&\displaystyle\lim_{q^{\mathfrak{M}}\rightarrow\infty}\mathcal{U}_{q^{\mathfrak{L}};q^{\mathfrak{M}};q^{\mathfrak{M}+\mathfrak{L}}/\gamma}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}\\\
&\quad=(-1)^{C_{n}}q^{\varphi(\overline{\bm{D}},\overline{\bm{C}})+\binom{C_{n}+1}{2}+\overline{c}\mathsf{d}+\mathfrak{L}C_{n}}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}\cdot\mathbbm{1}_{\overline{\bm{A}}+\overline{\bm{B}}=\overline{\bm{C}}+\overline{\bm{D}}}\cdot\mathbbm{1}_{\mathsf{b}-a=\mathsf{d}-c}\\\
&\qquad\times\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}-C_{n}}}\displaystyle\frac{(q^{-c-\mathfrak{L}}\gamma;q)_{\mathsf{d}}}{(q;q)_{C_{n}}}\displaystyle\frac{(q^{\mathsf{B}_{n}-\mathfrak{L}};q)_{C_{n}}(\gamma;q)_{\infty}}{(q^{\mathfrak{L}+c-C_{n}-\overline{p}+1}\gamma^{-1};q)_{C_{n}}(q^{-\mathfrak{L}}\gamma;q)_{\infty}}\\\
&\qquad\times\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}(-1)^{\overline{p}}q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\overline{\bm{P}})}\displaystyle\frac{(q^{\mathsf{d}-c-\mathfrak{L}}\gamma;q)_{\overline{p}}}{(q^{-\mathfrak{L}-c}\gamma;q)_{\overline{p}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}\\\
&\qquad\qquad\quad\times
q^{\overline{p}(\mathsf{b}-\mathsf{d}+1)+\binom{\overline{p}}{2}}\cdot{{}_{3}\varphi_{2}}\bigg{(}\begin{array}[]{cc}q^{-C_{n}};q^{-A_{n}},q^{\mathfrak{L}-C_{n}+c-\overline{p}+1}\gamma^{-1}\\\
q^{\mathsf{d}-C_{n}+1},q^{\mathfrak{L}-\mathsf{B}_{n}-C_{n}+1}\end{array}\bigg{|}q,q\bigg{)}.\end{aligned}$

###### Proof.

We assume throughout this proof that
$\overline{\bm{A}}+\overline{\bm{B}}=\overline{\bm{C}}+\overline{\bm{D}}$ and
$\mathsf{b}-a=\mathsf{d}-c$, for otherwise both sides of (A.15) are equal to
$0$ by A.3. Then letting $q^{\mathfrak{N}}$ and $q^{\mathfrak{M}}$ tend to
$\infty$, while keeping
$q^{\mathfrak{M}-\mathfrak{N}}=q^{-\mathfrak{L}}\gamma$ fixed, in A.3 yields

(A.16) $\displaystyle\begin{aligned}
&\displaystyle\lim_{q^{\mathfrak{M}}\rightarrow\infty}\mathcal{U}_{q^{\mathfrak{L}};q^{\mathfrak{M}};q^{\mathfrak{M}+\mathfrak{L}}/\gamma}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}\\\
&\quad=(-1)^{C_{n}}q^{\varphi(\overline{\bm{D}},\overline{\bm{C}})+\binom{C_{n}+1}{2}+\overline{c}\mathsf{d}+\mathfrak{L}C_{n}}\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}}}\displaystyle\frac{(q^{-c-\mathfrak{L}}\gamma;q)_{\mathsf{d}}}{(q;q)_{C_{n}}}\displaystyle\frac{(q^{\mathsf{D}_{n}-C_{n}-\mathfrak{L}};q)_{C_{n}}(\gamma;q)_{\infty}}{(q^{-\mathfrak{L}}\gamma;q)_{\infty}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}\\\
&\qquad\times\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}(-1)^{\overline{p}}q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\overline{\bm{P}})}\displaystyle\frac{(q^{\mathsf{d}-c-\mathfrak{L}}\gamma;q)_{\overline{p}}}{(q^{-\mathfrak{L}-c}\gamma;q)_{\overline{p}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}\\\
&\qquad\qquad\quad\times
q^{\overline{p}(\mathsf{b}-\mathsf{d}+1)+\binom{\overline{p}}{2}}\cdot{{}_{3}\varphi_{2}}\bigg{(}\begin{array}[]{cc}q^{-C_{n}};q^{\mathsf{B}_{n}-\mathfrak{L}},q^{\mathsf{d}-c+\overline{p}-\mathfrak{L}}\gamma\\\
q^{\overline{p}-\mathfrak{L}-c}\gamma,q^{\mathsf{D}_{n}-C_{n}-\mathfrak{L}}\end{array}\bigg{|}q,q\bigg{)}.\end{aligned}$

Next recall from the Sears identity [45, Equation (3.2.9)] that, for any
integer $m\geq 0$ and complex numbers $A,B,C,D,E,F\in\mathbb{C}$ with
$q^{1-m}ABC=DEF$, we have

$\displaystyle{{}_{4}\varphi_{3}}\bigg{(}\begin{array}[]{cccc}q^{-m};A,B,C\\\
D,E,F\end{array}\bigg{|}q,q\bigg{)}=\displaystyle\frac{(B;q)_{m}\big{(}\frac{DE}{AB};q\big{)}_{m}\big{(}\frac{DE}{BC};q\big{)}_{m}}{(D;q)_{m}(E;q)_{m}\big{(}\frac{DE}{ABC};q\big{)}_{m}}\cdot{{}_{4}\varphi_{3}}\bigg{(}\begin{array}[]{cccc}q^{-m};\frac{D}{B},\frac{E}{B},\frac{DE}{ABC}\\\
\frac{DE}{AB},\frac{DE}{BC},q^{1-m}B^{-1}\end{array}\bigg{|}q,q\bigg{)}.$

Letting $A$ and $E$ tend to $0$ in such a way that
$\frac{A}{E}=q^{m-1}\frac{DF}{BC}$ remains fixed, we deduce (under no
restrictions on $B,C,D,F\in\mathbb{C}$) that

$\displaystyle{{}_{3}\varphi_{2}}\bigg{(}\begin{array}[]{cccc}q^{-m};B,C\\\
D,F\end{array}\bigg{|}q,q\bigg{)}=\displaystyle\frac{(B;q)_{m}(q^{1-m}CF^{-1};q)_{m}}{(D;q)_{m}(q^{1-m}F^{-1};q)_{m}}\cdot{{}_{3}\varphi_{2}}\bigg{(}\begin{array}[]{cccc}q^{-m};\frac{D}{B},q^{1-m}F^{-1}\\\
q^{1-m}CF^{-1},q^{1-m}B^{-1}\end{array}\bigg{|}q,q\bigg{)}.$

Taking $m=C_{n}$ and
$(B,C;D,F)=(q^{\mathsf{B}_{n}-\mathfrak{L}},q^{\mathsf{d}-c+\overline{p}-\mathfrak{L}}\gamma;q^{\mathsf{D}_{n}-C_{n}-\mathfrak{L}},q^{\overline{p}-\mathfrak{L}-c}\gamma)$
(and using the fact that $\mathsf{B}_{n}-A_{n}=\mathsf{D}_{n}-C_{n}$, by (A.3)
and the equalities $\mathsf{b}-a=\mathsf{d}-c$ and
$\overline{\bm{A}}+\overline{\bm{B}}=\overline{\bm{C}}+\overline{\bm{D}}$), we
obtain

$\displaystyle{{}_{3}\varphi_{2}}\bigg{(}\begin{array}[]{cc}q^{-C_{n}};q^{\mathsf{B}_{n}-\mathfrak{L}},q^{\mathsf{d}-c+\overline{p}-\mathfrak{L}}\gamma\\\
q^{\overline{p}-\mathfrak{L}-c}\gamma,q^{\mathsf{D}_{n}-C_{n}-\mathfrak{L}}\end{array}\bigg{|}q,q\bigg{)}$
$\displaystyle=\displaystyle\frac{(q^{\mathsf{B}_{n}-\mathfrak{L}};q)_{C_{n}}(q^{\mathsf{d}-C_{n}+1};q)_{C_{n}}}{(q^{\mathsf{D}_{n}-C_{n}-\mathfrak{L}};q)_{C_{n}}(q^{\mathfrak{L}+c-C_{n}-\overline{p}+1}\gamma^{-1};q)_{C_{n}}}$
$\displaystyle\qquad\times{{}_{3}\varphi_{2}}\bigg{(}\begin{array}[]{cc}q^{-C_{n}};q^{-A_{n}},q^{\mathfrak{L}-C_{n}+c-\overline{p}+1}\gamma^{-1}\\\
q^{\mathsf{d}-C_{n}+1},q^{\mathfrak{L}-C_{n}-\mathsf{B}_{n}+1}\end{array}\bigg{|}q,q\bigg{)}.$

Inserting this into (A.16), and using the fact that
$(q^{\mathsf{d}-C_{n}+1};q)_{C_{n}}(q;q)_{\mathsf{d}}^{-1}=(q;q)_{\mathsf{d}-C_{n}}^{-1}$,
gives the lemma. ∎

We next take the further limit in A.4 as $q^{\mathfrak{L}}$ tends to $\infty$.
The below definition provides this limit; see the lemma that follows.

######  Definition A.5.

Adopting the notation of Definition A.1, define for any $\gamma\in\mathbb{C}$
the _$q$ -discrete polymer weight_
$\mathcal{U}_{\gamma}^{\operatorname{qp};n}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}=\mathcal{U}_{\gamma}^{\operatorname{qp}}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}$
by

(A.17) $\displaystyle\begin{aligned}
\mathcal{U}_{\gamma}^{\operatorname{qp}}&\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}\\\
&=q^{\varphi(\overline{\bm{D}},\overline{\bm{C}})+\overline{c}(\mathsf{B}_{n}-A_{n}-\overline{d})}(\gamma;q)_{\infty}\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{B}_{n}-A_{n}-\overline{d}}}\cdot\mathbbm{1}_{\overline{\bm{A}}+\overline{\bm{B}}=\overline{\bm{C}}+\overline{\bm{D}}}\cdot\mathbbm{1}_{\mathsf{b}-a=\mathsf{d}-c}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}}}{(q;q)_{C_{j}}(q;q)_{D_{j}}}\\\
&\qquad\times\displaystyle\sum_{k=0}^{C_{n}}\displaystyle\frac{\gamma^{C_{n}-k}}{(q;q)_{C_{n}-k}}\displaystyle\frac{(q^{A_{n}-k+1};q)_{k}}{(q;q)_{k}(q^{\mathsf{B}_{n}-A_{n}-\overline{d}+1};q)_{k}}q^{k(\mathsf{B}_{n}-A_{n}+\overline{c}+k)}\\\
&\qquad\quad\times\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}q^{\overline{p}(A_{n}-k+1)+\varphi(\overline{\bm{P}},\overline{\bm{D}}-\overline{\bm{B}})}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q^{-B_{j}};q)_{P_{j}}(q^{-C_{j}};q)_{P_{j}}}{(q;q)_{P_{j}}(q^{-C_{j}-D_{j}};q)_{P_{j}}}.\end{aligned}$

######  Remark A.6.

If $n=1$, then the $\mathcal{U}^{\operatorname{qp}}$ weights from Definition
A.5 coincide with those of the geometric $q$-PushTASEP introduced in [68,
Section 6.3], which degenerates to the log-gamma polymer [68, Theorem 8.7].
This is the reason behind the term, “$q$-discrete polymer weight” in
Definition A.5.

######  Lemma A.7.

Under the notation of A.4, we have

$\displaystyle\displaystyle\lim_{q^{\mathfrak{L}}\rightarrow\infty}\bigg{(}\displaystyle\lim_{q^{\mathfrak{M}}\rightarrow\infty}\mathcal{U}_{q^{\mathfrak{L}};q^{\mathfrak{M}};q^{\mathfrak{M}+\mathfrak{L}}/\gamma}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}\bigg{)}=\mathcal{U}_{\gamma}^{\operatorname{qp}}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}.$

###### Proof.

Throughout this proof, we assume that
$\overline{\bm{A}}+\overline{\bm{B}}=\overline{\bm{C}}+\overline{\bm{D}}$ and
$\mathsf{b}-a=\mathsf{d}-c$, for otherwise the lemma holds by A.4 and
Definition A.5. Since
$\lim_{a\rightarrow\infty}a^{-k}(ab;q)_{k}=(-b)^{k}q^{\binom{k}{2}}$ for any
$b\in\mathbb{C}$ and $k\in\mathbb{Z}_{\geq 0}$, we have

$\displaystyle\displaystyle\lim_{q^{\mathfrak{L}}\rightarrow\infty}(-1)^{C_{n}}q^{\mathfrak{L}C_{n}+\binom{C_{n}+1}{2}}(q^{\mathfrak{L}+c-C_{n}-\overline{p}+1}\gamma^{-1};q)_{C_{n}}^{-1}=q^{C_{n}(C_{n}-c+\overline{p})}\gamma^{C_{n}};$
$\displaystyle\displaystyle\lim_{q^{\mathfrak{L}}\rightarrow\infty}{{}_{3}\varphi_{2}}\bigg{(}\begin{array}[]{cc}q^{-C_{n}};q^{-A_{n}},q^{\mathfrak{L}-C_{n}+c-\overline{p}+1}\gamma^{-1}\\\
q^{\mathsf{d}-C_{n}+1},q^{\mathfrak{L}-\mathsf{B}_{n}-C_{n}+1}\end{array}\bigg{|}q,q\bigg{)}={{}_{2}\varphi_{1}}\bigg{(}\begin{array}[]{cc}q^{-C_{n}};q^{-A_{n}}\\\
q^{\mathsf{d}-C_{n}+1}\end{array}\bigg{|}q,q^{\mathsf{B}_{n}+c-\overline{p}+1}\gamma^{-1}\bigg{)},$

where in the second statement we used (A.6). Inserting these (with the fact
that $C_{n}-c=-\overline{c}$), into the limit as $q^{\mathfrak{L}}$ tends to
$\infty$ of (A.4), we find

(A.18) $\displaystyle\begin{aligned}
\displaystyle\lim_{q^{\mathfrak{L}}\rightarrow\infty}&\bigg{(}\displaystyle\lim_{q^{\mathfrak{M}}\rightarrow\infty}\mathcal{U}_{q^{\mathfrak{L}};q^{\mathfrak{M}};q^{\mathfrak{M}+\mathfrak{L}}/\gamma}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}\bigg{)}\\\
&\quad=q^{\varphi(\overline{\bm{D}},\overline{\bm{C}})+\overline{c}(\mathsf{d}-C_{n})}\gamma^{C_{n}}\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}-C_{n}}(q;q)_{C_{n}}}(\gamma;q)_{\infty}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}\\\
&\qquad\times\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}(-1)^{\overline{p}}q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\overline{\bm{P}})}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}\\\
&\qquad\qquad\quad\times
q^{\overline{p}(\mathsf{b}-\mathsf{d}+1)+\binom{\overline{p}}{2}}q^{\overline{p}C_{n}}\cdot{{}_{2}\varphi_{1}}\bigg{(}\begin{array}[]{cc}q^{-C_{n}},q^{-A_{n}}\\\
q^{\mathsf{d}-C_{n}+1}\end{array}\bigg{|}q,q^{\mathsf{B}_{n}+c-\overline{p}+1}\gamma^{-1}\bigg{)}.\end{aligned}$

Next we separate the $P_{j}$ subscripts in the $q$-Pochhammer symbols on the
right side of (A.18) from the other ones. To that end, observe since
$(q;q)_{m-k}=(-1)^{k}q^{\binom{k}{2}-mk}(q;q)_{m}(q^{-m};q)_{k}^{-1}$ that we
have

(A.19)
$\displaystyle\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{B_{j}-P_{j}}}=(-1)^{P_{j}}q^{P_{j}(B_{j}-D_{j})-\binom{P_{j}}{2}}\displaystyle\frac{(q;q)_{C_{j}+D_{j}}}{(q;q)_{C_{j}}(q;q)_{B_{j}}}\displaystyle\frac{(q^{-C_{j}};q)_{P_{j}}(q^{-B_{j}};q)_{P_{j}}}{(q^{-C_{j}-D_{j}};q)_{P_{j}}}.$

We also have

(A.20) $\displaystyle\begin{aligned}
\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\overline{\bm{P}})+\overline{p}(\mathsf{b}-\mathsf{d}+C_{n}+1)+\binom{\overline{p}}{2}+\displaystyle\sum_{j=1}^{n-1}\Bigg{(}&P_{j}(B_{j}-D_{j})-\binom{P_{j}}{2}\Bigg{)}\\\
&=\overline{p}(A_{n}+1)+\varphi(\overline{\bm{P}},\overline{\bm{D}}-\overline{\bm{B}}),\end{aligned}$

since

$\displaystyle\binom{\overline{p}}{2}=\varphi(\overline{\bm{P}},\overline{\bm{P}})+\displaystyle\sum_{j=1}^{n-1}\binom{P_{j}}{2};\qquad\varphi(\overline{\bm{B}}-\overline{\bm{D}},\overline{\bm{P}})+\overline{p}(\overline{d}-\overline{b})+\displaystyle\sum_{j=1}^{n-1}P_{j}(B_{j}-D_{j})=\varphi(\bm{\overline{P}},\bm{\overline{D}}-\bm{\overline{B}});$
$\displaystyle\overline{p}(\overline{b}-\overline{d})+\overline{p}(\mathsf{b}-\mathsf{d})+\overline{p}C_{n}=\overline{p}(\mathsf{B}_{n}-\mathsf{D}_{n})+\overline{p}C_{n}=\overline{p}A_{n}.$

Inserting (A.19) and (A.20) into (A.18) yields

$\displaystyle\displaystyle\lim_{q^{\mathfrak{L}}\rightarrow\infty}$
$\displaystyle\bigg{(}\displaystyle\lim_{q^{\mathfrak{M}}\rightarrow\infty}\mathcal{U}_{q^{\mathfrak{L}};q^{\mathfrak{M}};q^{\mathfrak{M}+\mathfrak{L}}/\gamma}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}\bigg{)}$
$\displaystyle=q^{\varphi(\overline{\bm{D}},\overline{\bm{C}})+\overline{c}(\mathsf{d}-C_{n})}\gamma^{C_{n}}\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}-C_{n}}(q;q)_{C_{n}}}(\gamma;q)_{\infty}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}}}{(q;q)_{C_{j}}(q;q)_{D_{j}}}$
$\displaystyle\qquad\times\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}q^{\overline{p}(A_{n}+1)+\varphi(\overline{\bm{P}},\overline{\bm{D}}-\overline{\bm{B}})}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q^{-C_{j}};q)_{P_{j}}(q^{-B_{j}};q)_{P_{j}}}{(q^{-C_{j}-D_{j}};q)_{P_{j}}(q;q)_{P_{j}}}$
$\displaystyle\qquad\qquad\quad\times{{}_{2}\varphi_{1}}\bigg{(}\begin{array}[]{cc}q^{-C_{n}},q^{-A_{n}}\\\
q^{\mathsf{d}-C_{n}+1}\end{array}\bigg{|}q,q^{\mathsf{B}_{n}+c-\overline{p}+1}\gamma^{-1}\bigg{)}.$

Thus, using (A.6) to write the ${{}_{2}\varphi_{1}}$ basic hypergeometric
series as a sum, we obtain

$\displaystyle\displaystyle\lim_{q^{\mathfrak{L}}\rightarrow\infty}$
$\displaystyle\bigg{(}\displaystyle\lim_{q^{\mathfrak{M}}\rightarrow\infty}\mathcal{U}_{q^{\mathfrak{L}};q^{\mathfrak{M}};q^{\mathfrak{M}+\mathfrak{L}}/\gamma}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}\bigg{)}$
$\displaystyle=q^{\varphi(\overline{\bm{D}},\overline{\bm{C}})+\overline{c}(\mathsf{d}-C_{n})}(\gamma;q)_{\infty}\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}-C_{n}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}}}{(q;q)_{C_{j}}(q;q)_{D_{j}}}$
$\displaystyle\qquad\times\displaystyle\sum_{k=0}^{C_{n}}\displaystyle\frac{\gamma^{C_{n}-k}}{(q;q)_{C_{n}}}\displaystyle\frac{(q^{-C_{n}};q)_{k}(q^{-A_{n}};q)_{k}}{(q;q)_{k}(q^{\mathsf{d}-C_{n}+1};q)_{k}}q^{k(\mathsf{B}_{n}+c+1)}$
$\displaystyle\qquad\qquad\times\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}q^{\overline{p}(A_{n}-k+1)+\varphi(\overline{\bm{P}},\overline{\bm{D}}-\overline{\bm{B}})}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q^{-B_{j}};q)_{P_{j}}(q^{-C_{j}};q)_{P_{j}}}{(q;q)_{P_{j}}(q^{-C_{j}-D_{j}};q)_{P_{j}}}.$

Together with the facts that

$\displaystyle(q^{-C_{n}};q)_{k}(q;q)_{C_{n}}^{-1}=(-1)^{k}q^{\binom{k}{2}-C_{n}k}(q;q)_{C_{n}-k}^{-1};\qquad(q^{-A_{n}};q)_{k}=(-1)^{k}q^{\binom{k}{2}-A_{n}k}(q^{A_{n}-k+1};q)_{k},$

that $c-C_{n}=\overline{c}$, and that
$\mathsf{d}-C_{n}=\mathsf{D}_{n}-\overline{d}-C_{n}=\mathsf{B}_{n}-A_{n}-\overline{d}$,
this yields the lemma. ∎

######  Remark A.8.

By 6.2, A.3, A.4, A.7, and uniqueness of analytic continuation (with the fact
that the $\mathcal{U}^{\operatorname{qp}}$ weights are rational in
$q^{\mathfrak{L}}$, to extend from $\mathfrak{L}\in\mathbb{Z}_{\geq 0}$ to
$\mathfrak{L}\in\mathbb{C}$) the $\mathcal{U}^{\operatorname{qp}}$ weights are
stochastic, in the sense that for any $\bm{A}\in\mathbb{Z}_{\geq 0}^{n}$ and
$(\overline{\bm{B}},\mathsf{B}_{n})\in\mathbb{Z}_{\geq
0}^{n-1}\times\mathbb{Z}_{\geq 0}$, we have
$\sum\mathcal{U}_{\gamma}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}=1$,
where we sum over all $\bm{C}\in\mathbb{Z}_{\geq 0}^{n}$ and
$(\overline{\bm{D}},\mathsf{D}_{n})\in\mathbb{Z}_{\geq
0}^{n-1}\times\mathbb{Z}_{\geq 0}$.

