Classification of boundary fluctuation universality classes for the six-vertex model with domain-wall boundary conditions near the critical anisotropy $\Delta=1$ and in degenerate zero-weight limits

For the symmetric six-vertex model with domain-wall boundary conditions (DWBC) on the n×n square, the joint statistics of the monotone-triangle rows that encode configurations near the boundary of the square are now classified away from the critical anisotropy $\Delta=1$: for $\Delta<1$ they have fluctuations of order $n^{1/2}$ converging to the GUE-corners process of random matrix theory, for $\Delta>1$ they have bounded-order fluctuations converging to the stochastic six-vertex model in a quadrant, and for $c=0$ they converge to q-exchangeable random permutations governed by the infinite Mallows measure. Every one of these theorems excludes $\Delta=1$, and the neighborhood of $\Delta=1$ has been analyzed only through the partition function. Gorin and Liechty, who proved the three known regimes, anticipate a rich family of scaling limits as $n\to\infty$ with $\Delta\to1$ — analogous to the KPZ and stochastic telegraph limits of the stochastic six-vertex model — and pose the comprehensive classification of boundary fluctuation universality classes near $\Delta=1$ and for degenerate zero-weight asymptotics ($a\to0$, $b\to0$, or $a,b\to0$, possibly simultaneously with $n\to\infty$) as an open goal. This problem asks for that classification: the map of all double-scaling regimes, their fluctuation exponents, limiting objects, and convergence theorems.

Background

The six-vertex model is a classical two-dimensional lattice model of paths on the square grid: at each vertex exactly two of the four incident edges are occupied, which leaves six possible local configurations (vertex types), carrying weights $a$, $b$, $c$ (two types for each weight) in the symmetric version of the model. Its anisotropy parameter is $\Delta=(a^2+b^2-c^2)/(2ab)$, and the model's macroscopic behavior is organized by $\Delta$: $|\Delta|<1$ is the disordered phase, $\Delta>1$ the ferroelectric (ordered) phase, and $\Delta<-1$ the antiferroelectric phase. Under domain-wall boundary conditions (DWBC), the model is placed on an $n\times n$ square with $n$ paths entering along the bottom and left sides and exiting along the top and right sides. In much of the phase diagram the interior then contains a random separation (arctic) curve dividing a disordered central region from frozen corners; near most of the boundary of the square the configurations are frozen, so the model interacts with the boundary only through the special points where this curve touches it, which makes the statistics of configurations near the boundary a natural and central observable.

A convenient encoding: for $k=1,\dots,n$ let $1\le\lambda^k_1<\lambda^k_2<\dots<\lambda^k_k\le n$ be the positions of the vertical path segments between rows $k$ and $k+1$; the interlacing array $(\lambda^k_i)$ is a random monotone triangle, and boundary statistics refers to the joint distribution of the rows with fixed $k$ as $n\to\infty$, i.e., the configuration near the boundary of the square. Gorin and Liechty (preprint 2023; published in Communications on Pure and Applied Mathematics, 2025) proved a sharp dichotomy for these statistics away from the critical value $\Delta=1$. For $\Delta<1$, after centering at $m(a,b,c)\,n$ and scaling by $s(a,b,c)\sqrt{n}$ with explicit constants, the rows converge, in finite-dimensional distributions, to the GUE-corners process — the interlacing array of eigenvalues of the top-left corners of a Hermitian random matrix from the Gaussian Unitary Ensemble (GUE), a universal object of random matrix theory — so boundary fluctuations are of order $n^{1/2}$. For $\Delta>1$ (with $a>b$; the other cases follow by the symmetries of the square), the rows converge without any rescaling to the monotone triangle of the stochastic six-vertex model in the quadrant $\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$ — the Markovian version of the model in which vertices are sampled sequentially with stochastic weights satisfying $b_1+c_1=b_2+c_2=1$ — so the fluctuations are of bounded order. In the degenerate case $c=0$ (which forces $\Delta\ge 1$, with equality only when $a=b$), the limit is governed by a $q$-exchangeable random bijection of the positive integers with $q=b^2/a^2$ — an infinite Mallows measure on permutations, weighting a permutation $\tau$ proportionally to $q^{\mathrm{inv}(\tau)}$, where $\mathrm{inv}(\tau)$ is the number of inversions.

