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Harmonic measure of critical curves and CFT

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Ising model Percolation Focus on one domain ... Global moments Local moments fractal dimension Ergodicity ... Here Write as a two-step ... – PowerPoint PPT presentation

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Title: Harmonic measure of critical curves and CFT


1
Harmonic measure of critical curves and CFT
  • Ilya A. Gruzberg
  • University of Chicago
  • with

E. Bettelheim, I. Rushkin, and P. Wiegmann
2
2D critical models
Ising model
Percolation
3
Critical curves
  • Focus on one domain wall using certain boundary
    conditions
  • Conformal invariance systems in simple domains.
  • Typically, upper half plane

4
Critical curves geometry and probabilities
  • Fractal dimensions
  • Multifractal spectrum of harmonic measure
  • Crossing probability
  • Left vs. right passage probability
  • Many more

5
Harmonic measure on a curve
  • Probability that a Brownian particle
  • hits a portion of the curve
  • Electrostatic analogy charge on the
  • portion of the curve (total charge one)
  • Related to local behavior of electric field
  • potential near wedge of angle

6
Harmonic measure on a curve
  • Electric field of a charged cluster

7
Multifractal exponents
  • Lumpy charge distribution on a cluster boundary
  • Cover the curve by small discs
  • of radius
  • Charges (probabilities) inside discs
  • Moments
  • Non-linear is the hallmark of a
    multifractal
  • Problem find for critical curves

8
Conformal multifractality
  • Originally obtained by quantum gravity

B. Duplantier, 2000
  • For critical clusters with central charge
  • We obtain this and more using traditional CFT
  • Our method is not restricted to

9
Moments of harmonic measure
  • Global moments

fractal dimension
  • Local moments
  • Ergodicity

10
Harmonic measure and conformal maps
  • Harmonic measure is conformally invariant
  • Multifractal spectrum is related to derivative
  • expectation values connection with SLE.
  • Use CFT methods

11
Various uniformizing maps
(1)
(2)
(4)
(3)
12
Correlators of boundary operators
13
Correlators of boundary operators
M. Bauer, D. Bernard
  • Two step averaging

14
Correlators of boundary operators
  • Insert probes of harmonic measure
  • primary operators of dimension
  • Need only -dependence in the limit
  • LHS fuse
  • RHS statistical independence

15
Conformal invariance
  • Map exterior of to by that
    satisfies
  • Primary field
  • Last factor does not depend on
  • Put everything together

16
Mapping to Coulomb gas
L. Kadanoff, B. Nienhuis, J. Kondev
  • Stat mech models loop models height
    models
  • Gaussian free field (compactified)

17
Coulomb gas
  • Parameters
  • Phases (similar to SLE)
  • Central charge

18
Coulomb gas fields and correlators
  • Vertex electromagnetic operators
  • Charges
  • Holomorphic dimension
  • Correlators and neutrality

19
Curve-creating operators
B. Nienhuis
  • To create curves choose

20
Curve-creating operators
  • In traditional CFT notation
  • - the boundary curve operator is

with charge
- the bulk curve operator is
with charge
21
Multifractal spectrum on the boundary
  • One curve on the boundary
  • The probe

22
Generalizations boundary
  • Several curves on the boundary
  • Higher multifractailty many curves and points

23
Higher multifractality on the boundary
  • Need to find
  • Consider
  • Here

24
Higher multifractality on the boundary
  • Exponents are
    dimensions of

primary boundary operators with
  • Comparing two expressions for , get

25
Generalizations bulk
  • Several curves in the bulk

26
Open questions
  • Spatial structure of harmonic measure on
    stochastic curves
  • Prefactor in
  • related to structure constants in CFT
  • Stochastic geometry in critical systems with
    additional
  • symmetries Wess-Zumino models, W-algebras,
    etc.
  • Stochastic geometry of growing clusters DLA,
    etc
  • no conformal invariance
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