###  A.3. An Additional Degenerated Weight

In this section we provide an additional degeneration of the complemented
$U_{z;r,s}$-weight, which will eventually serve as boundary weights for the
polymer model. We begin with the following lemma from [20] providing a
specialization of the $U_{z;r,s}$-weights from Definition 6.1.

######  Lemma A.9 ([20, Lemma 6.8]).

For any integer $\mathrm{L}\geq 1$ and $n$-tuples
$\bm{A},\bm{C},\bm{D}\in\mathbb{Z}_{\geq 0}^{n}$, we have

$\displaystyle\displaystyle\lim_{s\rightarrow
0}U_{z/s^{2};q^{-\mathrm{L}/2},s}(\bm{A},\mathrm{L}\bm{e}_{n};\bm{C},\bm{D})=\displaystyle\frac{(zq^{\mathrm{L}})^{|\bm{D}|}}{(z;q)_{\mathrm{L}}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{|\bm{D}|}}{(q;q)_{|\bm{D}|}}\cdot\mathbbm{1}_{\bm{A}+\mathrm{L}\cdot\bm{e}_{n}=\bm{C}+\bm{D}}\cdot\mathbbm{1}_{\bm{D}=|\bm{D}|\cdot\bm{e}_{n}}.$

We will next implement the complementation procedure explained in Section A.1
on the weights specialized as in A.9. The below definition provides the
eventual form they will take, which is shown in the corollary following it.

######  Definition A.10.

For any complex numbers $z,\mathfrak{L}\in\mathbb{C}$ and integer $k\geq 0$,
define the _$q$ -discrete boundary weight_

$\displaystyle\mathcal{U}_{z;q^{\mathfrak{L}}}^{\operatorname{b};n}(k)=\mathcal{U}_{z;q^{\mathfrak{L}}}^{\operatorname{b}}(k)=z^{-k}\displaystyle\frac{(z^{-1};q)_{\infty}}{(q^{-\mathfrak{L}}z^{-1};q)_{\infty}}\displaystyle\frac{(q^{-\mathfrak{L}};q)_{k}}{(q;q)_{k}}.$

We further set

$\displaystyle\mathcal{U}_{z}^{\operatorname{b};n}(k)=\mathcal{U}_{z}^{\operatorname{b}}(k)=\mathcal{U}_{z;\infty}^{\operatorname{b}}(k)=z^{-k}\displaystyle\frac{(z^{-1};q)_{k}}{(q;q)_{k}}.$

Observe that both the
$\mathcal{U}_{z;q^{\mathfrak{L}}}^{\operatorname{b}}$-weights and the
$\mathcal{U}_{z}^{\operatorname{b}}$-weights are stochastic, by the
$q$-binomial theorem.

######  Corollary A.11.

Adopt the notation of A.9, and set $\mathsf{D}_{n}=\mathrm{L}-|\bm{D}|$. We
have

(A.21) $\displaystyle\displaystyle\lim_{s\rightarrow
0}U_{z/s^{2};q^{-\mathrm{L}/2},s}(\bm{A},\mathrm{L}\bm{e}_{n};\bm{C},\bm{D})=\mathcal{U}_{z/q;q^{\mathrm{L}}}^{\operatorname{b}}(\mathsf{D}_{n})\cdot\mathbbm{1}_{\bm{A}+\mathsf{D}_{n}\cdot\bm{e}_{n}=\bm{C}}\cdot\mathbbm{1}_{\bm{D}=(\mathrm{L}-\mathsf{D}_{n})\cdot\bm{e}_{n}}.$

###### Proof.

We may assume throughout this proof that
$\bm{A}+\mathrm{L}\bm{e}_{n}=\bm{C}+\bm{D}$ and that
$\bm{D}=|\bm{D}|\bm{e}_{n}=(\mathrm{L}-\mathsf{D}_{n})\bm{e}_{n}$, for
otherwise both sides of (A.21) are equal to $0$ (by A.9). Then, combining the
equalities

$\displaystyle(zq^{\mathrm{L}})^{|\bm{D}|}=(zq^{\mathrm{L}})^{\mathrm{L}-\mathsf{D}_{n}};\qquad(z;q)_{\mathrm{L}}=(-z)^{\mathrm{L}}q^{\binom{\mathrm{L}}{2}}(q^{1-\mathrm{L}}z^{-1};q)_{\mathrm{L}};$
$\displaystyle(q^{-\mathrm{L}};q)_{\mathrm{L}-\mathsf{D}_{n}}=(-1)^{\mathrm{L}-\mathsf{D}_{n}}q^{\binom{\mathsf{D}_{n}+1}{2}-\binom{\mathrm{L}+1}{2}}\displaystyle\frac{(q;q)_{\mathrm{L}}}{(q;q)_{\mathsf{D}_{n}}};\qquad(q;q)_{\mathrm{L}-\mathsf{D}_{n}}=(-1)^{\mathsf{D}_{n}}q^{\binom{\mathsf{D}_{n}}{2}-\mathrm{L}\mathsf{D}_{n}}\displaystyle\frac{(q;q)_{\mathrm{L}}}{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}}},$

with A.11, we deduce

$\displaystyle\displaystyle\lim_{s\rightarrow
0}U_{z/s^{2};q^{-\mathrm{L}/2},s}(\bm{A},\mathrm{L}\bm{e}_{n};\bm{C},\bm{D})=\displaystyle\frac{(zq^{\mathrm{L}})^{\mathrm{L}-\mathsf{D}_{n}}}{(z;q)_{\mathrm{L}}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathrm{L}-\mathsf{D}_{n}}}{(q;q)_{\mathrm{L}-\mathsf{D}_{n}}}=\displaystyle\frac{(qz^{-1})^{\mathsf{D}_{n}}}{(q^{1-\mathrm{L}}z^{-1};q)_{\mathrm{L}}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}}}{(q;q)_{\mathsf{D}_{n}}}.$

This, with Definition A.10, yields the corollary. ∎

###  A.4. The $q\rightarrow 1$ Limit

In this section we analyze the limit as $q$ tends to $1$ of the $q$-discrete
polymer weights $\mathcal{U}^{\operatorname{qp}}$ from Definition A.5 (and
their boundary counterparts $\mathcal{U}^{\operatorname{b}}$ from Definition
A.10). Throughout this section, we fix a real number $\theta>0$ and $n$-tuples
of nonnegative real numbers
$(\mathfrak{a}_{1},\mathfrak{a}_{2},\ldots,\mathfrak{a}_{n})$ and
$(\mathfrak{b}_{1},\mathfrak{b}_{2},\ldots,\mathfrak{b}_{n})$. Moreover,
$\varepsilon\in(0,1)$ will be a parameter that we view as tending to $0$. If
for some functions $f$ and $g$ of $\varepsilon$ we have
$\lim_{\varepsilon\rightarrow
0}\big{|}f(\varepsilon)-g(\varepsilon)\big{|}=0$, then we write $f\sim g$. We
further let $\mathsf{B}_{n}=\mathsf{B}_{n}^{\varepsilon}\in\mathbb{Z}_{\geq
0}$ be an integer,
$\bm{\overline{B}}=\bm{\overline{B}}^{\varepsilon}\in\mathbb{Z}_{\geq
0}^{n-1}$ be an $(n-1)$-tuple, and
$\bm{A}=\bm{A}^{\varepsilon}\in\mathbb{Z}_{\geq 0}^{n}$ be an $n$-tuple (all
dependent on $\varepsilon$), with

(A.22) $\displaystyle\varepsilon
A_{n}-\log\varepsilon^{-1}\sim\mathfrak{a}_{n};\qquad\varepsilon
A_{j}\sim\mathfrak{a}_{j};\qquad\varepsilon\mathsf{B}_{n}-\log\varepsilon^{-1}=\mathfrak{b}_{n};\qquad\varepsilon
B_{j}\sim\mathfrak{b}_{j},$

for each $j\in\llbracket 1,n-1\rrbracket$. Also define
$q=q_{\varepsilon}\in(0,1)$ and $\gamma=\gamma_{\varepsilon,\theta}\in(0,1)$
by setting

(A.23) $\displaystyle
q_{\varepsilon}=e^{-\varepsilon};\qquad\gamma=q^{\theta}.$

Recalling from A.8 that the $\mathcal{U}^{\operatorname{qp}}$ weights are
stochastic, the following proposition describes the limiting law of the
$n$-tuple $\bm{C}$ sampled191919We do not verify here that the
$\mathcal{U}_{\gamma}^{\operatorname{qp}}$-weights are all nonnegative for
fixed $q\in(0,1)$. However, it can quickly be deduced from the proof of A.18
below that the sum of the absolute values of all negative
$\mathcal{U}_{\gamma}^{\operatorname{qp}}$-weights tends to $0$, as
$\varepsilon$ tends to $0$. Together with the stochasticity of the
$\mathcal{U}^{\operatorname{qp}}$ weights, this implies that one can view
$\mathcal{U}^{\operatorname{qp}}$ as prescribing a probability measure, as
$\varepsilon$ tends to $0$. according to
$\mathcal{U}_{\gamma}^{\operatorname{qp}}(\bm{A},\bm{B};\bm{C},\bm{D})$ as
$\varepsilon$ tends to $0$; it indicates that this law can be expressed
through a single $\mathrm{Gamma}(\theta)$ random variable $\mathfrak{Y}$.

######  Proposition A.12.

Let $q=q_{\varepsilon}\in(0,1)$ and
$\gamma=\gamma_{\varepsilon,\theta}\in(0,1)$ be as in (A.23); also let
$\mathsf{B}_{n}\in\mathbb{Z}_{\geq 0}$, $\overline{\bm{B}}\in\mathbb{Z}_{\geq
0}^{n-1}$, and $\bm{A}\in\mathbb{Z}_{\geq 0}^{n}$ be as in (A.22). Sample
$\big{(}\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}\in\mathbb{Z}_{\geq
0}^{n}\times\mathbb{Z}_{\geq 0}^{n-1}\times\mathbb{Z}_{\geq 0}$ with
probability
$\mathcal{U}_{\gamma}^{\operatorname{qp}}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}$,
and denote

$\displaystyle\mathfrak{c}_{n}=\varepsilon
C_{n}-\log\varepsilon^{-1},\quad\text{and}\quad\mathfrak{c}_{j}=\varepsilon
C_{j},\qquad\text{for each $j\in\llbracket 1,n-1\rrbracket$}.$

The joint law of $(\mathfrak{c}_{[j,n]})_{j\in\llbracket 1,n\rrbracket}$
converges to that of
$\big{(}\log(e^{\mathfrak{a}_{[j,n]}-\mathfrak{b}_{n}+\mathfrak{b}_{[j,n-1]}}+1)-\log\mathfrak{Y}\big{)}_{j\in\llbracket
1,n\rrbracket}$ as $\varepsilon$ tends to $0$, where
$\mathfrak{Y}\in\mathbb{R}_{>0}$ is a $\mathrm{Gamma}(\theta)$ random
variable.

######  Remark A.13.

Denoting
$(\mathfrak{d}_{1},\mathfrak{d}_{2},\ldots,\mathfrak{d}_{n})=(\varepsilon
D_{1},\varepsilon D_{2},\ldots,\varepsilon
D_{n-1},\varepsilon\mathsf{D}_{n}-\log\varepsilon^{-1})$ in A.12, the joint
limiting law of $(\mathfrak{d}_{[j,n]})_{j\in\llbracket 1,n\rrbracket}$ is
determined by that of $(\mathfrak{c}_{[j,n]})_{j\in\llbracket 1,n\rrbracket}$
by arrow conservation.

To establish A.12, we will express the right side of (A.17) as a convolution
of the following two functions, which we will then analyze separately.

######  Definition A.14.

Define the function $\Phi_{\varepsilon}:\mathbb{Z}_{\geq
0}\rightarrow\mathbb{R}$ by setting, for any integer $\ell\geq 0$,

(A.24)
$\displaystyle\Phi_{\varepsilon}(\ell)=(\gamma;q)_{\infty}\cdot\displaystyle\frac{\gamma^{\ell}}{(q;q)_{\ell}}.$

Also define the function $\Psi_{\varepsilon}:\mathbb{Z}_{\geq
0}\times\mathbb{Z}_{\geq 0}^{n-1}$ by setting, for any integer $k\geq 0$ and
$(n-1)$-tuple $\overline{\bm{C}}\in\mathbb{Z}_{\geq 0}^{n-1}$,

(A.25) $\displaystyle\begin{aligned}
\Psi_{\varepsilon}(k;\overline{\bm{C}})&=q^{\varphi(\overline{\bm{D}},\overline{\bm{C}})+\overline{c}(\mathsf{B}_{n}-A_{n}-\overline{d})}\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{B}_{n}-A_{n}-\overline{d}}}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}}}{(q;q)_{C_{j}}(q;q)_{D_{j}}}\cdot
q^{k(\mathsf{B}_{n}-A_{n}+\overline{c}+k)}\\\
&\quad\times\displaystyle\frac{(q^{A_{n}-k+1};q)_{k}}{(q;q)_{k}(q^{\mathsf{B}_{n}-A_{n}-\overline{d}+1};q)_{k}}\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}q^{\overline{p}(A_{n}-k+1)+\varphi(\overline{\bm{P}},\overline{\bm{D}}-\overline{\bm{B}})}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q^{-B_{j}};q)_{P_{j}}(q^{-C_{j}};q)_{P_{j}}}{(q;q)_{P_{j}}(q^{-C_{j}-D_{j}};q)_{P_{j}}},\end{aligned}$

where
$\overline{\bm{D}}=\overline{\bm{A}}+\overline{\bm{B}}-\overline{\bm{C}}$.

######  Remark A.15.

By Definition A.5, we have

(A.26)
$\displaystyle\mathcal{U}_{\gamma}^{\operatorname{qp}}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}=\displaystyle\sum_{k=0}^{C_{n}}\Phi_{\varepsilon}(C_{n}-k)\cdot\Psi_{\varepsilon}(k;\overline{\bm{C}}).$

Hence, to sample $\bm{C}$ as in A.12, we may first sample $\ell$ and
$(k;\overline{\bm{C}})$ under the density functions $\Phi_{\varepsilon}$ and
$\Psi_{\varepsilon}$ (the fact that these give stochastic weights is due to
A.16 below), respectively, and then set $\bm{C}=(\overline{\bm{C}},k+\ell)$.

######  Lemma A.16.

We have

(A.27)
$\displaystyle\displaystyle\sum_{\ell=0}^{\infty}\Phi_{\varepsilon}(\ell)=1;\qquad\displaystyle\sum_{\begin{subarray}{c}k\geq
0\\\ \overline{\bm{C}}\in\mathbb{Z}_{\geq
0}^{n-1}\end{subarray}}\Psi_{\varepsilon}(k;\overline{\bm{C}})=1.$

###### Proof.

The first statement in (A.27) holds by the $q$-binomial theorem. The second
holds since

$\displaystyle\displaystyle\sum_{\begin{subarray}{c}k\geq 0\\\
\overline{\bm{C}}\in\mathbb{Z}_{\geq
0}^{n-1}\end{subarray}}\Psi_{\varepsilon}(k;\overline{\bm{C}})=\displaystyle\sum_{\begin{subarray}{c}k,\ell\geq
0\\\ \overline{\bm{C}}\in\mathbb{Z}_{\geq
0}^{n-1}\end{subarray}}\Phi_{\varepsilon}(\ell)\cdot\Psi_{\varepsilon}(k;\overline{\bm{C}})=\displaystyle\sum_{\begin{subarray}{c}C_{n}\geq
0\\\ \overline{\bm{C}}\in\mathbb{Z}_{\geq 0}^{n-1}\\\
\bm{D}\in\mathbb{Z}_{\geq
0}^{n}\end{subarray}}\mathcal{U}_{\gamma}^{\operatorname{qp}}\big{(}\bm{A},(\mathsf{B}_{n},\overline{\bm{B}});(\overline{\bm{C}},C_{n}),(\mathsf{D}_{n},\overline{\bm{D}})\big{)}=1,$

where in the first equality we used the first statement of (A.27); in the
second we used (A.26); and in the third we used the stochasticity of
$\mathcal{U}^{\operatorname{qp}}$ (from A.8). ∎

We next have the following two lemmas that describe the probability measures
arising as the limit when $\varepsilon$ tends to $0$ of those with
distribution functions $\Phi_{\varepsilon}$ and $\Psi_{\varepsilon}$. The
first shows that the former (as well as the
$\mathcal{U}_{z}^{\operatorname{b}}$ weights from Definition A.10) gives rise
to the $\mathrm{Gamma}(\theta)$ random variable from A.12; it is due to [40].
The second shows that the latter is given by the delta distribution at a
specific pair of real number and $(n-1)$-tuple; we outline its proof (which is
similar to previous results, such as [68, Lemma 8.17] and [20, Proposition
6.19]) in Section A.5 below.

######  Lemma A.17 ([40, Lemma 2.1]).

The following two statements hold.

  1. (1)

Sample $\ell_{\varepsilon}\in\mathbb{Z}_{\geq 0}$ according to the
distribution $\mathbb{P}[\ell_{\varepsilon}=\ell]=\Phi_{\varepsilon}(\ell)$.
Then, $\varepsilon^{-1}\cdot e^{-\varepsilon\ell_{\varepsilon}}$ converges in
distribution to a $\mathrm{Gamma}(\theta)$ random variable, as $\varepsilon$
tends to $0$.

  2. (2)

Sample $\ell_{\varepsilon}\in\mathbb{Z}_{\geq 0}$ according to the
distribution
$\mathbb{P}[\ell_{\varepsilon}=\ell]=\mathcal{U}_{1/\gamma}^{\operatorname{b}}(\ell)$.
Then, $\varepsilon^{-1}\cdot e^{-\varepsilon\ell_{\varepsilon}}$ converges in
distribution to a $\mathrm{Gamma}(\theta)$ random variable, as $\varepsilon$
tends to $0$.

######  Lemma A.18.

Sample an $(n-1)$-tuple and integer $(\overline{\bm{C}};k)\in\mathbb{Z}_{\geq
0}^{n-1}\times\mathbb{Z}_{\geq 0}$ with probability
$\Psi_{\varepsilon}(\overline{\bm{C}};k)$; denote
$\bm{\mathfrak{c}}=(\mathfrak{c}_{1},\mathfrak{c}_{2},\ldots,\mathfrak{c}_{n-1})=\varepsilon\cdot\overline{\bm{C}}\in\mathbb{R}_{\geq
0}^{n-1}$ and $\mathfrak{k}=\varepsilon k\in\mathbb{R}$. As $\varepsilon$
tends to $0$, we have that

(A.28) $\displaystyle
e^{\mathfrak{k}+\mathfrak{c}_{[j,n-1]}}\quad\text{converges in probability to
the constant}\quad
e^{\mathfrak{a}_{[j,n]}-\mathfrak{b}_{n}+\mathfrak{b}_{[j,n-1]}}+1,\quad\text{for
each $j\in\llbracket 1,n\rrbracket$}.$

Now we can quickly establish A.12.

###### Proof of A.12.

This follows from A.15, A.17, and A.18. ∎

###  A.5. Proof Outline of A.18

In this section we outline the proof of A.18.

###### Proof of A.18 (Outline).

First, it is quickly verified that the contribution to the second sum in
(A.27) coming from $k\geq(2\varepsilon^{-1})\log\varepsilon^{-1}$ becomes
negligible as $\varepsilon$ tends to $0$, due to the factor of
$q^{k(k+\mathsf{B}_{n}-A_{n}+\overline{c})}$ on the right side of (A.25)
(namely, the fact that the exponent of $q$ is quadratic in $k$). We may
therefore restrict our attention to when
$k\leq(2\varepsilon)^{-1}\log\varepsilon^{-1}$, in which case it can be
confirmed that

(A.29) $\displaystyle\displaystyle\lim_{\varepsilon\rightarrow
0}\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}q^{\overline{p}(A_{n}-k+1)+\varphi(\overline{\bm{P}},\overline{\bm{D}}-\overline{\bm{B}})}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q^{-B_{j}};q)_{P_{j}}(q^{-C_{j}};q)_{P_{j}}}{(q;q)_{P_{j}}(q^{-C_{j}-D_{j}};q)_{P_{j}}}=1.$

Indeed, for sufficiently small $\varepsilon$, we have
$q^{\overline{p}(A_{n}-k+1)}\leq\varepsilon^{\overline{p}/3}$ (recalling
(A.22), (A.23), and the fact that
$k\leq(2\varepsilon)^{-1}\log\varepsilon^{-1}$), while the remaining factors
in (A.29) grow at most exponentially in $\overline{p}$ (with a uniformly
bounded base). Therefore the sum on the left side of (A.29) is asymptotically
supported on the term $\bm{P}=\bm{e}_{0}$, for which $p=0$, which gives
(A.29).

