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

Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles