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We design an FPRAS to count the number of bases of any matroid given by anindependent set oracle, and to estimate the partition function of the randomcluster model of any matroid in the regime where $0<q<1$. Consequently, we cansample random spanning forests in a graph and (approximately) compute thereliability polynomial of any matroid. We also prove the thirty year oldconjecture of Mihail and Vazirani that the bases exchange graph of any matroidhas expansion at least 1. One of our key observations is a close connectionbetween pure simplicial complexes and multiaffine homogeneous polynomials.Specifically, if $X$ is a pure simplicial complex with positive weights on itsmaximal 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 $p_{X}$.

Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid