Noise Sensitivity

1
Noise Sensitivity The case of Percolation

• Gil Kalai
• Institute of Mathematics
• Hebrew University
• HU HEP seminar, 25 April 2007

2

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

3
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

4
Planar Percolation

• The infinite model we have an infinite lattice
  grid
• in the plane. Every edge (bond) is open with
• probability 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.

5
Noise sensitivity

• Primal 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.

6
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.

7
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.

8
Critical Percolation
9
Critical Percolation problems and progress

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

10
Kesten Critical probability 1/2

• Kestens 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 gt 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)

11
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,
  agt0, as n tends to infinity.
• (Sounds almost obvious, yet very difficult to
  prove)
• Note we have moved from infinite models to
  finite ones.

12
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.)

13
Limits Conjectures and conformal Invariance
14
Schramm SLE

• Oded Schramm defined a one parameter planar
  stochastic models SLE(?). 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.

15
SLE and PercolationGrey/white Interface
16
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 Cardys formulas
  in this case found by Carleson was of
  importance.)

17
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.)

18
Noise Sensitivity The Primal description

• We consider a BOOLEAN FUNCTION
• f -1,1n ? -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. f1 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

19
Noise Sensitivity The Primal description (cont.)

• Let C(ft) 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 tgt0,
  C(fn,t) tends to zero with n.

20
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) hypercontractivity
• This argument applies to very general cases.

21
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!

22
Fourier-Walsh expansion

• Given a Boolean function f -1,1n ? -1,1, we
  write f(x) as a sum of multilinear (square free)
  monomials.
• f(x) Sfˆ(S)W(S), where W(S) ?xs s ? 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.

23
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 -1,1n ? -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.

24
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)

25
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.

26
Dynamic Percolation
27
Fourier Description of Crossing events of
Percolation

• Benjamini, Kalai, and Schramm Most Fourier
  Coefficients are above log n
• Schramm and Steif Most Fourier coefficients are
  above nb (bgt0)
• 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(1o(1)). ()
• () proved only for models where Smirnovs
  result apply.
• In summary Scaling limit for the spectral
  distribution of percolation is described by
  Cantor sets of dimension ¾.

28
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.)

29
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.

30
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 model
• Noise 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.)

31
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...

32
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.

33
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

34
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
• Quantum probabilities
• Symmetries are not Z/2Z but other fixed groups
  like U(1), SU(2) and SU(3).
• Noise sensitivity assumes a representation via
  independent random variables.

35
Required extensions for Noise sensitivity

• 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.

36

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

37
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.

38
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, isnt 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.)

39
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.
• Isnt 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).

40
III. Is the spectral distribution for percolation
something like strings? (cont.)

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

41
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.

42
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 D31.
• (And what about black holes?? ?)

43
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.

44
THANK YOU FOR HAVING ME!????
45
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.

46
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.

47
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.

48
HU-hep seminar participants Comments (continued)

1. Ido Ben-Dayan mentioned a work by Malomed and
   coauthors on sensitivity to noise of a classic
   two beams experiment.
2. 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.

49
HU-hep seminar participants Comments (continued)

1. 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.)

50
HU-hep seminar participants Comments (continued)

• 7. Shmuel Elitzur summarized the lectures
  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.

51

• More remarks are welcomed !