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 $0<q<1$. Consequently, we can
sample random spanning forests in a graph and (approximately) compute 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 expansion at least 1. One of our key observations is a close connection
between pure simplicial complexes and multiaffine homogeneous polynomials.
Specifically, if $X$ is a pure simplicial complex with positive weights on its
maximal faces, we can associate with $X$ 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 $p_{X}$.

本篇论文设计了一种 FPRAS 来计算基于独立集先验的任何线性族的基数，估算处于 $0<q<1$ 区间的任何线性族的随机团模型的分区函数，并证明了任何线性族的基交换图具有至少 1 的扩张，并研究了纯单纯复合物和多项式的密切联系。