2.1 Vertex weights

Fix the number of colors $n\ge 1$ . The higher spin $U_q(\widehat {\mathfrak {sl}}_{n+1})$ stochastic vertex weights $L_{s,x}(\mathbf {A},k;\mathbf {B},\ell )$ are indexed by the following data.

• – The quantum parameter $q\in [0,1)$ , which is fixed throughout this section;.

• – The spectral parameter $x$ and the spin parameter $s$ , which may depend on the vertex.

• – The configurations of incoming and outgoing arrows $(\mathbf {A},k;\mathbf {B},\ell )$ , where $k,\ell \in \{ 0,1,\ldots ,n \}$ , and $\mathbf {A}, \mathbf {B}$ are $n$ -tuples $(A_1,\ldots ,A_n ),(B_1,\ldots ,B_n )$ , where $A_i,B_j\in \mathbb {Z}_{\ge 0}$ . Here $k\ge 1$ represents an arrow of color $k$ entering from the left, and $k=0$ corresponds to no arrows entering from the left; similarly, $\ell$ encodes the exiting arrows to the right. Each $A_i$ is the number of arrows of color $i$ entering from the bottom, and $B_j$ is the number of arrows of color $j$ exiting from the top.

The arrow counts $(\mathbf {A},k;\mathbf {B},\ell )$ must satisfy the arrow conservation property (a higher spin analog of the ice rule), with the understanding that all arrows go in the up or right direction. Let $\mathbf {e}_1,\ldots ,\mathbf {e}_n$ be the standard basis in $\mathbb {Z}^n$ , then, the arrow conservation is

\begin{equation*} \mathbf {A}+\mathbf {e}_k\ \mathbf {1}_{k\ge 1}= \mathbf {B}+\mathbf {e}_{\ell }\ \mathbf {1}_{\ell \ge 1}. \end{equation*}

Here and throughout the paper, $\mathbf {1}_{E}$ denotes the indicator of the event or condition $E$ . For $1\le k,\ell \le n$ , define

\begin{equation*} \mathbf {A}^+_{k} := \mathbf {A} + \mathbf {e}_k, \quad{\kern-1pt} \mathbf {A}^-_{k} := \mathbf {A} - \mathbf {e}_k, \quad{\kern-1pt} \mathbf {A}^{+-}_{k\ell } := \mathbf {A} + \mathbf {e}_k - \mathbf {e}_\ell , \quad{\kern-1pt} |\mathbf {A}| := \sum \nolimits _{k=1}^{n} A_k, \quad{\kern-1pt} A_{[k,\ell ]} := \sum \nolimits _{i=k}^{\ell } A_k. \end{equation*}

The values of all the vertex weights $L_{s,x}(\mathbf {A},k;\mathbf {B},\ell )$ are listed in the table in Figure 5. In [Reference Borodin and WheelerBW22a, § 2], they are denoted by $\tilde {L}_{s,x}(\mathbf {A},k;\mathbf {B},\ell )$ , and they differ from [Reference Borodin and WheelerBW22a, (2.2.2)] by the factor $(-s)^{\mathbf {1}_{\ell \gt 0}}$ . However, in this paper, we work with stochastic weights from the beginning, and remove the tilde from the notation.

Figure 5.

Colored stochastic higher spin vertex weights $L_{s,x}$ . Here $1\le k\lt \ell \le n$ , and all other values of $L_{s,x}$ are zero.

Proposition 2.1. The vertex weights $L_{s,x}$ satisfy the sum-to-one property

(2.1) \begin{equation} \sum _{\mathbf {B}\in \mathbb {Z}_{\ge 0}^{n}}\sum _{\ell =0}^{n} L_{s,x}(\mathbf {A},k;\mathbf {B},\ell )=1 \end{equation}

for any fixed $\mathbf {A},k$ . Moreover, if $q\in [0,1)$ ,

(2.2) \begin{equation} -sx\gt 0, \quad -s^2\ge 0,\quad -sx+ s^2\ge 0, \end{equation}

then all the vertex weights $L_{s,x}(\mathbf {A},k;\mathbf {B},\ell )$ are nonnegative.

Conditions (2.2) mean that $s$ and $x$ must be purely imaginary numbers. Observe that the weights $L_{s,x}$ contain $s^2$ and $-sx$ as two independent parameters, and they are more convenient to formulate the nonnegativity.

A natural point of view is to interpret the vertex weights $L_{s, x}(\mathbf {A}, k; \mathbf {B}, \ell )$ as matrix elements of a linear operator $\mathbb {C}^{n+1} \otimes V \rightarrow \mathbb {C}^{n+1} \otimes V$ , where $\mathbb {C}^{n+1}$ has the standard basis $\{| j \rangle \}_{j=0}^n$ , and $V$ has the basis $\{| \mathbf {A} \rangle \}_{\mathbf {A} \in \mathbb {Z}_{\geq 0}^n}$ . For either space, we denote vectors of the dual basis by $\langle \mathbf {v} |$ , and for tensor products of vectors (or dual vectors), we use the notation $|\mathbf {v}, \mathbf {A} \rangle := | \mathbf {v} \rangle \otimes | \mathbf {A} \rangle$ . For the tensor product $\mathbb {C}^{n+1} \otimes V$ we use the basis $\{|j, \mathbf {A} \rangle \}_{j \in \{ 0,\ldots , n\}, \mathbf {A} \in \mathbb {Z}_{\geq 0}^n}$ , and for its dual we use the basis $\{\langle j, \mathbf {A} | \}_{j \in \{ 0,\ldots , n\}, \mathbf {A} \in \mathbb {Z}_{\geq 0}^n}$ . In these bases the operator $\mathscr{L}_{s, x}$ corresponding to $L_{s, x}$ acts as

(2.3) \begin{align} \langle k, \mathbf {A} | \ \mathscr{L}_{s, x} \ |\ell , \mathbf {B} \rangle = L_{s, x}(\mathbf {A}, k; \mathbf {B}, \ell ). \end{align}

In this way, pairs of dual and primal basis vectors of $\mathbb {C}^{n+1} \otimes V$ with nonzero $\mathscr{L}_{s, x}$ matrix entries correspond precisely to the allowed local configurations of paths at a vertex, displayed in Figure 5.

2.2 Yang–Baxter equation

Let us define the following cross vertex weights $R_{z}(i,j;k,\ell )$ , originally due to [Reference JimboJim86] and [Reference BazhanovBaz85]:

(2.4) \begin{align} &\;\ R_z(i,i;i,i) := 1, \quad i \in \{0,1,\ldots ,n \}, \nonumber\\[-6pt]&\nonumber\\&\\[-18pt]& \left . \begin{array}{l@{\quad}l} R_z(j,i;j,i) := \dfrac {q(1-z)}{1-qz}, & R_z(i,j;i,j) := \dfrac {1-z}{1-qz} \\[6pt] R_z(j,i;i,j) := \dfrac {1-q}{1-qz}, & R_z(i,j;j,i) := \dfrac {(1-q)z}{1-qz} \end{array} \right \} \quad i,j \in \{0,1,\ldots ,n \}, \quad i\lt j.\nonumber \end{align}

These weights also satisfy the sum-to-one property: $\displaystyle \sum \nolimits _{k,\ell =0}^{n} R_{z}(i,j;k,\ell )=1$ for any $i,j$ . They are nonnegative if $0\le z\le 1$ .

The vertex weights $R_{z}(i,j;k,\ell )$ can also be regarded as matrix elements of an operator $\mathscr{R}_z$ acting in $\mathbb {C}^{n+1} \otimes \mathbb {C}^{n+1}$ , namely, $\langle j,i | \mathscr{R}_z | \ell ,k \rangle = R_{z}(i,j;k,\ell )$ .

One can check that $R_z$ is the spin- $\frac {1}{2}$ reduction of $L_{s,x}$ (as in Remark 2.2):

(2.5) \begin{equation} R_{z}(i,j;k,\ell )= L_{q^{-1/2},\ z^{-1}q^{-1/2}}(\mathbf {e}_i \mathbf {1}_{i\ge 1},j;\mathbf {e}_k \mathbf {1}_{k\ge 1},\ell ),\quad i,j,k,\ell \in \{ 0,1,\ldots ,n \}. \end{equation}

In the right-hand side, if $i=0$ , then $\mathbf {e}_i \mathbf {1}_{i\ge 1}=( 0,\ldots ,0 )$ ( $n$ zeroes), which corresponds to no arrows at an edge.

The weights $L_{s,x}$ and $R_{z}$ satisfy the following Yang–Baxter equation.

Proposition 2.3 [Reference Borodin and WheelerBW22a, (2.3.1) and Corollary B.4.3]. For any fixed $i_1,i_2,j_1,j_2\in \{ 0,1,\ldots ,n \}$ and $\mathbf {A}, \mathbf {B}\in \mathbb {Z}_{\ge 0}^{n}$ , we have

(2.6) \begin{equation} \begin{split} &\sum _{\mathbf {K}\in \mathbb {Z}_{\ge 0}^{n}} \ \sum _{k_1,k_2=0}^{n} L_{s,y}(\mathbf {A},i_2;\mathbf {K},k_2)\ L_{s,x}(\mathbf {K},i_1;\mathbf {B},k_1)\ R_{y/x}(k_2,k_1;j_2,j_1) \\[-3pt]&\quad= \sum _{\mathbf {K}\in \mathbb {Z}_{\ge 0}^{n}} \ \sum _{k_1,k_2=0}^{n} R_{y/x}(i_2,i_1;k_2,k_1)\ L_{s,y}(\mathbf {K},k_2;\mathbf {B},j_2)\ L_{s,x}(\mathbf {A},k_1;\mathbf {K},j_1). \end{split} \end{equation}

See Figure 6 for an illustration. Note that the summations in both sides of (2.6) are actually finite.

Figure 6.

An illustration of the Yang–Baxter equation (2.6) in Proposition 2.3, where the sums in both sides are taken over all $k_1,k_2\in \{ 0,1,\ldots ,n \}$ and $\mathbf {K}\in \mathbb {Z}_{\ge 0}^n$ .

2.3 Fused weights

The vertex weights $L_{s,x}(\mathbf {A},k;\mathbf {B},\ell )$ are higher spin (that is, they allow multiple arrows per edge) in the vertical direction. For $s=q^{-1/2}$ , they reduce to the fundamental $R$ -matrix $R_z(a,k;b,\ell )$ , where $a,k,b,\ell \in \{ 0,1,\ldots ,n \}$ ; see (2.5). The inverse procedure for constructing $L_{s,x}$ from $R_z$ is called fusion and dates back to [Reference Kulish, Reshetikhin and SklyaninKRS81]. In the uncolored case, it was put into probabilistic context in [CP16, Reference Borodin and PetrovBP18a]. The colored fusion is described in, e.g., [Reference Borodin and WheelerBW22a, Appendix B]. The formula for the fully fused stochastic colored vertex weights $W_{x,\mathsf {L},\mathsf {M}}(\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})$ coming from $U_q(\widehat {\mathfrak {sl}}_{n+1})$ is obtained in [Reference Bosnjak and MangazeevBM16]; see also [Reference Borodin and WheelerBW22a, Appendix C]. Here we recall the stochastic vertex weights fused in both the horizontal and vertical directions.

We need the standard $q$ -Pochhammer symbols notation:

\begin{equation*} (a ; q)_k:= (1-a)(1-a q), \ldots, (1-a q^{k-1}), \quad k \in \mathbb {Z}_{\geq 0}, \end{equation*}

and $(z ; q)_{\infty }:= \prod _{i=0}^{\infty }(1-z q^i)$ is a convergent infinite product because $q\in [0,1)$ .

For $\mathbf {A},\mathbf {B}\in \mathbb {Z}_{\ge 0}^{n}$ such that $A_i\le B_i$ for all $i$ , define

(2.7) \begin{equation} \Phi (\mathbf {A}, \mathbf {B} ; x, y):= \frac {(x ; q)_{|\mathbf {A}|}(y / x ; q)_{|\mathbf {B}-\mathbf {A}|}}{(y ; q)_{|\mathbf {B}|}}\ (y / x)^{|\mathbf {A}|} \ q^{\sum _{1\le i\lt j\le n}(B_i-A_i) A_j} \prod _{i=1}^n \frac {(q;q)_{B_i}}{(q;q)_{A_i}(q;q)_{B_i-A_i}}. \end{equation}

For any fixed $\mathbf {B}\in \mathbb {Z}_{\ge 0}^{n}$ , we have

(2.8) \begin{equation} \sum _{\mathbf {A}\in \mathbb {Z}_{\ge 0}^n \colon A_i\le B_i\ \text {for all}\ i} \Phi (\mathbf {A}, \mathbf {B} ; x, y)=1. \end{equation}

With this notation, if $\mathbf {A},\mathbf {B},\mathbf {C},\mathbf {D} \in \mathbb {Z}_{\ge 0}^{n}$ , we define the vertex weights

(2.9) \begin{equation} \begin{split} W_{x,\mathsf {L},\mathsf {M}}(\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})& := \mathbf {1}_{\mathbf {A}+\mathbf {B}=\mathbf {C}+\mathbf {D}} \cdot x^{|\mathbf {D}-\mathbf {B}|} \ (q^{\mathsf {L}})^{|\mathbf {A}|} \ (q^{\mathsf {M}})^{-|\mathbf {D}|} \\ &\quad \times \sum _{\mathbf {P}} \Phi (\mathbf {C}-\mathbf {P}, \mathbf {C}+\mathbf {D}-\mathbf {P} ; q^{\mathsf {L}-\mathsf {M}} x, q^{-\mathsf {M}} x ) \Phi ( \mathbf {P}, \mathbf {B} ; q^{-\mathsf {L}} / x, q^{-\mathsf {L}} ). \end{split} \end{equation}

The sum in (2.9) is finite and is taken over all $\mathbf {P}\in \mathbb {Z}_{\ge 0}^{n}$ such that $0\le P_i\le \min ( B_i,C_i )$ for all $i$ .

The weights $W_{x,\mathsf {L},\mathsf {M}}$ (2.9) satisfy a version of the Yang–Baxter equation, see formulas (C.1.2)–(C.1.3) in [Reference Borodin and WheelerBW22a]. Graphically, this equation is similar to the one illustrated in Figure 6, but the weights $L_{s,x},L_{s,y}$ , and $R_{y/x}$ must be replaced with, respectively, $W_{\frac {x}{z},\mathsf {L},\mathsf {N}}$ , $W_{\frac {y}{z},\mathsf {M},\mathsf {N}}$ , and $W_{\frac {x}{y},\mathsf {L},\mathsf {M}}$ . The summation in the Yang–Baxter equation for the $W$ goes over triples of elements from $\mathbb {Z}_{\ge 0}^n$ .

The weights (2.9) sum to one [Reference Borodin and WheelerBW22a, (C.1.5)],

(2.10) \begin{equation} \sum \nolimits _ {\mathbf {C},\mathbf {D}\in \mathbb {Z}_{\ge 0}^n} W_{x,\mathsf {L},\mathsf {M}}( \mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D} )=1, \quad \mathbf {A},\mathbf {B}\in \mathbb {Z}_{\ge 0}^n, \end{equation}

and reduce to the weights $L_{s,x}$ (spin- $\frac {1}{2}$ in the horizontal direction and higher spin in the vertical direction) and $R_z$ (spin- $\frac {1}{2}$ in both directions) from §§ 2.1 and 2.2 as follows [Reference Borodin and WheelerBW22a, Proposition C.1.4 and formula (C.2.2)]:

(2.11) \begin{align} R_{z}(i,j;k,\ell ) &= W_{z^{-1},1,1} (\mathbf {e}_i\mathbf {1}_{i\ge 1}, \mathbf {e}_j\mathbf {1}_{j\ge 1}; \mathbf {e}_k\mathbf {1}_{k\ge 1}, \mathbf {e}_\ell \mathbf {1}_{\ell \ge 1} ), \nonumber\\[-6pt] &\\[-6pt]L_{s,x}(\mathbf {A},b;\mathbf {C},d) &= W_{x/s,1,\mathsf {N}} ( \mathbf {A},\mathbf {e}_b\mathbf {1}_{b\ge 1}; \mathbf {C},\mathbf {e}_d\mathbf {1}_{d\ge 1} ) \vert _{q^{\mathsf {N}}=s^{-2}},\nonumber \end{align}

where $i,j,k,\ell ,b,d\in \{ 0,1,\ldots ,n \}$ and $\mathbf {A},\mathbf {C}\in \mathbb {Z}_{\ge 0}^{n}$ . In the second line in (2.11), the parameter $s=q^{-\mathsf {N}/2}$ is real for $q\in [0,1)$ and $\mathsf {N}\in \mathbb {Z}_{\ge 1}$ . However, for the parameters $(s,x)$ to satisfy (2.2), we let $\mathsf {N}$ not to be an integer, and treating $q^{\mathsf {N}}$ as a generic complex parameter, see Remark 2.4.

