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The Completeness of the Bethe Ansatz for the Asymmetric Simple Exclusion Process

Eric Brattain

Jan 1, 20160 citations

Bethe Ansatz Results for the Partially Asymmetric Exclusion Process with Open Boundaries

Fabian H. L. Eßler, Jan de Gier

Jan 1, 20110 citations

Spectral Degeneracies in the Totally Asymmetric Exclusion Process

O. Golinelli, Kirone Mallick

arXiv:cond-mat/0412462Journal of Statistical PhysicsSep 1, 200528 citations

Spectrum of the totally asymmetric simple exclusion process on a periodic lattice-first excited states

Sylvain Prolhac

We consider the spectrum of the totally asymmetric simple exclusion process on a periodic lattice of L sites. The first eigenstates have an eigenvalue with real part scaling as for large L with finite density of particles. Bethe ansatz shows that these eigenstates are characterized by four finite sets of positive half-integers, or equivalently by two integer partitions. Each corresponding eigenvalue is found to be equal to the value at its saddle point of a function indexed by the four sets. Our derivation of the large L asymptotics relies on a version of the Euler–Maclaurin formula with square root singularities at both ends of the summation range.We consider the spectrum of the totally asymmetric simple exclusion process on a periodic lattice of L sites. The first eigenstates have an eigenvalue with real part scaling as for large L with finite density of particles. Bethe ansatz shows that these eigenstates are characterized by four finite sets of positive half-integers, or equivalently by two integer partitions. Each corresponding eigenvalue is found to be equal to the value at its saddle point of a function indexed by the four sets. Our derivation of the large L asymptotics relies on a version of the Euler–Maclaurin formula with square root singularities at both ends of the summation range.

arXiv:1404.1315Journal of Physics A Mathematical and TheoreticalAug 29, 201426 citations

Extrapolation methods and Bethe ansatz for the asymmetric exclusion process

Sylvain Prolhac

Abstract 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 10th order in the asymmetry.Abstract 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 10th order in the asymmetry.

arXiv:1604.08843Journal of Physics A Mathematical and TheoreticalOct 14, 201618 citations

Spectrum of the totally asymmetric simple exclusion process on a periodic lattice—bulk eigenvalues

Sylvain Prolhac

We consider the totally asymmetric simple exclusion process (TASEP) on a periodic one-dimensional lattice of L sites. Using Bethe ansatz, we derive parametric formulas for the eigenvalues of its generator in the thermodynamic limit. This allows to study the curve delimiting the edge of the spectrum in the complex plane. A functional integration over the eigenstates leads to an expression for the density of eigenvalues in the bulk of the spectrum. The density vanishes with an exponent 2/5 close to the eigenvalue 0.We consider the totally asymmetric simple exclusion process (TASEP) on a periodic one-dimensional lattice of L sites. Using Bethe ansatz, we derive parametric formulas for the eigenvalues of its generator in the thermodynamic limit. This allows to study the curve delimiting the edge of the spectrum in the complex plane. A functional integration over the eigenstates leads to an expression for the density of eigenvalues in the bulk of the spectrum. The density vanishes with an exponent 2/5 close to the eigenvalue 0.

arXiv:1306.2263Journal of Physics A Mathematical and TheoreticalSep 27, 201317 citations

Asymptotics for the Norm of Bethe Eigenstates in the Periodic Totally Asymmetric Exclusion Process

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Asymptotics for the Norm of Bethe Eigenstates in the Periodic Totally Asymmetric Exclusion Process

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