrbracket,$ (4.19)

and, for $j\in\\{1,2\\}$,

$\displaystyle A_{j}(u):=\mathrm{VertCt}_{j}(\bm{v},\bm{w},u-1).$ (4.20)

In the following, we will be reading vertex weights from Figure 10. Now, since
$w_{y}=1$, $\mathrm{Wgt}_{y}(\bm{v},\bm{w})=\tilde{z}(y)q^{A_{2}(y)}$ or
$\tilde{z}(y)(1-q^{A_{1}(y)})q^{A_{2}(y)}$, depending on whether $v_{y}=1$ or
$0$ respectively. Since $A_{2}(y)\geq m$ as shown in (4.13), both
possibilities for the weight are upper bounded by $\tilde{z}(y)q^{m}$.
Similarly since $w_{y}^{\prime}=2$ and
$\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{\prime},y-1)=\mathrm{VertCt}_{2}(\bm{v},\bm{w},y-1)=A_{2}(y)$,
$\mathrm{Wgt}_{y}(\bm{v},\bm{w}^{\prime})=\smash{\tilde{z}(y)(1-q^{A_{2}(y)})}\geq\tilde{z}(y)(1-q^{m})$.
Thus

$\displaystyle\frac{\mathrm{Wgt}_{y}(\bm{v},\bm{w})}{\mathrm{Wgt}_{y}(\bm{v},\bm{w}^{\prime})}\leq\frac{\tilde{z}(y)q^{m}}{\tilde{z}(y)(1-q^{m})}=\frac{q^{m}}{1-q^{m}}.$

Next we turn to the $y^{\prime}$ vertex. Here, $w_{y^{\prime}}=2$ and
$A_{2}(y^{\prime})=\lceil K_{0}\log N\rceil$ (see before (4.17)), so
$\mathrm{Wgt}_{y^{\prime}}(\bm{v},\bm{w})=\tilde{z}(y^{\prime})(1-q^{\lceil
K_{0}\log N\rceil})$ independently of $v_{y^{\prime}}\in\\{0,1\\}$, while
$\mathrm{Wgt}_{y^{\prime}}(\bm{v},\bm{w}^{\prime})=\tilde{z}(y^{\prime})q^{\lceil
K_{0}\log N\rceil-1}$ or
$\tilde{z}(y^{\prime})(1-q^{\mathrm{VertCt}_{1}(\bm{v},\bm{w}^{\prime},y^{\prime}-1)})q^{\lceil
K_{0}\log N\rceil-1}$ (depending on $v_{y^{\prime}}=1$ or $0$ respectively,
since $v_{y^{\prime}}\neq 2$), which is lower bounded by
$\tilde{z}(y^{\prime})(1-q)q^{\lceil K_{0}\log N\rceil-1}$, since, as noted in
(4.18),
$\mathrm{VertCt}_{2}(\bm{v},\bm{w}^{\prime},y^{\prime}-1)=\mathrm{VertCt}_{2}(\bm{v},\bm{w},y^{\prime}-1)-1=\lceil
K_{0}\log N\rceil-1$. So

$\displaystyle\frac{\mathrm{Wgt}_{y^{\prime}}(\bm{v},\bm{w})}{\mathrm{Wgt}_{y^{\prime}}(\bm{v},\bm{w}^{\prime})}\leq\frac{\tilde{z}(y^{\prime})(1-q^{\lceil
K_{0}\log N\rceil})}{\tilde{z}(y^{\prime})(1-q)q^{\lceil K_{0}\log
N\rceil-1}}=\frac{1-q^{\lceil K_{0}\log N\rceil}}{(1-q)q^{\lceil K_{0}\log
N\rceil-1}}.$

Finally we turn to looking at the weight of vertex $u$ for $u\in\llbracket
y+1,y^{\prime}-1\rrbracket$. Recall that $w_{u}=w^{\prime}_{u}$ for this range
of $u$. If $w_{u}=0$, the vertex weights are $1$ for both $\bm{w}$ and
$\bm{w}^{\prime}$. If $w_{u}=1$, then, using (4.18), and with $\tilde{z}(u)$
as in (4.19) and $A_{j}(u)$ as in (4.20),

$\frac{\mathrm{Wgt}_{u}(\bm{v},\bm{w})}{\mathrm{Wgt}_{u}(\bm{v},\bm{w}^{\prime})}=\begin{cases}\displaystyle\frac{\tilde{z}(u)(1-q^{A_{1}(u)})q^{A_{2}(u)}}{\tilde{z}(u)(1-q^{A_{1}(u)+1})q^{A_{2}(u)-1}}&v_{u}=0\\\\[12.0pt]
\displaystyle\frac{\tilde{z}(u)q^{A_{2}(u)}}{\tilde{z}(u)q^{A_{2}(u)-1}}&v_{u}=1,\end{cases}$

which is at most $1$. Finally, we consider the case $w_{u}=2$. Recall that
$A_{2}(u)\geq\lceil K_{0}\log N\rceil$ by (4.17). If $v_{u}=2$,
$\mathrm{Wgt}_{u}(\bm{v},\bm{w})/\mathrm{Wgt}_{u}(\bm{v},\bm{w}^{\prime})=1$,
while if $v_{u}=0$ or $1$,

$\displaystyle\frac{\mathrm{Wgt}_{u}(\bm{v},\bm{w})}{\mathrm{Wgt}_{u}(\bm{v},\bm{w}^{\prime})}=\frac{\tilde{z}(u)(1-q^{A_{2}(u)})}{\tilde{z}(u)(1-q^{A_{2}(u)-1})}=1+q^{A_{2}(u)-1}\cdot\frac{1-q}{1-q^{A_{2}(u)-1}}\leq
1+q^{\lceil K_{0}\log N\rceil-1}.$

Combining the above inequalities, and using that the number of locations $u$
where $w_{u}\neq 0$ is $N$ (and that if $w_{u}=0$ the ratio is 1), we obtain
that

$\displaystyle\frac{\mathrm{Wgt}(\bm{v},\bm{w})}{\mathrm{Wgt}(\bm{v},\bm{w}^{\prime})}=\prod_{u=y}^{y^{\prime}}\frac{\mathrm{Wgt}_{u}(\bm{v},\bm{w})}{\mathrm{Wgt}_{u}(\bm{v},\bm{w}^{\prime})}$
$\displaystyle\leq\frac{q^{m}}{1-q^{m}}\cdot\frac{1-q^{\lceil K_{0}\log
N\rceil}}{q^{\lceil K_{0}\log N\rceil-1}}\left(1+q^{K_{0}\log N-1}\right)^{N}$
$\displaystyle\leq q^{m-\lceil K_{0}\log N\rceil}\left(1+q^{\lceil K_{0}\log
N\rceil-1}\right)^{N}.$

This implies (4.16), completing the proof. ∎

##  5\. Properties of ASEP and S6V

In this section we collect some properties of the colored S6V model and ASEP
that will be useful in future sections, in particular, a form of stationarity
and one-point tightness.

###  5.1. Colored stochastic six-vertex model

We work with packed boundary condition for the colored S6V model; recall the
definition from Section 2.3. In particular, we have arrows entering
horizontally at every site along $\\{1\\}\times\llbracket-N,N\rrbracket$, with
the color of the arrow entering $(1,k)$ being $k$. Also recall that the color
of the arrow exiting horizontally from $(t,k)$ is $j_{(t,k)}$ and from (2.17)
that

$\displaystyle h^{\mathrm{S6V}}(x,0;y,t)=\\#\left\\{k>y:j_{(t,k)}\geq
x\right\\}$

for $x\in\llbracket-N,N\rrbracket$ and $y\in\llbracket-N,\infty\Rparen$. It is
easy to check from this description that, for any
$k\in\llbracket-N,N\rrbracket$ fixed,

$\displaystyle h^{\mathrm{S6V}}(x,0;y,t)\stackrel{{\scriptstyle
d}}{{=}}h^{\mathrm{S6V}}(x+k,0;y+k,t)+k$ (5.1)

as processes in $(x,y,t)$ on the domain
$\llbracket\max(-N-k,-N),\min(N-k,N)\rrbracket\times\llbracket\max(-N-k,-N),\infty\Rparen\times\mathbb{N}$;
this uses that, when $y\geq x$, $h^{\mathrm{S6V}}(x,0;y,t)-(N-y)$ is the
number of arrows starting at a site $(1,i)$ for some $i\in\llbracket
x,y\rrbracket$ which horizontally exit a vertex $(t,i)$ for some
$i\in\llbracket y+1,\infty\Rparen$, which is a quantity whose distribution is
invariant on shifting $x$ and $y$ by the same constant.