2.4 Queue specialization

Here we define a procedure which we call the queue specialization of the fully fused stochastic colored vertex weights (2.9). This specialization depends on an integer $1\le m\le n$ and on three parameters $u,s_1,s_2$ , and proceeds in the following manner.

• – First, encode the parameters $\mathsf {L}$ and $\mathsf {M}$ through $s_1,s_2\in \mathbb {C}$ as $q^{-\mathsf {L}}=s_1^2$ , $q^{-\mathsf {M}}=s_2^2$ , and let the spectral parameter be $x=s_1s_2^{-1}u$ . This is just a change of variables (as in Remark 2.4), which will be useful in subsequent computations.

• – After that, take the limit of $W_{s_1s_2^{-1}u,\mathsf {L},\mathsf {M}} (\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})$ as $A_m,C_m\to +\infty$ such that all other coordinates of $\mathbf {A},\mathbf {C}$ , as well as the whole tuples $\mathbf {B},\mathbf {D}$ are fixed, and $A_m-C_m=D_m-B_m$ is also fixed.Footnote ² The latter condition follows from the arrow conservation.

Lemma 2.5. The limit of the vertex weights $W_{s_1s_2^{-1}u,\mathsf {L},\mathsf {M}} (\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})$ described above exists. It is given by

(2.12) \begin{align} & \lim _{A_m,C_m\to +\infty } W_{s_1s_2^{-1}u,\mathsf {L},\mathsf {M}} (\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})\nonumber\\[4pt]&\quad = \mathbf {1}_{\mathbf {A}+\mathbf {B}=\mathbf {C}+\mathbf {D}} \cdot \mathbf {1}_{D_1=\ldots =D_{m-1}=0} \cdot \frac {(s_1^{-1}s_2 u; q)_{\infty }}{(s_1s_2u ; q)_{\infty }} \nonumber\\&\\[-8pt]&\qquad \times \sum _{\mathbf {P}} \frac {(s_1s_2/u ; q)_{|\mathbf {P}|} (s_1u/s_2 ; q)_{|\mathbf {B}-\mathbf {P}|}} {(s_1^2 ; q)_{|\mathbf {B}|}}\ q^{\sum _{1\le i\lt j\le n}(B_i-P_i) P_j} \prod _{i=1}^n \frac {(q;q)_{B_i}}{(q;q)_{P_i}(q;q)_{B_i-P_i}} \nonumber\\[4pt]&\qquad \times \Bigl (\frac {s_1s_2}{u} \Bigr )^{|\mathbf {B}|-|\mathbf {P}|} \Bigl (\frac {us_2}{s_1} \Bigr )^{|\mathbf {D}|} \ q^{\sum _{m\le i\lt j\le n} D_i (C_j-P_j)} \ \frac { (s_1^2 ; q)_{|\mathbf {D}|}} {(q;q)_{D_m}} \prod _{i=m+1}^n \frac {(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}},\nonumber \end{align}

where the sum is over $\mathbf {P}\in \mathbb {Z}_{\ge 0}^n$ with $0\le P_i\le \min (B_i,C_i)$ for all $i$ .

Proof. Set $x = s_1 s_2^{-1} u$ . Since $\mathbf {B}$ stays fixed, the second factor

\begin{equation*} \Phi ( \mathbf {P}, \mathbf {B} ; q^{-\mathsf {L}} / x, q^{-\mathsf {L}} ) =\Phi ( \mathbf {P}, \mathbf {B} ; s_1s_2 / u, s_1^2 ) \end{equation*}

in the sum in (2.9) is also fixed. Moreover, the summation multi-index $\mathbf {P}$ belongs to a fixed finite set where $P_i\le B_i$ for all $i$ . Thus, it remains to consider the limit of

(2.13) \begin{align} &(q^{\mathsf {L}})^{|\mathbf {A}|}\ \Phi (\mathbf {C}-\mathbf {P}, \mathbf {C}+\mathbf {D}-\mathbf {P} ; q^{\mathsf {L}-\mathsf {M}} x, q^{-\mathsf {M}} x ) \nonumber\\&\quad= (q^{\mathsf {L}})^{|\mathbf {A}|}\ \frac {(q^{\mathsf {L}-\mathsf {M}} x ; q)_{|\mathbf {C}-\mathbf {P}|}(q^{-\mathsf {L}} ; q)_{|\mathbf {D}|}} {(q^{-\mathsf {M}} x ; q)_{|\mathbf {C}-\mathbf {P}+\mathbf {D}|}}\ (q^{-\mathsf {L}})^{|\mathbf {C}-\mathbf {P}|} q^{\sum _{1\le i\lt j\le n} D_i (C_j-P_j)} \prod _{i=1}^n \frac {(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}}.\nonumber \\&\quad= s_1^{-2|\mathbf {D}-\mathbf {B}+\mathbf {P}|}\ \frac {(s_1^{-1}s_2 u; q)_{|\mathbf {C}-\mathbf {P}|} (s_1^2 ; q)_{|\mathbf {D}|}} {(s_1s_2u ; q)_{|\mathbf {C}-\mathbf {P}+\mathbf {D}|}}\ q^{\sum _{1\le i\lt j\le n} D_i (C_j-P_j)} \prod _{i=1}^n \frac {(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}}. \end{align}

We have $C_m-P_m\to +\infty$ , so (2.13) converges to zero unless $D_i=0$ for all $i\lt m$ (as $q \in [0, 1)$ ). This leads to the indicator $\mathbf {1}_{D_1=\cdots =D_{m-1}=0}$ in (2.12). Next, if $D_i=0$ for all $i\lt m$ , then all other factors in (2.13) behave well, and the desired limit of the whole vertex weight exists. Taking the limit as $C_m \rightarrow +\infty$ , we immediately obtain (2.12). This completes the proof.

The next lemma states the independence of the queue vertex weights under $B_m$ , too, provided that no lower colors are present.

Proof. By Remark 2.8, we may set $A_i=C_i=D_i=0$ for all $i\lt m$ . Set, for simplicity, $B_m=b$ , $B_{m+1}+\cdots +B_n=b'$ , $P_m=p$ , $P_{m+1}+\cdots +P_n =p'$ , and $D_m=d$ . Then the part of (2.12) which depends only on $b$ and $p$ has the form

\begin{align*} \begin{split} &\sum _{p=0}^{b} \frac {(s_1s_2/u;q)_{p+p'}(s_1u/s_2;q)_{b+b'-p-p'}} {(s_1^2;q)_{b+b'}} \frac {(q;q)_{b}}{(q;q)_p(q;q)_{b-p}} \ (s_1s_2/u)^{b-p} q^{(b-p)p'} \\&\quad= \frac {(s_1s_2/u;q)_{p'} (s_1u/s_2;q)_{b'-p'} }{(s_1^2;q)_{b'}} \sum _{p=0}^{b} \frac {(q^{p'} s_1s_2/u;q)_{p} (q^{b'-p'} s_1u/s_2; q)_{b-p} } {(q^{b'} s_1^2;q)_{b}}\\&\qquad\times \frac {(q;q)_{b}}{(q;q)_p(q;q)_{b-p}} \ (q^{p'} s_1s_2/u )^{b-p} \\&\quad= \frac {(s_1s_2/u;q)_{p'} (s_1u/s_2;q)_{b'-p'} }{(s_1^2;q)_{b'}}. \end{split} \end{align*}

In the last equality we used the sum-to-one property (2.8) for the single-color case $n=1$ (with $x = q^{b'-p'} s_1u/s_2$ , $y = q^{b'} s_1^2$ , and $\mathbf {B} = b$ ). We see that the resulting expression does not depend on $b$ , as desired.

Proposition 2.10. For any $m\in \{1,\ldots ,n\}$ , the queue vertex weights ${\mathbb {W}}_{s_1,s_0, {u_1}/{u_0}}^{(-m)}$ , ${\mathbb {W}}_{s_2,s_0,{u_2}/{u_0}}^{(-m)}$ (2.12), and the fused cross vertex weight $W_{{s_1u_1}/{s_2u_2},\mathsf {L},\mathsf {M}}$ (2.9), where $q^{-\mathsf {L}}=s_1^2$ and $q^{-\mathsf {M}}=s_2^2$ , satisfy the Yang–Baxter equation given in Figure 7. In symbols, for all fixed $\mathbf {A},\mathbf {I}_1, \mathbf {I}_2,\mathbf {B}, \mathbf {J}_1,\mathbf {J}_2$ with $A_m,B_m=+\infty$ , we have

(2.14) \begin{equation} \begin{split} & \sum _{\mathbf {K}_1,\mathbf {K}_2,\mathbf {K}_3}\!\! {\mathbb {W}}_{s_2,s_0, {u_2}/{u_0}}^{(-m)}(\mathbf {A}, \mathbf {I}_2 ; \mathbf {K}_3, \mathbf {K}_2) {\mathbb {W}}_{s_1,s_0, {u_1}/{u_0}}^{(-m)}(\mathbf {K}_3, \mathbf {I}_1; \mathbf {B}, \mathbf {K}_1) W_{{s_1u_1}/{s_2u_2},\mathsf {L},\mathsf {M}} (\mathbf {K}_2, \mathbf {K}_1; \mathbf {J}_2, \mathbf {J}_1) \\ &\quad = \sum _{\mathbf {K}_1,\mathbf {K}_2,\mathbf {K}_3}\!\! W_{{s_1u_1}/{s_2u_2},\mathsf {L},\mathsf {M}} (\mathbf {I}_2, \mathbf {I}_1; \mathbf {K}_2, \mathbf {K}_1) {\mathbb {W}}_{s_1,s_0,{u_1}/{u_0}}^{(-m)}(\mathbf {A}, \mathbf {K}_1; \mathbf {K}_3, \mathbf {J}_1) {\mathbb {W}}_{s_2,s_0,{u_2}/{u_0}}^{(-m)}(\mathbf {K}_3, \mathbf {K}_2 ; \mathbf {B}, \mathbf {J}_2). \end{split} \end{equation}

Figure 7.

Yang–Baxter equation for the queue specialization. The sums in both sides are taken over all $\mathbf {K}_1,\mathbf {K}_2,\mathbf {K}_3\in \mathbb {Z}_{\ge 0}^n$ , and the inputs and outputs $\mathbf {I}_1,\mathbf {I}_2,\mathbf {J}_1,\mathbf {J}_2,\mathbf {A},\mathbf {B}\in \mathbb {Z}_{\ge 0}^n$ are fixed.

Recall that $0\le q\lt 1$ . Let us define the following two subsets of the parameters $(s_1,s_2,u)$ in the queue vertex weights.

(2.15) \begin{align} &- \text { (Higher horizontal spin) } s_1, s_2 \in [-1,1] \text { such that } 0 \leq s_1 s_2 \lt u \lt \min ({s_1}/{s_2}, {s_2}/{s_1},\qquad \nonumber\\[-3pt]&\quad\;\ {1}/{s_1 s_2}). \text{ The number of paths on a horizontal edge is not restricted. }\qquad\nonumber\\[-6pt]&\qquad\\[-6pt]&- \text { (Finite horizontal spin) } s_1 = q^{-{\mathsf {L}}/{2}} \text { for some } \mathsf {L} \in \mathbb {Z}_{\ge 1} \text { and } u = q^{{\mathsf {L}}/{2}} u', \text { with purely }\qquad \nonumber\\[-3pt]&\quad\;\ \text{imaginary } u',s_2 \text{ satisfying } s_2 u' \leq s_2^2 \leq 0 \text { and } s_2/u'\ge q^{\mathsf {L}}. \text { The number of paths on}\qquad\nonumber\\[-3pt]&\quad\;\ \text{a horizontal edge is at most } \mathsf {L}.\qquad\nonumber \end{align}

These subsets present convenient sufficient nonnegativity conditions.

Proposition 2.11. Under (2.15), the queue vertex weights ${\mathbb {W}}^{(-m)}_{s_1, s_2, u}$ (2.12) are nonnegative, and

(2.16) \begin{equation} \sum _{C_{m+1},C_{m+2},\ldots ,C_n, D_{m},D_{m+1},\ldots ,D_n=0}^{\infty } {\mathbb {W}}^{(-m)}_{s_1, s_2, u}(\mathbf {A}, \mathbf {B}; \mathbf {C}, \mathbf {D})=1, \end{equation}

where $\mathbf {A},\mathbf {B}$ are fixed, $A_m,C_m=+\infty$ , and $C_i=A_i+B_i$ , $D_i=0$ for all $i\lt m$ .

Proof. Let us first consider the higher horizontal spin case of (2.15). One can check that all arguments of the $q$ -Pochhammer symbols in (2.12) are in $[0,1)$ . Moreover, the total sign coming from the powers $ s_1^{|\mathbf {B}|-|\mathbf {D}|-|\mathbf {P}|} s_2^{|\mathbf {B}|+|\mathbf {D}|-|\mathbf {P}|} u^{|\mathbf {D}|-|\mathbf {B}|+|\mathbf {P}|}$ is always nonnegative. Thus, we get the nonnegativity of the queue vertex weights.

To see that they sum to one, we need to take the limit of the sum-to-one identity (2.10) for the fully fused weights $W_{x,\mathsf {L},\mathsf {M}}$ as $A_m,C_m\to +\infty$ . One can check that under our Conditions (2.15), the weight $W_{x,\mathsf {L},\mathsf {M}} (\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})$ (with $x=s_1u/s_2$ , $q^{-\mathsf {L}}=s_1^2$ , $q^{-\mathsf {M}}=s_2^2$ ) decays exponentially fast when $|\mathbf {D}|\to +\infty$ due to the presence of the power $(us_2/s_1)^{|\mathbf {D}|}$ . Thus, since our queue specialization requires $|\mathbf {D}|$ to stay fixed, this decay ensures that identity (2.10) yields (2.16).

Let us now consider the finite horizontal spin case of (2.15). In this case, $|\mathbf {D}|$ must stay finite (cf. Remarks 2.2 and 2.4), so the sum-to-one Property (2.16) is automatic from (2.10). To see the nonnegativity, observe that since $s_1 = q^{-{\mathsf {L}}/{2}} \geq 1$ , the term $(s_1^2;q)_{|\mathbf {D}|}$ produces the sign $(-1)^{|\mathbf {D}|}$ , which is compensated by $(us_2/s_1)^{|\mathbf {D}|}$ . The other combined powers $ (s_1s_2/u)^{|\mathbf {B}|-|\mathbf {P}|}$ are always nonnegative. Next, we have

\begin{equation*} \frac {(s_1^{-1}s_2 u; q)_{\infty }}{(s_1s_2u ; q)_{\infty }} = \frac {1}{(s_2u';q)_{\mathsf {L}}}\ge 0. \end{equation*}

In the remaining $q$ -Pochhammer symbols, we have

\begin{equation*} s_1s_2/u=q^{-\mathsf {L}}s_2/u'\ge 1,\quad s_1u/s_2=u'/s_2\ge 1,\quad s_1^2=q^{-\mathsf {L}}\ge 1. \end{equation*}

Thus, the quantities $(s_1s_2/u ; q)_{|\mathbf {P}|} (s_1u/s_2 ; q)_{|\mathbf {B}-\mathbf {P}|} (s_1^2 ; q)_{|\mathbf {B}|}^{-1}$ are nonnegative for all $\mathbf {P}$ provided that $|\mathbf {B}|\le \mathsf {L}$ . This completes the proof.

