  The probability distribution of the current in the asymmetric simple
exclusion process is expected to undergo a phase transition in the regime of
weak asymmetry of the jumping rates. This transition was first predicted by
Bodineau and Derrida using a linear stability analysis of the hydrodynamical
limit of the process and further arguments have been given by Mallick and
Prolhac. However it has been impossible so far to study what happens after the
transition. The present paper presents an analysis of the large deviation
function of the current on both sides of the transition from a Bethe ansatz
approach of the weak asymmetry regime of the exclusion process.
