Concentration and influences |
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Authors: | Michel Talagrand |
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Institution: | 1. Equipe d’Analyse, Université Paris VI, Bo?te 186, 4, place Jussieu, 75252, Paris Cedex 5, France
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Abstract: | Consider the discrete cube Ω={0,1}
N
, provided with the uniform probabilityP. We denote byd(x, A) the Hamming distance of a pointx of Ω and a subsetA of Ω. We define the influenceI(A) of theith coordinate onA as follows. Forx in Ω, consider the pointT
i
(x) obtained by changing the value of theith coordinate. Then
We prove that we always have
Since it is easy to see that
, this recovers the well known fact that ∫Ω
d(x, A)dP(x) is at most of order
whenP(A)≥1/2. The new information is that ∫Ω
d(x, A)dP(x) can be of order
only ifA reassembles the Hamming ball {x; ∑1≤N
x
i
≥N/2}. |
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Keywords: | |
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