2.5 Queue vertex model on the cylinder

Let us fix the number $n$ of colors, and another integer $N\ge 1$ which is the size of the cylinder. In this subsection we define a linear operator $\mathfrak {Q}$ whose matrix elements $\langle \emptyset | \mathfrak {Q} | \mathbf {V} \rangle$ are partition functions coming from the queue vertex weights on the cylinder $\{-n, \ldots , -1\} \times \mathbb {Z}/N \mathbb {Z}$ . The vertices on the cylinder are indexed by $(-m,j)$ , $m=1,\ldots ,n$ , $j=1,\ldots ,N$ , see Figure 8 for an illustration. No paths enter from the left. The paths exiting horizontally from the right are encoded by the integer tuples $\mathbf {V}(1),\ldots , \mathbf {V}(N) \in \mathbb {Z}_{\ge 0}^n$ . Define the vector space for the states on the cylinder,

(2.17) \begin{equation} V^{\otimes N} := \mathop {\mathrm {Span}} ( \{ | \mathbf {V} \rangle = | (\mathbf {V}(1),\ldots , \mathbf {V}(N)) \rangle : \mathbf {V}(j) \in \mathbb {Z}_{\geq 0}^n \ \forall j =1,\ldots , N\}), \end{equation}

and similarly let $\langle \mathbf {V} |$ denote the dual basis in $V^{\otimes N}$ . We also need the subspace $V^{\otimes N}_{\mathrm {full}}$ of $V^{\otimes N}$ spanned by the vectors $| \mathbf {V} \rangle$ satisfying

\begin{equation*} \sum \nolimits _{j=1}^{N}\mathbf {V}(j)_k \gt 0 \quad \text {for all $k=1,\ldots ,n $} \end{equation*}

(where $\mathbf {V}(j)_k$ denotes the $k$ th coordinate of $\mathbf {V}(j)$ ). In words, each state $| \mathbf {V} \rangle \in V^{\otimes N}_{\mathrm {full}}$ must contain at least one arrow of each of the $n$ colors.

Definition 2.12. Fix complex parameters

(2.18) \begin{equation} \mathbf {u} =(u_1,\ldots , u_N) \in \mathbb {C}^N, \quad \mathbf {s}^{(h)} = (s^{(h)}_1,\ldots , s^{(h)}_N), \quad \mathbf {v} = (v_1,\ldots ,v_n), \quad \mathbf {s}^{(v)} = (s^{(v)}_1,\ldots , s^{(v)}_n), \end{equation}

The linear operator $\mathfrak {Q}=\mathfrak {Q} ( \mathbf {u}; \mathbf {s}^{(h)}; \mathbf {v}; \mathbf {s}^{(v)})$ on $V^{\otimes N}$ , called the queue transfer matrix, is defined via its matrix elements $\langle \mathbf {V}' | \mathfrak {Q} | \mathbf {V} \rangle$ as follows. First, if $| \mathbf {V} \rangle \notin V^{\otimes N}_{\mathrm {full}}$ , we set $\langle \mathbf {V}' | \mathfrak {Q} | \mathbf {V} \rangle =0$ . Otherwise, $\langle \mathbf {V}' | \mathfrak {Q} | \mathbf {V} \rangle$ is the partition function of the queue vertex weights on the cylinder $\{-n,\ldots ,-1 \}\times \mathbb {Z}/N\mathbb {Z}$ with the following data.

• – The entering arrow configurations $\mathbf {V}'(j)$ along the left horizontal edges $(-n-1,j)\to (-n,j)$ , $j=1,\ldots ,N$ .

• – The terminal arrow configurations $\mathbf {V}(j)$ along the right horizontal edges $(-1,j)\to (0,j)$ , $j=1,\ldots ,N$ .

• – Queue vertex weights ${\mathbb {W}}^{(-m)}_{s^{(h)}_j, s^{(v)}_m, u_j / v_{m}}$ at each vertex $(-m,j)$ in the cylinder, where $j=1,\ldots ,N$ , $m=1,\ldots ,n$ , and $u_j,s_j^{(h)}$ and $v_m,s^{(v)}_m$ are the horizontal and the vertical parameters, respectively.

Figure 8.

A vertex model on the cylinder whose partition function (indexed by the tuples $\mathbf {V}(1),\ldots ,\mathbf {V}(N)$ ) is equal to $\langle \emptyset | \mathfrak {Q} | \mathbf {V} \rangle$ . We identify the top and the bottom boundaries (dashed lines), and sum over all possible tuples $\mathbf {M}(-n),\ldots ,\mathbf {M}(-1)\in \mathbb {Z}^{n}_{\ge 0}$ encoding the paths which wind around the cylinder an arbitrary number of times. The left vector $\langle \emptyset |$ is empty, which corresponds to no paths entering from the left.

In detail, the partition function $\langle \mathbf {V}' | \mathfrak {Q} | \mathbf {V} \rangle$ is equal to the sum

\begin{equation*} \sum _{ \mathbf {M} } \ \sum _{\mathscr{C}\in \mathcal {P}_{\mathbf {M},\mathbf {V}',\mathbf {M},\mathbf {V}}} \ \prod _{m=1}^{n}\prod _{j=1}^{N} {\mathbb {W}}^{(-m)}_{s^{(h)}_j, s^{(v)}_m, u_j / v_{m}}(\mathbf {A}(m,j),\mathbf {B}(m,j);\mathbf {C}(m,j),\mathbf {D}(m,j)), \end{equation*}

where $\mathbf {M}=(\mathbf {M}(-n),\ldots ,\mathbf {M}(-1))$ encodes the paths winding around the cylinder. The sum over $\mathbf {M}$ , by definition, has

(2.19) \begin{equation} \mathbf {M}(-m)_m=+\infty \quad \text {and}\quad \mathbf {M}(-m)_i=0,\quad i\lt m \text{ for all}\ m=1,\ldots ,n. \end{equation}

For each $\mathbf {M}$ , the sum over $\mathscr{C}$ runs over all path configurations in the rectangle with the boundary conditions $\mathbf {M},\mathbf {V}',\mathbf {M},\mathbf {V}$ at the bottom, left, top, and right, respectively. The tuples $\mathbf {A}(m,j)$ , $\mathbf {B}(m,j)$ , $\mathbf {C}(m,j)$ , and $\mathbf {D}(m,j)$ encode the arrow configurations at each vertex $(-m,j)$ of the rectangle.

See Figure 8 for an example when $\mathbf {V}' = (\mathbf {0}, \ldots , \mathbf {0})$ .

Observe that the partition function $\langle \mathbf {V}' | \mathfrak {Q} | \mathbf {V} \rangle$ in Definition 2.12 involving the sum over $\mathbf {M}(-n),\ldots ,\mathbf {M}(-1)$ cannot be interpreted as a probability in a stochastic vertex model because we are summing over input path configurations at vertices, as well as over the output ones. Moreover, this sum may even be divergent. Therefore, we need to make sure that the quantities $\langle \mathbf {V}' | \mathfrak {Q} | \mathbf {V} \rangle$ are well defined.

Remark 2.14. The condition that $| \mathbf {V} \rangle \in V^{\otimes N}_{\mathrm {full}}$ is essential for Lemma 2.13. Indeed, for $n=3$ , we have

\begin{equation*} {\mathbb {W}}^{(-2)}_{s_1,s_2,u} (\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})= \frac {(s_2u/s_1;q)_\infty }{(s_1s_2u;q)_{\infty }}, \end{equation*}

where $\mathbf {A}=\mathbf {C}=(0,\infty ,k)$ and $\mathbf {B}=\mathbf {D}=(0,0,0)$ . This expression is independent of $k$ . For $N=1$ , we need to sum it over all $k$ , which leads to divergence. However, if $\mathbf {D}=(0,d,0)$ , the weight ${\mathbb {W}}^{(-2)}_{s_1,s_2,u} (\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})$ contains the power $q^{kd}$ , eliminating this problem of divergence.

The partition functions $\langle \mathbf {V}' | \mathfrak {Q} | \mathbf {V} \rangle$ are essentially independent of the entrance state $\langle \mathbf {V}'|$ .

Remark 2.16 (Trace formula for queue partition functions). The queue vertex model partition function can be interpreted through a product of the following $N$ operators, where $N$ is the size of the ring in the cross-section of the cylinder:

(2.20) \begin{equation} \langle \emptyset | \mathfrak {Q} | \mathbf {V} \rangle = \mathop {\mathrm {Trace}}\nolimits ^\bullet ( \mathscr{X}_{\mathbf {V}(1)}(u_1,s_1^{(h)})\cdots \mathscr{X}_{\mathbf {V}(N)}(u_N,s_N^{(h)}) ). \end{equation}

Here each $\mathscr{X}_{\mathbf {V}}(u,s)$ , $\mathbf {V}\in \mathbb {Z}_{\ge 0}^n$ , acts in the $n$ -fold tensor product $V_{-n}\otimes \cdots \otimes V_{-1}$ , where $V_{-m}$ has basis $|\mathbf {M}(-m)\rangle$ , $\mathbf {M}(-m)\in \mathbb {Z}_{\ge 0}^{n}$ , $m=1,\ldots ,n$ . The matrix elements of $\mathscr{X}_{\mathbf {V}}(u,s)$ are partition functions of the queue weights on the lattice $\{1\}\times \{-n,\ldots ,-1 \}$ . See Figure 9 for an illustration. The operation $\mathop {\mathrm {Trace}}\nolimits ^\bullet$ in (2.20) means that we restrict the summation to the tuples $\mathbf {M}$ satisfying (2.19). One can turn $\mathop {\mathrm {Trace}}\nolimits ^\bullet$ into a genuine trace by suitably modifying the definition of the spaces $V_{-m}$ .

Figure 9.

The matrix element $\large \langle \mathbf {M}(-n),\ldots ,\mathbf {M}(-1) \large | \ \mathscr{X}_{\mathbf {V}}(u,s)\ \large | \mathbf {M}'(-n),\ldots ,\mathbf {M}'(-1) \large \rangle$ of one of the operators $\mathscr{X}_{\mathbf {V}}(u,s)$ entering the trace formula (2.20). At the vertex indexed by $-m$ , $m=1,\ldots ,n$ , we have the queue vertex weight ${\mathbb {W}}^{(-m)}_{s, s^{(v)}_m, u / v_{m}}$ .

3.1 Twisted cylinder Markov operator

Recall that $n$ is the number of colors, and $N$ is the size of the ring in the vertical cross-section of the cylinder $\{-n,\ldots ,-1 \}\times \mathbb {Z}/N\mathbb {Z}$ carrying the queue transfer matrix $\mathfrak {Q}=\mathfrak {Q} ( \mathbf {u}; \mathbf {s}^{(h)}; \mathbf {v}; \mathbf {s}^{(v)})$ of Definition 2.12. For our first relation, we take the spin- $\frac {1}{2}$ specialization in the horizontal direction. That is, the horizontal spin parameters in (2.18) are all equal to $q^{-1/2}$ :

\begin{equation*} \mathbf {s}^{(h)} = (s^{(h)}_1,\ldots , s^{(h)}_N) = \mathbf {s}^{(h)}_{\frac 12} := (q^{-1/2},\ldots ,q^{-1/2} ). \end{equation*}

We can take each tensor component in the space $V^{\otimes N}$ (2.17) to be $V=\mathbb {C}^{n+1}$ .

We need some extra notation. Let $R_{z}$ be the $R$ -matrix (2.4), and denote by $\check {\mathscr{R}}_z$ the operator in $V\otimes V$ with matrix elements (see the end of § 2.1 for basis vector notations)

(3.1) \begin{equation} \langle i,j | \ \check {\mathscr{R}}_z \ | \ell ,k \rangle := R_{z}(i,j;k,\ell ). \end{equation}

When $\check {\mathscr{R}}_z$ acts on the $k$ th and the $\ell$ th tensor components of $V^{\otimes N}$ , we denote it by $\check {\mathscr{R}}_z^{(k\ell )}$ .

Definition 3.1. Fix two spectral parameters $u,u_1$ . Let the twisted cylinder Markov operator $\mathfrak {T}(u,u_1)$ be the linear operator on $V^{\otimes N}$ defined as

(3.2) \begin{equation} \mathfrak {T}(u, u_1) := \check {\mathscr{R}}_{u_1 u^{-1}}^{(1 2)} \ \check {\mathscr{R}}_{u_1 u^{-1}}^{(2 3)} \cdots \check {\mathscr{R}}_{u_1 u^{-1}}^{(N1)}, \end{equation}

Pictorially, $\mathfrak {T}(u, u_1)$ is given in Figure 10. The product in (3.2) is interpreted as a product of Markov operators, that is, we first apply $\check {\mathscr{R}}_{u_1 u^{-1}}^{(1 2)}$ to a fixed configuration of arrows on the cylinder, get a random configuration, then apply $\check {\mathscr{R}}_{u_1 u^{-1}}^{(2 3)}$ to the new configuration, and so on.

Figure 10.

The configuration of vertices whose partition function is the matrix element of the twisted cylinder Markov operator $\langle \mathbf {V} | \mathfrak {T}(u, u_1) | \mathbf {V}' \rangle$ , where $\mathbf {V}=(i_1,i_2,i_3)$ , $\mathbf {V}'=(i_1',i_2',i_3')$ , and $i_k,i_k'\in \{0,\ldots ,n \}$ . The size of the ring is $N=3$ . After all the $N$ crossings, the spectral parameters attached to the strands on the right are the same as on the left. Note that at each crossing, we use the vertex weights $R_{u_1u^{-1}}$ in the same way as in the Yang–Baxter equation in Figure 6; see also Remark 3.2 for a discussion of the notation.

Iterating the sum-to-one property of the $R$ -matrix (2.4), we see that for any initial configuration $\mathbf {V}' = (i_1,\ldots ,i_N)$ , where $i_k \in \{0,\ldots ,n\}$ , we have

(3.3) \begin{equation} \sum _{\mathbf {V}=(i_1',\ldots ,i_N')\in \{0,1,\ldots ,n \}^N} \langle \mathbf {V}' | \mathfrak {T}(u, u_1) | \mathbf {V} \rangle =1. \end{equation}

If the matrix elements of $\mathfrak {T}(u, u_1)$ are nonnegative, then it is a Markov operator. However, considering $\mathfrak {T}(u, u_1)$ as a formal Markov operator with the sum-to-one property suffices for this section.

We have the following stationarity of the queue vertex model under the action of the twisted cylinder Markov operator.

Theorem 3.3. Let the parameters $u,u_1$ of the twisted cylinder Markov operator $\mathfrak {T}$ (3.2), as well as the parameters $\mathbf {v} = (v_1,\ldots , v_n)$ and $\mathbf {s}^{(v)} = (s^{(v)}_1,\ldots , s^{(v)}_n)$ in the queue transfer matrix $\mathfrak {Q}$ of Definition 2.12 be arbitrary. Denote $\mathbf {u} := (u_1,u,\ldots , u)$ . Then we have

(3.4) \begin{equation} \langle \emptyset | \ \mathfrak {Q} (\mathbf {u}; \mathbf {s}_{\frac {1}{2}}^{(h)}; \mathbf {v}; \mathbf {s}^{(v)}) \ \mathfrak {T}(u, u_1) = \langle \emptyset | \ \mathfrak {Q} (\mathbf {u}; \mathbf {s}_{\frac {1}{2}}^{(h)}; \mathbf {v}; \mathbf {s}^{(v)}) . \end{equation}

Proof. We apply the Yang–Baxter equation specialized to the queue vertex weights; see Proposition 2.10. In this equation, the cross vertex weights have the form $ W_{\frac {u}{u_1},1,1}$ , which are encoded by the operator $\check {\mathscr{R}}_{u_1u^{-1}}$ ; see (2.11). We illustrate the argument diagrammatically for $N = 3$ in Figure 11.

