Multispecies inhomogeneous $t$-PushTASEP with general capacity

Abstract

We study an $n$-species $t$-PushTASEP, an integrable long-range stochastic process, on a one-dimensional periodic lattice with inhomogeneities $x_1,\ldots,x_L$ and arbitrary capacity $l$ at each lattice site. The Markov matrix is identified with an alternating sum of commuting transfer matrices over all fundamental representations of $U_t(\widehat{sl}_{n+1})$. Stationary probabilities are expressed in a matrix product form involving a fusion of quantized corner transfer matrices for the strange five-vertex model introduced by Okado, Scrimshaw, and the second author. The resulting partition function, which serves as the normalization factor of the stationary probabilities, is obtained from the $l=1$ case by a finite plethystic substitution of length $l$.

Figures (4)

• Figure 1: Schematic plot of a transition $\vec{\boldsymbol{\sigma}} \rightarrow \vec{\boldsymbol{\sigma}}'$ in Table \ref{['tab1']} for the case $0 \le h_0 < \cdots < h_{g=9} \le n$. The vertical axis ranges from $0$ to $n$, corresponding to the components of arrays in $\mathscr{B}_l$ defined in \ref{['vl']}, while the lattice sites $1,\ldots, L$ are aligned along the horizontal axis with periodic boundary conditions. At each point $(j,h) \in \{1,\ldots, L\} \times [0,n]$, the symbol $\oplus_h$, $\ominus_h$, or blank is placed according to $\sigma'_{j,h} - \sigma_{j,h} = 1$, $-1$, or $0$, respectively, where $\vec{\boldsymbol{\sigma}} = (\boldsymbol{\sigma}_1,\ldots, \boldsymbol{\sigma}_L)$ with $\boldsymbol{\sigma}_j = (\sigma_{j,0},\ldots, \sigma_{j,n}) \in \mathscr{B}_l$, and $\vec{\boldsymbol{\sigma}}' = (\boldsymbol{\sigma}'_1,\ldots, \boldsymbol{\sigma}'_L)$ with $\boldsymbol{\sigma}'_j = (\sigma'_{j,0},\ldots, \sigma'_{j,n}) \in \mathscr{B}_l$ in the multiplicity representation. Only the sites where the local state changes are depicted and labeled by $o, j_1, \ldots, j_4$; all other sites are omitted in accordance with the notion of a reduced diagram introduced in Section \ref{['ss:rd']}. For instance, $\boldsymbol{\sigma}'_{o} - \boldsymbol{\sigma}_{o} = {\bf e}_{r_1} - {\bf e}_{r_2} + {\bf e}_{r_3} - {\bf e}_{r_4}$ for some $0 \le r_1 < \cdots < r_4 \le n$, consistent with \ref{['rr1']}, and $\boldsymbol{\sigma}'_{j_2} - \boldsymbol{\sigma}_{j_2} = -{\bf e}_{t_1} + {\bf e}_{t_2}$ for some $0 \le t_1 < t_2 \le n$, as in \ref{['rr2']}. A segment of the form $\ominus_p \overset{h_q}{} \oplus_{p'}$ is understood to assume $h_q = p = p'$, and represents the movement of a particle of species $h_q$ from the site with $\ominus_p$ to that with $\oplus_{p'}$, under the periodic boundary condition. In column $o$, a vertical line entering the diagram from the top and an arrow exiting downward are added, so that it may be viewed as a path descending through the system, possibly wrapping around it. This viewpoint will be useful for the combinatorial description of the transition rates in Section \ref{['ss:cd']}.
• Figure 2: Weights of local configurations of blue paths at the cell containing $a \in [0,l]$. In the leftmost case, the index $i \in [0,n]$ is the row to which the cell belongs, and $\mathbb{K}_i$ is defined in \ref{['Ki']}. In the third and the fourth cases, $a=0$ and $a=l$ are forbidden, respectively by \ref{['brule']}.
• Figure 3: Diagram representation of the matrix element $\langle \vec{\boldsymbol{\sigma}}'|T^k(z)| \vec{\boldsymbol{\sigma}}\rangle$. Each vertex here gives a diagrammatic representation of $S^{k}_{\;\,l} \left( \frac{z}{x_j} \right)^{{\mathbf a}_{j+1}, \boldsymbol{\sigma}'_j}_{{\mathbf a}_j, \boldsymbol{\sigma}_j}$ in \ref{['tke']}, where thick and thin arrows represent $V^k$ and $V_l$, respectively.
• Figure 4: An example of non-minimum carriers inducing the same transition as Table \ref{['tab1']}. Horizontal black line segments specify the minimum carriers as explained in Remark \ref{['re:md']}. The heights $f_1, f_2, f_3, f_4 \in [0,n]\setminus \{h_0,\ldots, h_9\}$ of the extra red lines correspond to the supplemented letters to them.

Theorems & Definitions (54)

• Proposition 1
• proof
• Remark 2
• Example 3
• Remark 4
• Theorem 5
• Lemma 6
• proof
• proof : Proof of Theorem \ref{['th:pc']}
• Example 7