Density profile and limit fluctuations of the stationary measure of the stochastic six-vertex model on a strip

The stochastic six-vertex model on a strip with two open boundaries was introduced and solved by Yang (Electronic Journal of Probability, 2024): its stationary measure on any down-right path is given by a matrix product ansatz, and on a horizontal path it reduces to a product tilting of the stationary measure of the open asymmetric simple exclusion process (open ASEP). For that tilted measure, only the limit of the mean particle density has been determined, producing a phase diagram that is a tilting of the open ASEP phase diagram, and only on the fan region AC<1. The paper explicitly leaves open the density profile and the limit fluctuations of the tilted stationary measure as the strip width N tends to infinity, the extension of the phase diagram to the shock region AC>1, and the corresponding analysis on non-horizontal down-right paths. A follow-up work of the author (higher-spin extension, 2023) resolved the mean-density limits along arbitrary down-right paths and showed all paths share a common open-ASEP-type phase diagram, and the density profile and fluctuations have been completed for open ASEP itself, including in its shock region; however, no published work determines the profile or the fluctuations for the six-vertex model on a strip. The open problem is to determine the N-to-infinity density profile and limit fluctuations of the strip stationary measures across the full phase diagram, including the shock region, and on general down-right paths. Two further open directions — applying Bethe-ansatz or symmetric-function techniques to the strip model, and establishing weak-asymmetry scaling limits of the strip model to open KPZ-type boundary-driven growth processes — are outside the scope of this problem.

Background

The six-vertex model places arrows on the edges of the square lattice subject to the ice rule — at every vertex exactly two arrows point in and two point out — which leaves six admissible vertex configurations. The stochastic six-vertex model assigns these six configurations positive weights that depend only on the incoming arrow directions; with such weights the arrow configuration along any monotone path evolves as a discrete-time Markov chain, and the model is a central exactly solvable member of the Kardar–Parisi–Zhang (KPZ) universality class of interface growth processes.

Yang (Electronic Journal of Probability, 2024) studied this model on a strip, the sub-lattice $\{(x,y)\in\mathbb{Z}^2: 0\le y\le x\le y+N\}$, with bulk vertex weights parameterized by $0<\theta_1<\theta_2<1$ and two open boundaries carrying weights $a,b,c,d$ (throughout, admissible parameters satisfy $a,b,c,d,\theta_1,\theta_2\in(0,1)$, $\theta_1<\theta_2$, and $b+d<1$). A down-right path is a path from the left to the right boundary taking unit steps downward or rightward; each such path has exactly $N$ outgoing up/right edges, and pushing the path through the lattice using the vertex weights defines an interacting particle system on those $N$ edges, with $\tau_i\in\{0,1\}$ denoting whether the $i$-th outgoing edge carries an arrow. The paper solves the stationary measure of every such system by a matrix product ansatz: the stationary probability of $(\tau_1,\dots,\tau_N)$ is a product of non-commuting matrices, one factor $(1-\tau_i)E+\tau_iD$ per site, sandwiched between boundary vectors, generalizing the celebrated Derrida–Evans–Hakim–Pasquier (DEHP) matrix product solution of the open asymmetric simple exclusion process (open ASEP) — the paradigmatic boundary-driven particle system on $\{1,\dots,N\}$ in which particles hop one site to the right with rate $R$ and to the left with rate $L$, a hop being forbidden when the target site is occupied, and which enter at the left end with rate $\alpha$, enter at the right end with rate $\delta$, exit from the left end with rate $\gamma$, and exit from the right end with rate $\beta$ (the source fixes $L=q$, $R=1$, so the asymmetry parameter is $q=L/R$); its stationary weights are matrix products of matrices $D,E$ satisfying a quadratic algebra.

On a horizontal path the strip stationary measure takes a striking form: it is a product tilting

$$\mu_N(\tau_1,\dots,\tau_N)=\frac{\prod_{i=1}^{N} r^{\tau_i}\,\pi_N(\tau_1,\dots,\tau_N)}{Z_N},\qquad r=\frac{1-\theta_2}{1-\theta_1},$$

of the stationary measure $\pi_N$ of open ASEP (with $q=\theta_1/\theta_2$ and the boundary rates $\alpha,\beta,\gamma,\delta$ explicit functions of $a,b,c,d,\theta_1,\theta_2$ given in the source paper), where $Z_N$ is the normalizing constant. For open ASEP, the large-$N$ stationary statistics are organized by a phase diagram in two effective boundary parameters

$$A=\kappa_+(\beta,\delta),\qquad C=\kappa_+(\alpha,\gamma),\qquad \kappa_\pm(u,v)=\frac{1}{2u}\left(1-q-u+v\pm\sqrt{(1-q-u+v)^2+4uv}\right),$$

namely a low-density phase, a high-density phase, a maximal-current phase, a fan region $AC<1$, a shock region $AC>1$, and a coexistence line — the first-order low-density/high-density transition where a microscopic shock wanders and the two phases coexist. For open ASEP essentially the full large-$N$ program is complete: the density profile (the limit of $\mathbb{E}\,\tau_{\lfloor uN\rfloor+1}$ for $u\in(0,1)$, determined by Uchiyama–Sasamoto–Wadati via Askey–Wilson polynomials, a four-parameter family of orthogonal polynomials), the limit fluctuations of the density (Bryc–Wang), and the extension to the shock region (Wang–Wesołowski–Yang, through Askey–Wilson signed measures — finite signed, i.e. not necessarily positive, measures that take over the role of the Askey–Wilson process law when $AC>1$). The modern route expresses stationary expectations in terms of the Askey–Wilson process, a Markov process whose moments are Askey–Wilson polynomial averages, with the quadratic-harness construction of Bryc–Wesołowski as its backbone (a quadratic harness is a process whose conditional mean is linear in the present value and whose conditional second moment is a quadratic function of it, a structural property that lets one reconstruct the process from its first two conditional moments).

