Dynamic percolation, exceptional times, and harmonic analysis of boolean functions Oded Schramm...
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Transcript of Dynamic percolation, exceptional times, and harmonic analysis of boolean functions Oded Schramm...
![Page 1: Dynamic percolation, exceptional times, and harmonic analysis of boolean functions Oded Schramm joint w/ Jeff Steif.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d255503460f949fb544/html5/thumbnails/1.jpg)
Dynamic percolation, exceptional times, and
harmonic analysis of boolean functions
Oded Schrammjoint w/ Jeff Steif
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Percolation
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Dynamic Percolation
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Infinite clusters?
• Harris (1960): There is no infinite cluster
• Kesten (1980): There is if we increase p
For static percolation:
For dynamic percolation:
• At most times there is no infinite cluster
• Can there be exceptional times?
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• Any infinite graph G has a pc:
• Above pc:
• Below pc:
• The latter at pc for Zd, d>18.
• Some (non reg) trees with exceptional times.• Much about dynamic percolation on trees…
Häggström, Peres, Steif (1997)
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Exceptional times exist
Theorem (SS): The triangular grid has exceptional times at pc.
This is the only transitive graph for which it is known that there are exceptional times at pc.
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Proof idea 0: get to distance R
• Set
• We show
Namely, with positive probability the cluster
of the origin is unbounded for t in [0,1].
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2nd moment argument
We show that
Then use Cauchy-Schwarz:
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2nd moment spelled out
Consequently, enough to show
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Interested in expressions of the form
Where is the configuration at time
t, and f is a function of a static configuration.
Rewrite
where
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Understanding Tt
Set
Then
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Set
Then
Write
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Theorem (BKS): When fn is the indicator function for crossing an n x n square in percolation, for all positive t
Equivalently, for all k>0 fixed
Noise sensitivity
Equivalently, for all t>0 fixed
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Need more quantitative
Conjectured (BKS):
with
We (SS) prove this (for Z2 and for the triangular grid).
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Estimating the Fourier weights
Theorem (SS): Suppose that there is a randomized algorithm for calculating f that examines each bit with probability at most . Then
Probably not tight.
?
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The of percolation
Not optimal, (simulations) [LSW,Sm][PSSW]
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Annulus case δ
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Annulus case δ
The for the algorithm calculating
is approximately
get to approx radius r
visit a particular hex
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Putting it together
Etc...
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What about Z2 ?
• The argument almost applies to Z2
Need
2. Improve (better algorithm)
1. Have
3. Improve Fourier theorem
4. Calculate exponents for Z2
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How small can be
• Example with
• Monotone example with
• These are balanced
• Best possible, up to the log terms
• The monotone example shows that the Fourier inequality is sharp at k=1.
BSW:
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Next
• Proof of Fourier estimate
• Further results
• Open problems
• End
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Theorem (SS): Suppose that there is a randomized algorithm for calculating f that examines each bit with probability at most δ. Then
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Fourier coefficients change under algorithm
When algorithm examines bit :
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Proof of Fourier Thm
Set
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QED
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Further results
Never 2 infinite clusters.
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Wedges
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Wedges
• Exceptional times with k different infinite clusters if
• HD:
• None if
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Cones
glue
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Cones
• Exceptional times with k>1 different infinite clusters if
• HD:
• None if
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Open problems
• Improve estimate for
• Improve Fourier Theorem
• Percolation Fourier coefficients
• Settle
• Correct Hausdorff dimension?
• Space-time scaling limit
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The End