Negatively Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids

Abstract.Mason’s Conjecture asserts that for an m-element rank r matroid $${\mathcal{M}}$$ the sequence$$ (I_k /\left(_k^m\right): 0 \leq k \leq r) $$ is logarithmically concave, in which Ik is the number of independent k-sets of $${\mathcal{M}}$$ . A related conjecture in probability theory implies these inequalities provided that the set of independent sets of $${\mathcal{M}}$$ 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 ω is a weight function on a set system $${\mathcal{Q}}$$ that satisfies the Rayleigh condition then $${\mathcal{Q}}$$ 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-sums of uniform matroids satisfies the Rayleigh condition. Numerous conjectures and auxiliary results are included.