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arXiv:1511.09168 · Journal of Integrable Systems

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

Atsuo Kuniba, Shouya Maruyama, Masato Okado

Published Sep 1, 2015• 11 citations
arXivHTMLPDF

Abstract

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

DOI
10.1093/integr/xyw002

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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 .
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 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 investigate one of the simplest multispecies generalizations of the asymmetric simple exclusion process on a ring. This process has a rich combinatorial spectral structure and a matrix product form for the stationary state. In the totally asymmetric case, operators that conjugate the dynamics of systems with different numbers of species were obtained by the authors and recently reported by Arita et al (2011 J. Phys. A: Math. Theor. 44 335004). The existence of such nontrivial operators was reformulated as a representation problem for a specific quadratic algebra (generalized matrix Ansatz). In this work, we construct the family of representations explicitly for the partially asymmetric case. This solution cannot be obtained by a simple deformation of the totally asymmetric case.
A multi-species generalization of the asymmetric simple exclusion process is studied with ordered sequential and sub-lattice parallel updating schemes. In this model, particles hop with their own specific probabilities to their rightmost empty site and fast particles overtake slow ones with a definite probability. Using the matrix product ansatz technique, we obtain the relevant algebra and study the uncorrelated stationary state of the model both for an open system and on a ring. A complete comparison between the physical results in these updates and those of random sequential introduced in Karimipour V 1999 A multi-species ASEP and its relation to traffic flow Phys. Rev. E 59 205 and Karimipour V 1998 A multi-species ASEP, steady state and correlation functions on a periodic lattice Preprint cond-mat/9809193, is made.