An inequality for the weights of two families of sets, their unions and intersections

Ahlswede, Rudolf; Daykin, D. E.

doi:10.1007/bf00536201

Cited by 137 publications

(109 citation statements)

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“…In this section we will prove the quite intuitive (2) by way of a generalization of the not very intuitive (1). Before giving this generalization, we need some further definitions and notation.…”

Section: Pr(s ↔ a T ↔ B | S ↔ T) ≤ Pr(s ↔ A | S ↔ T) Pr(t ↔ B | S ↔ T)mentioning

confidence: 93%

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Some conditional correlation inequalities for percolation and related processes

Berg

Häggström

Kahn

2005

Random Struct Algorithms

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsSome Conditional Correlation Inequalities for Percolation and Related Processes ABSTRACT Consider ordinary bond percolation on a finite or countably infinite graph. Let s, t, a and b be vertices. An earlier paper proved the (nonintuitive) result that, conditioned on the event that there is no open path from s to t, the two events ``there is an open path from s to a" and ``there is an open path from s to b" are positively correlated. In the present paper we further investigate and generalize the theorem of which this result was a consequence. This leads to results saying, informally, that, with the above conditioning, the open cluster of s is conditionally positively (self-)associated and that it is conditionally negatively correlated with the open cluster of t. We also present analogues of some of our results for (a) random-cluster measures, and (b) directed percolation and contact processes, and observe that the latter lead to improvements of some of the results in a paper of Belitsky, Ferrari, Konno and Liggett (1997). In the present paper we further investigate and generalize the theorem of which this result was a consequence. This leads to results saying, informally, that, with the above conditioning, the open cluster of s is conditionally positively (self-)associated and that it is conditionally negatively correlated with the open cluster of t.We also present analogues of some of our results for (a) randomcluster measures, and (b) directed percolation and contact processes, and observe that the latter lead to improvements of some of the results in a paper of Belitsky, Ferrari, Konno and Liggett (1997).

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Section: Pr(s ↔ a T ↔ B | S ↔ T) ≤ Pr(s ↔ A | S ↔ T) Pr(t ↔ B | S ↔ T)mentioning

confidence: 93%

“…So according to the Ahlswede-Daykin ("Four Functions") Theorem ( [1] or e.g. [4]), (3) will follow if we show that, for all S, T ⊆ N ,…”

Section: Theorem 11 Let a And B Be Increasing Events Determined By mentioning

confidence: 99%

Some conditional correlation inequalities for percolation and related processes

Berg

Häggström

Kahn

2005

Random Struct Algorithms

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“…For Y ⊆ Γ , let (5). Note that the FKG inequality follows from the even more general Four Functions Theorem [1,3]. The special case of (5) for the subset lattice already follows from [21].…”

Section: Upper Ideals In Distributive Latticesmentioning

confidence: 99%

Quest for Negative Dependency Graphs

Lü

Mohr

Székely

2012

Springer Proceedings in Mathematics &Amp; Statistics

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The Lovász Local Lemma is a well-known probabilistic technique commonly used to prove the existence of rare combinatorial objects. We explore the lopsided (or negative dependency graph) version of the lemma, which, while more general, appears infrequently in literature due to the lack of settings in which the additional generality has thus far been needed. We present a general framework (matchings in hypergraphs) from which many such settings arise naturally. We also prove a seemingly new generalization of Cayley's formula, which helps defining negative dependency graphs for extensions of forests into spanning trees. We formulate open problems regarding partitions and doubly stochastic matrices that are likely amenable to the use of the lopsided local lemma.

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“…We give a condition for a Borel measure on R [0,1] which is sufficient for the validity of an AD-type correlation inequality in the function space 1 …”

mentioning

confidence: 99%

“…In [1] was proved that if ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 are bounded real non negative measurable functions on the space with measure (R n , B, μ) which satisfy for allx,ȳ ∈ R n the following inequality…”

mentioning

confidence: 99%