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Lectures on Noise Sensitivity and Percolation - download pdf or read online

By Garban C., Steif J.E.

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N Here the ‘continuous’ hypercube is endowed with the uniform (Lebesgue) measure and for any k ∈ [n], Ik (f ) denotes the probability that f is not almost-surely constant on the fiber given by (xi )i=k . 48 CHAPTER V. HYPERCONTRACTIVITY AND ITS APPLICATIONS In other words, Ik (f ) = P Var f (x1 , . . , xn ) xi , i = k > 0 . 2 from the above theorem. If p ∈ [0, 1] and f : Ωn → {0, 1}, consider f¯p : [0, 1]n → {0, 1} defined by f¯p (x1 , . . , xn ) = f ((1xi

Appendix: sketch of proof for hypercontractivity The purpose of this appendix is to show that we are not using a giant “hammer” but rather that this needed inequality arising from Fourier analysis is understandable from first principles. In fact, historically, the proof by Gross of the Gaussian case first looked at the case of the hypercube and so we have the tools to obtain the Gaussian case should we want to. Before starting the proof, observe that for ρ = 0 (where 00 is defined to be 1), this simply reduces to | f | ≤ |f |.

Before discussing where such an estimate comes from, let us see that it indeed implies a polynomial decay for H(fn ). 8).

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Lectures on Noise Sensitivity and Percolation by Garban C., Steif J.E.


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