Positivity, unimodality, real zeros, and the free-energy limit for the polynomials of the three-colour model with domain wall boundary conditions

The three-colour model with domain wall boundary conditions — proper three-colourings of a square chessboard whose boundary colours are forced to follow the cyclic order, weighted per colour — is exactly solvable: Rosengren (2011) expressed its partition function through one-variable polynomials $p_n(\zeta)$ of degree $n(n+1)/2$ generated by an explicit recursion. The solution left the structure of $p_n$ conjectural: its coefficients appear positive and unimodal, exactly $\lfloor (n+1)/2\rfloor$ of its zeros appear real and confined to prescribed intervals, and a formal derivation predicts an explicit free energy per volume. Later work proved that $p_n$ is a Painlevé VI tau-polynomial satisfying the Bazhanov–Mangazeev recursions, but the positivity, unimodality, zero-location, and free-energy conjectures remain unproven. The problem is to establish these conjectures — or refute them — rigorously.

Background

The three-colour model (Baxter's three-colouring model) is the statistical mechanics of proper colourings of the faces of the square lattice with three colours $0,1,2$, adjacent faces receiving distinct colours and colour $i$ carrying weight $t_i$; it is the zero-temperature three-state Potts antiferromagnet, and via a classical bijection it is a close relative of the six-vertex (ice-type) model, whose configurations are arrow assignments on lattice edges with two arrows entering and two leaving each vertex. Baxter solved the periodic three-colour model exactly and computed its free energy. For the six-vertex model the most studied finite geometry uses domain wall boundary conditions (DWBC): all boundary arrows point into the square along two adjacent sides and out of it along the other two. Six-vertex DWBC configurations are in bijection with alternating sign matrices (square matrices of $0,\pm1$ whose non-zero entries alternate in sign along each row and column, with all row and column sums equal to $1$), and Kuperberg re-proved the enumeration of alternating sign matrices from the Izergin–Korepin determinant representation of the six-vertex DWBC partition function.

Rosengren (2011) extended this strategy to the three-colour model with DWBC. A state is a three-coloured chessboard: an $(n+1)\times(n+1)$ square properly three-coloured, with the north-west and south-east corner squares coloured $0$ and the boundary colours increasing in the cyclic order $0<1<2<0$ as one proceeds away from these corners; for $n=3$ there are seven such chessboards, and in general they are in bijection with the $n\times n$ alternating sign matrices. The model had been studied earlier by Razumov and Stroganov, who treated a nontrivial trigonometric limit by functional equations and found a determinant representation for the partial partition functions. Rosengren's technical route passes through the eight-vertex solid-on-solid (8VSOS) model — an elliptic, two-parameter extension of the six-vertex model, introduced by Baxter as a tool for solving the eight-vertex model; unlike the eight-vertex model itself, the 8VSOS model is an ice model, so for DWBC its states are again ice graphs, now carrying Boltzmann weights that are theta functions (doubly periodic entire functions, here the elliptic parameter of the model) of the spectral variables — and in the specialization of the parameters to cubic roots of unity the 8VSOS model degenerates to the three-colour model. The outcome is that the partition function $Z_n^{3C}(t_0,t_1,t_2)$ is expressed through explicit one-variable polynomials $p_n(\zeta)$ of degree $n(n+1)/2$ (with $\zeta$ an elliptic reparametrization of the colour-weight invariant $T=(t_0t_1+t_0t_2+t_1t_2)^3/(t_0t_1t_2)^2$), where $p_{-1}=p_0=1$, $p_1=1+3\zeta$, $p_2=1+7\zeta+15\zeta^2+5\zeta^3$, and $p_{n+1}$ is generated from $p_{n-1},p_n$ by the polynomial equation of Theorem 3.7 of that paper (stated in the problem below). This gives a fast computation of $Z_n^{3C}$, but the finer structure of $p_n$ was only observed numerically: all coefficients seem positive (verified for $n\le16$), the coefficient sequence seems unimodal (a finite sequence is unimodal when it increases up to some peak index and then decreases), and for the computed instances the zero set shows a striking pattern with exactly $\lfloor (n+1)/2\rfloor$ real zeros.

