Odeds work on Noise Sensitivity Christophe Garban Universit Paris - - PowerPoint PPT Presentation

oded s work on noise sensitivity
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Odeds work on Noise Sensitivity Christophe Garban Universit Paris - - PowerPoint PPT Presentation

Odeds work on Noise Sensitivity Christophe Garban Universit Paris Sud and ENS Oded Schramm Memorial conference C. Garban (ENS, Orsay) Odeds work on Noise Sensitivity 1 / 22 Sensitivity of Percolation We will see that Macroscopic


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SLIDE 1

Oded’s work on Noise Sensitivity

Christophe Garban Université Paris Sud and ENS

Oded Schramm Memorial conference

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 1 / 22

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SLIDE 2

Sensitivity of Percolation

We will see that Macroscopic properties of critical percolation are highly sensitive to perturbations.

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 2 / 22

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SLIDE 3

Sensitivity of Percolation

We will see that Macroscopic properties of critical percolation are highly sensitive to perturbations. This will correspond to the following phenomenon:

Property

In critical percolation, macroscopic events are of ‘High Frequency’.

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 2 / 22

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SLIDE 4

An illustration of this noise sensitivity

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 3 / 22

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SLIDE 5

An illustration of this noise sensitivity

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 3 / 22

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SLIDE 6

Large scale properties are encoded by Boolean functions of the ‘inputs’

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 4 / 22

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SLIDE 7

Large scale properties are encoded by Boolean functions of the ‘inputs’

b · n a · n

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 4 / 22

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SLIDE 8

Large scale properties are encoded by Boolean functions of the ‘inputs’

b · n a · n

Let fn : {−1, 1}O(1)n2 → {0, 1} be the Boolean function defined as follows

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 4 / 22

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SLIDE 9

Large scale properties are encoded by Boolean functions of the ‘inputs’

b · n a · n

Let fn : {−1, 1}O(1)n2 → {0, 1} be the Boolean function defined as follows fn(ω) := 1 if there is a left-right crossing

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 4 / 22

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SLIDE 10

Large scale properties are encoded by Boolean functions of the ‘inputs’

b · n a · n

Let fn : {−1, 1}O(1)n2 → {0, 1} be the Boolean function defined as follows fn(ω) := 1 if there is a left-right crossing else

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 4 / 22

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SLIDE 11

ω0:

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 5 / 22

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SLIDE 12

ω0 → ωt:

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 6 / 22

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SLIDE 13

Noise Sensitivity

We are interested in a fast decorrelation (or fast mixing) of macroscopic properties. This can be measured with the covariance Cov(fn(ω0), fn(ωt)) = E

  • fn(ω0)fn(ωt)
  • − E
  • fn

2 ,

  • r equivalently by

Var

  • E
  • fn(ωt)
  • ω0
  • .
  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 7 / 22

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SLIDE 14

Noise Sensitivity

We are interested in a fast decorrelation (or fast mixing) of macroscopic properties. This can be measured with the covariance Cov(fn(ω0), fn(ωt)) = E

  • fn(ω0)fn(ωt)
  • − E
  • fn

2 ,

  • r equivalently by

Var

  • E
  • fn(ωt)
  • ω0
  • .

If these quantities converge towards 0 when the size n of the system goes to infinity, then the macroscopic property is said to be (asymptotically) noise sensitive.

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 7 / 22

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SLIDE 15

Noise Sensitivity

We are interested in a fast decorrelation (or fast mixing) of macroscopic properties. This can be measured with the covariance Cov(fn(ω0), fn(ωt)) = E

  • fn(ω0)fn(ωt)
  • − E
  • fn

2 ,

  • r equivalently by

Var

  • E
  • fn(ωt)
  • ω0
  • .

If these quantities converge towards 0 when the size n of the system goes to infinity, then the macroscopic property is said to be (asymptotically) noise sensitive. Defined in this way, noise sensitivity is a non-quantitative property. We will need more detailed information on the speed at which the large scale system decorrelates.

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 7 / 22

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SLIDE 16

Harmonic Analysis of Boolean functions

We consider the larger space L2({−1, 1}n) of real-valued functions from n bits into R, endowed with the scalar product: f , g =

  • x1,...,xn

2−nf (x1, . . . , xn)g(x1, . . . , xn) = E

  • fg
  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 8 / 22

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SLIDE 17

Harmonic Analysis of Boolean functions

We consider the larger space L2({−1, 1}n) of real-valued functions from n bits into R, endowed with the scalar product: f , g =

  • x1,...,xn

2−nf (x1, . . . , xn)g(x1, . . . , xn) = E

  • fg
  • One has at our disposal a natural basis for this space isomorphic to R2n:

the so-called characters of the group {−1, 1}n. For any subset S ⊂ {1, . . . , n}, consider the function χS defined by χS(x1, . . . , xn) :=

  • i∈S

xi The set of these 2n functions forms an orthonormal basis of L2({−1, 1}n).

