Correlation Decay up to Uniqueness in Spin Systems

We give a complete characterization of the two-state anti-ferromagnetic spin systems whichare of strong spatial mixing on general graphs. We show that a two-state anti-ferromagneticspin system is of strong spatial mixing on all graphs of maximum degree at most ∆ if and onlyif the system has a unique Gibbs measure on infinite regular trees of degree up to ∆, where∆ can be either bounded or unbounded. As a consequence, there exists an FPTAS for thepartition function of a two-state anti-ferromagnetic spin system on graphs of maximum degreeat most ∆ when the uniqueness condition is satisfied on infinite regular trees of degree up to∆. In particular, an FPTAS exists for arbitrary graphs if the uniqueness is satisfied on allinfinite regular trees. This covers as special cases all previous algorithmic results for two-stateanti-ferromagnetic systems on general-structure graphs.Combining with the FPRAS for two-state ferromagnetic spin systems of Jerrum-Sinclair andGoldberg-Jerrum-Paterson, and the very recent hardness results of Sly-Sun and independently ofGalanis-S̆tefankovic̆-Vigoda, this gives a complete classification, except at the phase transitionboundary, of the approximability of all two-state spin systems, on either degree-bounded familiesof graphs or family of all graphs. This work was done when this author visited Microsoft Research Asia.Supported by the National Science Foundation of China under Grant No. 61003023 and No. 61021062.