On the special anisotropy slice ν=−k\nu=-k of the two-dimensional Ising model (and its companion ν=−1/k\nu=-1/k), Boukraa, Maillard and McCoy (2020) established that the two-point correlation C(M,N)C(M,N) with M≤NM\leq N at ν=−k\nu=-k (equivalently M≥NM\geq N at ν=−1/k\nu=-1/k) satisfies one of a small set of nonlinear ODEs in (M,N)(M,N), each reducible to an Okamoto sigma form of the sixth Painlevé equation, and that C(M,N)C(M,N) is an N×NN\times N Toeplitz determinant. Completing this Painlevé VI characterization raises four questions they left open: (1) no closed form or determination principle is known for the boundary-condition constants (the normalization constants KM,NK_{M,N}, KM,N(1)K^{(1)}_{M,N}, KM,N(2)K^{(2)}_{M,N}) selecting the physical correlation out of its one-parameter lambda-extension family of Painlevé VI solutions; (2) the identity expressing a row-correlation Painlevé VI sigma function as a sum of four such sigma functions lacks a general mechanism and classification; (3) the regime M≥NM\geq N, where the elliptic integral Π~(−νk,k)\tilde\Pi(-\nu k,k) is singular and the homogeneous-polynomial structure in K~\tilde K, E~\tilde E fails, has no derived ODE classification, and it is unknown whether closed (M,N)(M,N)-dependent formulae exist; (4) nonlinear ODEs with the Painlevé property at generic anisotropy ν\nu remain to be constructed. A 2022 follow-up by the same group resolved a fifth question of the source, showing that the individual factorization factors do possess the Painlevé property, reducing to Okamoto sigma-form Painlevé VI; the remaining four components remain open and constitute the open core formulated here.

Background

The two-dimensional Ising model with horizontal and vertical couplings EhE_h, EvE_v is the paradigm of exactly solved statistical mechanics. Its two-point spin correlation C(M,N)=⟨σ0,0σM,N⟩C(M,N)=\langle\sigma_{0,0}\sigma_{M,N}\rangle has carried much of the model's exact structure since Onsager's solution: Montroll, Potts and Ward expressed the correlations as Toeplitz determinants (determinants det⁡(aj−k)\det(a_{j-k}) whose entries are constant along diagonals, here built from a Fourier-Bessel series), and the search for closed differential-equation characterizations of C(M,N)C(M,N) as a function of temperature has been a driving question of the field ever since. The relevant nonlinear structure is of Painlevé type. The Painlevé equations are second-order nonlinear ordinary differential equations whose general solutions have the Painlevé property — no movable critical points, i.e. no branch points or essential singularities whose locations depend on integration constants — which makes their solutions single-valued transcendents generalizing elliptic functions. The sixth and most general, Painlevé VI (PVI), has elliptic-function coefficients; Okamoto's sigma form writes it for h(t)=t(t−1) dln⁡τ/dth(t)=t(t-1)\,d\ln\tau/dt with four parameters n1,n2,n3,n4n_1,n_2,n_3,n_4 (the Okamoto parameters). Jimbo and Miwa proved that the diagonal correlation C(N,N)C(N,N) of the isotropic model satisfies such an equation in the elliptic-modulus variable t=k2t=k^2 below the critical temperature TcT_c (with t=k−2t=k^{-2} above), where kk is the elliptic modulus of the Ising fermion representation. The physical correlation is the member λ=1\lambda=1 of a one-parameter family C(N,N;λ)C(N,N;\lambda) of solutions of the same equation — the lambda-extensions, generated by the form-factor expansion (a multi-integral excitation representation) of the correlation — and McCoy proved the connection formulas for these lambda-generalized diagonal correlations, thereby settling the boundary-selection (connection) problem in the diagonal case.

