Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems
ACM Transactions on Computation Theory
For anti-ferromagnetic 2-spin systems, a beautiful connection has been established, namely that the following three notions align perfectly: the uniqueness 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
... on, 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 [JS93] works even if spatial mixing fails, and for a fixed degree the uniqueness condition is non-monotone with respect to the external field, which seems to have no meaningful interpretation in its computational complexity. However, it is still intriguing whether there are some relationship underneath the apparent disparities among them. We provide some partial answers to this question. We use (β, γ) to denote the (+, +) and (−, −) edge interactions and λ the external field, where βγ > 1. For graphs with degree bound βγ−1 , regardless of the field (even inconsistent fields are allowed), correlation decay always holds and FPTAS exists. If all fields satisfy λ < λ c (assuming β ≤ γ), where λ c = (γ/β) ∆c +1 2 , then approximating the partition function is #BIS-hard. Interestingly, unless ∆ c is an integer, neither λ c nor λ int c is the tight bound in each own respect. We provide examples where correlation decay continues to hold after λ c , and irregular trees in which spatial mixing fails for some λ < λ int c .