# Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

## Abstract

We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti-ferromagnetic Ising model (with non-zero external field) on the Bethe lattice. Following a method due to Weitz [15], we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of degree at most \(d\), throughout the uniqueness region of the Gibbs measure on the infinite \(d\)-regular tree. By a standard correspondence, our results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. Subsequent to a preliminary version of this paper, Sly and Sun [13] have shown that our results are optimal in the sense that, under standard complexity theoretic assumptions, there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree \(d\) for parameters outside the uniqueness region. Taken together, the results of [13] and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems.

## Keywords

Phase transitions Complexity theory Approximation algorithms Decay of correlations Two-spin systems on trees## Notes

### Acknowledgments

We thank Prasad Tetali for providing a manuscript of [11]. We also thank Colin McQuillan, Dror Weitz, Yitong Yin and two anonymous referees for several helpful comments. Alistair Sinclair was supported in part by United States National Science Foundation (NSF) Grant CCF-1016896. Piyush Srivastava was supported by the Berkeley Fellowship for Graduate Study and by NSF grant CCF-1016896, and performed part of this work while he was a research intern at Microsoft Research India. Marc Thurley was supported in part by a postdoctoral fellowship of the German Academic Exchange Service (DAAD) and by Marie Curie Intra-European Fellowship 271959, and performed part of this work while he was a postdoctoral scholar at the University of California, Berkeley, and at the Centre de Recerca Mathemàtica, Bellaterra.

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