The correlation constant of a field

Abstract

We study the correlation of edges, vectors or elements to be in a randomly chosen spanning tree or a basis, respectively. Here we follow the guideline of Huh and Wang and introduce as a measure an invariant that is called the correlation constant of a graph, vector configuration, matroid or field. It follows from one of their results that these correlation constants are numbers between $0$ and $2$. Here, we show that the correlation constant of every field is at least $\frac{8}{7}$. In our proof we explicitly construct vector configurations and matroids with positively correlated elements.