Negatively correlated random variables and Mason's conjecture for independent sets in matroids

Mason's Conjecture asserts that for an m-element rank r matroid M the sequence (I-k = ((m)(k)) : 0 <= k <= 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 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 w 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.