  We consider the asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$
with an initial data such that in the large time particle density $\rho(\cdot)$
a discontinuity at the origin is created, where the value of $\rho$ jumps from
zero to one, but $\rho(-\varepsilon),1-\rho(\varepsilon) >0 $ for any
$\varepsilon>0$. We consider the position of a particle $x_{M}$ macroscopically
located at the discontinuity, and show that its limit law has a cutoff under
$t^{1/2}$ scaling. Inside the discontinuity region, we show that a discrete
product limit law arises, which bounds from above the limiting fluctuations of
$x_{M}$ in the general ASEP, and equals them in the totally ASEP. Note: This
preprint has been superseded by arXiv:1906.07711 and is no longer updated.
