Log-concave polynomials II: High-dimensional walks and an FPRAS for counting bases of a matroid | Annals of Mathematics

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 $p_{X}$.