The remaining factors on the right side of (A.25) are nonnegative, since
$q\in(0,1)$. Let us analyze how their $k$-dependent parts

(A.30) $\displaystyle
F_{\varepsilon}(k;\bm{\overline{C}})=q^{k(k+\mathsf{B}_{n}-A_{n}+\overline{c})}\displaystyle\frac{(q^{A_{n}-k+1};q)_{k}}{(q;q)_{k}(q^{\mathsf{B}_{n}-A_{n}-\overline{d}+1};q)_{k}},$

behave as $\varepsilon$ tends to $0$. Observe for any real numbers
$y,z\in\mathbb{R}$ (with $y\geq 0$) that

(A.31) $\displaystyle\varepsilon\cdot\log(q^{z/\varepsilon};q)_{\lfloor
y/\varepsilon\rfloor}=\varepsilon\displaystyle\sum_{j=0}^{\lfloor
y/\varepsilon\rfloor}\log(1-q^{z/\varepsilon+j})=\varepsilon\displaystyle\sum_{x\in[0,y]\cap\varepsilon\cdot\mathbb{Z}}\log(1-e^{-x-z})\sim\displaystyle\int_{0}^{y}\log(e^{-x-z})dx,$

as $\varepsilon$ tends to $0$. Applying this in (A.30), and setting
$\mathfrak{k}=\varepsilon k$ and
$\bm{\overline{\mathfrak{c}}}=(\mathfrak{c}_{1},\mathfrak{c}_{2},\ldots,\mathfrak{c}_{n-1})=\varepsilon\cdot\overline{\bm{C}}$,
we obtain

(A.32) $\displaystyle\varepsilon\cdot\log
F_{\varepsilon}(k;\overline{\bm{C}})$
$\displaystyle\sim\mathfrak{k}(\mathfrak{a}_{n}-\mathfrak{b}_{n}-\mathfrak{k}-\mathfrak{c}_{[1,n-1]})-\displaystyle\int_{0}^{\mathfrak{k}}\log(1-e^{-x})dx$
(A.33)
$\displaystyle\qquad-\displaystyle\int_{0}^{\mathfrak{k}}\log(1-e^{\mathfrak{a}_{[1,n]}+\mathfrak{b}_{[1,n-1]}-\mathfrak{b}_{n}-\mathfrak{c}_{[1,n-1]}-x})dx,$

where we used the facts that
$\overline{d}=\overline{a}+\overline{b}-\overline{c}$ and that
$\varepsilon\cdot\log(q^{A_{n}-k+1};q)_{k}\sim 0$ (as $\varepsilon A_{n}$
grows faster than any constant as $\varepsilon$ tends to $0$, by (A.22)).
Denoting the right side of (A.32) by $G(\mathfrak{k})$, we find

$\displaystyle
G^{\prime}(\mathfrak{k})=\mathfrak{a}_{n}-\mathfrak{b}_{n}-\mathfrak{c}_{[1,n-1]}-2\mathfrak{k}-\log(1-e^{-\mathfrak{k}})-\log(1-e^{\mathfrak{a}_{[1,n]}+\mathfrak{b}_{[1,n-1]}-\mathfrak{b}_{n}-\mathfrak{c}_{[1,n-1]}-\mathfrak{k}});$
$\displaystyle
G^{\prime\prime}(\mathfrak{k})=-2-\displaystyle\frac{e^{-\mathfrak{k}}}{1-e^{-\mathfrak{k}}}-\displaystyle\frac{e^{\mathfrak{a}_{[1,n]}+\mathfrak{b}_{[1,n-1]}-\mathfrak{b}_{n}-\mathfrak{c}_{[1,n-1]}-\mathfrak{k}}}{1-e^{\mathfrak{a}_{[1,n]}+\mathfrak{b}_{[1,n-1]}-\mathfrak{b}_{n}-\mathfrak{c}_{[1,n-1]}-\mathfrak{k}}}\leq-2<0.$

Due to the negativity of $G^{\prime\prime}$, we deduce that $G$ is maximized
at the solution $\mathfrak{k}$ to the equation $G^{\prime}(\mathfrak{k})=0$,
or equivalently to the unique nonnegative solution of

(A.34)
$\displaystyle(e^{\mathfrak{k}}-1)(e^{\mathfrak{k}}-e^{\mathfrak{a}_{[1,n]}-\mathfrak{b}_{n}+\mathfrak{b}_{[1,n-1]}-\mathfrak{c}_{[1,n-1]}})=e^{\mathfrak{a}_{n}-\mathfrak{b}_{n}-\mathfrak{c}_{[1,n-1]}}.$

Hence, as $\varepsilon$ tends to $0$, the function
$F_{\varepsilon}(k;\overline{\bm{C}})$, and thus
$\Psi_{\varepsilon}(k;\overline{\bm{C}})$ is maximized when $\varepsilon
k=\mathfrak{k}$. By the strict concavity of $G$, these functions decay
exponentially in $\varepsilon^{-1/2}\cdot|\varepsilon k-\mathfrak{k}|$. Hence,
sampling $(k;\overline{\bm{C}})$ under $\Psi$, we find that (A.34) must hold
as $\varepsilon$ tends to $0$.

We next analyze the terms in $\Psi_{\varepsilon}(k;\bm{\overline{C}})$ that
asymptotically depend on a given $C_{i}$. Denoting

$\displaystyle R=\mathsf{B}_{n}-A_{n}-\overline{a}-\overline{b}+k,$

this quantity is given by (using (A.29) with the fact that
$\overline{\bm{D}}=\overline{\bm{A}}+\overline{\bm{B}}-\overline{\bm{C}}$)

$\displaystyle
F_{\varepsilon;i}(k;\overline{\bm{C}})=q^{\varphi(\overline{\bm{A}}+\overline{\bm{B}}-\overline{\bm{C}},\overline{\bm{C}})+\overline{c}(R+\overline{c})}\displaystyle\frac{1}{(q;q)_{R+\overline{c}}}\displaystyle\frac{1}{(q;q)_{C_{i}}(q;q)_{A_{j}+B_{j}-C_{i}}}.$

Using (A.31), and setting

(A.35)
$\displaystyle\mathfrak{r}=\mathfrak{b}_{n}-\mathfrak{a}_{[1,n]}-\mathfrak{b}_{[1,n-1]}+\mathfrak{k}\sim\varepsilon
R;\qquad\mathfrak{p}_{j}=\mathfrak{a}_{j}+\mathfrak{b}_{j},$

for each $j\in\llbracket 1,n-1\rrbracket$, it follows that

(A.36) $\displaystyle\begin{aligned} \varepsilon\cdot\log
F_{\varepsilon;i}(k;\overline{\bm{C}})&\sim\displaystyle\sum_{1\leq j<h\leq
n-1}\mathfrak{c}_{j}\mathfrak{c}_{h}-\displaystyle\sum_{j=1}^{n-1}\mathfrak{p}_{[1,j-1]}\mathfrak{c}_{j}-\mathfrak{c}_{[1,n-1]}\mathfrak{r}-\mathfrak{c}_{[1,n-1]}^{2}-\displaystyle\int_{0}^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}}\log(1-e^{-x})dx\\\
&\qquad-\displaystyle\int_{0}^{\mathfrak{c}_{i}}\log(1-e^{-x})dx-\displaystyle\int_{0}^{\mathfrak{p}_{i}-\mathfrak{c}_{i}}\log(1-e^{-x})dx.\end{aligned}$

Denoting the right side of (A.36) (as a function in $\mathfrak{c}_{i}$) by
$G_{i}(\mathfrak{c}_{i})$, we find that

$\displaystyle G_{i}^{\prime}(\mathfrak{c}_{i})$
$\displaystyle=\displaystyle\sum_{j\neq
i}\mathfrak{c}_{i}+\mathfrak{p}_{[1,i-1]}-\mathfrak{r}-2\mathfrak{c}_{[1,n-1]}-\log(1-e^{-\mathfrak{c}_{i}})-\log(1-e^{-\mathfrak{r}-\mathfrak{c}_{[1,n-1]}})+\log(1-e^{\mathfrak{c}_{i}-\mathfrak{p}_{i}})$
$\displaystyle=\log(1-e^{\mathfrak{c}_{i}-\mathfrak{p}_{i}})-\mathfrak{p}_{[1,i-1]}-\log(e^{\mathfrak{c}_{i}}-1)-\log(e^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}}-1),$

and hence

$\displaystyle
G_{i}^{\prime\prime}(\mathfrak{c}_{i})=-\displaystyle\frac{e^{\mathfrak{c}_{i}-\mathfrak{p}_{i}}}{1-e^{\mathfrak{c}_{i}-\mathfrak{p}_{i}}}-\displaystyle\frac{e^{\mathfrak{c_{i}}}}{e^{\mathfrak{c}_{i}}-1}-\displaystyle\frac{e^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}}}{e^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}}-1}<-2.$

Due to this negativity of $G_{i}^{\prime\prime}$, we deduce that $G_{i}$ is
maximized at the solution $\mathfrak{c}_{i}$ to the equation
$G_{i}^{\prime}(\mathfrak{c}_{i})=0$, or equivalently the solution of

$\displaystyle\displaystyle\frac{\mathfrak{e}^{\mathfrak{p}_{i}}-e^{\mathfrak{c_{i}}}}{e^{\mathfrak{c}_{i}}-1}=e^{\mathfrak{p}_{[1,i]}}\cdot(e^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}}-1).$

Setting

(A.37)
$\displaystyle\eta=\displaystyle\frac{(e^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}}-1)e^{\mathfrak{p}_{[1,n-1]}}}{(e^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}}-1)e^{\mathfrak{p}_{[1,n-1]}}+1},\qquad\text{so
that}\qquad\displaystyle\frac{1-\eta}{\eta}=e^{-\mathfrak{p}_{[1,n-1]}}(e^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}}-1)^{-1},$

it follows that $\mathfrak{c}_{i}$ solves

(A.38)
$\displaystyle\displaystyle\frac{e^{\mathfrak{c}_{i}}-1}{e^{\mathfrak{p}_{i}}-e^{\mathfrak{c}_{i}}}=e^{\mathfrak{p}_{[i+1,n-1]}}\cdot\displaystyle\frac{1-\eta}{\eta},\qquad\text{or
equivalently}\qquad
e^{\mathfrak{c}_{i}}=\displaystyle\frac{\eta+e^{\mathfrak{p}_{[i,n-1]}}(1-\eta)}{\eta+e^{\mathfrak{p}_{[i+1,n-1]}}(1-\eta)}.$

Hence, as $\varepsilon$ tends to $0$, the function
$F_{\varepsilon;i}(k;\overline{\bm{C}})$, and thus
$\Psi_{\varepsilon}(k;\overline{\bm{C}})$, is maximized when $\varepsilon
C_{i}=\mathfrak{c}_{i}$ solves the equation (A.38). By the strict concavity of
$G_{i}$, these functions decay exponentially in
$\varepsilon^{-1/2}\cdot|\varepsilon C_{i}-\mathfrak{c}_{i}|$. Hence, sampling
$(k;\overline{\bm{C}})$ under $\Psi$, we find that (A.38) must hold for all
$i\in\llbracket 1,n-1\rrbracket$, as $\varepsilon$ tends to $0$; recall that
(A.34) also must hold.

Let us now solve these equations. Fixing some $j\in\llbracket
1,n-1\rrbracket$, and taking the product in (A.38) over all $i\in\llbracket
j,n-1\rrbracket$, we find that

(A.39) $\displaystyle
e^{\mathfrak{c}_{[j,n-1]}}=\eta+e^{\mathfrak{p}_{[j,n-1]}}(1-\eta),\qquad\text{for
all $j\in\llbracket 1,n-1\rrbracket$}.$

At $j=1$, (A.39) gives (using (A.37) and (A.35))

(A.40) $\displaystyle
e^{\mathfrak{c}_{[1,n-1]}}=\displaystyle\frac{e^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}+\mathfrak{p}_{[1,n-1]}}}{(e^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}}-1)e^{\mathfrak{p}_{[1,n-1]}}+1},\qquad\text{so}\qquad
e^{\mathfrak{b}_{n}-\mathfrak{a}_{n}+\mathfrak{k}}=e^{\mathfrak{r}+\mathfrak{p}_{[1,n-1]}}=\displaystyle\frac{e^{\mathfrak{p}_{[1,n-1]}}-1}{e^{\mathfrak{c}_{[1,n-1]}}-1}.$

Inserting this into (A.34) yields

$\displaystyle\bigg{(}\displaystyle\frac{e^{\mathfrak{a}_{n}-\mathfrak{b}_{n}}(e^{\mathfrak{p}_{[1,n-1]}}-1)}{e^{\mathfrak{c}_{[1,n-1]}}-1}-1\bigg{)}\cdot
e^{\mathfrak{a}_{n}-\mathfrak{b}_{n}}\bigg{(}\displaystyle\frac{e^{\mathfrak{p}_{[1,n-1]}}-1}{e^{\mathfrak{c}_{[1,n-1]}}-1}-e^{\mathfrak{p}_{[1,n-1]}-\mathfrak{c}_{[1,n-1]}}\bigg{)}=e^{\mathfrak{a}_{n}-\mathfrak{b}_{n}-\mathfrak{c}_{[1,n-1]}},$

and thus

$\displaystyle
e^{\mathfrak{a}_{n}-\mathfrak{b}_{n}}(e^{\mathfrak{p}_{[1,n-1]}}-1)(e^{\mathfrak{p}_{[1,n-1]}-\mathfrak{c}_{[1,n-1]}}-1)=(1-e^{-\mathfrak{c}_{[1,n-1]}})(e^{\mathfrak{p}_{[1,n-1]}}-1).$

Hence,

(A.41) $\displaystyle
e^{\mathfrak{c}_{[1,n-1]}}=\displaystyle\frac{e^{\mathfrak{a}_{n}-\mathfrak{b}_{n}+\mathfrak{p}_{[1,n-1]}}+1}{e^{\mathfrak{a}_{n}-\mathfrak{b}_{n}}+1},\qquad\text{so}\qquad
q^{\mathfrak{k}}=e^{\mathfrak{a}_{n}-\mathfrak{b}_{n}}+1.$

where in the last equality we applied (A.40). Together with (A.35), it follows
that

$\displaystyle
e^{\mathfrak{r}+\mathfrak{c}_{[1,n-1]}}=e^{\mathfrak{b}_{n}-\mathfrak{a}_{n}-\mathfrak{p}_{[1,n-1]}+\mathfrak{k}+\mathfrak{c}_{[1,n-1]}}=e^{\mathfrak{b}_{n}-\mathfrak{a}_{n}-\mathfrak{p}_{[1,n-1]}}+1,$

and hence (A.37) implies

$\displaystyle\eta=\displaystyle\frac{e^{\mathfrak{b}_{n}-\mathfrak{a}_{n}}}{e^{\mathfrak{b}_{n}-\mathfrak{a}_{n}}+1},\qquad\text{so}\qquad
e^{\mathfrak{c}_{[j,n-1]}}=\displaystyle\frac{e^{\mathfrak{a}_{n}-\mathfrak{b}_{n}+\mathfrak{p}_{[j,n-1]}}+1}{e^{\mathfrak{a}_{n}-\mathfrak{b}_{n}}+1},\quad\text{for
any $j\in\llbracket 1,n-1\rrbracket$},$

where in the last equality we used (A.39). This, together with (A.41) yields
(A.28). ∎

###  A.6. The $q\rightarrow 1$ Limiting Weights in Transitionary Rows

Recall that in Section A.4 we analyzed the limits of the
$\mathcal{U}_{\gamma}^{\operatorname{qp}}$ weight (from Definition A.5), under
the conditions (A.22) of its arguments. While (A.22) may hold in several rows
of our eventual fused vertex model converging to the log-gamma polymer, it
will not hold in all of them. In particular, there will be some rows in which
$A_{n}=0$, such as the lowest one in which an arrow of color $n$ enters the
system. Such rows may be viewed as “transitionary” regions where the “dominant
color” changes from $n-1$ to $n$, in the sense that vertices output about
$\varepsilon^{-1}\log\varepsilon^{-1}$ vertical arrows of color $n-1$ in the
previous row, but they output about $\varepsilon^{-1}\log\varepsilon^{-1}$
vertical arrows of color $n$ in this one.

In this section we analyze the limit as $q$ tends to $1$ of the
$\mathcal{U}_{\gamma}^{\operatorname{qp}}$ weight, under this regime
$A_{n}=0$. Throughout this section, we fix a real number $\theta>0$ and
nonnegative real numbers
$(\mathfrak{a}_{1},\mathfrak{a}_{2},\ldots,\mathfrak{a}_{n-1})$ and
$(\mathfrak{b}_{1},\mathfrak{b}_{2},\ldots,\mathfrak{b}_{n})$. Moreover, let
$\varepsilon\in(0,1)$ be a real number; set $q=q_{\varepsilon}$ and
$\gamma=\gamma_{\varepsilon,\theta}$ as in (A.23); let
$\mathsf{B}_{n}=\mathsf{B}_{n}^{\varepsilon}\in\mathbb{Z}_{\geq 0}$ be an
integer; and let $\overline{\bm{A}}=\overline{\bm{A}}^{\varepsilon}$ and
$\overline{\bm{B}}=\overline{\bm{B}}^{\varepsilon}\in\mathbb{Z}_{\geq
0}^{n-1}$ be $(n-1)$-tuples of integers such that for some (uniformly bounded)
integer $\beta\geq 1$ we have

(A.42) $\displaystyle\begin{aligned} &\qquad\qquad\qquad\qquad\quad\varepsilon
A_{j}\sim\mathfrak{a}_{j},\quad\text{and}\quad\varepsilon
B_{j}\sim\mathfrak{b}_{j},\quad\text{for each $j\in\llbracket
1,n-2\rrbracket$};\\\ &\varepsilon
A_{n-1}-\log\varepsilon^{-1}\sim\mathfrak{a}_{n-1};\quad
A_{n}=0;\quad\varepsilon
B_{n-1}-(\beta-1)\log\varepsilon^{-1}\sim\mathfrak{b}_{n-1};\quad\varepsilon\mathsf{B}_{n}-\beta\log\varepsilon^{-1}\sim\mathfrak{b}_{n}.\end{aligned}$

The following lemma describes the limiting law of the $n$-tuple $\bm{C}$
sampled according to the stochastic weight
$\mathcal{U}_{\gamma}^{\operatorname{qp}}$ (from Definition A.5), as
$\varepsilon$ tends to $0$.

######  Lemma A.19.

Let $q=q_{\varepsilon}\in(0,1)$ and
$\gamma=\gamma_{\varepsilon,\theta}\in(0,1)$ be as in (A.23); also let
$\mathsf{B}_{n}\in\mathbb{Z}_{\geq 0}$, $\overline{\bm{B}}\in\mathbb{Z}_{\geq
0}^{n-1}$, and $\bm{A}\in\mathbb{Z}_{\geq 0}^{n}$ be as in (A.42). Sample
$\big{(}\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}\in\mathbb{Z}_{\geq
0}^{n}\times\mathbb{Z}_{\geq 0}^{n-1}\times\mathbb{Z}_{\geq 0}$ with
probability
$\mathcal{U}_{\gamma}^{\operatorname{qp}}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}$,
and denote

$\displaystyle\mathfrak{c}_{n}=\varepsilon
C_{n}-\log\varepsilon^{-1},\quad\text{and}\quad\mathfrak{c}_{j}=\varepsilon
C_{j},\quad\text{for each $j\in\llbracket 1,n-1\rrbracket$}.$

The the joint law of $(\mathfrak{c}_{[j,n]})_{j\in\llbracket 1,n\rrbracket}$
converges to that of
$\big{(}\log(e^{\mathfrak{a}_{[j,n-1]}-\mathfrak{b}_{n}+\mathfrak{b}_{[j,n-1]}}+1)-\log\mathfrak{Y}\big{)}_{j\in\llbracket
1,n-1\rrbracket}$ as $\varepsilon$ tends to $0$, where
$\mathfrak{Y}\in\mathbb{R}_{>0}$ is a $\mathrm{Gamma}(\theta)$ random
variable.

######  Remark A.20.

Set
$(\mathfrak{d}_{1},\mathfrak{d}_{2},\ldots,\mathfrak{d}_{n})=\big{(}\varepsilon
D_{1},\varepsilon D_{2},\ldots,\varepsilon D_{n-2},\varepsilon
D_{n-1}-\beta\log\varepsilon^{-1},\varepsilon\mathsf{D}_{n}-(\beta+1)\log\varepsilon^{-1}\big{)}$
in A.19. Then, the joint limiting law of
$(\mathfrak{d}_{[j,n]})_{j\in\llbracket 1,n\rrbracket}$ is determined by that
of $(\mathfrak{c}_{[j,n]})_{j\in\llbracket 1,n\rrbracket}$ by arrow
conservation.

######  Remark A.21.

Observe in A.19 that the limiting law of
$(\mathfrak{c}_{[j,n]})_{j\in\llbracket 1,n\rrbracket}$ is independent of
$\beta$, which will be useful in the proof of A.23 below.

###### Proof of A.19 (Outline).

The proof of this lemma is similar to that of A.12, and so we only briefly
outline it. Recalling the notation from Definition A.14, A.15 indicates that,
to sample $\bm{C}$, we may first sample $\ell$ and $(k;\overline{\bm{C}})$
under the density functions $\Phi_{\varepsilon}$ and $\Psi_{\varepsilon}$,
respectively, and then set $\bm{C}=(\overline{\bm{C}};k+\ell)$. By A.17, the
law of $\varepsilon^{-1}\cdot e^{-\varepsilon\ell}$ converges to a
$\mathrm{Gamma}(\theta)$ random variable $\mathfrak{Y}$, as $\varepsilon$
tends to $0$. Hence, it suffices to show that $\varepsilon C_{[j,n-1]}$
converges to
$\log(e^{\mathfrak{a}_{[j,n-1]}-\mathfrak{b}_{n}+\mathfrak{b}_{[1,n-1]}}+1)$
as $\varepsilon$ tends to $0$, if
$\overline{\bm{C}}=(C_{1},C_{2},\ldots,C_{n-1})\in\mathbb{Z}_{\geq 0}^{n-1}$
is sampled according to $\Psi_{\varepsilon}$.

