Two-Point Convergence of the Stochastic Six-Vertex Model to the Airy Process

Abstract

In this paper we consider the stochastic six-vertex model in the quadrant started with step initial data. After a long time T, it is known that the one-point height function fluctuations are of order $$T^{1/3}$$ and governed by the Tracy–Widom distribution. We prove that the two-point distribution of the height function, rescaled horizontally by $$T^{2/3}$$ and vertically by $$T^{1/3}$$ , converges to the two-point distribution of the Airy process. The starting point of this result is a recent connection discovered by Borodin–Bufetov–Wheeler between the stochastic six-vertex model and the ascending Hall–Littlewood process (a certain measure on plane partitions). Using the Macdonald difference operators, we obtain formulas for two-point observables for the ascending Hall–Littlewood process, which for the six-vertex model give access to the joint cumulative distribution function for its height function. A careful asymptotic analysis of these observables gives the two-point convergence result under certain restrictions on the parameters of the model.