The following records a one-point convergence result on the limiting one-point
fluctuations of $h^{\mathrm{S6V}}$ in its rescaled form
$\mathcal{S}^{\mathrm{S6V},\varepsilon}$ as defined in (2.20) (which recall
involves a parameter $\alpha$ controlling the location in the rarefaction fan
where the height function is evaluated). It was originally essentially shown
in [BCG16], but proved in the slightly improved form we require in [CD18] .

######  Proposition 5.1 (One-point distributional limit of S6V).

Let $q\in$ $[0,1)$, $z\in(0,1)$, and $\alpha\in(z,z^{-1})$. Fix
$x,y\in\mathbb{R}$. Then, as $\varepsilon\to 0$,
$\smash{\mathcal{S}^{\mathrm{S6V},\varepsilon}(x;y)+(x-y)^{2}\stackrel{{\scriptstyle
d}}{{\to}}\mathrm{GUE}\text{-}\mathrm{TW}}$, the GUE Tracy-Widom distribution
(see end of Definition 2.1).

###### Proof.

By (5.1) and the expression defining $\mathcal{S}^{\mathrm{S6V},\varepsilon}$
in Definition 2.19, it follows that
$\mathcal{S}^{\mathrm{S6V},\varepsilon}(x;y)\stackrel{{\scriptstyle\smash{d}}}{{=}}\mathcal{S}^{\mathrm{S6V},\varepsilon}(0;y-x)$,
so it suffices to prove the case $x=0$. When $q\in(0,1)$, this case is an
immediate consequence of combining [CD18, Theorems 2.3 and 2.5] and
translating the notation. When $q=0$, we first note that by Proposition 3.12,
$\mathcal{S}^{\mathrm{S6V},\varepsilon}(0;\bm{\cdot})$ is a rescaled version
of the top line of the uncolored Hall-Littlewood line ensemble
$\smash{\bm{L}^{\mathrm{cHL},(1)}(\bm{\cdot})}$ from Section 3.2, which
consists of non-intersecting Bernoulli paths. The scaling limit of the latter
is proven in [DFF+21, Corollary 1.3] as well as in [DNV23, Theorem 1.5] to be
$\mathcal{P}$, from which the one-point statement we claim follows
immediately. ∎

###  5.2. Colored ASEP and relation to colored S6V

Next we recall the model of colored ASEP from Section 2.2. Because the
approximate LPP representation of Theorem 3.20 is in terms of the colored
Hall-Littlewood line ensemble associated to the colored $q$-Boson model which
has a direct distributional equivalence only with colored S6V, we need to
relate colored ASEP to colored S6V.

Consider the colored S6V model under packed boundary condition with

$N=\lfloor\delta^{-1}t\rfloor$ (5.2)

for $\delta,t>0$ with $q\in[0,1)$ and $z=\frac{1-\delta}{1-\delta q}$ (from
Figure 8, this means that the probability of the higher color arrow moving
straight through a vertex vertically is $q\delta$ and horizontally is
$\delta$). We will evaluate the colored height function on the vertical line
$\\{M\\}\times\llbracket-N,\infty\Rparen$ with $M=\lfloor\delta^{-1}t\rfloor$.
Consider colored ASEP with the usual jump rates of $1$ to the right and $q$ to
the left. It is known from [BCG16, Agg17] that by evaluating the height
function of the S6V model near
$(M,N)=(\lfloor\delta^{-1}t\rfloor,\lfloor\delta^{-1}t\rfloor)$ one obtains
the ASEP height function in the distributional limit as $\delta\to 0$.

Unfortunately, this description poses issues to use directly, namely, in the
$\delta\to 0$ limit the domain becomes infinite even for fixed $t$. This means
that estimates we wish to apply in the ASEP setting like Theorem 3.20
(probability bound on the deviation $m$ from the approximate LPP
representation), which only apply for $m\geq K(q)\log N$, would not hold.

####  5.2.1. Effective coupling

We instead make use of an effective coupling proven in [AB24] of colored S6V
with $N=\lfloor\delta^{-1}t\rfloor$ as in (5.2) and colored ASEP at time $t$
which allows to take $\delta\to 0$ at a particular rate with $t\to\infty$. We
will take $\delta=t^{-\theta}$ for some $\theta>0$ large enough. We define

$\displaystyle h^{\mathrm{ASEP},\theta}(x,0;y,t):=\lfloor
t^{1+\theta}\rfloor+x+1-h^{\mathrm{S6V}}(-x,0;\lfloor
t^{1+\theta}\rfloor-y-1,\lfloor t^{1+\theta}\rfloor);$ (5.3)

the first argument of $h^{\mathrm{S6V}}$ is $-x$ because
$h^{\mathrm{ASEP}}(x,0;\bm{\cdot},t)$ involves counts of particles of color
greater than or equal to $-x$ and not $x$ (recall (2.5)). As we will see, the
presence of negative signs in going between $h^{\mathrm{ASEP},\theta}$ and
$h^{\mathrm{S6V}}$ arises from the fact that the ASEP particle configuration
can be read off (under a coupling that we will introduce shortly) of a
horizontal line in the S6V model, but the S6V height function considers arrow
counts on a vertical line.

We note that (5.1) implies that

$h^{\mathrm{ASEP},\theta}(\bm{\cdot},0;\bm{\cdot},t)\stackrel{{\scriptstyle\smash{d}}}{{=}}h^{\mathrm{ASEP},\theta}(\bm{\cdot}+k,0;\bm{\cdot}+k,t)$
(5.4)

for any $k\in\mathbb{Z}$ fixed, matching the relation satisfied by
$h^{\mathrm{ASEP}}$ (it is easily seen from the latter’s definition (2.5) that
(5.4) holds with $h^{\mathrm{ASEP}}$ in place of $h^{\mathrm{ASEP},\theta}$).

The properties of the coupling between $h^{\mathrm{ASEP},\theta}$ and
$h^{\mathrm{ASEP}}$ that we need is as follows.

######  Lemma 5.2 (Effective coupling of ASEP with S6V).

There exists $C>0$ such that, for any $\theta>0$ and $t\geq 1$, there is a
coupling such that
$h^{\mathrm{ASEP},\theta}(x,0;y,t)=h^{\mathrm{ASEP}}(x,0;y,t)$ for all
$|x|,|y|\leq 2t$ with probability at least $1-Ct^{3-\theta/8}$.

In the remainder of the paper we will assume $\theta$ is a large fixed
constant such as $40$ so that, by the Borel-Cantelli lemma, Lemma 5.2 implies
that with probability one
$h^{\mathrm{ASEP},\theta}(x,0;y,t)=h^{\mathrm{ASEP}}(x,0;y,t)$ for all
$|x|,|y|\leq 2t$ and for all $t$ large enough. To prove Lemma 5.2, we will
need the following statements for colored ASEP and colored S6V, which are
essentially implications of the finite speed of propagation in the models. As
their proofs are straightforward, we defer them to Appendix B. Define

$\displaystyle\tilde{h}^{\mathrm{ASEP}}(x,0;y,t):=\\#\\{z\in\llbracket
y+1,\lfloor 4t\rfloor\rrbracket:\eta_{t}(z)\geq-x\\},$ (5.5)

where recall $(\eta_{t}(k))_{k\in\mathbb{Z}}$ records the state of the colored
ASEP at time $t$. In words, $\tilde{h}^{\mathrm{ASEP}}$ is a variant of the
colored ASEP height function (2.5) where particles beyond location $\lfloor
4t\rfloor$ are not counted.

