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 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 B 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 .
Cite this article
June Huh, Benjamin Schröter, Botong Wang, Correlation bounds for fields and matroids. J. Eur. Math. Soc. 24 (2022), no. 4, pp. 1335–1351DOI 10.4171/JEMS/1119