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IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 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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IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Harmonic measure of critical curves and CFT Harmonic measure of critical curves and CFT
Ilya A. GruzbergUniversity of Chicago
with
E. Bettelheim, I. Rushkin, and P. Wiegmann
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
2D critical models2D critical models
Ising model Percolation
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Critical curvesCritical curves
• Focus on one domain wall using certain boundary conditions
• Conformal invariance: systems in simple domains. Typically, upper half plane
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Critical curves: geometry and probabilitiesCritical curves: geometry and probabilities
• Fractal dimensions
• Multifractal spectrum of harmonic measure
• Crossing probability
• Left vs. right passage probability
• Many more …
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Harmonic measure on a curveHarmonic 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
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Harmonic measure on a curveHarmonic measure on a curve
• Electric field of a charged cluster
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Multifractal exponentsMultifractal exponents
• Lumpy charge distribution on a cluster boundary
• Non-linear is the hallmark of a multifractal
• Problem: find for critical curves
• Cover the curve by small discs of radius
• Charges (probabilities) inside discs
• Moments
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Conformal multifractalityConformal multifractality
B. Duplantier, 2000
• For critical clusters with central charge
• We obtain this and more using traditional CFT Our method is not restricted to
• Originally obtained by quantum gravity
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Moments of harmonic measureMoments of harmonic measure
• Global moments
• Local moments
fractal dimension
• Ergodicity
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Harmonic measure and conformal mapsHarmonic measure and conformal maps
• Harmonic measure is conformally invariant:
• Multifractal spectrum is related to derivative expectation values: connection with SLE.
• Use CFT methods
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Various uniformizing mapsVarious uniformizing maps
(1) (2)(3) (4)
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Correlators of boundary operatorsCorrelators of boundary operators
- partition function with modified BC
- boundary condition (BC) changing operator
- partition function
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Correlators of boundary operatorsCorrelators of boundary operators
• Two step averaging:
1. Average over microscopic degrees of freedom in the presence of a given curve
2. Average over all curves
M. Bauer, D. Bernard
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Correlators of boundary operatorsCorrelators of boundary operators
• Insert “probes” of harmonic measure: primary operators of dimension
• LHS: fuse
• RHS: statistical independence
• Need only -dependence in the limit
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Conformal invarianceConformal invariance
• Map exterior of to by that satisfies
• Primary field
• Last factor does not depend on
• Put everything together:
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Mapping to Coulomb gasMapping to Coulomb gas
• Stat mech models loop models height models Gaussian free field (compactified)
L. Kadanoff, B. Nienhuis, J. Kondev
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Coulomb gasCoulomb gas
• Parameters
• Phases (similar to SLE)
• Central charge
densedilute
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Coulomb gas: fields and correlatorsCoulomb gas: fields and correlators
• Vertex “electromagnetic” operators
• Charges
• Holomorphic dimension
• Correlators and neutrality
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Curve-creating operatorsCurve-creating operators
• Magnetic charge creates a vortex in the field
• To create curves choose
B. Nienhuis
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Curve-creating operatorsCurve-creating operators
• In traditional CFT notation
- the boundary curve operator is
- the bulk curve operator is
with charge
with charge
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Multifractal spectrum on the boundaryMultifractal spectrum on the boundary
• One curve on the boundary
• KPZ formula: is the gravitationally dressed dimension!
• The “probe”
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Generalizations: boundaryGeneralizations: boundary
• Several curves on the boundary
• Higher multifractailty: many curves and points
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Higher multifractality on the boundaryHigher multifractality on the boundary
• Consider
• Need to find
• Here
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Higher multifractality on the boundaryHigher multifractality on the boundary
• Write as a two-step average and map to UHP:
• Exponents are dimensions of
primary boundary operators with
• Comparing two expressions for , get
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Generalizations: bulkGeneralizations: bulk
• Several curves in the bulk
IPAM Workshop “Random Shapes, Representation Theory, and CFT” , March 26, 2007
Open questionsOpen questions
• Spatial structure of harmonic measure on stochastic curves
• Stochastic geometry in critical systems with additional symmetries: Wess-Zumino models, W-algebras, etc.
• Stochastic geometry of growing clusters: DLA, etc: no conformal invariance…
• Prefactor in related to structure constants in CFT