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Negatively Correlated Random Variables and Mason's Conjecture for Independent Sets in Matroids
David G. Wagner
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Annals of Combinatorics 12 (2) p.211-239 June, 2008
AMS Subject Classification: 05A20; 05B35, 60C05, 82B20
Mason's Conjecture asserts that for an m-element rank r matroid $\M$ the sequence (Ik/\binom{m}{k} 0‹ k‹ r) is logarithmically concave, in which Ik 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 $\omega$ 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.
Keywords: matroid, delta-matroid, unimodality, logarithmic concavity, Rayleigh monotonicity, Potts model, random cluster model


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