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：我们针对平面（不必为二分图）图中给定大小的匹配计数问题，或更一般地，单体-二聚体系统中的采样/计数问题，展示了全多项式时间随机近似方案（FPRAS）。尽管平面图中的完美匹配可在多项式时间内精确计数，但[Jer87]证明了非完美匹配计数是#P难的，同时也提出了是否存在高效近似计数方法的问题。我们通过证明单体-二聚体系统中单体集合上的多点格点动力学始终快速混合来肯定回答这一问题，且在完美匹配可计算的向下封闭图族中，该动力学可高效实现。作为结果的进一步应用，我们展示了如何利用多点格点动力学从分区约束的强瑞利分布和非对称行列式点过程中高效采样。为分析多点格点动力学的混合性质，我们针对离散集值分布生成多项式建立了两个概念：扇区稳定性与分数对数凹性。这些概念推广了实稳定性与对数凹性等被广泛研究的性质，但不同于后者，它们在分布的有用变换下能鲁棒地退化。我们将这些概念与底层分布中的成对相关性及[ALO20]引入的谱独立性概念相关联，提供了基于多项式几何建立谱独立性的新工具。作为技术的副产品，我们证明了避免复平面扇区内根的多项式必然满足我们所谓的分数对数凹性；这扩展了[Gar59]的经典结果，该结果展示了无半平面根的齐次多项式在正象限上必为对数凹。