The influence of variables in product spaces

Bourgain, Jean; Kahn, Jeff; Kalai, Gil; Katznelson, Yitzhak; Linial, Nathan

doi:10.1007/bf02808010

Cited by 121 publications

(172 citation statements)

References 5 publications

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“…The main result we shall use is a sharp-threshold result of Friedgut and Kalai [9], a simple consequence of a result of Kahn, Kalai and Linial [15] concerning the influences of coordinates in a product space. (See also [7].) Let X be a fixed ground set with N elements, and let X p be a random subset of X obtained by selecting each x ∈ X independently with probability p. For a family A ⊂ P(X) of subsets of X, let P X p (A) be the probability that X p ∈ A.…”

Section: Preliminariesmentioning

confidence: 99%

Sharp thresholds and percolation in the plane

Bollobás

Riordan

2006

Random Struct Algorithms

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Recently, it was shown in [4] that the critical probability for random Voronoi percolation in the plane is 1/2. As a by-product of the method, a short proof of the Harris-Kesten Theorem was given in [5]. The aim of this paper is to show that the techniques used in these papers can be applied to many other planar percolation models, both to obtain short proofs of known results, and to prove new ones.

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Section: Preliminariesmentioning

confidence: 99%

Sharp thresholds and percolation in the plane

Bollobás

Riordan

2006

Random Struct Algorithms

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“…The KKL inequality [KKL88] and Friedgut's threshold theorem [Fri99] are both mentioned in [Bou00] as notable consequences of the interaction between harmonic analysis and combinatorics. Let us also mention the more recent work of Mossel, O'Donnell, and Oleszkiewicz [MOO10] that studies Boolean functions where all variables have small influences and now is one of the main tools in the study of the hardness of approximation in theoretical computer science.…”

Section: Introductionmentioning

confidence: 99%

A structure theorem for Boolean functions with small total influences

Hatami

2012

Ann. Math.

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We show that on every product probability space, Boolean functions with small total influences are essentially the ones that are almost measurable with respect to certain natural sub-sigma algebras. This theorem in particular describes the structure of monotone set properties that do not exhibit sharp thresholds.Our result generalizes the core of Friedgut's seminal work on properties of random graphs to the setting of arbitrary Boolean functions on general product probability spaces and improves the result of Bourgain in his appendix to Friedgut's paper.

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“…the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval. More precise analysis of the size of the threshold interval is done in [7].…”

Section: Introduction and Definitionsmentioning

confidence: 99%