For anisotropic couplings, with ν=sinh⁡(2Eh/kBT)/sinh⁡(2Ev/kBT)\nu=\sinh(2E_h/k_BT)/\sinh(2E_v/k_BT), McCoy and Maillard showed that C(M,N)C(M,N) with M≤NM\leq N is a homogeneous polynomial in the normalized complete elliptic integrals K~(k)\tilde K(k) and E~(k)\tilde E(k) (first and second kind) and the third-kind integral Π~(−kν,k)\tilde\Pi(-k\nu,k), so that generically C(M,N)C(M,N) depends on two variables through Π~\tilde\Pi. On special anisotropy slices this third-kind integral degenerates. On the slice ν=−k\nu=-k (where k→0k\to0 as T→0T\to0, so the slice behaves as a low-temperature regime with complex, unphysical couplings) and the companion slice ν=−1/k\nu=-1/k (high-temperature analogue), Boukraa, Maillard and McCoy established, for all M≤NM\leq N at ν=−k\nu=-k (equivalently M≥NM\geq N at ν=−1/k\nu=-1/k), closed nonlinear ODEs — four two-parameter nonlinear ODEs distinguishing T≶TcT\lessgtr T_c and the parity of M+NM+N (displayed as three families, the low-temperature one containing an explicit (−1)M+N(-1)^{M+N} parity term); for example, below TcT_c with t=k2t=k^2 and σ=t(t−1) dln⁡C(M,N)/dt−t/4\sigma=t(t-1)\,d\ln C(M,N)/dt-t/4, the equation [t(t−1)σ′′]2+4{σ′(tσ′−σ)((t−1)σ′−σ)}−M2(tσ′−σ)2−N2σ′2+[M2+N2−12 (1+(−1)M+N)] σ′(tσ′−σ)=0[t(t-1)\sigma'']^{2}+4\{\sigma'(t\sigma'-\sigma)((t-1)\sigma'-\sigma)\}-M^{2}(t\sigma'-\sigma)^{2}-N^{2}\sigma'^{2}+[M^{2}+N^{2}-\frac{1}{2}\,(1+(-1)^{M+N})]\,\sigma'(t\sigma'-\sigma)=0 holds; every such family is reducible to an Okamoto sigma-form PVI equation with Okamoto parameters rational in (M,N)(M,N), and the two families that contain the diagonal reduce there, at M=NM=N, to the Jimbo–Miwa equation; they also showed that on these slices C(M,N)C(M,N) is an N×NN\times N Toeplitz determinant, placing it in the Forrester–Witte and Gamayun–Iorgov–Lisovyy family of PVI Toeplitz determinants. Two structural phenomena emerged from this work. First, for T<TcT<T_c with M+NM+N odd, C(M,N)C(M,N) factors into two multiplicative factors when M≠0M\neq0 and into four when M=0M=0. Second, for the row correlations C(0,N)C(0,N) with NN odd, a PVI sigma function with one set of Okamoto parameters and a specific boundary-condition constant equals the sum of four PVI sigma functions with a common, different set of Okamoto parameters and four specific boundary-condition constants — an additive decomposition reminiscent of addition formulae for elliptic functions (with a Painlevé V precedent found by Tracy and Widom). For generic anisotropy, where C(M,N)C(M,N) depends on two variables, the natural isomonodromic objects are expected to be Garnier systems — multivariable generalizations of PVI describing isomonodromic deformations of linear systems with several regular singularities: Witte proved that the diagonal correlations of the anisotropic triangular-lattice Ising model are tau-functions of a Garnier system; related isomonodromic-deformation results cover the next-to-diagonal square-lattice correlations. Landen transformations — the degree-two isogenies of elliptic curves, and more generally modular correspondences — were subsequently identified as the changes of variable that reduce the factor equations of the T<TcT<T_c, M+NM+N odd factorization to PVI form. The open questions formulated below ask what remains to complete this Painlevé VI characterization of C(M,N)C(M,N).

Problem Statement

The common objective is to complete the Painlevé VI characterization of the anisotropic square-lattice Ising two-point correlation C(M,N)C(M,N) — the nonlinear ODEs with the Painlevé property in every regime, the boundary data that select the physical solution, and the additive structure of the solution family — on and beyond the special anisotropy slices ν=−k\nu=-k and ν=−1/k\nu=-1/k. Boukraa, Maillard and McCoy (2020) established the closed (M,N)(M,N)-dependent Okamoto sigma-form Painlevé VI equations for M≤NM\leq N at ν=−k\nu=-k (equivalently M≥NM\geq N at ν=−1/k\nu=-1/k); the following four components are stated as open questions in that paper and remain open.

(1) Boundary data. Every C(M,N)C(M,N) on the slice is the λ=1\lambda=1 member of a one-parameter family of Painlevé VI solutions of its equation, but the boundary-condition data — the normalization constants of the t→0t\to0 expansions (KM,NK_{M,N} for T<TcT<T_c, where the arbitrary constant enters at order tN+1t^{N+1}; KM,N(1)K^{(1)}_{M,N} and KM,N(2)K^{(2)}_{M,N} for T>TcT>T_c, where it enters at order t3N/2+2t^{3N/2+2}) — can currently only be extracted order-by-order from the exact elliptic-integral or Toeplitz representations. Find a closed-form expression, or determination principle, for these constants for arbitrary (M,N)(M,N) on the slices.

(2) Additive sigma decompositions. For the low-temperature row correlations C(0,N)C(0,N) with NN odd, a Painlevé VI sigma function with one set of Okamoto parameters and a specific boundary-condition constant equals the sum of four Painlevé VI sigma functions with a common, different set of Okamoto parameters and four specific boundary-condition constants. Establish the general conditions and origin of such decompositions: for which Okamoto parameter sets, boundary constants, and correlation types does a Painlevé VI sigma function decompose into a finite sum of non-constant Painlevé VI sigma functions?

(3) The M≥NM\geq N regime. At ν=−k\nu=-k with M≥NM\geq N (equivalently M≤NM\leq N at ν=−1/k\nu=-1/k, by the symmetry C(M,N;ν=−k)=C(N,M;ν=−1/k)C(M,N;\nu=-k)=C(N,M;\nu=-1/k)), the elliptic integral Π~(−νk,k)\tilde\Pi(-\nu k,k) becomes singular and some correlations cease to be homogeneous polynomials in K~\tilde K and E~\tilde E, so the corresponding nonlinear ODEs are more involved. Two questions remain here: first, derive and classify these ODEs; second, determine whether they admit closed formulae depending only on (M,N)(M,N), analogous to the families (80), (82), (88) of the source.

(4) Generic anisotropy. Construct nonlinear ODEs with the Painlevé property for C(M,N)C(M,N) at generic anisotropy ν\nu — not restricted to the selected conditions ν=−k\nu=-k or ν=−1/k\nu=-1/k.

Components (1)-(4) share the same objects — the same correlation functions, the same (M,N)(M,N)-dependent Painlevé VI families, and the same one-parameter solution families — and jointly exhaust the outstanding structure of the 2020 characterization: which equations hold in which regimes, which member of each solution family is the physical correlation, and how the family decomposes. They are posed as four separate questions, and the characterization is complete only when all four have been resolved; resolving a strict subset is progress on that component, not completion of the open problem. (A fifth open question of the source — whether the two multiplicative factors of C(M,N)C(M,N) for T<TcT<T_c, M+NM+N odd, M≠0M\neq0 possess the Painlevé property — was resolved affirmatively in 2022 and is therefore not part of the open problem.)

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