Negative correlation and log‐concavity

Neiman, M.

doi:10.1002/rsa.20292

Cited by 21 publications

(38 citation statements)

References 37 publications

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“…The results in §6 are also useful in §7, as they allow us to show that the examples we construct there provide counterexamples to the series of conjectures on ultra log-concave rank sequences proposed in [65,72] to which we already alluded. We note that Corollary 6.6 in §6 was subsequently proved by different methods in [48], where it was additionally shown that almost exchangeable measures satisfy the (strong) Feder-Mihail property.…”

Section: (Nlc) µ(S)µ(t ) ≥ µ(S ∪ T )µ(S ∩ T ) For All S T ⊆ [N]mentioning

confidence: 80%

“…In §7 we construct the first counterexamples in the literature to all of these conjectures except Mason's, which once again confirms the delicate nature of negative dependence. After this work was made publicly available on www.arxiv.org, other counterexamples to some of these conjectures were reported [48,60]. Markström [60] proved that negative association does not imply unimodality, and Kahn-Neiman [48] later showed that not even strong negative association (CNA+) implies unimodality.…”

Section: (Nlc) µ(S)µ(t ) ≥ µ(S ∪ T )µ(S ∩ T ) For All S T ⊆ [N]mentioning

confidence: 99%

“…After this work was made publicly available on www.arxiv.org, other counterexamples to some of these conjectures were reported [48,60]. Markström [60] proved that negative association does not imply unimodality, and Kahn-Neiman [48] later showed that not even strong negative association (CNA+) implies unimodality. Note that the rank sequence of the CNA+ measure constructed in Counterexample 1 of §7 is neither ultra nor strong log-concave but it is log-concave.…”

Section: (Nlc) µ(S)µ(t ) ≥ µ(S ∪ T )µ(S ∩ T ) For All S T ⊆ [N]mentioning

confidence: 99%

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Negative dependence and the geometry of polynomials

Borcea

Brändén

Liggett

2008

J. Amer. Math. Soc.

200

312

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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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Section: (Nlc) µ(S)µ(t ) ≥ µ(S ∪ T )µ(S ∩ T ) For All S T ⊆ [N]mentioning

confidence: 80%

Section: (Nlc) µ(S)µ(t ) ≥ µ(S ∪ T )µ(S ∩ T ) For All S T ⊆ [N]mentioning

confidence: 99%

Section: (Nlc) µ(S)µ(t ) ≥ µ(S ∪ T )µ(S ∩ T ) For All S T ⊆ [N]mentioning

confidence: 99%

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Negative dependence and the geometry of polynomials

Borcea

Brändén

Liggett

2008

J. Amer. Math. Soc.

200

312

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“…Rayleigh measures were introduced in the context of matroid theory [38], but soon found their place in the modern theory of negative dependence for probability measures [23, 32]. The size‐increasing property can be viewed as a weak form of the so‐called normalized matching property, or as a strong form of the so‐called Feder–Mihail property (see [23] for definitions). Cavity‐monotone measures will play a major role in our study, for the following elementary reason.…”

Section: Introductionmentioning

confidence: 99%

Weighted enumeration of spanning subgraphs in locally tree‐like graphs

Salez

2012

Random Struct Algorithms

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Using the theory of negative association for measures and the notion of unimodularity for random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to local constraints in asymptotically tree‐like graphs. Specifically, the normalized logarithm of the associated partition function (free energy) is shown to converge along any sequence of graphs whose random weak limit is a tree, and the limit is directly expressed in terms of the unique solution to a limiting cavity equation. On a Galton–Watson tree, the latter simplifies into a recursive distributional equation which can be solved explicitly. As an illustration, we provide a new asymptotic formula for the maximum size of a b‐matching in the Erdős–Rényi random graph with fixed average degree and diverging size, for any \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}$b\in\mathbb{N}$\end{document}. To the best of our knowledge, this is the first time that correlation inequalities and unimodularity are combined together to yield a general proof of uniqueness of Gibbs measures on infinite trees. We believe that a similar argument is applicable to other Gibbs measures than those over spanning subgraphs considered here. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013

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“…† Very recently, counterexamples to the Big Conjecture have been found by Borcea, Brändén, and Liggett[4] and by Kahn and Neiman[26]. However, It is still reasonable to look for a hypothesis that excludes these examples and yet implies Mason's Conjecture for some restricted classes of matroids.…”

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confidence: 99%

Negatively Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids

Wagner

2008

Ann. Comb.

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Mason's Conjecture asserts that for an m-element rank r matroid M the sequence I k / m k : 0 ≤ k ≤ r is logarithmically concave, in which I k 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 ω is a weight function on a set system Q that satisfies the Rayleigh condition then Q is a convex delta-matroid and ω 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 twosum of uniform matroids satisfies the Rayleigh condition. Numerous conjectures and auxiliary results are included.

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Negative correlation and log‐concavity

Abstract: (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.

Cited by 21 publications

References 37 publications

Negative dependence and the geometry of polynomials

Negative dependence and the geometry of polynomials

Weighted enumeration of spanning subgraphs in locally tree‐like graphs

Negatively Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids

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