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Noise Sensitivity of Boolean Functions and Percolation

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Noise Sensitivity of Boolean Functions and Percolation Noise sensitivity of Boolean functions and percolation Christophe Garban1 Jeffrey E. Steif2 arXiv:1102.5761v3 [math.PR] 29 May 2012 1ENS Lyon, CNRS 2Chalmers University Contents Overview 5 I Boolean functions and key concepts 9 1 Boolean functions . .9 2 Some Examples . .9 3 Pivotality and Influence . 11 4 The Kahn, Kalai, Linial theorem . 12 5 Noise sensitivity and noise stability . 14 6 The Benjamini, Kalai and Schramm noise sensitivity theorem . 14 7 Percolation crossings: our final and most important example . 16 II Percolation in a nutshell 21 1 The model . 21 2 Russo-Seymour-Welsh . 22 3 Phase transition . 23 4 Conformal invariance at criticality and SLE processes . 23 5 Critical exponents . 25 6 Quasi-multiplicativity . 26 III Sharp thresholds and the critical point 27 1 Monotone functions and the Margulis-Russo formula . 27 2 KKL away from the uniform measure case . 28 3 Sharp thresholds in general : the Friedgut-Kalai Theorem . 28 2 1 4 The critical point for percolation for Z and T is 2 ............ 29 5 Further discussion . 30 IV Fourier analysis of Boolean functions 33 1 Discrete Fourier analysis and the energy spectrum . 33 2 Examples . 34 3 Noise sensitivity and stability in terms of the energy spectrum . 35 4 Link between the spectrum and influence . 36 5 Monotone functions and their spectrum . 37 1 2 CONTENTS V Hypercontractivity and its applications 41 1 Heuristics of proofs . 41 2 About hypercontractivity . 42 3 Proof of the KKL theorems . 44 4 KKL away from the uniform measure . 47 5 The noise sensitivity theorem . 49 Appendix on Bonami-Gross-Beckner 51 VI First evidence of noise sensitivity of percolation 57 1 Influences of crossing events . 57 2 The case of Z2 percolation . 61 3 Some other consequences of our study of influences . 64 4 Quantitative noise sensitivity . 66 VII Anomalous fluctuations 73 1 The model of first passage percolation . 73 2 State of the art . 75 3 The case of the torus . 75 4 Upper bounds on fluctuations in the spirit of KKL . 78 5 Further discussion . 78 VIII Randomized algorithms and noise sensitivity 83 1 BKS and randomized algorithms . 83 2 The revealment theorem . 83 3 An application to noise sensitivity of percolation . 87 4 Lower bounds on revealments . 89 5 An application to a critical exponent . 91 6 Does noise sensitivity imply low revealment? . 92 IX The spectral sample 97 1 Definition of the spectral sample . 97 2 A way to sample the spectral sample in a sub-domain . 99 3 Nontrivial spectrum near the upper bound for percolation . 101 X Sharp noise sensitivity of percolation 107 1 State of the art and main statement . 107 2 Overall strategy . 109 3 Toy model: the case of fractal percolation . 111 4 Back to the spectrum: an exposition of the proof . 118 5 The radial case . 128 CONTENTS 3 XI Applications to dynamical percolation 133 1 The model of dynamical percolation . 133 2 What's going on in high dimensions: Zd; d 19? . 134 3 d = 2 and BKS . .≥ . 135 4 The second moment method and the spectrum . 135 5 Proof of existence of exceptional times on T ............... 137 6 Exceptional times via the geometric approach . 140 Overview The goal of this set of lectures is to combine two seemingly unrelated topics: The study of Boolean functions, a field particularly active in computer science • Some models in statistical physics, mostly percolation • The link between these two fields can be loosely explained as follows: a percolation configuration is built out of a collection of i.i.d. \bits" which determines whether the corresponding edges, sites, or blocks are present or absent. In that respect, any event concerning percolation can be seen as a Boolean function whose input is precisely these \bits". Over the last 20 years, mainly thanks to the computer science community, a very rich structure has emerged concerning the properties of Boolean functions. The first part of this course will be devoted to a description of some of the main achievements in this field. In some sense one can say, although this is an exaggeration, that computer scientists are mostly interested in the stability or robustness of Boolean functions. As we will see later in this course, the Boolean functions which \encode" large scale properties of critical percolation will turn out to be very sensitive to small perturbations. This phenomenon corresponds to what we will call noise sensitivity. Hence, the Boolean functions one wishes to describe here are in some sense orthogonal to the Boolean