Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems

For anti-ferromagnetic 2-spin systems, a beautiful connection has been established, namely that the following three notions align perfectly: the uniqueness of Gibbs measures in infinite regular trees, the decay of correlations (also known as spatial mixing), and the approximability of the partition function. The uniqueness condition implies spatial mixing, and an FPTAS for the partition function exists based on spatial mixing. On the other hand, non-uniqueness implies some long range correlation, based on which NP-hardness reductions are built. These connections for ferromagnetic 2-spin systems are much less clear, despite their similarities to anti-ferromagnetic systems. The celebrated Jerrum-Sinclair Markov chain [8] works even if spatial mixing fails. Also, for a fixed degree the uniqueness condition is non-monotone with respect to the external field, which seems to have no meaningful interpretation in terms of computational complexity. However, it is still intriguing whether there are some relationship underneath the apparent disparities among them. We provide some answers to this question. Let β, γ be the (0, 0) and (1, 1) edge interactions respectively (βγ > 1), and λ the external field for spin “0”. For graphs with degree bound ∆ ≤ ∆c + 1 where ∆c = √ βγ+1 √ βγ−1 , regardless of the field (even inconsistent fields are allowed), correlation decay always holds and FPTAS exists. If all fields satisfy λ < λc (assuming β ≤ γ), where λc = (γ/β) ∆c+1 2 , then a weaker version of spatial mixing holds in all trees. Moreover, if β ≤ 1, then λ < λc is sufficient to guarantee strong spatial mixing and FPTAS. This improves the best previous algorithm, a Markov chain based FPRAS for λ ≤ γ/β [13]. The bound λc is almost optimal and can be viewed as a variant of the uniqueness condition with the degree d relaxed to be a real number instead of an integer. When β ≤ 1, uniqueness holds in all infinite regular trees, if and only if λ ≤ λ c , where λ c = (γ/β) d∆ce+1 2 . If we allow fields λ > λ c , where λ c ′ = (γ/β) b∆cc+2 2 , then approximating the partition function is #BIS-hard. Interestingly, unless ∆c is an integer, neither λc nor λ c is the tight bound in each own respect. We provide examples where correlation decay continues to hold in a small interval beyond λc, and irregular trees in which spatial mixing fails for some λ < λ c . 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.1 Combinatorics