Inhomogeneous discrete-time exclusion processes

Crampé, Nicolas; Mallick, Kirone; Ragoucy, É.; Vanicat, Matthieu

doi:10.1088/1751-8113/48/48/484002

Cited by 13 publications

(17 citation statements)

References 60 publications

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“…The conjectures about the correpondence may be solved by using other ideas such as the theory of divided difference operators. As for the wavefunction with reflecting boundary condition, we remark that there are several works on the XXZ model with reflecting boundary condition and its degeneration in [54,55,56,57,58], for example.…”

Section: Resultsmentioning

confidence: 99%

Dual wavefunction of the symplectic ice

Motegi

2017

Reports on Mathematical Physics

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The wavefunction of the free-fermion six-vertex model was found to give a natural realization of the Tokuyama combinatorial formula for the Schur polynomials by Bump-Brubaker-Friedberg. Recently, we studied the correspondence between the dual version of the wavefunction and the Schur polynomials, which gave rise to another combinatorial formula. In this paper, we extend the analysis to the reflecting boundary condition, and show the exact correspondence between the dual wavefunction and the symplectic Schur functions. This gives a dual version of the integrable model realization of the symplectic Schur functions by Ivanov. We also generalize to the correspondence between the wavefunction, the dual wavefunction of the six-vertex model and the factorial symplectic Schur functions by the inhomogeneous generalization of the model.Comment: 31 pages, 12 figure

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Section: Resultsmentioning

confidence: 99%

Dual wavefunction of the symplectic ice

Motegi

2017

Reports on Mathematical Physics

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“…Note also that there are connections with orthogonal polynomials: either in computing physical data using the fact that they are orthogonal polynomials, or to compute explicitly orthogonal polynomials starting from these physical data, see e.g. [17,18,19]. Surely, a generalisation of the matrix ansatz that would allow to get other (excited) states above the steady states is desirable.…”

Section: Discussionmentioning

confidence: 99%

Integrability in out-of-equilibrium systems

Ragoucy

2017

J. Phys.: Conf. Ser.

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This short note presents a summary of the articles arXiv: 1408.5357, arXiv:1412.5939, arXiv:1603.06796, arXiv:1606.01018, arXiv:1606.08148 that were done in collaboration with N. CRAMPE, M. EVANS, C. FINN, K. MALLICK and M. VANICAT. It presents an approach to the matrix ansatz of out-of-equilibrium systems with boundaries, within the framework of integrable systems, and was presented at ISQS24 (Prague, 2016).

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“…It would be interesting to investigate what is the physical interpretation, in terms of transition probabilities on the lattice, of all these models. In [12] the transfer matrix was used, without specifying any inhomogeneity parameters, to define a discrete time process in the specific case of the totally asymmetric exclusion process for both periodic and open boundary conditions. A physical interpretation in terms of a sequential update was provided.…”

Section: Definition Of the Processmentioning

confidence: 99%

Integrable Floquet dynamics, generalized exclusion processes and “fused” matrix ansatz

Vanicat

2018

Nuclear Physics B

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We present a general method for constructing integrable stochastic processes, with twostep discrete time Floquet dynamics, from the transfer matrix formalism. The models can be interpreted as a discrete time parallel update. The method can be applied for both periodic and open boundary conditions. We also show how the stationary distribution can be built as a matrix product state. As an illustration we construct parallel discrete time dynamics associated with the R-matrix of the SSEP and of the ASEP, and provide the associated stationary distributions in a matrix product form. We use this general framework to introduce new integrable generalized exclusion processes, where a fixed number of particles is allowed on each lattice site in opposition to the (single particle) exclusion process models. They are constructed using the fusion procedure of R-matrices (and K-matrices for open boundary conditions) for the SSEP and ASEP. We develop a new method, that we named "fused" matrix ansatz, to build explicitly the stationary distribution in a matrix product form. We use this algebraic structure to compute physical observables such as the correlation functions and the mean particle current. 1 matthieu.vanicat@fmf.uni-lj.si 1 Introduction.Systems of particles in interaction on a one-dimensional lattice have attracted lots of attention in the last decades. The reason is twofold: on one hand they seem to capture the essential physical features of out-of-equilibrium systems, and on the other hand they allow in some particular cases for exact computations of physical quantities. In particular the study of exclusion processes, for which particles experience a hard-core interaction (there is at most one particle on each site of the lattice), turned out to be very fruitful. The Asymmetric Simple Exclusion Process (ASEP) is a paradigmatic example of such model [11,14]. The phase diagram of the continuous time model with open boundaries was exactly computed using a matrix product construction of the stationary state [15,41]. The ASEP was also studied in the discrete time setting using various stochastic update rules [39] (see also for instance [1] and references therein for more recent developments and applications to traffic flow). The matrix product ansatz was successfully used to solve the ASEP with discrete time parallel update and sequential update [25,26,40,39,18,22,49].The reason behind the exact solutions of these models is their integrability: the Markov matrix M ruling their stochastic dynamics is part of a family of commuting operators generated by a transfer matrix. The transfer matrix is constructed from a R-matrix satisfying the Yang-Baxter equation [5]. It is well known that continuous time Markov matrices with local stochastic rules can be obtained by taking the first logarithmic derivative of an appropriate transfer matrix. This mechanism is quite general and relies only on minor assumptions on the R-matrix (essentially the regularity property). It has also been observed that the transfer matrix of the six vertex ...

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Inhomogeneous discrete-time exclusion processes

Cited by 13 publications

References 60 publications

Dual wavefunction of the symplectic ice

Dual wavefunction of the symplectic ice

Integrability in out-of-equilibrium systems

Integrable Floquet dynamics, generalized exclusion processes and “fused” matrix ansatz

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