Long time asymptotics of the totally asymmetric simple exclusion process

Motegi, Kohei; Sakai, Kazumitsu; Sato, Jun

doi:10.1088/1751-8113/45/46/465004

Cited by 12 publications

(25 citation statements)

References 46 publications

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“…The expression (4) follows directly from that of the generating function e sh(x,t) flat , s > 0 obtained in [39], equation (8) S , based on earlier works [44,85,86,87,54,45,88,53] on the Bethe ansatz solution of TASEP with periodic boundaries. One has e sh(x,t)…”

Section: Relation With the Generating Function Of The Heightmentioning

confidence: 99%

Riemann surfaces for KPZ with periodic boundaries

Prolhac

2020

SciPost Phys.

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The Riemann surface for polylogarithms of half-integer index, which has the topology of an infinite dimensional hypercube, is studied in relation to onedimensional KPZ universality in finite volume. Known exact results for fluctuations of the KPZ height with periodic boundaries are expressed in terms of meromorphic functions on this Riemann surface, summed over all the sheets of a covering map to an infinite cylinder. Connections to stationary large deviations, particle-hole excitations and KdV solitons are discussed.

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Section: Relation With the Generating Function Of The Heightmentioning

confidence: 99%

Riemann surfaces for KPZ with periodic boundaries

Prolhac

2020

SciPost Phys.

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“…The derivative with respect to y i in the determinant has to be computed before setting the y j 's equal to a solution of the Bethe equations (4). At q = 0, the determinant can be calculated explicitly [25], which allows the asymptotic analysis for large L, N [26]. The Gaudin determinant is a consequence of the Slavnov determinant [27] for the scalar product between a Bethe eigenstate ψ y with Bethe roots y j and a Bethe vector ψ w with arbitrary parameters w j not solution of Bethe equations,…”

Section: Scalar Product Of Bethe Eigenstatesmentioning

confidence: 99%

Extrapolation methods and Bethe ansatz for the asymmetric exclusion process

Prolhac¹

2016

J. Phys. A: Math. Theor.

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The one-dimensional asymmetric simple exclusion process (ASEP), where N hard-core particles hop forward with rate 1 and backward with rate q < 1, is considered on a periodic lattice of L site. Using KPZ universality and previous results for the totally asymmetric model q = 0, precise conjectures are formulated for asymptotics at finite density ρ = N/L of ASEP eigenstates close to the stationary state. The conjectures are checked with high precision using extrapolation methods on finite size Bethe ansatz numerics. For weak asymmetry 1 − q ∼ 1/ √ L, double extrapolation combined with an integer relation algorithm gives an exact expression for the spectral gap up to 10-th order in the asymmetry.

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“…This fact allows us to determine the factor K by evaluating the overlap for a particular particle configuration. In fact, the overlaps for some particular cases can be directly evaluated as in [30,31]. For instance, we find the following explicit expression for the case x j = j (1 ≤ j ≤ n):…”

Section: (J)mentioning

confidence: 99%

“…On the other hand, the overall factor is independent of the particle positions, and therefore we can determine this factor by considering the specific configuration: we explicitly calculate it with the help of the result for the overlap of the consecutive configuration (i.e. x j = j) obtained in [30,31].…”

Section: Wavefunctionsmentioning

confidence: 99%

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Vertex models, TASEP and Grothendieck polynomials

Motegi

Sakai

2013

J. Phys. A: Math. Theor.

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106

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We examine the wavefunctions and their scalar products of a one-parameter family of integrable five vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on/off-shell wavefunctions of the five vertex models are represented as a certain determinant form. Up to some normalization factors, we find the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of Grothendieck polynomials.

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Long time asymptotics of the totally asymmetric simple exclusion process

Cited by 12 publications

References 46 publications

Riemann surfaces for KPZ with periodic boundaries

Riemann surfaces for KPZ with periodic boundaries

Extrapolation methods and Bethe ansatz for the asymmetric exclusion process

Vertex models, TASEP and Grothendieck polynomials

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