Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1602.04574 · Journal of Integrable Systems

Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation

Atsuo Kuniba, Shouya Maruyama, Masato Okado

Published Sep 1, 2015• 9 citations
arXivHTMLPDF

Abstract

We consider a three-dimensional (3D) lattice model associated with the intertwiner of the quantized coordinate ring $A_q(sl_3)$, and introduce a family of layer to layer transfer matrices on $m\times n$ square lattice. By using the tetrahedron equation we derive their commutativity and bilinear relations mixing various boundary conditions. At $q=0$ and $m=n$, they lead to a new proof of the steady state probability of the $n$-species totally asymmetric zero range process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.

DOI
10.1093/integr/xyw008

Related papers

Multispecies TASEP and the tetrahedron equation

Atsuo Kuniba, Shouya Maruyama, Masato Okado

Abstract We introduce a family of layer to layer transfer matrices in a three-dimensional (3D) lattice model which can be viewed as partition functions of the q -oscillator valued six-vertex model on an m × n square lattice. By invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions. At q = 0 and m = n , they ultimately yield a new proof of the steady state formula for the n -species totally asymmetric simple exclusion process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.

arXiv:1509.09018•Journal of Physics A Mathematical and Theoretical•Feb 2, 2016•30 citations

Multispecies totally asymmetric zero range process: I. Multiline process and combinatorialR

Atsuo Kuniba, Shouya Maruyama, Masato Okado

We introduce an |$n$|-species totally asymmetric zero range process (⁠|$n$|-TAZRP) on one-dimensional periodic lattice with |$L$| sites. It is a continuous time Markov process in which |$n$| species of particles hop to the adjacent site only in one direction under the condition that smaller species ones have the priority to do so. Also introduced is an |$n$|-line process, a companion stochastic system having the uniform steady-state from which the |$n$|-TAZRP is derived as the image by a certain projection |$\pi$|⁠. We construct the |$\pi$| by a combinatorial |$R$| of the quantum affine algebra |$U_q(\widehat{sl}_L)$| and establish a matrix product formula of the steady state probability of the |$n$|-TAZRP in terms of corner transfer matrices of a |$q=0$|-oscillator-valued vertex model. These results parallel the recent reformulation of the |$n$|-species totally asymmetric simple exclusion process (⁠|$n$|-TASEP) by the authors, demonstrating that |$n$|-TAZRP and |$n$|-TASEP are the canonical sister models associated with the symmetric and the antisymmetric tensor representations of |$U_q(\widehat{sl}_L)$| at |$q=0$|⁠, respectively. Communicated by: Masatoshi Noumi

arXiv:1511.09168•Journal of Integrable Systems•Sep 1, 2015•11 citations

Integrable Structure of Multispecies Zero Range Process

Atsuo Kuniba, Masato Okado, Satoshi W. Watanabe

We present a brief review on integrability of multispecies zero range process in one dimension introduced recently.The topics range over stochastic R matrices of quantum affine algebra U q A (1) n , matrix product construction of stationary states for periodic systems, q-boson representation of Zamolodchikov-Faddeev algebra, etc.We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of R matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter.

arXiv:1701.07279•Symmetry Integrability and Geometry Methods and Applications•Jun 17, 2017•4 citations

Application to Multispecies TASEP

Atsuo Kuniba

This chapter is an exposition of a 3D approach to an integrable Markov process called the n species totally asymmetric simple exclusion process (n-TASEP). The main result is a matrix product formula of the stationary probability involving layer transfer matrices of the $$q=0$$ -oscillator-valued five-vertex model on an $$n \times n$$ lattice. The stationary condition is translated into their quadratic relations, the so-called Faddeev–Zamolodchikov algebra, which are highly non-local from the viewpoint of the five-vertex model. They are shown to be a far-reaching consequence of the single tetrahedron equation of type $$MMLL=LLMM$$ in Sect. 2.6 and its solution in Theorem 3.25 .

