Negatively correlated random variables and Mason's conjecture

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Mason's Conjecture asserts that for an m--element rank r matroid the sequence (Iₖ/mk: 0≤ k≤ r) is logarithmically concave, in which Iₖ is the number of independent k--sets of. A related conjecture in probability theory implies these inequalities provided that the set of independent sets of 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 ω is a weight function on a set system that satisfies the Rayleigh condition then is a convex delta--matroid and ω 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.