Generic-weight thermodynamics of the six-vertex model in an L-shaped domain

Colomo and Pronko (2015) computed the exact thermodynamic free energy of the six-vertex model with domain-wall boundary conditions on an L-shaped domain (a square with a rectangular cut-off corner) only at the free-fermion point, where the partition function reduces to a Meixner-type discrete Coulomb gas with two hard walls; they obtained the free-energy density in closed form, explicit resolvents and band end-point equations in all regimes, and found a third-order (Douglas–Kazakov-type) phase transition when the arctic ellipse meets the cut boundary, attributing the non-transparency of the soft-edge/hard-edge meeting to a finite slope of the confining potential at the right hard wall. The issue raised by that analysis which remains open is whether the closed-form thermodynamic structure admits an analogue for generic six-vertex Boltzmann weights away from the free-fermion condition, the needed large-N asymptotics of the generalized emptiness formation probability being unavailable beyond free fermion (explicitly acknowledged as a limitation by the same authors in 2019). Subsequent work treats free-fermion arctic curves on the L-shaped domain, pentagonal cut-offs, five-vertex degenerations, and continuous Coulomb-gas universality, but the generic-weight L-shaped thermodynamics has not been established.

Background

The six-vertex model places arrows on the edges of a square lattice subject to the ice rule (at every vertex, two arrows point in and two point out), leaving six allowed vertex configurations with Boltzmann weights $w_1,\dots,w_6$. Under domain-wall boundary conditions (DWBC), in which all external horizontal arrows point into the region and all external vertical arrows point out, the model is the standard combinatorial laboratory for alternating-sign matrices and domino tilings, and in the scaling limit it exhibits phase separation: a disordered ('liquid') region coexists with frozen regions separated by the arctic curve, the phase-separation boundary of the limit shape. For the square $N\times N$ domain the thermodynamic free energy is known both at free-fermion weights and beyond, through the Korepin–Zinn-Justin approach and the Riemann–Hilbert analysis of the Izergin–Korepin determinant by Bleher and Liechty.

Colomo and Pronko (2015) addressed a genuinely non-rectangular geometry: the L-shaped domain, obtained by cutting an $s\times(s+q)$ rectangle out of one corner of the $N\times N$ square, with DWBC inherited from the square. At the free-fermion point, where the weights satisfy $w_1w_2+w_3w_4=w_5w_6$, the partition function equals a Hankel determinant built from Meixner polynomials (orthogonal polynomials on the nonnegative integers with weight $w_q^\alpha(m)=\binom{q+m}{q}\alpha^m$), or equivalently a discrete Coulomb gas: charges on integer sites with logarithmic repulsion, confined by a non-polynomial external potential, and subject to two hard walls, i.e. infinite barriers beyond which charges cannot sit (one wall is the constraint $m<r$ coming from the lattice, the other is the lattice end). They solved the large-$N$ equilibrium-measure problem (the minimizer of the gas energy) exactly: the free-energy density $f(x,y)$, with $x,y$ the scaled sizes of the cut-off, is expressed through a function $\varphi(x,y)$ given in closed form; the resolvents (the Cauchy transforms of the equilibrium measure, which encode the charge density) are obtained explicitly in each regime, i.e. for each pattern of voids (empty intervals) and saturated regions (charge density equal to the maximal value of one charge per site) adjacent to the two walls; and the band end-points obey explicit parametric equations. A by-product is a third-order phase transition (the free energy and its first two derivatives are continuous, the third jumps), of the Douglas–Kazakov type familiar from random-matrix and gauge-theory large-$N$ transitions, occurring when the geometry or weight parameter is tuned so that the arctic ellipse touches the cut boundary; it is the boundary analogue of the pulled-to-pushed third-order transitions of random matrices with a hard wall.

One issue follows directly from this analysis and is not resolved by it: the derivation leans everywhere on the Hankel/Meixner determinantal structure, which exists only at free fermion; the natural extension to generic (non-free-fermion) weights has as its starting point only multiple-integral representations of the generalized emptiness formation probability (GEFP), introduced by Colomo–Pronko–Sportiello as the probability of full ferroelectric order in a prescribed subdomain, whose large-$N$ asymptotics at generic weights is not available (for the square domain the arctic curve at generic weights is known, but not the free-energy rate function). Relatedly, the mechanism of the observed singularity is stated in the source only as an ascription: near the left hard wall the derivative of the potential diverges, $V'(x)\simeq\log|x-x_0|$, giving a smooth ('transparent') soft-edge/hard-edge matching with no discontinuity in the free energy, whereas near the right hard wall the slope stays finite and the meeting is non-transparent, producing the third-order singularity. In the asymptotic theory of discrete orthogonal polynomials, where analogous discrete Coulomb gases with hard walls have been analyzed extensively, soft-edge/hard-edge meetings are 'usually transparent'; the Colomo–Pronko example shows this is not a theorem, but no general criterion demarcating the two behaviors is known.

Problem Statement

The problem arises from the analysis of Colomo and Pronko (2015), with the objective of completing the exact thermodynamic theory of the domain-wall six-vertex model on domains with a macroscopic cut-off corner, beyond the free-fermion point.

Consider the six-vertex model with domain-wall boundary conditions on the L-shaped domain, i.e. the $N\times N$ square with an $s\times(s+q)$ rectangle removed from one corner, with arbitrary positive Boltzmann weights $w_1,\dots,w_6$ not restricted by the free-fermion condition $w_1w_2+w_3w_4=w_5w_6$. Determine the thermodynamic-limit free energy as a function of the scaled geometry parameters and the weights, in particular establishing, for each of the following features of the free-fermion solution, whether an analogous statement holds at generic weights and producing that analogue wherever it does: the closed-form free-energy density (the analogue of the function $\varphi(x,y)$), the explicit resolvents of the equilibrium measure, the regime structure of voids and saturated regions at the two hard walls, and the parametric band end-point equations. The question includes the singularity structure: at which loci in the parameter space the free energy is non-analytic, and of which order.

Answers may start from any exact representation of the L-shaped partition function or of the generalized emptiness formation probability valid at generic weights (such as the available multiple-integral representations), but must control the large-$N$ asymptotics rather than only formal finite-$N$ expressions.

References

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