摘要：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 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 or uniqueness fails. We provide some partial answers. We use (β,γ) to denote the (+,+) and (−,−) edge interactions and λ the external field, where βγ>1. If all fields satisfy λ λ^int′_c, where λ^int′_c=(γ/β)(⌊Δc⌋+2)/2, then approximating the partition function is #BIS-hard.