By the factor of $(q^{A_{n}-k+1};q)_{k}$ on the right side of (A.25), and the
fact that $A_{n}=0$, we must have $k=0$ almost surely under
$\Psi_{\varepsilon}$. Then, recalling (A.3) and the fact that
$\overline{\bm{D}}=\overline{\bm{A}}+\overline{\bm{B}}-\overline{\bm{C}}$ in
(A.25), it follows that

(A.43) $\displaystyle\begin{aligned}
\Psi_{\varepsilon}(0;\overline{\bm{C}})&=q^{\varphi(\overline{\bm{A}}+\overline{\bm{B}}-\overline{\bm{C}},\overline{\bm{C}})+\overline{c}(\mathsf{B}_{n}-\overline{a}-\overline{b}+\overline{c})}\displaystyle\frac{(q;q)_{\mathsf{B}_{n}-\overline{b}}}{(q;q)_{\mathsf{B}_{n}-\overline{a}-\overline{b}+\overline{c}}}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{A_{j}+B_{j}}}{(q;q)_{C_{j}}(q;q)_{A_{j}+B_{j}-C_{j}}}\\\
&\qquad\times\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}q^{\overline{p}+\varphi(\overline{\bm{P}},\overline{\bm{D}}-\overline{\bm{B}})}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q^{-B_{j}};q)_{P_{j}}(q^{-C_{j}};q)_{P_{j}}}{(q;q)_{P_{j}}(q^{-A_{j}-B_{j}};q)_{P_{j}}}.\end{aligned}$

We first restrict to the case when
$\overline{c}=\mathcal{O}(\varepsilon^{-1})$, by showing that otherwise
$\big{|}\Psi_{\varepsilon}(0;\overline{\bm{C}})\big{|}$ would decay as faster
than $e^{-\varepsilon\overline{c}^{2}/3}$ and therefore become negligible.
Indeed, since $C_{j}\leq A_{j}+B_{j}=\mathcal{O}(\varepsilon^{-1})$ for
$j\in\llbracket 1,n-2\rrbracket$ by (A.42), it suffices to verify that
$\big{|}\Psi_{\varepsilon}(0;\overline{\bm{C}})\big{|}\leq
e^{\mathcal{O}(C_{n-1})-\varepsilon C_{n-1}^{2}/2}$ if $\varepsilon C_{n-1}$
is sufficiently large. To that end, first observe since
$\varepsilon(\mathsf{B}_{n}-\overline{a}-\overline{b})=\mathcal{O}(1)$ by
(A.42), we have
$q^{\overline{c}(\mathsf{B}_{n}-\overline{a}-\overline{b}+\overline{c})}\leq
q^{\overline{c}^{2}-\mathcal{O}(C_{n-1}/\varepsilon)}\leq
e^{\mathcal{O}(C_{n-1})-\varepsilon C_{n-1}^{2}}$. Moreover (since
$\big{|}(q^{-B_{j}};q)_{P_{j}}\big{|}<\big{|}(q^{-A_{j}-B_{j}};q)_{P_{j}}\big{|}$),
the remaining terms on the right side of (A.43) decay at most exponentially in
$\varepsilon^{-1}$, except for the term
$(q^{-C_{n-1}};q)_{P_{n-1}}(q;q)_{P_{n-1}}^{-1}$, which is at most
$e^{\mathcal{O}(C_{n-1})-\varepsilon C_{n-1}^{2}/2}$. Therefore,
$\big{|}\Psi_{\varepsilon}(0;\overline{\bm{C}})\big{|}\leq
e^{\mathcal{O}(C_{n-1})-\varepsilon C_{n-1}^{2}/2}$, which verifies the above-
mentioned decay.

Thus we may assume $\overline{c}=\mathcal{O}(\varepsilon^{-1})$, so
$\varepsilon(D_{n-1}-B_{n-1})=\varepsilon(A_{n-1}-C_{n-1})=\log\varepsilon^{-1}-\mathcal{O}(1)$.
Due to factor of
$q^{\varphi(\overline{\bm{P}},\overline{\bm{D}}-\overline{\bm{B}})}$ in the
sum on the right side of (A.43) (and the fact that the remaining terms in the
sum grow at most exponentially in $|\overline{\bm{P}}|$), one can verify that
this sum is asymptotically supported on the $\bm{P}\in\mathbb{Z}_{\geq
0}^{n-1}$ satisfying $P_{j}=0$ for $j\in\llbracket 1,n-2\rrbracket$. Moreover,
due to the fact that
$\big{|}(q^{-B_{n-1}};q)_{P_{n-1}}(q^{-C_{n-1}};q)_{P_{n-1}}(q;q)_{P_{n-1}}^{-1}(q^{-A_{n-1}-B_{n-1}};q)_{P_{n-1}}^{-1}\big{|}$
decays as $q^{P_{n-1}(A_{n-1}-C_{n-1})}$, the asymptotic support of this sum
also requires $P_{n-1}=0$, and hence $\bm{P}=\bm{e}_{0}$. Thus, to understand
the behavior of $(k;\overline{\bm{C}})=(0;\overline{\bm{C}})$ sampled from the
density function $\Psi_{\varepsilon}$, it suffices to understand the limiting
behavior as $\varepsilon$ tends to $0$ of

$\displaystyle\widetilde{\Psi}_{\varepsilon}(0;\overline{\bm{C}})$
$\displaystyle=q^{\varphi(\overline{\bm{A}}+\overline{\bm{B}}-\overline{\bm{C}},\overline{\bm{C}})+\overline{c}(\mathsf{B}_{n}-\overline{a}-\overline{b}+\overline{c})}\displaystyle\frac{(q;q)_{\mathsf{B}_{n}-\overline{b}}}{(q;q)_{\mathsf{B}_{n}-\overline{a}-\overline{b}+\overline{c}}}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{A_{j}+B_{j}}}{(q;q)_{C_{j}}(q;q)_{A_{j}+B_{j}-C_{j}}}.$

As in the proof of A.18, the $\overline{\bm{C}}$ that contribute to
$\widetilde{\Psi}_{\varepsilon}$ will asymptotically be supported on a single
value of
$\overline{\bm{\mathfrak{c}}}=(\mathfrak{c}_{1},\mathfrak{c}_{2},\ldots,\mathfrak{c}_{n-1})=\varepsilon\cdot\overline{\bm{C}}$.
To understand which value, we take the logarithm of the part of
$\widetilde{\Psi}_{\varepsilon}$ that depends on $\overline{\bm{C}}$, and
multiply by $\varepsilon$ to obtain (using (A.31) and the facts that
$\varepsilon(A_{n-1}+B_{n-1}-C_{n-1})\geq\log\varepsilon^{-1}-\mathcal{O}(1)$)
the function

$\displaystyle G_{\varepsilon}(\overline{\bm{\mathfrak{c}}})$
$\displaystyle=\mathfrak{c}_{[1,n-1]}\cdot(\mathfrak{p}_{[1,n-1]}-\mathfrak{b}_{n}-\mathfrak{c}_{[1,n-1]})-\displaystyle\sum_{j=1}^{n-1}\mathfrak{p}_{[1,j-1]}\mathfrak{c}_{j}-\displaystyle\sum_{j=1}^{n-1}\displaystyle\int_{0}^{\mathfrak{c}_{j}}\log(1-e^{-x})dx$
$\displaystyle\qquad+\displaystyle\sum_{1\leq j<h\leq
n-1}\mathfrak{c}_{j}\mathfrak{c}_{h}-\displaystyle\int_{0}^{\mathfrak{b}_{n}-\mathfrak{p}_{[1,n-1]}+\mathfrak{c}_{[1,n-1]}}\log(1-e^{-x})dx-\displaystyle\sum_{j=1}^{n-2}\displaystyle\int_{0}^{\mathfrak{p}_{j}-\mathfrak{c}_{j}}\log(1-e^{-x})dx,$

where we have denoted $\mathfrak{p}_{j}=\mathfrak{a}_{j}+\mathfrak{b}_{j}$ for
each $j\in\llbracket 1,n-1\rrbracket$. The $\overline{\bm{C}}$ that
contributes to $\widetilde{\Psi}_{\varepsilon}$ is then obtained at the
maximum of $G_{\varepsilon}$, which is the solution of
$\partial_{\mathfrak{c}_{i}}G_{\varepsilon}(\overline{\bm{\mathfrak{c}}})=0$
for each $i\in\llbracket 1,n-1\rrbracket$. These equations are given by

$\displaystyle\log(e^{\mathfrak{p}_{i}}-e^{\mathfrak{c}_{i}})-\log(e^{\mathfrak{c}_{i}}-1)-\mathfrak{p}_{[1,i]}-\log(e^{\mathfrak{b}_{n}+\mathfrak{c}_{[1,n-1]}-\mathfrak{p}_{[1,n-1]}}-1)=0,\qquad\text{for
$i\in\llbracket 1,n-2\rrbracket$};$
$\displaystyle-\log(e^{\mathfrak{c}_{n-1}}-1)-\log(e^{\mathfrak{c}_{[1,n-1]}+\mathfrak{b}_{n}-\mathfrak{p}_{[1.n-1]}}-1)-\mathfrak{p}_{[1,n-2]}=0.$

Denoting

(A.44)
$\displaystyle\eta=\displaystyle\frac{e^{\mathfrak{b}_{n}+\mathfrak{c}_{[1,n-1]}-\mathfrak{p}_{[1,n-1]}}}{e^{\mathfrak{b}_{n}+\mathfrak{c}_{[1,n-1]}-\mathfrak{p}_{[1,n-1]}}-1},\qquad\text{so
that}\qquad\displaystyle\frac{1}{\eta-1}=e^{\mathfrak{b}_{n}+\mathfrak{c}_{[1,n-1]}-\mathfrak{p}_{[1,n-1]}}-1,$

it follows that

$\displaystyle
e^{\mathfrak{c}_{j}}=e^{\mathfrak{p}_{j}}\cdot\displaystyle\frac{e^{\mathfrak{p}_{[1,j-1]}}+\eta-1}{e^{\mathfrak{p}_{[1,j]}}+\eta-1},\quad\text{for
each $j\in\llbracket 1,n-2\rrbracket$};\qquad
e^{\mathfrak{c}_{n-1}}=e^{-\mathfrak{p}_{[1,n-2]}}(\eta-1+e^{\mathfrak{p}_{[1,n-2]}}).$

Thus, for each $j\in\llbracket 1,n-1\rrbracket$, we have

$\displaystyle
e^{\mathfrak{c}_{[j,n-1]}}=e^{-\mathfrak{p}_{[1,j-1]}}\cdot(e^{\mathfrak{p}_{[1,j-1]}}+\eta-1),$

which taken at $j=1$ implies that $\eta=e^{\mathfrak{c}_{[1,n-1]}}$. Together
with (A.44), this yields
$\eta=e^{\mathfrak{c}_{[1,n-1]}}=e^{\mathfrak{p}_{[1,n-1]}-\mathfrak{b}_{n}}+1$,
and therefore

(A.45) $\displaystyle
e^{\mathfrak{c}_{[j,n-1]}}=e^{\mathfrak{p}_{[j,n-1]}-\mathfrak{b}_{n}}+1.$

Hence, if we sample $\overline{\bm{C}}$ according to $\Psi_{\varepsilon}$ and
set
$\overline{\mathfrak{c}}=(\mathfrak{c}_{1},\mathfrak{c}_{2},\ldots,\mathfrak{c}_{n-1})=\varepsilon\cdot\overline{\bm{C}}$,
then (A.45) holds. As mentioned previously, this (with A.15 and A.17) implies
the lemma. ∎

###  A.7. Convergence to the Log-Gamma Polymer

In this section we show convergence of a stochastic vertex model with weights
$\mathcal{U}_{\gamma}^{\operatorname{qp}}$ and
$\mathcal{U}_{1/\gamma}^{\operatorname{b}}$ (from Definition A.5 and
Definition A.10) to the log-gamma polymer. We begin by defining the latter;
throughout this section, we fix positive real parameters
$\bm{\theta}=(\theta_{1},\theta_{2},\ldots)$ and
$\bm{\theta}^{\prime}=(\theta_{1}^{\prime},\theta_{2}^{\prime},\ldots)$.

For each vertex $(i,j)\in\mathbb{Z}_{>0}^{2}$, let $\mathfrak{Y}_{ij}$ denote
a $\mathrm{Gamma}(\theta_{i}+\theta_{j}^{\prime})$ random variable, with all
$(\mathfrak{Y}_{ij})$ mutually independent over $(i,j)\in\mathbb{Z}_{>0}^{2}$.
For any vertices $u,v\in\mathbb{Z}_{>0}^{2}$ such that $v-u\in\mathbb{Z}_{\geq
0}^{2}$, a directed path
$\mathcal{P}=(w_{0},w_{1},\ldots,w_{k})\in\mathbb{Z}_{>0}$ from $u$ to $v$ is
a sequence of vertices starting at $w_{0}=u$, ending at $w_{k}=v$, and
satisfying $w_{i}-w_{i-1}\in\big{\\{}(1,0),(0,1)\big{\\}}$ for each
$i\in\llbracket 1,k\rrbracket$. The _polymer weight_ of such a directed path
is defined to be

(A.46)
$\displaystyle\mathfrak{Z}(\mathcal{P})=\displaystyle\prod_{(i,j)\in\mathcal{P}}\mathfrak{Y}_{ij}^{-1},$

and the _point-to-point partition function_ from $u$ to $v$ is defined to be

(A.47) $\displaystyle\mathfrak{Z}(u\rightarrow
v)=\displaystyle\sum_{\mathcal{P}}\mathfrak{Z}(\mathcal{P}),$

where the sum is over all directed paths
$\mathcal{P}\subset\mathbb{Z}_{>0}^{2}$ from $u$ to $v$.

Now let us describe the stochastic vertex model that converges to the log-
gamma polymer. To that end, let $\varepsilon\in(0,1)$ denote a real number,
and set

$\displaystyle
q=q_{\varepsilon}=e^{-\varepsilon},\quad\text{and}\quad\gamma_{ij}=q^{\theta_{i}+\theta_{j}^{\prime}},\quad\text{for
each $(i,j)\in\mathbb{Z}_{>0}^{2}$}.$

Fix (possibly infinite) integers $\mathrm{K}_{1},\mathrm{K}_{2},\ldots\geq 1$.
At any vertex $(i,j)\in\mathbb{Z}_{>0}^{2}$, we will sample a random colored
(complemented) arrow configuration
$\big{(}\bm{A}(i,j),(\overline{\bm{B}}(i,j),\mathsf{B}_{m}(i,j));\bm{C}(i,j),(\overline{\bm{D}}(i,j),\mathsf{D}_{m}(i,j))\big{)}$,
where $\mathsf{B}_{m}(i,j),\mathsf{D}_{m}(i,j)\in\mathbb{Z}_{\geq 0}$;
$\overline{\bm{B}}(i,j),\overline{\bm{D}}(i,j)\in\mathbb{Z}_{\geq 0}^{m-1}$;
and $\bm{A}(i,j),\bm{C}(i,j)\in\mathbb{Z}_{\geq 0}^{m}$. Here, $m=m(j)\geq 1$
is the (unique) positive integer such that $\mathrm{K}_{[1,m-1]}+1\leq
j\leq\mathrm{K}_{[1,m]}$. These arrow configurations will be consistent, in
the sense that $\bm{A}(i,j+1)=\bm{C}(i,j)$ and
$\big{(}\overline{\bm{B}}(i+1,j),\mathsf{B}_{m}(i+1,j)\big{)}=\big{(}\overline{\bm{D}}(i,j),\mathsf{D}_{m}(i,j)\big{)}$
for each $(i,j)\in\mathbb{Z}_{>0}^{2}$ (where if $j=\mathrm{K}_{[1,m]}$, the
$m$-tuple $\bm{C}(i,j)$ is interpreted as an $(m+1)$-tuple by setting its
$(m+1)$-th entry to $0$).

We will sample these arrow configurations recursively on triangles of the form
$\mathbb{T}_{N}=\big{\\{}(x,y)\in\mathbb{Z}_{>0}^{2}:x+y\leq N\big{\\}}$.
Given some integer $N\geq 1$, suppose that arrow configurations have been
assigned to all vertices in $\mathbb{T}_{N-1}$; we will explain how to sample
them on $\mathbb{T}_{N}$. Fix a vertex on the diagonal
$(i,j)\in\mathbb{D}_{N}=\mathbb{T}_{N}\setminus\mathbb{T}_{N-1}$. First, set
$\bm{A}(N-1,1)=\bm{e}_{0}$, $\overline{\bm{B}}(1,N-1)=\bm{e}_{0}$, and
$\mathsf{B}_{m(N-1)}(1,N-1)=0$. Then, if $i=1$ set
$\overline{\bm{D}}(1,j)=\bm{e}_{0}$, and inductively define
$\bm{C}(1,j)=\bm{A}(1,j)+\mathsf{D}_{m(j)}(1,j)\cdot\bm{e}_{m(j)}$, where
$\mathsf{D}_{m}(1,j)$ is sampled according to the probability (recalling
Definition A.10)

(A.48)
$\displaystyle\mathbb{P}\big{[}\mathsf{D}_{m}(1,j)=k\big{]}=\mathcal{U}_{1/\gamma_{1,j}}^{\operatorname{b};m}(k)$

If $i>1$, then the inputs $\bm{A}(i,j)=\bm{C}(i,j-1)$ and
$\big{(}\overline{\bm{B}}(i,j),\mathsf{B}_{m}(i,j)\big{)}=\big{(}\overline{\bm{D}}(i-1,j),\mathsf{D}_{m}(i-1,j)\big{)}$
at $(i,j)$ have already been assigned (as they arise from arrow configurations
at vertices in $\mathbb{T}_{N-1}$). Sample its outputs
$\big{(}\bm{C}(i,j),(\overline{\bm{D}}(i,j),\mathsf{D}_{m}(i,j))\big{)}$
according to the probability (recalling Definition A.5)

$\displaystyle\mathbb{P}\bigg{[}\Big{(}\bm{C}(i,j),\big{(}\overline{\bm{D}}($
$\displaystyle
i,j),\mathsf{D}_{m}(i,j)\big{)}\Big{)}\bigg{|}\Big{(}\bm{A}(i,j),\big{(}\overline{\bm{B}}(i,j),\mathsf{B}_{m}(i,j)\big{)}\Big{)}\bigg{]}$
$\displaystyle=\mathcal{U}_{\gamma_{ij}}^{\operatorname{b};m}\Big{(}\bm{A}(i,j),\big{(}\overline{\bm{B}}(i,j),\mathsf{B}_{m}(i,j)\big{)};\bm{C}(i,j),(\overline{\bm{D}}(i,j),\mathsf{D}_{m}(i,j))\Big{)}.$

This assigns a random arrow configuration to each vertex in $\mathbb{T}_{N}$;
letting $N$ tend to $\infty$ yields a random ensemble of arrow configurations
on the quadrant $\mathbb{Z}_{>0}^{2}$.

Observe in this way that, by (A.48), arrows of color $n$ begin to enter the
system through its $(\mathrm{K}_{[1,n-1]}+1)$-st row. As in Section A.6, we
will sometimes refer to such rows as _transitionary_ (as they will be where
the most frequent color transitions from $n-1$ to $n$). Moreover, due to the
forms (from Definition A.5) of the weights $\mathcal{U}^{\operatorname{qp}}$,
arrow conservation in the ensemble sampled above appears slightly different
from that in the colored fused path ensembles introduced in Section 6.1. Here,
it depends on the vertex $(i,j)\in\mathbb{Z}_{>0}^{2}$, and in particular on
the index202020This is related to the fact that we are implicitly
“complementing” the arrows of color $m(j)$ on the $j$-th row. $m(j)$
associated with its row; it is given by

(A.49)
$\displaystyle\overline{\bm{A}}(i,j)+\overline{\bm{B}}(i,j)=\overline{\bm{C}}(i,j)+\overline{\bm{D}}(i,j);\qquad\mathsf{B}_{m(j)}(i,j)-A_{m(j)}(i,j)=\mathsf{D}_{m(j)}(i,j)-C_{m(i,j)}(i,j).$

Still, we will explain how the above ensemble arises from the colored
stochastic fused vertex model (from Section 6.1) in A.24 below.

To state convergence of this model to the log-gamma polymer, we need to
prescribe an associated height function. Given an integer $c\geq 1$, define
the color (at least) $c$ height functions
$\mathfrak{h}_{c}^{\rightarrow},\mathfrak{h}_{\geq
c}^{\rightarrow}:\mathbb{Z}_{\geq 0}^{2}\rightarrow\mathbb{Z}$ by for any
$(i,j)\in\mathbb{Z}_{\geq 0}^{2}$ setting (where we let $X_{c}=0$ for any
$\bm{X}\in\mathbb{Z}^{n}$ and $c>n$)

(A.50)
$\displaystyle\mathfrak{h}_{c}^{\rightarrow}(i,j)=-\displaystyle\sum_{k=1}^{j}\mathbbm{1}_{c=m(k)}\cdot\mathsf{D}_{m(k)}(1,k)-\displaystyle\sum_{k=2}^{i}C_{c}(k,j);\qquad\mathfrak{h}_{\geq
c}^{\rightarrow}(i,j)=\displaystyle\sum_{c^{\prime}=c}^{\infty}\mathfrak{h}_{c^{\prime}}(i,j).$

######  Remark A.22.

The height functions are defined above by summing (negative) entries of arrow
configurations along the specific up-right path from $(1,1)$ to $(i,j)$ that
first proceeds north to $(1,j)$ and then east to $(i,j)$. Using arrow
conservation (A.49), one obtains the same result by replacing this by any up-
right path. For instance, by instead considering the path that first proceeds
east to $(i,1)$ and then north to $(i,j)$, we also have

$\displaystyle\mathfrak{h}_{c}^{\rightarrow}(i,j)=\displaystyle\sum_{k=1}^{j}\mathbbm{1}_{c<m(k)}\cdot
D_{c}(i,k)-\displaystyle\sum_{k=1}^{j}\mathbbm{1}_{c=m(k)}\cdot\mathsf{D}_{m(k)}(i,k)-\displaystyle\sum_{k=2}^{i}\mathbbm{1}_{c\leq
m(i)}\cdot C_{c}(k,1).$

Given this notation, we have the following theorem. It indicates the
convergence, as $\varepsilon$ tends to $0$, of the height functions for the
above vertex model to the point-to-point polymer partition functions of the
log-gamma polymer. Observe in this result that the colors are lost in the
polymer degeneration, and they instead track one endpoint of the polymer.

######  Theorem A.23.

Under the above setup, the random variables
$\mathfrak{X}_{c}^{\varepsilon}(i,j)=\varepsilon^{i+j}\cdot
e^{-\mathfrak{h}_{\geq c}^{\rightarrow}(i,j)}$ converge to the log-gamma
polymer partition functions
$\mathfrak{Z}\big{(}(1,\mathrm{K}_{[1,c-1]+1})\rightarrow(i,j)\big{)}$, as
$\varepsilon$ tends to $0$, jointly over all $(c,i,j)$ in compact subsets of
$\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$.

