Negative dependence and the geometry of polynomials

By Julius Borcea, Petter Brand'en, Thomas M. Liggett. 2007

306 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b6116589a0b462acc8df4f3f38ee939ba211cfc7

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Hyperbolicity and stable polynomials in combinatorics and probability (2012, background): “These closure properties turn out to have powerful implications for systems of negatively dependent random variables, as worked out in 2009 by Borcea, Brändén and Liggett [BBL09].”
Unimodality, log-concavity, real-rootedness and beyond (2015, background): “d from a zero to a one in . (D)A particle at site iis annihilated (dies) at rate d i: The conguration is unchanged unless i= 1, and then iis changed from a one to a zero in . It was proved in [12,104] that SEP(Q;b;d) preserves the family of strong Rayleigh measures. 32 PETTER BRAND EN Theorem 9.7.”
The Palm measure and the Voronoi tessellation for the Ginibre process (2006, unknown): “For the kernels K and K 0 deﬁned by formulas (1) and (2), we have, obviously, K ≥ K 0 thus Theorem 3 implies Theorem 2. of The proof Theorem 3 rests on an explicit description of the marginal laws of the process ψ , obtained in Section 1, which allows us to use a similar A. GOLDMAN result proved by ”
Analytic Combinatorics in Several Variables (2013, background): “For polynomials which are generating functions of joint distributions of binary variables, the property implies a number of negative dependence properties, and this has resulted in the solution of a number of outstanding conjectures in the theory of negative dependence (Borcea et al., 2009).”
Random Complexes and l^2-Betti Numbers (2008, background): “All our measures will be determinantal, whence they satisfy various strong properties such as negative associations (Lyons (2003), Borcea, Brändén, and Liggett (2009)).”
Negative Dependence in Sampling (2012, background): “This is in fact a special case of Borcea et al. (2009), theorem 4.20.”
Multivariate stable polynomials: theory and applications (2009, background): “tiane real polynomial satisfying this weaker condition is a Rayleigh polynomial. (This terminology is by analogy with the Rayleigh monotonicity property of electrical networks { see Denition 2.5 of [7] and the references cited there. Multiane real stable polynomials are also called strongly Rayleigh”
The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability (2008, methodology): “Multi-affine Polynomials and the Lieb-Sokal Lemma 9 2.2.”
Multivariate Pólya–Schur classification problems in the Weyl algebra (2006, methodology): “49]. Let us ﬁnally mention that real stable polynomials have also found remarkable applications in probability theory and interacting particle systems. Indeed, these polynomials were recently used in [11] to develop a theory of negative dependence for the class of strongly Rayleigh probability measu”
Negatively Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids (2006, background): “U Very recently, counterexamples to the Big Conjecture have been found by Borcea, Br¤ and· and Liggett [ 4 ] and by Kahn and Neiman [26].”
Probability on Trees and Networks (2017, background): “In the discrete case, the direction of Buser’s inequality is the easy one. Cheeger’s result was transferred to the discrete setting in various contexts of infinite graphs by Dodziuk (1984), Dodziuk and Kendall (1986), Varopoulos (1985a), Ancona (1988), Gerl (1988), Biggs, Mohar, and Shawe-Taylor (19”