functions one encounters, ideally, in computer science. Remarkably, it turns out that the tools developed by the computer science community to capture the properties and stability of Boolean functions are also suitable for the study of noise sensitive functions. This is why it is worth us first spending some time on the general properties of Boolean functions. One of the main tools needed to understand properties of Boolean functions is Fourier analysis on the hypercube. Noise sensitivity will correspond to our Boolean function being of \high frequency" while stability will correspond to our Boolean func- tion being of \low frequency". We will apply these ideas to some other models from statistical mechanics as well; namely, first passage percolation and dynamical percola- tion. Some of the different topics here can be found (in a more condensed form) in [Gar11]. 5 Acknowledgements We wish to warmly thank the organizers David Ellwood, Charles Newman, Vladas Sidoravicius and Wendelin Werner for inviting us to give this course at the Clay summer school 2010 in Buzios. It was a wonderful experience for us to give this set of lectures. We also wish to thank Ragnar Freij who served as a very good teaching assistant for this course and for various comments on the manuscript. Some standard notations In the following table, f(n) and g(n) are any sequences of positive real numbers. there exists some constant C > 0 such that f(n) f(n) g(n) C−1 C; n 1 ≤ g(n) ≤ 8 ≥ there exists some constant C > 0 such that f(n) O(g(n)) f(n) Cg(n) ; n 1 ≤ ≤ 8 ≥ there exists some constant C > 0 such that f(n) Ω(g(n)) f(n) Cg(n) ; n 1 ≥ ≥ 8 ≥ f(n) f(n) = o(g(n)) lim = 0 n!1 g(n) 8 CONTENTS Chapter I Boolean functions and key concepts 1 Boolean functions n Definition I.1. A Boolean function is a function from the hypercube Ωn := 1; 1 into either 1; 1 or 0; 1 . {− g {− g f g n 1 1 ⊗n Ωn will be endowed with the uniform measure P = P = ( 2 δ−1 + 2 δ1) and E will denote the corresponding expectation. At various times, Ωn will be endowed with the n ⊗n general product measure Pp = Pp = ((1 p)δ−1 + pδ1) but in such cases the p will − be explicit. Ep will then denote the corresponding expectations. An element of Ωn will be denoted by either ! or !n and its n bits by x1; : : : ; xn so that ! = (x1; : : : ; xn). Depending on the context, concerning the range, it might be more pleasant to work with one of 1; 1 or 0; 1 rather than the other and at some specific places in these lectures, we{− will eveng relaxf theg Boolean constraint (i.e. taking only two possible values). In these cases (which will be clearly mentioned), we will consider instead real-valued functions f :Ωn R. ! A Boolean function f is canonically identified with a subset Af of Ωn via Af := ! : f(!) = 1 . f g n n Remark I.1. Often, Boolean functions are defined on 0; 1 rather than Ωn = 1; 1 . This does not make any fundamental difference at allf but,g as we will see later, the{− choiceg of 1; 1 n turns out to be more convenient when one wishes to apply Fourier analysis on{− the hypercube.g 2 Some Examples We begin with a few examples of Boolean functions. Others will appear throughout this chapter. 9 10 CHAPTER I. BOOLEAN FUNCTIONS AND KEY CONCEPTS Example 1 (Dictatorship). DICTn(x1; : : : ; xn) := x1 The first bit determines what the outcome is. Example 2 (Parity). n PARn(x1; : : : ; xn) := xi i=1 Y This Boolean function tells whether the number of 1's is even or odd. − These two examples are in some sense trivial, but they are good to keep in mind since in many cases they turn out to be the \extreme cases" for properties concerning Boolean functions. The next rather simple Boolean function is of interest in social choice theory. Example 3 (Majority function). Let n be odd and define n MAJn(x1; : : : ; xn) := sign( xi) : i=1 X Following are two further examples which will also arise in our discussions. Example 4 (Iterated 3-Majority function). Let n = 3k for some integer k. The bits are indexed by the leaves of a rooted 3-ary tree (so the root has degree 3, the leaves have degree 1 and all others have degree 4) with depth k. One iteratively applies the previous example (with n = 3) to obtain values at the vertices at level k 1, then level k 2, etc. until the root is assigned a value. The root's value is then the output− of f. For example− when k = 2, f( 1; 1; 1; 1; 1; 1; 1; 1; 1) = 1. The recursive structure of this Boolean function− will enable− explicit− − computations− − for various properties of interest. n Example 5 (Clique containment). If r = 2 for some integer n, then Ωr can be identified with the set of labelled graphs on n vertices.
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