Uniqueness of limit shapes for the six-vertex model with rare corners and restricted Mallows permutations at general bias

The six-vertex model in the rare-corner limit $c\to0$ is, on rectilinear polygonal domains, equivalent to Mallows-distributed random permutations whose positions are restricted by a 0/1 array $I$. Gorin and Kenyon (2024; Probab. Theory Relat. Fields 2025) proved that the macroscopic limit shape is governed by a permuton variational principle whose Euler-Lagrange equation is the hyperbolic Liouville equation $(\log g)_{xy}=2rg$, reduced, for convex $I$, to a finite algebraic system; they proved the maximizer is unique at $r=0$ for all domains and, given a smooth Euler-Lagrange solution, for $|r|<r_0$ with $r_0$ depending on the data, and they conjectured (Conjecture 3.12) that the limiting density is uniformly bounded away from $0$ and $\infty$ on occupied rectangles. The open problem is whether this variational problem is well posed at arbitrary bias $r\neq0$ and general non-degenerate data: is the maximizer unique for all $r$ (i.e. $r_0=+\infty$), is every maximizer a smooth solution of the Euler-Lagrange equations, equivalently the algebraic system, and does the uniform-positivity conjecture hold? No resolution is known in the literature to date.

Background

The six-vertex model is a classical statistical-mechanical model on the square lattice in which each vertex is assigned one of six allowed configurations of two incoming and two outgoing edges, weighted by positive parameters $a$, $b$, $c$, with the paths allowed to touch but not to cross. Under domain-wall boundary conditions (all path entries on two adjacent sides of a square and exits on the other two) the model is central to integrable combinatorics. Its macroscopic behavior is captured by a limit shape: as the lattice size $N\to\infty$, the random height function, an integer-valued 1-Lipschitz function encoding the path configuration, converges in probability to a deterministic surface. For generic weights the density of $c$-type vertices (the local turns or 'corners' of paths) vanishes in frozen regions bounded by arctic curves, and complete limit-shape descriptions were known only in three situations: the free-fermion line (equivalent to random domino tilings), the five-vertex limit, and stochastic weights with free boundaries.

Gorin and Kenyon identified a fourth exactly solvable degeneration, the rare-corner regime $c\to0$. On rectilinear polygonal domains determined by subdivisions $0=X_0<\dots<X_k=N$ and $0=Y_0<\dots<Y_\ell=N$ of the sides together with a 0/1 array $I$ marking which rectangles of the induced grid are occupied (the array is convex when the 1s in every row and column form a contiguous segment), the $c\to0$ six-vertex model is equivalent, via a weight-preserving bijection, to the Mallows measure on permutations. The Mallows measure (introduced by Mallows in 1957) weights each permutation $\sigma$ by $q^{\mathrm{inv}(\sigma)}$, where $\mathrm{inv}(\sigma)$ counts inversions and $q=b^2/a^2>0$; here the positions of $\sigma$ are restricted by the array $I$. In the scaling regime $q=\exp(-r/N)$ with fixed real $r$, macroscopic limits are described by permutons, which are probability measures on $[0,1]^2$ with uniform marginals arising as scaling limits of large random permutation matrices. Writing $h$ for the rescaled height function, equivalent to a permuton through $h(x,y)=\mu([x,1]\times[0,y])$, the limit shape is the maximizer of the permuton energy

$$\mathfrak{E}(h)=\int_0^1\int_0^1\bigl[h_{xy}\ln(-h_{xy})-r\,h_xh_y\bigr]\,\mathrm{d}x\,\mathrm{d}y$$

over the admissible class $\mathfrak{F}^{\mathrm{x},\mathrm{y},I}$ of 1-Lipschitz height functions with the boundary values fixed by the rectilinear domain, satisfying the supermodularity inequality $h(x_1,y_1)+h(x_2,y_2)-h(x_1,y_2)-h(x_2,y_1)\le0$ (which is what makes the permuton density $g=-h_{xy}$ nonnegative), and such that the permuton mass of every grid rectangle with $I_{uv}=0$ vanishes; the data is called non-degenerate when there exists a positive array with row sums $x_u-x_{u-1}$ and column sums $y_v-y_{v-1}$ supported exactly on the 1s of $I$. The first term of $\mathfrak{E}$ is the entropy of the permuton density $g=-h_{xy}$ (nonnegative, since $-1\le h_x\le0$ and $0\le h_y\le1$), and the second term is the continuum inversion weight. The Euler-Lagrange equation of this variational problem is the hyperbolic Liouville equation $(\log g)_{xy}=2rg$, which is equivalent to the four-point relation (equation (15) of the paper): for all $0<x_1\le x_2<1$ and $0<y_1\le y_2<1$,

$$\ln\left(\frac{g(x_1-,y_1-)\,g(x_2+,y_2+)}{g(x_1-,y_2+)\,g(x_2+,y_1-)}\right)=2r\int_{y_1}^{y_2}\int_{x_1}^{x_2}g(x,y)\,\mathrm{d}x\,\mathrm{d}y,$$

where $g(x\pm,y\pm)$ denote one-sided limits: the logarithm of the cross-ratio of the values of $g$ at the corners of any rectangle equals $2r$ times the integral of $g$ over it. On each occupied rectangle the solutions take the Liouville form $g(x,y)=-\frac{1}{r}\frac{\phi'(x)\psi'(y)}{(\phi(x)-\psi(y))^2}$ with $\phi,\psi$ defined up to simultaneous Moebius transformations, so that limit shapes are typically only piecewise algebraic, with multiplicative jumps of $g$ across lines of discontinuity. For convex $I$ the boundary values $\phi(x_u)$, $\psi(y_v)$ satisfy a finite algebraic system (equations (45),(46) of the paper), which makes the limit shapes explicitly computable.

