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Noise Sensitivity – The case of Percolation

Gil KalaiInstitute of Mathematics

Hebrew UniversityHU – HEP seminar, 25 April 2007

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(We start with a one-slide summary of the lecture followed by a 4 slides very informal summary of its three main ingredients.)

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Plan of the talk

• Two dimensional percolation • Noise sensitivity – The primal description• Noise sensitivity - The Fourier description• How the spectrum looks like - Scaling limit – existence and

description• Other models with noise sensitivity• Questions and thoughts regarding models

from high-energy physics

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Planar PercolationThe infinite model: we have an infinite lattice gridin the plane. Every edge (bond) is open withprobability p. All these probabilities are statistically

independent.

Basic questions:

What is the probability of an infinite open cluster?

What is the probability of an infinite open cluster containing the origin?

Critical properties of percolation.

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Noise sensitivityPrimal description - Functions (random

variables) that are extremely sensitive to small random changes (which respect the overall underlying distribution.) Such functions cannot be measured by (even slightly) noisy measurements.

Dual description – Spectrum concentrated on “large sets”

Examples: Critical percolation, and many others

Basic insight: Noise sensitivity is common and forced in various general situations.

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Noise stability/Noise sensitivity Dichotomy

Familiar stochastic processes are “noise stable”.Their sensitivity to small amount of noise is small.Their spectrum is concentrated on small sets.

The notions of noise stability and noise sensitivity were introduced by Benjamini, Kalai and Schramm. Closely related notions (black noise; non-Fock models) were introduced by Tsirelson and Vershik.

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High energy physics

Are the basic models of high energy physics noise stable?

If this is indeed the case, does it reflect some law of physics or (more likely), will noise sensitivity allow additional modeling power.

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Critical Percolation

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Critical Percolation: problems and progress

• The critical probability• Limit conjectures and Conformal invariance• SLE and scaling limits• Noise sensitivity and spectral description

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Kesten: Critical probability 1/2

Kesten’s Theorem (1980): The critical probability for percolation in the plane is ½.

If the probability p for a bond to be open (or for a hexagon to be grey) is below ½ the probability for an infinite cluster is 0. If the probability for a bond to be open is > ½ then the probability for an infinite cluster is 1.

(Q: And when p is precisely ½?) (A: The probability for an infinite cluster is 0)

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Limit conjectures

Conjecture: The probability for the crossing event for an n by m rectangular grid tends to a limit if the ratio m/n tends to a real number a, a>0, as n tends to infinity.

(Sounds almost obvious, yet very difficult to prove)

Note: we have moved from infinite models to finite ones.

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Cardy, Aizenman, Langlands: Conformal invariance

conjectures

Conjecture: Crossing events in percolation are conformally invariant!!

Sounds very surprising. (But there is no case of a planar percolation model where the limit conjectures are proven and conformal invariance is not.)

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Limits Conjectures and conformal Invariance

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Schramm: SLE

Oded Schramm defined a one parameter planar stochastic models SLE(κ). Lawler, Schramm and Werner extensively studied the SLE processes, found relations to several planar processes, and computed various critical exponents. SLE(6) describes the scaling limit of percolation.

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SLE and Percolation:Grey/white Interface

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Smirnov: Conformal Invariance

Smirnov proved that for the model of site percolation on the triangular grid, equivalently

For the white/grey hexagonal model (simply HEX), the conformal invariance conjecture is correct!

(An incredibly simple form of Cardy’s formulas in this case found by Carleson was of importance.)

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Putting things together Combining Smirnov results with the work of

Lawler Schramm and Werner all critical exponents for percolation predicted by physicists and quite a few more were computed. (rigorously)

(For the model of bond percolation with square grid this is yet to be done.)

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Noise Sensitivity: The Primal description

We consider a BOOLEAN FUNCTION

f :{-1,1}n {-1,1}f(x1 ,x2,...,xn)

(For percolation, every hexagon corresponds to a variable. xi =-1 if the hexagon is white and xi =1 if it is grey. f=1 if there is a left to right grey crossing.)

Given x1 ,x2,...,xn we define y1 ,y2,...,yn as follows:

xi = yi with probability 1-t

xi = -yi with probability t

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Noise Sensitivity: The Primal description (cont.)

Let C(f;t) be the correlation between f(x1 , x2,...,xn) and f(y1,y2,...,yn)

A sequence of Boolean function (fn ) is (completely) noise-sensitive if for every t>0, C(fn,t) tends to zero with n.

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Percolation is Noise sensitive

Theorem [BKS]: The crossing event for critical planar percolation model is noise- sensitive

Basic argument: 1) Fourier description of noise sensitivity; 2) hypercontractivityThis argument applies to very general cases.

