Workshop on Discrete Harmonic Analysis Newton Institute, March 2011

Transcription

1 High frequency criteria for Boolean functions (with an application to percolation) Christophe Garban ENS Lyon and CNRS Workshop on Discrete Harmonic Analysis Newton Institute, March 2011 C. Garban (ENS Lyon) High frequency criteria for Boolean functions 1 / 23

2 Plan Spectrum of Boolean functions Motivations (percolation...) Randomized algorithms A theorem by Schramm and Steif Noise sensitivity of percolation Where exactly does the spectrum of percolation localize? (Work with G. Pete and O. Schramm) C. Garban (ENS Lyon) High frequency criteria for Boolean functions 2 / 23

3 Spectrum of a Boolean function Let f : Ω n = { 1, 1} n { 1, 1} be a Boolean function C. Garban (ENS Lyon) High frequency criteria for Boolean functions 3 / 23

4 Spectrum of a Boolean function Let f : Ω n = { 1, 1} n { 1, 1} be a Boolean function f = f (S) χ(s), with χ(s) := i S x i. S {1,...,n} C. Garban (ENS Lyon) High frequency criteria for Boolean functions 3 / 23

5 f 2 2 = S Spectrum of a Boolean function Let f : Ω n = { 1, 1} n { 1, 1} be a Boolean function with χ(s) := i S x i. f = S {1,...,n} f (S) χ(s), These functions form an othornormal basis for L 2 ({ 1, 1} n ) endowed with the uniform measure µ = (1/2δ 1 + 1/2δ 1 ) n. The Fourier coefficients f (S) satisfy f (S) := f, χs = E [ f χ S ] Parseval tells us f (S) 2 C. Garban (ENS Lyon) High frequency criteria for Boolean functions 3 / 23

6 Energy spectrum of a Boolean function For any Boolean or real-valued function f : { 1, 1} n {0, 1} or R, we define its energy spectrum E f to be E f (k) := f (S) 2 S: S =k 1 k n

7 Energy spectrum of a Boolean function For any Boolean or real-valued function f : { 1, 1} n {0, 1} or R, we define its energy spectrum E f to be E f (k) := f (S) 2 S: S =k 1 k n E f (k) k k = 1 k = 2 k = n

8 Energy spectrum of a Boolean function For any Boolean or real-valued function f : { 1, 1} n {0, 1} or R, we define its energy spectrum E f to be E f (k) := f (S) 2 S: S =k 1 k n E f (k) The total Spectral mass here is f(s) 2 = Var[f] S k k = 1 k = 2 k = n

9 Energy spectrum of Majority Let MAJ n (x 1,..., x n ) := sign( x i ).

10 Energy spectrum of Majority Let MAJ n (x 1,..., x n ) := sign( x i ). Its energy spectrum can be computed explicitly. It has the following shape: S =k MAJ n (S) n k

11 Energy spectrum of Majority Let MAJ n (x 1,..., x n ) := sign( x i ). Its energy spectrum can be computed explicitly. It has the following shape: S =k MAJ n (S) 2 1 k 3/ n k

12 Noise stability Let f : { 1, 1} n { 1, 1} be a Boolean function. For any ɛ > 0, if ω n = (x 1,..., x n ) is sampled uniformly in Ω n, let ω ɛ n be the noised configuration obtained out of ω n by resampling each bit with probability ɛ. The noise stability of f is given by S ɛ (f ) := P [ f (ω) = f (ω ɛ ) ] It is easy to check that S ɛ (f ) = f (S) 2 (1 ɛ) S S = E [ f ] 2 + E f (k)(1 ɛ) k k 1 C. Garban (ENS Lyon) High frequency criteria for Boolean functions 6 / 23

13 What can be said on the spectrum in general? C. Garban (ENS Lyon) High frequency criteria for Boolean functions 7 / 23

14 What can be said on the spectrum in general? Here is a deep result by Bourgain: Theorem (Bourgain, 2001) If f is a balanced Boolean function with low influences, then if k is not too large. S >k f (S) 2 1 k, C. Garban (ENS Lyon) High frequency criteria for Boolean functions 7 / 23