Theoretical and mathematical physics•Jan 1, 2022•0 citations

ELLIPTIC SOLUTION FOR MODIFIED TETRAHEDRON EQUATIONS

Vladimir V. Mangazeev, Yu. G. Stroganov

As is known, tetrahedron equations lead to the commuting family of transfer-matrices and provide the integrability of the corresponding three-dimensional lattice models. We present the modified version of these equations which give the commuting family of more complicated two-layer transfer-matrices. In the static limit we have succeeded in constructing the solution of these equations in terms of elliptic functions.

arXiv:hep-th/9305145•Modern Physics Letters A•Nov 30, 1993•19 citations

The matrix product solution of the multispecies partially asymmetric exclusion process

Sylvain Prolhac, M. R. Evans, Kirone Mallick

We find the exact solution for the stationary state measure of the partially asymmetric exclusion process on a ring with multiple species of particles. The solution is in the form of a matrix product representation where the matrices for a system of N species are defined recursively in terms of the matrices for a system of N − 1 species. A complete proof is given, based on the quadratic relations verified by these matrices. This matrix product construction is interpreted in terms of the action of a transfer matrix.

arXiv:0812.3293•Journal of Physics A Mathematical and Theoretical•Mar 27, 2009•84 citations
We consider a three-dimensional (3D) lattice model associated with the intertwiner of the quantized coordinate ring Aq(sl3)A_q(sl_3)Aq​(sl3​), and introduce a family of layer to layer transfer matrices on m×nm\times nm×n square lattice. By using the tetrahedron equation we derive their commutativity and bilinear relations mixing various boundary conditions. At q=0q=0q=0 and m=nm=nm=n, they lead to a new proof of the steady state probability of the nnn-species totally asymmetric zero range process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.
Abstract We introduce a family of layer to layer transfer matrices in a three-dimensional (3D) lattice model which can be viewed as partition functions of the q -oscillator valued six-vertex model on an m × n square lattice. By invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions. At q = 0 and m = n , they ultimately yield a new proof of the steady state formula for the n -species totally asymmetric simple exclusion process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.
We introduce an |nnn|-species totally asymmetric zero range process (⁠|nnn|-TAZRP) on one-dimensional periodic lattice with |LLL| sites. It is a continuous time Markov process in which |nnn| species of particles hop to the adjacent site only in one direction under the condition that smaller species ones have the priority to do so. Also introduced is an |nnn|-line process, a companion stochastic system having the uniform steady-state from which the |nnn|-TAZRP is derived as the image by a certain projection |π\piπ|⁠. We construct the |π\piπ| by a combinatorial |RRR| of the quantum affine algebra |Uq(sl^L)U_q(\widehat{sl}_L)Uq​(slL​)| and establish a matrix product formula of the steady state probability of the |nnn|-TAZRP in terms of corner transfer matrices of a |q=0q=0q=0|-oscillator-valued vertex model. These results parallel the recent reformulation of the |nnn|-species totally asymmetric simple exclusion process (⁠|nnn|-TASEP) by the authors, demonstrating that |nnn|-TAZRP and |nnn|-TASEP are the canonical sister models associated with the symmetric and the antisymmetric tensor representations of |Uq(sl^L)U_q(\widehat{sl}_L)Uq​(slL​)| at |q=0q=0q=0|⁠, respectively. Communicated by: Masatoshi Noumi
We present a brief review on integrability of multispecies zero range process in one dimension introduced recently.The topics range over stochastic R matrices of quantum affine algebra U q A (1) n , matrix product construction of stationary states for periodic systems, q-boson representation of Zamolodchikov-Faddeev algebra, etc.We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of R matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter.
This chapter is an exposition of a 3D approach to an integrable Markov process called the n species totally asymmetric simple exclusion process (n-TASEP). The main result is a matrix product formula of the stationary probability involving layer transfer matrices of the q=0q=0q=0 -oscillator-valued five-vertex model on an n×nn \times nn×n lattice. The stationary condition is translated into their quadratic relations, the so-called Faddeev–Zamolodchikov algebra, which are highly non-local from the viewpoint of the five-vertex model. They are shown to be a far-reaching consequence of the single tetrahedron equation of type MMLL=LLMMMMLL=LLMMMMLL=LLMM in Sect. 2.6 and its solution in Theorem 3.25 .
We find the exact solution for the stationary state measure of the partially asymmetric exclusion process on a ring with multiple species of particles. The solution is in the form of a matrix product representation where the matrices for a system of N species are defined recursively in terms of the matrices for a system of N − 1 species. A complete proof is given, based on the quadratic relations verified by these matrices. This matrix product construction is interpreted in terms of the action of a transfer matrix.