Negative Dependence Through the FKG Inequality

Dubhashi, Devdatt; Priebe, Volker; Ranjan, Desh

doi:10.7146/brics.v3i27.20008

Cited by 18 publications

(13 citation statements)

References 14 publications

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“…Note that we are using the usual convention that n m = 0 unless 0 ≤ m ≤ n. Since the summand above is not changed if the roles of i and j are interchanged and the roles of k and are interchanged, this can be written as (25) …”

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confidence: 99%

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Negative dependence and the geometry of polynomials

Borcea

Brändén

Liggett

2008

J. Amer. Math. Soc.

206

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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confidence: 99%

“…, so that f k reduces to the symmetrization used in the proof of Theorem 6.4, and (28) reduces to (25).…”

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confidence: 99%

Negative dependence and the geometry of polynomials

Borcea

Brändén

Liggett

2008

J. Amer. Math. Soc.

206

312

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“…Since µ(e) ∼ Bin(m, 2 n(n−1) ), we have Pr[µ(e i ) ≥ k i | µ(e j ) ≥ k j ∀e j ∈ A 0 ] ≤ Pr[µ(e i ) ≥ k i ] for every i with e i / ∈ A 0 . The inequality follows, for example, from Theorem 10 in [8] or from the main theorem in [22]. By Lemma 2.15, Pr[µ(e) ≥ k] ≤ 4em n(n−1) k ≤ 5em n 2 k .…”

Section: 1mentioning

confidence: 93%

Goldberg's conjecture is true for random multigraphs

Haxell

Krivelevich

Kronenberg

2019

Journal of Combinatorial Theory, Series B

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In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph G,We show that their conjecture (in a stronger form) is true for random multigraphs. Let M (n, m) be the probability space consisting of all loopless multigraphs with n vertices and m edges, in which m pairs from [n] are chosen independently at random with repetitions. Our result states that, for a given m := m(n), M ∼ M (n, m) typically satisfies χ ′ (G) = max{∆(G), ⌈ρ(G)⌉}. In particular, we show that if n is even and m := m(n), then χ ′ (M ) = ∆(M ) for a typical M ∼ M (n, m). Furthermore, for a fixed ε > 0, if n is odd, then a typical M ∼ M (n, m) has χ ′ (M ) = ∆(M ) for m ≤ (1 − ε)n 3 log n, and χ ′ (M ) = ⌈ρ(M )⌉ for m ≥ (1 + ε)n 3 log n. To prove this result, we develop a new structural characterization of multigraphs with chromatic index larger than the maximum degree.

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“…The following is a proof of Theorem 5 by the FKG inequality. (See Theorem 12 in Dubhashi, Priebe, and Ranjan (1996)).…”

Section: Fkg Inequalitymentioning

confidence: 99%

The dixie cup problem and FKG inequality

Flatto

2019

High Frequency

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Let Tmfalse(nfalse) be the number of purchases required to obtain m copies of n given items, each purchase choosing at random one of the n items. Emfalse(nfalse) is the expected value of the random variable Tmfalse(nfalse). The problem of obtaining a formula for Emfalse(nfalse) is known as the dixie cup problem. The problem is easy for m=1, but difficult for m > 1. Newman and Shepp solve the problem for all m, n. From the formula, they obtain the asymptotics of Emfalse(nfalse) for each fixed m and n tending to infinity. Later, Erdös and Rényi obtain the limit law for Tmfalse(nfalse), for each fixed m and n tending to infinity. From the limit law, they also derive and improve on the result of Newman and Shepp. The derivation is however incomplete, as they do not address the problem of estimating the tails of the distribution of Tmfalse(nfalse). In this paper, we provide the estimates. The estimates depend on notions concerning conditional probabilities. In particular, we use the FKG inequality, a correlation inequality which is a fundamental tool in statistical mechanics and probabilistic combinatorics.

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Negative Dependence Through the FKG Inequality

Cited by 18 publications

References 14 publications

Negative dependence and the geometry of polynomials

Negative dependence and the geometry of polynomials

Goldberg's conjecture is true for random multigraphs

The dixie cup problem and FKG inequality

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