Interpolation between the stochastic telegraph equation and KPZ fluctuation limits of the stochastic six-vertex model

The stochastic six-vertex model — an exactly solvable interacting system in the KPZ (Kardar–Parisi–Zhang) universality class — has two rigorously established fluctuation limits of opposite character. In the rare-turn regime of Borodin–Gorin, where the weights take the form $b_1=\exp(-\beta_1/L)$ and $b_2=\exp(-\beta_2/L)$ and both coordinates scale ballistically, $\sqrt L$-scaled fluctuations of the exponential height observable converge to the Gaussian solution of the linear hyperbolic stochastic telegraph equation. In the weak-asymmetry regime of Corwin–Ghosal–Shen–Tsai, where the weight ratio tends to one under diffusive scaling, height fluctuations converge to the non-Gaussian Hopf–Cole solution of the nonlinear parabolic KPZ equation. Borodin–Gorin explicitly asked for an interpolation between the two regimes. Later literature resolved an adjacent regularity question (Shen–Tsai), extended both endpoints to the higher-spin model (Lin), and proved SPDE limits for dynamic degenerations (Corwin–Ghosal–Matetski), but no scaling regime of the model interpolating between the hyperbolic and parabolic limits is known. This record poses the open problem: construct an explicitly parameterized family of scaling regimes of the stochastic six-vertex model, prove convergence of the height fluctuations throughout it, and identify the limiting SPDEs — Gaussian, KPZ-type, or of a genuinely new class — at the interior parameter values.

Background

The six-vertex model is a classical statistical-mechanical system on the square lattice whose configurations are families of non-crossing up-right paths: exactly two of the four edges around each vertex are occupied, giving six possible local vertex types. In the stochastic six-vertex model (introduced by Gwa and Spohn in 1992 and developed as an exactly solvable interacting particle system by Borodin, Corwin and Gorin in 2016), the six vertex weights are chosen so that the configuration in the quadrant can be generated sequentially — given the lines entering a vertex from the left and from below, the lines leaving it upward and to the right are drawn at random with probabilities summing to one. Vertical lines can then be read as particles that move right or pause, placing the model in the KPZ (Kardar–Parisi–Zhang) universality class of stochastic growth. The two parameters $b_1,b_2\in(0,1)$ govern the competition between a line continuing straight and turning between vertical and horizontal motion. The height function $H(x,y)$ — the number of lines passing below and to the left of the point $(x,y)$ — is the central observable.

Borodin–Gorin analyzed the rare-turn regime in which the mesh tends to zero while turns of the paths become rare:

$$b_1=\exp(-\beta_1/L),\qquad b_2=\exp(-\beta_2/L),\qquad \beta_1,\beta_2>0,\ \beta_1\ne\beta_2,$$

under the ballistic scaling $(x,y)\mapsto(Lx,Ly)$. They proved that the exponential height observable $q^{H(Lx,Ly)}$ (with $q$ tending to 1 as $L\to\infty$) converges to a deterministic limit shape solving the classical telegraph equation — the linear second-order hyperbolic PDE

$$\phi_{xy}+\beta_1\phi_y+\beta_2\phi_x=0,$$

closely related to the damped-wave (telegrapher's) equation of classical physics — and that the centered, $\sqrt L$-scaled fluctuations of $q^{H(Lx,Ly)}$ converge, for piecewise-$C^1$ monotone boundary data specified on the two axes, to a Gaussian field solving the stochastic telegraph equation: the same PDE with right-hand side two-dimensional white noise (a generalized random field with delta-correlated values) multiplied by an explicit function of the limit shape. This remains one of the few exactly solvable systems whose fluctuation field is governed by a hyperbolic SPDE rather than a parabolic one.

On the other side, Corwin–Ghosal–Shen–Tsai proved a weak-asymmetry limit of the same model: $b_1\in(0,1)$ fixed while the ratio $b_2/b_1=e^{-\sqrt\varepsilon}$ tends to 1, with spatial scale $\varepsilon^{-1}$ and temporal scale $\varepsilon^{-2}$. Under this diffusive scaling, centered height fluctuations at scale $\sqrt\varepsilon$ converge (for near-stationary and step initial data) to the Hopf–Cole solution of the KPZ equation — the nonlinear parabolic SPDE $\partial_t h=\nu\,\partial_x^2 h+\tfrac{\kappa}{2}(\partial_x h)^2+\sqrt D\,\xi$ with $\xi$ space-time white noise, the namesake of the KPZ universality class — meaning that the exponential of the height solves the stochastic heat equation with multiplicative noise. The limit is non-Gaussian, reflecting the strong fluctuations characteristic of that class.

