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

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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.

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Acknowledgements

The author would like to thank Alexei Borodin, Guillaume Barraquand and Ivan Corwin for useful comments on earlier drafts of this paper as well as Amol Aggarwal for stimulating conversations about computing \(L^2\) norms of Cauchy determinants. The author is partially supported by the Minerva Foundation Fellowship.

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    Evgeni Dimitrov

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Dimitrov, E. Two-Point Convergence of the Stochastic Six-Vertex Model to the Airy Process. Commun. Math. Phys. 398, 925–1027 (2023). https://doi.org/10.1007/s00220-022-04499-3

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