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In the Ising model, we consider the problem of estimating the covariance ofthe spins at two specified vertices. In the ferromagnetic case, it is easy toobtain an additive approximation to this covariance by repeatedly sampling fromthe relevant Gibbs distribution. However, we desire a multiplicativeapproximation, and it is not clear how to achieve this by sampling, given thatthe covariance can be exponentially small. Our main contribution is a fullypolynomial time randomised approximation scheme (FPRAS) for the covariance. Wealso show that that the restriction to the ferromagnetic case is essential ---there is no FPRAS for multiplicatively estimating the covariance of anantiferromagnetic Ising model unless RP = #P. In fact, we show that evendetermining the sign of the covariance is #P-hard in the antiferromagneticcase.

Approximating Pairwise Correlations in the Ising Model