15 What can be said on the spectrum in general? Here is a deep result by Bourgain: Theorem (Bourgain, 2001) If f is a balanced Boolean function with low influences, then if k is not too large. S >k f (S) 2 1 k, Theorem ( Majority is Stablest Mossel, O Donnell and Oleszkiewicz, 2005) If f is a balanced Boolean function with low influences, then S ɛ (f ) 2 arcsin(1 ɛ) π C. Garban (ENS Lyon) High frequency criteria for Boolean functions 7 / 23

16 Recognizing high frequency behavior In what follows, we will be interested in Boolean (or real-valued) functions which are highly sensitive to small perturbations (or small noise). Such functions are called noise sensitive. They are such that most of their Fourier mass is localized on high frequencies. In particular, their energy spectrum should look as follows S =k f(s) k

17 Question we wish to address in this talk

18 Question we wish to address in this talk You re given a Boolean function f : { 1, 1} n { 1, 1}

19 Question we wish to address in this talk You re given a Boolean function f : { 1, 1} n { 1, 1} Can you find an efficient criterion which ensures that its spectrum has the following shape?

20 Question we wish to address in this talk You re given a Boolean function f : { 1, 1} n { 1, 1} Can you find an efficient criterion which ensures that its spectrum has the following shape? k

21 Let us start with a visual experiment Let A be a subset of the square in the plane. Question: Can you tell whether the set A is noise sensitive or not? C. Garban (ENS Lyon) High frequency criteria for Boolean functions 10 / 23

22 Let us start with a visual experiment Let A be a subset of the square in the plane. Question: Can you tell whether the set A is noise sensitive or not? C. Garban (ENS Lyon) High frequency criteria for Boolean functions 10 / 23

23 Let us start with a visual experiment Let A be a subset of the square in the plane. Question: Can you tell whether the set A is noise sensitive or not? C. Garban (ENS Lyon) High frequency criteria for Boolean functions 10 / 23

24 Let us start with a visual experiment Let A be a subset of the square in the plane. Question: Can you tell whether the set A is noise sensitive or not? C. Garban (ENS Lyon) High frequency criteria for Boolean functions 10 / 23

25 Let us start with a visual experiment Let A be a subset of the square in the plane. Question: Can you tell whether the set A is noise sensitive or not? C. Garban (ENS Lyon) High frequency criteria for Boolean functions 10 / 23

26 Motivations for high spectra C. Garban (ENS Lyon) High frequency criteria for Boolean functions 11 / 23

27 Motivations for high spectra Influences of Boolean functions, Sharp thresholds.. C. Garban (ENS Lyon) High frequency criteria for Boolean functions 11 / 23

28 Motivations for high spectra Influences of Boolean functions, Sharp thresholds.. Dynamical percolation C. Garban (ENS Lyon) High frequency criteria for Boolean functions 11 / 23

29 Motivations for high spectra Influences of Boolean functions, Sharp thresholds.. Dynamical percolation Fluctuations for natural random metrics on Z d (First passage percolation). C. Garban (ENS Lyon) High frequency criteria for Boolean functions 11 / 23

30 Dynamical percolation C. Garban (ENS Lyon) High frequency criteria for Boolean functions 12 / 23

31 ω 0 : Dynamical percolation C. Garban (ENS Lyon) High frequency criteria for Boolean functions 12 / 23

32 ω 0 ω t : Dynamical percolation C. Garban (ENS Lyon) High frequency criteria for Boolean functions 13 / 23

33 Key step: percolation is noise sensitive C. Garban (ENS Lyon) High frequency criteria for Boolean functions 14 / 23

34 Key step: percolation is noise sensitive C. Garban (ENS Lyon) High frequency criteria for Boolean functions 14 / 23

35 Large scale properties are encoded by Boolean functions of the inputs C. Garban (ENS Lyon) High frequency criteria for Boolean functions 15 / 23

36 Large scale properties are encoded by Boolean functions of the inputs b n a n C. Garban (ENS Lyon) High frequency criteria for Boolean functions 15 / 23

37 Large scale properties are encoded by Boolean functions of the inputs b n Let f n : { 1, 1} O(1)n2 {0, 1} be the Boolean function defined as follows a n C. Garban (ENS Lyon) High frequency criteria for Boolean functions 15 / 23

