Negative correlation and log-concavity

We give counterexamples and a few positive results related to several conjectures of R. Pemantle (Pemantle, J Math Phys 41 (2000), 1371–1390) and D. Wagner (Wagner, Ann Combin 12 (2008), 211–239) concerning negative correlation and log-concavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. (Borcea et al., J Am Math Soc 22 (2009), 521–567). We also give short proofs of a pair of results from (Pemantle, J Math Phys 41 (2000), 1371–1390) and (Borcea et al., J Am Math Soc 22 (2009), 521–567); prove that “almost exchangeable” measures satisfy the “Feder-Mihail” property, thus providing a “non-obvious” example of a class of measures for which this important property can be shown to hold; and mention some further questions. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010