Correlation bounds for fields and matroids

TL;DR

This paper extends correlation bounds from graphic matroids to general matroids using Hodge theory, and proves Mason's conjecture on ultra-log-concavity of independent set counts.

Contribution

It introduces Hodge theory techniques to bound correlations in non-graphic matroids and proves Mason's conjecture on ultra-log-concavity.

Findings

Negative correlation bounds for edges in general matroids

Proof of Mason's conjecture on ultra-log-concavity

Application of Hodge theory to combinatorial probability

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 $e_1 \in T$ and $e_2 \in T$ 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 $e \in B$, 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 $k$.