  We consider the asymmetric simple exclusion processes (ASEP) on a ring
constrained to produce an atypically large flux, or an extreme activity. Using
quantum free fermion techniques we find the time-dependent conditional
transition probabilities and the exact dynamical structure factor under such
conditioned dynamics. In the thermodynamic limit we obtain the explicit scaling
form. This gives a direct proof that the dynamical exponent in the extreme
current regime is $z=1$ rather than the KPZ exponent $z=3/2$ which
characterizes the ASEP in the regime of typical currents. Some of our results
extend to the activity in the partially asymmetric simple exclusion process,
including the symmetric case.
