We show fully polynomial time randomized approximation schemes (FPRAS) for counting matchings of a given size, or more generally sampling/counting monomer-dimer systems in planar, not-necessarily-bipartite, graphs. While perfect matchings on planar graphs can be counted exactly in polynomial time, counting non-perfect matchings was shown by [Jer87] to be #P-hard, who also raised the question of whether efficient approximate counting is possible. We answer this affirmatively by showing that the multi-site Glauber dynamics on the set of monomers in a monomer-dimer system always mixes rapidly, and that this dynamics can be implemented efficiently on downward-closed families of graphs where counting perfect matchings is tractable. As further applications of our results, we show how to sample efficiently using multi-site Glauber dynamics from partition-constrained strongly Rayleigh distributions, and nonsymmetric determinantal point processes. In order to analyze mixing properties of the multi-site Glauber dynamics, we establish two notions for generating polynomials of discrete set-valued distributions: sector-stability and fractional log-concavity. These notions generalize well-studied properties like real-stability and log-concavity, but unlike them robustly degrade under useful transformations applied to the distribution. We relate these notions to pairwise correlations in the underlying distribution and the notion of spectral independence introduced by [ALO20], providing a new tool for establishing spectral independence based on geometry of polynomials. As a byproduct of our techniques, we show that polynomials avoiding roots in a sector of the complex plane must satisfy what we call fractional log-concavity; this extends a classic result established by [Gar59] who showed homogeneous polynomials that have no roots in a half-plane must be log-concave over the positive orthant.

翻译：我们展示了完全多项式时间随机近似方案(FPRAS)，用于计数给定大小的匹配或更广泛地在平面、不一定是二分图的图中采样/计数单体-二聚体系统。虽然平面图上的完美匹配可以在多项式时间内精确计算，但是计数非完美匹配在[Jer87]中被证明是#P难的，他还提出了有效的近似计数是否可能的问题。我们通过显示单体-二聚体系统中多点Glauber动力学总是快速混合，以及在计算完美匹配易于处理的向下封闭图族上可以有效的实现此动力学，肯定地回答了这个问题。作为我们结果的更进一步应用，我们展示了如何使用多点Glauber动力学有效地从分区受限的强Rayleigh分布和非对称行列式点过程中进行采样。为了分析多点Glauber动力学的混合特性，我们为离散集值分布构造了两个生成多项式的概念：扇形稳定和分数对数凸性。这些概念推广了广泛研究的性质，如实对称性和对数凸性，但与它们不同的是，在应用于分布的有用转换下，它们的可靠退化。我们将这些概念与底层分布中的成对相关性以及[ALO20]引入的谱独立性概念联系起来，提供了一个基于多项式几何的建立谱独立性的新工具。作为我们技术的副产品，我们展示了在平面复平面的一个扇形中避免根的多项式必须满足我们所说的分数对数凸性。这扩展了由[Gar59]建立的经典结果，他证明没有半平面根的齐次多项式必须在正定向上对数凸。