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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
• 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
• - 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