Abstract
We first give an example of a negatively associated measure which does not satisfy the van den Berg-Kesten inequality. Next we show that the class of measures satisfying the van den Berg-Kesten inequality is not closed under either of conditioning, introduction of external fields or convex combinations. Finally we show that this class also includes measure which have rank sequence which is not logconcave.
Highlights
In 2009 Petter Brändén gave lecture series on the results of [BBL09] at the Newton Institute
Note that all measures satisfying the BK-inequality are negatively associated [DR98]. This had been stated as an open problem already in 1998 by Dubhashi and Ranjan [DR98]
The second aim of this paper is to construct a measure which shows that the class of measurers satisfying the BK-inequality is not closed under conditioning or external fields
Summary
The second aim of this paper is to construct a measure which shows that the class of measurers satisfying the BK-inequality is not closed under conditioning or external fields. A probability measure μ on Bn is negatively associated if every pair of increasing functions f : Bn → R+ , g : Bn → R+ such that f and g depend on disjoint sets of variables satisfies. We need to define the concept that two events A and B occur disjointly, denoted AB This is most done by interpreting the the boolean lattice. This result has proven to be one of the most important tools in the study of both percolation and random graphs Those authors conjectured that for product measures the inequality holds for all pairs of events, not just increasing ones. One of the most fundamental such question is the conjecture by Dubhashi and Ranjan that the class of BK-measures is closed under direct products [DR98]
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