Stochastic six-vertex model

TL;DR

This paper analyzes the stochastic six-vertex model, demonstrating its convergence to a deterministic shape and GUE Tracy-Widom fluctuations, confirming its place in the KPZ universality class.

Contribution

It provides a rigorous proof of the limit shape and fluctuation distribution for the stochastic six-vertex model, linking it to KPZ universality.

Findings

Convergence to a deterministic limit shape

GUE Tracy-Widom distribution for fluctuations

Model belongs to KPZ universality class

Abstract

We study the asymmetric six-vertex model in the quadrant with parameters on the stochastic line. We show that the random height function of the model converges to an explicit deterministic limit shape as the mesh size tends to 0. We further prove that the one-point fluctuations around the limit shape are asymptotically governed by the GUE Tracy-Widom distribution. We also explain an equivalent formulation of our model as an interacting particle system, which can be viewed as a discrete time generalization of ASEP started from the step initial condition. Our results confirm an earlier prediction of Gwa and Spohn (1992) that this system belongs to the KPZ universality class.