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arXiv:1201.0039

A SYSTEMATIC WAY TO FIND AND CONSTRUCT EXACT FINITE DIMENSIONAL MATRIX-PRODUCT STATIONARY STATES

Yasuhiro Hieida, T. SASAMOTO

Published Jul 16, 2012• 0 citations
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Abstract

We explain how to construct matrix product stationary states which are composed of finite-dimensional matrices. Our construction explained in this article was first presented in a part of [Hieida and Sasamoto:J. Phys. A: Math. Gen. 37 (2004) 9873] for general models. In this article, we give more details on the treatment than in the above-mentioned reference, for one-dimensional asymmetric simple exclusion process(ASEP).

DOI
10.1142/9789814407199_0013

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We explain how to construct matrix product stationary states which are composed of finite-dimensional matrices. Our construction explained in this article was first presented in a part of [Hieida and Sasamoto:J. Phys. A: Math. Gen. 37 (2004) 9873] for general models. In this article, we give more details on the treatment than in the above-mentioned reference, for one-dimensional asymmetric simple exclusion process(ASEP).
Stationary states of stochastic models, which have NNN states per site, in matrix product form are considered. First we give a necessary condition for the existence of a finite MMM-dimensional matrix product state for any N,M{N,M}N,M. Second, we give a method to construct the matrices from the stationary states of small size system when the above condition and N≤MN\le MN≤M are satisfied. Third, the method by which one can check that the obtained matrices are valid for any system size is presented for the case where M=NM=NM=N is satisfied. The application of our methods is explained using three examples: the asymmetric exclusion process, a model studied in [F. H. Jafarpour: J. Phys. A: Math. Gen. 36 (2003) 7497] and a hybrid of both of the models.
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 .