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 8 / 22

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SLIDE 18

Fourier-Walsh expansion

Thus, any Boolean function f : {−1, 1}n → {0, 1} can be decomposed as f =

  • S⊂[n]
  • f (S) χS

where f (S) are the Fourier-Walsh coefficients of f . They satisfy

  • f (S) = f , χS = E
  • f χS
  • Note in particular that the coefficient

f (∅) = E

  • f
  • corresponds to the

mean E

  • f
  • .
  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 9 / 22

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SLIDE 19

Why is it any helpful ?

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 10 / 22

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SLIDE 20

Why is it any helpful ?

The correlation between f (ω0) and f (ωt) has a very simple form in terms

  • f the Fourier coefficients

f (S). Indeed:

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 10 / 22

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SLIDE 21

Why is it any helpful ?

The correlation between f (ω0) and f (ωt) has a very simple form in terms

  • f the Fourier coefficients

f (S). Indeed: E

  • f (ω0) f (ωt)
  • =

E

  • S1
  • f (S1)χS1(ω0)
  • S2
  • f (S2)χS2(ωt)
  • =
  • S
  • f (S)2 E
  • χS(ω0)χS(ωt)
  • =
  • S
  • f (S)2 e−t |S|

Therefore our covariance can be written

E

  • f (ω0) f (ωt)
  • − E
  • f (ω)

2 =

  • S=∅
  • f (S)2 e−t|S|
  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 10 / 22

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SLIDE 22

Energy spectrum of a Boolean function

If f : {−1, 1}n → {0, 1} is a Boolean function, its “sensitivity” is controlled by its Energy Spectrum:

  • |S|=k

f(S)2 k . . . . . . k = 1 k = 2 k = n

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 11 / 22

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SLIDE 23

Energy spectrum of a Boolean function

If f : {−1, 1}n → {0, 1} is a Boolean function, its “sensitivity” is controlled by its Energy Spectrum:

  • |S|=k

f(S)2 k . . . . . . k = 1 k = 2 k = n The total Spectral mass here is

  • |S|=0
  • f(S)2 = Var[f]
  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 11 / 22

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SLIDE 24

The Energy Spectrum of macroscopic events

Recall our above left-right crossing events corresponding to the Boolean functions fn, n ≥ 1.

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 12 / 22

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SLIDE 25

The Energy Spectrum of macroscopic events

Recall our above left-right crossing events corresponding to the Boolean functions fn, n ≥ 1. One is interested in the shape of their Energy Spectrum

  • |S|=k

fn(S)2 k . . . . . .

?

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 12 / 22

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SLIDE 26

The Energy Spectrum of macroscopic events

Recall our above left-right crossing events corresponding to the Boolean functions fn, n ≥ 1. One is interested in the shape of their Energy Spectrum

  • |S|=k

fn(S)2 k . . . . . .

?

At which speed does the Spectral mass “spread” as the scale n goes to infinity ?

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 12 / 22

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SLIDE 27

Energy Spectrum of Majority

Let Φn be the majority function

  • n {−1, 1}n (n being odd)

Φn(x1, . . . , xn) := sign(

i xi)

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 13 / 22

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SLIDE 28

Energy Spectrum of Majority

Let Φn be the majority function

  • n {−1, 1}n (n being odd)

Φn(x1, . . . , xn) := sign(

i xi)

The Energy Spectrum of Φn has the following shape:

. . .

  • |S|=k

Φn(S)2 1 5 k n 3

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 13 / 22

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SLIDE 29

Three (very different !) approaches to Localize the Spectrum

  • |S|=k

fn(S)2 k . . . . . .

?

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 14 / 22

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SLIDE 30

Three (very different !) approaches to Localize the Spectrum

  • |S|=k

fn(S)2 k . . . . . .

?

  • Hypercontractivity, 1998

Benjamini, Kalai, Schramm

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 14 / 22

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SLIDE 31

Three (very different !) approaches to Localize the Spectrum

  • |S|=k

fn(S)2 k . . . . . .

?

  • Hypercontractivity, 1998

Benjamini, Kalai, Schramm

  • Randomized Algorithms, 2005

Schramm , Steif

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 14 / 22

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SLIDE 32

Three (very different !) approaches to Localize the Spectrum

  • |S|=k

fn(S)2 k . . . . . .

?

  • Hypercontractivity, 1998

Benjamini, Kalai, Schramm

  • Randomized Algorithms, 2005

Schramm , Steif

  • Geometric study of the

‘frequencies’, 2008 G., Pete, Schramm

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 14 / 22

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SLIDE 33

Three (very different !) approaches to Localize the Spectrum

  • |S|=k

fn(S)2 k . . . . . .

?