Subsequent work resolved the integrable-systems part of the original problem list. Bazhanov and Mangazeev, after seeing a preprint, conjectured much faster (Toda-type, i.e. three-term) recursions for $p_n$ and the related two-variable polynomials $P_n$, published in their own paper on eigenvectors of the eight-vertex model; Rosengren's later series on supersymmetric eight-vertex polynomials then constructed a master family of symmetric polynomials specializing both to the three-colour polynomials and to Bazhanov–Mangazeev's, proved the conjectured recursions and the conjectured differential equation, and identified the whole family with tau functions of the Painlevé VI equation (a classical integrable ordinary differential equation; its tau functions are, roughly, functions whose logarithmic derivatives encode its Hamiltonians) obtained from Picard's algebraic solution (one of the few explicitly known algebraic solutions of the equation) by Bäcklund transformations (symmetries mapping solutions of the equation to other solutions). In particular, the three-colour $p_n$ is now rigorously a Painlevé VI tau-polynomial. None of this, however, touches the coefficient, zero, or thermodynamic conjectures.

The remaining analytic target is the free energy per volume, $f=\lim_{n\to\infty}n^{-2}\log Z_n^{3C}(t_0,t_1,t_2)$ for positive weights. Rosengren gave a formal derivation of an explicit expression for $f$: assuming that $\lim n^{-2}\log p_n(\zeta)$ exists as a formal power series together with certain factorization and stability properties, the recursion of Theorem 3.7 forces a closed formula, which is consistent with the known alternating-sign-matrix asymptotics at $t_0=t_1=t_2=1$ (where $Z_n^{3C}(1,1,1)=A_n$, the number of $n\times n$ alternating sign matrices, with $A_n\sim C\,n^{-5/36}(3\sqrt3/4)^{n^2}$), with the symmetry $\zeta\mapsto\zeta^{-1}$, and with numerical experiment. It is, however, manifestly different from Baxter's free energy for the periodic three-colour model (both have the form $f=\frac13\log(t_0t_1t_2)+\log W$, and the two are related by $W_{\mathrm{DWBC}}(\zeta)=2/\sqrt{W_{\mathrm{per}}(\zeta)W_{\mathrm{per}}(1/\zeta)}$), so the domain-wall problem is genuinely new. Turning the formal argument into a rigorous limit is the analogue, for the three-colour model, of the rigorous six-vertex DWBC analyses of Bleher–Fokin and successors, which started from a determinant representation; for the three-colour model, determinantal representations of the 8VSOS DWBC partition function (Galléas) and the proved Painlevé VI tau-function identification are available starting points, but the corresponding asymptotic analysis has never been carried out.

Problem Statement

Let $p_n(\zeta)$, $n\ge-1$, be the polynomials defined by $p_{-1}=p_0=1$ together with the recursion of Rosengren (Theorem 3.7 of the source). Write $\tilde p(x)=x^{\deg p}p(1/x)$ for the coefficient-reversed polynomial and $\chi(\cdot)$ for the indicator function. Given $p_{n-1}$ and $p_n$, the polynomial $p_{n+1}$ is the solution $X$ (with some polynomial $Y$ of degree at most $\deg A$) of the linear equation

$$A(\zeta)X(\zeta)-B(\zeta)Y(\zeta)=C(\zeta),$$

where

$$A=(\zeta+1)^2p_{n-1}(\zeta),\qquad B=\zeta^{n+1}\tilde p_n(\zeta),$$ $$C=(1+2\zeta)^{1+\chi(n\text{ even})}\left(1+\frac{\zeta}{2}\right)^{1+\chi(n\text{ odd})}p_n(\zeta)^2,$$

the solution being normalized by the leading term prescribed in Proposition 8.9 of the source: $p_m$ has degree $m(m+1)/2$ and leading coefficient $2^{-\lfloor (m+2)/2\rfloor}\binom{2m+2}{m+1}$ (so the leading term of $p_{n+1}$ is $2^{-\lfloor (n+3)/2\rfloor}\binom{2n+4}{n+2}\zeta^{(n+1)(n+2)/2}$). The first instances are

$$p_1=1+3\zeta,\qquad p_2=1+7\zeta+15\zeta^2+5\zeta^3,$$ $$p_3=\frac{1}{2}\left(35\zeta^6+231\zeta^5+504\zeta^4+398\zeta^3+147\zeta^2+27\zeta+2\right).$$

The partition function $Z_n^{3C}(t_0,t_1,t_2)$ of the three-colour model with domain wall boundary conditions is an explicit combination of $p_{n-1}$ and $\tilde p_{n-1}$ with theta-function coefficients (Corollary 9.4 of the source). The common objective is a rigorous structure theory and thermodynamics for this polynomial family. Establish, or rigorously refute, the following conjectures, each stated in the source and verified there only numerically (Conjectures 3.8–3.9 up to $n=16$):

1. (Positivity, Conjecture 3.8) All coefficients of $p_n(\zeta)$ are positive.

