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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
dgwagner@math.uwaterloo.ca
Annals of Combinatorics 12 (2) p.211-239 June, 2008
AMS Subject Classification: 05A20; 05B35, 60C05, 82B20
Abstract:
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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