Negatively correlated random variables and Mason's conjecture

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.