###### Proof.

For each $(i,j)\in\mathbb{Z}_{>0}^{2}$, let
$\mathfrak{Y}_{ij}=\mathfrak{Y}_{i,j}$ denote a
$\mathrm{Gamma}(\theta_{i}+\theta_{j}^{\prime})$ random variable, with all
$(\mathfrak{Y}_{ij})$ mutually independent over $(i,j)\in\mathbb{Z}_{>0}^{2}$.
Further set
$\mathfrak{Z}_{c}(i,j)=\mathfrak{Z}\big{(}(1,\mathrm{K}_{[1,c-1]}+1)\rightarrow(i,j)\big{)}$
for each $c\in\mathbb{Z}_{>0}$ and $(i,j)\in\mathbb{Z}_{>0}^{2}$. Then, it
follows from (A.46) and (A.47) that for $c>0$ the $\mathfrak{Z}_{c}(i,j)$ are
determined by the recursive relations

(A.51) $\displaystyle\begin{aligned}
&\mathfrak{Z}_{c}(i,j)=\mathfrak{Y}_{ij}^{-1}\cdot\big{(}\mathfrak{Z}_{c}(i-1,j)+\mathfrak{Z}_{c}(i,j-1)\big{)},\quad\text{for
$(i,j)\in\mathbb{Z}_{>1}\times\mathbb{Z}_{>\mathrm{K}_{[1,c-1]+1}}$};\\\
&\mathfrak{Z}_{c}(1,j)=\mathfrak{Y}_{1j}^{-1}\cdot\mathfrak{Z}_{c}(1,j-1),\quad\text{for
$j\geq\mathrm{K}_{[1,c-1]}+1$};\\\
&\mathfrak{Z}_{c}(i,\mathrm{K}_{[1,c-1]}+1)=\mathfrak{Y}_{i,\mathrm{K}_{[1,c-1]}+1}^{-1}\cdot\mathfrak{Z}_{c}(i-1,\mathrm{K}_{[1,c-1]}+1),\quad\text{for
$i\geq 2$};\\\
&\mathfrak{Z}_{c}(1,\mathrm{K}_{[1,c-1]}+1)=\mathfrak{Y}_{1,\mathrm{K}_{[1,c-1]}+1}^{-1}.\end{aligned}$

It therefore suffices to show that $\mathfrak{X}_{c}^{\varepsilon}(i,j)$
satisfies the same recursion, as $\varepsilon$ tends to $0$.

To that end let us analyze the behavior, as $\varepsilon$ tends to $0$, of the
dynamics for the arrow configurations
$\big{(}\bm{A}(i,j),(\overline{\bm{B}}(i,j),\mathsf{B}_{m(j)}(i,j));\bm{C}(i,j),(\overline{\bm{D}}(i,j),\mathsf{D}_{m(j)}(i,j))\big{)}$.
First observe from (A.48) and the second part of A.17 that for each $j\geq 1$
we have $B_{c}(1,j)=D_{c}(1,j)=0$ for $c\neq m(j)$, and
$\mathsf{B}_{m(j)}(1,j)=0$ and
$\mathsf{D}_{m(j)}(1,j)=\varepsilon^{-1}\log\varepsilon^{-1}+\mathcal{O}(\varepsilon^{-1})$;
more specifically, we have

(A.52) $\displaystyle\varepsilon\cdot
e^{\varepsilon\mathsf{D}_{m(j)}(1,j)}\qquad\text{converges in distribution
to}\qquad\mathfrak{Y}_{1j}^{-1},\qquad\text{as $\varepsilon$ tends to $0$}.$

We will next understand the behavior of these arrow configurations for
$(i,j)\in\mathbb{Z}_{>1}\times\mathbb{Z}_{>0}$. We will see that their
behavior depends on whether $j$ is a transitionary row, that is, if
$j=\mathrm{K}_{[1,m(j)-1]}+1$. In particular, we will show the following by
induction on the lexicographic pair $(j,i)$; here, we abbreviate $m=m(j)$, and
$\big{(}\mathfrak{z}_{1}(i,j),\mathfrak{z}_{2}(i,j),\ldots,\mathfrak{z}_{m}(i,j)\big{)}$
are some real numbers that are bounded by a random number independent of
$\varepsilon$, for each index
$\mathfrak{z}\in\\{\mathfrak{a},\mathfrak{b},\mathfrak{c},\mathfrak{d}\\}$. If
$j=\mathrm{K}_{[1,m-1]}+1$ indexes a transitionary row, then we will show

(A.53) $\displaystyle\begin{aligned}
&A_{m-1}(i,j)=\varepsilon^{-1}\log\varepsilon^{-1}+\varepsilon^{-1}\mathfrak{a}_{m-1}(i,j);\qquad
B_{m-1}(i,j)=(i-2)\cdot\varepsilon^{-1}\log\varepsilon^{-1}+\varepsilon^{-1}\mathfrak{b}_{m-1}(i,j);\\\
&C_{m-1}(i,j)=\varepsilon^{-1}\mathfrak{c}_{m-1}(i,j);\qquad\qquad\qquad\qquad
D_{m-1}(i,j)=(i-1)\cdot\varepsilon^{-1}\log\varepsilon^{-1}+\varepsilon^{-1}\mathfrak{d}_{m-1}(i,j);\\\
&A_{m}(i,j)=0;\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mathsf{B}_{m}(i,j)=(i-1)\cdot\varepsilon^{-1}\log\varepsilon^{-1}+\varepsilon^{-1}\mathfrak{b}_{m}(i,j);\\\
&C_{m}=\varepsilon^{-1}\log\varepsilon^{-1}+\varepsilon^{-1}\mathfrak{c}_{m}(i,j);\qquad\qquad\qquad\mathsf{D}_{m}(i,j)=i\cdot\varepsilon^{-1}\log\varepsilon^{-1}+\varepsilon^{-1}\mathfrak{d}_{m}(i,j).\end{aligned}$

In this way, the transitionary row “absorbs” most color $m-1$ arrows from the
previous row, and also “emits” arrows of color $m$; due to the arrow
conservation (A.49), this leads to an accumulation of arrows of both colors
$m-1$ and $m$ in this row. If instead $j>\mathrm{K}_{[1,m-1]}+1$ does not
index a transitionary row, then we will show

(A.54) $\displaystyle\begin{aligned}
&Z_{m-1}(i,j)=\varepsilon^{-1}\mathfrak{z}_{m-1}(i,j),\qquad\qquad\qquad\text{if
$Z\in\\{A,B,C,D\\}$};\\\
&Z_{m}(i,j)=\varepsilon^{-1}\log\varepsilon^{-1}+\varepsilon^{-1}\mathfrak{z}_{m}(i,j),\qquad\text{if
$Z\in\\{A,C\\}$};\\\
&\mathsf{Z}_{m}(i,j)=\varepsilon^{-1}\log\varepsilon^{-1}+\varepsilon^{-1}\mathfrak{z}_{m}(i,j),\qquad\text{if
$\mathsf{Z}\in\\{\mathsf{B},\mathsf{D}\\}$}.\end{aligned}$

In both cases for $j$, we will also show for any index $Z\in\\{A,B,C,D\\}$
that

(A.55) $\displaystyle
Z_{c}(i,j)=\varepsilon^{-1}\mathfrak{z}_{c}(i,j),\qquad\text{if $c\leq m-2$}.$

Let us first verify that (A.53) and (A.55) both hold if
$j=\mathrm{K}_{[1,m-1]}+1$ indexes a transitionary row. To that end, observe
by the inductive hypothesis (and the previous discussion on the first column,
if $i=2$) that these statements for the entrance data $A_{k}(i,j)$,
$B_{k}(i,j)$, and $\mathsf{B}_{k}(i,j)$ hold. The fact that it also holds for
the exit data $C_{k}(i,j)$, $D_{k}(i,j)$, and $\mathsf{D}_{m}(i,j)$ then
follows from (the $\beta=i-1$ case of) A.19, together with A.21 and arrow
conservation (A.49) (recall A.20). We next verify that (A.54) and (A.55) hold
if $j>\mathrm{K}_{[1,m-1]}+1$ does not index a transitionary row. In this
case, again the inductive hypothesis (together with the previous discussion on
the first column, if $i=2$, and (A.53) if $j=\mathrm{K}_{[1,m-1]}+2$ indexes a
row directly above a transitionary one) verifies these statements for the
entrance data $A_{k}(i,j)$ and $B_{k}(i,j)$. The fact that it also holds for
the exit data $C_{k}(i,j)$, $D_{k}(i,j)$, and $\mathsf{D}_{m}(i,j)$ then
follows from A.12, together with arrow conservation (A.49). This verifies
(A.53), (A.54), and (A.55); by A.12 and A.19, it also shows that

(A.56) $\displaystyle e^{\mathfrak{c}_{[k,m]}(i,j)}\qquad\text{converges in
distribution
to}\qquad\mathfrak{Y}_{ij}^{-1}\cdot(e^{\mathfrak{a}_{[k,m]}(i,j)-\mathfrak{b}_{m}(i,j)+\mathfrak{b}_{[k,m]}(i,j)}+1),$

as $\varepsilon$ tends to $0$ (where we have set $\mathfrak{a}_{n}(i,j)=0$ if
$j=\mathrm{K}_{[1,m-1]}+1$ indexes a transitionary row, due to the fact that
$A_{m}=0$ in (A.53)).

Now let us analyze the height function for this model. First observe for any
$(i,j)\in\mathbb{Z}_{>1}\times\mathbb{Z}_{>0}$ and $c\in\llbracket
1,m-1\rrbracket$, where $m=m(j)$, that the definition (A.50) of the height
function, consistency of the ensemble, arrow conservation (A.49), and A.22
together imply that

(A.57) $\displaystyle\begin{aligned}
\mathfrak{h}_{c}^{\rightarrow}(i,j-1)=\mathfrak{h}_{c}^{\rightarrow}(i-1&,j-1)-A_{c}(i,j);\qquad\mathfrak{h}_{c}^{\rightarrow}(i-1,j)=\mathfrak{h}_{c}^{\rightarrow}(i-1,j-1)+B_{c}(i,j);\\\
&\mathfrak{h}_{c}^{\rightarrow}(i,j)=\mathfrak{h}_{c}^{\rightarrow}(i-1,j)-C_{c}(i,j),\end{aligned}$

and that

(A.58) $\displaystyle\begin{aligned}
\mathfrak{h}_{m}^{\rightarrow}(i,j-1)=\mathfrak{h}_{m}^{\rightarrow}(i-1&,j-1)-A_{m}(i,j);\qquad\mathfrak{h}_{m}^{\rightarrow}(i-1,j)=\mathfrak{h}_{m}(i-1,j-1)-\mathsf{B}_{m}(i,j);\\\
&\mathfrak{h}_{m}^{\rightarrow}(i,j)=\mathfrak{h}_{m}^{\rightarrow}(i-1,j)-C_{m}(i,j).\end{aligned}$

Hence, for any integer $c\geq 1$, we have

(A.59) $\displaystyle\mathfrak{h}_{\geq
c}^{\rightarrow}(i-1,j)=\mathfrak{h}_{\geq
c}^{\rightarrow}(i,j-1)+A_{[c,m]}(i,j)+B_{[c,m-1]}(i,j)-\mathsf{B}_{m}(i,j).$

Thus, setting $\mathfrak{X}_{c}^{0}(i,j)=\lim_{\varepsilon\rightarrow
0}\mathfrak{X}_{c}^{\varepsilon}(i,j)$, we have if $i>1$ and
$j>\mathrm{K}_{[1,c-1]}+1$ that

(A.60) $\displaystyle\begin{aligned}
\mathfrak{X}_{c}^{0}(i,j)&=\displaystyle\lim_{\varepsilon\rightarrow
0}\varepsilon^{i+j}e^{-\varepsilon\mathfrak{h}_{\geq c}^{\rightarrow}(i,j)}\\\
&=\displaystyle\lim_{\varepsilon\rightarrow
0}\varepsilon^{i+j}e^{-\varepsilon\mathfrak{h}_{\geq
c}^{\rightarrow}(i-1,j)}\cdot e^{\varepsilon C_{[c,m]}(i,j)}\\\
&=\displaystyle\lim_{\varepsilon\rightarrow
0}\varepsilon^{i+j-1}e^{-\varepsilon\mathfrak{h}_{\geq
c}^{\rightarrow}(i-1,j)}\cdot e^{\mathfrak{c}_{[c,m]}(i,j)}\\\
&=\displaystyle\lim_{\varepsilon\rightarrow
0}\varepsilon^{i+j-1}e^{-\varepsilon\mathfrak{h}_{[c,m]}^{\rightarrow}(i-1,j)}\cdot(e^{\mathfrak{a}_{[j,m]}(i,j)+\mathfrak{b}_{[j,m-1]}(i,j)-\mathfrak{b}_{m}(i,j)}+1)\cdot\mathfrak{Y}_{ij}^{-1}\\\
&=\displaystyle\lim_{\varepsilon\rightarrow
0}\varepsilon^{i+j-1}e^{-\varepsilon\mathfrak{h}_{\geq
c}^{\rightarrow}(i-1,j)}\cdot(e^{\varepsilon A_{[c,m]}(i,j)+\varepsilon
B_{[c,m-1]}(i,j)-\varepsilon\mathsf{B}_{m}(i,j)}+1)\cdot\mathfrak{Y}_{ij}^{-1}\\\
&=\displaystyle\lim_{\varepsilon\rightarrow
0}\varepsilon^{i+j-1}(e^{-\varepsilon\mathfrak{h}_{\geq
c}^{\rightarrow}(i,j-1)}+e^{-\varepsilon\mathfrak{h}_{\geq
c}^{\rightarrow}(i-1,j)})\cdot\mathfrak{Y}_{ij}^{-1}=\mathfrak{Y}_{ij}^{-1}\cdot(\mathfrak{X}_{c}^{0}(i,j-1)+\mathfrak{X}_{c}^{0}(i-1,j)\big{)},\end{aligned}$

where the first equality follows from the definition of
$\mathfrak{X}_{c}^{\varepsilon}(i,j)=\varepsilon^{i+j}\cdot
e^{-\mathfrak{h}_{\geq c}(i,j)}$; the second from the third statements in
(A.57) and (A.58); the third from the definition of $\mathfrak{c}_{k}$ in
terms of $C_{k}$ from (A.53), (A.54), and (A.55); the fourth from (A.56); the
fifth from the definitions of $\mathfrak{a}_{k}$ and $\mathfrak{b}_{k}$ in
terms of $A_{k}$ and $B_{k}$, respectively, from (A.53), (A.54), and (A.55);
the sixth from (A.59); and the seventh again from the definition of
$\mathfrak{X}_{c}^{\varepsilon}(i,j)$. If instead $i>1$ and
$j=\mathrm{K}_{[1,c-1]}+1$ (meaning that $c=m$ and $j$ indexes a transitionary
row), then following (A.60) we obtain

$\displaystyle\mathfrak{X}_{c}^{0}(i,j)$
$\displaystyle=\displaystyle\lim_{\varepsilon\rightarrow
0}\varepsilon^{i+j-1}e^{-\varepsilon\mathfrak{h}_{[c,m]}^{\rightarrow}(i-1,j)}\cdot(e^{\mathfrak{a}_{[j,m]}(i,j)+\mathfrak{b}_{[j,m-1]}(i,j)-\mathfrak{b}_{m}(i,j)}+1)\cdot\mathfrak{Y}_{ij}^{-1}$
$\displaystyle=\displaystyle\lim_{\varepsilon\rightarrow
0}\varepsilon^{i+j-1}e^{-\varepsilon\mathfrak{h}_{\geq
c}^{\rightarrow}(i-1,j)}\cdot(\varepsilon e^{\varepsilon
A_{[c,m]}(i,j)+\varepsilon
B_{[c,m-1]}(i,j)-\varepsilon\mathsf{B}_{m}(i,j)}+1)\cdot\mathfrak{Y}_{ij}^{-1}$
$\displaystyle=\displaystyle\lim_{\varepsilon\rightarrow
0}\varepsilon^{i+j-1}e^{-\varepsilon\mathfrak{h}_{\geq
c}^{\rightarrow}(i-1,j)}\cdot\mathfrak{Y}_{ij}^{-1}=\mathfrak{Y}_{ij}^{-1}\cdot\mathfrak{X}_{c}^{0}(i-1,j),$

where the second statement follows from (A.53); similarly, it is quickly
verified that
$\mathfrak{X}_{c}^{0}(1,j)=\mathfrak{Y}_{1j}^{-1}\cdot\mathfrak{X}_{c}^{0}(1,j-1)$
for $j>\mathrm{K}_{[1,c-1]}+1$ and
$\mathfrak{X}_{c}^{0}(1,\mathrm{K}_{[1,c-1]}+1)=\mathfrak{Y}_{1,\mathrm{K}_{[1,c-1]}}^{-1}$,
using (A.52). These, with (A.60) confirm that $\mathfrak{X}_{c}^{0}$ satisfies
the same recursion (A.51) determining the $\mathfrak{Z}_{c}(i,j)$. It follows
that $\mathfrak{X}_{c}^{0}(i,j)=\mathfrak{Z}_{c}(i,j)$, which establishes the
theorem. ∎

######  Remark A.24.

Let us explain how the random ensemble described in this section arises as a
specialization of the colored stochastic fused vertex model introduced in
Section 6.1. To that end, let $\mathfrak{M}\in\mathbb{R}$ be a real number and
$\mathrm{L}\geq 1$ be an integer (that we will analytically continue in). The
parameters $(x_{j},r_{j})$ associated with the $j$-th row of the model, and
those $(y_{i},s_{i})$ associated with the $i$-th column, will be given by

$\displaystyle(x_{j},r_{j})=(q^{1-\theta_{j}^{\prime}},q^{-\mathrm{L}/2}),\quad\text{for
each $j\geq
1$};\qquad(y_{i},s_{i})=(q^{\theta_{i}-\mathfrak{M}},q^{-\mathfrak{M}/2}),\quad\text{for
each $i\geq 1$}.$

Consider a colored stochastic fused vertex model on $\mathbb{Z}_{>0}^{2}$ with
these parameters, and with $\mathrm{L}$ arrows of color $m(j)$ entering
through row212121Since $m=m(j)$ is defined so that
$j\in\llbracket\mathrm{K}_{[1,m-1]}+1,\mathrm{K}_{[1,m]}\rrbracket$, this
means that there are $\mathrm{K}_{1},\mathrm{K}_{2},\ldots$ rows inputting
arrows of colors $1,2,\ldots$ into the model, respectively. $j$, for each
$j\geq 1$. Let $\big{(}\bm{A}(i,j),\bm{B}(i,j);\bm{C}(i,j),\bm{D}(i,j)\big{)}$
denote the colored fused arrow configuration in this model at any vertex
$(i,j)\in\mathbb{Z}_{>0}^{2}$. We next “complement” arrows of color $m(j)$ in
the $j$-th row, by setting
$\bm{B}(i,j)=\big{(}\overline{\bm{B}}(i,j),\mathrm{L}-\mathsf{B}_{m(j)}(i,j)\big{)}$
and
$\bm{D}(i,j)=\big{(}\overline{\bm{D}}(i,j),\mathrm{L}-\mathsf{D}_{m(j)}(i,j)\big{)}$
for each $i\geq 1$. We then track the complemented arrow configuration
$\big{(}\bm{A}(i,j),(\overline{\bm{B}}(i,j),\mathsf{B}_{m(j)}(i,j));\bm{C}(i,j),(\overline{\bm{D}}(i,j),\mathsf{D}_{m(j)}(i,j))\big{)}$
over $(i,j)\in\mathbb{Z}_{>0}^{2}$.

Now analytically continue in $\mathrm{L}$, replacing it with a real number
$\mathfrak{L}\geq 0$. Let $\mathfrak{M}$ tend to $\infty$, and then let
$\mathfrak{L}$ tend to $\infty$. Since
$x_{j}y_{i}^{-1}=q^{1-\theta_{i}-\theta_{j}^{\prime}+\mathfrak{M}}=q^{\mathfrak{M}+1}\gamma_{ij}^{-1}$,
A.3, A.4, and A.7 (with the $q^{\mathfrak{N}}$ in the first two equal to
$q^{\mathfrak{M}+\mathrm{L}}\gamma_{ij}^{-1}$ here, so that
$x_{j}y_{i}^{-1}=q^{\mathfrak{N}-\mathrm{L}+1}$) implies that this procedure
yields at $(i,j)\in\mathbb{Z}_{>0}^{2}$ the
$\mathcal{U}_{\gamma}^{\operatorname{qp};m(j)}$-weights from A.7. Since
$x_{j}(y_{i}s_{i}^{2})^{-1}=q\gamma_{ij}^{-1}$, it also at any vertex $(1,j)$
(in the first column) yields the
$\mathcal{U}_{1/\gamma_{1j}}^{b;m(j)}$-weights from Definition A.10. Hence,
this gives rise to the vertex model described above A.23.

######  Remark A.25.

Let us briefly (and informally) explain the reason behind our parameter
choices when degenerating the colored fused stochastic vertex model to the
log-gamma polymer in A.24. In the uncolored case $n=1$, [27, Proposition 7.26]
describes how to specialize the $U_{q}(\widehat{\mathfrak{sl}}_{2})$
stochastic fused vertex model to the $q$-Hahn PushTASEP introduced in [37].
However, the arrows in that vertex model are reversed relative to here; they
are directed up-left instead of up-right [27, Figure 19]. To remedy this, one
must complement those arrow configurations by tracking how many arrows its
horizontal edges are from being saturated (as in (A.2)); this directs its
paths up-right (and also imposes a change in the spectral parameters involved
in the vertex weights). The vertex weights obtained in this way precisely
coincide with the $n=1$ case of the complemented ones provided in Definition
A.1 (and A.3). The subsequent limits taken in Section A.2 correspond to
degenerating the $q$-Hahn PushTASEP to the $q$-PushTASEP; those taken in
Section A.4 correspond to degenerating the latter to the log-gamma polymer (as
described in [68]).

In the colored case $n>1$, the setup is similar, though it admits a few
differences. The main one is that we only complement arrows of the largest
color when defining the complemented weights in Secion A.1 (a choice that is
essentially forced by the step type boundary conditions we consider). This
leads to a seemingly new presence of “transitionary rows,” where the largest
color in the model changes, and the limiting behavior of the weights in these
rows must be addressed in Section A.4.