Figure 11.

The commutation in the proof of Theorem 3.3 for $N=3$ . Here $i_1,i_2,i_3 \in \{0,1,\ldots , n\}$ are arbitrary.

Applying the Yang–Baxter equation successively at columns $-n, -n+1, \ldots ,-2, -1$ of the queue vertex model, we get the intertwining relation

(3.5) \begin{equation} \mathfrak {Q}((u_1, u, \ldots , u);\mathbf {s}_{\frac {1}{2}}^{(h)}; \mathbf {v}; \mathbf {s}^{(v)})\ \check {\mathscr{R}}_{u_{1} u^{-1}}^{(1 2)} = \check {\mathscr{R}}_{u_1 u^{-1}}^{(1 2)}\ \mathfrak {Q}((u, u_1, \ldots , u); \mathbf {s}_{\frac {1}{2}}^{(h)}; \mathbf {v} ; \mathbf {s}^{(v)}) \end{equation}

Continuing inductively with $\check {\mathscr{R}}_{u_{1} u^{-1}}^{(2 3)}, \ldots , \check {\mathscr{R}}_{u_{1} u^{-1}}^{(N-1,N)}$ turns the horizontal spectral parameter sequence in $\mathfrak {Q}$ into $(u, \ldots , u, u_1)$ . Finally, after applying the last operator $\check {\mathscr{R}}_{u_{1} u^{-1}}^{(N,1)}$ , we get back to the original sequence. Here we used the periodicity in the vertical direction of the vertex model defining $\mathfrak {Q}$ , which is crucial to complete the commutation. Therefore,

\begin{equation*} \mathfrak {Q} (\mathbf {u}; \mathbf {s}_{\frac {1}{2}}^{(h)}; \mathbf {v}; \mathbf {s}^{(v)}) \ \mathfrak {T}(u, u_1) = \mathfrak {T}(u, u_1) \ \mathfrak {Q} (\mathbf {u}; \mathbf {s}_{\frac {1}{2}}^{(h)}; \mathbf {v}; \mathbf {s}^{(v)}). \end{equation*}

It remains to observe that $ \langle \emptyset | \check {\mathscr{R}}_{u_1 u^{-1}}^{(i,j)} = \langle \emptyset |$ for all $i,j$ . This completes the proof.

3.2 Straight cylinder Markov operator

Here we define another Markov operator on the cylinder of size $N+1$ which commutes with the queue transfer matrix on this cylinder when applied to the empty configuration $\emptyset$ at the left boundary of the cylinder. It acts in the space $\mathbb {C}^{n+1} \otimes V^{\otimes N}$ , where the first factor $\mathbb {C}^{n+1}$ is an auxiliary space which we identify with the index $0$ in the superscripts. The tensor components of the space $V^{\otimes N}$ are indexed by $j=1,2,\ldots , N$ , and have basis vectors $|\mathbf {V}(j)\rangle$ , where $\mathbf {V}(j) \in \mathbb {Z}_{\geq 0}^n$ encodes the arrow configuration on site $j$ . That is, in contrast with § 3.1, we return to the higher spin setting. Later in §§ 5 and 6 we will take limits in which the marginal process corresponding to the factor $V^{\otimes N}$ will lead to the colored $q$ -Boson or the colored $q$ -PushTASEP on the ring of size $N$ .

Recall the Markov operator $\mathscr{L}_{s,z}$ acting in $\mathbb {C}^{n+1}\otimes V$ ; see (2.3).

Definition 3.4. Fix spectral parameters $(x,\mathbf {u})=(x,u_1,u_2,\ldots , u_N)\in \mathbb {C}^{N+1}$ and horizontal spin parameters $\mathbf {s}^{(h)}=(s_1^{(h)},\ldots , s_N^{(h)})\in \mathbb {C}^N$ . The straight cylinder Markov operator denoted by $\mathfrak {S}=\mathfrak {S}(x,\mathbf {u};\mathbf {s}^{(h)})$ acts in $\mathbb {C}^{n+1}\otimes V^{\otimes N}$ as follows:

(3.6) \begin{equation} \mathfrak {S}(x,\mathbf {u};\mathbf {s}^{(h)}) := \mathscr{L}^{\,\,(0N)}_{s^{(h)}_N,xu_{N}^{-1}}\ \mathscr{L}^{\,\,(0,N-1)}_{s^{(h)}_{N-1},xu_{N-1}^{-1}} \ldots \mathscr{L}^{\,\,(01)}_{s^{(h)}_1,xu_{1}^{-1}}. \end{equation}

Here, by $\mathscr{L}^{\,(0,j)}_{s,z}$ we denote the operator $\mathscr{L}_{s,z}$ acting on $\mathbb {C}^{n+1}\otimes V^{\otimes N}$ in the auxiliary space $\mathbb {C}^{n+1}$ and the $j$ th tensor component of $V^{\otimes N}$ . The operator $\mathfrak {S}$ is illustrated in Figure 12.

Figure 12.

The straight cylinder Markov operator $\mathfrak {S}=\mathfrak {S}(x,\mathbf {u};\mathbf {s}^{(h)})$ , applied three times. The line with spectral parameter $x$ has spin parameter $q^{-\frac {1}{2}}$ , so edges along this line can only be occupied by at most one path (hence $k,\ell \in \{0,1,\ldots , n \}$ ). The partition function of the displayed configuration is the matrix element $\langle k, \mathbf {U} |\ \mathfrak {S}^3 \ |\ell , \mathbf {V} \rangle$ .

Remark 3.5. In Definition 3.4 and throughout this subsection, we index the tensor factors of the space $\mathbb {C}^{n+1} \otimes V^{\otimes N}$ by $0,1,\ldots ,N$ , but put the horizontal strand corresponding to the $0$ th factor on top of the cylinder in Figures 12 and 13. This way of indexing is consistent with the action of the operators $\mathscr{L}_{s,z}$ in $\mathbb {C}^{n+1}\otimes V$ in (3.6), while the location of the $0$ th strand on top just below the cylinder’s cut is convenient for illustrations.

Figure 13.

An illustration of the proof of Theorem 3.6. Each move is justified by the Yang–Baxter equation (Proposition 2.10). As usual, the equality of pictures means the equality of the corresponding partition functions with the fixed boundary conditions.

The operator $\mathfrak {S}(x,\mathbf {u};\mathbf {s}^{(h)})$ satisfies the sum-to-one property similarly to (3.3), and thus is a formal Markov operator. We have the following stationarity of the queue vertex model under $\mathfrak {S}$ .

Theorem 3.6. Let us take the parameters $\mathbf {u},\mathbf {s}^{(h)},\mathbf {v},\mathbf {s}^{(v)}$ as in (2.18), and consider the queue vertex transfer matrix on the ring of size $N+1$ with the parameters

\begin{equation*} \mathfrak {Q}=\mathfrak {Q}( (xq^{ 1/2},\mathbf {u});(q^{-{1}/{2}},\mathbf {s}^{(h)}) ; \mathbf {v};\mathbf {s}^{(v)}). \end{equation*}

Then we have

\begin{equation*} \langle \emptyset |\ \mathfrak {Q}\ \mathfrak {S} = \langle \emptyset |\ \mathfrak {Q}, \end{equation*}

where $\mathfrak {S}=\mathfrak {S}(x,\mathbf {u};\mathbf {s}^{(h)})$ has parameters compatible with those in $\mathfrak {Q}$ .

Proof. We employ the Yang–Baxter equation of Proposition 2.10. Let us match our parameters to this equation. The queue vertex model at rows $0$ and $j$ has the vertex weights

\begin{equation*} {\mathbb {W}}^{(-m)}_{q^{-1/2},s_m^{(v)},{xq^{1/2}}/{v_m}} ,\quad {\mathbb {W}}^{(-m)}_{s_j^{(h)},s_m^{(v)},{u_j}/{v_m}}, \end{equation*}

respectively. The operator $\mathfrak {S}$ has the vertex weight $L_{s_j^{(h)},xu_j^{-1}}$ , which, by (2.11), is the same as the fused weight $W_{x u_j^{-1}/s_j^{(h)},1,\mathsf {M}}$ , where $q^{-\mathsf {M}/2}=s_j^{(h)}$ . We see that these three weights indeed satisfy the Yang–Baxter equation illustrated in Figure 7.

Applying this Yang–Baxter equation with parameters $(v_1,s_1^{(v)}), \ldots ,(v_n,s_n^{(v)})$ (that is, at the columns $-1,\ldots ,-n$ of the queue vertex model), we obtain the following relation between operators acting in the space $\mathbb {C}^{n+1} \otimes V^{\otimes N}$ , applied to the empty configuration $\emptyset$ at the left boundary of the cylinder:

(3.7) \begin{equation} \begin{split} &\langle \emptyset |\ \mathfrak {Q}( (xq^{\frac 12},u_1,\ldots ,u_N );(q^{-\frac {1}{2}},s_1^{(h)},\ldots ,s_N^{(h)}) ; \mathbf {v};\mathbf {s}^{(v)}) \mathscr{L}^{\,(0N)}_{s_N^{(h)},xu_{N}^{-1}} \\&\hspace {10pt} = \langle \emptyset |\ P^{(0,N)}\ \mathfrak {Q}( (u_N,u_1,\ldots ,u_{N-1},xq^{\frac 12} );(s_N^{(h)},s_1^{(h)},\ldots ,s_{N-1}^{(h)}, q^{-\frac {1}{2}}); \mathbf {v};\mathbf {s}^{(v)}) P^{(0,N)} . \end{split} \end{equation}

Here the permutation operator $P^{(0,N)}$ swaps the $0$ th and the $N$ th tensor components in the space $\mathbb {C}^{n+1}\otimes V\otimes \cdots \otimes V$ , and is required since the operator $\mathfrak {Q}$ in the right-hand side acts in $V\otimes \cdots \otimes V\otimes \mathbb {C}^{n+1}$ . Identity (3.7) represents the first step of the transformations illustrated in Figure 13. In the next step, the action of $\mathscr{L}^{\,(0,N-1)}_{s^{(h)}_{N-1},xu_{N-1}^{-1}}$ results in the following identity:

\begin{align*} \begin{split} & \langle \emptyset |\ P^{(0,N)}\ \mathfrak {Q}((u_N,u_1,\ldots ,u_{N-1},xq^{\frac 12});(s_N^{(h)},s_1^{(h)},\ldots ,s_{N-1}^{(h)}, q^{-\frac {1}{2}}) ; \mathbf {v};\mathbf {s}^{(v)}) P^{(0,N)} \mathscr{L}^{\,(0,N-1)}_{s^{(h)}_{N-1},xu_{N-1}^{-1}} \\&\quad = \langle \emptyset |\ P^{(N,N-1)}P^{(0,N)} \mathfrak {Q}( (u_N,u_1,\ldots ,u_{N-2},xq^{\frac 12},u_{N-1});\\ &\qquad\qquad\qquad\qquad\qquad\qquad\ (s_N^{(h)},s_1^{(h)},\ldots ,s_{N-2}^{(h)}, q^{-\frac {1}{2}},s_{N-1}^{(h)}) ; \mathbf {v};\mathbf {s}^{(v)}) P^{(0,N)}P^{(N,N-1)} . \end{split} \end{align*}

Iterating the action of the other operators $\mathscr{L}$ , after $N$ total steps the horizontal parameter sequences $\mathbf {u}$ and $\mathbf {s}^{(h)}$ return back to their original states $(xq^{\frac 12},\mathbf {u});(q^{-\frac {1}{2}},\mathbf {s}^{(h)})$ . Here we employed the periodicity of the vertex model defining $\mathfrak {Q}$ to complete the commutation. This establishes the desired stationarity relation.

4.1 Multi-species ASEP on the ring

Recall that $n$ is the number of particle species (also called ‘types’ or ‘colors’), and $N$ is the size of the ring $\mathbb {Z}/N\mathbb {Z}$ . The state space of the multi-species ASEP consists of particle configurations $\eta =(\eta _1,\ldots ,\eta _N )$ on the ring, where $\eta _k\in \{ 0,1,\ldots ,n \}$ encodes the type of the particle at site $k$ . The type $0$ corresponds to the empty site.

For a configuration $\eta$ and each pair of neighboring sites $(k,k+1)$ (including $(N,1)$ for $k=N$ ), denote by $\eta ^{k,k+1}$ the configuration $(\eta _1,\ldots ,\eta _{k+1},\eta _k ,\ldots ,\eta _N )$ . That is, $\eta ^{k,k+1}$ is obtained from $\eta$ by swapping the types at sites $k$ and $k+1$ .

Definition 4.1. The multi-species ASEP (mASEP) is a continuous time Markov chain on the space of particle configurations on the ring, with the following transition rates:

(4.1) \begin{equation} \text {Rate}(\eta \rightarrow \eta ^{k,k+1}) = \begin{cases} q, & \eta _k \gt \eta _{k+1}; \\ 1, & \eta _k \lt \eta _{k+1}, \end{cases} \end{equation}

where $k$ runs over $1,2,\ldots ,N$ , and in (4.1) we assume that $\eta _k\ne \eta _{k+1}$ . The multi-species ASEP depends on a single parameter $q\in [0,1)$ .

Observe that mASEP preserves the number of particles of each type. We denote these type counts by

(4.2) \begin{equation} N_m:= \sum_{j=1}^N \mathbf {1}_{\eta _j = m}, \quad m=1,\ldots ,n. \end{equation}

We have $N_1+\cdots +N_n\le N$ . In addition, throughout this section, we assume that $N_m\ge 1$ for all $m=1,\ldots ,n$ . This assumption is very natural since if there are no particles of a given type, then the evolution of the $n$ -type mASEP is the same as that of a $(n-1)$ -type mASEP, where the missing type is removed entirely. Note also that at the level of queue vertex models, a violation of the assumption that $N_m\ge 1$ for all $m$ leads to convergence issues when summing over windings around the cylinder; see Lemma 2.13 and Remark 2.14.

It is natural to encode the states $\eta$ as basis vectors in $(\mathbb {C}^{n+1})^{\otimes N}$ :

\begin{equation*} |\eta \rangle = |\eta _1,\ldots ,\eta _N \rangle = |\eta _1 \rangle \otimes \cdots \otimes |\eta _N \rangle . \end{equation*}

Since all jumps under mASEP are nearest neighbor, the infinitesimal generator of mASEP can be written as a sum of local rate matrices as follows. Consider two possible configurations of particles $i i'$ , $j j'$ at adjacent lattice sites, say, $k$ and $k+1$ . Define an operator $\mathscr{M}_{\textrm{loc}}$ in $(\mathbb {C}^{n+1})^{\otimes 2}$ such that

(4.3) \begin{align} \langle i,i' | \mathscr{M}_{\textrm{loc}} | j,j' \rangle &= ( \mathscr{M}_{\textrm{loc}} )_{i i', j j'}\nonumber\\& := \mathbf {1}_{(i, i') = (j', j)} ( \text {jump rate $i i' \rightarrow j j'$} ) - \mathbf {1}_{(i, i') = (j, j')} ( \text {jump rate $i i' \rightarrow j' j$} ). \end{align}

The matrix element (4.3) is nonzero if and only if $(i,i')=(j,j')$ or $(i,i')=(j',j)$ . The infinitesimal Markov generator of mASEP then has the form

(4.4) \begin{equation} \mathfrak {M}_{\mathrm {mASEP}}= \sum _{l=1}^{N} ( \mathscr{M}_{\textrm{loc}})^{l,l+1}, \end{equation}

where $( \mathscr{M}_{\textrm{loc}})^{l,l+1}$ acts as $\mathscr{M}_{\textrm{loc}}$ on tensor factors of sites $l, l+1$ , and as the identity on all other factors. Denote by $\{\mathfrak {P}_{\mathrm {mASEP}}(t)\}_{t\in \mathbb {R}_{\ge 0}}$ , the continuous-time Markov semigroup generated by (4.4). The passage from the infinitesimal generator to this semigroup is straightforward, as the process lives on a finite state space. See the left side of Figure 14 for an illustration of the process.

