IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure...

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


2D critical models2D critical models

Ising model Percolation


Critical curvesCritical curves

• Focus on one domain wall using certain boundary conditions

• Conformal invariance: systems in simple domains. Typically, upper half plane


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 …


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


Harmonic measure on a curveHarmonic measure on a curve

• Electric field of a charged cluster


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


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


Moments of harmonic measureMoments of harmonic measure

• Global moments

• Local moments

fractal dimension

• Ergodicity


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


Various uniformizing mapsVarious uniformizing maps

(1) (2)(3) (4)


Correlators of boundary operatorsCorrelators of boundary operators

- partition function with modified BC

- boundary condition (BC) changing operator

- partition function



• 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



• Insert “probes” of harmonic measure: primary operators of dimension

• LHS: fuse

• RHS: statistical independence

• Need only -dependence in the limit


Conformal invarianceConformal invariance

• Map exterior of to by that satisfies

• Primary field

• Last factor does not depend on

• Put everything together:


Mapping to Coulomb gasMapping to Coulomb gas

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

L. Kadanoff, B. Nienhuis, J. Kondev


Coulomb gasCoulomb gas

• Parameters

• Phases (similar to SLE)

• Central charge

densedilute


Coulomb gas: fields and correlatorsCoulomb gas: fields and correlators

• Vertex “electromagnetic” operators

• Charges

• Holomorphic dimension

• Correlators and neutrality


Curve-creating operatorsCurve-creating operators

• Magnetic charge creates a vortex in the field

• To create curves choose

B. Nienhuis


Curve-creating operatorsCurve-creating operators

• In traditional CFT notation

- the boundary curve operator is

- the bulk curve operator is

with charge

with charge


Multifractal spectrum on the boundaryMultifractal spectrum on the boundary

• One curve on the boundary

• KPZ formula: is the gravitationally dressed dimension!

• The “probe”


Generalizations: boundaryGeneralizations: boundary

• Several curves on the boundary

• Higher multifractailty: many curves and points


Higher multifractality on the boundaryHigher multifractality on the boundary

• Consider

• Need to find

• Here


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


Generalizations: bulkGeneralizations: bulk

• Several curves in the bulk


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

IPAM Workshop “Random Shapes, Representation Theory, and CFT”, March 26, 2007 Harmonic measure...

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