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Correlation bounds for fields and matroids
Let be a finite connected graph, and let be a spanning tree of
chosen uniformly at random. The work of Kirchhoff on electrical networks can be
used to show that the events and are negatively
correlated for any distinct edges and . 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 ,
where is a randomly chosen basis of a matroid. As an application, we prove
Mason's conjecture that the number of -element independent sets of a matroid
forms an ultra-log-concave sequence in .Comment: 16 pages. Supersedes arXiv:1804.0307
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