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

60 pages, 11 figures

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• Asymptotics in ASEP with Step Initial Condition
• Fluctuation properties of the TASEP with periodic initial configuration
• The one-dimensional KPZ equation and its universality class
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Cited by in corpus (8)

• Stochastic six-vertex model in a half-quadrant and half-line open ASEP
• Uniform convergence to the Airy line ensemble
• Half-space Macdonald processes
• Limits and fluctuations of $p$-adic random matrix products
• Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model
• Tightness of discrete Gibbsian line ensembles
• Characterization of $H$-Brownian Gibbsian line ensembles
• The stochastic six-vertex model speed process