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/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 similar to 1 / root L, double extrapolation combined with an integer relation algorithm gives an exact expression for the spectral gap up to 10th order in the asymmetry.