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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Negative dependence and the geometry of polynomials


Authors: Julius Borcea, Petter Brändén and Thomas M. Liggett
Journal: J. Amer. Math. Soc. 22 (2009), 521-567
MSC (2000): Primary 62H20; Secondary 05B35, 15A15, 15A22, 15A48, 26C10, 30C15, 32A60, 60C05, 60D05, 60E05, 60E15, 60G55, 60K35, 82B31
Published electronically: September 19, 2008
MathSciNet review: 2476782
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Abstract: We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures and uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence, and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons' recent results on determinantal measures, and we construct counterexamples to several conjectures of Pemantle and Wagner on negative dependence and ultra log-concave rank sequences.


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Additional Information

Julius Borcea
Affiliation: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
Email: julius@math.su.se

Petter Brändén
Affiliation: Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Email: pbranden@math.kth.se

Thomas M. Liggett
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555
Email: tml@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-08-00618-8
PII: S 0894-0347(08)00618-8
Keywords: Negative association, stable polynomials, hyperbolic polynomials, determinants, matrices, spanning trees, matroids, probability measures, stochastic domination, interacting particle systems, exclusion processes
Received by editor(s): July 17, 2007
Published electronically: September 19, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.