The two regimes differ in every structural feature: ballistic versus diffusive coordinate scaling; linear versus nonlinear limit equation; Gaussian versus non-Gaussian limit. They are also proved by mutually incompatible techniques: Borodin–Gorin rely on an exact stochastic four-point relation (a martingale-type integral identity) for the observable $q^H$, while Corwin–Ghosal–Shen–Tsai rely on a microscopic Hopf–Cole transform together with one- and two-particle self-duality estimates. Borodin–Gorin close their comparison of the two regimes with the remark: "It would be interesting to try to find an interpolation between our results and those of [CGST]."

Both endpoints have moreover been extended to the stochastic higher-spin six-vertex model of Corwin and Petrov — a fused generalization allowing up to $I$ and $J$ parallel lines per edge, containing the six-vertex model at $I=J=1$ — where Lin proved the KPZ limit under weak asymmetry and the stochastic telegraph limit under rare turns, the latter with a functional central limit theorem. No scaling regime of either model producing a limiting fluctuation field strictly between the two is known. (A third parameter, the dynamic parameter $\alpha$ of the dynamic six-vertex model, deforms the weights so that they depend on the height itself; Borodin–Gorin's general-boundary fluctuation theorems are proved at $\alpha=0$, which is the homogeneous stochastic model discussed throughout.)

Problem Statement

Construct and rigorously analyze an interpolating family of scaling limits of the stochastic six-vertex model connecting its two known fluctuation regimes.

1. Hyperbolic endpoint (Borodin–Gorin): the rare-turn scaling $b_1=\exp(-\beta_1/L)$, $b_2=\exp(-\beta_2/L)$ with fixed $\beta_1\ne\beta_2>0$, under ballistic scaling $(x,y)\mapsto(Lx,Ly)$, in which the centered, $\sqrt L$-scaled fluctuations of the exponential height observable $q^{H(Lx,Ly)}$ converge to the Gaussian solution of the stochastic telegraph equation $\phi_{xy}+\beta_1\phi_y+\beta_2\phi_x=\dot W\,g$, with $\dot W$ two-dimensional white noise and $g$ an explicit functional of the limit shape.

2. Parabolic endpoint (Corwin–Ghosal–Shen–Tsai): the weak-asymmetry scaling with $b_1\in(0,1)$ fixed, $b_2/b_1=e^{-\sqrt\varepsilon}$, spatial scale $\varepsilon^{-1}$ and temporal scale $\varepsilon^{-2}$, in which centered height fluctuations at scale $\sqrt\varepsilon$ converge to the Hopf–Cole solution of the KPZ equation.

A complete answer should provide: (i) an explicitly parameterized family of scalings of the weights and coordinates of the stochastic six-vertex model (by a parameter $\theta\in[0,1]$, say), depending continuously on $\theta$, whose endpoint values reproduce, up to reparameterization, the two regimes above and whose interior values give scaling regimes genuinely distinct from both endpoints (not reparameterizations of either endpoint scaling); (ii) for at least one class of initial/boundary data treated uniformly across the whole family (for example step, or near-stationary), a proof of convergence of the height-function fluctuations for interior values of the parameter, at the level of finite-dimensional distributions or stronger; and (iii) identification of the limiting object for each interior parameter value — its equation, its Gaussian or non-Gaussian character, and its well-posedness — together with a description of the crossover mechanism: whether interior limits degenerate to one of the endpoints under further rescaling, or form a genuinely new class, such as a nonlinear second-order SPDE with white noise.

Work carried out for the stochastic higher-spin six-vertex model is admissible provided the six-vertex case (spin $I=J=1$) is included; interpolations between different models (for example degenerations that never realize the telegraph endpoint) do not answer the question. A piecewise family whose interior parameter values merely reproduce one endpoint scaling restates the two known theorems and does not interpolate between the two regimes. Heuristic variational arguments or numerical simulations without convergence theorems do not constitute a solution.

References

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