2. (Unimodality, Conjecture 3.9) Writing $p_n(\zeta)=\sum_{k=0}^{n(n+1)/2}a_k\zeta^k$, the coefficient sequence $(a_k)$ is strictly unimodal, i.e. $a_0<a_1<\cdots<a_N>a_{N+1}>\cdots>a_{n(n+1)/2}$ for some $N$ (for $n\le16$ the peak was observed at $N=n(n+2)/4$ for even $n$, and at $N=(n+1)^2/4$ for odd $n\le7$, $N=(n-1)(n+3)/4$ for odd $9\le n\le15$).

3. (Real zeros, Conjectures 3.10–3.11) $p_n$ has exactly $\lfloor (n+1)/2\rfloor$ real zeros; in particular $p_n$ does not vanish on the interval $-2<\zeta<-1/2$ (the latter follows from the former via Proposition 3.12 of the source, which also shows that Conjecture 3.10 forces all real zeros to be simple and to lie in $-1/2<\zeta<0$ for odd $n$ and in $\zeta<-2$ for even $n$, with the interlacing pattern stated there).

4. (Free energy, Conjecture 3.14) For positive parameters $t_0,t_1,t_2$, the free energy per volume

$$f(t_0,t_1,t_2)=\lim_{n\to\infty}\frac{\log Z_n^{3C}(t_0,t_1,t_2)}{n^2}$$

exists and, for $\zeta$ any positive solution of

$$T=\frac{(t_0t_1+t_0t_2+t_1t_2)^3}{(t_0t_1t_2)^2}=\frac{2(\zeta^2+4\zeta+1)^3}{\zeta(\zeta+1)^4},$$

equals

$$f=\frac{1}{3}\log(t_0t_1t_2)+\log\left(\frac{(\zeta+2)^{3/4}(2\zeta+1)^{3/4}}{2^{2/3}\zeta^{1/12}(\zeta+1)^{4/3}}\right).$$

The source obtains this formula through a formal derivation whose assumptions are that (10.1) the limit $\lim_{n\to\infty}n^{-2}\log p_n(\zeta)=\log g(\zeta)$ exists in the algebra of formal power series in $\zeta$, (10.2) $p_n(\zeta)$ factors as $g(\zeta)^{n^2}\phi_n(\zeta)$, and (10.3) $\lim_{n\to\infty}\phi_n(\zeta)^2/(\phi_{n+2}(\zeta)\phi_{n-2}(\zeta))=1$ in the sense of formal power series; these hypotheses belong to that derivation, which the source explicitly describes as non-rigorous, and are not part of Conjecture 3.14 itself. The right-hand side is invariant under $\zeta\mapsto\zeta^{-1}$, so the two positive solutions of the parametrizing equation give the same value. The four items above are independent targets: each may be established or refuted on its own, and a resolution of one settles that conjecture while leaving the others open. Where one item implies another the stronger statement is the one to establish; in particular, strict unimodality (item 2) forces every coefficient to be positive, so establishing item 2 also establishes item 1.

Two arithmetic remarks recorded in the same source lie outside this list of conjectures: the belief, stated after Theorem 3.7, that the polynomials $A$ and $B$ of the recursion are always relatively prime — indeed, that $p_n$ is irreducible over $\mathbb{Q}$ for $n\ge2$ — and the observation that the divisibility $2^{\mu_k}a_k\in\mathbb{Z}$ of Corollary 9.8 of the source, where $\mu_k=\min(\lfloor n(n+2)/12\rfloor,\,k,\,n(n+2)/2-k)$ for even $n$ and $\mu_k=\min(\lfloor (n+1)^2/12\rfloor,\,k,\,(n+1)^2/2-k)$ for odd $n$, seems far from sharp.

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