##  Appendix B Effective Convergence of the Six-Vertex Model to ASEP

In Section 4 we described colored line ensembles for the stochastic six-vertex
model. It was shown in [18, 1] that under a certain limit degeneration that
the latter, with weights $(b_{1},b_{2})$ as depicted in Figure 23, converges
to the asymmetric simple exclusion process (ASEP), with left jump rate $L$ and
right jump rate $R$. This degeneration takes $(b_{1},b_{2})=(\varepsilon
L,\varepsilon R)$, scales the vertical coordinate by $\varepsilon^{-1}$, lets
$\varepsilon$ tend to $0$, and observes the vertically exiting arrows along
the main diagonal of $\mathbb{Z}^{2}$; this in particular makes time (the
vertical coordinate) continuous and space (the horizontal coordinate)
infinite. A similar limit can be taken on the associated colored line
ensemble, and most of its relevant properties would be preserved, including
its height function match 4.7 (in this case, to the colored ASEP) and its
Gibbs property 4.8. However, the domain of this colored line ensemble would be
infinite, and therefore its boundary conditions would be lost.

When analyzing line ensembles, it is at times (including in the forthcoming
work [6]) useful to keep these boundary conditions in tact. Thus, to
understand the colored ASEP, it can be helpful not to directly study its line
ensemble by letting $\varepsilon$ tend to $0$ in the colored stochastic six-
vertex one, but instead to analyze the latter at $\varepsilon>0$ (where its
boundary conditions are present) and then let $\varepsilon$ tend to $0$
afterwards. This can require effective convergence rates of the stochastic
six-vertex model to the ASEP, which were not proven in [1].

In this section we provide such a convergence result. We state it in Section
B.2 after introducing the colored ASEP in Section B.1; its proof is then given
in Section B.3 and Section B.4. Throughout this section, we fix real
numbers222222The ASEP usually imposes the asymmetry condition $R\neq L$, but
we will not require that here. $R,L\geq 0$. The constants below might
implicitly depend on $R$ and $L$, even when not stated explicitly.

###  B.1. Properties of the Colored ASEP

The _colored ASEP_ is a continuous time Markov process that can be described
as follows. Particles are initially, at time $0$, placed on $\mathbb{Z}$ in
such a way that exactly one particle occupies any site. Assigned to each
particle is a color, which is a nonnegative integer label that informally
measures the “priority” of the particle (those of a larger color are viewed as
having higher priority than those of a smaller one). Denote the color of the
particle at site $x\in\mathbb{Z}$ and time $t\in\mathbb{R}_{\geq 0}$ by
$\eta_{t}(x)\in\mathbb{Z}_{\geq 0}$; further denote the full state of the
process at time $t$ by
$\bm{\eta}_{t}=\big{(}\eta_{t}(x)\big{)}_{x\in\mathbb{Z}}$.

Associated with each site $x\in\mathbb{Z}$ are two independent exponential
clocks, a “left” one of rate $L$ and a “right” one of rate $R$. If the left
clock of $x$ rings at some time $s\in\mathbb{R}_{>0}$, then the particle at
site $x$ switches places with the one at site $x-1$ if
$\eta_{s^{-}}(x)>\eta_{s^{-}}(x-1)$ (and does nothing otherwise). Similarly,
if the right clock of $x$ rings at time $s$, then the particle at site $x$
switches places with the one at site $x+1$ if
$\eta_{s^{-}}(x)>\eta_{s^{-}}(x+1)$ (and does nothing otherwise).

We next recall from [52, 53] a graphical representation for the colored ASEP.
For any $x\in\mathbb{Z}$, let
$\bm{\mathsf{S}}(x)=\big{(}\mathsf{S}_{1}(x),\mathsf{S}_{2}(x),\ldots\big{)}$
and
$\bm{\mathsf{T}}(x)=\big{(}\mathsf{T}_{1}(x),\mathsf{T}_{2}(x),\ldots\big{)}$
denote the ringing times (in increasing order) for the left and right clocks
associated with site $x$, respectively. For each integer $i\geq 1$, draw a
directed arrow on $\mathbb{Z}\times\mathbb{R}_{\geq 0}$ from
$\big{(}x,\mathsf{S}_{i}(x)\big{)}$ to $\big{(}x-1,\mathsf{S}_{i}(x)\big{)}$
and from $\big{(}x,\mathsf{T}_{i}(x)\big{)}$ to
$\big{(}x+1,\mathsf{T}_{i}(x)\big{)}$. The union of these arrows from a
directed graph $\mathcal{G}$, which we call the ASEP time graph; its
horizontal and vertical directions index space and time, respectively. Given
$\mathcal{G}$, the dynamics of the colored ASEP are defined by having each
particle remain at its site $x$ until it reaches a time $t$ at which there is
an arrow in $\mathcal{G}$ connecting $(x,t)$ to some $(y,t)$ (for
$y\in\\{x-1,x+1\\}$). At this time $t$, the particle at site $x$ switches
locations with the one at site $y$ if either $\eta_{t^{-}}(x)>\eta_{t^{-}}(y)$
and the edge is directed from $(x,t)$ to $(y,t)$, or if
$\eta_{t^{-}}(x)<\eta_{t^{-}}(y)$ and the edge is directed from $(y,t)$ to
$(x,t)$; otherwise, the particle at $x$ stays in place.

Before proceeding, let us record the following lemma, which is sometimes known
as a finite speed of discrepancy bound. It states that, if two colored ASEPs
initially agree on a (sufficiently long) interval, then with high probability
their dynamics can be coupled so as to agree on a shorter interval, up until a
given time.

######  Lemma B.1.

There exists a constant $C>1$ such that the following holds. Let $T\geq 0$ and
$K>1$ be real numbers; $U\leq V$ be integers; and
$\bm{\xi}=\big{(}\xi_{t}(x)\big{)}$ and $\bm{\eta}=\big{(}\eta_{t}(x)\big{)}$
denote two colored ASEPs whose initial data satisfy $\xi_{0}(x)=\eta_{0}(x)$
for each $x\in\llbracket U-CKT,V+CKT\rrbracket$. Then it is possible to couple
$\bm{\xi}$ and $\bm{\eta}$ such that, with probability at least
$1-Ce^{-K(T+1)}$, we have $\xi_{t}(x)=\eta_{t}(x)$ for each
$(x,t)\in\llbracket U,V\rrbracket\times[0,T]$.

###### Proof.

Let $C>1$ be a constant to be fixed later. Couple $\bm{\xi}$ and $\bm{\eta}$
under the same time graph $\mathcal{G}$. Then
$\xi_{t_{0}}(x_{0})\neq\eta_{t_{0}}(x_{0})$ holds for some
$(x_{0},t_{0})\in\llbracket U,V\rrbracket\times[0,T]$ only if there exists
some $(x_{1},t_{1})\in\big{(}\mathbb{Z}\setminus\llbracket
U-CKT,V+CKT\rrbracket\big{)}\times[0,T]$, such that a particle at site $x_{1}$
at time $t_{1}$ could (for some trajectories of the remaining particles) enter
the interval $\llbracket U,V\rrbracket$ sometime during $[0,T]$ under
$\mathcal{G}$; this is contained in the union of two events. The first is that
there exists a sequence of times $0\leq r_{1}\leq r_{2}\leq\ldots\leq
r_{\lfloor CKT\rfloor}\leq T$ for which there exists an arrow in $\mathcal{G}$
connecting $\big{(}U-\lfloor CKT\rfloor+i-1,r_{i}\big{)}$ and
$\big{(}U-\lfloor CKT\rfloor+i,r_{i}\big{)}$ for each $i\in\llbracket
1,CKT\rrbracket$; the second is that there exists a sequence of times $0\leq
r_{1}^{\prime}\leq r_{2}^{\prime}\leq\cdots\leq r_{\lfloor
CKT\rfloor}^{\prime}\leq T$ for which there exists an arrow in $\mathcal{G}$
connecting $\big{(}V+\lfloor CKT\rfloor-i+1,r_{i}^{\prime})$ and
$\big{(}V+\lfloor CKT\rfloor-i,r_{i}^{\prime}\big{)}$ for each $i\in\llbracket
1,CKT\rrbracket$.

Now recall that the set of times at which arrows in $\mathcal{G}$ enter or
exit a given column $\\{x\\}\times\mathbb{R}_{\geq 0}$ has the same law as
that of the ringing times of an exponential clock of rate $R+L$. Hence, the
probabilty of each of the above events is bounded by the probability that of a
sum of $T$ exponential random variables with parameter $R+L$ travels a
distance of at least $CKT$. A Chernoff bound implies that the latter is at
most $Ce^{-K(T+1)}$ if $C$ is sufficiently large, establishing the lemma. ∎

###  B.2. Properties of the Colored Stochastic Six-Vertex Model

In this section we state the effective convergence of the stochastic six-
vertex model to the colored ASEP, and also provide several properties of the
former. Let $b_{1},b_{2}\in[0,1]$ denote real numbers; set
$q=b_{1}b_{2}^{-1}$; consider the colored stochastic six-vertex model (as
defined in Section 1.2) on the quadrant $\mathbb{Z}_{>0}^{2}$, with spectral
parameter $z_{i,j}=(1-b_{2})(1-b_{1})^{-1}$ at each
$(i,j)\in\mathbb{Z}_{>0}^{2}$; and assume that the largest color in this
system is at most some integer $n\geq 0$. See Figure 23 for the stochastic
weights of this model. For any integer $(x,y)\in\mathbb{Z}_{>0}^{2}$, let
$\eta_{y}(x)\in\llbracket 0,n\rrbracket$ denote the color of the arrow
vertically exiting $(x,y)$ in this model; also let
$\bm{\eta}_{y}=\big{(}\eta_{y}(x)\big{)}_{x>0}$ denote the full state of the
process at vertical coordinate $y$.

$i$$i$$j$$j$$i$$j$$i$$j$$i$$i$$i$$j$$i$$i$$j$$i$$j$$i$$j$$i$$i$$j$$i$$j$$i$$i\in\llbracket
0,n\rrbracket$$0\leq i<j\leq
n$$(i,i;i,i)$$(j,i;j,i)$$(i,j;i,j)$$(j,i;i,j)$$(i,j;j,i)$$1$$b_{1}$$b_{2}$$1-b_{1}$$1-b_{2}$
Figure 23.  The weights for the colored stochastic six-vertex model in Section
B.2 are depicted above.

Next we state the effective232323We made no effort to optimize the exponent
$\varepsilon^{1/8}$ in the probability in B.2. convergence result; it will be
proven in Section B.3 below. In what follows, given a function
$\varpi:\mathbb{Z}\rightarrow\llbracket 0,n\rrbracket$, we say that the
colored stochastic six-vertex model $\bm{\eta}$ on the quadrant
$\mathbb{Z}_{>0}^{2}$ has boundary data $\varpi$ if the below holds. For each
integer $x\leq 0$, an arrow of color $\varpi(x)$ horizontally enters the
quadrant through $(0,1-x)$ and, for each integer $x>0$, an arrow of color
$\varpi(x)$ vertically enters the quadrant through $(x,0)$. We say that the
boundary data for this model matches the initial condition for a colored ASEP
$\bm{\xi}=\big{(}\xi_{t}(x)\big{)}$ if $\varpi=\xi_{0}$.

######  Proposition B.2.

There exists a constant $C>1$ such that the following holds. Let
$\varepsilon\in(0,1)$ and $T\geq 0$ be real numbers; let $U\leq V$ be
integers; let $\mathcal{T}\subset[0,T]$ denote a finite set of real numbers;
and let $\bm{\xi}=\big{(}\xi_{t}(x)\big{)}$ denote a colored ASEP with at most
$n$ colors. Further let $\bm{\eta}=\big{(}\eta_{y}(x)\big{)}$ denote a colored
stochastic six-vertex model on $\mathbb{Z}_{>0}^{2}$ with parameters
$(b_{1},b_{2})=(\varepsilon L,\varepsilon R)$, whose boundary data matches the
initial data of $\bm{\xi}$. It is possible to couple $\bm{\xi}$ and
$\bm{\eta}$ so that, with probability at least
$1-C(V-U+1)(T+1)^{2}|\mathcal{T}|\varepsilon^{1/8}$, we have

(B.1) $\displaystyle\xi_{t}(x)=\eta_{\lfloor
t/\varepsilon\rfloor}\big{(}x+\lfloor
t\varepsilon^{-1}\rfloor\big{)},\qquad\text{for each $(x,t)\in\llbracket
U,V\rrbracket\times\mathcal{T}$ with $x+\lfloor t\varepsilon^{-1}\rfloor>0$}.$

To prove B.2, it will be useful to introduce notation for the randomness
defining the colored stochastic six-vertex model (this discussion also appears
in a slightly different, though equivalent, form in [3, Section 2.3] and [64,
Section 2.2]). To that end, for any vertex $(x,y)\in\mathbb{Z}_{\geq 0}$, we
associate Bernoulli random variables $\chi_{x,y},\psi_{x,y}\in\\{0,1\\}$ with
$\mathbb{P}[\chi_{x,y}=1]=b_{1}$ and $\mathbb{P}[\psi_{x,y}=1]=b_{2}$. Given
all of these random variables, the dynamics of the colored stochastic six-
vertex model are defined as follows. Fix a vertex
$(x,y)\in\mathbb{Z}_{>0}^{2}$, and suppose that colored six-vertex arrow
configurations have been assigned to each
$(x^{\prime},y^{\prime})\in\mathbb{Z}_{>0}^{2}$ with
$x^{\prime}+y^{\prime}<x+y$; this fixes the colors $a=a(x,y)$ and $b=b(x,y)$
of the arrows vertically and horizontally entering $(x,y)$, respectively. Then
define the colors $c=c(x,y)$ and $d=d(x,y)$ of the arrows vertically and
horizontally exiting $(x,y)$, respectively, according to the below procedure.

  1. (1)

If $a=b$, then set $c=d=a=b$.

  2. (2)

If $a>b$, then set $(c,d)=(a,b)$ if $\chi_{x,y}=1$ and $(c,d)=(b,a)$ if
$\chi_{x,y}=0$.

  3. (3)

If $a<b$, then set $(c,d)=(a,b)$ if $\psi_{x,y}=1$ and $(c,d)=(b,a)$ if
$\psi_{x,y}=0$.

This provides a way of sampling the colored stochastic six-vertex model
defined above. In what follows, for any $x\in\mathbb{Z}$, define the
increasing integer sequences
$\bm{\mathfrak{s}}(x)=\big{(}\mathfrak{s}_{1}(x),\mathfrak{s}_{2}(x),\ldots)$
and $\bm{\mathfrak{t}}(x)=\big{(}t_{1}(x),t_{2}(x),\ldots\big{)}$, such that
$\mathfrak{s}\in\bm{\mathfrak{s}}(x)$ if and only if
$\chi_{x+\mathfrak{s}-1,\mathfrak{s}}>0$, and
$\mathfrak{t}\in\bm{\mathfrak{t}}(x)$ if and only if
$\psi_{x+\mathfrak{t},\mathfrak{t}}>0$ (here, the offsets $x+\mathfrak{s}$ and
$x+\mathfrak{t}$ are introduced to match with (B.1)). Observe that the
$\big{(}\bm{\mathfrak{s}}(x)\big{)}$ and $\big{(}\bm{\mathfrak{t}}(x)\big{)}$
(over all $x\in\mathbb{Z}$) together determine all
$(\chi_{x,y},\psi_{x,y})_{x,y>0}$.

Before proceeding, we record the following lemma that provides a finite speed
of discrepancy bound for the colored stochastic six-vertex model (and is
analogous to B.1).

######  Lemma B.3.

There exists a constant $C>1$ such that the following holds. Let $T\geq 0$,
$K>1$, and $\varepsilon\in(0,1)$ be real numbers; set
$(b_{1},b_{2})=(\varepsilon L,\varepsilon R)$; and let $U\leq V$ be integers.
Further let $\bm{\xi}_{y}=\big{(}\xi_{y}(x)\big{)}_{x>0}$ and
$\bm{\eta}_{y}=\big{(}\eta_{y}(x)\big{)}_{x>0}$ denote two colored stochastic
six-vertex models with parameters $(b_{1},b_{2})$ and boundary data
$\varpi^{\bm{\xi}}:\mathbb{Z}\rightarrow\llbracket 0,n\rrbracket$ and
$\varpi^{\bm{\eta}}:\mathbb{Z}\rightarrow\llbracket 0,n\rrbracket$,
respectively. Assume that $\varpi^{\bm{\xi}}(x)=\varpi^{\bm{\eta}}(x)$ for
each $x\in\llbracket U-CKT,V+CKT\rrbracket$. Then it is possible to couple
$\bm{\xi}$ and $\bm{\eta}$ such that, with probability at least
$1-Ce^{-K(T+1)}$, we have $\xi_{y}(x)=\eta_{y}(x)$ for each
$(x,y)\in\mathbb{Z}_{>0}\times\llbracket 1,T\varepsilon^{-1}\rrbracket$
satisfying $U\leq x-y\leq V$.

###### Proof.

Couple $\bm{\xi}$ and $\bm{\eta}$ under the same processes
$\big{(}\bm{\mathfrak{s}}(x)\big{)}$ and $\big{(}\bm{\mathfrak{t}}(x)\big{)}$,
and define the strip

$\displaystyle\mathcal{R}=\big{\\{}(x,y)\in\mathbb{Z}_{>0}\times\llbracket
1,T\varepsilon^{-1}\rrbracket:U\leq x-y\leq V\big{\\}}.$

Then $\xi_{y_{0}}(x_{0})\neq\eta_{y_{0}}(x_{0})$ holds for some
$(x_{0},y_{0})\in\mathcal{R}$ only if there exists some $(x_{1},y_{1})$ on the
$x$-axis or $y$-axis with $U-CKT\leq x_{1}-y_{1}\leq V+CKT$, such that a
colored path entering the quadrant through $(x_{1},y_{1})$ could (for some
trajectories of the remaining colored paths) enter $\mathcal{R}$ under the
above randomness $\big{(}\bm{\mathfrak{s}}(x)\big{)}$ and
$\big{(}\bm{\mathfrak{t}}(x)\big{)}$. This is contained in the event on which
there exists an integer $k\geq 0$; a sequence of integers
$w_{0},w_{1},\ldots,w_{k}$ with $|w_{i}-w_{i-1}|=1$ for each $i\in\llbracket
1,k\rrbracket$; and a sequence of positive integers
$0\leq\mathfrak{r}_{0}\leq\mathfrak{r}_{1}\leq\ldots\leq\mathfrak{r}_{k}\leq
T\varepsilon^{-1}$ such that
$\mathfrak{r}_{i}\in\bm{\mathfrak{s}}(w_{i})\cup\bm{\mathfrak{t}}(w_{i})$ for
each $i\in\llbracket 1,k\rrbracket$, and one of the following two
possibilities holds. The first is that $w_{0}\leq U-CKT\leq U\leq w_{k}$; the
second is that $w_{0}\geq V+CKT\geq V\geq w_{k}$.

Now, observe that the entry differences of any
$\bm{\mathfrak{s}}(x)\cup\bm{\mathfrak{t}}(x)$ are distributed as independent
geometric random variables with parameter $1-(1-\varepsilon L)(1-\varepsilon
R)\leq\varepsilon(L+R)$. Hence, the probabilty of each of the above
possibilities is bounded by the probability that of a sum of $\lfloor
T\varepsilon^{-1}\rfloor$ geometric random variables with parameter
$\varepsilon(R+L)$ travels a distance of at least $CKT$. A Chernoff bound
implies that the latter is at most $Ce^{-K(T+1)}$ if $C$ is sufficiently
large, establishing the lemma. ∎

###  B.3. Proof of B.2

In this section we establish B.2. Its proof will use the below lemma, which is
shown in Section B.4 below. It indicates that the processes
$\bm{\mathsf{S}}(x)$ and $\bm{\mathsf{T}}(x)$ from Section B.1 can be coupled
to nearly coincide (after scaling) with the $\bm{\mathfrak{s}}(x)$ and
$\bm{\mathfrak{t}}(x)$ from Section B.2, on a long interval, with high
probability. In what follows, we recall the notation from those sections,
assocating the parameters $(L;R)$ with the $(\bm{\mathsf{S}};\bm{\mathsf{T}})$
processes and $(\varepsilon L;\varepsilon R)$ with the
$(\bm{\mathfrak{s}};\bm{\mathfrak{t}})$ ones.

######  Lemma B.4.

There exists a coupling between
$\big{(}\bm{\mathsf{S}}(x);\bm{\mathsf{T}}(x)\big{)}_{x\in\mathbb{Z}}$ and
$\big{(}\bm{\mathfrak{s}}(x);\bm{\mathfrak{t}}(x)\big{)}_{x\in\mathbb{Z}}$,
and a constant $C>1$ such that, for any real numbers $U_{0}\leq V_{0}$,
$\varepsilon\in(0,1)$, and $T\geq 0$, the following statements all hold with
probability at least $1-C(V_{0}-U_{0}+1)(T+1)\varepsilon^{1/4}$.

  1. (1)

For any index pair
$(\mathsf{R},\mathfrak{r})\in\big{\\{}(\mathsf{S},\mathfrak{s}),(\mathsf{T},\mathfrak{t})\big{\\}}$
and each pair $(i,x)\in\mathbb{Z}_{\geq 1}\times\llbracket
U_{0},V_{0}\rrbracket$ such that $\mathsf{R}_{i}(x)\leq T$ or
$\mathfrak{r}_{i}(x)\leq T\varepsilon^{-1}$, we have
$\varepsilon\mathfrak{r}_{i}(x)\leq\mathsf{R}_{i}(x)\leq\varepsilon\mathfrak{r}_{i}(x)+\varepsilon^{1/2}<T$.

  2. (2)

For distinct triples
$(\mathsf{R},i,x),(\mathsf{R}^{\prime},i^{\prime},x^{\prime})\in\\{\mathsf{S},\mathsf{T}\\}\times\mathbb{Z}_{\geq
1}\times\llbracket U_{0},V_{0}\rrbracket$ such that
$\mathsf{R}_{i}(x),\mathsf{R}_{i^{\prime}}^{\prime}(x^{\prime})<T$, we have
$\big{|}\mathsf{R}_{i}(x)-\mathsf{R}_{i^{\prime}}^{\prime}(x^{\prime})\big{|}>2\varepsilon^{1/2}$.

###### Proof of B.2.

We will assume for notational convenience that the set of times
$\mathcal{T}=\\{T\\}$ (from which the proof of the proposition for general
$\mathcal{T}$ quickly follows by a union bound). Set
$K=\lceil\varepsilon^{-1/8}\rceil$, and let $C_{0}>1$ denote the maximum of
the constants $C$ from B.1, B.3, and B.4.

