Extrapolation methods and Bethe ansatz for the asymmetric exclusion process

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 $\rho =N\mathrm{/}L$ 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 $1-q\sim 1\mathrm{/}\sqrt{L}$, double extrapolation combined with an integer relation algorithm gives an exact expression for the spectral gap up to $10$-th order in the asymmetry.