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The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems

The diagonal spin-spin correlations $ \langle σ_{0,0}σ_{N,N} \rangle $ of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of the sixth Painlevé system, namely a Garnier system. This identification, which is accomplished using the theory of bi-orthogonal polynomials on the unit circle with regular semi-classical weights, has an additional consequence whereby the correlations are characterised by a simple system of coupled, nonlinear recurrence relations in the spin separation $ N \in \mathbb{Z}_{\geq 0} $. These later recurrence relations are an example of the discrete Garnier equations which, in turn, are extensions to the "discrete Painlevé V" system.