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

Abstract

The diagonal spin–spin correlations 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 . The later recurrence relations are an example of discrete Garnier equations which, in turn, are extensions to a ‘discrete Painlevé V’ system.