All of these theorems exclude $\Delta=1$, where the weights satisfy $c=|a-b|$ and the model sits on the critical line between the disordered and ferroelectric phases. The neighborhood of $\Delta=1$ has been studied only through the partition function $Z_n$: Bleher and Liechty (2009) derived the large-$n$ asymptotics of $Z_n$ on this critical line, and Bleher and Bothner (2014) computed the constant factor in those asymptotics by a Riemann–Hilbert double-scaling analysis (a steepest-descent technique of asymptotic analysis that recasts the problem as a matrix factorization problem in the complex plane), in which $n$ and the critical-line parameter tend jointly to infinity. For comparison, in the stochastic six-vertex model (which lives entirely at $\Delta>1$) two scaling limits near special points are known: convergence to a Kardar–Parisi–Zhang (KPZ)-type stochastic PDE — the stochastic heat equation, the universal field theory of one-dimensional random growth — by Corwin, Ghosal, Shen, and Tsai (2020), and convergence to a stochastic telegraph equation (a second-order-in-time stochastic PDE) by Borodin–Gorin (2019) and by Shen–Tsai (2019). Gorin and Liechty anticipate an analogous rich family of scaling limits for the DWBC model as $n\to\infty$ with $\Delta\to1$, and prove only one relatively simple result in this direction (their Proposition 1.10): with $c=0$ and $n\ln(b^2/a^2)\to\theta$, so that $a/b\to1$ and $\Delta\to1^+$, the scaled boundary rows $(\lambda^k_i/n)$ converge, in finite-dimensional distributions, to reorderings of i.i.d. random variables with density $\theta e^{\theta x}/(e^{\theta}-1)$ on $[0,1]$. The classification of all remaining limit regimes near $\Delta=1$, and of the degenerate limits in which $a\to0$, $b\to0$, or $a,b\to0$ possibly simultaneously with $n\to\infty$, is left open; in these degenerate directions the only subsequent progress (Gorin–Kenyon, 2025, on the related $c\to0$ rare-corners regime) concerns macroscopic limit shapes, not boundary fluctuations.

Problem Statement

Classify the boundary statistics of the symmetric six-vertex model with domain-wall boundary conditions on the $n\times n$ square — the joint distributions of the fixed rows $(\lambda^k_i)_{1\le i\le k}$ of the random monotone triangle, after whatever centering and rescaling is appropriate — in the sector of the parameter space not covered by the theorems of Gorin–Liechty, namely:

(1) The critical anisotropy $\Delta=1$ and its neighborhood. For every way in which the weights $(a,b,c)$ can depend on $n$ so that $n\to\infty$ while $\Delta\to1$ — at fixed $\Delta=1$ (where $c=|a-b|$), from the disordered side $\Delta<1$, or from the ferroelectric side $\Delta>1$, including degenerate paths such as $c\to0$ with $a/b\to1$ as in Proposition 1.10 — determine the scaling limit of the boundary rows: the fluctuation scale, the limiting random array or field, and the mode of convergence. Identify which limiting objects arise in which double-scaling windows — the known universal objects (GUE-corners process, stochastic six-vertex monotone triangle in the quadrant, $q$-exchangeable/Mallows arrays, the exponential reorderings of Proposition 1.10), KPZ-type or stochastic-telegraph-type limits as anticipated by the source, or genuinely new objects — prove the corresponding convergence theorems, and establish the sharp boundaries between windows.

(2) The degenerate zero-weight limits. Do the same for limits in which $a\to0$, $b\to0$, or $a,b\to0$ simultaneously (possibly combined with $n\to\infty$ and possibly with $c\to0$), determining whether non-trivial boundary statistics arise in each degenerate direction and, if so, classifying them.

The single overarching objective is a comprehensive classification of boundary fluctuation universality classes for the six-vertex model with DWBC: a complete map of the double-scaling diagram covering all regimes in (1) and (2), the two sectors jointly completing the universality picture begun by the fixed-weight theorems. Here the objects to be classified are the admissible weight families themselves — all $(a_n,b_n,c_n)_{n\ge1}$ with $n\to\infty$ and $\Delta_n\to1$ in (1), and the zero-weight families of (2) — and a regime is a set of such families on which the fluctuation scale, limiting object, and mode of convergence of the boundary rows are constant, so that the double-scaling diagram is the partition of the admissible families into regimes over the parameter space coordinatized by the side and the rate at which $\Delta_n\to1$ and by the limiting direction of the normalized weights $(a_n:b_n:c_n)$, with the degenerate zero-weight directions of (2) as the boundary of this space. Partial classifications (individual windows or directions) are admissible progress, but the problem is completed only when the classification is delivered as an explicit, proved partition: the regimes are listed, each with its fluctuation scale, limiting object, and mode of convergence, every admissible family is shown to lie in one of them, and each path separating neighbouring regimes is proved sharp, so that no regime in (1) or (2) remains unclassified.

References

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