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Theoretical and mathematical physics

Application to Multispecies TASEP

Atsuo Kuniba

Published Jan 1, 2022• 0 citations
Publisher

Abstract

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 .

DOI
10.1007/978-981-19-3262-5_18

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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 .
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 consider a multispecies generalization of the totally asymmetric simple exclusion process (TASEP) with the simple hopping rule: for the α and βth-class particles (α < β), the transition αβ → βα occurs with a rate independent from the values α and β. Ferrari and Martin (2007 Ann. Prob. 35 807) obtained the stationary state of this model thanks to a combinatorial algorithm, which was subsequently interpreted as a matrix product representation by Evans et al (2009 J. Stat. Phys. 135 217). This 'matrix ansatz' shows that the stationary state of the multispecies TASEP with N classes of particles ( N -TASEP) can be constructed algebraically by the action of an operator on the ( N − 1)-TASEP stationary state. Besides, Arita et al ( 2009 J. Phys. A. Math Theor. 45 345002 ) analyzed the spectral structure of the Markov matrix: they showed that the set of eigenvalues of the N -TASEP contains those of the ( N − 1)-TASEP and that the various spectral inclusions can be encoded in a hierarchical set-theoretic structure known as the Hasse diagram. Inspired by these works, we define nontrivial operators that allow us to construct eigenvectors of the N -TASEP by lifting the eigenvectors of the ( N − 1)-TASEP. This goal is achieved by generalizing the matrix product representation and the Ferrari–Martin algorithm. In particular, we show that the matrix ansatz is not only a convenient tool to write the stationary state but in fact intertwines Markov matrices of different values of N .
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.
In recent years, one-dimensional driven diffusion models consisting of particles on a lattice have been studied. The most famous topics of these models are solvability and phase transitions. For example, the exact stationary-state solution of the asymmetric exclusion process (ASEP) with an open boundary condition can be obtained as a matrix product form and its physical quantities exhibit boundary-induced phase transitions in the thermodynamic limit. There are three regimes, namely, maximal-current (MC) phase, low-density (LD) phase and high-density (HD) phase in the phase diagram. The two-species models or three-state models are also interesting. The stationary-state solutions of the two-species totally asymmetric exclusion process (TASEP) with a periodic boundary condition and an open boundary condition can be obtained as matrix product forms. The model in ref. 6 exhibits two phases in which symmetry breaks spontaneously. Furthermore, the model in ref. 7 exhibits phase separation. In this note, the two-species TASEP with a new open boundary condition different from the condition in ref. 6 is introduced and we obtain the stationary-state solution exactly as a matrix product form and calculate the physical quantities in the thermodynamic limit. Let us consider a stochastic process on an L-site chain with two-species particles. We note j 1⁄4 0 if site j is empty, j 1⁄4 1 if site j is occupied by the first-class particle and j 1⁄4 2 if site j is occupied by the second-class particle. The exchange rule in the bulk of the chain is expressed as 10 ! 01 with a rate 1; ð1Þ 20 ! 02 with a rate 1; ð2Þ 12 ! 21 with a rate 1: ð3Þ
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.