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

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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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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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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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An illustration of this noise sensitivity

C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 3 / 22

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An illustration of this noise sensitivity

C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 3 / 22

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

C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 4 / 22

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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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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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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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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ω0:

C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 5 / 22

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ω0 → ωt:

C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 6 / 22

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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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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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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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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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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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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

SLIDE 19

Why is it any helpful ?

C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 10 / 22

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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

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

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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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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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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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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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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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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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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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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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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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

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

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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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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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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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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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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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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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

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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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

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

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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(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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(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

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

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

SLIDE 48

Examples

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

Oded’s work on Noise Sensitivity 19 / 22

SLIDE 49

Examples

For the Majority function Φn:

δ ≈ 1

C. Garban (ENS, Orsay)

Oded’s work on Noise Sensitivity 19 / 22

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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