######  Lemma 5.3.

There exist $C,c>0$ such that, for all $t>0$, with probability at least
$1-C\exp(-ct)$,

$\displaystyle\left(h^{\mathrm{ASEP}}(x,0;y,t)\right)_{|x|,|y|\leq
2t}=\bigl{(}\tilde{h}^{\mathrm{ASEP}}(x,0;y,t)\bigr{)}_{|x|,|y|\leq 2t},$
(5.6)

######  Lemma 5.4.

Let $\delta$, $t>0$, $N=\lfloor\delta^{-1}t\rfloor$, and
$z=\frac{1-\delta}{1-\delta q}$. There exist absolute constants $C,c>0$ such
that, with probability at least $1-C\exp(-ct)$, it holds that, for all
$|x|\leq 2t$ and $z\geq\lfloor 4t\rfloor$,

$\displaystyle
h^{\mathrm{S6V}}(x,0;\lfloor\delta^{-1}t\rfloor-z,\lfloor\delta^{-1}t\rfloor)=\lfloor\delta^{-1}t\rfloor-x+1,$

where note the righthand side is the total number of arrows of color at least
$x$ in the system.

###### Proof of Lemma 5.2.

Let $\delta>0$. Recall
$h^{\mathrm{S6V}}(-x,0;\lfloor\delta^{-1}t\rfloor-y-1,\lfloor\delta^{-1}t\rfloor)$
is the number of arrows of color greater than or equal to $-x$ passing
strictly above $(\lfloor\delta^{-1}t\rfloor,\lfloor\delta^{-1}t\rfloor-y-1)$,
and let $\tilde{h}^{\mathrm{ASEP}}$ be as in (5.5). Also recall the packed
boundary and initial conditions for colored S6V and colored ASEP,
respectively. We note that in both colored S6V and colored ASEP, color merging
(Remarks 2.6 and 2.18) implies that the distribution of
$(\tilde{h}^{\mathrm{ASEP}}(x,0;y,t))_{|x|,|y|\leq 2t}$ and
$\smash{(h^{\mathrm{S6V}}(x,0;y,t))_{|x|,|y|\leq 2t}}$ are unchanged by
merging all particles/arrows of color strictly greater than $\lfloor
2t\rfloor$ to have color $\lfloor 2t\rfloor+1$, and merging all
particles/arrows of color strictly smaller than $-\lfloor 2t\rfloor$ to have
color $-\lfloor 2t\rfloor-1$. We work with these color merged versions of the
packed initial conditions (this is required as the result we invoke from
[AB24] states that only finitely many colors exist in the systems).

We work with the coupling whose existence is asserted in [AB24, Proposition
B.2]. Then we know that, on an event $\mathsf{E}_{1}$ with probability at
least $1-C\delta^{1/8}t^{3}$ (and we work on this event in the remainder of
the proof), for all $|x|\leq 2t$, $|y|\leq 4t$, there is an arrow of color $x$
exiting vertically from
$(\lfloor\delta^{-1}t\rfloor+y,\lfloor\delta^{-1}t\rfloor)$ (i.e., along the
horizontal line at height $\lfloor\delta^{-1}t\rfloor$) if and only if there
is a particle of color $x$ at location $y$ in colored ASEP at time $t$.
However, the S6V height function we defined in (2.17) is in terms of arrow
counts on a vertical and not horizontal line.

To handle this, we claim that, with probability at least $1-32\delta t^{2}$,
the color of the arrow exiting horizontally from
$(\lfloor\delta^{-1}t\rfloor,\lfloor\delta^{-1}t\rfloor-y)$ is the same as
that of the arrow exiting vertically from
$(\lfloor\delta^{-1}t\rfloor+y,\lfloor\delta^{-1}t\rfloor)$ for all
$y\in\llbracket-\lfloor 4t\rfloor,\lfloor 4t\rfloor\rrbracket$; call the event
that this occurs for a given $y$ by $\mathsf{E}_{2}(y)$ and define
$\mathsf{E}_{2}=\cap_{y\in\llbracket-\lfloor 4t\rfloor,\lfloor
4t\rfloor\rrbracket}\mathsf{E}_{2}(y)$. Let
$\bm{v}_{t}=(\lfloor\delta^{-1}t\rfloor,\lfloor\delta^{-1}t\rfloor)$ and
$\ell_{t}=\lfloor 4t\rfloor$, and define the squares $S_{1}$ and $S_{2}$ with
corners given by
$\\{\bm{v}_{t},\bm{v}_{t}+(-\ell_{t},0),\bm{v}_{t}+(0,\ell_{t}),\bm{v}_{t}+(-\ell_{t},\ell_{t})\\}$
and
$\\{\bm{v}_{t},\bm{v}_{t}+(\ell_{t},0),\bm{v}_{t}+(0,-\ell_{t}),\bm{v}_{t}+(\ell_{t},-\ell_{t})\\}$,
respectively. To see that our claim holds, note first that for
$\mathsf{E}_{2}(y)$ to not hold for some $y\in\llbracket-\lfloor
4t\rfloor,\lfloor 4t\rfloor\rrbracket$, it must be the case that at least one
vertex in the squares $S_{1}$ and $S_{2}$ (for the cases of $y\leq 0$ and
$y>0$ respectively) allows at least one arrow to pass straight through
vertically or horizontally; the probability of this happening at any given
vertex is at most $\delta$. A union bound over the $32t^{2}$ vertices in the
two squares proves the claim that $\mathbb{P}(\mathsf{E}_{2}^{c})\leq 32\delta
t^{2}$.

Let $\mathsf{E}_{3}$ and $\mathsf{E}_{4}$ be the events that
$\tilde{h}^{\mathrm{ASEP}}(x,0;y,t)=h^{\mathrm{ASEP}}(x,0;y,t)$ and
$h^{\mathrm{S6V}}(x,0;\lfloor\delta^{-1}t\rfloor-\lfloor
4t\rfloor,\lfloor\delta^{-1}t\rfloor)=\lfloor\delta^{-1}t\rfloor-x+1$ hold for
all $|x|,|y|\leq 2t$. By Lemmas 5.3 and 5.4,
$\mathbb{P}((\mathsf{E}_{3}\cup\mathsf{E}_{4})^{c})\leq C\exp(-ct)$ for some
$C,c>0$.

