We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where $0\lt q\lt 1$. Consequently, we can sample random spanning forests in a graph and estimate the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has edge expansion at least 1.

Our algorithm and proof build on the recent results of Dinur, Kaufman, Mass and Oppenheim who show that a high-dimensional walk on a weighted simplicial complex mixes rapidly if for every link of the complex, the corresponding localized random walk on the 1-skeleton is a strong spectral expander. One of our key observations is that a weighted simplicial complex $X$ is a $0$-local spectral expander if and only if a naturally associated generating polynomial $p_{X}$ is strongly log-concave. More generally, to every pure simplicial complex $X$ with positive weights on its maximal faces, we can associate a multiaffine homogeneous polynomial $p_{X}$ such that the eigenvalues of the localized random walks on $X$ correspond to the eigenvalues of the Hessian of derivatives of $p_{X}$.

我们设计了一个 FPRAS 来计算由独立集合预言机给出的任何拟阵的碱基数量，并估计 $0\lt q\lt 1$ 范围内任何拟阵的随机聚类模型的配分函数。因此，我们可以在图中对随机生成森林进行采样，并估计任何拟阵的可靠性多项式。我们还证明了 Mihail 和 Vazirani 三十年前的猜想，即任何拟阵的碱基交换图的边扩展至少为 1。

我们的算法和证明建立在 Dinur、Kaufman、Mass 和 Oppenheim 的最新结果的基础上，他们表明，如果对于复形的每个链接，1-骨架上相应的局部随机游走，则加权单纯复形上的高维游走会快速混合是一种强大的光谱扩展器。我们的关键观察之一是，当且仅当自然关联的生成多项式 $p_{X}$ 是强对数凹时，加权单纯复形 $X$ 是 $0$ 局部谱扩展器。更一般地，对于每个最大面上具有正权重的纯单纯复形 $X$，我们可以将一个多重仿射齐次多项式 $p_{X}$ 关联起来，使得 $X$ 上的局部随机游走的特征值对应于$p_{X}$ 的导数的 Hessian 矩阵。