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Percolation is Noise sensitive

Imagine two separate pictures of n by n hexagonal models for percolation. A hexagon is grey with probability ½. If the grey and white hexagons are independent in the two pictures the probability for crossing in both is ¼.If for each hexagon the correlation between its colors in the two pictures is 0.99, still the probability for crossing in both pictures is very close to ¼ as n grows! If you put one drawing on top of the other you will hardly notice a difference!

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Fourier-Walsh expansion

Given a Boolean function f :{-1,1}n {-1,1}, we write f(x) as a sum of multilinear (square free) monomials. f(x) = Σfˆ(S)W(S), where W(S) =∏{xs : s є S}.

f^(S) is the Fourier-Walsh coefficient corresponding to S.

Used by Kahn, Kalai and Linial (1988) to settle a conjecture by Ben-Or and Linial on “influences”.

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Noise sensitivity – the dual Description

The spectral distribution of f is a probability distribution assigning to a subset S the probability (f^(S))2

For a sequence of Boolean function fn :{-1,1}n {-1,1}

(fn) is (completely) noise sensitive if for every k the overall spectral probability for non empty sets of size at most k tends to 0 as n tends to 0.

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The motivations

This was an attempt towards limits and conformal invariance conjectures. (Second attempt for Oded and Itai.)

Understanding the spectrum of percolation looked interesting; One critical exponent (correlation length) has a simple description.

(Late) Percolation on certain random planar graphs arise here naturally. (KPZ)

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An application: Dynamic percolation

Dynamic percolation was introduced and first studied by Häggström, Peres and Steif (1997). The model was introduced independently by Itai Benjamini. Häggström, Peres and Steif proved that above the critical probability we have infinite clusters at all times, and below the critical probability there are infinite clusters at no times.

Schramm and Steif proved that for dynamic percolation on the HEX model there are exceptional times. The proof is based on their strong versions of noise sensitivity for planar percolation.

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Dynamic Percolation

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Fourier Description of Crossing events of

PercolationBenjamini, Kalai, and Schramm: Most Fourier Coefficients

are above log nSchramm and Steif: Most Fourier coefficients are above nb

(b>0)Schramm and Smirnov: Scaling limit for spectral

distribution for Percolation exists (*)Garban, Pete and Schramm (yet unwritten) : Spectral

distributions concentrated on sets of size n3/4(1+o(1)). (*)(*) – proved only for models where Smirnov’s result apply.In summary: Scaling limit for the spectral distribution of

percolation is described by Cantor sets of dimension ¾.

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Diversion: Simulating and computing the spectrum for

percolation Can we sample according (approximately) to

the spectral distribution of the crossing event of percolation?

This is unknown and it might be hard on digital computers.

But... it is known to be easy for... quantum computers. For every Boolean function where f is computable in polynomial time. (Quantum computers are hypothetical devices based on QM which allow superior computational power.)

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Noise sensitivity, and non-classical stochastic

processes; black noise

Closly related notions to “noise sensitivity” were studied by Tsirelson and Vershik . In their terminology “noise sensitivity” translates to “non Fock processes”, “black noise”, and “non-classical stochastic processes”. Their motivation is closer to mathematical quantum physics.

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Tsirelson and Vershik: Non Fock spaces; black noise; non classical stochastic processes

(cont)The terminology is confusing but here is the dictionary:Noise stable – White noise; classical stochastic process; Fock modelNoise sensitive – Black noise; non classical stochastic process; non-Fock model.Tsirelson and Vershik pointed out a connection between noise sensitivity and non-linearity. (Well within the realm of QM.)

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Other cases of noise sensitivity

First Passage Percolation (Benjamini, Kalai, Schramm)

A recursive example by Ben-Or and Linial

Eigenvalues of random Gaussian matrices (Essentially follows from the work of Tracy-Widom) Here, we leave the Boolean setting.

Examples related to random walks (required replacing the discrete cube by trees) and more...

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Questions about HEP models

•Are current HEP models noise stable?•Or perhaps there is some internal inconsistency about their noise stability

•The naive idea is this: Hep models describe a (quantum) stochastic state. Is this state necessarily noise stable?

•(Less naively, according to Tsirelson): “Noise sensitivity means that the very idea of `the field operator at a point' (on the level of operator-valued Schwartz distributions (or something like that)) will fail.”

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Questions about HEP models

Tsireslon constructed a toy non-Fock model

in hep-th/9912031

My thoughts on the matter can be found in hep-th/0703092

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Are basic models from High energy physics noise stable?

Remark: In order to properly ask the question we need to extend the notion of noise sensitivity:

1)Quantum probabilities2)Symmetries are not Z/2Z but other fixed groups like U(1), SU(2) and SU(3). 3)Noise sensitivity assumes a representation via independent random variables.

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Required extensions for Noise sensitivity

1)Quantum probabilities: This appears not to pose difficulties. Was studied by Tsirelson.

2) Symmetries are not Z/2Z but other fixed groups like U(1), SU(2) and SU(3). The notions of noise sensitivity extends. Interesting new phenomena occurred even when moving from Z/2Z to to Z/3Z and more are expected in the non-Abelian case.

