Correlation bounds for fields and matroids

Source 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\rm T and e_2 \in\rm 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\rm 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 .