  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.
