Negatively correlated random variables and Mason's conjecture

TL;DR

This paper explores the relationship between negative correlation properties in matroids, specifically the Rayleigh condition, and Mason's conjecture, establishing new links between probabilistic inequalities and matroid theory.

Contribution

It demonstrates that the Rayleigh condition implies a matroid structure and shows preservation of this condition under two-sums, connecting probabilistic properties with combinatorial matroid theory.

Findings

Rayleigh condition implies matroid structure.

Two-sums of matroids preserve the Rayleigh condition.

Potts model of iterated two-sums satisfies the Rayleigh condition.

Abstract

Mason's Conjecture asserts that for an $m$--element rank $r$ matroid $\M$ the sequence $(I_k/\binom{m}{k}: 0\leq k\leq r)$ is logarithmically concave, in which $I_k$ is the number of independent $k$--sets of $\M$. A related conjecture in probability theory implies these inequalities provided that the set of independent sets of $\M$ satisfies a strong negative correlation property we call the \emph{Rayleigh condition}. This condition is known to hold for the set of bases of a regular matroid. We show that if $\omega$ is a weight function on a set system $\Q$ that satisfies the Rayleigh condition then $\Q$ is a convex delta--matroid and $\omega$ is logarithmically submodular. Thus, the hypothesis of the probabilistic conjecture leads inevitably to matroid theory. We also show that two--sums of matroids preserve the Rayleigh condition in four distinct senses, and hence that the Potts model of an iterated two--sum of uniform matroids satisfies the Rayleigh condition. Numerous conjectures and auxiliary results are included.