38 Large scale properties are encoded by Boolean functions of the inputs b n Let f n : { 1, 1} O(1)n2 {0, 1} be the Boolean function defined as follows a n { 1 if there is a left-right crossing f n (ω) := C. Garban (ENS Lyon) High frequency criteria for Boolean functions 15 / 23

39 Large scale properties are encoded by Boolean functions of the inputs b n Let f n : { 1, 1} O(1)n2 {0, 1} be the Boolean function defined as follows a n { 1 if there is a left-right crossing f n (ω) := 0 else C. Garban (ENS Lyon) High frequency criteria for Boolean functions 15 / 23

40 The energy spectrum of macroscopic events Question: how does the energy spectrum of the above Boolean functions f n, n 1 look? C. Garban (ENS Lyon) High frequency criteria for Boolean functions 16 / 23

41 The energy spectrum of macroscopic events Question: how does the energy spectrum of the above Boolean functions f n, n 1 look? S =k f n (S) 2? k C. Garban (ENS Lyon) High frequency criteria for Boolean functions 16 / 23

42 The energy spectrum of macroscopic events Question: how does the energy spectrum of the above Boolean functions f n, n 1 look? S =k f n (S) 2? At which speed does the Spectral mass spread as the scale n goes to infinity? k C. Garban (ENS Lyon) High frequency criteria for Boolean functions 16 / 23

43 Sub-Gaussian fluctuations in First-passage-percolation Informal definition (First Passage Percolation) Let 0 < a < b. Define the random metric on the graph Z d as follows: for each edge e E d, fix its length τ e to be a with probability 1/2 and b with probability 1/2. C. Garban (ENS Lyon) High frequency criteria for Boolean functions 17 / 23

44 Sub-Gaussian fluctuations in First-passage-percolation Informal definition (First Passage Percolation) Let 0 < a < b. Define the random metric on the graph Z d as follows: for each edge e E d, fix its length τ e to be a with probability 1/2 and b with probability 1/2. It is well-known that the random ball B ω (R) := {x Z d, dist ω (0, x) R} has an asymptotic shape. C. Garban (ENS Lyon) High frequency criteria for Boolean functions 17 / 23

45 Sub-Gaussian fluctuations in First-passage-percolation Informal definition (First Passage Percolation) Let 0 < a < b. Define the random metric on the graph Z d as follows: for each edge e E d, fix its length τ e to be a with probability 1/2 and b with probability 1/2. It is well-known that the random ball B ω (R) := {x Z d, dist ω (0, x) R} has an asymptotic shape. Question What are the fluctuations around this asymptotic shape? C. Garban (ENS Lyon) High frequency criteria for Boolean functions 17 / 23

46 Three different approaches to localize the Spectrum S =k f n (S) 2? k C. Garban (ENS Lyon) High frequency criteria for Boolean functions 18 / 23

47 Three different approaches to localize the Spectrum S =k f n (S) 2 Hypercontractivity, 1998 Benjamini, Kalai, Schramm? k C. Garban (ENS Lyon) High frequency criteria for Boolean functions 18 / 23

48 Three different approaches to localize the Spectrum S =k f n (S) 2? Hypercontractivity, 1998 Benjamini, Kalai, Schramm Randomized Algorithms, 2005 Schramm, Steif k C. Garban (ENS Lyon) High frequency criteria for Boolean functions 18 / 23

49 Three different approaches to localize the Spectrum S =k f n (S) 2? Hypercontractivity, 1998 Benjamini, Kalai, Schramm Randomized Algorithms, 2005 Schramm, Steif k Geometric study of the frequencies, 2008 G., Pete, Schramm C. Garban (ENS Lyon) High frequency criteria for Boolean functions 18 / 23

50 Three different approaches to localize the Spectrum S =k f n (S) 2? k Hypercontractivity, 1998 Benjamini, Kalai, Schramm Gil s talk Randomized Algorithms, 2005 Schramm, Steif Rocco s talk Geometric study of the frequencies, 2008 G., Pete, Schramm Ryan s talk C. Garban (ENS Lyon) High frequency criteria for Boolean functions 18 / 23