Figure 14.

Left: An illustration of a state in mASEP on a ring of size $N = 9$ , with all possible jump rates indicated. Right: The interpretation of the mASEP hopping rates as limits of the operators $\check {\mathscr{R}}_{1-\epsilon }$ . Throughout the figure, the colors (types) of the particles are indicated by the numbers.

The next statement identifies mASEP as a Poisson-type limit of the twisted cylinder Markov operators.

Proposition 4.4. For any $u\in \mathbb {R}$ , we have the convergence of Markov operators in $(\mathbb {C}^{n+1})^{\otimes N}$ :

(4.5) \begin{equation} \lim _{\epsilon \to 0} \mathfrak {T}(u,\ u(1-\epsilon ))^{\lfloor (1-q) t/\epsilon \rfloor } = \mathfrak {P}_{\mathrm {mASEP}}(t), \quad t\in \mathbb {R}_{\ge 0}. \end{equation}

Here $\mathfrak {T}$ is the twisted cylinder Markov operator (3.2). See the right side of Figure 14 for an illustration of the limiting jump rates.

Recall that $q\in [0,1)$ . One readily sees that for any $u\in \mathbb {R}$ and $\epsilon \in (0,1)$ , both sides of the limiting relation (4.5) are Markov operators with nonnegative matrix elements.

Proof outline of Proposition 4.4 . This is a standard limit of the stochastic six-vertex model leading to the ASEP, see [Reference Borodin, Corwin and GorinBCG16, Reference AggarwalAgg17] and also [Reference Borodin and WheelerBW22a, § 12.3] for its colored version. In short, in the regime $\epsilon \to 0$ , all paths want to follow a ‘staircase’ motion, with occasional deviations that occur in continuous time according to independent exponential clocks. Because of how the cross vertices are organized to form the twisted cylinder Markov operator (Figure 10), the staircase motion corresponds to particles staying in place. For convenience, let us reproduce the main computation.

First, consider the limit of the local operators $\check {\mathscr{R}}_{z}$ (3.1), where $z=u_1u^{-1}=1-\epsilon$ because $u_1=u(1-\epsilon )$ . We have for the matrix elements (2.4):

\begin{equation*} \frac {1-z}{1-qz}=\frac {\epsilon }{1-q}+O(\epsilon ^2), \quad 1-\frac {1-z}{1-qz}=1- \frac {\epsilon }{1-q} + O(\epsilon ^2), \end{equation*}

and similarly for $q ({1-z})/({1-qz})$ and $1-q({1-z})/({1-qz})$ . Therefore, the local infinitesimal generator (4.3) of mASEP has the following form:

(4.6) \begin{equation} \mathscr{M}_{\textrm{loc}} = (1-q) \left .\frac {\partial }{\partial \epsilon }\right |_{\epsilon =0} \check {\mathscr{R}}_{1-\epsilon }. \end{equation}

See the right part of Figure 14 for an illustration of how we interpret the operators $\check {\mathscr{R}}_{1-\epsilon }$ in terms of the hopping rates.

In the $\epsilon \to 0$ limit, the product of the operators $\check {\mathscr{R}}_{1-\epsilon }$ over all pairs of neighboring lattice sites (which is equal to the twisted cylinder operator (3.2)) behaves as $\textrm{Id} + ({1}/({1-q})) \epsilon \; \mathfrak {M}_{\mathrm {mASEP}}$ , where $\textrm{Id}$ is the identity matrix, and $\mathfrak {M}_{\mathrm {mASEP}}$ is defined in (4.4). This leads to the desired statement about the convergence to the mASEP Markov semigroup.

4.2 Vertex model for the mASEP stationary distribution

Let us now apply Theorem 3.3 to mASEP. Recall that by $\langle \emptyset | \ \mathfrak {Q} \ | \mathbf {V} \rangle$ we denote the partition function of the queue vertex model on the cylinder introduced in Definition 2.12. For mASEP, the queue transfer matrix $\mathfrak {Q} = \mathfrak {Q}(\mathbf {u};\mathbf {s}^{(h)}_{\frac 12} ;\mathbf {v};\mathbf {s}^{(v)})$ has the parameters

\begin{equation*} \mathbf {u}=(u,\ldots ,u ), \quad \mathbf {s}^{(h)}_{\frac 12}=(q^{-1/2},\ldots ,q^{-1/2} ), \end{equation*}

where $\mathbf {v},\mathbf {s}^{(v)}\in \mathbb {C}^n$ are arbitrary. The $N$ -tuple $ \mathbf {V}=\mathbf {V}_\eta := (\mathbf {e}_{\eta _1} \mathbf {1}_{\eta _1 \ge 1},\ldots , \mathbf {e}_{\eta _N} \mathbf {1}_{\eta _N \ge 1})$ encodes the same information as the mASEP state $\eta =(\eta _1,\ldots ,\eta _N )$ . Indeed, this is because each of the subsets, $\mathbf {V}_\eta (j)\subset \mathbb {Z}_{\ge 0}^n$ , $j=1,\ldots ,N$ , must have at most one element thanks to the spin- $\frac 12$ reduction coming from $\mathbf {s}^{(h)}_{\frac 12}$ .

We split this subsection into two parts. First, we show in Proposition 4.5 that the normalized partition functions of the queue vertex model produce the mASEP stationary distribution. Then we present in Proposition 4.9 a slightly modified vertex model for which all partition functions on the cylinder with right boundary $\eta$ are positive without normalization.

Proposition 4.5. With the above notation and for generic complex parameters $u,\mathbf {v},\mathbf {s}^{(v)}$ , the queue vertex model partition function with boundary $\eta$ and vertex weights ${\mathbb {W}}_{q^{-1/2},s_m^{(v)},u/v_m}^{(-m)}$ is proportional to the stationary probability for the multi-species ASEP on the ring:

(4.7) \begin{equation} \mathop {\mathrm {Prob}}\nolimits ^{\mathrm {mASEP}}_{N_1,\ldots ,N_n }(\eta ) = \frac {1}{Z^{\mathrm {mASEP}}_{N_1,\ldots ,N_n }(u ;\mathbf {v};\mathbf {s}^{(v)})}\langle \emptyset | \ \mathfrak {Q}(\mathbf {u};\mathbf {s}^{(h)}_{\frac 12} ;\mathbf {v};\mathbf {s}^{(v)}) \ | \mathbf {V}_\eta \rangle . \end{equation}

The normalizing constant $Z^{\mathrm {mASEP}}_{N_1,\ldots ,N_n }(u ;\mathbf {v};\mathbf {s}^{(v)})$ may depend on the type counts $(N_1,\ldots ,N_n)$ (4.2), but not on the state $\eta$ within the sector determined by $(N_1,\ldots ,N_n)$ .

In Proposition 4.5, by ‘generic’ we mean that the parameters must ensure that $\langle \emptyset | \ \mathfrak {Q} \ | \mathbf {V}_\eta \rangle$ is finite and nonzero for all $\eta$ with type counts $N_m\ge 1$ , $m=1,\ldots ,n$ . For fixed $N$ and $n$ , genericity is ensured by excluding zero sets of finitely many polynomials in $u,\mathbf {v}$ , and $\mathbf {s}^{(v)}$ from the parameter space. Later in Proposition 4.9, we present concrete conditions on the parameters producing stationary measures for all $N$ and $n$ .

Proof of Proposition 4.5. Iterating Theorem 3.3, we have

\begin{equation*} \langle \emptyset | \ \mathfrak {Q} ( (u_1,u,\ldots ,u ); \mathbf {s}_{\frac {1}{2}}^{(h)}; \mathbf {v}; \mathbf {s}^{(v)}) \ \mathfrak {T}(u, u_1)^{\lfloor (1-q)t / \epsilon \rfloor } = \langle \emptyset | \ \mathfrak {Q} ( (u_1,u,\ldots ,u ); \mathbf {s}_{\frac {1}{2}}^{(h)}; \mathbf {v}; \mathbf {s}^{(v)}). \end{equation*}

Setting $u_1=u(1-\epsilon )$ and sending $\epsilon \to 0$ turns the power of the twisted cylinder Markov operator in the left-hand side into $\mathfrak {P}_{\mathrm {mASEP}}(t)$ ; see Proposition 4.4. The vertex weights in the queue vertex model with horizontal spin $\frac 12$ are given in Figure 15 (recall that $\mathfrak {Q}$ involves the weights ${\mathbb {W}}_{q^{-1/2},s_m^{(v)},u/v_m}^{(-m)}$ ). These vertex weights are continuous in the spectral parameter $u$ for generic parameters. Therefore, we can simply take the limit $u_1\to u$ in the queue transfer matrix, which implies that $\langle \emptyset | \ \mathfrak {Q} ( (u,u,\ldots ,u ); \mathbf {s}_{\frac {1}{2}}^{(h)}; \mathbf {v}; \mathbf {s}^{(v)})$ is the left (row) eigenvector of the Markov semigroup $\mathfrak {P}_{\mathrm {mASEP}}(t)$ with eigenvalue $1$ (where here the exchange of the limit and the implicit summation in both sides of the above equation is justified by the fact that the summation defining each partition function is uniformly bounded by a geometric series, and the sum over $\mathbf {V}_{\eta }$ on the left hand side has a bounded number of terms, independently of $\epsilon$ ). Thus, it is proportional to the row vector representing the stationary distribution of mASEP, as desired.

Figure 15.

Weights ${\mathbb {W}}_{q^{-1/2},s,uq^{1/2}}^{(-m)}(\mathbf {A},k;\mathbf {C},\ell )$ entering the spin- $\frac 12$ queue vertex model which represents the stationary distribution of mASEP. Here $m\lt k\lt \ell \le n$ , and it suffices to consider only vertices with no colors $\le m$ entering from the left. Recall that $A_m=+\infty$ , so the vertices in the last column also satisfy the arrow conservation property.

Definition 4.7. For $s\ne 0$ let us define the mASEP gauge transformation of the queue vertex weights:

(4.8) \begin{equation} {\mathbb {W}}_{q^{-1/2},s,uq^{1/2}}^{(-m),\mathrm {mASEP}_+}(\mathbf {A},k;\mathbf {C},\ell ) := (-1/s)^{\mathbf {1}_{\ell \ge 1}}\ {\mathbb {W}}_{q^{-1/2},s,uq^{1/2}}^{(-m)}(\mathbf {A},k;\mathbf {C},\ell ). \end{equation}

That is, we remove the factor $(-s)$ from the weights in the right three columns in Figure 15. The resulting weights make sense for $s=0$ , too. The notation ` $+$ ’ in (4.8) indicates that we will impose conditions on the parameters under which these weights are nonnegative.

Also, denote by

(4.9) \begin{equation} \frac {\langle \emptyset | \ \mathfrak {Q}^{\mathrm {mASEP}_+} \ | \mathbf {V}_\eta \rangle }{ Z^{\mathrm {mASEP}_+}_{N_1,\ldots ,N_n }} \end{equation}

the corresponding normalized partition function of the queue vertex model on the cylinder with the right boundary $\eta$ .

Proposition 4.9. Fix the type counts $(N_1,\ldots ,N_n)$ with $N_m\ge 1$ for all $m$ . Let the parameters satisfy

(4.10) \begin{equation} 0 \le s_m^{(v)} \lt \frac {u}{v_m q^{1/2}}, \quad s_m^{(v)} \frac {u}{v_m q^{1/2}} \lt 1, \quad m = 1,\ldots ,n. \end{equation}

Then

\begin{equation*} \mathop {\mathrm {Prob}}\nolimits ^{\mathrm {mASEP}}_{N_1,\ldots ,N_n }(\eta ) = \frac { \langle \emptyset | \ \mathfrak {Q}^{\mathrm {mASEP}_+} \ | \mathbf {V}_\eta \rangle }{ Z^{\mathrm {mASEP}_+}_{N_1,\ldots ,N_n }} \end{equation*}

is the mASEP stationary distribution, and $\langle \emptyset | \ \mathfrak {Q}^{\mathrm {mASEP}_+} \ | \mathbf {V}_\eta \rangle \gt 0$ for all $\eta$ .

Note that conditions (4.10) are written for $u/v_m$ entering the vertex weights as ${\mathbb {W}}_{q^{-1/2},s_m^{(v)},u/v_m}^{(-m),\mathrm {mASEP}_+}$ , while for displaying the weights in Figure 15 it was convenient to mulitply the spectral parameter by $q^{1/2}$ .

Proof of Proposition 4.9 . Assume first that $s_m^{(v)}\gt 0$ for all $m$ . Replacing the queue vertex weights with their gauge transformed versions (4.8) multiplies the partition function $\langle \emptyset | \ \mathfrak {Q} \ | \mathbf {V}_\eta \rangle$ by

\begin{equation*} \prod \nolimits _{m=1}^{n}(-1/s_m^{(v)})^{N_m+\cdots +N_n }, \end{equation*}

which depends only on the sector, but not on the configuration $\eta$ . Therefore, the gauge transformation may be incorporated into the normalizing constant $Z^{\mathrm {mASEP}_+}_{N_1,\ldots ,N_n }$ . One readily sees that under Conditions (4.10) and when $s_m^{(v)}\gt 0$ for all $m$ , all vertex weights (4.8) are positive (see Figure 15). This completes the proof in the case when all the $s_m^{(v)}$ are positive.

Setting some (or all) of the $s_m^{(v)}$ to zero in the weights (4.8) is allowed, and leads to a well-defined partition function $\langle \emptyset | \ \mathfrak {Q}^{\mathrm {mASEP}_+} \ | \mathbf {V}_\eta \rangle$ . In this partition function, some of the vertex weights in Figure 15 vanish. To show that $\langle \emptyset | \ \mathfrak {Q}^{\mathrm {mASEP}_+} \ | \mathbf {V}_\eta \rangle$ is still positive and not merely nonnegative for all $\eta$ , first notice that there exists $\eta$ for which $\langle \emptyset | \ \mathfrak {Q}^{\mathrm {mASEP}_+} \ | \mathbf {V}_\eta \rangle \ne 0$ (this verification is straightforward, and we omit it). Next, observe that

(4.11) \begin{equation} \sum \nolimits _{\eta } \langle \emptyset | \ \mathfrak {Q}^{\mathrm {mASEP}_+} \ | \mathbf {V}_\eta \rangle \langle \mathbf {V}_\eta | \end{equation}

is a nonzero left (row) eigenvector with eigenvalue $1$ of the mASEP semigroup $\mathfrak {P}_{\mathrm {mASEP}}(t)$ corresponding to an irreducible continuous-time Markov process on a finite state space. Therefore, (4.11) is proportional to the Perron–Frobenius eigenvector of $\mathfrak {P}_{\mathrm {mASEP}}(t)$ , which has all components positive. This completes the proof.

4.3 Matching to multiline queues

4.3.1 Original multiline queues

First, we connect the queue vertex model $\mathfrak {Q}^{\mathrm {mASEP}_+}$ with special parameters with multiline queues introduced by Martin [Reference MartinMar20]. He showed that the output of the latter produces the stationary distribution of mASEP on the ring.