Set $\delta=t^{-\theta}$. Under the coupling above and on
$\mathsf{E}:=\mathsf{E}_{1}\cap\mathsf{E}_{2}\cap\mathsf{E}_{3}\cap\mathsf{E}_{4}$,
we will show that
$h^{\mathrm{ASEP},\theta}(x,0;y,t)=h^{\mathrm{ASEP}}(x,0;y,t)$ for all
$|x|,|y|\leq 2t$. First, by the definition of $\tilde{h}^{\mathrm{ASEP}}$, on
$\mathsf{E}_{1}\cap\mathsf{E}_{2}$ and for $|x|,|y|\leq 2t$,

$\displaystyle\tilde{h}^{\mathrm{ASEP}}(x,0;y,t)$
$\displaystyle=\\#\left\\{k\in\llbracket y+1,\lfloor
4t\rfloor\rrbracket:j_{(\lfloor\delta^{-1}t\rfloor,\lfloor\delta^{-1}t\rfloor-k)}\geq-x\right\\}$
$\displaystyle=h^{\mathrm{S6V}}(-x,0;\lfloor\delta^{-1}t\rfloor-\lfloor
4t\rfloor-1,\lfloor\delta^{-1}t\rfloor)-h^{\mathrm{S6V}}(-x,0;\lfloor\delta^{-1}t\rfloor-y-1,\lfloor\delta^{-1}t\rfloor)$
$\displaystyle=\lfloor\delta^{-1}t\rfloor+x+1-h^{\mathrm{S6V}}(-x,0;\lfloor\delta^{-1}t\rfloor-y-1,\lfloor\delta^{-1}t\rfloor)=h^{\mathrm{ASEP},\theta}(x,0;y,t),$

the penultimate equality on $\mathsf{E}_{3}$. The proof is completed by the
fact that $\tilde{h}^{\mathrm{ASEP}}(x,0;y,t)=h^{\mathrm{ASEP}}(x,0;y,t)$ on
$\mathsf{E}_{4}$ and noting that it holds for all large enough $C$ and all
$t\geq 1$ that $\mathbb{P}(\mathsf{E}^{c})\leq C(\exp(-ct)+\delta
t^{2}+\delta^{1/8}t^{3})\leq C\delta^{1/8}t^{3}$ (since $\delta=t^{-\theta}$).
∎

####  5.2.2. One-point tightness

The one-point tightness for $h^{\mathrm{ASEP},\theta}$ that we require follows
from Lemma 5.2 and the following one-point convergence result for
$h^{\mathrm{ASEP}}$. Recall the ASEP sheet
$\mathcal{S}^{\mathrm{ASEP},\varepsilon}$ from (2.8) and the parameter
$\alpha$ controlling the macroscopic location around which the height function
$h^{\mathrm{ASEP}}$ is evaluated.

######  Proposition 5.5.

Let $q\in[0,1)$ and $\alpha\in(-1,1)$. Fix $x,y\in\mathbb{R}$. Then, as
$\varepsilon\to 0$,
$\mathcal{S}^{\mathrm{ASEP},\varepsilon}(x;y)+(x-y)^{2}\stackrel{{\scriptstyle
d}}{{\to}}\mathrm{GUE}\text{-}\mathrm{TW}$, the GUE Tracy-Widom distribution.

###### Proof.

Again by stationarity of $h^{\mathrm{ASEP}}$ (as noted after (5.4)) it
suffices to prove the case $x=0$. This follows from [TW09, Theorem 3], and is
also stated in a form closer to ours in [BO17, Theorem 11.3]. ∎

We can combine this result with the coupling in Lemma 5.2 to obtain a one-
point tightness statement for $h^{\mathrm{ASEP},\theta}$. The rescaled version
$\mathcal{S}^{\mathrm{ASEP},\theta,\varepsilon}$ we work with is defined by
(2.8) with $h^{\mathrm{ASEP},\theta}$ in place of $h^{\mathrm{ASEP}}$ with
$\theta=40$, i.e., for $\alpha\in(-1,1)$ fixed and recalling $\mu(\alpha)$,
$\sigma(\alpha)$, and $\beta(\alpha)$ from (2.6) and (2.7),

$\begin{split}&\mathcal{S}^{\mathrm{ASEP},\theta,\varepsilon}(x;y):=\sigma(\alpha)^{-1}\varepsilon^{1/3}\Bigl{(}\mu(\alpha)2\varepsilon^{-1}+\mu^{\prime}(\alpha)\beta(\alpha)(y-x)\varepsilon^{-2/3}\\\
&-h^{\mathrm{ASEP},\theta}\bigl{(}\beta(\alpha)x\varepsilon^{-2/3},0;2\alpha\varepsilon^{-1}+\beta(\alpha)y\varepsilon^{-2/3},2\gamma^{-1}\varepsilon^{-1}\bigr{)}\Bigr{)}.\end{split}$
(5.7)

######  Corollary 5.6.

Fix $\theta=40$. Let $q\in[0,1)$, $\alpha\in(-1,1)$ and $x,y\in\mathbb{R}$.
Then, as $\varepsilon\to 0$,
$\mathcal{S}^{\mathrm{ASEP},\theta,\varepsilon}(x;y)+(x-y)^{2}\stackrel{{\scriptstyle
d}}{{\to}}\mathrm{GUE}\text{-}\mathrm{TW}$, the GUE Tracy-Widom distribution.

###### Proof.

This follows immediately from Lemma 5.2 and Proposition 5.5. ∎

##  6\. Preliminaries for line ensemble tightness

In this section we start by introducing some line ensemble notions that will
be necessary in the proof of Theorem 3.18 on tightness of line ensembles. Then
in Section 6.2 we introduce the Brownian Gibbs property and give the proof of
Theorem 3.19 (on the convergence of line ensembles to the parabolic Airy line
ensemble), assuming Theorem 3.18 and that all subsequential limits possess the
Brownian Gibbs property, which will be proven in later sections. In Section
6.3 we will relate the colored Hall-Littlewood line ensemble, with appropriate
centerings, to colored ASEP (in the approximate form (7.7)) and colored S6V
such that the former’s top curves are the height functions of the respective
models, in such a way that the corresponding uncolored line ensembles satisfy
the assumptions introduced in Section 3.3.4. Finally in Section 6.4 we collect
some ingredient statements needed to prove Theorem 3.18 and give the latter’s
proof assuming them; the ingredient statements will be proved in later
sections.

###  6.1. The Strong Gibbs property

We will need at one or two places a stronger version of the Gibbs property
which allows us to resample on certain random domains known as stopping
domains, analogous to the strong Markov property and stopping times.

######  Definition 6.1 (Stopping domain).

Fix $k\in\mathbb{N}$ and let $\bm{L}$ be a discrete line ensemble (Definition
3.6). A pair of random variables $\mathfrak{l},\mathfrak{r}\in\mathbb{Z}$
forms a stopping domain $\llbracket\mathfrak{l},\mathfrak{r}\rrbracket$ with
respect to $(L_{1},\ldots,L_{k})$ if, for all $\ell,r\in\mathbb{Z}$ with
$\ell<r$,

$\displaystyle\\{\mathfrak{l}\leq\ell,\mathfrak{r}\geq
r\\}\in\mathcal{F}_{\mathrm{ext}}(k,\llbracket\ell,r\rrbracket,\bm{L}).$

We define the $\sigma$-algebra
$\mathcal{F}_{\mathrm{ext}}(k,\llbracket\mathfrak{l},\mathfrak{r}\rrbracket,\bm{L})$
to be the collection of all events $A$ such that
$A\cap\\{\mathfrak{l}\leq\ell,\mathfrak{r}\geq
r\\}\in\mathcal{F}_{\mathrm{ext}}(k,\llbracket\ell,r\rrbracket,\bm{L})$ (as in
Definition 3.14) for all $\ell<r$.

######  Proposition 6.2 (Strong Gibbs property).

Let $k\in\mathbb{N}$ and let $\llbracket\mathfrak{l},\mathfrak{r}\rrbracket$
be a stopping domain. Let $F:\\{(\ell,r,f_{1},\ldots,f_{k}):\ell<r\text{ and
}(f_{1},\ldots,f_{k})\in\mathbb{R}^{\llbracket
1,k\rrbracket\times\llbracket\ell,r\rrbracket}\\}\to\mathbb{R}$ be bounded. If
$\bm{L}$ is a discrete line ensemble that has the Hall-Littlewood Gibbs
property, it holds that, almost surely,

$\displaystyle\mathbb{E}\left[F(\mathfrak{l},\mathfrak{r},L_{1},\ldots,L_{k})\mid\mathcal{F}_{\mathrm{ext}}(k,\llbracket\mathfrak{l},\mathfrak{r}\rrbracket,\bm{L})\right]=\frac{\mathbb{E}[F(\mathfrak{l},\mathfrak{r},B_{1},\ldots,B_{k})W(\bm{B},L_{k+1})\mid\mathcal{F}_{\mathrm{ext}}(k,\llbracket\mathfrak{l},\mathfrak{r}\rrbracket,\bm{L})]}{\mathbb{E}[W(\bm{B},L_{k+1})\mid\mathcal{F}_{\mathrm{ext}}(k,\llbracket\mathfrak{l},\mathfrak{r}\rrbracket,\bm{L})]},$

where $\bm{B}=(B_{1},\ldots,B_{k})$ is a collection of $k$ independent
Bernoulli random walk bridges with $B_{i}(x)=L_{i}(x)$ for
$x\in\\{\mathfrak{l},\mathfrak{r}\\}$ and $i=1,\ldots,k$.

