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 ϵ T and e2 ϵ 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 ϵ 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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Hodge theory