Harmonic measure of critical curves and CFT

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Harmonic measure of critical curves and CFT

• Ilya A. Gruzberg
• University of Chicago
• with

E. Bettelheim, I. Rushkin, and P. Wiegmann
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2D critical models
Ising model
Percolation
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Critical curves

• Focus on one domain wall using certain boundary
  conditions
• Conformal invariance systems in simple domains.
  
• Typically, upper half plane

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Critical curves geometry and probabilities

• Fractal dimensions
• Multifractal spectrum of harmonic measure
• Crossing probability
• Left vs. right passage probability
• Many more

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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

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Harmonic measure on a curve

• Electric field of a charged cluster

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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

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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

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Moments of harmonic measure

• Global moments

fractal dimension

• Local moments

• Ergodicity

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Harmonic measure and conformal maps

• Harmonic measure is conformally invariant

• Multifractal spectrum is related to derivative
• expectation values connection with SLE.
• Use CFT methods

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Various uniformizing maps
(1)
(2)
(4)
(3)
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Correlators of boundary operators
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Correlators of boundary operators
M. Bauer, D. Bernard

• Two step averaging

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Correlators of boundary operators

• Insert probes of harmonic measure
• primary operators of dimension

• Need only -dependence in the limit

• LHS fuse
• RHS statistical independence

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Conformal invariance

• Map exterior of to by that
  satisfies

• Primary field

• Last factor does not depend on

• Put everything together

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Mapping to Coulomb gas
L. Kadanoff, B. Nienhuis, J. Kondev

• Stat mech models loop models height
  models
• Gaussian free field (compactified)

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Coulomb gas

• Parameters
• Phases (similar to SLE)
• Central charge

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Coulomb gas fields and correlators

• Vertex electromagnetic operators
• Charges
• Holomorphic dimension
• Correlators and neutrality

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Curve-creating operators
B. Nienhuis

• To create curves choose

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Curve-creating operators

• In traditional CFT notation
• - the boundary curve operator is

with charge
- the bulk curve operator is
with charge
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Multifractal spectrum on the boundary

• One curve on the boundary

• The probe

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Generalizations boundary

• Several curves on the boundary

• Higher multifractailty many curves and points

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Higher multifractality on the boundary

• Need to find

• Consider

• Here

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Higher multifractality on the boundary

• Exponents are
  dimensions of

primary boundary operators with

• Comparing two expressions for , get

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Generalizations bulk

• Several curves in the bulk

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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