  • Hypercontractivity, 1998

Benjamini, Kalai, Schramm

  • Randomized Algorithms, 2005

Schramm, Steif

  • Geometric study of the

‘frequencies’, 2008 G., Pete, Schramm

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 14 / 22

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SLIDE 34

First Approach [BKS, 98] Hypercontractivity

k . . .

  • |S|=k

fn(S)2

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 15 / 22

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SLIDE 35

First Approach [BKS, 98] Hypercontractivity

k . . .

≈ n3/4

  • |S|=k

fn(S)2

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 15 / 22

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SLIDE 36

First Approach [BKS, 98] Hypercontractivity

k . . .

≈ n3/4 ≍ log n

  • |S|=k

fn(S)2

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 15 / 22

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SLIDE 37

First Approach [BKS, 98] Hypercontractivity

k . . .

≈ n3/4 ≍ log n

Thm [BKS98]:

O(1) log n

  • k=1
  • fn(S)2 −

n→∞ 0

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 15 / 22

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SLIDE 38

Second Approach [SS, 05] Randomized Algorithms

k . . .

≈ n3/4 ≍ log n

  • |S|=k

fn(S)2

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 16 / 22

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SLIDE 39

Second Approach [SS, 05] Randomized Algorithms

k . . .

≈ n3/4 ≍ log n ≈ n1/8

  • |S|=k

fn(S)2

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 16 / 22

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SLIDE 40

Second Approach [SS, 05] Randomized Algorithms

k . . .

≈ n3/4 ≍ log n ≈ n1/8

  • |S|=k

fn(S)2

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 16 / 22

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SLIDE 41

Third Approach [GPS 08] Geometric Study of the ‘frequencies’

k . . .

≈ n3/4 ≍ log n ≈ n1/8

  • |S|=k

fn(S)2

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 17 / 22

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SLIDE 42

Third Approach [GPS 08] Geometric Study of the ‘frequencies’

k . . .

≈ n3/4 ≍ log n ≈ n1/8 ≈ n3/4

  • |S|=k

fn(S)2

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 17 / 22

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SLIDE 43

Third Approach [GPS 08] Geometric Study of the ‘frequencies’

k . . .

≈ n3/4 ≍ log n ≈ n1/8 ≈ n3/4

  • |S|=k

fn(S)2

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 17 / 22

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SLIDE 44

(Markovian) Randomized Algorithms

Consider a Boolean function f : {−1, 1}n → {0, 1}.

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 18 / 22

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SLIDE 45

(Markovian) Randomized Algorithms

Consider a Boolean function f : {−1, 1}n → {0, 1}. A (markovian) 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 = JA ⊂ [n] be the random set

  • f bits that are examined along the algorithm.
  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 18 / 22

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SLIDE 46

(Markovian) Randomized Algorithms

Consider a Boolean function f : {−1, 1}n → {0, 1}. A (markovian) 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 = JA ⊂ [n] be the random set

  • f bits that are examined along the algorithm.

We are looking for algorithms which examine the least possible number of bits.

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 18 / 22

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SLIDE 47

(Markovian) Randomized Algorithms

Consider a Boolean function f : {−1, 1}n → {0, 1}. A (markovian) 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 = JA ⊂ [n] be the random set

  • f 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 = δA := sup

i∈[n]

P

  • i ∈ J
  • .
  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 18 / 22

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SLIDE 48

Examples

  • For the Majority function Φn:
  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 19 / 22

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SLIDE 49

Examples

  • For the Majority function Φn:

δ ≈ 1

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 19 / 22

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SLIDE 50

Examples

  • For the Majority function Φn:

δ ≈ 1

  • Recursive Majority:
  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 19 / 22

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SLIDE 51

Percolation is very suitable to randomized algorithms

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 52

Percolation is very suitable to randomized algorithms

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 53

Percolation is very suitable to randomized algorithms

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 54

Percolation is very suitable to randomized algorithms

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 55

Percolation is very suitable to randomized algorithms

?

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 56

Percolation is very suitable to randomized algorithms

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 57

Percolation is very suitable to randomized algorithms

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 58

Percolation is very suitable to randomized algorithms

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 59

Percolation is very suitable to randomized algorithms

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 60

Percolation is very suitable to randomized algorithms

fn = 0

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 61

Percolation is very suitable to randomized algorithms

fn = 0 fn = 1

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 20 / 22

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SLIDE 62

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 fn

  • f order

δn ≈ n−1/4

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 21 / 22

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SLIDE 63

How is it related with the Fourier expansion of f ?

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 22 / 22

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SLIDE 64

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, Orsay)

Oded’s work on Noise Sensitivity 22 / 22

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SLIDE 65

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 = δA.

  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 22 / 22

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SLIDE 66

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 = δA. Then, for any k = 1, 2, . . . the Fourier coefficients of f satisfy

  • |S|=k
  • f (S)2 ≤ k δ f 2
  • C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 22 / 22