Similar statements have been proven for continuous line ensembles, e.g.,
[CH14, Lemma 2.5]. The proof amounts to decomposing based on the exact value
of $\mathfrak{l}$ and $\mathfrak{r}$ (which is straightforward here as
everything is discrete) and using the Hall-Littlewood Gibbs property on each
of the resulting events; we do not record the details here.

###  6.2. The Brownian Gibbs property

To introduce the Brownian Gibbs property, we will need notation for the
$\sigma$-algebra we condition on for continuous line ensembles. We first
precisely define the meaning of a continuous line ensemble.

######  Definition 6.3 (Continuous line ensemble).

Fix a (possibly infinite) interval $\Lambda\subseteq\mathbb{R}$. A _$\Lambda$
-indexed continuous line ensemble_
$\bm{\mathfrak{L}}=(\mathfrak{L}_{1},\mathfrak{L}_{2},\ldots)$ is a random
variable defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$,
taking values in $\mathcal{C}(\mathbb{N}\times\Lambda,\mathbb{R})$, such that
(again writing the first argument of $\bm{\mathfrak{L}}$ as a subscript, i.e.,
$\mathfrak{L}_{i}(\bm{\cdot}):=\bm{\mathfrak{L}}(i,\bm{\cdot})$)
$\mathfrak{L}_{i}(y)\geq\mathfrak{L}_{i+1}(y)$ for any $y\in\Lambda$ and
$i\in\mathbb{N}$. Here $\mathbb{N}\times\Lambda$ is given the product topology
of the discrete topology and the Euclidean topology, and
$\mathcal{C}(\mathbb{N}\times\Lambda,\mathbb{R})$ has the topology of uniform
convergence on compact sets.

Recall again the $\sigma$-algebra $\mathcal{F}_{\mathrm{ext}}$ in the case of
discrete line ensembles from Definition 3.14. In the case of a continuous line
ensemble $\bm{\mathfrak{L}}:\mathbb{N}\times\mathbb{R}\to\mathbb{R}$, we will
write, for $a,b\in\mathbb{R}$ and $j,k\in\mathbb{N}$ with $a<b$ and $j<k$,
$\smash{\mathcal{F}_{\mathrm{ext}}(\llbracket
j,k\rrbracket,[a,b],\bm{\mathfrak{L}})}$ to be the $\sigma$-algebra generated
by the collection of random variables

$\bigl{\\{}\mathfrak{L}_{i}(x):(i,x)\in\mathbb{N}\times\mathbb{R}\setminus(\llbracket
j,k\rrbracket\times[a,b])\bigr{\\}}.$

(Compared to Definition 3.14, we have replaced $\llbracket a+1,b-1\rrbracket$
with $[a,b]$ in the second bullet point, so that $\mathfrak{L}_{i}(a)$ and
$\mathfrak{L}_{i}(b)$ are still conditioned on, for each $i\in\llbracket
j,k\rrbracket$). We will again often adopt the shorthand
$\mathcal{F}_{\mathrm{ext}}(k,[a,b],\bm{L})=\mathcal{F}_{\mathrm{ext}}(\llbracket
1,k\rrbracket,[a,b],\bm{L})$.

######  Definition 6.4 (Brownian Gibbs property).

A line ensemble
$\bm{\mathfrak{L}}=(\mathfrak{L}_{1},\mathfrak{L}_{2},\ldots):\mathbb{N}\times\mathbb{R}\to\mathbb{R}$
is said to satisfy the Brownian Gibbs property if the following holds for any
$\llbracket j,k\rrbracket\subseteq\mathbb{N}$ and $[a,b]\subseteq\mathbb{R}$.
Conditional on $\mathcal{F}_{\mathrm{ext}}(\llbracket
j,k\rrbracket,[a,b],\bm{\mathfrak{L}})$, the distribution of
$\bm{\mathfrak{L}}|_{\llbracket j,k\rrbracket\times[a,b]}$ is that of $k-j+1$
independent rate two Brownian bridges
$B^{\mathrm{Br}}_{j},\ldots,B^{\mathrm{Br}}_{k}$, with
$B^{\mathrm{Br}}_{i}(x)=\mathfrak{L}_{i}(x)$ for each $x\in\\{a,b\\}$ and
$i\in\llbracket j,k\rrbracket$, conditioned on
$\mathfrak{L}_{j-1}(x)>B^{\mathrm{Br}}_{j}(x)>\ldots>B^{\mathrm{Br}}_{k}(x)>\mathfrak{L}_{k+1}(x)$
for all $x\in[a,b]$ (with $\mathfrak{L}_{0}\equiv\infty$ by convention).

Figure 15. A depiction of a special case, with $k=1$, of the Brownian Gibbs
property: in the left panel is the original line ensemble $\bm{\mathfrak{L}}$,
in the middle panel we condition on the data which is depicted, essentially
forgetting what has been erased, and in the right panel we resample the
forgotten data, which is conditionally distributed as a rate two Brownian
bridge between the endpoints, conditioned to avoid the second curve
$\mathfrak{L}_{2}$.

Recall now the framework for line ensemble convergence in Section 3.3.4. In
particular, Theorem 3.18 asserts that for any sequence of discrete line
ensembles $\bm{L}^{N}$ satisfying Assumptions 1 and 2, the rescaled line
ensembles $\bm{\mathfrak{L}}^{N}$ defined in (3.13) forms a tight sequence,
and Theorem 3.19 characterizes the subsequential limits and says that, under
the additional Assumption 3, the original sequence converges to
$\bm{\mathcal{P}}$.

Theorem 3.19 is an immediate consequence of knowing that all subsequential
limit points of $\bm{\mathfrak{L}}^{N}$ satisfy the Brownian Gibbs property,
on combining with a recent characterization result for $\bm{\mathcal{P}}$
[AH23a]. These two statements are given precisely next. Proposition 6.5 is
proved in Section 12.

######  Proposition 6.5.

Let $\bm{L}^{N}$ satisfy Assumptions 1 and 2, let $\bm{\mathfrak{L}}^{N}$ be
defined as in (3.13), and let $\bm{\mathfrak{L}}$ be any subsequential weak
limit of $\bm{\mathfrak{L}}^{N}$ as $N\to\infty$. Then $\bm{\mathfrak{L}}$ has
the Brownian Gibbs property.

######  Proposition 6.6 ([AH23a, Corollary 2.11 (2)]).

Suppose $\bm{\mathfrak{L}}$ is a line ensemble with the Brownian Gibbs
property such that, for any $\varepsilon>0$, there exists a constant
$C=C(\varepsilon)>0$ such that

$\displaystyle\mathbb{P}\left(|\mathfrak{L}_{1}(x)+x^{2}|>\varepsilon|x|+C\right)\leq\varepsilon$

for all $x\in\mathbb{R}$. Then there exists a parabolic Airy line ensemble
$\bm{\mathcal{P}}$ and an independent random variable $\mathfrak{c}$ such that
$\mathfrak{L}_{i}(x)=\mathcal{P}_{i}(x)+\mathfrak{c}$ for all $i\in\mathbb{N}$
and $x\in\mathbb{R}$.