3) Noise sensitivity assumes a representation via independent random variables. This is the most serious and interesting concern.

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The next few slides consider some critical comments concerning the relevance and novelty of noise-sensitive models.

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I. Does noise sensitivity just reflects “wrong

scaling?”

Perhaps, in some cases. (And if it does it may give an interesting mathematical setting for such scaling problems/renormalization.) It is known that for Boolean functions at the “wrong scale” noise sensitivity is forced. However, in some cases, like the case of percolation, noise sensitivity occurs at all scales.

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II...But Percolation is a model arising in CFT

The percolation model is a very basic example in CFT (conformal field theory). Since noise sensitivity occurs in a rather special case of a very familiar physics model, isn’t it just an “artifact” of the way we look here at percolation?

Maybe. But it does not seem to be the case. (Perhaps the non-rigorous physics theories

treat noise sensitive processes as being noise stable.)

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III. Do strings already capture the idea of noise sensitivity?

(and also QCD(N) and other familiar models...)

The conjecture that Hep-model are describing noise stable processes is also based on the description of point particles and their (mainly) pair-wise interactions. We think about particles as living in the spectrum world.

Isn’t noise sensitivity just a very primitive version of ideas that come to play in various current models from physics (which are really relevant to physics).

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III. Is the spectral distribution for percolation something like

strings? (cont.) (and also QCD(N) and other familiar models...)

Isn’t noise sensitivity a very primitive version of ideas that come to play in various current models from physics (which are really relevant to physics)? Do strings already capture the idea of noise-sensitivity?

Well, if indeed the scaling limit for the spectrum of percolation is a rigorous mathematical model of something “like” strings this can be of interest.

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III. Do strings capture the idea of noise sensitivity (cont.) ?

Yes, maybe, but there are things that look quite different:

A) The geometry of the scaling limit object for percolation is not that of a string; the

scaling limits are Cantor sets.

(It is an interesting problem to find a case of a noise sensitive process (“black noise)” with

connected spectral scaling limit.)

B) For the case of percolation the spectral objects themselves appear to represent non-

classical stochastic objects. (unlike strings which themselves looks as classical

stochastic objects)

C) The mathematical framework for strings still looks (in part) as assuming some sort of

noise-stability.

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Other physics speculations

“Black Noise/Noise sensitivity occurred at the Big Bang (Tsirelson and Vershik) this was a motivating idea behind their paper but it is not mentioned there.

Dark Energy is a black noise (noise-sensitive process).

Noise sensitivity models may allow string or string-like models in D=3+1.

(And what about black holes?? )

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Conclusion

If noise sensitivity is an option, noise sensitive models may allow modeling power beyond those of existing models.

If noise stability is a law of physics this is also interesting.

Noise sensitivity (and our notions of “pixelwise” Fourier expansions) may be relevant to mathematical foundations of current successful models from high energy physics (QED, QCD).

Noise sensitive (black) perturbations of other PDE from physics and related notions of generalized solutions, can also be of interest.

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THANK YOU FOR HAVING ME!

תודה

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Comments by HU-HEP seminar participants

1. (Major, rather justified critique.) Jumping from the model of percolation to hep-ph models was not justified. I was not specific about what is it precisely that I suspect to be noise sensitive. Also the analogy made in the lecture between the Z/2Z symmetry in the percolation model and symmetries in hep models may be “by name only”.

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HU-hep seminar participants Comments (continued)

2. Shmuel Elitzur asked if we can do something similar for the Ising model. In sort of defense against the previous critique he pointed out that, in some sense, general field theories can be “built up” from copies of the Ising model.

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HU-hep seminar participants Comments (continued)

3. There was a long discussion if (and how) noise sensitivity can be tested experimentally. (Not just by simulations.) Initially I thought the answer is negative but was convinced by the others otherwise.

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HU-hep seminar participants Comments (continued)

4. Ido Ben-Dayan mentioned a work by Malomed and coauthors on sensitivity to noise of a classic two beams experiment.

5. Merav Stern commented that these notions are more suitable to condensed matter physics and speculated/suggested to think about noise-sensitive models in connection with high temperature superconductivity.

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HU-hep seminar participants Comments (continued)

6. Matteo Cardella asked about noise sensitivity of the actual paths exhibiting the crossing events in percolation. So, in the two pictures with 0.99 correlation is it true that even if there is crossing in both pictures, the paths exhibiting the crossing are in some sense uncorrelated. (Oded Schramm pointed out that indeed this is the case.)

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HU-hep seminar participants Comments (continued)

7. Shmuel Elitzur summarized the lecture’s massage in a very sweet way : In addition to “classical part” which is described by current HEP models the lecture proposes that there is another component which represents a different kind of stochastic behavior.

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More remarks are welcomed !