Correlation bounds for fields and matroids

Let G be a finite connected graph, and let T spanning tree of chosen uniformly at random. The work Kirchhoff on electrical networks can used to show that the events $e\_1 \in\rm T$ $e\_2 are negatively correlated for any distinct edges $e\_1$ $e\_2$. What said such when underlying matroid is not necessarily graphic? We use Hodge theory matroids bound correlation between $e B$, where B randomly basis matroid. As an application, we prove Mason’s conjecture number $k$-element independent sets forms ultra-log-concave sequence in $k$.