We first use the finite speed of discrepancy bounds (B.1 and B.3) to “cut off”
the initial data for $\bm{\xi}$ and $\bm{\eta}$. To that end, let
$\bm{\xi}^{\prime}=\big{(}\xi_{t}^{\prime}(x)\big{)}$ denote a colored ASEP
with initial data
$\xi_{0}^{\prime}(x)=\xi_{0}(x)\cdot\mathbbm{1}_{x\in[U-C_{0}KT,V+C_{0}KT]}$,
and let $\bm{\eta}^{\prime}=\big{(}\eta_{y}^{\prime}(x)\big{)}$ denote a
colored stochastic six-vertex model with parameters
$(b_{1},b_{2})=(\varepsilon L,\varepsilon R)$, whose boundary data matches the
initial data of $\bm{\xi}^{\prime}$. By B.1 and B.3, we may couple $\bm{\xi}$
and $\bm{\xi}^{\prime}$, and also $\bm{\eta}$ and $\bm{\eta}^{\prime}$, so
that with probability at least $1-2C_{0}e^{-K}\geq 1-16C_{0}\varepsilon$ we
have $\xi_{t}(x)=\xi_{t}^{\prime}(x)$ for each $(x,t)\in\llbracket
U,V\rrbracket\times[0,T]$, and $\eta_{y}(x)=\eta_{y}^{\prime}(x)$ for each
$(x,y)\in\mathbb{Z}_{>0}\times\llbracket 1,T\varepsilon^{-1}\rrbracket$ with
$U\leq x-y\leq V$. Hence, we may replace $\bm{\xi}$ and $\bm{\eta}$ by
$\bm{\xi}^{\prime}$ and $\bm{\eta}^{\prime}$, respectively, so we will assume
in what follows that $\bm{\xi}=\bm{\xi}^{\prime}$ and
$\bm{\eta}=\bm{\eta}^{\prime}$.

Next define the process $\bm{\zeta}=\big{(}\zeta_{y}(x)\big{)}$ from
$\bm{\eta}$ by setting $\zeta_{y}(x)=\eta_{y}(x+y)$ (see (B.1)) for each
$(x,y)\in\mathbb{Z}_{>0}^{2}$. Recall from Section B.1 and Section B.2 that
$\bm{\xi}$ and $\bm{\zeta}$ are determined from the processes
$\big{(}\bm{\mathsf{S}}(x);\bm{\mathsf{T}}(x)\big{)}_{x\in\mathbb{Z}}$ and
$\big{(}\bm{\mathfrak{s}}(x);\bm{\mathfrak{t}}(x)\big{)}_{x\in\mathbb{Z}}$,
respectively; it will be useful to restrict these processes to a bounded
subset of $x\in\mathbb{Z}$. To that end, recall for any $x\in\mathbb{Z}$ that
$\bm{\mathsf{S}}(x)\cup\bm{\mathsf{T}}(x)$ is given by the ringing times of an
exponential clock of rate $L+R$. Hence, the trajectory of the rightmost
particle of nonzero color in $\bm{\xi}$ is stochastically dominated by a
random walk starting at $V+C_{0}KT$, that jumps one space to the right
whenever an exponential clock of rate $L+R$ rings. Therefore, a Chernoff bound
implies (after increasing $C_{0}$ if necessary) that this particle remains
left of $V+2C_{0}KT$ with probability at least $1-C_{0}e^{-K}\geq
1-8C_{0}\varepsilon$. Similarly, the leftmost particle of nonzero color in
$\bm{\xi}$ remains right of $U-2C_{0}KT$ with probability at least
$1-8C_{0}\varepsilon$.

We may apply analogous reasoning to $\bm{\zeta}$. In particular, for any
$x\in\mathbb{Z}$, the differences of the entries in
$\bm{\mathfrak{s}}(x)\cup\bm{\mathfrak{t}}(x)$ are given by mutually
independent geometric random variables of parameter $1-(1-\varepsilon
L)(1-\varepsilon R)\leq\varepsilon(L+R)$. Hence, recalling the diagonal shifts
in the definitions of $\bm{\mathfrak{s}}$ and $\bm{\mathfrak{t}}$ from Section
B.2, the trajectory of the rightmost path (of nonzero color) in $\bm{\zeta}$
is stochastically dominated by a random walk starting at $V+C_{0}KT$ that
jumps to the right according to a geometric random variable of parameter
$\varepsilon(L+R)$. Therefore, a Chernoff bound implies (after increasing
$C_{0}$ if necessary) that this path remains to the left of $V+2C_{0}KT$ with
probability at least $1-C_{0}e^{-K}\geq 1-8C_{0}\varepsilon$. Similarly, the
leftmost path of nonzero color in $\bm{\zeta}$ remains right of $U-2C_{0}KT$
with probability at least $1-8C_{0}\varepsilon$. Together, these facts yield
$\mathbb{P}[\mathscr{E}]\geq 1-32C_{0}\varepsilon$, where
$\mathscr{E}=\mathscr{E}_{1}\cap\mathscr{E}_{2}$ and we have denoted the
events

$\displaystyle\mathscr{E}_{1}=\big{\\{}\xi_{t}(x)=0,\quad\text{for all
$x\notin\llbracket U-2C_{0}KT,V+2C_{0}KT\rrbracket$ and
$t\in[0,T]$}\big{\\}};$
$\displaystyle\mathscr{E}_{2}=\big{\\{}\zeta_{y}(x)=0,\quad\text{for all
$(x,y)\in\llbracket U-2C_{0}KT,V+2C_{0}KT\rrbracket\times\llbracket
0,T\varepsilon^{-1}\rrbracket$ with $x+y>0$}\big{\\}}.$

Now, B.4 gives a coupling between
$\big{(}\bm{\mathsf{S}}(x);\bm{\mathsf{T}}(x)\big{)}$ and
$\big{(}\bm{\mathfrak{s}}(x);\bm{\mathfrak{t}}(x)\big{)}$ such that the
following holds on an event $\mathscr{F}$ with $\mathbb{P}[\mathscr{F}]\geq
1-C_{0}(V-U+4C_{0}KT+1)(T+1)\varepsilon^{1/4}$. First, for any index pair
$(\mathsf{R},\mathfrak{r})\in\big{\\{}(\mathsf{S},\mathfrak{s}),(\mathsf{T},\mathfrak{t})\big{\\}}$
and each pair $(i,x)\in\mathbb{Z}_{\geq 1}\times\llbracket
U-2C_{0}KT,V+2C_{0}KT\rrbracket$ with $\mathsf{R}_{i}(x)\leq T$, we have

(B.2)
$\displaystyle\varepsilon\mathfrak{r}_{i}(x)\leq\mathsf{R}_{i}(x)\leq\varepsilon\mathfrak{r}_{i}(x)+\varepsilon^{1/2}<T.$

Second, for any distinct triples
$(\mathsf{R},i,x),(\mathsf{R}^{\prime},i^{\prime},x^{\prime})\in\\{\mathsf{S},\mathsf{T}\\}\times\mathbb{Z}_{\geq
1}\times\llbracket U-2C_{0}KT,V+2C_{0}KT\rrbracket$ with
$\mathsf{R}_{i}(x),\mathsf{R}_{i^{\prime}}^{\prime}(x^{\prime})\leq T$, we
have

(B.3)
$\displaystyle\big{|}\mathsf{R}_{i}(x)-\mathsf{R}_{i^{\prime}}^{\prime}(x^{\prime})\big{|}>2\varepsilon^{1/2}.$

We claim on $\mathscr{E}\cap\mathscr{F}$ that the ASEP and stochastic six-
vertex configurations coincide, namely,

(B.4) $\displaystyle\xi_{T}(x)=\zeta_{\lfloor
T/\varepsilon\rfloor}(x),\qquad\text{for each $x\in\mathbb{Z}$ with $x+\lfloor
T\varepsilon^{-1}\rfloor>0$}.$

To verify this, first observe on $\mathscr{E}$ that $(\bm{\zeta}_{t})_{t\leq
T}$ and $(\bm{\xi}_{y})_{y\leq T/\varepsilon}$ only depend on
$\big{(}\bm{\mathsf{S}}(x),\bm{T}(x)\big{)}$ and
$\big{(}\bm{\mathfrak{s}}(x),\bm{\mathfrak{t}}(x)\big{)}$ for $x\in\llbracket
U-2C_{0}KT,V+2C_{0}KT\rrbracket$, respectively. Let
$\mathsf{R}_{1}<\mathsf{R}_{2}<\cdots<\mathsf{R}_{m}$ and
$\mathfrak{r}_{1}<\mathfrak{r}_{2}<\cdots<\mathfrak{r}_{m^{\prime}}$ be such
that

$\displaystyle[0,T]\cap\bigcup_{x=\lceil U-2C_{0}KT\rceil}^{\lfloor
V-2C_{0}KT\rfloor}\big{(}\bm{\mathsf{S}}(x)\cup\bm{T}(x)\big{)}=\\{\mathsf{R}_{1},\mathsf{R}_{2},\ldots,\mathsf{R}_{m}\\};$
$\displaystyle\llbracket 1,T\varepsilon^{-1}\rrbracket\cap\bigcup_{x=\lceil
U-2C_{0}KT\rceil}^{\lfloor
V-2C_{0}KT\rfloor}\big{(}\bm{\mathfrak{s}}(x)\cup\bm{\mathfrak{t}}(x)\big{)}=\\{\mathfrak{r}_{1},\mathfrak{r}_{2},\ldots,\mathfrak{r}_{m^{\prime}}\\}.$

For each $i\in\llbracket 1,m\rrbracket$, let $w_{i}\in\llbracket
U-2C_{0}KT,V+2C_{0}KT\rrbracket$ be such that
$\mathsf{R}_{i}\in\bm{\mathsf{S}}(w_{i})\cup\bm{\mathsf{T}}(w_{i})$. Then, by
(B.2) and (B.3), we have that $m^{\prime}=m$ and for each $i\in\llbracket
1,m\rrbracket$ that
$\mathfrak{r}_{i}\in\bm{\mathfrak{s}}(w_{i})\cup\bm{\mathfrak{t}}(w_{i})$ and

(B.5)
$\displaystyle\mathsf{R}_{1}-\varepsilon^{1/2}\leq\mathfrak{r}_{1}\leq\mathsf{R}_{1}<\mathsf{R}_{2}-\varepsilon^{1/2}\leq\mathfrak{r}_{2}\leq\mathsf{R}_{2}<\cdots\leq\mathsf{R}_{m}-\varepsilon^{1/2}\leq\mathfrak{r}_{m}\leq\mathsf{R}_{m}<T-\varepsilon^{1/2}.$

Now, observe if
$y\notin\\{\mathfrak{r}_{1},\mathfrak{r}_{2},\ldots,\mathfrak{r}_{m}\\}$ then
$\eta_{y+1}(x+1)=\eta_{y}(x)$ holds for all $x\in\mathbb{Z}$, as all paths in
$\bm{\eta}$ proceed one step horizontally and one step vertically; thus,
$\bm{\zeta}_{y+1}=\bm{\zeta}_{y}$. Similarly, if
$t\notin\\{\mathsf{R}_{1},\mathsf{R}_{2},\ldots,\mathsf{R}_{m}\\}$ then
$\bm{\xi}_{t}=\bm{\xi}_{t^{-}}$. Hence, to show (B.4), it suffices to show for
each $(i,x)\in\llbracket 1,m\rrbracket\times\mathbb{Z}$ that
$\xi_{\mathsf{R}_{i}}(x)=\zeta_{\lfloor\mathsf{R}_{i}/\varepsilon\rfloor}(x)$.

We do this by induction on $i\in\llbracket 0,m\rrbracket$, where we set
$\mathsf{R}_{0}=0$. It holds it $i=0$, since the boundary data of $\bm{\eta}$
matches that of $\bm{\xi}$. Hence, let us assume it holds for all $i\leq
m_{0}-1$ for some integer $m_{0}\in\llbracket 1,m\rrbracket$ and verify it
holds for $i=m_{0}$. By (B.5) and the inductive hypothesis (with the above
discussion that $\bm{\zeta}_{y+1}=\bm{\zeta}_{y}$ if
$y\notin\\{\mathfrak{r}_{1},\mathfrak{r}_{2},\ldots,\mathfrak{r}_{m}\\}$ and
$\bm{\xi}_{t}=\bm{\xi}_{t^{-}}$ if
$t\notin\\{\mathsf{R}_{1},\mathsf{R}_{2},\ldots,\mathsf{R}_{m}\\}$), we have
$\xi_{\mathsf{R}_{m_{0}}^{-}}(x)=\xi_{\mathsf{R}_{m_{0}-1}}(x)=\zeta_{\lfloor\mathsf{R}_{m_{0}-1}/\varepsilon\rfloor}(x)=\bm{\zeta}_{\lfloor\mathsf{R}_{m_{0}}/\varepsilon-1\rfloor}(x)$
for each $x\in\mathbb{Z}$. Next, if
$\mathfrak{r}_{m_{0}}\in\bm{\mathfrak{s}}(w_{m_{0}})$, then it follows from
the discussion in Section B.2 that
$\eta_{\mathfrak{r}_{m_{0}}}(w_{m_{0}}+\mathfrak{r}_{m_{0}}-1)=\eta_{\mathfrak{r}_{m_{0}}-1}(w_{m_{0}}+\mathfrak{r}_{m_{0}}-1)$,
as then the path of color
$\eta_{\mathfrak{r}_{m_{0}}-1}(w_{m_{0}}+\mathfrak{r}_{m_{0}}-1)$ entering
$(w_{m_{0}}+\mathfrak{r}_{m_{0}}-1,\mathfrak{r}_{m_{0}})$ proceeds one step
vertically and no steps horizontally; thus,

(B.6)
$\displaystyle\zeta_{\lfloor\mathsf{R}_{m_{0}-1}/\varepsilon\rfloor}(w_{m_{0}})=\zeta_{\mathfrak{r}_{m_{0}}-1}(w_{m_{0}})=\zeta_{\mathfrak{r}_{m_{0}}}(w_{m_{0}}-1)=\zeta_{\lfloor\mathsf{R}_{m_{0}}/\varepsilon\rfloor}(w_{m_{0}}-1),$

where in the first and last equality we again used (B.5). By (B.5) and (B.3),
we then also have $\mathsf{R}_{m_{0}}\in\bm{\mathsf{S}}(w_{m_{0}})$, and so
$\zeta_{\lfloor\mathsf{R}_{m_{0}-1}/\varepsilon\rfloor}(w_{m_{0}})=\xi_{\mathsf{R}_{m_{0}-1}}(w_{m_{0}})=\xi_{\mathsf{R}_{m_{0}}^{-}}(w_{m_{0}})=\xi_{\mathsf{R}_{m_{0}}}(w_{m_{0}}-1)$.
This coincides with (B.6); the proof that
$\zeta_{\lfloor\mathsf{R}_{m_{0}}/\varepsilon\rfloor}(x)=\xi_{\mathsf{R}_{m_{0}}}(x)$
for all other $x\in\mathbb{Z}$ is entirely analogous. This verifies the
statement at $i=m_{0}$ in this case.

If instead $\mathfrak{r}_{m_{0}}\in\bm{\mathfrak{t}}(w_{m_{0}})$, then
$\eta_{\mathfrak{r}_{m_{0}}}(w_{m_{0}}+\mathfrak{r}_{m_{0}}+1)=\eta_{\mathfrak{r}_{m_{0}}-1}(w_{m_{0}}+\mathfrak{r}_{m_{0}}-1)$,
as then the path of color
$\eta_{\mathfrak{r}_{m_{0}}-1}(w_{m_{0}}+\mathfrak{r}_{m_{0}}-1)$ at
$(w_{m_{0}}+\mathfrak{r}_{m_{0}}-1,\mathfrak{r}_{m_{0}}-1)$ proceeds two steps
horizontally and one step vertically. So,

(B.7)
$\displaystyle\zeta_{\lfloor\mathsf{R}_{m_{0}-1}/\varepsilon\rfloor}(w_{m_{0}})=\zeta_{\mathfrak{r}_{m_{0}}-1}(w_{m_{0}})=\zeta_{\mathfrak{r}_{m_{0}}}(w_{m_{0}})=\zeta_{\lfloor\mathsf{R}_{m_{0}}/\varepsilon\rfloor}(w_{m_{0}}+1).$

By (B.5) and (B.3), we then also have
$\mathsf{R}_{m_{0}}\in\bm{\mathsf{T}}(w_{m_{0}})$. Hence,
$\zeta_{\lfloor\mathsf{R}_{m_{0}-1}/\varepsilon\rfloor}(w_{m_{0}})=\xi_{\mathsf{R}_{m_{0}-1}}(w_{m_{0}})=\xi_{\mathsf{R}_{m_{0}}^{-}}(w_{m_{0}})=\xi_{\mathsf{R}_{m_{0}}}(w_{m_{0}}+1)$.
This again coincides with (B.7), and the proof that
$\zeta_{\lfloor\mathsf{R}_{m_{0}}/\varepsilon\rfloor}(x)=\xi_{\mathsf{R}_{m_{0}}}(x)$
for all other $x\in\mathbb{Z}$ is entirely analogous. This confirims (B.4).

Thus, (B.1) holds on $\mathscr{E}\cap\mathscr{F}$. Together with the fact that

$\displaystyle\mathbb{P}[\mathscr{E}\cap\mathscr{F}]$ $\displaystyle\geq
1-32C_{0}\varepsilon-C_{0}(V-U+4C_{0}KT+1)(T+1)\varepsilon^{1/4}$
$\displaystyle\geq 1-40C_{0}^{2}(V-U+1)(T+1)^{2}K\varepsilon^{1/4}\geq
1-80C_{0}^{2}(V-U+1)(T+1)^{2}\varepsilon^{1/8},$

this establishes the proposition. ∎

###  B.4. Proof of B.4

In this section we establish B.4 as a quick consequence of the two lemmas
below.

######  Lemma B.5.

For any real number $A>0$, there exists a constant $C=C(A)>1$ such that, if
$\bm{R}=(r_{1},r_{2},\ldots)$ denotes the ringing times in increasing order
for an exponential clock with parameter $A$, then the following two statements
hold.

  1. (1)

For any real numbers $T\geq 0$ and $K\geq 1$, we have $\mathbb{P}[r_{\lfloor
CK(T+1)\rfloor}\geq T]\geq 1-Ce^{-K(T+1)}$.

  2. (2)

For any real numbers $0<\delta\leq 1\leq B$, we have
$\mathbb{P}\big{[}\min_{1\leq i\leq B}(r_{i+1}-r_{i})\geq\delta\big{]}\geq
1-AB\delta$.

###### Proof.

First observe that the event on which $r_{\lfloor C(T+1)\rceil}\geq T$ is that
on which the sum of $\big{\lfloor}CK(T+1)\big{\rfloor}$ independent
exponential random variables is at least $T$. By a Chernoff bound, this is at
least $1-Ce^{-K(T+1)}$ if $C=C(A)>1$ is sufficiently large, which verifies the
first statement of the lemma. Further observe for any integer $i\geq 1$ that,
since $r_{i+1}-r_{i}$ is an exponential random variable of parameter $A$, we
have $\mathbb{P}[r_{i+1}-r_{i}\leq\delta]=1-e^{-A\delta}\leq A\delta$. This,
together with a union bound over $i\in\llbracket 1,B\rrbracket$, implies the
second statement of the lemma. ∎

######  Lemma B.6.

Let $A>0$ and $\delta\in(0,1)$ be real numbers; let
$\mathfrak{g}\in\mathbb{Z}_{\geq 0}$ denote a geometric random variable with
$\mathbb{P}[\mathfrak{g}=k]=A\delta(1-A\delta)^{k}$ for each
$k\in\mathbb{Z}_{\geq 0}$; and let $\mathfrak{e}\in\mathbb{R}_{\geq 0}$ denote
an exponential random variable with $\mathbb{P}[\mathfrak{e}>x]=e^{-Ax}$ for
each $x\in\mathbb{R}_{\geq 0}$. It is possible to couple $\mathfrak{e}$ and
$\mathfrak{g}$ such that

$\displaystyle\mathbb{P}\big{[}\mathfrak{g}=\lfloor\delta^{-1}\mathfrak{e}\rfloor\big{]}\geq
1-12A\delta.$

###### Proof.

We may assume that $\delta<(2A)^{-1}$, for otherwise $1-12A\delta<0$. It
suffices to show that

(B.8) $\displaystyle
e^{-A^{2}S\delta^{2}}\cdot\mathbb{P}\big{[}S\delta\leq\mathfrak{e}<(S+1)\delta\big{]}\leq\mathbb{P}[\mathfrak{g}=S]\leq(1+2A\delta)\cdot\mathbb{P}\big{[}S\delta\leq\mathfrak{e}<(S+1)\delta\big{]},$

for any integer $S\geq 0$, or equivalently that

(B.9) $\displaystyle
e^{-A^{2}S\delta^{2}}\leq\displaystyle\frac{A\delta(1-A\delta)^{S}}{e^{-AS\delta}(1-e^{-A\delta})}\leq
1+2A\delta.$

Indeed, given (B.8), it would follow that

$\displaystyle\displaystyle\sum_{S=0}^{\infty}\Big{|}\mathbb{P}\big{[}S\delta\leq\mathfrak{e}<(S+1)\delta\big{]}-\mathbb{P}[\mathfrak{g}=S]\Big{|}$
$\displaystyle\leq
2\displaystyle\sum_{S=0}^{\infty}(1-e^{-A^{2}S\delta^{2}}+A\delta)\cdot
e^{-AS\delta}(1-e^{-A\delta})$ $\displaystyle\leq
2A^{2}\delta^{2}\displaystyle\sum_{S=0}^{\infty}(AS\delta+1)e^{-AS\delta}$
$\displaystyle\leq\displaystyle\frac{2A^{2}\delta^{2}}{1-e^{-A\delta}}+\displaystyle\frac{2A^{3}\delta^{3}}{(1-e^{-A\delta})^{2}}\leq
12A\delta,$

where in the first equality we used the explicit probability distribution for
$\mathfrak{e}$; in the second we used the facts that
$1-e^{-A^{2}S\delta^{2}}\leq A^{2}S\delta^{2}$ and that $1-e^{-A\delta}\leq
A\delta$; in the third we bounded the sums
$\sum_{i=0}^{\infty}r^{i}=(1-r)^{-1}$ and
$\sum_{i=0}^{\infty}ir^{i}=r(1-r)^{-2}\leq(1-r)^{-2}$ at
$r=e^{-A\delta}\in(0,1)$; and in the fourth we bounded $1-e^{-A\delta}\geq
A\delta/2$ (as $\delta<(2A)^{-1}$). This implies that it is possible to couple
$\mathfrak{e}$ and $\mathfrak{g}$ so that
$\mathfrak{g}\leq\delta^{-1}\mathfrak{e}<\mathfrak{g}+1$, or equivalently that
$\mathfrak{g}=\lfloor\delta^{-1}\mathfrak{e}\rfloor$, with probability at
least $1-12A\delta$.

