The stochastic six vertex model and discrete orthogonal polynomial ensembles

TL;DR

This paper develops a refined asymptotic analysis of discrete orthogonal polynomial ensembles to understand moderate deviations in the stochastic six-vertex model, revealing universal crossover behaviors and providing sharp tail probability estimates.

Contribution

It introduces a novel Riemann-Hilbert approach with a parameter-dependent local parametrix to analyze critical scaling regimes in dOPEs, connecting these results to KPZ tail behaviors.

Findings

Derived universal crossover asymptotics between Airy, Painlevé II, and Bessel regimes.

Obtained sharp moderate deviation estimates for the height function tails.

Developed a new RHP analysis technique with a parameter-dependent local parametrix.

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, Painlev\'e II, 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 employs a novel parameter-dependent local parametrix, which requires 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.