On uniqueness of the maximizer the following is known. At $r=0$ the functional is strictly concave and Gorin and Kenyon prove uniqueness for any domain, with the maximizer computable by a Sinkhorn-type iterative proportional fitting algorithm. For small bias, their Theorem 3.15 states that a smooth solution of the Euler-Lagrange equations, meaning one whose density $g$ extends continuously to each occupied closed rectangle and is bounded inside fixed constants $0<c<C$, is the unique maximizer, but only in a regime $|r|<r_0$ where $r_0$ depends on the data and on the class $[c,C]$. For unrestricted Mallows permutations (the single-cell array $I=(1)$) uniqueness for all $r$ follows from work of Starr and of Starr and Walters. For 'simple' convex arrays, those reducible to a single cell by repeatedly deleting rows or columns with a single 1 or merging adjacent completely filled rows or columns, Theorem 4.16 of Gorin and Kenyon gives uniqueness, up to Moebius transformations, of the solution of the algebraic system for every $r\neq0$. Beyond these regimes nothing is known: the functional is not convex for $r\neq0$, so stationarity need not imply global maximality and multiple stationary points may exist; no general uniqueness of admissible Euler-Lagrange solutions is available for non-simple convex arrays or non-convex arrays; and the positivity of the density itself is conjectural, a genuinely special feature of the $c\to0$ degeneration since for generic six-vertex weights the density of $c$-type vertices vanishes across frozen (arctic) regions.

Problem Statement

Determine whether the permuton variational problem of the rare-corner six-vertex model (equivalently, of Mallows permutations with restricted positions) is well posed at arbitrary bias. For non-degenerate data $(\mathrm{x},\mathrm{y},I)$, where $0=\mathrm{x}_0<\mathrm{x}_1<\dots<\mathrm{x}_k=1$ and $0=\mathrm{y}_0<\mathrm{y}_1<\dots<\mathrm{y}_\ell=1$ are subdivisions and $I$ is an arbitrary 0/1 array, convex or not, and for arbitrary real $r$, consider the permuton energy

$$\mathfrak{E}(h)=\int_0^1\int_0^1\bigl[h_{xy}\ln(-h_{xy})-r\,h_xh_y\bigr]\,\mathrm{d}x\,\mathrm{d}y$$

over the admissible class $\mathfrak{F}^{\mathrm{x},\mathrm{y},I}$.

(1) Prove, or disprove by an explicit counterexample, that $\mathfrak{E}$ has a unique maximizer over $\mathfrak{F}^{\mathrm{x},\mathrm{y},I}$ for every real $r$: that is, remove the small-$|r|$ restriction of Gorin-Kenyon Theorem 3.15 and settle whether $r_0=+\infty$ for all non-degenerate data, as suggested by their Remark 3.16 in analogy with the unrestricted Mallows case.

(2) Prove, or disprove, that every maximizer is a smooth solution of the Euler-Lagrange equations, the four-point relation for the density $g=-h_{xy}$ stated explicitly in the Background, so that for convex $I$ the limit shape is always computed by the algebraic boundary system (45),(46) of Gorin-Kenyon through the Liouville-form solution $g(x,y)=-\frac{1}{r}\frac{\phi'(x)\psi'(y)}{(\phi(x)-\psi(y))^2}$; in particular, settle whether the Euler-Lagrange system (for convex $I$, equivalently the algebraic system (45),(46) with its positivity and marginal constraints) has a unique admissible solution for non-simple convex arrays and for non-convex arrays, where no uniqueness statement is currently known.

(3) Settle Gorin-Kenyon Conjecture 3.12: for non-degenerate data, does every maximizer have a permuton density $g=-h_{xy}$ with $0<c<g(x,y)<C<\infty$ almost everywhere on the occupied rectangles of the domain, with the constants depending on the data? (This is a corollary of (2), since a smooth Euler-Lagrange solution is by definition bounded on occupied rectangles; establishing (2) therefore settles (3), while exhibiting a maximizer with unbounded density refutes both.)

A complete answer establishes (1)-(3) in the stated generality, all non-degenerate data and all real $r$, or refutes any of them by exhibiting specific non-degenerate data and a value of $r$, together with two distinct maximizers, a non-smooth or non-unique admissible Euler-Lagrange solution, or a maximizer whose density degenerates to $0$ or $\infty$ on a set of positive measure, in sufficient detail that admissibility and extremality can be verified.

References

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