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

## TL;DR

This paper proves the KPZ $T^{2/3}$ spatial scaling for the height functions of ASEP and stochastic six vertex models, using connections to Hall-Littlewood processes and Gibbsian line ensembles.

## Contribution

It establishes the $T^{2/3}$ spatial fluctuation scaling for these models, confirming KPZ predictions through novel connections to Hall-Littlewood processes.

## Key findings

- Proves tightness of height functions with $T^{2/3}$ spatial scaling.
- Shows Brownian absolute continuity of subsequential limits.
- Extends tightness from the top curve to entire line ensembles.

## Abstract

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.