Correlation bounds for fields and matroids

Abstract

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 $e_1\in \mathrm{T}$ and $e_2\in \mathrm{T}$ are negatively correlated for any distinct edges $e_1$ and $e_2$. 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 $e\in \mathrm{B}$, 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 $k$.