Isomonodromic Deformation Theory and the Next-to-diagonal Correlations of the Anisotropic Square Lattice Ising Model

In 1980 Jimbo and Miwa evaluated the diagonal two-point correlation function of the square lattice Ising model as a τ -function of the sixth Painlevé system by constructing an associated isomonodromic system within their theory of holonomic quantum fields. More recently an alternative isomonodromy theory was constructed based on bi-orthogonal polynomials on the unit circle with regular semi-classical weights, for which the diagonal Ising correlations arise as the leading coefficient of the polynomials specialised appropriately. Here we demonstrate that the next-to-diagonal correlations of the anisotropic Ising model are evaluated as one of the elements of this isomonodromic system or essentially as the Cauchy-Hilbert transform of one of the bi-orthogonal polynomials. For the square lattice Ising model on the infinite lattice an unpublished result of Onsager (see [13]) gives that the diagonal spin-spin correlation 〈σ0,0σN,N〉 has the Toeplitz determinant form (1) 〈σ0,0σN,N〉 = det(ai−j(k))1≤i,j≤N , where the elements are given by (2) an = ∫ π −π dθ 2π k cosnθ − cos(n−1)θ √ k2 + 1− 2k cos θ . A significant development occurred when Jimbo and Miwa [10, 11] identified (1) as the τ -function of a PVI system. This identification has the consequence of allowing (1) to be characterised in terms of a solution of the σ-form of the Painlevé VI equation, a second order second degree ordinary differential equation with respect to t := k with parameter N , or as the solution of coupled recurrence relations in N with parameter t, which were subsequently shown to be equivalent to the discrete Painlevé V equation. This was derived from the monodromy preserving deformation of a certain linear system as a particular example of their general theory of holonomic quantum fields [14, 17, 12], however the theoretical machinery employed there was never put to use on related problems arising from the Ising model. See the forthcoming monograph [16] on recent progress utilising this viewpoint. In a recent work [6] Forrester and the present author identified (1) as a τ -function in the Okamoto theory of PVI [15] and subsequently developed an alternative isomonodromic theory [8] founded on bi-orthogonal systems on the unit circle with regular semi-classical weights. We remark that a result of Borodin [4] can also be used for the same purpose. 2000 Mathematics Subject Classification. 82B20,34M55,33C45. 1