This paper formalizes connections between stability of polynomials and convergence rates of Markov Chain Monte Carlo (MCMC) algorithms. We prove that if a (multivariate) partition function is nonzero in a region around a real point $\lambda$ then spectral independence holds at $\lambda$. As a consequence, for Holant-type problems (e.g., spin systems) on bounded-degree graphs, we obtain optimal $O(n\log n)$ mixing time bounds for the single-site update Markov chain known as the Glauber dynamics. Our result significantly improves the running time guarantees obtained via the polynomial interpolation method of Barvinok (2017), refined by Patel and Regts (2017). There are a variety of applications of our results. In this paper, we focus on Holant-type (i.e., edge-coloring) problems, including weighted edge covers and weighted even subgraphs. For the weighted edge cover problem (and several natural generalizations) we obtain an $O(n\log{n})$ sampling algorithm on bounded-degree graphs. The even subgraphs problem corresponds to the high-temperature expansion of the ferromagnetic Ising model. We obtain an $O(n\log{n})$ sampling algorithm for the ferromagnetic Ising model with a nonzero external field on bounded-degree graphs, which improves upon the classical result of Jerrum and Sinclair (1993) for this class of graphs. We obtain further applications to antiferromagnetic two-spin models on line graphs, weighted graph homomorphisms, tensor networks, and more.

翻译：本文形式化地建立了多项式稳定性与马尔可夫链蒙特卡洛（MCMC）算法收敛速率之间的联系。我们证明：若（多元）配分函数在实数点$\lambda$的某个邻域内非零，则在$\lambda$处谱独立性成立。作为推论，对于有界度图上的Holant型问题（例如自旋系统），我们为单点更新的马尔可夫链（即Glauber动力学）获得了最优的$O(n\log n)$混合时间界。我们的结果显著改进了通过Barvinok（2017年）提出并经Patel与Regts（2017年）改进的多项式插值方法所获得的运行时间保证。我们的结果具有多种应用。本文聚焦于Holant型（即边着色）问题，包括加权边覆盖与加权偶子图问题。对于加权边覆盖问题（及其若干自然推广），我们在有界度图上得到了$O(n\log{n})$采样算法。偶子图问题对应于铁磁Ising模型的高温展开。对于有界度图上具有非零外场的铁磁Ising模型，我们获得了$O(n\log{n})$采样算法，这改进了Jerrum与Sinclair（1993年）对该类图的经典结果。我们在线图的反铁磁二自旋模型、加权图同态、张量网络等领域获得了进一步的应用。