The Stochastic Six Vertex model and discrete Orthogonal Polynomial ensembles

Abstract

Stochastic growth models in the Kardar-Parisi-Zhang (KPZ) universality class exhibit remarkable fluctuation phenomena. While a variety of powerful methods have led to a detailed understanding of their typical fluctuations or large deviations, much less is known about behavior on intermediate, or moderate deviation, scales. Addressing this problem requires refined asymptotic control of the integrable structures underlying KPZ models. Motivated by this perspective, we study multiplicative statistics of discrete orthogonal polynomial ensembles (dOPEs) in different scaling regimes, with a particular focus on applications to tail probabilities of the height function in the stochastic six-vertex model. For a large class of dOPEs, we obtain robust singular asymptotic estimates for multiplicative statistics critically scaled near a saturated-to-band transition. These asymptotics exhibit universal crossover behavior, interpolating between Airy, Painlevé XXXIV, and Bessel-type regimes. Our proofs employ the Riemann-Hilbert Problem (RHP) approach to obtain asymptotics for the correlation kernel of a deformed version of the dOPE across the critical scaling windows. These asymptotics are then used on a double integral formula relating this kernel to partition function ratios, which may be of independent interest. At the technical level, the RHP analysis requires a novel parameter-dependent local parametrix, which needs a separate asymptotic analysis of its own. Using these results, together with a known identity relating a Laplace-type transform of the stochastic six-vertex model height function to a multiplicative statistic of the Meixner point process, we derive moderate deviation estimates for the height function in both the upper and lower tail regimes, with sharp exponents and constants.

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