Approximating Pairwise Correlations in the Ising Model
ACM Transactions on Computation Theory
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In the Ising model, we consider the problem of estimating the covariance of the spins at two specified vertices. In the ferromagnetic case, it is easy to obtain an additive approx- imation to this covariance by repeatedly sampling from the relevant Gibbs distribution. However, we desire a multiplicative approximation, and it is not clear how to achieve this by sampling, given that the covariance can be exponentially small. Our main contribution is a fully polynomial time randomised approximation scheme (FPRAS) for the covariance in the ferromagnetic case. We also show that that the restriction to the ferromagnetic case is essential — there is no FPRAS for multiplicatively estimating the covariance of an antiferromagnetic Ising model unless RP = #P. In fact, we show that even determining the sign of the covariance is #P-hard in the antiferromagnetic case.
AuthorsJerrum, M; Goldberg, LA
- Mathematics