The Stochastic Six Vertex model and discrete Orthogonal Polynomial ensembles

TL;DR

This work analyzes moderate deviations for height fluctuations in the stochastic six-vertex model by exploiting a deep link to discrete orthogonal polynomial ensembles, notably the Meixner ensemble, via the Borodin–Olshanski identity. A novel Riemann–Hilbert framework is developed, featuring an $N$-dependent local model problem and a deformed-weights deformation formula that ties multiplicative statistics to partition functions. The main contributions are universal asymptotics for multiplicative statistics that interpolate between Airy (soft edge), Painlevé XXXIV (critical), and Bessel (hard edge) regimes, and precise tail bounds for the height function in both the upper and lower tails, valid across moderate deviations scales. The results provide sharp exponents and constants for KPZ-type tails in positive-temperature settings and establish a robust bridge between integrable probability, RHP techniques, and KPZ universality through the Meixner-Meixner-S6V correspondence.

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.

Figures (11)

• Figure 1: On the left, the six possible configurations of individual vertices in the six-vertex model in its classical formulation. On the right, the corresponding configurations of vertices used in the representation as a random path configuration.
• Figure 2: A configuration of the stochastic six-vertex model in the random path representation.
• Figure 3: A simulation of the stochastic six-vertex model with narrow wedge initial condition, in a grid of size $200\times 200$, performed with the software Mathematica. Dark (blue) lines correspond to the paths in the model. The region filled with paths in the upper left corner, as well as the empty region in the lower right corner, correspond to the frozen regions. In these regions, the behavior of paths is determined with probability exponentially close to $1$. In the roughly conic region between the frozen ones we see a true stochastic behavior of paths. This region corresponds to the liquid region.
• Figure 4: Markovian evolution of upright paths in $\mathbb{Z}^2_{\geq}$, starting from the narrow wedge initial condition.
• Figure 5: Spatial decomposition for analyzing the error term $\mathsf R$. The summation domain is divided into regions based on distance from the band edge $\mathsf a$. Far regions (saturated, gap and band, depicted in blue and green) contribute exponentially small terms $O(\mathop{\mathrm{\rm e}}\nolimits^{-\eta N})$ via Euler-Lagrange inequalities. The outer region (orange) near $\mathsf a$ yields exponential decay through crude bounds on the model RHP. The inner region (red) requires precise asymptotics from Part 3, but remains subleading to the main term $\mathsf S$. The conformal map $\varphi$ and the model problem from Part 2 are essential technical tools.

Theorems & Definitions (129)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem 2.1: Upper tail for the stochastic six-vertex model
• Theorem 2.2: Lower tail for the stochastic six-vertex model
• Remark 2.3: Extension up to the region $\mathsf{h} \leq \mathsf h_0^{-1}N^{\tfrac{1}{6}}$
• Proposition 2.6: Deformation formula
• Theorem 2.8: Localized asymptotics for multiplicative statistics
• Theorem 2.9: Universal global asymptotics