We give the proof of Theorem 3.19 given these two statements and Theorem 3.18
now.

###### Proof of Theorem 3.19.

By Theorem 3.18 and Proposition 6.5, we know that
$\\{\bm{\mathfrak{L}}^{N}\\}_{N=1}^{\infty}$ is tight and that all
subsequential limits have the Brownian Gibbs property. Let $\bm{\mathfrak{L}}$
be such a subsequential limit. By the characterization theorem (Proposition
6.6), it follows that $\bm{\mathfrak{L}}$ equals the parabolic Airy line
ensemble up to an independent random constant shift. Then Assumption 3
guarantees that the constant shift is zero, completing the proof. ∎

######  Remark 6.7.

In Section 7, we will apply the strong characterization Proposition 6.6 of
[AH23a] (indirectly, through Theorem 3.19) to the line ensembles coming from
S6V and ASEP. In the case of ASEP for $\alpha=0$, we can circumvent
Proposition 6.6. Indeed, in Appendix B.3, we will give an alternate proof of
that case using [QS22, Theorem 2.2] (along with Lemma 5.2) to infer that
$\mathfrak{L}^{N}_{1}$ converges to the parabolic Airy2 process and the
characterization theorem from [DM21] (which asserts that if two Brownian
Gibbsian line ensembles have the same top curve, in law, then the entire line
ensembles are equal in law) to conclude that that limiting line ensemble is
the parabolic Airy line ensemble.

In the next section we will define centered versions of the colored Hall-
Littlewood line ensembles which are associated to S6V and ASEP, and verify
that they satisfy the assumptions from Section 3.3.4.

###  6.3. Verifying the assumptions for colored S6V and colored ASEP

####  6.3.1. S6V

Recall that we work in the domain $\mathbb{Z}_{\geq
1}\times\llbracket-N,\infty\Rparen$ and that we evaluate the colored height
function on the vertical line $x=\lfloor\varepsilon^{-1}\rfloor$. Fix
$q\in[0,1)$, $z\in(0,1)$, and $\alpha\in(z,z^{-1})$ and recall $\mu(\alpha)$
from (2.18) and $\smash{L^{\mathrm{cHL},(j)}_{i}}$ from (3.6) with parameters
$q$, $z$, $N$, and $M=\lfloor\varepsilon^{-1}\rfloor$. Define
$\bm{L}^{\mathrm{S6V},(j),\varepsilon}$ for $j\in\llbracket 1,N\rrbracket$ by,
for
$y\in\llbracket-\lfloor\alpha\varepsilon^{-1}\rfloor,N+M-\lfloor\alpha\varepsilon^{-1}\rfloor\rrbracket$,

$\displaystyle
L^{\mathrm{S6V},(j),\varepsilon}_{i}(y):=L^{\mathrm{cHL},(j)}_{i}(\lfloor\alpha\varepsilon^{-1}\rfloor+y)+j-N-\lfloor\mu(\alpha)\varepsilon^{-1}\rfloor.$
(6.1)

Observe that, since
$h^{\mathrm{S6V}}(j,0;\bm{\cdot},\lfloor\varepsilon^{-1}\rfloor)\stackrel{{\scriptstyle
d}}{{=}}L^{\mathrm{cHL},(j)}_{1}(\bm{\cdot})$ as functions on $\llbracket
1,N-1\rrbracket$ from Proposition 3.12, it follows that, as functions on
$\llbracket-\lfloor\alpha\varepsilon^{-1}\rfloor+1,N-1-\lfloor\alpha\varepsilon^{-1}\rfloor\rrbracket$
and jointly over $j\in\llbracket 1,N\rrbracket$,

$\displaystyle
L^{\mathrm{S6V},(j),\varepsilon}_{1}(\bm{\cdot})\stackrel{{\scriptstyle
d}}{{=}}h^{\mathrm{S6V}}(j,0;\lfloor\alpha\varepsilon^{-1}\rfloor+\bm{\cdot},\lfloor\varepsilon^{-1}\rfloor)+j-N-\lfloor\mu(\alpha)\varepsilon^{-1}\rfloor.$
(6.2)

######  Lemma 6.8.

$\bm{L}^{\mathrm{S6V},(1),\varepsilon}$ satisfies Assumptions 1, 2, and 3 with
$p=\mu^{\prime}(\alpha)$ and $\lambda=\frac{1}{2}\mu^{\prime\prime}(\alpha)$
(as defined in (2.18)) and with $\varepsilon^{-1}$ in place of $N$.

###### Proof.

Since $\bm{L}^{\mathrm{cHL},(1)}$ marginally has the Hall-Littlewood Gibbs
property (Proposition 3.17), and since the Gibbs property is not changed by
deterministic constant shifts of the curves, it follows that
$\bm{L}^{\mathrm{S6V},(1),\varepsilon}$ has the Hall-Littlewood Gibbs
property. That Assumptions 2 and 3 hold follows immediately from combining
Propositions 3.12 and 5.1 with (6.2). ∎

####  6.3.2. ASEP

Fix $q\in[0,1)$, $\alpha\in(-1,1)$ and define $\gamma=1-q$. Let $\theta=40$
and

$K_{\varepsilon,\theta}=\lfloor(2\gamma^{-1}\varepsilon^{-1})^{1+\theta}\rfloor-\lfloor
2\alpha\varepsilon^{-1}\rfloor-1.$

Next we define line ensembles $\bm{L}^{\mathrm{ASEP},(j),\theta,\varepsilon}$
for $j\in\llbracket
1,\lfloor(2\gamma^{-1}\varepsilon^{-1})^{1+\theta}\rfloor\rrbracket$. Recall
$\mu(\alpha)=\smash{\frac{1}{4}(1-\alpha)^{2}}$ from (2.6). They are specified
by setting, for $y\in\llbracket
1-K_{\varepsilon,\theta},N+M-K_{\varepsilon,\theta}\rrbracket$, and where the
parameters for the righthand side are $z=\frac{1-\varepsilon}{1-\varepsilon
q}$ and $M=N=\smash{\lfloor(2\gamma^{-1}\varepsilon^{-1})^{1+\theta}\rfloor}$,

$\begin{split}L^{\mathrm{ASEP},(j),\theta,\varepsilon}_{i}(y)&:=L^{\mathrm{cHL},(j)}_{i}\left(K_{\varepsilon,\theta}+y\right)-\lfloor(2\gamma^{-1}\varepsilon^{-1})^{1+\theta}\rfloor+j-1+\lfloor\tfrac{1}{2}(1-\alpha)^{2}\varepsilon^{-1}\rfloor.\end{split}$
(6.3)

The relation between $L^{\mathrm{ASEP},(j),\theta,\varepsilon}_{1}$ and
$h^{\mathrm{ASEP},\theta}$ (recall (5.3) and
$h^{\mathrm{S6V}}(j,0;\bm{\cdot},\lfloor(2\gamma^{-1}\varepsilon^{-1})^{1+\theta}\rfloor)\stackrel{{\scriptstyle
d}}{{=}}L^{\smash{\mathrm{cHL},(j)}}_{1}(\bm{\cdot})$) is

$\displaystyle
L^{\mathrm{ASEP},(j),\theta,\varepsilon}_{1}(\bm{\cdot})\stackrel{{\scriptstyle
d}}{{=}}-h^{\mathrm{ASEP},\theta}(-j,0;\lfloor
2\alpha\varepsilon^{-1}\rfloor-\bm{\cdot},\lfloor
2\gamma^{-1}\varepsilon^{-1}\rfloor)+\lfloor\tfrac{1}{2}(1-\alpha)^{2}\varepsilon^{-1}\rfloor.$
(6.4)

######  Lemma 6.9.

$\bm{L}^{\mathrm{ASEP},(1),\theta,\varepsilon}$ satisfies Assumptions 1, 2,
and 3 with $p=-\frac{1}{2}(1-\alpha)$ and $\lambda=\frac{1}{8}$ and with
$\varepsilon^{-1}$ in place of $N$.

