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In 1980 Jimbo and Miwa evaluated the diagonal two-point correlation functionof the square lattice Ising model as a $\tau$-function of the sixth Painlev\'esystem by constructing an associated isomonodromic system within their theoryof holonomic quantum fields. More recently an alternative isomonodromy theorywas constructed based on bi-orthogonal polynomials on the unit circle withregular semi-classical weights, for which the diagonal Ising correlations ariseas the leading coefficient of the polynomials specialised appropriately. Herewe demonstrate that the next-to-diagonal correlations of the anisotropic Isingmodel are evaluated as one of the elements of this isomonodromic system oressentially as the Cauchy-Hilbert transform of one of the bi-orthogonalpolynomials.

Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model