The correlation constant of a field

TL;DR

This paper introduces the correlation constant of a field, an invariant measuring the likelihood of elements being in a basis or spanning tree, and proves it is at least 8/7 for all fields, with explicit constructions.

Contribution

It establishes a lower bound of 8/7 for the correlation constant of any field and provides explicit examples of configurations with positive correlation.

Findings

Correlation constants are between 0 and 2.

The correlation constant of every field is at least 8/7.

Explicit constructions of positively correlated configurations are provided.

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.