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The one-dimensional asymmetric simple exclusion process (ASEP), where $N$hard-core particles hop forward with rate $1$ and backward with rate $q<1$, isconsidered on a periodic lattice of $L$ site. Using KPZ universality andprevious results for the totally asymmetric model $q=0$, precise conjecturesare formulated for asymptotics at finite density $\rho=N/L$ of ASEP eigenstatesclose to the stationary state. The conjectures are checked with high precisionusing extrapolation methods on finite size Bethe ansatz numerics. For weakasymmetry $1-q\sim1/\sqrt{L}$, double extrapolation combined with an integerrelation algorithm gives an exact expression for the spectral gap up to $10$-thorder in the asymmetry.

Extrapolation methods and Bethe ansatz for the asymmetric exclusion process