On the special anisotropy slice ν=−k\nu=-k of the square-lattice Ising model, every two-point correlation C(M,N)C(M,N) with M≤NM\leq N satisfies an Okamoto sigma-form Painlevé VI equation whose parameters are explicit functions of (M,N)(M,N), and each physical correlation sits inside a one-parameter (λ\lambda-extension) family of solutions of its equation; the constant selecting the physical member is known only case-by-case. One of the source's open sub-questions — whether the multiplicative factors of C(M,N)C(M,N) for T<TcT<T_c, M+NM+N odd have the Painlevé property — was resolved affirmatively by Boukraa–Cosgrove–Maillard–McCoy in 2022, which also clarified the origin of the sum-of-sigma-function decompositions for the specific instances known so far; the general origin and extent of the phenomenon remain open. The surviving open core, stated here as one coherent problem, is: a general closed-form principle selecting the boundary/monodromy constant for arbitrary (M,N)(M,N) on the slice; a classification of the sum-of-sigma decompositions; the nonlinear ODEs governing the M≥NM\geq N regime at ν=−k\nu=-k, where some correlations are no longer homogeneous polynomials in the elliptic integrals K~\tilde K and E~\tilde E; and nonlinear equations with the Painlevé property for generic anisotropy ν\nu.

Background

The Ising model on a square lattice with horizontal and vertical nearest-neighbor couplings EhE_h and EvE_v is the paradigmatic exactly solved statistical system. Beyond the free energy, its central objects are the two-point spin correlations C(M,N)=⟨σ0,0σM,N⟩C(M,N)=\langle\sigma_{0,0}\sigma_{M,N}\rangle the two spins being separated by MM lattice steps vertically and NN horizontally. For M≤NM\leq N these correlations are known in closed form as homogeneous polynomials in three complete elliptic integrals K~(k)\tilde K(k), E~(k)\tilde E(k) and Π~(−νk,k)\tilde\Pi(-\nu k,k) of an elliptic modulus kk: the first two are the complete elliptic integrals of the first and second kind, the third-kind integral Π~\tilde\Pi carries the dependence on the anisotropy ratio ν=sinh⁡(2Eh/kBT)/sinh⁡(2Ev/kBT)\nu=\sinh(2E_h/k_BT)/\sinh(2E_v/k_BT), and a tilde denotes the standard normalization with a 2/π2/\pi prefactor (so K~(0)=1\tilde K(0)=1). The squared modulus t=k2t=k^2 is the standard temperature-like variable, with k=1k=1 at the critical temperature TcT_c (so k<1k<1 for T<TcT<T_c); for T>TcT>T_c the natural modulus is instead the dual modulus k>=1/kk_>=1/k on the dual side of the Kramers–Wannier high/low-temperature self-duality of the model, with t=k>2t=k_>^2 and the integrals evaluated at k>k_>.

A nonlinear ODE has the Painlevé property when its general solutions have no movable critical points, i.e. no branch points whose locations depend on the constants of integration; the six Painlevé equations are the canonical second-order ODEs with this property. The sixth (Painlevé VI, or PVI) carries four parameters and, in Okamoto's Hamiltonian formulation, is equivalent to a nonlinear second-order equation for a sigma variable σ=t(t−1) d(ln⁡τ)/dt+⋯\sigma=t(t-1)\,d(\ln\tau)/dt+\cdots built from the tau function τ\tau (whose logarithmic derivative generates the Hamiltonian of an isomonodromic deformation — a deformation of a linear differential system that preserves its monodromy); the four entries of this sigma-form equation, the Okamoto parameters, are linear combinations of the four PVI parameters. Wu, McCoy, Tracy and Barouch (1976) first obtained Painlevé-type equations for Ising correlations in the scaling region, and Jimbo and Miwa (1980) proved that the diagonal correlations C(N,N)C(N,N) satisfy a PVI equation, with the physical solution selected by monodromy data. Form-factor (expansions in particle excitations) representations of row and diagonal correlations (Lyberg and McCoy, 2007) and the conformal-block representation of the PVI tau function (Gamayun, Iorgov and Lisovyy, 2013; conformal blocks are correlation functions of two-dimensional conformal field theory, which in this case reproduce the Ising tau functions) each embed a physical correlation in a one-parameter family of solutions of the same nonlinear equation, called a λ\lambda-extension: with λ\lambda the parameter (in a form-factor normalization it multiplies successive form factors, C(λ)=∑nλ2nfnC(\lambda)=\sum_n\lambda^{2n}f_n), the physical correlation corresponds to one specific value of λ\lambda. The same PVI tau functions arise as Toeplitz determinants (determinants whose entries depend only on the difference of their indices) in random matrix theory (Forrester and Witte, 2004), which is the bridge through which Ising results connect to that field.