###### Proof.

As in the case of $\bm{L}^{\mathrm{S6V},(1),\varepsilon}$, it is immediate
from Proposition 3.17 that $\bm{L}^{\mathrm{ASEP},(1),\theta,\varepsilon}$ has
the Hall-Littlewood Gibbs property. It follows from Corollary 5.6 that
Assumptions 2 and 3 hold. ∎

Next, we set up some of the ingredients for the proof of Theorem 3.18, which
will also be useful in Section 7 on the tightness and limit of
$\mathcal{S}^{\mathrm{S6V},\varepsilon}$ and
$\mathcal{S}^{\mathrm{ASEP},\varepsilon}$. In particular, we are working in
the setup of Theorem 3.18, i.e., we have a sequence of discrete line ensembles
$\bm{L}^{N}$ indexed by $N$ and defined on $\mathbb{N}\times\mathbb{Z}$ that
satisfy Assumptions 1 and 2.

###  6.4. Ingredients for proof of tightness

Proving tightness of the path measures requires us to establish one-point
tightness, as well as control on the modulus of continuity, for all curves. We
state the estimates we need and then use them to give the proof of Theorem
3.18.

####  6.4.1. One-point tightness of all curves

Note that from Assumption 2 we have one-point tightness for $L^{N}_{1}$ and,
since line ensembles which satisfy the Hall-Littlewood Gibbs property must by
definition be ordered, we therefore have control on the upper tail of
$L^{N}_{k}$ for every $k$. The lower tail for the $k$th curve for $k\geq 2$ is
more delicate, however, and we record a statement for it now.

######  Theorem 6.10.

Fix $k\in\mathbb{N}$, $T>0$, and $\varepsilon>0$. Then there exist
$N_{0}=N_{0}(k,T,\varepsilon)$ and $R=R(k,T,\varepsilon)>0$ such that, for
$N\geq N_{0}$,

$\displaystyle\mathbb{P}\left(\inf_{sN^{-2/3}\in[-T,T]}\left(L^{N}_{k}(s)-ps+\lambda
s^{2}N^{-1}\right)\leq-RN^{1/3}\right)<\varepsilon.$

We will prove Theorem 6.10 in Sections 9 and 10. As mentioned in Sections
1.3.3 and 3.3.3, the proof inductively interweaves uniform lower tail control
of the $k$th curve over an interval with control over uniform separation
between the $(k-1)$st and $k$th curves, both on the fluctuation scale
$N^{1/3}$. Obtaining control on these two quantities forms the bulk of the
technical work of the proof of Theorem 3.18.

####  6.4.2. Modulus of continuity

To prove tightness of path measures, we also need control on the modulus of
continuity of $L^{N}_{k}$. As in earlier works establishing tightness for line
ensembles with Gibbs properties, this comes down to controlling the partition
function of the Gibbs property, in order to transfer modulus of continuity
estimates from the base path measure (in this case, Bernoulli random walk
bridges) to the line ensemble. For an interval $[-TN^{2/3},TN^{2/3}]$, recall
from Definition 3.16 that the partition function of the top $k$ curves is

$\displaystyle
Z^{k,[-T,T],\bm{L}^{N}(-TN^{2/3}),\bm{L}^{N}(T^{2/3}),L^{N}_{k+1}}_{N}:=\mathbb
E_{\mathcal{F}}[W(B_{1},\ldots,B_{k},L^{N}_{k+1})],$ (6.5)

where $\mathcal{F}=\mathcal{F}_{\mathrm{ext}}(k,\llbracket-
TN^{2/3},TN^{2/3}\rrbracket,\bm{L}^{N})$ and $(B_{1},\ldots,B_{k})$ is a
collection of $k$ independent Bernoulli random walk bridges with $B_{i}$ from
$(-TN^{2/3},L^{N}_{i}(-TN^{2/3}))$ to $(TN^{2/3},L^{N}_{i}(TN^{2/3}))$.

######  Remark 6.11.

Note here that we have written certain quantities, such as $\pm TN^{2/3}$,
without floors, though they are meant to be integers to be completely precise.
Here as in the rest of the paper we will often drop the requisite floor
symbols to ease the notation, since this will not affect our arguments.

The control we need on the partition function is the following. It is proved
in Section 11.

######  Proposition 6.12.

Let $T\geq 2$ and $k\in\mathbb{N}$. Then for any $\varepsilon>0$ there exist
$\delta=\delta(k,T,\varepsilon)>0$ and $N_{0}=N_{0}(k,T,\varepsilon)$ such
that, for $N\geq N_{0}$,

$\displaystyle\mathbb{P}\left(Z^{k,[-T,T],\bm{L}^{N}(-TN^{2/3}),\bm{L}^{N}(TN^{2/3}),L^{N}_{k+1}}_{N}>\delta\right)\geq
1-\varepsilon.$

The control we need on the base path measure of Bernoulli random walk bridges
is as follows. As its proof is quite standard, we defer it to Appendix B.

######  Lemma 6.13.

There exist $C$ and $c>0$ such that, for $n\in\mathbb{N}$, $p\in(-1,0)$, $B$ a
Bernoulli random walk bridge from $(0,0)$ to $(n,pn)$, all $0\leq y<z\leq n$,
and $M>0$,

$\displaystyle\mathbb{P}\left(\sup_{x\in[y,z]}|B(x)-B(y)-p(x-y)|>M|z-y|^{1/2}\right)\leq
C\exp(-cM^{2}).$

At several locations, we will also need to use normal or Brownian bridge
approximations for Bernoulli random walk bridges. We will require the Brownian
bridges to have an explicit variance coming from the increment of the
Bernoulli random walk bridge being approximated. For this we make use of the
following Komlós-Major-Tusnády (KMT) coupling result for Bernoulli random walk
bridges, which is obtained by combining [CD18, Theorem 4.5] with the
exponential Markov inequality.

######  Lemma 6.14 (KMT coupling, [CD18, Theorem 4.5]).

Let $p\in(-1,0)$ and $\\{z_{N}\\}_{N\in\mathbb{N}}\subseteq\mathbb{R}$. For
each $N$, let $B^{N}$ be a Bernoulli random walk bridge from $(0,0)$ to
$(N,z_{N})$. There exist positive constants $C,c<\infty$ (depending on $p$)
such that there is a coupling between $B^{N}$ and a Brownian bridge
$B^{\mathrm{Br}}$ with variance $p(1-p)$ from $(0,0)$ to $(N,z_{N})$ with

$\displaystyle\mathbb{P}\left(\sup_{0\leq t\leq
N}|B^{\mathrm{Br}}(t)-B^{N}(t)|>R\right)\leq C\exp\Bigl{(}-cR+C(\log
N)^{2}+N^{-1}|z_{N}-pN|^{2}\Bigr{)}.$

Note that the variance of the Brownian bridge being $p(1-p)$ can be understood
by the fact that the Bernoulli random walk bridge, if it has the endpoint
$(N,z_{N})\approx(N,pN)$, will behave like a Bernoulli random walk with
parameter $p$, whose increments indeed have variance $p(1-p)$.

Assuming Proposition 6.12, we can give the following corollary on the modulus
of continuity of $L^{N}_{k}$. Recall from Section 3.3.4 that we view
$L^{N}_{k}$ as a $\mathbb{R}$-valued function via linear interpolation.

