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Lectures on noise sensitivity and percolation Christophe Garban and - - PowerPoint PPT Presentation

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Lectures on noise sensitivity and percolation Christophe Garban and Jeffrey E. Steif Clay summer school, Buzios 2010 Boolean functions Definition A Boolean function is a function f : { 1 , 1 } n { 0 , 1 } OR { 1 , 1 } Boolean

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Lectures on noise sensitivity and percolation

Christophe Garban and Jeffrey E. Steif

Clay summer school, Buzios 2010

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Boolean functions

Definition

A Boolean function is a function f : {−1, 1}n → {0, 1} OR {−1, 1}

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Boolean functions

Definition

A Boolean function is a function f : {−1, 1}n → {0, 1} OR {−1, 1} Example: Majority f (x1, . . . , xn) = sign(

n

  • i=1

xi)

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Boolean functions

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Boolean functions computer sci- ence

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Boolean functions computer sci- ence Probability theory

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Boolean functions computer sci- ence Analytical tools (Fourier analysis) Probability theory

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Boolean functions computer sci- ence Analytical tools (Fourier analysis) Probability theory Statistical physics (percolation)

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A concrete situation : VOTING SCHEMES

Imagine one has n people labelled 1, . . . , n which are deciding between candidates A and B according to a certain procedure or voting scheme. This procedure can be represented by a Boolean function f : {−1, 1}n → {0, 1}

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A concrete situation : VOTING SCHEMES

Imagine one has n people labelled 1, . . . , n which are deciding between candidates A and B according to a certain procedure or voting scheme. This procedure can be represented by a Boolean function f : {−1, 1}n → {0, 1} For instance, you may think of    A = Al Gore B = Bush n ≈ 108

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Noise stability

Suppose the election is “well-balanced” between A and B. One may thus consider the actual configuration of votes as a random ω = (x1, . . . , xn) ∈ {−1, 1}n , sampled according to the uniform measure. The outcome of the election should be f (ω).

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Noise stability

Suppose the election is “well-balanced” between A and B. One may thus consider the actual configuration of votes as a random ω = (x1, . . . , xn) ∈ {−1, 1}n , sampled according to the uniform measure. The outcome of the election should be f (ω). In fact due to inevitable errors in the recording of the votes, the outcome is f (ωǫ) instead. Here ωǫ is a “slight perturbation” of ω.

Informal definition

Noise stability corresponds to P

  • f (ω) = f (ωǫ)
  • being “small” .
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Case of the majority function

If f (ω) = sign( xi),

ω n

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Case of the majority function

If f (ω) = sign( xi),

ω ωǫ n

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Percolation

Sub-critical (p < pc) critique (pc) Super-critical (p > pc) δZ2

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Percolation

Sub-critical (p < pc) critique (pc) Super-critical (p > pc) δZ2

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Percolation

Sub-critical (p < pc) Critical (pc) Super-critical (p > pc) δZ2 δZ2

Question

How does critical percolation “react” to perturbations ?

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ω:

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ω → ωǫ:

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Large clusters are very sensitive to “noise”

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Large clusters are very sensitive to “noise”

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

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

b · n a · n

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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

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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

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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

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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

Informal definition

Noise sensitivity corresponds to fn(ω) and fn(ωǫ) being very little correlated (i.e. Cov(fn(ω), fn(ωǫ)) being very small).

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Applications to dynamical percolation

Informal definition

This is a very simple (stationary) dynamics on percolation configurations.

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Applications to dynamical percolation

Informal definition

This is a very simple (stationary) dynamics on percolation configurations. Each hexagon (or edge) switches color at the times of a Poisson Point Process.

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How is it related to Noise Sensitivity ? t

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How is it related to Noise Sensitivity ? t ωt ωt+ǫ n

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Applications to Sub-Gaussian fluctuations

Informal definition (First Passage Percolation)

Let 0 < a < b. Define the random metric on the graph Zd as follows: for each edge e ∈ Ed, fix its length τe to be a with probability 1/2 and b with probability 1/2.

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Applications to Sub-Gaussian fluctuations

Informal definition (First Passage Percolation)

Let 0 < a < b. Define the random metric on the graph Zd as follows: for each edge e ∈ Ed, fix its length τe to be a with probability 1/2 and b with probability 1/2. It is well-known that the random ball Bω(R) := {x ∈ Zd, dist

ω (0, x) ≤ R}

has an asymptotic shape.

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Applications to Sub-Gaussian fluctuations

Informal definition (First Passage Percolation)

Let 0 < a < b. Define the random metric on the graph Zd as follows: for each edge e ∈ Ed, fix its length τe to be a with probability 1/2 and b with probability 1/2. It is well-known that the random ball Bω(R) := {x ∈ Zd, dist

ω (0, x) ≤ R}

has an asymptotic shape.

Question

What are the fluctuations around this asymptotic shape ?

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What will be our main tools ?

  • Some concepts which arised in computer science: influence of a

variable, etc

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What will be our main tools ?

  • Some concepts which arised in computer science: influence of a

variable, etc

  • Discrete Fourier analysis
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What will be our main tools ?

  • Some concepts which arised in computer science: influence of a

variable, etc

  • Discrete Fourier analysis

           In the same way as a function f : R/Z → R can be decomposed into Fourier series, we will see that a Boolean function f : {−1, 1}n → {0, 1} can be naturally decomposed into f =

  • S

ˆ f (S)χS           

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What will be our main tools ?

  • Some concepts which arised in computer science: influence of a

variable, etc

  • Discrete Fourier analysis

           In the same way as a function f : R/Z → R can be decomposed into Fourier series, we will see that a Boolean function f : {−1, 1}n → {0, 1} can be naturally decomposed into f =

  • S

ˆ f (S)χS           

Fact

f being noise sensitive will correspond to f being of “High frequency”.

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What will be our main tools ?

  • Some concepts which arised in computer science: influence of a

variable, etc

  • Discrete Fourier analysis
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What will be our main tools ?

  • Some concepts which arised in computer science: influence of a

variable, etc

  • Discrete Fourier analysis
  • Hypercontractivity
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What will be our main tools ?

  • Some concepts which arised in computer science: influence of a

variable, etc

  • Discrete Fourier analysis
  • Hypercontractivity
  • Randomized algorithms
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What will be our main tools ?

  • Some concepts which arised in computer science: influence of a

variable, etc

  • Discrete Fourier analysis
  • Hypercontractivity
  • Randomized algorithms
  • Viewing the “frequencies of percolation” as random fractals of the

plane.