Let G be a finite connected graph, and let T be a spanning tree of G
chosen uniformly at random. The work of Kirchhoff on electrical networks can be
used to show that the events e1∈T and e2∈T are negatively
correlated for any distinct edges e1 and e2. What can be said for such
events when the underlying matroid is not necessarily graphic? We use Hodge
theory for matroids to bound the correlation between the events e∈B,
where B is a randomly chosen basis of a matroid. As an application, we prove
Mason's conjecture that the number of k-element independent sets of a matroid
forms an ultra-log-concave sequence in k.Comment: 16 pages. Supersedes arXiv:1804.0307