Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs

In a seminal paper [12], Weitz gave a deterministic fully polynomial approximation scheme for counting exponentially weighted independent sets (equivalently, approximating the partition function of the hard-core model from statistical physics) on graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the infinite d-regular tree. More recently Sly [10] showed that this is optimal in the sense that if there is an FPRAS for the hard-core partition function on graphs of maximum degree d for activities larger than the critical activity on the infinite d-regular tree then NP = RP. In this paper, we extend Weitz's approach to derive a deterministic fully polynomial approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of maximum degree d, up to the corresponding critical point on the d-regular tree. The main ingredient of our result is a proof that for two-state anti-ferromagnetic spin systems on the d-regular tree, weak spatial mixing implies strong spatial mixing. This in turn uses a message-decay argument which extends a similar approach proposed recently for the hard-core model by Restrepo et al [9] to the case of the anti-ferromagnetic Ising model with arbitrary field. By a standard correspondence, these results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints.