For general antiferromagnetic 2-spin systems, including the hardcore model and the antiferromagnetic Ising model, there is an $\mathsf{FPTAS}$ for the partition function on graphs of maximum degree $\Delta$ when the infinite regular tree lies in the uniqueness region by Li et al. (2013). Moreover, in the tree non-uniqueness region, Sly (2010) showed that there is no $\mathsf{FPRAS}$ to estimate the partition function unless $\mathsf{NP}=\mathsf{RP}$. The algorithmic results follow from the correlation decay approach due to Weitz (2006) or the polynomial interpolation approach developed by Barvinok (2016). However the running time is only polynomial for constant $\Delta$. For the hardcore model, recent work of Anari et al. (2020) establishes rapid mixing of the simple single-site Markov chain known as the Glauber dynamics in the tree uniqueness region. Our work simplifies their analysis of the Glauber dynamics by considering the total pairwise influence of a fixed vertex $v$ on other vertices, as opposed to the total influence on $v$, thereby extending their work to all 2-spin models and improving the mixing time. More importantly our proof ties together the three disparate algorithmic approaches: we show that contraction of the tree recursions with a suitable potential function, which is the primary technique for establishing efficiency of Weitz's correlation decay approach and Barvinok's polynomial interpolation approach, also establishes rapid mixing of the Glauber dynamics. We emphasize that this connection holds for all 2-spin models (both antiferromagnetic and ferromagnetic), and existing proofs for correlation decay or polynomial interpolation immediately imply rapid mixing of Glauber dynamics. Our proof utilizes that the graph partition function divides that of Weitz's self-avoiding walk trees, leading to new tools for analyzing influence of vertices.

翻译：对于普通反电磁2级系统,包括硬分子模型和反电磁Ising模型,在最大度的图形中,当无穷的普通树位于Li等人的独特性区域时,在最大度的图形中,有1美元=Delta$的分区功能。此外,在树非独特性区域,Sly (2010) 显示没有$\mathfsf{FTRAS}$来估计分区功能, 除非 $\mathsf{NP ⁇ mathsf{RP}$。 在最大度的图形中, 有1美元=mathsf{FFPFTAS}。 在最大度的常规树平面树平面的图形中, 有1美元=Delta$。 Sly (2010) 显示,在树特度区域的简单单点的马可变电离子电离子电离子中, 有1美元基电离子电流的电流法, 用于树氏度区域的Glabelfertal-deal macial-deal macideal-lational-lations, 我们的工作对Glabuilcideal-lational-lations 3 matical matical matial max max matis max max max max max max max, 。