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Multispecies inhomogeneous $t$-PushTASEP from antisymmetric fusion

Arvind Ayyer, Atsuo Kuniba

TL;DR

The paper delivers a Baxter-type construction for the inhomogeneous $n$-species $t$-PushTASEP by expressing the Markov generator as a signed derivative of a family of commuting transfer matrices built from antisymmetric representations of $U_t(\,\widehat{sl}_{n+1}\,)$. This framework relies on the antisymmetric fusion to construct $S^{k,1}(z)$ and the associated transfer matrices $T^k(z)$, whose alternating sum extracts the PushTASEP dynamics while canceling forbidden channels. In the homogeneous limit, PushTASEP and ASEP become sister models sharing eigenstates and stationary distributions, with a matrix-product state description for stationary probabilities tied to a Zamolodchikov–Faddeev algebra. The results illuminate a deep integrable structure for long-range multispecies stochastic processes and suggest avenues toward higher-dimensional formulations via three-dimensional integrability.

Abstract

We investigate the recently introduced inhomogeneous $n$-species $t$-PushTASEP, a long-range stochastic process on a periodic lattice. A Baxter-type formula is established, expressing the Markov matrix as an alternating sum of commuting transfer matrices over all the fundamental representations of $U_t(\widehat{sl}_{n+1})$. This superposition acts as an inclusion-exclusion principle, selectively extracting the sequential particle transitions characteristic of the PushTASEP, while canceling forbidden channels. The homogeneous specialization connects the PushTASEP to ASEP, showing that the two models share eigenstates and a common integrability structure.

Multispecies inhomogeneous $t$-PushTASEP from antisymmetric fusion

TL;DR

The paper delivers a Baxter-type construction for the inhomogeneous -species -PushTASEP by expressing the Markov generator as a signed derivative of a family of commuting transfer matrices built from antisymmetric representations of . This framework relies on the antisymmetric fusion to construct and the associated transfer matrices , whose alternating sum extracts the PushTASEP dynamics while canceling forbidden channels. In the homogeneous limit, PushTASEP and ASEP become sister models sharing eigenstates and stationary distributions, with a matrix-product state description for stationary probabilities tied to a Zamolodchikov–Faddeev algebra. The results illuminate a deep integrable structure for long-range multispecies stochastic processes and suggest avenues toward higher-dimensional formulations via three-dimensional integrability.

Abstract

We investigate the recently introduced inhomogeneous -species -PushTASEP, a long-range stochastic process on a periodic lattice. A Baxter-type formula is established, expressing the Markov matrix as an alternating sum of commuting transfer matrices over all the fundamental representations of . This superposition acts as an inclusion-exclusion principle, selectively extracting the sequential particle transitions characteristic of the PushTASEP, while canceling forbidden channels. The homogeneous specialization connects the PushTASEP to ASEP, showing that the two models share eigenstates and a common integrability structure.

Paper Structure

This paper contains 22 sections, 4 theorems, 120 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multispecies $t$-PushTASEP
  3. Definition of $n$ species inhomogeneous $t$-PushTASEP
  4. The matrix $S(z)$
  5. Space $V^k$ with base labeled with $\mathscr{B}^k$ and $\mathscr{T}^k$
  6. $S^{k,1}(z)$ from antisymmetric fusion
  7. Transfer matrix $T^k(z)$
  8. Definition
  9. Basic properties
  10. ${H_{\text{PushTASEP} }}$ from transfer matrices
  11. Diagonal elements
  12. Reduced diagram and its depth
  13. Wanted terms
  14. Unwanted terms
  15. Further properties of $t$-PushTASEP
  16. ...and 7 more sections

Key Result

Theorem 3.1

Figures (6)

  • Figure 1: All transitions going in and out of a configuration in $\mathcal{S}(1,2,1)$. The moves for the incoming transitions are shown with blue arrows.
  • Figure 2: A typical matrix element.
  • Figure 3: Nonzero elements of $S^{1,1}(z)$ with $0 \le i \neq j \le n$. Horizontal arrows, corresponding to $k=1$, are depicted as ordinary thin ones.
  • Figure 4: The weight of the element in \ref{['Sf']}, where $b_r$ is assigned to the vertical edge between $A_r$ and $A_{r+1}$. Note that $I_1 < I_2 < \dots < I_k$ and $A_1 < A_2 < \dots < A_k$ by \ref{['tk']}.
  • Figure 5: Diagram representation of the matrix element $\langle \boldsymbol{\sigma}'|T^k(z)| \boldsymbol{\sigma}\rangle$.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Example 2.1
  • Theorem 3.1
  • Example 4.1
  • Theorem 5.1
  • Example 5.2
  • Lemma 5.3
  • proof
  • Example 5.4
  • Example 5.5
  • Example 6.1
  • ...and 3 more