51 Randomized Algorithm (or randomized decision tree) Consider a Boolean function f : { 1, 1} n {0, 1}. C. Garban (ENS Lyon) High frequency criteria for Boolean functions 19 / 23

52 Randomized Algorithm (or randomized decision tree) Consider a Boolean function f : { 1, 1} n {0, 1}. A randomized algorithm for f is an algorithm which examines the intput bits one at a time until it finds what the output of f is. If A is such a randomized algorithm, let J = J A [n] be the random set of bits that are examined along the algorithm. C. Garban (ENS Lyon) High frequency criteria for Boolean functions 19 / 23

53 Randomized Algorithm (or randomized decision tree) Consider a Boolean function f : { 1, 1} n {0, 1}. A randomized algorithm for f is an algorithm which examines the intput bits one at a time until it finds what the output of f is. If A is such a randomized algorithm, let J = J A [n] be the random set of bits that are examined along the algorithm. We are looking for algorithms which examine the least possible number of bits. C. Garban (ENS Lyon) High frequency criteria for Boolean functions 19 / 23

54 Randomized Algorithm (or randomized decision tree) Consider a Boolean function f : { 1, 1} n {0, 1}. A randomized algorithm for f is an algorithm which examines the intput bits one at a time until it finds what the output of f is. If A is such a randomized algorithm, let J = J A [n] be the random set of bits that are examined along the algorithm. We are looking for algorithms which examine the least possible number of bits. This can be quantified by the revealment: δ = δ A := sup P [ i J ]. i [n] C. Garban (ENS Lyon) High frequency criteria for Boolean functions 19 / 23

55 Examples For the Majority function Φ n : C. Garban (ENS Lyon) High frequency criteria for Boolean functions 20 / 23

56 Examples For the Majority function Φ n : δ 1 C. Garban (ENS Lyon) High frequency criteria for Boolean functions 20 / 23

57 Examples For the Majority function Φ n : δ 1 Recursive Majority: C. Garban (ENS Lyon) High frequency criteria for Boolean functions 20 / 23

58 Percolation is very suitable to randomized algorithms C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

59 Percolation is very suitable to randomized algorithms C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

60 Percolation is very suitable to randomized algorithms C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

61 Percolation is very suitable to randomized algorithms C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

62 Percolation is very suitable to randomized algorithms? C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

63 Percolation is very suitable to randomized algorithms C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

64 Percolation is very suitable to randomized algorithms C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

65 Percolation is very suitable to randomized algorithms C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

66 Percolation is very suitable to randomized algorithms C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

67 Percolation is very suitable to randomized algorithms f n = 0 C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

68 Percolation is very suitable to randomized algorithms f n = 1 f n = 0 C. Garban (ENS Lyon) High frequency criteria for Boolean functions 21 / 23

69 Revealment for percolation Proposition (Schramm, Steif, 2005) On the triangular lattice, a slight modification of the above randomized algorithm gives a small revealment for the left-right Boolean functions f n of order δ n n 1/4 C. Garban (ENS Lyon) High frequency criteria for Boolean functions 22 / 23

70 How is it related with the Fourier expansion of f? C. Garban (ENS Lyon) High frequency criteria for Boolean functions 23 / 23

71 How is it related with the Fourier expansion of f? Theorem (Schramm, Steif, 2005) Let f : { 1, 1} n R be a real-valued function. C. Garban (ENS Lyon) High frequency criteria for Boolean functions 23 / 23

72 How is it related with the Fourier expansion of f? Theorem (Schramm, Steif, 2005) Let f : { 1, 1} n R be a real-valued function. Let A be a randomized algorithm computing f whose revealment is δ = δ A. C. Garban (ENS Lyon) High frequency criteria for Boolean functions 23 / 23

73 How is it related with the Fourier expansion of f? Theorem (Schramm, Steif, 2005) Let f : { 1, 1} n R be a real-valued function. Let A be a randomized algorithm computing f whose revealment is δ = δ A. Then, for any k = 1, 2,... the Fourier coefficients of f satisfy f (S) 2 k δ f 2 S =k C. Garban (ENS Lyon) High frequency criteria for Boolean functions 23 / 23