For the strip six-vertex model, Yang proved that under the tilted measure $\mu_N$ the mean particle density $\frac1N\sum_{i=1}^{N}\tau_i$ converges as $N\to\infty$ (parameters fixed) with limit $\sqrt r/(1+\sqrt r)$ in the maximal-current phase ($A\le 1/\sqrt r$, $C\le\sqrt r$), $Ar/(1+Ar)$ in the high-density phase ($A>1/\sqrt r$, $AC<1$), and $r/(r+C)$ in the low-density phase ($C>\sqrt r$, $AC<1$) — a phase diagram that is a tilting of the open ASEP diagram — but only on the fan region $AC<1$ and under a technical condition ($|A\sqrt r|,|B\sqrt r|,|C/\sqrt r|,|D/\sqrt r|\ne q^{-l}$ for $l=0,1,2,\dots$, where $B=\kappa_-(\beta,\delta)$ and $D=\kappa_-(\alpha,\gamma)$) that excludes countably many parameter values. The paper also proved (Theorem 2.3) that when all bulk and boundary parameters are scaled to zero linearly in a small parameter, the strip dynamics converges to open ASEP, so the strip model is a genuine refinement of it. A follow-up work of the author (arXiv:2309.04897) extended the model by fusion — the $R$-matrix procedure that merges several adjacent lattice lines into one, so that up to $I$ arrows may occupy a single edge (spin $I/2$) — solved the stationary measure on arbitrary down-right paths by a fused matrix product ansatz, characterized it via the Askey–Wilson process, and determined the mean-density limits along any down-right path in the fan region: all down-right paths share a common open-ASEP-type phase diagram after reparameterization (though with path-dependent density limits), and the zig-zag path realizes the two-step Floquet dynamics introduced by Vanicat (a discrete-time exclusion process updated by alternating two deterministic sub-steps). Again, only the mean density and only the fan region were obtained.

What has not been done for the strip model is everything beyond the mean density: the spatial density profile in each phase and on the phase boundaries, the limit fluctuations of the density field, the extension to the shock region $AC>1$ where the Askey–Wilson process degenerates to signed measures, and the analogues of these questions for non-horizontal down-right paths. For open ASEP this program is complete, and the source paper explicitly expected its techniques to transfer to the strip model; no published work has carried out the transfer.

Problem Statement

For the stochastic six-vertex model on the strip $\{(x,y)\in\mathbb{Z}^2: 0\le y\le x\le y+N\}$ with bulk parameters $0<\theta_1<\theta_2<1$ and admissible boundary parameters (i.e. $a,b,c,d,\theta_1,\theta_2\in(0,1)$, $\theta_1<\theta_2$, $b+d<1$), consider the stationary measures given by the matrix product ansatz of Yang (Electronic Journal of Probability, 2024); on a horizontal path this is the tilted open-ASEP measure $\mu_N(\tau)=Z_N^{-1}\prod_{i=1}^{N} r^{\tau_i}\pi_N(\tau)$ with $r=(1-\theta_2)/(1-\theta_1)$ and $q=\theta_1/\theta_2$, and on other down-right paths it is the matrix-product stationary measure of the same paper (Theorem 2.8 there), extended to fused higher-spin weights and characterized through the Askey–Wilson process by the 2023 follow-up. Determine the asymptotics of these stationary measures as $N\to\infty$ with all parameters fixed, specifically:

(1) Density profile: for $u\in(0,1)$, the limit of $\mathbb{E}_{\mu_N}[\tau_{\lfloor uN\rfloor+1}]$ in each phase of the tilted phase diagram — the low-density phase ($C>\sqrt r$, $AC<1$), the high-density phase ($A>1/\sqrt r$, $AC<1$), and the maximal-current phase ($A\le 1/\sqrt r$, $C\le\sqrt r$) — and on the phase boundaries, identifying where the profile is flat, where boundary layers or shocks occur, and the limiting values on each side of every transition. Here $A=\kappa_+(\beta,\delta)$ and $C=\kappa_+(\alpha,\gamma)$ with $\kappa_\pm(u,v)=\frac{1}{2u}(1-q-u+v\pm\sqrt{(1-q-u+v)^2+4uv})$, and $(\alpha,\beta,\gamma,\delta)$ are the open-ASEP boundary rates corresponding to $(a,b,c,d,\theta_1,\theta_2)$ via the source paper's formulas.

(2) Limit fluctuations: the scaling limit and fluctuations of the density field about the profile — or of the stationary height function (the cumulative particle count $\sum_{i\le\lfloor uN\rfloor}\tau_i$ viewed as a function of $u\in[0,1]$) — including the centering, the fluctuation scale, the limit law and its explicit parameters in each phase, and on the coexistence line (the first-order low-density/high-density boundary).

(3) Shock region: the extension of items (1)–(2) and of the mean-density phase diagram to the shock region $AC>1$, where the Askey–Wilson process representation degenerates to signed measures.

(4) Other down-right paths: the analogues of items (1)–(2) for the stationary measures on general down-right paths (for example the zig-zag path), including the dependence of the answers on the limiting shape of the path.

Any rigorous technique — Askey–Wilson process or signed-measure asymptotics, Bethe ansatz, symmetric functions, or otherwise — is admissible; the completion boundary is the level of description already achieved for open ASEP, where profile, fluctuations, and the shock region are all determined.

References

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