######  Corollary 6.15.

Let $T\geq 2$ and $k\in\mathbb{N}$. For any $\varepsilon>0$ and $\rho>0$,
there exist $\delta=\delta(k,T,\varepsilon,\rho)>0$ and
$N_{0}=N_{0}(k,T,\varepsilon,\rho)$ such that, for $N\geq N_{0}$,

$\displaystyle\mathbb{P}\left(\sup_{|x|,|y|\leq TN^{2/3},|x-y|\leq\delta
N^{2/3}}|L^{N}_{k}(x)-L^{N}_{k}(y)-p(x-y)|\geq\rho
N^{1/3}\right)\leq\varepsilon.$

######  Remark 6.16.

Before giving the proof of Corollary 6.15, we point out a notational practice
that will be used many times throughout the rest of the paper: we will often
introduce events (e.g., $\mathsf{A}(X)$) whose definition involves an abstract
$\mathbb{R}$-valued process $X$ and then in the course of the proof write
different processes (e.g., $L^{N}_{k}$) in place of $X$ to mean the event
whose definition is that of $\mathsf{A}(X)$ with the process in question in
place of $X$. In spite of a random process being part of the notation for the
event, the event itself is fixed, i.e., does not depend on the instantiation
of the process. This convention will be convenient because when applying the
Hall-Littlewood Gibbs property we will often be working with events whose only
difference is that a collection of Bernoulli random walk bridges is in place
of a collection of curves of the line ensemble $\bm{L}^{N}$.

###### Proof of Corollary 6.15.

For a random process $X:[-TN^{2/3},TN^{2/3}]\to\mathbb{R}$, define

$\mathsf{ModCon}_{N}(\delta,\rho,X):=\left\\{\sup_{|x|,|y|\leq
TN^{2/3},|x-y|\leq\delta N^{2/3}}|X(x)-X(y)-p(x-y)|\geq\rho N^{1/3}\right\\}.$

So $\mathsf{ModCon}_{N}(\delta,\rho,L^{N}_{k})$ is the event whose probability
we are trying to upper bound. Let
$\mathcal{F}=\mathcal{F}_{\mathrm{ext}}(k,[-TN^{2/3},TN^{2/3}],\bm{L}^{N})$,

$\displaystyle
Z_{N}:=Z^{k,[-T,T],\bm{L}^{N}(-TN^{2/3}),\bm{L}^{N}(TN^{2/3}),L^{N}_{k+1}}_{N},$

and $\mathsf{Fav}(\delta^{\prime},M)$ be the event defined by

$\displaystyle\mathsf{Fav}(\delta^{\prime},M):=\left\\{Z_{N}>\delta^{\prime}\right\\}\cap\left\\{|L^{N}_{1}(\pm
TN^{2/3})|,|L^{N}_{k}(\pm TN^{2/3})|\leq MN^{1/3}\right\\};$

we pick $\delta^{\prime}$ and $M$ such that
$\mathbb{P}(\mathsf{Fav}(\delta^{\prime},M))\geq 1-\varepsilon$, which is
possible by Proposition 6.12, one-point tightness of $L^{N}_{1}$ from
Assumption 2, and Theorem 6.10. Note that this event is
$\mathcal{F}$-measurable.

Now by using this bound on $\mathbb{P}(\mathsf{Fav}(\delta^{\prime},M))$, the
Hall-Littlewood Gibbs property, and the fact that $W\leq 1$ ($W$ being from
Definition 3.15), the probability we are bounding equals

$\displaystyle\mathbb{E}\left[\mathbb
P_{\mathcal{F}}\left(\mathsf{ModCon}_{N}(\delta,\rho,L^{N}_{k})\right)\right]$
$\displaystyle\leq\mathbb{E}\left[\mathbb
P_{\mathcal{F}}\left(\mathsf{ModCon}_{N}(\delta,\rho,L^{N}_{k})\right)\mathbbm{1}_{\mathsf{Fav}(\delta^{\prime},M)}\right]+\varepsilon$
$\displaystyle=\mathbb{E}\left[Z_{N}^{-1}\mathbb
E_{\mathcal{F}}\left[\mathbbm{1}_{\mathsf{ModCon}_{N}(\delta,\rho,B_{k})}W\left(\bm{B},L^{N}_{k+1}\right)\right]\mathbbm{1}_{\mathsf{Fav}(\delta^{\prime},M)}\right]+\varepsilon$
$\displaystyle\leq\mathbb{E}\left[Z_{N}^{-1}\mathbb
P_{\mathcal{F}}\left(\mathsf{ModCon}_{N}(\delta,\rho,B_{k})\right)\mathbbm{1}_{\mathsf{Fav}(\delta^{\prime},M)}\right]+\varepsilon,$

where $\bm{B}=(B_{1},\ldots,B_{k})$ is a collection of $k$ independent
Bernoulli random walk bridges with $B_{i}$’s endpoints given by
$L^{N}_{i}(-TN^{2/3})$ and $L^{N}_{i}(TN^{2/3})$. Since we have a lower bound
of $\delta^{\prime}$ on $Z_{N}$ on the event
$\mathsf{Fav}(\delta^{\prime},M)$, we obtain that the righthand side of the
previous display is upper bounded by

$\displaystyle(\delta^{\prime})^{-1}\cdot\mathbb{E}\left[\mathbb
P_{\mathcal{F}}\left(\mathsf{ModCon}_{N}(\delta,\rho,B_{k})\right)\mathbbm{1}_{\mathsf{Fav}(\delta^{\prime},M)}\right]+\varepsilon.$

By the modulus of continuity estimates for Bernoulli random walks from Lemma
6.13, the probability inside the expectation in the previous line can be made
smaller than $\delta^{\prime}\varepsilon$ for an appropriately small choice of
$\delta$. Relabeling $\varepsilon$ completes the proof. ∎

Assuming Theorem 6.10 on the lower tail and Proposition 6.12 (therefore
Corollary 6.15), we may quickly give the proof of tightness.

###### Proof of Theorem 3.18.

Recall that $\bm{\mathfrak{L}}^{N}$ is defined by (3.13). By an easy extension
of [Bil13, Theorem 7.3] (see also [DFF+21, Lemma 2.4]), to show tightness of
$\bm{\mathfrak{L}}^{N}$ it suffices to show, for every $k\in\mathbb{N}$:

  1. (1)

$\lim_{R\to\infty}\lim_{N\to\infty}\mathbb{P}\left(|\mathfrak{L}^{N}_{k}(0)|\geq
R\right)=0$ and

  2. (2)

For all $\varepsilon>0$ and $M\in\mathbb{N}$,

$\displaystyle\lim_{\delta\to
0}\lim_{N\to\infty}\mathbb{P}\left(\sup_{|x|,|y|\leq
M,|x-y|\leq\delta}|\mathfrak{L}^{N}_{k}(x)-\mathfrak{L}^{N}_{k}(y)|\geq\varepsilon\right)=0.$

The first item is immediate from Theorem 6.10 and Assumption 2, using the
ordering of the curves, while (2) follows from Corollary 6.15. ∎

##  7\. Airy sheet and tightness of S6V sheet

In this section we address tightness and convergence to the Airy sheet of the
colored height functions of our models. As in Section 5, we restrict to packed
boundary or initial conditions.

###  7.1. Convergence to the Airy sheet

We will give a single proof of Theorems 2.8 and 2.20 using a unified notation
for the objects arising from both S6V and ASEP, as follows. We will use
$\bm{L}^{(j),N}$ for line ensembles corresponding to different colors in the
two models. The S6V case will correspond to identifying $\bm{L}^{(j),N}$ with
$\bm{L}^{\mathrm{S6V},(j),\varepsilon}$, and the ASEP case to identifying the
same with $\bm{L}^{\mathrm{ASEP},(j),\theta,\varepsilon}$, as defined in (6.1)
and (6.3), and in both cases under the identification of $N$ with
$\varepsilon^{-1}$. Next recall the scaling operator $T_{p,\lambda,N}$ from
(3.11). For a sequence (indexed by $N$) of collections of line ensembles
$\\{\bm{L}^{(j),N}\\}_{j}$ such that $\smash{\bm{L}^{(1),N}}$ satisfies