Setting $u=q^{1/2}$ and $s_m^{(v)}=0$ , $v_m=1$ for all $m=1,\ldots ,N$ in the vertex weights (4.8) leads to the weights given in Figure 16 which we denote by

(4.12) \begin{equation} {\mathbb {W}}^{(-m),\mathrm {mq}} := {\mathbb {W}}_{q^{-1/2},0,q^{1/2}}^{(-m),\mathrm {mASEP}_+}. \end{equation}

By Proposition 4.9, these weights produce positive partition functions of the queue vertex model on the cylinder.

Figure 16.

Weights ${\mathbb {W}}^{(-m),\mathrm {mq}}$ (4.12). In Proposition 4.10 we match them to probabilities under multiline queues of [Reference MartinMar20]. Here $m\lt k\lt \ell \le n$ , and recall (Remark 4.2) that our ordering of colors is reversed compared to particle types in [Reference MartinMar20].

Let us connect the queue vertex model $\mathfrak {Q}^{\mathrm {mASEP}_+}$ with these particular parameters to multiline queue diagrams. These diagrams were defined in [Reference MartinMar20, §§ 1.1 and 3.6], and the vertex model interpretation follows from formula (3.9) in [Reference MartinMar20]. For the reader’s convenience, in the rest of § 4.3.1 we reproduce the main definitions and the matching of queues to our vertex models.

Let us recast [Reference MartinMar20, Algorithm 2] (called the Martin algorithm in what follows) in the language of one column of a vertex model on the cylinder. Fix $m=1,\ldots ,n$ , type counts $N_m,N_{m+1},\ldots ,N_n$ , and assume that we have an arbitrary fixed configuration $\eta$ of paths of colors strictly larger than $m$ entering the column $(-m)$ from the left. The configuration $\eta$ has $N_i$ paths of color $i$ , $i\gt m$ . The Martin algorithm (for color $m$ ) samples a random new configuration $\eta '$ of paths exiting the column $(-m)$ to the right. The configuration $\eta '$ has $N_i$ paths of color $i \ge m$ ; it is constructed sequentially, through the following three steps.

1. (1) Start with the empty configuration $\eta '=\{0,\ldots ,0 \}$ ( $N$ zeros). In addition, sample a uniformly random subset $\mathcal {J}\subset \{1,\ldots ,N \}$ of sites on the cylinder of cardinality $N_m+\cdots +N_n$ . Next, we randomly update $\eta '$ such that in the end $\eta _j'\gt 0$ if and only if $j \in \mathcal {J}$ .

2. (2) For each color $i=n,n-1,\ldots ,m+1$ (in this order), let $a_1^i\lt \cdots \lt a^i_{N_i}$ be the locations of paths of color $i$ in $\eta$ .

   1. (2a) For $j=1,\ldots ,N_i$ , if $\eta '_{a_j^i}=0$ and $a_j^i \in \mathcal {J}$ , set $\eta '_{a_j^i}=i$ (if a path of color $i$ can come straight through, it does so).

   2. (2b) Otherwise, the $j$ th path of color $i$ starts from $\eta _{a_j^i}$ and randomly chooses an exit site among yet unoccupied sites in $\mathcal {J}$ as follows. Let $a_j^i\lt p_1\lt p_2\lt \cdots \lt p_l$ (here, $p \lt p'$ means that as we read upwards starting from $p$ , possibly wrapping around in the vertical direction, we observe $p'$ before getting back to $a_{j}^i$ ), where $(p_1,\ldots ,p_l )$ are all sites in $\mathcal {J}$ for which at this point we have $\eta '_{p_t}=0$ , $t=1,\ldots ,l$ . Then, set $\eta '_{p_t}=i$ with probability proportional to $q^{t-1}$ .

   3. (2b^′) Equivalently, instead of step (2b), one can think that the color $i$ path starting from site $a_j^i$ goes up the cylinder and sequentially with probability $1-q$ picks an unoccupied site from $\mathcal {J}$ to exit, or with probability $q$ skips this site (accepts or declines the service, in queueing terminology). The path continues the motion up the cylinder until its exit, and can go around the cylinder an arbitrary number of times. If the path exits at $p_t$ , then after normalization, this produces the same probability proportional to $q^{t-1}$ .

3. (3) Once all paths of all colors strictly larger than $m$ are processed, we have $N_m$ unoccupied sites in $\mathcal {J}$ left. We set $\eta '_j=m$ for all these remaining sites.

To obtain the mASEP stationary distribution $\mathop {\mathrm {Prob}}\nolimits ^{\mathrm {mASEP}}_{N_1,\ldots ,N_n }$ (Definition 4.3), one needs to apply the Martin algorithm for color $n$ with input $\eta (0)=\emptyset$ , and get a random output $\eta (1)$ . Then apply the algorithm for color $n-1$ with input $\eta (1)$ , get an output $\eta (2)$ , and so on. The final output $\eta (n)$ of the algorithm for color $1$ is the random configuration distributed according to the mASEP stationary distribution.

The following statement matches the output of the Martin algorithm to vertex models and essentially coincides with [Reference MartinMar20, Theorem 3.4]. For convenience, we reproduce it here.

Proposition 4.10. In each sector determined by the fixed type counts $(N_1,\ldots ,N_n)$ , the output $\eta$ of the Martin algorithm has the same distribution as the output of our queue vertex model on the cylinder with the weights ${\mathbb {W}}^{(-m),\mathrm {mq}}$ given in Figure 16 .

Idea of proof. This follows by matching the vertex weights in Figure 16 to the weights $w_i(Q | A, S)$ given in [Reference MartinMar20, (3.9)]. The translation from the queueing language to vertex models is straightforward and we omit it.

In [Reference MartinMar20], the stationarity of the output $\eta$ of the Martin algorithm under mASEP follows from the Matrix Product Ansatz. The connection between the algorithm and the Matrix Product Ansatz is essentially equivalent to Proposition 4.10. We link queue vertex models to Matrix Product Ansatz in § 4.4 below.

Remark 4.11. While the Martin algorithm and the queue vertex model produce the same output $\eta$ (in distribution, in each sector), it remains unclear whether one can define appropriate ‘states’ of the queueing system under the Martin algorithm such that these states are in a weight-preserving bijection (possibly up to a common proportionality constant) with states of the queue vertex model. Indeed, tracking each particle’s choices as in step (2b^′) of the Martin algorithm with the input as in Figure 17 involves the following information.

• – Track how many times the path of color $4$ wraps around the cylinder.

• – Pick a bijection between the color $3$ inputs and outputs (there are $2!$ choices in Figure 17).

• – For each of the two bijections, track how many times each of the two color $3$ paths wraps around the cylinder.

Figure 17.

The Martin algorithm from [Reference MartinMar20] with $N=6$ , $n=4$ , in the column $(-m)$ , where $m=2$ . The set $\mathcal {J}$ is $\{ 1,4,5,6 \}$ . Given this $\mathcal {J}$ , the conditional probability of the configuration in the figure (according to the description with (2b^′)) is proportional to $q^5$ (the color $4$ path skips five possibilities) times $q^2$ (the top color $3$ path skips two possibilities) times $1$ (the bottom color $3$ path selects the first available possibility). Colors of the arrows are indicated by the numbers and by different dashing. The boxed numbers on the right indicate the ring sites.

Under the queue vertex model, we do not choose a bijection, and the wrapping arrows of colors $m+1,\ldots ,n$ are encoded by the tuple $\mathbf {M}(-m)$ . Here $\mathbf {M}(-m)_k$ is the total number of arrows of color $k$ that wrap around the ring (that is, go between the sites $N$ and $1$ ).

To show that summing over all data in the queueing system produces the desired distribution of the output $\eta$ , one seems to require the intricate argument with compatible queue-length processes; see [Reference MartinMar20, § 4.2] for details.

4.3.2 Alternative multiline queues and an interpolation

Let us consider a different specialization of the queue vertex model $\mathfrak {Q}^{\mathrm {mASEP}_+}$ introduced in § 4.2. Namely, set $u=1$ and $s = s_m^{(v)}=q$ , $v_m=1$ for all $m=1,\ldots ,N$ in (4.8), and clear the common denominators $1-q$ . The resulting weights $ {\mathbb {W}}^{(-m),\mathrm {mq\,alt}} := (1-q)\ {\mathbb {W}}_{q^{-1/2},q,q^{1/2}}^{(-m),\mathrm {mASEP}_+}$ are given in Figure 18. By Proposition 4.9, they lead to positive partition functions on the cylinder.

Figure 18.

Weights ${\mathbb {W}}^{(-m),\mathrm {mq\,alt}}$ for the alternative multiline queue model.

Just as by Proposition 4.10, the weights ${\mathbb {W}}^{(-m),\mathrm {mq}}$ from Figure 16 produce the same output $\eta$ (in distribution) as the Martin algorithm, the new weights ${\mathbb {W}}^{(-m),\mathrm {mq\,alt}}$ should be related to the alternative multiline queue model introduced in [Reference MartinMar20, § 7].

By definition, the alternative algorithm for color $m$ consists of the same steps as the algorithm described in § 4.3.1 above, except step (2a). Instead, if $\eta '_{a^i_j}=0$ and $a^i_j\in \mathcal {J}$ , then the entering color $i$ path ( $i\gt m$ ) from site $a^i_j$ still has probability $q$ to go up the cylinder and not exit immediately through this site.

Proposition 4.12. In each sector determined by the fixed type counts $(N_1,\ldots ,N_n)$ , the output $\eta$ of the alternative multiline queue algorithm has the same distribution as the output of the queue vertex model on the cylinder with the weights ${\mathbb {W}}^{(-m),\mathrm {mq\,alt}}$ given in Figure 18 .

Idea of proof. This is established very similarly to [Reference MartinMar20, § 4.2], by using the notion of compatibility of a queue-length process with ‘marks’. Here marks encode geometrically distributed choices of how many times a service is rejected at a site before being accepted. These choices may be done beforehand. The queue-length process corresponds to the vertical paths in the queue vertex model. If one interprets the randomness in the alternative model in terms of marks, then one can see the vertex weights ${\mathbb {W}}^{(-m),\mathrm {mq\,alt}}$ . We omit the technical details of this proof.

Corollary 4.13 (Conjecture from [Reference MartinMar20, § 7]). The output $\eta$ of the alternative multiline queue algorithm is stationary under the mASEP dynamics on the ring.

Proof. We already know that the output of the queue vertex model with the weights ${\mathbb {W}}^{(-m),\mathrm {mq\,alt}}$ is stationary under the mASEP dynamics on the ring, thanks to our general Propositions 4.5 and 4.9 which ultimately rely on the Yang–Baxter equation for the twisted cylinder (Theorem 3.3). Applying Proposition 4.12, we arrive at the desired statement.

Figure 19.

Weights $(1-s)\ {\mathbb {W}}_{q^{-1/2},s,q^{1/2}}^{(-m),\mathrm {mASEP}_+}$ interpolating between the vertex weights related to the original and the alternative multiline queues of [Reference MartinMar20].

If we do not specialize the parameter $s$ to $0$ or $q$ , we obtain a family of queue vertex weights depending on $q$ and $s$ (see Figure 19). The output of the vertex model with these weights produces the mASEP stationary distribution (this again follows from Propositions 4.5 and 4.9). The $(q,s)$ -dependent weights should be related to a new multiline queue model that interpolates between the original and the alternative multiline queues.

Definition 4.14 (Interpolating multiline queues). Let $s\in [0,1)$ , and modify the Martin algorithm (for a given color $m$ ) by changing step (2a) as follows. If a color $i$ path ( $i\gt m$ ) enters at $a^i_j$ and $\eta '_{a^i_j}=0$ (that is, a service is immediately available), then the path exits (accepts the service) with probability $1-s$ . With probability $s$ , the path turns up the cylinder and skips every successive available exit (service) with probability $q$ (as prescribed by step (2b^′)). All other parts of the algorithm remain the same.

For $q=0$ (when mASEP becomes the multi-species TASEP), the interpolating model produces a multiline queue model with random service assignment. Note that for $q=0$ , both the original and the alternative multiline queues become the same and are deterministic. For two colors, this deterministic model was constructed in [Reference AngelAng06] to describe the stationary distribution of the two-color TASEP. It was generalized to $n$ colors in [Reference Ferrari and MartinFM07].

Similarly to the argument in [Reference MartinMar20, § 4.2], it should be possible to identify the output of the interpolating multiline queues with that of the $(q,s)$ -dependent queue vertex model on the cylinder. In § 4.4 below, we outline a possible connection of the latter with the Matrix Product Ansatz.

4.4 Connection to Matrix Product Ansatz

Prior to the multiline queue realization of the mASEP stationary distribution $\mathop {\mathrm {Prob}}\nolimits ^{\mathrm {mASEP}}_{N_1,\ldots ,N_n }$ in [Reference MartinMar20], Prolhac, Evans and Mallick [Reference Prolhac, Evans and MallickPEM09] showed that $\mathop {\mathrm {Prob}}\nolimits ^{\mathrm {mASEP}}_{N_1,\ldots ,N_n }$ can be expressed in a matrix product form. For processes on the ring, this expression has the same format as our general trace formula (2.20), see (4.14) below, and includes matrices $\mathscr{X}^{{\kern2pt}\mathrm {MPA}}_{m}$ , $m=0,1,\ldots ,n$ , indexed by available colors. Matrix product ansatz representations for stationary probabilities of stochastic interacting particle systems date back to [Reference Derrida, Evans, Hakim and PasquierDEHP93]. In the single-species case, the stationary distribution on the ring is uniform, and so the Matrix Product Ansatz becomes nontrivial only for ASEP on an open interval, in which particles can hop in and out at the endpoints. This case was considered in [Reference Derrida, Evans, Hakim and PasquierDEHP93]. For the two- and three-species ASEP on the ring, the matrix product approach was employed, respectively, in [Reference Derrida, Janowsky, Lebowitz and SpeerDJLS93] and [Reference Mallick, Mallick and RajewskyMMR99]. A full multi-species solution on the ring appeared about ten years later in [Reference Prolhac, Evans and MallickPEM09]. See also [Reference Blythe and EvansBE07] for an earlier survey of Matrix Product Ansatz applications to particle systems.

In the multi-species case, the matrices $\mathscr{X}^{{\kern2pt}\mathrm {MPA}}_{m}$ entering the product ansatz are constructed by recursive tensoring from a few single-species building blocks $A,D$ , and $E$ satisfying quadratic relations

(4.13) \begin{equation} AD-qDA= EA-qAE=(1-q)A, \quad ED-qDE=(1-q)(E+D). \end{equation}

The matrices $A,D,E$ , as well as the $\mathscr{X}_m^{{\kern2pt}\mathrm {MPA}}$ , are infinite-dimensional, and their products, as well as the trace of $A$ times a finite product of $D$ and $E$ matrices must be well defined. The tensoring construction of $\mathscr{X}^{{\kern2pt}\mathrm {MPA}}_{m}$ resembles the process of horizontally stacking the vertices in columns $-n,-n+1,\ldots ,-1$ as in Figure 9. We refer to [Reference Prolhac, Evans and MallickPEM09, (24)–(33)] or [Reference MartinMar20, § 2] for details on the tensoring construction, and omit them here.

