Integral Formulas of ASEP and q -TAZRP on a Ring
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Communications in
Mathematical Physics
Integral Formulas of ASEP and q -TAZRP on a Ring Zhipeng Liu1 , Axel Saenz2 , Dong Wang3 1 Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA.
E-mail: [email protected]
2 Mathematics Department, Tulane University, New Orleans, LA 70118, USA.
E-mail: [email protected]
3 Department of Mathematics, National University of Singapore, Singapore 119076, Singapore.
E-mail: [email protected] Received: 21 October 2019 / Accepted: 26 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: In this paper, we obtain the transition probability formulas for the asymmetric simple exclusion process and the q-deformed totally asymmetric zero range process on the ring by applying the coordinate Bethe ansatz. We also compute the distribution function for a tagged particle with general initial condition. 1. Introduction We investigate the transition probabilities of 1-dimensional interacting particles systems with spatial periodicity (i.e. on a ring), by using the coordinate Bethe ansatz. We concentrate on two models: the Asymmetric Simple Exclusion Process (ASEP) and the q-deformed Totally Asymmetric Zero Range Process (q-TAZRP). We explicitly give the transition probability function and the one-point function for the ASEP and the q-TAZRP on a ring as a sum of nested contour integrals. The ASEP and the q-TAZRP are continuous time Markov processes on a discrete one-dimensional lattice. In particular, we assume that the models are defined on a one dimensional periodic lattice such as Z/LZ. The state of the Markov process is determined by the (random) location of N particles. For the case of the infinite lattice Z, the ASEP and the q-TAZRP have been well-studied resulting in exact transition probability formulas and asymptotic computations [Joh00,TW08,TW09,KL14,LW19,BCPS15]. These exact transition probability formulas are the L → ∞ limit of our formulas for the periodic lattice Z/LZ. (see Sects. 5 and 7). In the ASEP, each site may be occupied by at most one particle and particles move independently and asymmetrically to the left or the right, unless they are blocked by a neighboring particle. In the q-TAZRP, more than one particle may occupy a site and only the top particle at a site may move to the right, regardless of the neighboring particles. For the precise definition of the models, see Sect. 1.1. The coordinate Bethe ansatz was introduced in [Bet31] in the context of spin chains (see [Sut04,Gau14] for a modern review). In the work of Spohn and Gwa [GS92], its
Z. Liu, A. Saenz, D. Wang
application to interacting particle systems was observed, and Schütz [Sch97] applied it to the Totally Asymmetric Simple Exclusion Process (TASEP).1 For one dimensional integrable interacting particle systems on the lattice Z, the coordinate Bethe ansatz has been applied to wide class of models, most notably the ASEP [TW08] and the q-TAZRP [KL14,WW16]; see also [BCPS15]. For one dimensional periodic integrable interacting particle systems on the lattice Z/LZ,
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