通过稳定性实现谱独立及其在 Holant 型问题中的应用
Spectral Independence via Stability and Applications to Holant-Type Problems
AI总结：
本文证明多项式稳定性可推出谱独立，从而为有界度图上的 Holant 型问题（如带权边覆盖、铁磁 Ising 模型等）获得最优 O(n log n) 混合时间的 Glauber 动力学采样算法，显著改进了此前基于多项式插值方法的运行时间保证。
摘要：
本文形式化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，如果一个（多元）配分函数在实点 $λ$ 周围的某个区域内非零，那么在 $λ$ 处谱独立成立。作为推论，对于有界度图上的 Holant 型问题（例如自旋系统），我们为称为 Glauber 动力学的单点更新马尔可夫链获得了最优的 $O(n\log n)$ 混合时间界。我们的结果显著改善了通过 Barvinok（2017）提出的多项式插值方法（并由 Patel 和 Regts（2017）加以改进）所获得的运行时间保证。我们的结果有多种应用。在本文中，我们重点关注 Holant 型（即边着色）问题，包括带权边覆盖和带权偶子图。对于带权边覆盖问题（以及若干自然推广），我们在有界度图上获得了 $O(n\log{n})$ 的采样算法。偶子图问题对应于铁磁 Ising 模型的高温展开。我们在有界度图上为具有非零外场的铁磁 Ising 模型获得了 $O(n\log{n})$ 的采样算法，这改进了 Jerrum 和 Sinclair（1993）针对该类图的经典结果。我们还进一步获得了在线图上的反铁磁双自旋模型、带权图同态、张量网络等方面的应用。
英文摘要：
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 $λ$ then spectral independence holds at $λ$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