Once the matrix product probability distribution is defined through the trace in an appropriate space as

(4.14) \begin{equation} \mathop {\mathrm {Prob}}\nolimits ^{\mathrm {mASEP}}_{N_1,\ldots ,N_n } (\eta )= \frac {\mathop {\mathrm {Trace}} \left ( \mathscr{X}^{{\kern2pt}\mathrm {MPA}}_{\eta _1}\cdots \mathscr{X}^{{\kern2pt}\mathrm {MPA}}_{\eta _N} \right )}{Z^{\mathrm {MPA}}_{N_1,\ldots ,N_n }} , \quad \eta =(\eta _1,\ldots ,\eta _N ), \end{equation}

one must independently check that it is stationary under the mASEP dynamics. A key property in the argument is the existence of the so-called hat matrices $\skew5\widehat {\mathscr{X}}{}_m^{{\kern2.5pt}\mathrm {MPA}}$ , $m=0,1,\ldots ,N$ , satisfying quadratic relations [Reference Prolhac, Evans and MallickPEM09, (68)]:

(4.15) \begin{equation} \sum _{i,i'=0}^{n} \mathscr{X}_i^{{\kern2pt}\mathrm {MPA}} \mathscr{X}_{i'}^{{\kern2pt}\mathrm {MPA}} ( \mathscr{M}_{\textrm{loc}} )_{i i', j j'} = \mathscr{X}_j^{{\kern2pt}\mathrm {MPA}} \skew5\widehat {\mathscr{X}}{}_{j'}^{{\kern2.5pt}\mathrm {MPA}} - \skew5\widehat {\mathscr{X}}{}_{j}^{{\kern2.5pt}\mathrm {MPA}} \mathscr{X}_{j'}^{{\kern2pt}\mathrm {MPA}} , \end{equation}

where $\mathscr{M}_{\textrm{loc}}$ are the local infinitesimal rates of the mASEP, see (4.3). These hat matrices are also constructed in [Reference Prolhac, Evans and MallickPEM09] by recursive tensoring procedures. Note that our notation differs from [Reference Prolhac, Evans and MallickPEM09] by a transposition (e.g. comparing (4.15) with formula (66) in [Reference Prolhac, Evans and MallickPEM09]). See also [Reference Arita, Ayyer, Mallick and ProlhacAAMP12] for a more general family of matrices $\mathscr{X}_j^{{\kern2pt}\mathrm {MPA}}$ , $\skew5\widehat {\mathscr{X}}{}_{j}^{{\kern2.5pt}\mathrm {MPA}}$ satisfying (4.15).

Let us explain how the construction of the hat matrices, and identity (4.15) for vertex model partition functions directly follow from the Yang–Baxter equation. We take a concrete realization of the Matrix Product Ansatz matrices $\mathscr{X}_m^{{\kern2pt}\mathrm {MPA}}$ using our vertex models.

Namely, let $\mathscr{X}_j^{{\kern2pt}\mathrm {MPA}}(u)$ , $j=0,1,\ldots ,n$ , be operators in $V_{-n}\otimes \cdots \otimes V_{-1}$ , where $V_{-m}$ has basis $|\mathbf {M}(-m)\rangle$ , $\mathbf {M}(-m)\in \mathbb {Z}_{\ge 0}^{n}$ , $m=1,\ldots ,n$ . By definition, the matrix elements of $\mathscr{X}^{{\kern2pt}\mathrm {MPA}}_j$ are partition functions (with weights ${\mathbb {W}}_{q^{-1/2},s,u}$ ) on the one-row lattice $\{1\}\times \{-n,\ldots ,-1 \}$ with boundary conditions $0$ and $j$ on the left and right, respectively. See Figure 9 for an illustration.

Let $\epsilon \gt 0$ be small. By the Yang–Baxter equation (Proposition 2.10), the matrices $\mathscr{X}^{{\kern2pt}\mathrm {MPA}}_i(u)$ and $\mathscr{X}^{{\kern2pt}\mathrm {MPA}}_{i'}(u(1-\epsilon ))$ satisfy the following identity involving the elements of $R_{1-\epsilon }$ (2.4):

(4.16) \begin{equation} \sum _{i,i'=0}^{n} \mathscr{X}^{{\kern2pt}\mathrm {MPA}}_i(u) \ \mathscr{X}^{{\kern2pt}\mathrm {MPA}}_{i'}(u(1-\epsilon )) \cdot R_{1-\epsilon }(i,i';j',j)= \mathscr{X}^{{\kern2pt}\mathrm {MPA}}_{j}(u(1-\epsilon )) \ \mathscr{X}^{{\kern2pt}\mathrm {MPA}}_{j'}(u). \end{equation}

As in the proof of Proposition 4.4, let us differentiate (4.16) with respect to $\epsilon$ at $\epsilon =0$ . Denote

\begin{equation*} \skew5\widehat {\mathscr{X}}{}^{{\kern2.5pt}\mathrm {MPA}}_j(u):= (1-q)\ u\ \frac {\partial }{\partial u}\ \mathscr{X}^{{\kern2pt}\mathrm {MPA}}_j(u),\quad j=0,1,\ldots ,n . \end{equation*}

Our realization of the matrices $\mathscr{X}_j^{{\kern2pt}\mathrm {MPA}}$ , $\skew5\widehat {\mathscr{X}}{}_{j}^{{\kern2.5pt}\mathrm {MPA}}$ satisfying the hat relation (4.15) as vertex model partition functions highlights the Yang–Baxter structure of the Matrix Product Ansatz for the multi-species ASEP which was previously less transparent (see, however, [Reference Cantini, de Gier and WheelerCdGW15] and [Reference Crampe, Ragoucy and VanicatCRV14] for earlier connections between the two structures).

Proof of Proposition 4.15 . The $\epsilon$ -derivative at $\epsilon =0$ of the right-hand side of (4.16) is equal to

\begin{equation*}- (1-q)^{-1}\skew5\widehat {\mathscr{X}}{}_j^{{\kern2.5pt}\mathrm {MPA}}(u) \mathscr{X}_{j'}^{{\kern2pt}\mathrm {MPA}}(u).\end{equation*}

In the left-hand side, differentiating $\mathscr{X}_{i'}^{{\kern2pt}\mathrm {MPA}}$ and noticing that $R_{1}(i,i';j',j)=\mathbf {1}_{i=j}\mathbf {1}_{i'=j'}$ , we obtain $- (1-q)^{-1} \mathscr{X}^{{\kern2pt}\mathrm {MPA}}_j(u) \skew5\widehat {\mathscr{X}}{}^{{\kern2.5pt}\mathrm {MPA}}_{j'}(u)$ . This yields the second summand in the right-hand side of (4.15). Finally, differentiating $R_{1-\epsilon }$ and using (4.6), we recover the local infinitesimal rates of mASEP, also multiplied by $(1-q)^{-1}$ . This completes the proof.

Let us also note that one can ‘recognize’ the matrix product building blocks $A,D,E$ in the vertex weights of the queue vertex model on the cylinder. This allows us to insert two extra parameters into the matrices. Namely, in [Reference MartinMar20], two examples of $A,D,E$ are presented, for the original and for the alternative multiline queue models discussed in § 4.3 above. Both of these examples satisfy the relations (4.13). In these two examples, the matrices $A, D , E$ are equal to the $n = 2$ species versions of the corresponding $\mathscr{X}^{{\kern2pt}\mathrm {MPA}}$ matrices. Informed by this, and now keeping track of the parameters $u$ and $s$ in the vertex weights, let us define

(4.17) \begin{equation} \begin{split} & A := \!\left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & s & 0 & \ldots \\ 0 & q & q s & \ldots \\ 0 & 0 & q^2 & \ldots \\ \vdots & \vdots & \vdots & \ddots \\ \end{array} \!\!\!\!\right )\!, \;\;\;\; D:= u^{-1}\! \left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} u-s & 0 & 0 & \ldots \\ 1-q & u- sq & 0 & \ldots \\ 0 & 1-q^2 & u-sq^2 & \ldots \\ \vdots & \vdots & \vdots & \ddots \\ \end{array}\!\!\!\! \right )\!,\;\;\;\; E:= \!\left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & u & 0 & \ldots \\ 0 & 1 & u & \ldots \\ 0 & 0 & 1 & \ldots \\ \vdots & \vdots & \vdots & \ddots \\ \end{array}\! \!\!\!\right )\!\!. \end{split} \end{equation}

These matrices can be obtained as one-row partition functions of the two-color queue-specialized vertex model as shown in Figure 21. The relevant vertex weights ${\mathbb {W}}_{q^{-1/2},s,uq^{1/2}}^{(-m),\mathrm {mASEP}_+}$ are shown in Figure 20. We use the same parameter $u$ in both columns.

Figure 20.

Table of weights ${\mathbb {W}}_{q^{-1/2},s,uq^{1/2}}^{(-m),\mathrm {mASEP}_+}$ (4.8) with common denominators $(1-su)$ cleared, which are used in the construction of the matrices $A, D, E$ satisfying (4.13).

Figure 21.

The $(M,M')$ martrix elements of the matrices $A, D, E$ are obtained from the queue-specialized vertex model. The symbol $\infty _2$ represents $(0, +\infty )$ (infinitely many color $2$ paths, and no color $1$ paths). Similarly, $(\infty , M)$ means a path configuration with $M$ color $2$ paths and $+\infty$ color $1$ paths.

One can directly check that the matrices (4.17) satisfy (4.13) for all $u,s$ . Alternatively, relations (4.13) can be obtained from the Yang–Baxter equation, satisfied by $A, u^2 D, E$ and the $R$ matrix, in the same way as described in the proof of Proposition 4.15. When $u=1$ and $s=0$ , the matrices (4.17) become the matrix product building blocks of [Reference Prolhac, Evans and MallickPEM09], and lead to the original multiline queue model [Reference MartinMar20, (2.5)]. For $u=1$ and $s=q$ , they correspond to the alternative model from [Reference MartinMar20, § 7]. However, for the alternative queue, it is not known whether the recursive tensoring of the matrices $A,D,E$ (with $u=1$ , $s=q$ ) like in [Reference Prolhac, Evans and MallickPEM09] produces a matrix product ansatz for mASEP. We do not pursue this question here.

We remark that matrices similar to (4.17) have occurred in the product ansatz for the open ASEP on a bounded interval (for example, see [Reference Derrida, Evans, Hakim and PasquierDEHP93]). There, the extra parameters are tied to the boundary rates.

5.1 Colored $q$ -Boson process on the ring

Let us fix the size of the ring $N$ and the number of colors $n$ . Also let us fix the type counts $(N_1,\ldots ,N_n )$ , where $N_i\ge 1$ stands for the number of particles of color $i$ in the system. The state space of the colored stochastic $q$ -Boson process consists of configurations of particles at sites of the ring, where at each site there can be an arbitrary number of particles. The configurations are encoded by

\begin{equation*} \mathbf {V}=(\mathbf {V}(1),\ldots ,\mathbf {V}(N) ), \quad \mathbf {V}(j)\in \mathbb {Z}_{\ge 0}^n. \end{equation*}

Here $\mathbf {V}(j)_i$ denotes the number of particles of color $i$ at site $j$ , and $N_i=\sum _{j=1}^{N}\mathbf {V}(j)_i$ .

Definition 5.1. The stochastic colored $q$ -Boson process depends on parameters $q\in [0,1)$ and $u_1,\ldots ,u_N\gt 0$ , and evolves in continuous time as follows. A particle of color $i$ hops from site $k$ to site $k-1$ (cyclically mod $N$ ) according to an independent exponential clock with rate

\begin{equation*} u_k^{-1}(1-q^{\mathbf {V}(k)_i})\ q^{\mathbf {V}(k)_{[i+1,n]}}. \end{equation*}

Here we used the usual notation $\mathbf {V}(k)_{[i+1,n]}=\sum _{r=i+1}^n\mathbf {V}(k)_r$ . Denote by $\mathfrak {P}_{\mathrm {qBos}}(t)$ , $t\in \mathbb {R}_{\ge 0}$ , the continuous-time Markov semigroup of this stochastic process.

The colored $q$ -Boson process evolution is of zero-range kind, that is, the jump from site $k$ depends only on the state of the system at site $k$ . In [Reference Ayyer, Mandelshtam and MartinAMM22] it is referred to as the multi-species totally asymmetric zero-range process, or mTAZRP.

The $q$ -Boson process preserves the type counts $(N_1,\ldots ,N_n )$ . For a fixed vector of type counts, this continuous-time Markov chain evolves on a finite state space and is clearly irreducible. Thus, it has a unique stationary distribution. We denote it by $\mathop {\mathrm {Prob}}\nolimits ^{\mathrm {qBos}}_{N_1,\ldots ,N_n}(\mathbf {V})$ .

Following [Reference Borodin and WheelerBW22a, § 12.4.3], we can identify $\mathfrak {P}_{\mathrm {qBos}}(t)$ as a certain Poisson-type continuous-time limit of the straight cylinder formal Markov operator $\mathfrak {S}(x,\mathbf {u};\mathbf {s}^{(h)})$ defined in § 3.2. Recall that $\mathfrak {S}$ has $N+1$ sites on the ring. However, in the degeneration to the $q$ -Boson process, the distinguished site corresponding to spectral parameter $x$ (cf. Figure 12) will be empty with probability $1$ , and the dynamics can be restricted to $N$ sites.

Fix small $\epsilon \gt 0$ , and set the horizontal spin parameters of $\mathfrak {S}$ to $s_j^{(h)}=\epsilon$ . Also let $x=-1$ . With this specialization, the matrix elements of the operators $\mathscr{L}_{s_j^{(h)},xu_j^{-1}}$ entering the definition of $\mathfrak {S}$ (formula (3.6), see also Figure 5) become as in Figure 22. The operator $\mathfrak {S}$ is a formal Markov operator acting on states of the form $(c, \mathbf {V})$ , where $\mathbf {V}\in \mathbb {Z}_{\ge 0}^{n}$ is a state of the color $q$ -Boson process, and $c \in \{ 0,1,\ldots ,n \}$ corresponds to the auxiliary line (i.e., the one with the spectral parameter $x$ and the spin parameter $q^{-1/2}$ ; see Figure 12).

Figure 22.

The weights $\mathscr{L}_{\epsilon ,-u_j^{-1}}$ employed in the approximation of the colored $q$ -Boson process, Taylor expanded to $O(\epsilon ^2)$ or $1+O(\epsilon )$ depending on whether they go to $0$ or $1$ as $\epsilon \to 0$ . Here, as usual, $1\le k\lt \ell \le n$ .

Proposition 5.2. Fix $t\in \mathbb {R}_{\ge 0}$ . With the parameter specialization as above, have

\begin{equation*} \lim _{\epsilon \to 0}\, \langle 0, \mathbf {V}| \ \mathfrak {S}(-1,\mathbf {u};(\epsilon , \ldots ,\epsilon ))^{\lfloor t/\epsilon \rfloor } \ |0, \mathbf {V}'\rangle = \langle \mathbf {V}| \ \mathfrak {P}_{\mathrm {qBos}}(t) \ |\mathbf {V}'\rangle , \quad \mathbf {V},\mathbf {V}'\in \mathbb {Z}_{\ge 0}^n. \end{equation*}

Moreover, for any $c\ge 1$ we have $ \lim _{\epsilon \to 0} \langle 0, \mathbf {V}| \ \mathfrak {S}(-1,\mathbf {u};(\epsilon , \ldots ,\epsilon ))^{\lfloor t/\epsilon \rfloor } \ |c, \mathbf {V}'\rangle =0$ .

Not all matrix elements of $\mathfrak {S}$ are nonnegative before the $\epsilon \to 0$ limit. This is not a problem because the limiting semigroup $\mathfrak {P}_{\mathrm {qBos}}(t)$ is a nonnegative Markov semigroup, and the stationarity result (which we prove in Proposition 5.5 below) is a purely algebraic statement.

Proof of Proposition 5.2. Both statements follow from the expansions in Figure 22, after the identification of the vertices in the cylinder (in Figure 12) with the stochastic $q$ -Boson transitions via

where in the right-hand side the time is continuous and increases in the upward direction.

We see that the auxiliary line may become occupied with probability $O(\epsilon )$ , and then instantaneously becomes free again with probability $1+O(\epsilon )$ . This means that the vertices of the type $(\mathbf {A},k;\mathbf {A},k)$ are not present in the limit. All other probabilities of order $O(\epsilon )$ in Figure 22 give rise to the corresponding colored $q$ -Boson transitions, which leads to the first claim.

