Ground state energy of the δ\delta-Bose and Fermi gas at weak coupling from double extrapolation

We consider the ground state energy of the Lieb-Liniger gas with $\delta$ interaction in the weak coupling regime $\gamma\to0$. For bosons with repulsive interaction, previous studies gave the expansion $e_{\text{B}}(\gamma)\simeq\gamma-4\gamma^{3/2}/3\pi+(1/6-1/\pi^{2})\gamma^{2}$. Using a numerical solution of the Lieb-Liniger integral equation discretized with $M$ points and finite strength $\gamma$ of the interaction, we obtain very accurate numerics for the next orders after extrapolation on $M$ and $\gamma$. The coefficient of $\gamma^{5/2}$ in the expansion is found approximately equal to $-0.00158769986550594498929$, accurate within all digits shown. This value is supported by a numerical solution of the Bethe equations with $N$ particles followed by extrapolation on $N$ and $\gamma$. It was identified as $(3\zeta(3)/8-1/2)/\pi^{3}$ by G. Lang. The next two coefficients are also guessed from numerics. For balanced spin $1/2$ fermions with attractive interaction, the best result so far for the ground state energy was $e_{\text{F}}(\gamma)\simeq\pi^{2}/12-\gamma/2+\gamma^{2}/6$. An analogue double extrapolation scheme leads to the value $-\zeta(3)/\pi^{4}$ for the coefficient of $\gamma^{3}$.Comment: 11 pages, 2 figures, 3 table

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

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.Comment: 40 page

Tree structures for the current fluctuations in the exclusion process

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.Comment: 43 page

A combinatorial solution for the current fluctuations in the exclusion process

We conjecture an exact expression for the large deviation function of the stationary state current in the partially asymmetric exclusion process with periodic boundary conditions. This expression is checked for small systems using functional Bethe Ansatz. It generalizes a previous result by Derrida and Lebowitz for the totally asymmetric exclusion process, and gives the known values for the three first cumulants of the current in the partially asymmetric model. Our result is written in terms of tree structures and provides a new example of a link between integrable models and combinatorics.Comment: 7 page