It therefore remains to confirm (B.9). To that end, since $1-e^{-A\delta}\leq
A\delta$ and $1-e^{-A\delta}\geq A\delta(1-A\delta)\geq
A\delta(1+2A\delta)^{-1}$ (the latter as $\delta<(2A)^{-1}$), observe that

(B.10) $\displaystyle 1\leq\displaystyle\frac{A\delta}{1-e^{-A\delta}}\leq
1+2A\delta.$

We also have since $\log(1-x)\leq-x$ for each $x\geq 0$, since
$\log(1-x)\geq-x(1+x)$ for $x\in(0,1/2)$, and since $\delta<(2A)^{-1}$ that

(B.11) $\displaystyle e^{-A^{2}S\delta^{2}}\leq
e^{AS\delta}(1-A\delta)^{S}\leq 1.$

Combining (B.10) and (B.11) yields (B.9) and thus the lemma. ∎

###### Proof of B.4.

Let $C_{0}$ denote the constant $C$ from B.5; we will assume throughout this
proof that $\varepsilon<C_{0}^{-4}(T+1)^{-4}$, for otherwise
$1-C(V_{0}-U_{0}+1)(T+1)\varepsilon^{1/4}\leq 0$ for any $C>C_{0}$. Let us
first bound the cardinalities of $\bm{\mathsf{S}}(x)\cap[0,T]$ and
$\bm{T}(x)\cap[0,T]$, with high probability. To that end, observe by the first
part of B.5 (with $A\in\\{L,R\\}$ and
$K=C_{0}^{-1}(T+1)^{-1}\varepsilon^{-1/4}\geq 1$) that there exist constants
$c>0$ and $C_{1}>1$ such that
$\mathbb{P}\big{[}\mathsf{S}_{\lfloor\varepsilon^{-1/4}\rfloor}(x)\geq
T\big{]}\geq 1-c^{-1}e^{-c\varepsilon^{-1/4}}\geq 1-C_{1}\varepsilon$, and
similarly
$\mathbb{P}\big{[}\mathsf{T}_{\lfloor\varepsilon^{-1/4}\rfloor}(x)\geq
T\big{]}\geq 1-C_{1}\varepsilon$, both hold for any $x\in\mathbb{Z}$. Together
with a union bound, it follows that $\mathbb{P}[\mathscr{A}_{1}]\geq
1-2C_{1}\varepsilon(V_{0}-U_{0}+1)$, where

(B.12) $\displaystyle\mathscr{A}_{1}=\bigcap_{x=\lceil U_{0}\rceil}^{\lfloor
V_{0}\rfloor}\Big{\\{}\\#\big{\\{}\bm{\mathsf{S}}(x)\cap[0,T]\big{\\}}\leq\varepsilon^{-1/4}\Big{\\}}\cap\Big{\\{}\\#\big{\\{}\bm{\mathsf{T}}(x)\cap[0,T]\big{\\}}\leq\varepsilon^{-1/4}\Big{\\}}.$

We may therefore restrict to the event $\mathscr{A}_{1}$ in what follows.

On $\mathscr{A}_{1}$, there are at most $\varepsilon^{-1/4}$ entries in any
$\bm{\mathsf{S}}(x)\cap[0,T]$ or $\bm{\mathsf{T}}(x)\cap[0,T]$; their
differences are exponential random variables with parameters $L$ and $R$,
respectively. Moreover, the differences between consecutive entries of any
$\bm{\mathfrak{s}}(x)$ and $\bm{\mathfrak{t}}(x)$ are geometric random
variables of parameters $b_{1}=1-\varepsilon L$ and $b_{2}=1-\varepsilon R$,
respectively. Hence, fixing an integer $x\in\llbracket U,V\rrbracket$ and an
index
$(\mathsf{R},\mathfrak{r})\in\big{\\{}(\mathsf{S},\mathfrak{s}),(\mathsf{T},\mathfrak{t})\big{\\}}$,
and applying B.6 to the at most $\varepsilon^{-1/4}$ differences in
$\bm{\mathsf{R}}(x)$, it follows that we may couple
$\bm{\mathsf{R}}(x)\cap[0,T]=\big{(}\mathsf{R}_{1}(x),\mathsf{R}_{2}(x),\ldots,\mathsf{\mathsf{R}}_{m}(x)\big{)}$
with $\bm{\mathfrak{r}}(x)\cap\llbracket
1,T\varepsilon^{-1}\rrbracket=\big{(}\mathfrak{r}_{1}(x),\mathfrak{r}_{2}(x),\ldots,\mathfrak{r}_{m^{\prime}}(x)\big{)}$
so that with probability $1-12(R+L)\varepsilon^{3/4}$ we have $m\leq
m^{\prime}$ and
$\varepsilon\mathfrak{r}_{i}(x)\leq\mathsf{R}_{i}(x)\leq\varepsilon\big{(}\mathfrak{r}_{i}(x)+\varepsilon^{-1/4}\big{)}\leq\varepsilon\mathfrak{r}_{i}(x)+\varepsilon^{1/2}$,
for each $i\in\llbracket 1,m\rrbracket$. After increasing $C_{0}$ if
necessary, we further have that
$\varepsilon\mathfrak{r}_{m^{\prime}}(x)<T-\varepsilon^{1/2}$ with probability
at least $1-C_{0}\varepsilon^{1/2}$ (since the probability that
$\mathfrak{r}_{m^{\prime}}(x)$ lies in any fixed interval at size
$\varepsilon^{-1/2}+1$ is at most
$(R+L)\varepsilon\cdot(\varepsilon^{-1/2}+1)=2(R+L)\varepsilon^{1/2}$), and so
$m=m^{\prime}$ and
$\varepsilon\mathfrak{r}_{i}(x)\leq\mathsf{R}_{i}(x)\leq\varepsilon\mathfrak{r}_{i}(x)+\varepsilon^{1/2}<T$
for all $i\in\llbracket 1,m\rrbracket$. Applying a union bound over
$x\in\llbracket U_{0},V_{0}\rrbracket$ gives the first statement of the lemma.

To establish the second, observe that the law of $\bigcup_{x=\lceil
U_{0}\rceil}^{\lfloor
V_{0}\rfloor}\big{(}\bm{\mathsf{S}}(x)\cup\bm{\mathsf{T}}(x)\big{)}$ coincides
with that of the ringing times of an exponential clock of rate
$(R+L)\cdot\big{(}\lfloor V_{0}\rfloor-\lceil
U_{0}\rceil+1\big{)}\leq(V_{0}-U_{0}+1)(R+L)$. Hence the second part of B.5
implies, for any real number $B\geq 1$, that with probability at least
$1-2(V_{0}-U_{0}+1)(R+L)B\varepsilon^{1/2}$ we have
$\big{|}\mathsf{R}_{i}(x)-\mathsf{R}_{i^{\prime}}^{\prime}(x^{\prime})\big{|}>2\varepsilon^{1/2}$,
for any distinct triples
$(\mathsf{R},i,x),(\mathsf{R}^{\prime},i^{\prime},x^{\prime})\in\\{\mathsf{S},\mathsf{T}\\}\times\llbracket
1,B\rrbracket\times\llbracket U_{0},V_{0}\rrbracket$. Applying this at
$B=\varepsilon^{-1/4}$, using our restriction to $\mathscr{A}_{1}$, and
applying a union bound then yields the second part of the lemma. ∎

## References

  * [1] A. Aggarwal.  Convergence of the stochastic six-vertex model to the ASEP: stochastic six-vertex model and ASEP.  Math. Phys. Anal. Geom., 20(2):Paper No. 3, 20, 2017.
  * [2] A. Aggarwal.  Current fluctuations of the stationary ASEP and six-vertex model.  Duke Math. J., 167(2):269–384, 2018.
  * [3] A. Aggarwal.  Limit shapes and local statistics for the stochastic six-vertex model.  Comm. Math. Phys., 376(1):681–746, 2020.
  * [4] A. Aggarwal, A. Borodin, and A. Bufetov.  Stochasticization of solutions to the Yang-Baxter equation.  Ann. Henri Poincaré, 20(8):2495–2554, 2019.
  * [5] A. Aggarwal, A. Borodin, and M. Wheeler.  Colored fermionic vertex models and symmetric functions.  Comm. Amer. Math. Soc., 3:400–630, 2023.
  * [6] A. Aggarwal, I. Corwin, and M. Hegde.  In preparation.
  * [7] A. Aggarwal and J. Huang.  Strong characterization for the Airy line ensemble.  Preprint, arXiv:2308.11908.
  * [8] A. Aggarwal, M. Nicoletti, and L. Petrov.  Colored interacting particle systems on the ring: Stationary measures from Yang–Baxter equation.  Preprint, arXiv:2309.11865.
  * [9] G. Barraquand, A. Borodin, I. Corwin, and M. Wheeler.  Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process.  Duke Math. J., 167(13):2457–2529, 2018.
  * [10] G. Barraquand, I. Corwin, and S. Das.  KPZ exponents for the half-space log-gamma polymer.  Preprint, arXiv:2310.10019.
  * [11] G. Barraquand, I. Corwin, and E. Dimitrov.  Spatial tightness at the edge of Gibbsian line ensembles.  Comm. Math. Phys., 397(3):1309–1386, 2023.
  * [12] Y. Baryshnikov.  GUEs and queues.  Probab. Theory Related Fields, 119(2):256–274, 2001.
  * [13] V. V. Bazhanov.  Trigonometric solutions of triangle equations and classical Lie algebras.  Phys. Lett. B, 159(4-6):321–324, 1985.
  * [14] A. Borodin.  On a family of symmetric rational functions.  Adv. Math., 306:973–1018, 2017.
  * [15] A. Borodin, A. Bufetov, and M. Wheeler.  Between the stochastic six vertex model and Hall-Littlewood processes.  Preprint, arXiv:1611.09486.
  * [16] A. Borodin and I. Corwin.  Macdonald processes.  Probab. Theory Related Fields, 158(1-2):225–400, 2014.
  * [17] A. Borodin and I. Corwin.  Discrete time $q$-TASEPs.  Int. Math. Res. Not. IMRN, (2):499–537, 2015.
  * [18] A. Borodin, I. Corwin, and V. Gorin.  Stochastic six-vertex model.  Duke Math. J., 165(3):563–624, 2016.
  * [19] A. Borodin and P. L. Ferrari.  Anisotropic growth of random surfaces in $2+1$ dimensions.  Comm. Math. Phys., 325(2):603–684, 2014.
  * [20] A. Borodin, V. Gorin, and M. Wheeler.  Shift-invariance for vertex models and polymers.  Proc. Lond. Math. Soc. (3), 124(2):182–299, 2022.
  * [21] A. Borodin and L. Petrov.  Nearest neighbor Markov dynamics on Macdonald processes.  Adv. Math., 300:71–155, 2016.
  * [22] A. Borodin and L. Petrov.  Higher spin six vertex model and symmetric rational functions.  Selecta Math. (N.S.), 24(2):751–874, 2018.
  * [23] A. Borodin and M. Wheeler.  Observables of coloured stochastic vertex models and their polymer limits.  Probab. Math. Phys., 1(1):205–265, 2020.
  * [24] A. Borodin and M. Wheeler.  Spin $q$-Whittaker polynomials.  Adv. Math., 376:Paper No. 107449, 50, 2021.
  * [25] A. Borodin and M. Wheeler.  Colored stochastic vertex models and their spectral theory.  Astérisque, (437):ix+225, 2022.
  * [26] G. Bosnjak and V. V. Mangazeev.  Construction of $R$-matrices for symmetric tensor representations related to $U_{q}(\widehat{sl_{n}})$.  J. Phys. A, 49(49):495204, 19, 2016.
  * [27] A. Bufetov, M. Mucciconi, and L. Petrov.  Yang-Baxter random fields and stochastic vertex models.  Adv. Math., 388:Paper No. 107865, 94, 2021.
  * [28] A. Bufetov and L. Petrov.  Yang-Baxter field for spin Hall-Littlewood symmetric functions.  Forum Math. Sigma, 7:Paper No. e39, 70, 2019.
  * [29] O. Busani, T. Seppäläinen, and E. Sorensen.  Scaling limit of multi-type invariant measures via the directed landscape.  Preprint, arXiv:2310.09824.
  * [30] O. Busani, T. Seppäläinen, and E. Sorensen.  The stationary horizon and semi-infinite geodesics in the directed landscape.  Preprint, arXiv:2203.13242.
  * [31] L. Cantini.  Algebraic Bethe ansatz for the two species ASEP with different hopping rates.  J. Phys. A, 41(9):095001, 16, 2008.
  * [32] Z. Chen, J. de Gier, I. Hiki, T. Sasamoto, and M. Usui.  Limiting current distribution for a two species asymmetric exclusion process.  Comm. Math. Phys., 395(1):59–142, 2022.
  * [33] I. Corwin and E. Dimitrov.  Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles.  Comm. Math. Phys., 363(2):435–501, 2018.
  * [34] I. Corwin, P. Ghosal, and A. Hammond.  KPZ equation correlations in time.  Ann. Probab., 49(2):832–876, 2021.
  * [35] I. Corwin and A. Hammond.  Brownian Gibbs property for Airy line ensembles.  Invent. Math., 195(2):441–508, 2014.
  * [36] I. Corwin and A. Hammond.  KPZ line ensemble.  Probab. Theory Related Fields, 166(1-2):67–185, 2016.
  * [37] I. Corwin, K. Matveev, and L. Petrov.  The $q$-Hahn PushTASEP.  Int. Math. Res. Not. IMRN, (3):2210–2249, 2021.
  * [38] I. Corwin, N. O’Connell, T. Seppäläinen, and N. Zygouras.  Tropical combinatorics and Whittaker functions.  Duke Math. J., 163(3):513–563, 2014.
  * [39] I. Corwin and L. Petrov.  Stochastic higher spin vertex models on the line.  Comm. Math. Phys., 343(2):651–700, 2016.
  * [40] I. Corwin, T. Seppäläinen, and H. Shen.  The strict-weak lattice polymer.  J. Stat. Phys., 160(4):1027–1053, 2015.
  * [41] D. Dauvergne, J. Ortmann, and B. Virág.  The directed landscape.  Acta Math., 229(2):201–285, 2022.
  * [42] N. Elkies, G. Kuperberg, M. Larsen, and J. Propp.  Alternating-sign matrices and domino tilings. II.  J. Algebraic Combin., 1(3):219–234, 1992.
  * [43] S. Ganguly, M. Hedge, and L. Zhang.  Brownian bridge limit of path measures in the upper tail of KPZ models.  Preprint, arXiv:2311.12009.
  * [44] S. Ganguly and M. Hegde.  Sharp upper tail estimates and limit shapes for the KPZ equation via the tangent method.  Preprint, arXiv:2208.08922.
  * [45] G. Gasper and M. Rahman.  Basic hypergeometric series, volume 96 of Encyclopedia of Mathematics and its Applications.  Cambridge University Press, Cambridge, second edition, 2004.  With a foreword by Richard Askey.
  * [46] L.-H. Gwa and H. Spohn.  Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian.  Phys. Rev. Lett., 68(6):725–728, 1992.
  * [47] J. Haglund, S. Mason, and J. Remmel.  Properties of the nonsymmetric Robinson-Schensted-Knuth algorithm.  J. Algebraic Combin., 38(2):285–327, 2013.
  * [48] A. Hammond.  Modulus of continuity of polymer weight profiles in Brownian last passage percolation.  Ann. Probab., 47(6):3911–3962, 2019.
  * [49] A. Hammond.  A patchwork quilt sewn from Brownian fabric: regularity of polymer weight profiles in Brownian last passage percolation.  Forum Math. Pi, 7:e2, 69, 2019.
  * [50] A. Hammond.  Exponents governing the rarity of disjoint polymers in Brownian last passage percolation.  Proc. Lond. Math. Soc. (3), 120(3):370–433, 2020.
  * [51] A. Hammond.  Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation.  Mem. Amer. Math. Soc., 277(1363):v+133, 2022.
  * [52] T. E. Harris.  Nearest-neighbor Markov interaction processes on multidimensional lattices.  Adv. Math., 9:66–89, 1972.
  * [53] T. E. Harris.  Additive set-valued Markov processes and graphical methods.  Ann. Probability, 6(3):355–378, 1978.
  * [54] J. He.  Boundary current fluctuations for the half space ASEP and six vertex model.  Preprint, arXiv:2303.16335.
  * [55] J. He.  Shift invariance of half space integrable models.  Preprint, arXiv:2205.13029.
  * [56] M. Jimbo.  Quantum $R$ matrix for the generalized Toda system.  Comm. Math. Phys., 102(4):537–547, 1986.
  * [57] W. Jockusch, J. Propp, and P. Shor.  Random domino tilings and the arctic circle theorem.  Preprint, arXiv:9801068, 1998.
  * [58] K. Johansson.  Discrete polynuclear growth and determinantal processes.  Comm. Math. Phys., 242(1-2):277–329, 2003.
  * [59] S. G. G. Johnston and N. O’Connell.  Scaling limits for non-intersecting polymers and Whittaker measures.  J. Stat. Phys., 179(2):354–407, 2020.
  * [60] M. Kardar, G. Parisi, and Y.-C. Zhang.  Dynamic scaling of growing interfaces.  Phys. Rev. Lett., 56(9):889, 1986.
  * [61] A. N. Kirillov.  Introduction to tropical combinatorics.  In Physics and combinatorics, 2000 (Nagoya), pages 82–150. World Sci. Publ., River Edge, NJ, 2001.
  * [62] P. P. Kulish, N. Y. Reshetikhin, and E. K. Sklyanin.  Yang-Baxter equations and representation theory. I.  Lett. Math. Phys., 5(5):393–403, 1981.
  * [63] A. Kuniba, V. V. Mangazeev, S. Maruyama, and M. Okado.  Stochastic $R$ matrix for $U_{q}(A_{n}^{(1)})$.  Nuclear Phys. B, 913:248–277, 2016.
  * [64] Y. Lin.  Classification of stationary distributions for the stochastic vertex models.  Electron. J. Probab., 28:Paper No. 1, 2023.
  * [65] V. V. Mangazeev.  On the Yang-Baxter equation for the six-vertex model.  Nuclear Phys. B, 882:70–96, 2014.
  * [66] S. Mason.  A decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth algorithm.  Sém. Lothar. Combin., 57:Art. B57e, 24, 2006/08.
  * [67] K. Matetski, J. Quastel, and D. Remenik.  The KPZ fixed point.  Acta Math., 227(1):115–203, 2021.
  * [68] K. Matveev and L. Petrov.  $q$-randomized Robinson-Schensted-Knuth correspondences and random polymers.  Ann. Inst. Henri Poincaré D, 4(1):1–123, 2017.
  * [69] M. Mucciconi and L. Petrov.  Spin $q$-Whittaker polynomials and deformed quantum Toda.  Comm. Math. Phys., 389(3):1331–1416, 2022.
  * [70] M. Nica.  Intermediate disorder limits for multi-layer semi-discrete directed polymers.  Electron. J. Probab., 26:Paper No. 62, 50, 2021.
  * [71] M. Noumi and Y. Yamada.  Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions.  In Representation theory of algebraic groups and quantum groups, volume 40 of Adv. Stud. Pure Math., pages 371–442. Math. Soc. Japan, Tokyo, 2004.
  * [72] N. O’Connell.  A path-transformation for random walks and the Robinson-Schensted correspondence.  Trans. Amer. Math. Soc., 355(9):3669–3697, 2003.
  * [73] N. O’Connell.  Directed polymers and the quantum Toda lattice.  Ann. Probab., 40(2):437–458, 2012.
  * [74] N. O’Connell and J. Warren.  A multi-layer extension of the stochastic heat equation.  Comm. Math. Phys., 341(1):1–33, 2016.
  * [75] N. O’Connell and M. Yor.  Brownian analogues of Burke’s theorem.  Stochastic Process. Appl., 96(2):285–304, 2001.
  * [76] A. Okounkov and N. Reshetikhin.  Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram.  J. Amer. Math. Soc., 16(3):581–603, 2003.
  * [77] M. Prähofer and H. Spohn.  Scale invariance of the PNG droplet and the Airy process.  _J. Statist. Phys._ 108(5):1071–1106, 2002.  Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays.
  * [78] J. Quastel and S. Sarkar.  Convergence of exclusion processes and the KPZ equation to the KPZ fixed point.  J. Amer. Math. Soc., 36(1):251–289, 2023.
  * [79] H. Rost.  Non-equilibrium behaviour of a many particle process: Density profile and local equilibria.  Z. Wahrsch. Verw. Gebiete, 58(1):41–53, 1981.
  * [80] T. Sasamoto and M. Wadati.  Exact results for one-dimensional totally asymmetric diffusion models.  J. Phys. A, 31(28):6057–6071, 1998.
  * [81] T. Seppäläinen.  Scaling for a one-dimensional directed polymer with boundary conditions.  Ann. Probab., 40(1):19–73, 2012.
  * [82] J. Warren.  Dyson’s Brownian motions, intertwining and interlacing.  Electron. J. Probab., 12:no. 19, 573–590, 2007.
  * [83] X. Wu.  Brownian regularity for the KPZ line ensemble.  Preprint, arXiv:2106.08052.
  * [84] X. Wu.  The KPZ equation and the directed landscape.  Preprint, arXiv:2301.00547.
  * [85] X. Wu.  Tightness of discrete Gibbsian line ensembles with exponential interaction Hamiltonians.  Ann. Inst. Henri Poincaré Probab. Stat., 59(4):2106–2150, 2023\.
  * [86] Z. Yang.  Stationary measures for higher spin vertex models on a strip.  arXiv:2309.04897.