To get the second claim, observe that with probability going to $1$ in the limit as $\epsilon \to 0$ , the auxiliary line is not occupied. This completes the proof.

5.2 Vertex model for the $q$ -Boson stationary distribution

The convergence of Proposition 5.2 together with the general stationarity result (Theorem 3.6) allows us to express the stationary distribution $\mathop {\mathrm {Prob}}\nolimits ^{\mathrm {qBos}}_{N_1,\ldots ,N_n}(\mathbf {V})$ of the colored $q$ -Boson process as a vertex model partition function.

To get a queue vertex model on the cylinder with nonnegative vertex weights, we take a certain limit in the vertical parameters $\mathbf {v},\mathbf {s}^{(v)}$ . As a first step, let us consider the following degeneration of the queue vertex weights ${\mathbb {W}}_{s_1,s_2,u}^{(-m)}$ (2.12).

Lemma 5.3. We have

(5.1) \begin{equation} \begin{split} & ( u; q)_{\infty }^{-1}\cdot \lim _{s_1\to 0} {\mathbb {W}}_{s_1,s,us_1/s}^{(-m)}(\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})\\&\quad = \mathbf {1}_{\mathbf {A}+\mathbf {B}=\mathbf {C}+\mathbf {D}} \cdot \mathbf {1}_{D_1=\cdots =D_{m-1}=0} \\&\qquad\times \sum _{\mathbf {P}} (s^2/u ; q)_{|\mathbf {P}|} (s^2)^{|\mathbf {B}|-|\mathbf {P}|} u^{|\mathbf {D}|-|\mathbf {B}|+|\mathbf {P}|} \ q^{\sum _{1\le i\lt j\le n} (B_i-P_i) P_j} \prod _{i=1}^n \frac {(q;q)_{B_i}}{(q;q)_{P_i}(q;q)_{B_i-P_i}} \\&\qquad\times \ q^{\sum _{m\le i\lt j\le n} D_i (C_j-P_j)} \ \frac { 1} {(q;q)_{D_m}} \prod _{i=m+1}^n \frac {(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}}, \end{split} \end{equation}

where the sum is over $\mathbf {P}\in \mathbb {Z}_{\ge 0}^n$ with $0\le P_i\le \min (B_i,C_i)$ for all $i$ , and

(5.2) \begin{equation} \begin{split} & ( u; q)_{\infty }^{-1}\cdot \lim _{s\to 0}\big (\lim _{s_1\to 0} {\mathbb {W}}_{s_1,s,us_1/s}^{(-m)}(\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D}) \big )\\&\quad = \mathbf {1}_{\mathbf {A}+\mathbf {B}=\mathbf {C}+\mathbf {D}} \cdot \mathbf {1}_{D_1=\cdots =D_{m-1}=0}\cdot \prod _{i=1}^n\mathbf {1}_{B_i\le C_i} \\&\qquad \times u^{|\mathbf {D}|}\ q^{\sum _{m\le i\lt j\le n} D_i (C_j-B_j)} \ \frac { 1} {(q;q)_{D_m}} \prod _{i=m+1}^n \frac {(q;q)_{C_i-B_i+D_i}}{(q;q)_{C_i-B_i}(q;q)_{D_i}}. \end{split} \end{equation}

Proof. We have from (2.12):

\begin{align*} \begin{split} & {\mathbb {W}}_{s_1,s,us_1/s}^{(-m)}(\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})\\&\quad = \mathbf {1}_{\mathbf {A}+\mathbf {B}=\mathbf {C}+\mathbf {D}} \cdot \mathbf {1}_{D_1=\cdots =D_{m-1}=0} \cdot \frac {( u; q)_{\infty }}{(s_1^2 u ; q)_{\infty }} \\&\qquad\times \sum _{\mathbf {P}} \frac {(s^2/u ; q)_{|\mathbf {P}|} (s_1^2u/s^2 ; q)_{|\mathbf {B}-\mathbf {P}|}} {(s_1^2 ; q)_{|\mathbf {B}|}}\ \ q^{\sum _{1\le i\lt j\le n} (B_i-P_i) P_j} \prod _{i=1}^n \frac {(q;q)_{B_i}}{(q;q)_{P_i}(q;q)_{B_i-P_i}} \\&\qquad\times (s^2)^{|\mathbf {B}|-|\mathbf {P}|} u^{|\mathbf {D}|-|\mathbf {B}|+|\mathbf {P}|} \ q^{\sum _{m\le i\lt j\le n} D_i (C_j-P_j)} \ \frac { (s_1^2 ; q)_{|\mathbf {D}|}} {(q;q)_{D_m}} \prod _{i=m+1}^n \frac {(q;q)_{C_i-P_i+D_i}}{(q;q)_{C_i-P_i}(q;q)_{D_i}}. \end{split} \end{align*}

Sending $s_1\to 0$ immediately leads to (5.1). Further letting $s\to 0$ , we see that $\mathbf {P}=\mathbf {B}$ , for otherwise the factor $(s^2)^{|\mathbf {B}|-|\mathbf {P}|}$ vanishes. This eliminates the summation over $\mathbf {P}$ and produces the desired expression (5.2) together with the indicator that $B_i\le C_i$ for all $i$ .

We denote the right-hand side of (5.1) by ${\mathbb {W}}_{s,u}^{(-m),\mathrm {qBos}}(\mathbf {A},\mathbf {B};\mathbf {C},\mathbf {D})$ . The right-hand side of (5.2) is the $s=0$ degeneration of (5.1). The weights ${\mathbb {W}}_{s,u}^{(-m),\mathrm {qBos}}$ are nonnegative for $q\in [0,1)$ and $u\gt s^2\ge 0$ .

Definition 5.4. Fix parameters $\mathbf {u}=(u_1,\ldots ,u_N )$ , $\mathbf {y}=(y_1,\ldots ,y_n)$ , and $\mathbf {s}^{(v)}=(s_1^{(v)},\ldots ,s_n^{(v)} )$ such that

(5.3) \begin{equation} 0\le (s_m^{(v)})^2\lt u_iy_m,\quad i=1,\ldots ,N {,}\ m=1,\ldots ,n . \end{equation}

Let $\mathfrak {Q}^{\mathrm {qBos}}(\mathbf {u};\mathbf {y};\mathbf {s}^{(v)})$ denote the queue transfer matrix on the $n\times N$ cylinder as in Figure 8, where the vertex weight at each site $(-m,j)$ is ${\mathbb {W}}_{s_m^{(v)},u_jy_m}^{(-m),\mathrm {qBos}}$ .

The vertex model of Definition 5.4 has nonnegative weights. Note that its partition functions $\langle \emptyset |\ \mathfrak {Q}^{\mathrm {qBos}}(\mathbf {u};\mathbf {y};\mathbf {s}^{(v)})\ |\mathbf {V}\rangle$ involve infinite sums over paths winding around the cylinder. Similarly to Lemma 2.13, we see that these sums are convergent when $\mathbf {V}$ has all types appearing at least once.

Proposition 5.5. For any type counts $(N_1,\ldots ,N_n )$ , $N_i\ge 1$ , and the parameters $\mathbf {y},\mathbf {s}^{(v)}$ satisfying (5.3), the stationary distribution of the colored $q$ -Boson process with parameters $\mathbf {u}$ has the form

(5.4) \begin{equation} \mathop {\mathrm {Prob}}\nolimits ^{\mathrm {qBos}}_{N_1,\ldots ,N_n}(\mathbf {V}) = \frac {\langle \emptyset | \ \mathfrak {Q}^{\mathrm {qBos}}(\mathbf {u};\mathbf {y};\mathbf {s}^{(v)}) \ | \mathbf {V}\rangle }{Z_{N_1,\ldots ,N_n }^{\mathrm {qBos}}}. \end{equation}

The normalizing constant $Z_{N_1,\ldots ,N_n }^{\mathrm {qBos}}$ depends on the parameters and the type counts, but not on the state $\mathbf {V}$ within the sector determined by $(N_1,\ldots ,N_n)$ .

Proof. We use Theorem 3.6 (in particular, recall the queue vertex model on the cylinder interacting with the straight cylinder Markov operator as illustrated in Figure 13). Let us choose the parameters of the queue vertex model

\begin{equation*} \mathfrak {Q}=\mathfrak {Q}( (xq^{\frac 12},\mathbf {u});(q^{-\frac {1}{2}},\mathbf {s}^{(h)}) ; \mathbf {v};\mathbf {s}^{(v)}) \end{equation*}

as

\begin{equation*} x=-1,\quad s_j^{(h)}=\epsilon \to 0, \quad v_m=\frac {s_m^{(v)}}{\epsilon \ y_m} \end{equation*}

for all $1\le j\le N$ , $1\le m\le n$ . By Lemma 5.3, sending $\epsilon \to 0$ turns the weight at each site $(-m,j)$ of this queue vertex model on the cylinder into $(u_jy_m;q)_\infty {\mathbb {W}}_{s_m^{(v)},u_j y_m}^{(-m),\mathrm {qBos}}$ . The overall factor $\prod _{j=1}^{N}\prod _{m=1}^n(u_jy_m;q)_{\infty }$ is absorbed into the normalizing constant, and thus we can ignore it.

At the sites $(-m,0)$ , before the limit we have the weights ${\mathbb {W}}_{q^{-1/2},s_m^{(v)},-\epsilon q^{1/2}y_m/s_m^{(v)}}^{(-m)}$ . Up to reparametrization, these are the same weights as in Figure 15. Sending $\epsilon \to 0$ (that is, $-su\to 0$ in the notation Figure 15), we see that

\begin{equation*} {\mathbb {W}}_{q^{-1/2},s_m^{(v)},-\epsilon q^{1/2}y_m/v_m^{(v)}}^{(-m)} (\mathbf {A},0;\mathbf {A},0)\to 1, \quad {\mathbb {W}}_{q^{-1/2},s_m^{(v)},-\epsilon q^{1/2}y_m/v_m^{(v)}}^{(-m)} (\mathbf {A},0;\mathbf {A}_k^-,k)\to 0. \end{equation*}

Since the auxiliary line (i.e., the one with the spin parameter $q^{-1/2}$ ) begins as initially unoccupied, these convergences imply that this auxiliary line remains unoccupied in the cylindrical queue vertex model under this limit (more specifically, any configuration in which it is occupied has weight going to $0$ as $\epsilon \to 0$ ). Therefore, we can remove this auxiliary line from the model on the cylinder as follows:

(5.5) \begin{equation} \lim _{\epsilon \to 0}\ \langle 0,\emptyset | \ \mathfrak {Q} ( (xq^{\frac 12},\mathbf {u});(q^{-\frac {1}{2}},\mathbf {s}^{(h)}) ; \mathbf {v};\mathbf {s}^{(v)}) \ | 0, \mathbf {V} \rangle = \langle \emptyset |\ \mathfrak {Q}^{\mathrm {qBos}} (\mathbf {u};\mathbf {y};\mathbf {s}^{(v)}) \ | \mathbf {V} \rangle , \end{equation}

where $\mathfrak {Q}^{\mathrm {qBos}}(\mathbf {u};\mathbf {y})$ is defined before the proposition.

Arguing as in the proof of Proposition 4.5, we can take the limit as $\epsilon \to 0$ simultaneously in the queue vertex model and in the straight cylinder Markov operator. Before the limit, these operators satisfy the general stationarity relation of Theorem 3.6. By Proposition 5.2, the straight cylinder Markov operator converges as $\epsilon \to 0$ (in the Poisson-type continuous-time limit) to the Markov semigroup $\mathfrak {P}_{\mathrm {qBos}}(t)$ . The limit of the general stationarity relation yields

\begin{align*}& \langle \emptyset |\ \mathfrak {Q}^{\mathrm {qBos}} (\mathbf {u};\mathbf {y}; \mathbf {s}^{(v)}) \mathfrak {P}_{\mathrm {qBos}}(t) \ | \mathbf {V} \rangle \\ &\quad = \lim _{\epsilon \to 0}\, \langle 0,\emptyset | \ \mathfrak {Q}( (xq^{\frac 12},\mathbf {u});(q^{-\frac {1}{2}},\mathbf {s}^{(h)}) ; \mathbf {v};\mathbf {s}^{(v)}) \ \mathfrak {S}(-1,\mathbf {u};(\epsilon , \ldots ,\epsilon ))^{\lfloor t/\epsilon \rfloor } \ |0, \mathbf {V}\rangle \\ &\quad = \lim _{\epsilon \to 0}\, \langle 0,\emptyset | \ \mathfrak {Q}( (xq^{\frac 12},\mathbf {u});(q^{-\frac {1}{2}},\mathbf {s}^{(h)}); \mathbf {v};\mathbf {s}^{(v)}) |0, \mathbf {V}\rangle \\ &\quad = \langle \emptyset |\ \mathfrak {Q}^{\mathrm {qBos}} (\mathbf {u};\mathbf {y}; \mathbf {s}^{(v)}) | \mathbf {V} \rangle . \end{align*}

Here, the first equality holds by Proposition 5.2 and equation (5.5); the second holds by Theorem 3.6; and the third holds again by (5.5).

This completes the proof.

5.3 Matching to multiline queues

In [Reference Ayyer, Mandelshtam and MartinAMM22, § 8], a multiline queue model for the stationary distribution of the colored $q$ -Boson process is presented. Let us match this model to our queue vertex model $\mathfrak {Q}^{\mathrm {qBos}}$ on the cylinder, specialized to $s_m^{(v)}=0$ , $1\le m\le n$ (that is, with the simpler product-form weights (5.2)).

As in [Reference Ayyer, Mandelshtam and MartinAMM22], we restrict our attention to the simpler strict case when, by definition, there is at most one particle of each color. First, we recall the definition of a $q$ -Boson multiline queue and its weight. We replace the parameter $t$ from [Reference Ayyer, Mandelshtam and MartinAMM22] by our $q$ , and adjust the notation of integer indices, spectral parameters, and the direction of the ring to match the conventions used throughout our paper.

Definition 5.7 [Reference Ayyer, Mandelshtam and MartinAMM22]. A multiline diagram is an assignment of the labels from $\{1,\ldots ,n \}$ to the vertices of a cylinder $\{-n,\ldots ,-1 \}\times (\mathbb {Z}/N\mathbb {Z})$ , satisfying the following.

• – Each vertex $(-m,j)$ is assigned a multiset of labels.

• – In column $(-m)$ , all labels are from $\{m,m+1,\ldots ,n \}$ .

• – The combined multiset of all labels in column $(-m)$ is obtained from the multiset of labels in column $-(m+1)$ , together with some new labels of type $m$ .

• – (Strict condition.) Each label $m$ , $1\le m\le n$ , appears at most once in each of the columns $-m,-(m-1),\ldots ,-1$ .

The weight of a multiline diagram is, by definition, $q^{\mathfrak {R}} u_1^{c_1}\ldots u_N^{c_N}$ , where $c_j$ is the total number of labels assigned to the row $j$ , and $\mathfrak {R}$ is the refusal statistic defined as follows. Let

(5.6) \begin{equation} R_{m}:= \sum _{m-1\le k\lt \ell \le n} \mathbf {1}_{p_\ell (-(m-1))\gt p_k(-(m-1))\gt p_\ell (-m)}, \end{equation}

where $p_r(-m)$ is the position of the label $r$ in column $-m$ , and the event $a\gt b\gt c$ means that, reading along the ring in the downward direction (corresponding to decreasing positions $j$ ), the label $b$ is strictly between $a$ and $c$ . This includes the case $a=c\ne b$ ; what this means for a corresponding term in the sum is $p_{\ell }(-m) = p_\ell (-(m-1))$ , and we think of this as $p_\ell$ making a full loop around the ring to get to its position at $-(m-1)$ . Then we set $\mathfrak {R}:= \sum _{m=2}^{n}R_{m}$ . See Figure 23 for an illustration.