# Correlation bounds for fields and matroids

### June Huh

Princeton University, USA### Benjamin Schröter

KTH Royal Institute of Technology, Stockholm, Sweden### Botong Wang

University of Wisconsin-Madison, USA

## 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 $k$.

## Cite this article

June Huh, Benjamin Schröter, Botong Wang, Correlation bounds for fields and matroids. J. Eur. Math. Soc. 24 (2022), no. 4, pp. 1335–1351

DOI 10.4171/JEMS/1119