  We consider the ASEP and the stochastic six vertex model started with step
initial data. After a long time, $T$, it is known that the one-point height
function fluctuations for these systems are of order $T^{1/3}$. We prove the
KPZ prediction of $T^{2/3}$ scaling in space. Namely, we prove tightness (and
Brownian absolute continuity of all subsequential limits) as $T$ goes to
infinity of the height function with spatial coordinate scaled by $T^{2/3}$ and
fluctuations scaled by $T^{1/3}$. The starting point for proving these results
is a connection discovered recently by Borodin-Bufetov-Wheeler between the
stochastic six vertex height function and the Hall-Littlewood process (a
certain measure on plane partitions). Interpreting this process as a line
ensemble with a Gibbsian resampling invariance, we show that the one-point
tightness of the top curve can be propagated to the tightness of the entire
curve.
