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

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

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

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

An illustration of this noise sensitivity C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 3 / 22

An illustration of this noise sensitivity C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 3 / 22

Large scale properties are encoded by Boolean functions of the ‘inputs’ C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 4 / 22

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

Large scale properties are encoded by Boolean functions of the ‘inputs’ Let f n : {− 1 , 1 } O ( 1 ) n 2 → { 0 , 1 } b · n be the Boolean function defined as follows a · n C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 4 / 22

Large scale properties are encoded by Boolean functions of the ‘inputs’ Let f n : {− 1 , 1 } O ( 1 ) n 2 → { 0 , 1 } b · n be the Boolean function defined as follows a · n � 1 if there is a left-right crossing f n ( ω ) := C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 4 / 22

Large scale properties are encoded by Boolean functions of the ‘inputs’ Let f n : {− 1 , 1 } O ( 1 ) n 2 → { 0 , 1 } b · n 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, Orsay) Oded’s work on Noise Sensitivity 4 / 22

ω 0 : C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 5 / 22

ω 0 → ω t : C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 6 / 22

Noise Sensitivity We are interested in a fast decorrelation (or fast mixing) of macroscopic properties. This can be measured with the covariance � � � � 2 , − E Cov ( f n ( ω 0 ) , f n ( ω t )) = E f n ( ω 0 ) f n ( ω t ) f n or equivalently by � � � �� � ω 0 Var E f n ( ω t ) . C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 7 / 22

Noise Sensitivity We are interested in a fast decorrelation (or fast mixing) of macroscopic properties. This can be measured with the covariance � � � � 2 , − E Cov ( f n ( ω 0 ) , f n ( ω t )) = E f n ( ω 0 ) f n ( ω t ) f n or equivalently by � � � �� � ω 0 Var E f n ( ω t ) . 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

Noise Sensitivity We are interested in a fast decorrelation (or fast mixing) of macroscopic properties. This can be measured with the covariance � � � � 2 , − E Cov ( f n ( ω 0 ) , f n ( ω t )) = E f n ( ω 0 ) f n ( ω t ) f n or equivalently by � � � �� � ω 0 Var E f n ( ω t ) . 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

Harmonic Analysis of Boolean functions We consider the larger space L 2 ( {− 1 , 1 } n ) of real-valued functions from n bits into R , endowed with the scalar product: � 2 − n f ( x 1 , . . . , x n ) g ( x 1 , . . . , x n ) � f , g � = x 1 ,..., x n � � = E fg C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 8 / 22

Harmonic Analysis of Boolean functions We consider the larger space L 2 ( {− 1 , 1 } n ) of real-valued functions from n bits into R , endowed with the scalar product: � 2 − n f ( x 1 , . . . , x n ) g ( x 1 , . . . , x n ) � f , g � = x 1 ,..., x n � � = E fg One has at our disposal a natural basis for this space isomorphic to R 2 n : the so-called characters of the group {− 1 , 1 } n . For any subset S ⊂ { 1 , . . . , n } , consider the function χ S defined by � χ S ( x 1 , . . . , x n ) := x i i ∈ S The set of these 2 n functions forms an orthonormal basis of L 2 ( {− 1 , 1 } n ) . C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 8 / 22

Fourier-Walsh expansion Thus, any Boolean function f : {− 1 , 1 } n → { 0 , 1 } can be decomposed as � � f = f ( S ) χ S S ⊂ [ n ] where � f ( S ) are the Fourier-Walsh coefficients of f . They satisfy � � � f ( S ) = � f , χ S � = E f χ S � � 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

Why is it any helpful ? C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 10 / 22

Why is it any helpful ? The correlation between f ( ω 0 ) and f ( ω t ) has a very simple form in terms of the Fourier coefficients � f ( S ) . Indeed: C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 10 / 22

Why is it any helpful ? The correlation between f ( ω 0 ) and f ( ω t ) has a very simple form in terms of the Fourier coefficients � f ( S ) . Indeed: ��� ��� �� � � � � f ( ω 0 ) f ( ω t ) = f ( S 1 ) χ S 1 ( ω 0 ) f ( S 2 ) χ S 2 ( ω t ) E E S 1 S 2 � � � f ( S ) 2 E � = χ S ( ω 0 ) χ S ( ω t ) S � f ( S ) 2 e − t | S | � = S Therefore our covariance can be written � � � � � 2 = f ( S ) 2 e − t | S | � − E E f ( ω 0 ) f ( ω t ) f ( ω ) S � = ∅ C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 10 / 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 = n k = 1 k = 2 C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 11 / 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: The total Spectral � | S | = k � f ( S ) 2 mass here is � f ( S ) 2 = Var[ f ] � | S |� =0 k . . . . . . k = n k = 1 k = 2 C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 11 / 22

The Energy Spectrum of macroscopic events Recall our above left-right crossing events corresponding to the Boolean functions f n , n ≥ 1. C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 12 / 22

The Energy Spectrum of macroscopic events Recall our above left-right crossing events corresponding to the Boolean functions f n , n ≥ 1. One is interested in the shape of their Energy Spectrum � | S | = k � f n ( S ) 2 ? k . . . . . . C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 12 / 22

The Energy Spectrum of macroscopic events Recall our above left-right crossing events corresponding to the Boolean functions f n , n ≥ 1. One is interested in the shape of their Energy Spectrum � | 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, Orsay) Oded’s work on Noise Sensitivity 12 / 22

Energy Spectrum of Majority Let Φ n be the majority function on {− 1 , 1 } n ( n being odd) Φ n ( x 1 , . . . , x n ) := sign ( � i x i ) C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 13 / 22

Energy Spectrum of Majority � | S | = k � Φ n ( S ) 2 Let Φ n be the majority function on {− 1 , 1 } n ( n being odd) Φ n ( x 1 , . . . , x n ) := sign ( � i x i ) The Energy Spectrum of Φ n has the following shape: . . . k 1 3 5 n C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 13 / 22

Three (very different !) approaches to Localize the Spectrum � | S | = k � f n ( S ) 2 ? k . . . . . . C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 14 / 22

Three (very different !) approaches to Localize the Spectrum � | S | = k � f n ( S ) 2 • Hypercontractivity, 1998 Benjamini, Kalai, Schramm ? k . . . . . . C. Garban (ENS, Orsay) Oded’s work on Noise Sensitivity 14 / 22

Three (very 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, Orsay) Oded’s work on Noise Sensitivity 14 / 22