[en] The Bethe ansatz solution of periodic TASEP is formulated in terms of a ramified covering from a Riemann surface to the sphere. The joint probability distribution of height fluctuations at n distinct times has in particular a relatively simple expression as a function of n variables on the Riemann surface built from exponentials of Abelian integrals, traced over the ramified covering and integrated on n nested contours in the complex plane. (paper)

[en] We consider the totally asymmetric simple exclusion process (TASEP) with open boundaries, at the edge of the maximal current (MC) phase. Using analytic continuations from the known stationary eigenvalue, we obtain exact expressions for the spectral gaps in the limit of large system size. The underlying Riemann surface, generated by modified Lambert functions, interpolates between the one for periodic TASEP and the one for open TASEP in the MC phase. (paper)

[en] We study the fluctuations of the total current for a partially asymmetric exclusion process in the scaling of a weak asymmetry (asymmetry of order the inverse of the size of the system) using the Bethe ansatz. Starting from the functional formulation of the Bethe equations, we obtain for all the cumulants of the current both the leading and next-to-leading contribution in the size of the system

[en] 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 L^−3/2 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. (paper)

[en] We study spectral gaps of the one-dimensional totally asymmetric simple exclusion process (TASEP) with open boundaries in the maximal current phase. Earlier results for the model with periodic boundaries suggest that the gaps contributing to the universal KPZ regime may be understood as points on an infinite genus Riemann surface built from a parametric representation of the cumulant generating function of the current. We perform explicit analytic continuations from the known large deviations of the current for open TASEP, and confirm the results for the gaps by an exact Bethe ansatz calculation, with additional checks using high precision extrapolation numerics. (paper)

[en] We consider the asymmetric simple exclusion process (ASEP) with forward hopping rate 1, backward hopping rate q and periodic boundary conditions. We show that the Bethe equations of ASEP can be decoupled, at all order in perturbation in the variable q, by introducing a formal Laurent series mapping the Bethe roots of the totally asymmetric case q = 0 (TASEP) to the Bethe roots of ASEP. The probability of the height for ASEP is then written as a single contour integral on the Riemann surface on which symmetric functions of TASEP Bethe roots live. (paper)

[en] We consider the weakly asymmetric exclusion process with $N=L/2$ particles on a periodic lattice of L sites and hopping rates 1 and $q=1-<!--\; ???\; -->\mathrm{\mu <!--\; ??\; -->}/\sqrt{L}$ in the forward and in the backward direction respectively. Using the Bethe ansatz, we obtain a systematic perturbative expansion of the spectral gap near $\mathrm{\mu <!--\; ??\; -->}=0$ by solving a simple functional equation order by order. A key point is that when $\mathrm{\mu <!--\; ??\; -->}\to <!--\; ???\; -->0$, Bethe roots at a distance $1/\sqrt{L}$ from the edge of the Fermi sea should not be considered as a continuum but converge instead at large L to the complex zeroes of $1+\mathrm{e}\mathrm{r}\mathrm{f}(x)$ after a rescaling by $\sqrt{L}$. (paper)

[en] We consider the asymmetric simple exclusion process on a ring, with an arbitrary asymmetry between the hopping rates of the particles. Using a functional formulation of the Bethe equations of the model, we derive exact expressions for all the cumulants of the current in the stationary state. These expressions involve tree structures with composite nodes. In the thermodynamic limit, three regimes can be observed for the current fluctuations depending on how the asymmetry scales with the size of the system.

[en] 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 $\mathrm{\rho <!--\; ??\; -->}=N/L$ 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 $1-<!--\; ???\; -->q\sim <!--\; ???\; -->1/\sqrt{L}$, double extrapolation combined with an integer relation algorithm gives an exact expression for the spectral gap up to 10th order in the asymmetry. (paper)

[en] We use the Bethe ansatz to derive analytical expressions for the current statistics in the asymmetric exclusion process with both forward and backward jumps. The Bethe equations are highly coupled and this fact has impeded their use in deriving exact results for finite systems. We overcome this technical difficulty by reformulating Bethe equations into a one-variable polynomial problem, akin to the functional Bethe ansatz. The perturbative solution of this equation leads to the cumulants of the current. We calculate here the first two orders and derive exact formulae for the mean value of the current and its fluctuations