The source paper (Boukraa, Maillard and McCoy, 2020) studies the special anisotropy slice on which the anisotropy ratio equals minus the elliptic modulus of the elliptic representation: ν=−k\nu=-k for T<TcT<T_c and ν=−k>=−1/k\nu=-k_>=-1/k for T>TcT>T_c (referred to shorthand as ν=−k\nu=-k below). On this slice the third-kind integral degenerates, Π~(k2,k)=E~(k)/(1−k2)\tilde\Pi(k^2,k)=\tilde E(k)/(1-k^2), so the correlations become homogeneous polynomials in K~\tilde K and E~\tilde E alone; and by the interchange symmetry C(M,N;k,ν)=C(N,M;k,1/ν)C(M,N;k,\nu)=C(N,M;k,1/\nu) of the horizontal and vertical couplings, the case M≥NM\geq N at ν=−k\nu=-k is equivalent to the case M≤NM\leq N at ν=−1/k\nu=-1/k. Combining computer-algebra guessing (validated against exact elliptic-polynomial expressions) with structural arguments, the paper establishes that for all M≤NM\leq N the correlations on this slice satisfy four two-parameter families of Okamoto sigma-form PVI equations — one family for each combination of low/high temperature and M+NM+N even/odd — whose Okamoto parameters are explicit functions of (M,N)(M,N); for T>TcT>T_c with M+NM+N odd the correlation itself vanishes on the slice and the equation governs the rescaled correlation C~(M,N)=lim⁡ν→−k>(1+ν/k>)−1/2C(M,N)\tilde C(M,N)=\lim_{\nu\to-k_>}(1+\nu/k_>)^{-1/2}C(M,N). The paper also shows that C(M,N)C(M,N) is an N×NN\times N Toeplitz determinant on this slice; exhibits an identity in which a single PVI sigma function equals the sum of four PVI sigma functions sharing the same Okamoto parameters (for the low-temperature row correlations C(0,N)C(0,N) with NN odd); finds that for T<TcT<T_c, M+NM+N odd and M≠0M\neq0 the correlation always factors into two multiplicative factors; and determines, case by case from series expansions at t→0t\to0 (in the spirit of Kaneko's classification of PVI solutions that are meromorphic at a fixed singular point), which member of each one-parameter solution family is the physical correlation. The questions posed below are those left open by the source that subsequent work has not since settled; components already settled by later work are excluded.

Problem Statement

Complete the Painlevé/isomonodromic characterization of the two-point correlations of the anisotropic square-lattice Ising model beyond the results available on the special slice ν=−k\nu=-k with M≤NM\leq N. With the conventions of the background (t=k2t=k^2 below TcT_c and t=k>2t=k_>^2 above; the four temperature/parity sectors; the rescaled C~(M,N)\tilde C(M,N) for T>TcT>T_c with M+NM+N odd), resolve the following four linked questions, which together constitute one objective — a closed nonlinear-equation theory of C(M,N)C(M,N) in which the physical solution is selected from microscopic data:

(P1) Boundary-condition selection. Find a general principle — a closed formula, or an equivalent closed characterization — that gives, for arbitrary non-negative integers M≤NM\leq N on the slice ν=−k\nu=-k in all four sectors, the value of the one-parameter constant (the λ\lambda of the λ\lambda-extension, or equivalently the remaining boundary/monodromy constant of the Okamoto sigma-form Painlevé VI solution) that selects the physical Ising correlation. Determinations by matching series expansion case-by-case in (M,N)(M,N) do not by themselves answer the question; the principle must be uniform in (M,N)(M,N).

(P2) Sum-of-sigma decompositions. Determine the general conditions under which a Painlevé VI sigma function satisfying one set of Okamoto parameters and one boundary constant equals a sum of Painlevé VI sigma functions with the same (or explicitly related) Okamoto parameters and other boundary constants — in particular, which correlation types and parameter choices admit such additive decompositions — and explain the origin of the phenomenon. The known instances to be covered are the four-term decompositions for the low-temperature row correlations C(0,N)C(0,N) with NN odd and the additive decompositions of the sigma functions associated with the factorization of C(M,N)C(M,N) for T<TcT<T_c, M+NM+N odd (two terms for M≠0M\neq0, four for M=0M=0).

(P3) The M≥NM\geq N regime at ν=−k\nu=-k. Derive and classify the nonlinear ODEs with the Painlevé property satisfied by C(M,N)C(M,N) for M≥NM\geq N at ν=−k\nu=-k (equivalently M≤NM\leq N at ν=−1/k\nu=-1/k), where the elliptic integral Π~(−νk,k)\tilde\Pi(-\nu k,k) becomes singular and some correlations are no longer homogeneous polynomials in K~\tilde K and E~\tilde E; in particular, either exhibit closed formulae depending only on (M,N)(M,N) analogous to the four known families, or prove that no such closed (M,N)(M,N)-formulae exist for these cases.

(P4) Generic anisotropy. For generic ν\nu (off the special slices ν=−k\nu=-k and ν=−1/k\nu=-1/k), construct nonlinear ODEs with the Painlevé property satisfied by C(M,N)C(M,N) for arbitrary (M,N)(M,N); the answer is not restricted to single Painlevé VI equations, and larger isomonodromic systems are admissible (for example Garnier systems, the multivariable generalizations of Painlevé VI obtained as isomonodromic deformations of linear systems with additional regular singular points).

Throughout (P3) and (P4), a nonlinear ODE with the Painlevé property satisfied by C(M,N)C(M,N) is required to be of structural isomonodromic type — an isomonodromic deformation of a linear system whose monodromy data select the physical correlation, or an Okamoto sigma-form reduction of such a deformation; equations whose coefficients are constructed from C(M,N)C(M,N) itself (for instance a first-order equation fitted to the known correlation) do not count.

A complete answer resolves (P1)–(P4) at this level of generality; resolutions of individual components are progress but do not by themselves close the problem. The sub-question of whether the multiplicative factors of C(M,N)C(M,N) (T<TcT<T_c, M+NM+N odd, M≤NM\leq N) possess the Painlevé property, listed as open in the source paper, was resolved affirmatively by later work and is not part of the target.

References

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